Methods of Experimental Physics VOLUME 5
NUCLEAR PHYSICS PART B
METHODS OF
EXPERIMENTAL PHYSICS: 1. Marton, Editor-in-Chief Claire Marton, Assistant Editor
1. Classical Methods, 1959 Edited by lmmanuel Estermann
2. Electronic Methods, in preparation Edited by E. Bleuler and R. 0. Haxby 3. Molecular Physics, 1961 Edited by Dudley Williams
4. Atomic and Electron Physics, in preparation Edited by Vernon W. Hughes and Howard L. Schultz 5. Nuclear Physics (in two parts), 1961 and 1963 Edited by Luke C. L. Yuan and Chien-Shiung Wu
6. Solid State Physics (in fwo parts), 1959 Edited by K. Lark-Horovitz and Vivian A. Johnson
Volume 5
Nuclear Physics Edited by
LUKE C. 1. YUAN Brookhaven National Laboratory Upfon, New York
CHIEN-SHIUNG WU Columbia University New York, New York
PART B
1963
@
ACADEMIC PRESS New York and London
Copyright @ 1963, by
ACADEMIC PRESS INC. ALL RIGHTS RESERVED NO P A R T OF THIS BOOK MAY B E REPRODUCED I N ANY FORM B Y PHOTOSTAT, MICROFILM, OR A N Y O T H E R MEANS, WITHOUT WRITTEN PERMISSION FROM T H E PUBLISHERS
ACADEMIC PRESS INC. 111 FIFTHAVENUE NEW YORK3, N. Y.
United Kingdom Edilon Published by ACADEMIC PRESS INC. (LONDON) LTD.
BERKELEY SQUARE HOUSE,LONDON W. 1
Library of Congress Catalog Card Number 61-17860 PRINTED I N T H E UNITED STATES O F AMERICA
CONTRIBUTORS TO VOLUME 5, PART B E. AMBLER,Low Temperature Section, National Bureau of Standards, Washington, D. C.I Page 162. F. AJZENBERG-SELOVE, Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania. Page 339. M. BLAU,Institut f u r Radiumforschung, Vienna, Austria. Page 37. M. H. BLEWETT,Brookhaven National Laboratory, Upton, New York. Pages 580 and 623. 0. CHAMBERLAIN, Department of Physics, University of California, Berkeley, California. Page 485. B. CORK,Lawrence Radiation Laboratory, University of California, Berkeley, California. Page 747. H. DANIEL, M a x Planck Institute for Nuclear Physics, Heidelberg, Germany. Page 275. M. DEUTSCH, Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts. Page 303. H. E. DUCKWORTH, Department of Physics, McMaster University, Hamilton, Ontario, Canada. Page 1. R. D. EVANS,Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts. Page 761. G. FEHER,Department of Solid State Physics, University of California, L a Jolla, California. Page 127. H. FRAUENFELDER, Department of Physics and Coordinated Science Laboratory, University of Illinois, Urbana, Illinois. Pages 129 and 214. W. B. FRETTER, Department ifPhysics, University of California, Berkeley, California. Page 35. W. GENTNER, M a x Planck Institute for Nuclear Physics, Heidelberg, Germany. Page 275. S. GESCHWIND, Bell Laboratories, Murray Hill, New Jersey. Page 20. J. G. HIRSCHBERG, Project Matterhorn, Princeton University, Princeton, New Jersey. Page 44. J. C. HUBBS,E-H Research Laboratories, Inc., Oakland, California. Page 58. 1
Present address: 1504 Summit Road, Berkeley, California. V
vi
CONTRIBUTORS TO VOLUME
5,
PART B
C. D. JEFFRIES,Department of Physics, University of California, Berkeley, California. Page 104. C. K. JEN, Applied Physics Laboratory, The Johns Hopkins University, Silver Spring, Maryland. Page 85. H. W. KOCH,Betatron Laboratories, National Bureau of Standards, Washington, D. C. Page 508. H. KOUTS,Brookhaven National Laboratory, Upton, New York. Page 590. D. W. MILLER,Department of Physics, Indiana University, Bloomington, Indiana. Page 366. W. A. NIERENBERG, Department of Physics, University of California, Berkeley, California. Page 58. G. D. O'KELLEY,Oak Ridge National Laboratory, Oak Ridge, Tennessee. Page 555. J. S. PRUITT, Betatron Laboratories, National Bureau of Standards, Washington, D. C. Page 508. S . REED,O$ce of Naval Research, Washington, D. C . Page 807. L. ROSEN,Department of Physics, Los Alamos Scientific Laboratory, Los Alamos, New Mexico. Page 366. A. ROSSI,Department of Physics, University of Illinois, Urbana, IElinois.2 Page 214. R. M. STERNHEIMER, Brookhaven National Laboratory, Upton, New York. Pages 691 and 821. A. H. WAPSTRA,Instztute voor Kernphysisch Onderzoek, Amsterdam, Holland. Page 152. W. WHALING, Department of Physics, California Institute of Technology, Pasadena, California. Page 9.
* Present address: Sorin, Saluggia (Vercelli), Italy.
CONTENTS, VOLUME 5, PART B CONTRIBUTORS TO VOLUME 5, PARTB . . . . . . . . . . . . .
v
CONTRIBUTORS TO VOLUME 5, PART A . . . . . . . . . . . . . xiii 5, PARTA . . . . . . . . . . . . . . . . xv CONTENTS, VOLUME
2. Methods for the Determination of Fundamental Physical Quantities (Continued from Vol. 5, Part A )
2.3. Determination of Mass of Nuclei and of Individual Particles. . . . . . . . . . . . . . . . . . . . . 2.3.1. Mass Spectroscopy. . . . . . . . . . . . . . . by H. E. DUCKWORTH
1 1
2.3.2. Determination of Atomic Masses from Nuclear Reaction and Nuclear Disintegration Energies . . . by WARDWHALING
9
2.3.3. Atomic Mass Determination from Microwave Spectra. . . . . . . . . . . . . . . . . . . 20 by S. GESCHWIND 2.3.4. Measurement of Mass with Cloud Chambers and Bubble Chambers . . . . . . . . . . . . . . 35 by W. B. FRETTER 2.3.5. Determination of Mass of Nucleons in Emulsions by MARIETTA BLAU 2.4. Determination of Spin, Parity, and Nuclear Moments . 2.4.1. Spectroscopic Methods . . . . . . . . . . . . 2.4.1.1. Optical and Ultraviolet Spectroscopy . . by JOSEPH G. HIRSCHBERG
.
37
.
44
. 44 . 44
2.4.1.2. The Investigation of Short-Lived Radionuclei by Atomic Beam Methods. . . . by JOHN C. HUBBS and WILLIAMA. NIERENBERG
58
2.4.1.3. Microwave Method. . . . . . . . . . . 85 by C. K. JEN 2.4.1.4. Nuclear Magnetic and Quadrupole Resonance . . . . . . . . . . . . . . . . 104 by C. D. JEFFRIES vii
...
CONTENTS. VOLUME
Vlll
5.
PART B
2.4.1.4.7.3. The Electron Nuclear Double Resonance (ENDOR) Technique . . . . . . . 127 by G . FEHER 2.4.2. Indirect Methods . . . . . . . . . . 2.4.2.1. Angular Correlation . . . . . by HANSFRAUENFELDER
. . . . . 129
. . . . . 129
2.4.2.2. Conversion Coefficients . . . . . . . . . 152 by A. H . WAPSTRA 2.4.2.3. Nuclear Orientat,ion. . . . . . . . . . . 162 by E . AMBLER 2.5. Determination of the Polarization of Electrons and Photons 214 by H . FRAUENFELDER and A. ROSSI 2.5.1. Introduction . . . . . . . . . . . . . . . . . 214 2.5.2. Description of Polarized Beams . . . . . . . . . 216 2.5.3. Motion of Electrons in Electromagnetic Fields . . 223 2.5.4. Polarization Transfer . . . . . . . . . . . . . . 233 2.5.5. Detection of Electron Polarization . . . . . . . . 239 2.5.6. Detection of Photon Polarization . . . . . . . . 264 2.6. Determination of Life-Time . . . . . . . . . . . . . . 275 2.6.1. Long Life-Time . . . . . . . . . . . . . . . . 275 by H . DANIEL and W . GENTNER 2.6.2. Short Lifetimes . . . . . . . . . . . . . . . . 303 by M . DEUTSCH 2.7. Determination of Nuclear Reactions . . . . . . . . . . 339 2.7.1. Determination of the Q Value for Nuclear Reactions 339 2.7.2. Determination of Nuclear Energy Levels from Reaction Energies . . . . . . . . . . . . . . . 352 by FAYAJZENBERB-SELOVE 2.7.3. 2.7.4. 2.7.5. 2.7.6. 2.7.7.
Total Interaction Cross Sections . . . . . Nonelastic Neutron Cross Sections . . . . . Differential Interaction Cross Sections . . . Differential Elastic-Scattering Cross Sections Supplementary Remarks . . . . . . . . . by LOUISROSENand DANW . MILLER
2.8. Determination of Flux and Densities . . . . . . . 2.8.1. Determination of Flux of Charged Particles . by 0. CHAMBERLAIN
. . . . . . . . .
. . . . . .
366 397 411 472 482
. . . 485 . . . 485
CONTENTSJ VOLUME
5,
ix
PART B
2.8.2. Determination of Differential X-ray Photon Flux and Total Beam Energy . . . . . . . . , . . 508 by J. S. PRUITT and H. W. KOCH
3. Sources of Nuclear Particles and Radiations 3.1. Radioactive Sources . . . . . . . . . . . . .
.
.
. .
by G. D. O J K ~ ~ ~ ~ ~ 3.2. Artificial Sources . . . . . . . . . . . . . . . . . . 3.2.1. Low-Energy Sources . . . . . . . . . . . . . . 3.2.1.1. Cascade Rectifiers . . , . . . . . . . . 3.2.1.2. The Electrostatic (Van de Graaff) Generator by M. H. BLEWETT 3.2.1.3. Nuclear Reactors. . . . . . . . . . . . by H. KOUTS 3.2.2. Medium- and High-Energy Sources . . . . . . . by M. H. BLEWETT
4. Beam Transport Systems 4.1. Introduction , . . . . . . 4.2.
. . . . . . . . . . . . . Beam Bending and Focusing Systems . . , . . . . . .
by R. M. STERNHEIMER 4.2.1. The Trajectories of Particles in a Strong Focusing (Quadrupole) Magnet . . . . . . . . . . . . 4.2.2. The Lens Equations for a Single Quadrupole Magnet and for a Two-Magnet System. Conditions of Double Focusing by a Two-Magnet System . . 4.2.3. The Matrix Method of Calculation for Strong Focusing Magnet Systems . . , . . . . . . . 4.2.4. The Focusing Equations for Deflecting WedgeShaped Magnets with Finite Field Index n . . . 4.3. Beam Separators . . . . . . . . . . . . . by BRUCECORK 4.3.1. Degrader Type Separation . . . . . . 4.3.2. Electromagnetic Separators . . . . . . 4.3.3. Radiofrequency Separators . . . . . . 4.3.4. Separation by Nuclear Interactions . .
555 580 580 581 584 590 623
691 692
694
696 719 731
. . .
747
. . . . . . . . . . . . . . . . . . . . . . . .
747 748 750 751
. . . .
4.4. Some Examples of Beam Transport Systems 752 by BRUCECORK 4.4.1. High Momentum Beams Using Counters as Detectors . . . . . . . . . . . . . . . . , . 752
.
CONTENTS. VOLUME
X
5.
PART B
4.4.2. Separated Beams for Bubble Chambers . 4.4.3. Special Beams . . . . . . . . . . . .
. . . . . 755 . . . . . 759
.
5 Statistical Fluctuations in Nuclear Processes by ROBLEY D . EVANS 5.1. Frequency Distributions . . . . . . 5.1.1. The Binomial Distribution . . 5.1.2. The Multinomial Distribution . 5.1.3. The Normal Distribution . . . 5.1.4. The Poisson Distribution . . . 5.1.5. The Interval Distribution . . . 5.2. Statistical Characterization of Data 5.2.1. Mean Value . . . . . . . . 5.2.2. Variance . . . . . . . . . 5.2.3. Sample Variance . . . . . . 5.2.4. Standard Deviation . . . . 5.2.5. Standard Error . . . . . . 5.2.6. Probable Error . . . . . . 5.2.7. Dimensions . . . . . . . . 5.2.8. Precision vs . Accuracy . . .
. . . . . . . . . 761 . . . . . . . . . 761 . . . . . . . . . 762 . . . . . . . . . 763 . . . . . . . . . 764 . . . . . . . . . 767
. . . . . . . . . . 771 . . . . . . . . . . 772 . . . . . . . . . . 773 . . . . . . . . . . 774 . . . . . . . . . . 774 . . . . . . . . . . 776 . . . . . . . . . . 778 . . . . . . . . . . 778 . . . . . . . . . . 779
5.3. Composite Distributions . . . . . . . . . . . . . . . 780 5.3.1. Combined Probabilities . . . . . . . . . . . . . 780 5.3.2, Superposition of Several Independent Random Processes . . . . . . . . . . . . . . . . . . 781 5.3.3. Propagation of Errors . . . . . . . . . . . . . 783 5.3.4. Difference of Two Mean Values . . . . . . . . . 784 5.3.5. Significance Levels and Confidence Intervals . . . 785 5.4. Tests for Goodness of Fit . . . . . . . . . . . . . . . 786 5.4.1. Pearson’s Chi-square Test . . . . . . . . . . . 786 5.4.2. Poisson Index of Dispersion . . . . . . . . . . . 789 5.4.3. Confidence Interval for the Standard Deviation of a Normal Distribution . . . . . . . . . . . . 791 5.5. Applications of Poisson Statistics to Some Instruments Used in Nuclear Physics . . . . . . . . . . . . . . 792 5.5.1. Effects of Resolving Time in Single-Channel Counting . . . . . . . . . . . . . . . . . . . . . 792 5.5.2. Effects of Resolving Time in Coincidence and Anticoincidence Circuits . . . . . . . . . . . . . 794 5.5.3. Scaling Circuits . . . . . . . . . . . . . . . . 795 5.5.4. Counting-Rate Meters . . . . . . . . . . . . . 799
CONTENTS. VOLUME
5.
xi
PART B
5.6. Useful Inefficient Statistics . . . . . . . . . 5.6.1. Estimate of the Mean Value . . . . 5.6.2. Estimate of Standard Deviation . . . . 5.6.3. Estimate of Standard Error . . . . . . 5.6.4. Estimate of x2. . . . . . . . . . . . 5.6.5. Examples . . . . . . . . . . . . . .
. . . . . 803 . . . . . 803 . . . . . 803 . . . . . 803 . . . . . 804 . . . . . 806
Appendix 1 by SIDNEYREED 1 . Evaluation of Measurement . . . . . . . . . . . . . . . 807 1.1. General Rules . . . . . . . . . . . . . . . . . . . 807 2 . Errors . . . . . . . . . . . . . . . . . . . . . . . . . 808 2.1. Systematic Errors, Accuracy . . . . . . . . . . . . . 808 2.2. Accidental Errors, Precision . . . . . . . . . . . . . 808 3. Statistical Methods . . . . . . . . . . . . . . . . . . . 809 3.1. Mean Value and Variance . . . . . . . . . . . . . . 809 3.2. Statistical Control of Measurements . . . . . . . . . . 810 4 . Direct Measurements . . . . . . 4.1. Errors of Direct Measurements 4.2. Rejection of Data . . . . . . 4.3. Significance of Results . . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
811 812 813 814
5 . Indirect Measurement . . . . . . . . . . . . . . . . . . 814 5.1. Propagation of Errors . . . . . . . . . . . . . . . . 814 6. Prcliminary Estimation . . . . . . . . . . . . . . . . . 819
7. Errors of Computation . . . . . . . . . . . . . . . . . . 819
.
Appendix 2 Kinematics by R . M . STERNHEIMER 1 . Equations for the Lorentz Transformation from the Laboratory System to Center-of-Mass System; The Jacobian for Particle Reactions . . . . . . . . . . . . . . . . . . . . . . 821 2 . Maximum Angle Criterion for the Identification of Outcoming
Particles in Fundamental Collisions . . . . . . . . . . . 828 3 . Kinematics of Two-Body Decay . . . . . . . . . . . . . 831 4 . Threshold Energies for Associated Production of Unstable
Particles . . . . . . . . . . . . . . . . . . . . . . .
833
5 . Phase Space Factors for Two- and Three-Particle Final States 838
xii
CONTENTS. VOLUME
5.
PART B
.
Appendix 3 Properties of Elementary Particles and Particle Resonance States Table 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 845
Table 2 . . . . . . . . . . . . . . . . . . . . . . . . . .
846
Table3 . . . . . . . . . . . . . . . . . . . . . . . . . .
847
AUTHORINDEX. . . . . . . . . . . . . . . . . . . . . . .
849
SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . .
874
CONTRIBUTORS TO VOLUME 5, PART A D. E. ALBURGER, Brookhaven National Laboratory, Upton, New York M. BLAU,Institut f u r Radiumforschung, Vienna, Austria J. E. BROLLEY, JR., Los Alamos Scientijc Laboratory, Los Alamos, New
Mexico
B. CORK, Lawrence Radiation Laboratory, University of California, Berkeley, California J. W. M. DUMOND,Department of Physics, California Institute of Technology, Pasadena, California
R. D. EVANS,Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts H. FRAUENFELDER, Department of Physics, University of Illinois, Urbana, Illinois
W. B. FRETTER, Department of Physics, University of California, Berkeley, California S. S. FRIEDLAND, Solid State Radiations, Inc., Culver City, California
T. R. GERHOLM, Institute of Physics, University of Uppsala, Uppsala, Sweden
W. W. HAVENS,P u p i n Physics Laboratory, Columbia University, New York, New York R. HOFSTADTER, Physics Department, Stanford University, Stanford, California D. J. HUGHES,* Brookhaven National Laboratory, Upton, New York S. J. LINDENBAUM, Brookhaven National Laboratory, Upton, New York G. C. MORRISON, Atomic Energy Establishment, Harwelt, Berkshire, England G. D. 0’KELLEY,Oak Ridge National Laboratory, Oak Ridge, Tennessee F. REINES,Department of Physics, Case Institute of Technology, Cleveland, Ohio G. T. REYNOLDS, Princeton University, Princeton, New Jersey A. SILVERMAN, Department of Physics, Cornell University, Ithaca, New York
* Deceased. xiii
xi v
CONTRIBUTORS TO VOLUME
5,
PART A
R. M. STERNHEIMER, Brookhaven National Laboratory, Upton, New York R. W. WILLIAMS,Department of Physics, University of Washington, Seattle, Washington L. C. L. YUAN,Brookhaven National Laboratory, Upton, New York F. P. ZIEMBA, Solid State Radiations, Inc., Culver City, California
CONTENTS. VOLUME 5. PART A
.
1 Fundamental Principles and Methods of Particle Detection 1.1. Interaction of Radiation with Matter . . . . . . . by R . M . STERNHEIMER 1.1.1. Introduction . . . . . . . . . . . . . . . 1.1.2. The Ionization Loss d R / d x of Charged Particles 1.1.3. Range-Energy Relations . . . . . . . . . . 1.1.4. Scattering of Heavy Particles by Atoms . . . 1.1.5. Passage of Electrons through Matter . . . . . 1.1.6. Multiple Scattering of Charged Particles . . . 1.1.7. Penetration of Gamma Rays . . . . . . . .
. .
1
. . .
1 4 . . 44 . . 55 . . 56 . . 73 . . 76 .
I .2. Ionization Chambers . . . . . . . . . . . . . . . . . 89 by ROBERT W . WILLIAMS 1.2.1. General Considerations . . . . . . . . . . . . . 89 1.2.2. Pulse Formation . . . . . . . . . . . . . . . . 95 1.2.3. Quantitative Operation and Some Practical Considerations . . . . . . . . . . . . . . . . . 100 1.2.4. Amount of Ionization Liberated . . . . . . . . . 103 1.2.5. Noise: Practical Limit of Energy Loss Measurable . 105 1.2.6. Some Types of Pulse Ionization Chambers . . . . 107 1.2.7. Current Ionization Chambers and Integrating Chambers . . . . . . . . . . . . . . . . . 109 1.3. Gas-Filled Counters . . . . . . . . . . . . . . . . . 110 by ROBERTW. WILLIAMS 1.3.1. Gas Multiplication; Proportional Counters . . . . 110 1.3.2. Geiger Counters and Other Breakdown Counters . 118 I .4. Scintillation Counters and Luminescent Chambers . . . . 120 by GEORGE T. REYNOLDS and F. REINES 1.4.1. Scintillation Counters . . . . . . . . . . . . . 120 1.4.2. Solid Luminescent Chambers . . . . . . . . . . 159
1.5. Cerenkov Counters . . . . . . . . . . . . by S. J . LINDENBAUM and LUKEC . L. YUAN 1.5.1. Introduction . . . . . . . . . . . . 1.5.2. Focusing Cerenkov Counters . . . . . 1.5.3. Nonfocusing Counters . . . . . . . . XV
. . . . . 162
. . . . . 162 . . . . . 168 . . . . . 186
xvi
CONTENTS. VOLUME
5.
PART A
1.5.4. Total Shower Absorption eerenkov Counters for Photons and Electrons . . . . . . . . . . . . 189 1.5.5. Other Applications . . . . . . . . . . . . . . . 191 1.6. Cloud Chambers and Bubble Chambers . . . . . . . . 194 by W . B. FRETTER 1.6.1. Cloud Chambers . . . . . . . . . . . . . . . . 194 1.6.2. Bubble Chambers . . . . . . . . . . . . . . . 203 1.7. Photographic Emulsions . . . . . . . . . . . . . . . 208 by M . BLAU 1.7.1, Introduction . . . . . . . . . . . . . . . . . 208 1.7.2. Sensitivity of Nuclear Emulsions . . . . . . . . 210 1.7.3. Processing of Nuclear Emulsions . . . . . . . . . 216 1.7.4. Optical Equipment and Microscopes . . . . . . . 224 1.7.5. Range of Particles in Nuclear Emulsions . . . . . 226 1.7.6. Ionization Measurements in Emulsions . . . . . . 240 1.7.7. Ionization Parameters . . . . . . . . . . . . . 245 1.7.8. Photoelectric Method . . . . . . . . . . . . . 264 1.8. Special Detectors . . . . . . . . . . . . . 1.8.1. The Semiconductor Detector . . . . . and F. P. ZIEMBA by S. S. FRIEDLAND 1.8.2. Spark Chambers . . . . . . . . . . . by BRUCECORK
. . . . . 265 . . . . . 265
. . . . . 281
.
2 Methods for the Determination of Fundamental Physical Quantities 2.1. Determination of Charge and Size . . . . . . . . . . . 2.1.1. Charge of Atomic Nuclei and Particles . . . . . . 2.1.1.1 Rutherford Scattering . . . . . . . . . . . . . 2.1.1.2. Characteristic X-ray Spectra . . . . . . . . . by ROBLEY D . EVANS 2.1.1.3. Charge Determination of Particles in Photographic Emulsions . . . . . . . . . . . . . by M . BLAU 2.1.2. Principal Methods of Measuring Nuclear Size . . . by ROBERTHOFSTADTER
289 289 289 293
298 307
2.2. Determination of Momentum and Energy . . . . . . . 341 2.2.1. Charged Particles . . . . . . . . . . . . . . . 341 2.2.1.1. Measurement of Momentum . Electric and Magnetic Analysis . . . . . . . . . . . . . . . 341 by T. R . GERHOLM
CONTENTS, VOLUME
5,
PART A
xvii
2.2.1.1.4. Measurement of Momentum with Cloud Chambers or Bubble Chambers . . . . . . . . . 375 by W. B. FRETTER 2.2.1.1.5. Momentum Measurement in Nuclear Emulsions 388 by M. BLAU 2.2.1.2. Determination of Energy . . . . . . . . . . . 409 2.2.1.2.1. Energy Measurement with Ionization Chambers 409 by R. W. WXLLIAMS 2.2.1.2.2, Scintillation Spectrometry of Charged Particles 411 by G. D. O'KELLEY 2.2.1.2.3. Measurement of Range and Energy with Cloud Chambersand Bubble Chambers. . . . . . 436 by W. B. FRETTER 2.2.1.3. Determination of Velocity. . . . . . . . . . . 438 2.2.1.3.1. Time-of-Flight Method . . . . . . . . . . . 438 by LUKEG. L. YUANand S. J. LINDENBAUM 2.2.1.3.2. Measurement of Velocity . . . . . . . . . . 444 by W. B. FRETTER 2.2.1.3.3. Measurement of Velocity Using cerenkov Counters . . . . . . . . . . . . . . . . 454 by LUKEC. L. YUANand S. J. LINDENBAUM 2.2.2. Neutrons . . . . . . . . . . . . . . . . . . . 461 2.2.2.1. Recoil Techniques for the Measurement of Neutron Flux, Energy, Linear and Spin Angular Momentum . . . . . . . . . . . . . . . . 461 E. BROLLEY, JR. by JOHN 2.2.2.2. Time-of-Flight Method . . . . . . . . . . . . 495 by W. W. HAVENS,JR. 2.2.2.3. Crystal Diffraction. . . . . . . . . . . . . . 566 by D. J. HUGHES 2.2.2.4. Determination of Momentum and Energy of Neutrons with Her Neutron Spectrometer. . . . . 570 by G. C. MORRISON 2.2.3. Gamma-Rays . . . . . . . . . . . . . . . . . 582 2.2.3.1. Internal and External Conversion Lines. . . . . 582 by T. R. GERHOLM 2.2.3.2. Determination of Momentum and Energy of Gamma Rays with the Curved Crystal Spectrometer . . . . . . . . . . . . . . . 599 by J. W. M. DUMOND
xviii
CONTENTS, VOLUME
5,
PART A
2.2.3.3. Gamma-Ray Scintillation Spectrometry. . . . . 616 by G. D. O’KELLEY 2.2.3.4. Determination of the Momentum and Energy of Gamma Rays with Pair Spectrometers . . . . 641 by D. E. ALBURGER 2.2.3.5. Shower Detectors . . . . . . . . . . . . . . 652 by R. HOFSTADTER 2.2.3.6. Gamma-Ray Telescopes. . . . . . . . . . . . 668 by A. SILVERMAN 2.2.3.7. Measurement of ?-Ray Energy by Absorption. . 671 by ROBLEY D. EVANS 2.2.3.8. Detection and Measurement of Gamma Rays in Photographic Emulsions . . . . . . . . . . 676 by M. BLAU 2.2.4. Neutrino . . . . . . . . . . . . . . . . . . . 682 2.2.4.1. Neutrino Reactions. . . . . . . . . . . . . . 682 by F. REINES
AUTHORINDEX . . . . . . . . . . . . . . . . . . . . . . .
699
SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . .
718
2. METHODS FOR THE DETERMINATION OF FUNDAMENTAL PHYSICAL QUANTITIES (Continued) 2.3. Determination of Mass of Nuclei and of Individual Particles 2.3.1. Mass Spectroscopy1 *
t
2.3.1 .l. Introduction. The first mass spectroscopes were invented by Thomsoq2 A ~ t o nand , ~ Dempster4 in the second decade of this century for the purpose of investigating isotopy among stable atoms. In these instruments, and in most of the mass spectroscopes of the present day, the specific charge ( e / M ) of a positive ion is deduced by observing the deflection experienced by the ion as it passes through electric and/or magnetic fields. The mass of the ion is readily calculated from this as the charge carried by the ion is generally known. Within the precision of the first experiments the masses of all atoms appeared to be integral multiples of that of HI. Small divergences from this “whole number rule” were discovered, however, in 1923 and these have since been the subject of intensive investigation by mass spectroscopists because of the information they yield concerning nuclear stability. The reader will recall in this connection that the mass of an atom is less than the combined masses of its constituent nucleons and electrons, and that it is this difference between deduced and observed masses, known as the binding energy, which accounts for the stability of the atom. Prior to September, 1960, atomic masses were expressed in atomic mass units (OIE = 16 amu), but since September, 1960 they have been expressed in relative nuclidic mass units, abbreviated u, so defined that C12 = 12u. On these scales the masses of all atoms are nearly integral, the nearest integer being in each case the mass number (number of
t See also Vol. 4, A, Section 4.1.1. K. T. Bainbridge, Charged particle dynamics and optics, relative isotopic abundances of the elements, atomic masses. I n “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 1. Wiley, New York, 1953;H. Ewald and H. Hintenberger, ‘‘ Methoden und Anwendungen der Massenspectroskopie.” Verlag Chemie, Weinheim, Germany, 1953; M. G. lnghram and R. J. Hayden, “ A Handbook on Mass Spectroscopy,” Nuclear Energy Series, Report No. 14. National Academy of Sciences, Washington, D.C., 1954;and H. E. Duckworth, “Mass Spectroscopy.” Cambridge Univ. Press, London and New York, 1958. * J. J. Thomson, “Rays of Positive Electricity.” Longmans, Green, London, 1913. F. W. Aston, Phil. Mag. [6]38, 709 (1919). ‘A. J. Dempster, P h y . Rev. 11, 316 (1918). 1
* Section 2.3.1 is by H. E.
Duckworth. 1
2
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
nucleons) of the atom in question, Aston and many since him have expressed the exact mass of an atom as zM*= A ( l f), where f is the “packing fraction” and A and Z are the mass number and atomic number (number of protons), respectively. Thus
+
packing fraction, f = ( M
- A)/A.
An algebraically small packing fraction indicates an atom whose mass is numerically small with respect to its mass number and whose stability is consequently high. It is more common to express this stability in terms of the “binding energy per nucleon,” defined as follows: b.e./nucleon =
ZH’
+ ( A - 2)n - zMA A
where H1and n are the masses of the hydrogen atom and neutron, respectively. The term b.e./nucleon implies that the binding energy is associated with the nucleus, as indeed moat of it is. The binding energies of the orbital electrons, here practically neglected, are not only small but increase with 2 in a gradual manner so that the b.e./nucleon gives a significant picture of the variations and trends in nuclear stability. Modern precision mass spectroscopy is concerned with the study of these variations and trends. 2.3.1.2. Mass Spectroscopes for the Determination of Atomic Mass. The mass spectroscopes employed in the determination of atomic masses are necessarily high resolution instruments. With a single exception those under construction or now in use are instruments in which the ions are deflected by a linear combination of magnetic and electric fields as, for example, in the instrument shown in Fig. 1. The essential parts of such an apparatus are (a) a source of positive ions, (b) a region in which the ions are accelerated, (c) a system of slits for collimating the ion beam, (d) an electromagnetic analyzer which separates the beam into its several specific charge groups, and (e) a device for detecting the ions after analysis. Depending upon whether photographic or electrical detection is employed, “lines” or “peaks” are recorded by the detector, each representing a particular group of ions present in the initial ion beam. The electrical detector of Fig. 1 could be replaced, if desired, by a photographic plate located at Saand lying in the direction SaS4. These instruments possess the property of focusing at the final detector ions which emerge from the collimating system with an angular spread in the plane of the paper and/or with a velocity spread. They are, therefore, said to be “double focusing,’’ a term which has unfortunately been
2.3.
MASS O F NUCLEI AND INDIVIDUAL PARTICLES
3
abrogated by beta-ray spectroscopists to describe instruments which achieve angular focusing in two directions. The original double focusing instruments of Dempster, Bainbridge and Jordan,B and Mattauch,7 and the majority of their lineal descendants, are based on first-order focusing theory. In these the final image
90" ELECTROSTATIC ELECTROS 90"
ELECTRON MULTIPLIER
FIG. 1. Schematic diagram of the Nier double focusing mass spectrometer (from reference 10).
breadth contains aberration terms involving second-order and higher powers of the angular and velocity spread in the transmitted ion beam. Notable exceptions are the mass spectroscope of Nier and his collaborators,*-1° shown in Fig. 1, which possesses second-order direction A. J. Dempster, Proc. Am. Phil. SOC.76, 755 (1935). K. T. Bainbridge and E. B. Jordan, Phys. Rev. 60,282 (1936). 7 J . Mattauch, Phys. Rev. 60, 617 (1936). * A. 0. Nier and T. R. Roberts, Phys. Rev. 81, 507 (1951). E. G. Johnson and A . 0. Nier, Phys. Rev. 91, 10 (1953). l o K. S. Quisenberry, T. T. Scolman, and A . 0. Nier, Phys. Rev. 102, 1071 (1956). 6
6
4
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
focusing plus first-order double focusing, and the new instrument a t the Max Planck Institute for Chemistry11v12which will possess first-order double focusing along the entire photographic plate and second-order direction focusing in the middle of the plate. The resolution of a mass spectroscope is written as A M / M , which implies that two ions of mass M and M AM, respectively, can be just resolved. In instruments of the type we have been discussing, the resolution improves linearly with the radius of curvature of the electrostatic analyzer and also as the widths of the defining slits (Sz, for photographic detection; Sz and 54 for electrical detection) are reduced. With mass spectroscopes employing photographic detection the grain size of the cm) sets the lower limit to the actual line photographic plate width with the result that the resolution cannot be reduced indefinitely by reducing the width of the object slit Sz. Because of this grain size limitation in the case of photographic detection instruments and also because of the difficulty, whatever the method of detection] of adjusting and aligning the diminutive slits required to secure high resolution in any small scale apparatus, the instruments currently (1960) under coiistruction for the purpose of precision mass determinations exceed their predecessors in size by about an order of magnitude. That is, high resolution is being sought by increasing the radius of curvature a, of the electrostatic analyzer (and, of course, the size of the magnetic analyzer, which is related to it). Included in this category are new instruments at Osaka University13*14(a,= 1.09 m), Harvard Universityl6 (ae = 2.14 m), the Max Planck Institute for Chemistry11v12(ae = 5.43m), and McMaster University16 (a,= 2.75 m). Whereas the first double focusing instruments6-’ possessed resolutions ranging from 1/2500 - 1/10,000, the values for these new instruments should be considerably better than 1/100,000.
+
11 H. H. Hintenberger, H. Wende, and L. A. Konig, 2.Naturfursch. lOa, 605 (1 055) ; laa, 88 (1957). H. Hintenberger, J. Mattauch, W. Miiller-Warmuth, H. Voshage, and H. Wende, in “Proceedings of the International Conference on Nuclidic Masses” (H. E. Duckworth, ed.). Univ. of Toronto Press, Toronto, 1960. 18 K. Ogata and H. Matsuda, 2.Naturforsch. 10a, 843 (1955). 14 K. Ogata and H. Matsuda, in “Nuclear Masses and Their Determination” (H. Hintenberger, ed.). Pergamon, London, 1957. 16X. T. Bainbridge, R. C. Barber, H. E. Duckworth, and P. E. Moreland, Jr., in “Proceedings of the International Conference on Nuclidic Masses” (H. E. Duckworth, ed.). Univ. of Toronto Press, Toronto, 1960. 16 N. R. Isenor, R. C. Barber, and H. E. Duckworth, in “Proceedings of the International Conference on Nuclidic Masses” (H. E. Duckworth, ed.). Univ. of Toronto Press, Toronto, 1960.
2.3.
MASS OF NUCLEI AND INDIVIDUAL
PARTICLES
5
Smith, 17-19 a t the Brookhaven National Laboratory, has employed a completely different type of mass spectroscope known as the “mass synchrometer.” Here the ions describe circular paths in a homogeneous magnetic field, the frequency of the motion being f = Be/27rM. This cyclotron frequency is independent of the velocity of the ion but linearly dependent upon its specific charge. Measurements are therefore made of the cyclotron frequencies of the ions under study. The resolution is adjustable electrically between 1/10,000 and 1/25,000. 2.3.1.3. T h e D o u b l e t M e t h o d of Mass Comparison. In theory, in a typical mass spectroscope, one could measure the potential through which an ion is accelerated, ascertain the deflection produced by an analyzer of known characteristics, and from these data calculate the ion mass. This procedure, however, involves a knowledge of certain absolute values which it is not yet possible to determine to the desired precision. Instead, in practice, the mass of an unknown atom is found by comparing it to one whose mass is known. As a rule the known and unknown ions have specific charges which are nearly equal, with the result that the two peaks or lines appear on the mass spectrum at the same mass number. For example, singly charged CL2 ( =12u) and doubly charged Mg24 (=23.9850446 f 19u) both appear a t mass number 12, where they are slightly displaced with respect to one another by virtue of the one part in 1600 mass difference between them. These two constitute a “doublet” and it is the concern of the mass spectroscopist to ascertain, in terms of mass, this doublet spacing. In the main, three factors determine the precision with which mass differences can be determined. These are (a) the accuracy with which the dispersion of the mass spectroscope is known, and (b) the accuracy with which the lines or peaks belonging to the atoms under study can be located, and (c) the extent to which the lack of perfect velocity and angular focusing in the mass spectroscope introduces aberrations in the image. OF DISPERSION LAW,The theoretical dispersion 2.3.1.3.1. UNIFORMITY law governing a mass spectroscope is never realized experimentally because of lack of homogeneity in the analyzer fields. Thus, although it is possible in principle to compare the masses of atoms even if their mass spectral lines are wide apart on a photographic plate, it is difficult in practice to secure a uniform dispersion law over the large region involved. It is much less difficult to do so over the region occupied by a doublet. This consideration has led to the almost exclusive use of the doublet method in mass spectroscopes that employ photographic detection. G. Smith, Phys. Rev. 81, 295 (1951). G. Smith and C . C . Damm, Phys. Rev. 90, 324 (1953). 19L.G. Smith and C. C . Demm, Rev. Sci. Instr. 27,638 (1956). 17L.
18L.
2.
6
DETERMINATION O F FUNDAMENTAL QUANTITIES
In the case of electrical detection the fact that the ions, a t the time of detection, have each traveled identical paths makes the achievement of a uniform dispersion law much easier than in the case of photographic detection. The doublet members are brought in turn to the collector by varying some circuit parameter as, for example, in the case of the mass spectroscope of Fig. 1, a resistance that controls the voltage across the electrostatic analyzer. The mass change is expected to be proportional to the resistance change and is so in practice to a very close approximation. This has been verified by direct determination of the H* mass as, for example, from the CaHs-CaH, mass difference. This, it will be realized, is an extraordinarily stringent test, as these two ions do not constitute a doublet but are spaced one mass unit apart. Twenty-four such determinationsZ0of the H1 mass yielded a mean of H’ = 1.0078239 13u, as compared with the best value from the same laboratory, based on doublet comparisons, of H’ = 1.0078247 jz 2u. The extensive linearity of this dispersion law makes feasible, therefore, the determination of mass-unit differences with a precision which is not much inferior to that associated with doublet comparisons. 2.3.1.3.2. LOCATION OF MASSSPECTRAL LINESOR PEAKS. The precision with which a given mass difference can be determined is dependent upon the accuracy with which the mass spectral lines or peaks may be located, Let us designate the mass width of a line or peak by AM, a quantity which is directly proportional to the resolution of the mass spectroscope. I n the case of photographic recording, the position of a mass spectral line may be determined to some fraction of its width, say, for an observer who is neither unduly optimistic nor unduly conservative, Thus, if a resolution of 1/20,000 be available, a mass difference can be determined with a precision of one part in a million. With electrical recording it has been demonstrated that it is possible to locate a peak with a precision of & of its width, and perhaps even more closely. At least a tenfold improvement over the photographic case is thus obtained. This is done by a “peak-matching” technique introduced by Smith and Damm17 and also employed, in a modified form, by Nierlo and several others. In this technique, by taking advantage of rapidly responding detector systems, the two doublet peaks are made to appear on an oscilloscope screen on alternate sweeps. These two peaks are then brought into coincidence by adjustment of some variable (in Smith’s case, a frequency; in Nier’s case, a resistance) whose value gives the doublet mass difference. The peaks are thus “matched” by the human eye, an organ which can discern lack of coincidence with exceptionally keen discrimination. Two peaks which have almost been brought into coincidence by this method are shown in Fig. 2.
*
10
K. S. Quisenberry, C. F. Giese, and J. L. Benson, Phys. Rev. 107, 1664 (1957).
2.3.
MASS
OF NUCLEI A N D INDIVIDUAL
PARTICLES
7
2.3.1.3.3. IMAGE ABERRATIONS RESULTING FROM IMPERFECT FOCUSING. It has been mentioned that at least two of the modern precision mass
spectroscopes7-~*provide second-order angular focusing. Actually, even for first-order angular focusing instruments, the lack of perfect angular focusing is often not serious for the reason that the ions are strongly collimated during their acceleration (25-50 kev). The lack of perfect velocity focusing, on the other hand, had led in the past t o serious systematic errors in cases where the velocity spread associated with one of the doublet members differs from that associated with the other. This will be true if one of the ions and not the other arises from the dissociation of a molecule. The one may then possess significant initial kinetic energy as, for example, the S32-ion (from HZS or SO,) in the
FIQ. 2. Peak matching of two doublet members. The degree of mismatch shown (as indicated by the lateral displacement of one peak with respect to the other) corresponds to a mass difference of approximately one part in 500,000. Resolution is 1/40,000. (Courtesy of Professor A. 0. C. Nier and Dr. K. S. Quisenberry.) 02-S32 doublet. As i t possesses no velocity focusing feature, the mass synchrometer of Smith1?-l9 is particularly vulnerable in this respect and Smith has, therefore, taken pains to study doublets in which both members are nondissociation or “parent” ions. I n this way, for example, masses for OX6and D2 can be obtainedz1from the two doublets C8Ds-C4D2 and CDd-D20. I n the usual deflection-type mass spectroscope the slit S3 in Fig. 1, located between the electrostatic and magnetic analyzers, serves the purpose of limiting the velocity band which enters the magnetic analyzer. This slit should be narrow enough to ensure that it is completely illuminated by both doublet members. 2.3.1.4. Accuracy of Mass Spectroscopic Mass Values. Although “best” values have a habit of changing, and any listed here will in all probability be out-of-date a t the time of publication of this text, it is 21
L. G. Smith, Bull. Am. Phys. SOC.[2] 2, 223 (1957).
8
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
nevertheless instructive to list a few, if for no other reason than to indicate the degree of precision of modern mass spectroscopy. The values for three key light atoms, shown in the tabulation, are the best available in the autumn of 1960. Value
Reference 20
H’ DZ
1.0078247 k 2u
0’8
15.9949142 -t 5u
Reference 21 2.01409964 f 3u 15.99491550 i 15u
Thus, the error associated with H1 is 200 ev, with D2, 30 ev and with 0 1 6 , 130-500 ev. In the case of D2 the accuracy approaches the point where one must take into account differences in the ionization potential of the two ions forming a doublet. The errors quoted above are statistical ones, based on the internal consistency of the experimental data, and therefore represent the actual errors only if systematic errors be absent. Systematic errors will manifest themselves as discrepancies between the values obtained in laboratories that employ different experimental methods; otherwise there is no basis for suspecting their existence. As mentioned above, the mass of 0l6has been determined to a high degree of precision both a t the University of Minnesota20 and a t Brookhaven National Laboratory, 2 1 using radically different types of mass spectroscopes. The discrepancy between the two values is -1.3 5- 0.6 micro-u, that is, somewhat greater than twice the probable error, suggesting that systematic errors are of more importance than statistical ones. This is a fairly typical example; if anything, the discrepancy is rather larger here than usual. On the whole it appears that the actual probable errors associated with mass spectroscopic mass values are 1.5-2 times the stated statistical errors, with the limits of error being larger again by an additional factor of two. Finally, a word will be said concerning the accuracy as a function of mass. At the time of writing, the actual probable errors in the various mass regions that have been studied with modern techniques are as follows: A = 32, -1.5 kev;20 A = 48, -2 kev;22 A = 64, -4 k e ~ ; ~ ~ A = 132, -5 kev;Z4 A = 150, -0.14 and A = 200, -25 kev.26 22
C. F. Giese and J. L. Benson, Bull. Am. Phys. SOC.[2] 2, 223 (1957).
K. S. Quisenberry, T. T. Scolman, and A. 0. Nier, Phys. Rev. 104, 461 (1956). R. R. Ries, R. A. Damerow, and W. H. Johnson, Jr., in “Proceedings of the International Conference on Nuclidic Masses” (H. E. Duckworth, ed.). Univ. of Toronto Press, Toronto, 1960. 26 V. B. Bhanot, W. H. Johnson, Jr., and A. 0. Nier, Phys. Rev. 120, 235 (1960). *a
24
2.3. MASS
OF NUCLEI AND INDIVIDUAL PARTICLES
9
Further details concerning these and other mass regions can be found in published tab~lations2~,~~,27 of mass spectroscopic atomic mass data.
2.3.2. Determination of Atomic Masses from Nuclear Reaction and Nuclear Disintegration Energies*
The energy balance in a nuclear reaction is expressed in terms of the
Q value of the reaction, the difference between the mass of the target and
+ + +
projectile atoms before the reaction Ma M1, and the mass of the atoms that are produced in the reaction M z M,. The Q value is customarily expressed in energy units, Q = c2(Mo M 1- M z- Ma). For decay processes the Q value is the difference between the mass of the radioactive atom and the daughter atom, in the case of @-decay, or the daughter atom plus a helium atom in the case of a-decay. The electron mass does not appear explicitly as an emitted particle in 6-decay because the daughter atom has just one more, or less, electron than the radioactive atom, and the use of atomic masses rather than nuclear masses automatically includes the emitted electron. However, in order that the Q value for positron decay include all of the energy released in the transition, one must add 2m0c2,the energy which eventually appears as annihilation radiation, to the beta endpoint energy. The atomic masses are customarily expressed in atomic mass units. Prior to 1960, the mass unit universally used in precise studies of atomic masses was OL6/16.In September 1960, the International Union of Pure and Applied Physics adopted C12 as the mass standard and C12/12 as the atomic mass unit in order to take advantage of the fact that mass spectroscopic doublets containing carbon provide the most convenient measurement of the mass of heavy atoms. The new unit CI2/12 is larger than 016/16 by a factor 1.000 317 917 k 17,' and all atomic masses expressed in the new units are thereby decreased, but the mass to energy conversion factor is increased to (C12/12) c2 = 931.441 Mev so that any mass difference expressed in energy units is independent of the mass unit chosen. * B H. E. Duckworth, B. G . Hogg, and E. M. Pennington, Revs. Modern Phys. 26, 463 (1954). 27 H. E. Duckworth, Revs. Modern Phys. 29, 767 (1957). F. Everling, L. A. Konig, J. H. E. Mattauch, and A. H. Wapstra, Nuclear Phys. 16, 342 (1960).
* Section 2.3.2 is by Ward Whaling.
10
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
The following discussion will deal with the mass of atoms in their ground state, and the Q values will always refer to ground-state transitions. The experimental measurement of a Q value is based on the fact that the kinetic energy of the particles produced in the reaction exceeds the kinetic energy of the reacting atoms by just the amount of the Q value, and the experimental measurement consists of the determination of the kinetic energy of all of the atoms involved.* If the particles produced in the reaction carry away some of the energy released in the reaction as internal energy of excitation, either nuclear or atomic, the kinetic energy of the outgoing particles is thereby reduced, and the measured Q value does not represent the true mass difference. Nuclear excitation energy is usually so large that it is not easily overlooked, but it is much more difficult to detect, or otherwise make allowance for, atomic excitation of the residual atom. An estimate2 of the amount of energy that might be expected to appear as electronic excitation indicates that it is of the order of a kilovolt for reactions of atoms with atomic number in the neighborhood of ten. For reactions among nuclei with large atomic number, this atomic excitation energy may constitute a significant correction to the measured Q value3 and must be taken into account. From the mathematical point of view the experimental Q value is equivalent to a linear equation relating two or more masses. By proper choice of the &-value equations, one obtains a set of simultaneous equations which may be solved to find the unknown masses. It will be recognized that this procedure is identical with that used to determine masses from mass-spectroscopic doublet measurements. Indeed, a mass spectroscopic doublet is also a linear relation between unknown masses and hence is exactly equivalent to a Q-value measurement. I n selecting the set of simultaneous equations for solution, one may include mass spectroscopic doublets as well as reaction Q values. The mass tables published in the past few years (1957) have been based either on mass spectrographic doublets or Q values, and the two types of measurements have not been combined. The purpose behind this artificial separation of the data from the two sources is to discover possible inconsistencies between the different methods of measurement. As more accurate measurements remove the sources of disagreement, the information provided by both * A wide variety of techniques have been developed for measuring Q values, of which the most precise are based on electric and magnetic analysis (see Sections 2.2.1.1.1 and 2.7.1). ¶ A . B. Brown,C. W. Snyder, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 82, 159 (1951).
'R. Serber and H. Snyder, Phys. Rev. 87, 152 (1952).
2.3.
MASS OF NUCLEI AND INDIVIDUAL
PARTICLES
11
types of experiment will be combined to provide the most accurate information about atomic masses. The procedure used in determining masses from Q values differs from that used with mass doublets only in that a system of equations involving Q values is necessarily large, much larger than the minimum system of equations that can be used when treating mass doublets. 2.3.2.1. Formulation of Input Data. The large number of equations introduces difficulties, in addition to the obvious mathematical complexity, since it becomes necessary to combine values measured in different laboratories by different methods, values which may be based on inconsistent energy calibration standards. All of the input data must first be examined and, if necessary, corrected to correspond to best values of the energy calibration standards. It is convenient to consider the mass excess M , expressed in energy units, rather than the atomic mass M. If A is the mass number, then M = (M - A)$, and the Q value of a reaction is a simple combination of the mass excesses of the atoms taking part in the reaction. The set of input Q value equations would have the form
2
AikMk =
Qi
AQt
i
=
1, 2, 3,
. . . ,n
(2.3.2.1)
k=l
where A i k is a n integer, usually i 1 or 0, the Jfk are the unknown mass excesses which one wishes to find, Qi is an experimental Q value, and AQ, is the standard deviation in the experimental measurement. It will be assumed throughout the following discussion that AQiare standard errors.* This set of equations, called conditional equations in the mathematical literature, may be solved to find Mk if the following conditions are satisfied. (1) The determinant of the Aik must not be zero. If the set of equations (2.3.2.1)is large, evaluation of the determinant may be quite tedious, and the following graphical method determines immediately whether this condition is satisfied. Let us consider, for example, the set of Q value equations listed in Table I. This set of equations is represented graphically in Fig. 1, in which the nuclides are located as a function of A - 2 and 2. 4 link connecting two nuclides represents a mass difference equation relating the two nuclides. The mass difference equation given by a
* In practice, the errors quoted by various experimental physicists are sometimes standard errors, sometimes probable errors, and are frequently not identified at all. Statistical analyses of experimental data from many laboratories have indicated that the internal consistency of the measurements is such that, on the average, the quoted error is usually greater than the probable error and smaller than the standard error.
2.
12
DETERMINATION O F FUNDAMENTAL QUANTITIES
TABLEI. A Set of 40 Experimental Q Values Expressed as Linear Equations Which Relate the Masses of 20 Nuclides between n and N16 In these equations the symbol for a nuclide represents the mass excess of the atom, expressed in units of MeV.&
(1) n - HI = 0.783 f 0.013 Mev (2) n HI - HZ = 2.227 L- 0.002 = 6.251 f 0.008 (3) n t Ha - H3 (4) 2Ha - n - He3 = 3.267 f 0.007 = 0.0185 f 0.0002 (5) H 3 - He3 (6) He3 n - H1 - H3 = 0.7637 f 0.001 (7) Lie H' - He4 - He3 = 4.023 f 0.002 (8) Lie Ha - Li7 - H1 = 5.027 f 0.003 (9) 2 H a - H' - HS = 4.038 f 0.005 = 1.6449 f 0.0004 (10) Be7 n - H1 - Li7 HI - 2He4 = 17.346 0.010 (11) Li7 (12) Be8 - 2He4 = 0.0941 rt 0.007 (13) Be8 n - Be9 = 1.665 f 0.0014 = 6.816 f 0.006 (14) Be9 n - Bela (15) Be9 H' - HZ - Be8 = 0.559 f 0.001 (16) Be9 H' - He4 - Lie = 2.126 f 0.002 (17) Be9 HZ - H1 - Belo = 4.587 f 0.005 (18) Be9 H Z - H a - Be8 = 4.598 f 0.012 (19) Be9 Ha - He4 - Li7 = 7.153 f 0.003 = 0.557 f 0.004 (20) Belo - BIO (21) BO ' n - He4 - Li7 = 2.786 f 0.008 (22) BIO H' - He4 - Be7 = 1.148 f 0.002 (23) B'O Ha - H' - B" = 9.229 f 0.005 (24) BI1 H1 - He4 - Be8 = 8.585 rt 0.006 (25) B11 Ha - He4 - Be9 = 8.024 rt 0.004 = 4.949 f 0.006 (26) CI2 n - CI3 (27) N13 + n - H2 - C" = 0.281 f 0.003 (28) C'a H* - H' - CIa = 2.721 f 0.002 (29) NIS + n - HI - CI3 = 3.003 f 0.003 (30) C" + H a - H' - CI4 = 5.943 f 0.003 (31) C" Ha - H3 - CIa = 1.310 f 0.003 (32) C13 + H a - He4 - B1' = 5.164 f 0.004 (33) C'4 - N'4 = 0.156 0.001 0.005 (34) N14 + n - HI - CI4 = 0.6264 = 2.23 k 0.010 (35) N'3 - C'* (36) N14 n - Nx6 = 10.832 f 0.008 (37) N" + H a - H' - N" = 8.614 f 0.007 (38) N16 H1 - He4 - CI2 = 4.961 f 0.003 He - He4 - CIa = 7.681 f 0.009 (39) N16 = 3.115 f 0.0025 (40) 0 1 6 + H e - He4 - "4
+
+ + +
+
+
*
+ + + + + + + + + + + + + + +
+
*
+ + +
o This particular set of equations is taken from reference 8. The experimental Q values are t a b n from D. M. Van Patter and W. Whaling, Revs. Modem Phys. 26,
402 (1954)'.
2.3.
MASS OF NUCLEI AND INDIVIDUAL
13
PARTICLES
nuclear reaction Q value is represented by a link connecting the target and residual nuclei in the reaction; e.g., the line connecting Li6 and Li7 might represent the Li6(d,p)Li7reaction, or the equation Li6
+ H* - Li7 - H1 = &.
The legend in the corner of the figure indicates the links established by several common types of reactions.
a 7
6 5 z
4
3 2
I
0
0
I
2
3
4
5
6
7
8
A-Z FIG.1. Graphical representation of the set of &-value equations in Table I. A link connecting two nuclides represents a nuclear reaction or decay process in which one of the nuclides is the target, the other is the residual nucleus. The links established by common reactions are indicated in the lower corner of t h e figure.
In order that the determinant of the set of equations represented by the figure have a nonzero determinant, it is necessary that the links in the figure form a continuous path connecting C12, or some other atom of known mass, with every other nuclide that appears in the set of equations. From the figure it is apparent that the set of equations in Table I satisfies this requirement. However, if the C13(d,a)B" reaction were removed
14
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
from Table I, the remaining set could not be solved. Furthermore, the subset formed by reactions (28)-(40) in Table I can be solved for this group of masses connected to CI2 only if the p , n, d, I, and a masses, the bombarding and emitted particles in these reactions, were furnished as auxiliary information. (2) The mass standard C12,or some other atom of known mass, must appear in a t least one of the equations in the set. This requirement follows from the fact that the conditional equations are restricted to those in which the mass number, had it been included in the expression for the mass, would cancel out of the equation. For example, suppose the first equation in set (2.3.2.1) were n - H’ = QI (we may neglect the +AQi in this discussion). Since the difference n - H’ is the only combination of these masses which cancels the mass numbers, any other equation containing n and H1 must also contain at least one additional mass (e.g., n H1 - H2 = Q 2 ) . Each additional equation adds at least one additional unknown mass, and any system of s equations must contain therefore at least s 1 unknown masses. However, if to this system we add the (a 1)th equation, in which the only new mass appearing is C12,the set of equations now becomes soluble. If our second equation had been 6n 6 H - C12= Q2, the two equations would determine the n and H1 masses. Mattauch has shown that certain linear combinations of the masses, such as the nuclear binding energy, may be determined from a set of equations which does not contain the mass standard.4 2.3.2.2. Reaction Cycles. Since the set of equations (2.3.2.1) usually contains many more equations than unknowns, n > rn, there exist n - rn independent equations relating the Q values in the set. These equations, usually called reaction cycles, are of the form Z, a,&, = 0, where a. is a n integer, usually ? 1. The reaction cycles are useful in screening the input data for experimental Q values that contain gross errors. From the experimental values Q, k AQ, one computes the value of the cycle sum, and the uncertainty in the sum. If the value differs from zero by several times the standard deviation of the sum, it is likely that a t least one of the experimental Q values in the cycle is wrong. One then examines one at a time each Q value in the original cycle by summing other cycles which contain one of the original Q values. I n this way, it is possible to uncover experimental Q values which appear to be in error and eliminate them from the original set before proceeding with the solution.
+
+ +
+
J. Mattauch, The masses of light nuclides according to an adjustment of nuclear data and a comparison with masses based on mass spectroscopic doublets. I n “NUdear Masses and Their Determination’’ (H. Hintenberger, ed.), p. 123. Pergamon, New York, 1957.
2.3.
MASS OF NUCLEI AND INDIVIDUAL PARTICLES
15
TABLE 11. Reaction Cycles These are linear relationships which must be satisfied by the Q values in Table I by virtue of the fact that there are more equations in Table I than unknown masses.'
Table I1 contains 20 reaction cycles that relate the Q values of Table I. The first cycle, Q 1 6 - Q 2 4 &26 may be written out and evaluated
+
+ p - Be8 - d -& - p + Be8 + a Qza B11 + d - Be9 - a QI6 = Be9
-B11
24
=
0
= = =
0.559 k 0.002 -8.570 k 0.009 8.018 k 0.007 0.007 k 0.012
I n this example the experimental Q values are consistent with a cycle sum of zero. Cycles (6)-(11) all contain Q1= n - H1. If the cycle sum is set equal to zero, each cycle (6)-(11) determines a value of the mass difference n - HI, and the values determined in this way may have errors smaller than the direct determinations of the mass difference. For example, cycle (6) gives the value n - H1 = 0.7822 5 0.001 MeV, whereas the direct measurement of the beta-decay endpoint energy gives the experimental value listed in Table I, 0.783 jz 0.013 MeV. The weighted average of the several values of n - H1 obtained from the cycles and the directly measured value, should be substituted in the cycles (6)-(11) to reduce
16
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
the error in the cycle sum. Similarly, cycles (12)-(16) determine values of n H1 - H2 = Q2, cycles (18)-(21) contain the mass difference 2 H Z - He4. In principle, the reaction cycles could be discovered by eliminating the unknown masses from the original set of n equations. However, in practice, it is much easier to construct the cycles by examining the closed figures formed by the links in Fig. 1 . Since each mass must appear at least twice in a reaction cycle, the links representing the reactions in the cycle form a closed polygon in the graphical representation. The sum or difference of any pair of cycles is also a cycle, and the choice of the cycles is therefore somewhat arbitrary. For the purpose of screening input data, the shortest cycles and those with the smallest errors in the sum are most useful. Mattauchs discusses limitations that should be observed in order to obtain independent cycles; he also considers the significance of the deviation from zero relative to the probable error of the sum. 2.3.2.3. Solution of the Equations. The method of least squares is the common method of solving the set of equations (2.3.2.1) to find the most probable values of t'he mass excesses, defined as those values M k for which the sum
+
m
has its minimum value, or those masses M k for which d S / d M k = 0. It is evident from the form of the sum S that an experimental observation Qi f AQi is to be given a weight ( l / A Q i ) 2 .This weighting of the observations may be accomplished conveniently by multiplying the ith equation of the set (2.3.2.1) by a factor (l/AQi). If aik = (l/A&i)Aik, and qi = (l/AQi)Qi, after multiplication the set of equations (2.3.2.1) becomes
and the least squares condition is n
m
for all Mk. Setting the derivative with respect to each of the m d u e s 6
J. Mattauch, L. Waldmann, R. Bieri, and F. Everling, Ann. Rev. Nuclear Sci. 6,
179 (1956).
2.3.
of
Mk
17
MASS OF NUCLEI AND INDIVIDUAL PARTICLES
equal to zero leads to the m equations of the form n
m
C2
n
astaaiMi
s=l i
-
C
astq8= 0,
t = 1, 2, 3,
. . . , m.
(2.3.2.3)
8
The equations (2.3.2.3) are called the normal equations of the least squares solutions. I n vector form Eqs. (2.3.2.3)may be written aaa =
aq
(2.3.2.4)
where a is the matrix whose elements are aik, and 6 is the transpose of 6,&k = ski. The solution of Eq. (2.3.2.4) is =
(2.3.2.5)
(6a)-l&j = bQ
where (iia)-' is the inverse of the matrix &a,and the matrix b is defined by Eq. (2.3.2.5).Inversion of the square matrix 6a may be a very extensive computation if the number of unknown masses is large. Table 111 TABLE 111. Atomic Masses of the First Few Nuclides Appearing in Table I These are most probable values obtained by a least squmes solution of the set of equations in Table I. The values below were computed using the 0l6mass standard; the results are also expressed in terms of the C1*/12 mass unit.= 0 1 8 Mass standard atomic mass unit = 016/16
Nuclide
Mass excess (Mev)
n H1 H* Ha Hea He4
8.3622 f 11 7.5811 f 14 13.7189 f 20 15.8228 f 37 15.8044 f 37 3.6047 f 15
Atomic mass (am4 1.008 1.008 2.014 3.016 3.016 4.003
9806 1417 7334 993 973 8713
f 12 f 15 f 21 f 4
f 4 f 16
C12 Mass standard atomic mass unit = CL*/12
Atomic-mass (&mu) 1.008 1.007 2.014 3.016 3.016 4.002
6599 8213 0931 034 014 5988
Mass excess (MeV)
f 11 8.0662 f 12 f 14 7.2851 f 15 f 20 13.1269 L- 21 f 37 14.935 f 4 f 37 14.916 L- 4 5~ 15 2.4206 f I6
~~
For mass values computed from more recent and accurate Q values, see reference 1.
lists the first few mass values obtained by the least squares method of solution of the observational equations listed in Table I. 2.3.2.4. Accuracy of the Solution. According to Eq. (2.3.2.5), M i is a linear combination of the experimental Q values,
2.
18
DETERMINATION O F FUNDAMENTAL QUANTITIES
and the standard deviation AMi may be computed from the experimental AQ's n
(AMi)2 =
2 n
[(5) AQ.]'
=
aQ8
b:8 =
(bb)ii =
(6~);~.
(2.3.2.6)
8
8
Equation (2.3.2.6) follows from the assumption that all of the AQ's are random errors, independent of each other. Although this assumption is not true in practice, since many Q-value measurements contain a common source of error in the energy calibration standards used, no one has yet attempted to trace the truly independent errors though the vast amount of data involved in the input equations, and the assumption leading to Eq. (2.3.2.6) is usually made. One attractive feature of the least-squares method of solution is that the matrix (&a)-l which contains the AIMi information is computed during the course of the solution. Table IV lists the first few elements of this matrix (&a)-' obtained in the solution of the observational equations in Table I. The square root of the diagonal elements are the errors which appear in Table 111. The off-diagonal elements of the matrix (&a)-l are used to find the error AC in a linear combination of masses, C = ZlclMl, such as a Q value that one might compute from the masses: (AC)z
=
(2.3.2.7)
c~c~(&u),'.
Since the sum in Eq. (2.3.2.7) may contain negative terms, the value of AC may be much smaller than any of the AMt appearing in the expression for C. For example, from Table I11 we find that the value of H3 - He3 is 15.8228 - 15.8044 = 0.0184 MeV. The error in this value of the beta decay energy can be found by Eq. (2.3.2.7) with the matrix elements in Table IV.
A2(H3 - HeS) = aEs,H8
+
U E ~ S , H~ S2
=
13.76
~ ~ 8 , ~ ~ :
- 2(13.77)
+ 13.82
= 0.04; AC = 0.2 kev.
The error in this mass difference is far smaller than the error in H8 or Hea, and reflects the small error in the fourth observational equation in Table I. In the least squares solutions that have been published, Mattauchs and Drummonda have also published the matrix from which the errors can be computed. 8
J. E. Drummond, Phys. Rev. 97,
1004 (1955).
2.3.
19
MASS OF NUCLEI AND INDIVIDUAL PARTICLES
TABLE IV. The Matrix (&z)-~ in Units of (kev)2 The standard error in the neutron mass is ( ( E U ) ; ; ) ~ / * = (1.277 kev*)”a = 1.13 kev. These matrix elements were computed using the 0 1 6 ma68 standard.
n H‘ HZ
Hs Hea He4
n
H’
H2
Ha
Hea
He
1.277 1.191 1.860 2.927 2.924 1.170
1.191 1.903 2.381 3.845 3.874 1.549
1.860 2.381 3.960 5.943 5.964 2.431
2.927 3.845 5.943 13.76 13.77 3.597
2.924 3.874 5.964 13.77 13.82 3.612
1.170 1.549 2.431 3.597 3.612 2.152
2.3.2.5. Other Methods of Solution. From the original set of n equations (2.3.2.1) containing m unknowns, one may choose a particular subset of m equations and solve them to find the m unknown masses. The mass values obtained in this way will, of course, depend on the particular subset chosen, unless the original set of equations (2.3.2.1)is self-consistent, i.e., unless the original equations satisfy the auxiliary conditions imposed by the reaction cycles. This follows from the fact that any particular subset can be converted into any other subset by adding or subtracting reaction cycles; if the cycle sums are zero, the solutions to the modified subset will be the same as the solutions to the initial subset. Hence another method of solution, employed by Tollestrup, Fowler, and Lauritsen17is to adjust the experimental Q values so that one obtains a set of Q values which satisfy all of the reaction cycles. The increment SQi = Q:di - Qf””, the amount which is added to the ith experimental Q value to produce a Q;dj in the self-consistent set, is made directly proportional to the weight of the experimental Q value, SQi a (l/AQi)z. Li et aZ.* describe a solution of this type for the equations listed in Table I in which the adjustment process was iterated until each cycle sum was within 1 kev of zero. Once the set of self-consistent Q values is determined, the solution of any m equations of the original set for the m unknowns is straightforward, as is the calculation of AM from the errors in the adjusted Q values. Whittaker and Robinsong describe the method of computing the error in the adjusted Q values. I n order to compute the error in a linear combination of the masses, it is necessary to express the desired combination as a sum of the adjusted Q values, then compute A. V. Tollestrup, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 78, 372 (1950). 8 C . W. Li, W. Whaling, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 83, 512 (1951). 9 E. Whittaker and G. Robinson, “The Calculus of Observt~tions,”p. 252. Blackie, Glmgow, 1924.
2.
20
DETERMINATION OF FUNDAMENTAL QUANTITIES
the error in the sum from the errors in the adjusted Q values. This method of solutions has the advantage, a t least for hand computation, that in the iterative procedure of adjusting the Q values, one sees a t each stage of the solution which of the input experimental values requires the largest adjustment in order to bring it into consistency with the remaining data. The examination of the input data by means of the reaction cycles is thus continued throughout the solution, and one has the opportunity of altering the choice of input data if that should appear to be desirable.
2.3.3. Atomic Mass Determination from Microwave Spectra* 2.3.3.1. Introduction. In microwave spectroscopy, t information on relative atomic masses is obtained by correlating the shift in frequency of a pure rotational absorption line of a molecule with isotopic mass substitution for one of the atoms in the molecule. The rotational frequencies of molecules fall in the microwave range (1 to 300 mm in wavelength), a region of the electromagnetic spectrum which only became accessible in the last two decades. Prior to the advent of microwave techniques the rotational spectra of molecules were observed as a fine structure on their vibrational spectra which fall in the infrared region. Historically there was one very notable infrared mass measurement of this type at its time. In 1932 Hardy et al.‘ measured the deuterium mass t o an accuracy of 0.1 mmu by comparing the spacing of the vibrational lines in the DCl vibrational spectrum with those of HC1. The most general class of molecule suitable for mass measurements is the symmetric-top molecule shown in Fig. 1 whose rotational energy levels are given byz-* Wg where B
=
==
hBJ(J
+ 1) + h ( A - B ) K z
(2.3.3.1)
h/87r21B and A = h/8?r21A.Here I B and I A are the moments
t See also Vol. 2, Chapter 10.6.1; Vol. 3, Chapter 2.1; and Vol. 4, Chapter 4.1.2. J. D . Hardy, E. F. Barker, and D. M. Dennison, Phys. Rev. 42, 279 (1932). Diatomic Molecules.” Van Nostrand, Princeton, New Jersey, 1950. * G . Herzberg, “Infrared and Raman Spectra.” Van Nostrand, Princeton, New Jersey, 1945. 4 C. H. Townes and A. L. Schawlow, “Microwave Spectroscopy.” McGraw-Hill, New York, 1955. 1
* G. Hemberg, “Spectra of
* Section 2.3.3 is by S. Geschwind.
2.3.
MASS OF NUCLEI AND INDIVIDUAL PARTICLES
21
of inertria of the molecule perpendicular and parallel respectively to the symmetry axis of the molecule; [ J ( J 1)]*% is the total angular momentum of the molecule and K h is its projection along the symmetry axis. If the molecule possesses a permanent electric dipole moment it will have a pure rotational absorption spectrum corresponding to the transition J - 1 -+J with AK = 0. The pure rotational absorption frequencies are then v = 2BJ. (2.3.3.2)
+
In many cases the rotational energy levels of a molecule are split by interactions of the rotation with the nuclear electric quadrupole moment
L-SYMMETRY AXIS 1
OF MOLECULE
I
H
FIQ.1. Symmetric-top molecule.
and nuclear magnetic moment. The theory of these effects are thoroughly explored in reference 4,for example. It has been worked out in sufficient detail so that the value of the rotational frequency as it would be in the absence of these effects can be precisely determined, and will be assumed to have been corrected for in all that follows. For a diatomic molecule B in Eq. (2.3.3.2) is inversely proportional t o the reduced mass which for DC1 is about twice that of HC1 so that the isotopic shift in this case is quite large, i.e., a factor of two in the rotational spacings. However, for heavier atoms such as C1 these shifts are, of course, considerably smaller. For example, in IC1 the shift in rotational frequency for the two isotopes C136and C13’ is only about 5% and, because of the
22
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
limited resolution of infrared techniques, mass measurements in the infrared region are completely unfavorable. However, in the microwave range, due t o the monochromatic oscillator sources available along with refined techniques of frequency measurement, the situation for accurate mass determination becomes extremely favorable. For example, the above-mentioned shift of 5% in the IC1 rotational spectrum corresponds t o a frequency shift of 1200 Mc in the 24,000 Mc region which in turn was measured to an accuracy of 0.01 Mc so that this change of two mass units in C1 can be experimentally measured to an accuracy of 0.05 mmu, and even more accurately (to 0.02 mmu) in other alkali halides. Even more striking is the relative measurement of the carbon and oxygen isotopes in the CO molecule to 2.5 micromass units by Rosenblum et aLs I n general, isotopic frequency shifts may vary from a few Mc to a few thousand Mc per mass unit in the 24 kMc region. In the middle mass region around Se, typical shifts are 100 Mc/mass unit in the 24 kMc region, which measured to an accuracy of 0.002 Mc correspond to a relative mass measurement 0.02 mmu, which compares favorably with other techniques of measurement. The major contributions t o this technique of mass determination have been made by Professor C. H. Townes and his students and reference to their work may be found in the review articles cited which contain tables of all masses measured by this technique.&* The accuracy of experimental measurement of isotopic frequency shift will be determined by absorption signal-to-noise ratio combined with absorption linewidth. These considerations will be outlined in the next section dealing with the various spectrometers. Secondly, the accuracy of mass determination will depend on small molecular corrections needed t o correlate experimentally determined isotopic frequency shifts of rotational spectra with mass change and this will be treated in Section 2.3.3.3. The general requirements for relative mass determination of isotopes of the same atom from microwave spectra may be summarized as follows. (1) The atom should be contained in a molecule with a permanent electric dipole moment (requirement for pure rotational spectrum). (2) The molecule should be in a gaseous state (lo-$ to mm of Hg), or in a molecular beam. B. Rosenblum, A. H. Nethercot, Jr., and C. H. Townes, Phys. Rev. 109,400 (1958). 5. Geschwind, G. R. Gunther-Mohr, and C. H. Townes, Revs. Modern Phys. 26, 444 (1954). 7 S. Geschwind, in “Handbuch der Physik-Encyclopedia of Physics” (S. Flilgge, ed.), Vol. 38, Part 1, p. 38. Springer, Berlin, 1958. 8B. Rosenblum, C. H. Townes, and 5. Geschwind, Revs. Modern Phys. SO, 409 (1958). 5
f)
2.3.
MASS OF NUCLEI A N D INDIVIDUAL PARTICLES
23
(3) The molecule should have a reasonable isotopic shift of its rotational frequency for the atom in question (generally realizable and almost characteristic of microwave spectra). (4) The molecule should have a symmetry no lower than a symmetric top, as the spectra of asymmetric rotors are too complicated for mass determination.
Requirement 2 implies the need for only a very small amount of substance and only gm of S3swas present in the form of OCS in the gas absorption cell in the S3‘jmass determination by W. A. Hardy (private communication). In the case of radioactive isotopes much larger amounts may be needed, however, only to do the chemistry. The list of radioactive isotopes whose masses have been measured by microwave spectra may be found in earlier 2.3.3.2. Experimental Methods for Measuring Microwave Absorption Lines. Detailed accounts of the techniques of microwave spectroscopy may be found in the l i t e r a t ~ r e , ~ so s ~that - ~ ~only the salient features of the different types of spectrometers which have been used in mass determinations will be outlined here, It is characteristic of microwave gas absorption spectra that over many decades of pressure peak absorption intensity is constant, as long as the linewidths are determined by collision broadening. Thus, while the total number of molecules is proportional to pressure the peak absorption of an individual molecule is inversely proportional to the mean time between collisions or pressure, so that the peak absorption intensity of the gas is constant. As narrower lines are obviously desirable for more accurate frequency measurement, one therefore tries to work at as low a pressure as possible to lo-* mm of Hg) before other sources of line broadening set in, after which any further reduction in pressure would mean a loss in intensity. The different sources of line broadening are listed in Table I and are evaluated for the case of the linear molecule OCS. This table is pertinent to an understanding of the factors that lead to the selection of a particular spectrometer as described below. The factors contributing to the peak absorption coefficient, a! (the fractional absorption per centimeter), such as the dipole moment, matrix element for the transition, etc., are fully discussed in Chapter 13 of the work by Townes and S c h a ~ l o wThese . ~ authors also include various tables in the appendices which are helpful in calculating a! for the different molecules. The value of a! may range from to cm-’ or less in the M. W. P. Strandberg, “Microwave Spectroscopy.” Wiley, New York, 1954. W. Gordy, W. V. Smith, and R. F. Trambarulo, “Microwave Spectroscopy.” Wiley, New York, 1953. l1 “Microwave Spectroscopy.” Ann. N . Y . Acad. Sci. 66 (November, 1952). 9
lo
24
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
TABLE I. Contributions to Line Width of Microwave Gas Spectra Evaluated 2(Av) for OCS at room temperature
Half line width, AV (half-half width)d Doppler effect
A V / V = 3.58 X l O - T z / T l i i ?
Collision broadening (10-1 mm Hg)
Avo0ll
Power saturation and collision broadening.
120 kc
= 1 / 2 ~
” ‘v AvcO1l
38 kc at 24 kMc
-
4’
~?FPI~.&,,, 280 kc for P = 200pw -I- hic A ~ ; ~ , , and Avcall = 120 kc
Stark-modulation broadeningb
120 kc for f = 120 kc
dT
-
Wall collisionsc
Av
‘v
Av ,
0.6 X lo9
M
17 kc for 24 kMc wavcguide
R. Karplus and J. Schwinger, Phys. Rev. 73, 1020 (1948). R. Karplus, Phys. Rev. 73, 1027 (1948). M. Danos and 5. Geschwind, Phys. Rev. 91, 1159 (1953). KEY: T = temperature in degrees Kelvin; M = molecular weight; T = mean time between collisions; P = microwave power flow in ergs/sec cma; Ip.t,lmsx = largest dipole moment matrix element between the rotational levels a and b; f = Stark modulation frequency; A / V = wall area to volume ratio of gas absorption cell in cm-1. a
b
24 kMc region. A discussion of the limiting sensitivity of gas absorption cell spectrometers may be found in many place^.^*^^^^ 2.3.3.2.1. GAS ABSORPTION CELLS. 2.3.3.2.1.1 Basic Gas Absorption Cell (Crystal Video Detection). An outline of the waveguide gas absorption cell is shown in Fig. 2. The molecule under examination is contained in the hollow waveguide cell, in gaseous form at a pressure of approximately to mm. The microwave klystron oscillator is swept in frequency by application of a sawtooth voltage t o the repeller electrode. This same sawtJoothvoltage is simultaneously applied t o the horizontal plates of an oscilloscope to provide a time base proportional to frequency. As the frequency of the oscillator sweeps through a molecular absorption in the gas cell, the resultant reduction in the power reaching the microwave crystal detector, after passage through a video amplifier, is displayed on the vertical plates of the oscilloscope. The strength of these absorptions expressed in terms of the fractional absorption per centimeter, a, ranges from lo-* t o cm-l or less in the 24,000 Mc region. These absorption coefficients vary with the third power of frequency so 12
C. H. Townes and 6. Geschwind, J . Appl. Phys. 19, 795L.
2.3. MASS
25
OF NUCLEI AND INDIVIDUAL PARTICLES
that for very light molecules such as CO whose first rotational transition is in the 100 kMc region, the absorption becomes as large as 50% in a 6-foot length of waveguide. The frequency of an absorption line is measured very precisely by comparing the signal oscillator frequency with the harmonics of a standard quartz crystal oscillator which have been obtained by multiplication up to microwave f r e q ~ e n c i e s . ~ * ' ~ WAVEGUIDE GAS. ABSORPTION CELL
GAS INLET
TO VACUUM PUMP
\
\
M SAW TOOTH VOLTAGE GENERATOR
AMPLIFIER CROSS SECTION
FIG.2. Basic gas absorption cell microwave spectrometer.
2.3.3.2.1.2. Stark-Modulation Spectrometer. In practice these very small absorptions are masked by spurious signals from frequency sensitive guide reflections and discontinuities in the mode pattern of the microwave source klystron. To separate the true absorption from these spurious effects, the characteristic gas absorption frequencies are Stark modulated by applying a time varying (100 kc) square wave of voltage between the guide proper and a metal septum insulated from the guide as shown in Fig. 3. This technique was first introduced by Hughes and Wilson.14 The 100 kc component of the absorption is detected by a crystal rectifier, amplified in a tuned 100 kc amplifier and then detected in a phasesensitive detector. This scheme of detection greatly reduces spurious signals, since the amplitude of noise or spurious signals which fall in the pass band of the amplifier is small. Widths of absorption lines, as measl*S. Geschwind, Ann. N . Y . Acad. Sci. 66, 751 (1952). I4 R. H, Hughes and E. B. Wilson, Jr., Phys. Rev. 71, 562 (1947).
26
2.
DETERMINATION OR FUNDAMENTAL QUANTITIES
ured between points of half-maximum intensity, will vary from about 150 to 300 kc/sec in this type of spectrometer. This width results from the combined effects of Doppler broadening, pressure broadening, collisions with walls, Stark modulation, and power saturation (see Table I). Power saturation and collision broadening are major contributors to this width. A minimum of several hundred microwatts of power must impinge upon the silicon crystal detector for optimum detection sensitivity. However, this amount of power will saturate the absorption lines
SIGNAL OSCl LLATOR
I d l
-i
I
- VARIABLE -ATTENUATOR
I 1
I
1
INSULATING ,*STRIPS STARK WAVE GU IDE SECT I ON
i00 KC SQUARE WAVE GENERATOR
'I '
CRYSTAL DETECTOR
TUNED 100 KC AMPLIFIER
I
1 SAWTOOTH GEN ERATOR
-
FIG.3. Stark-Modulation spectrometer.
if the gas pressure is reduced too far, and so one is constrained by this power requirement for good detection sensitivity to work at pressures where collision broadening is a factor. 2.3.3.2.1.3. Balanced Microwave Bridge with Superheterodyne Detection.'* If one could use lower power levels without a reduction in sensitivity and consequently reduce the gas pressure, then linewidths would be limited by Doppler effect and wall collisions. This can be done by using a balanced microwave bridge with superheterodyne detection (shown in Fig. 4). The optimum power necessary for sensitive detection is furnished by the beat oscillator rather than the signal oscillator whose power can now be kept low enough (a few microwatts) t o avoid saturation with
2.3.
MASS OF NUCLEI AND INDIVIDUAL PARTICLES
27
the reduced gas sample pressure of 10-3 to 10-4 mm. In the initial work with this spectrometer, the Stark modulation was dispensed with t o avoid distortion or displacement of the absorption line by Stark effects. Even without Stark modulation, amplitude fluctuations and mode discontinuities in the signal klystron and reflections in the arms of the bridge were canceled in first order by use of a symmetrical bridge. Linewidth of 60 kc (for OCS) and a sensitivity of a 3X cm-' were achieved with this spectrometer. The limit in sensitivity was here imposed by uncancelled second-order effects of reflections in the bridge arms.
-
SAW TOOTH
FIG.4. Balanced microwave bridge spectrometer with superheterodyne detection.
An improvement of sensitivity of a factor of ten in the balanced bridge scheme was realized by adding low frequency, 1 kc Stark modulation.16 However, great care must be exercised to accurately clamp the Stark voltage on zero to avoid displacement of the absorption line. 2.3.3.2.1.4. High Temperature Spectrometer. A noteworthy achievement was the extension of microwave gas absorption spectroscopy t o very Such a high temperahigh temperatures (25Oo-930"C) by Stitch et aZ.16v17 ture is necessary to obtain a sufficient vapor pressure for the alkali halide diatomic molecules, many of whose pure rotational absorption spectrum were studied by this technique yielding accurate mass information for the C1, Br, Li, K, and Rb isotope^.'^ A standard 100 kc Stark modulation spectrometer was used in which formidable difficulties were overcome in extending it to this high temperature range. l a G . R. Gunther-Mob, R. L. White, A. L. Schawlow, W. E. Good, and D. K. Coles, Phys. Rev. 94, 1184 (1954). 16 M. L. Stitch, A. Honig, and C. H. Townes, Rev. Sci. Instr. 26, 759 (1954). A. Honig, M. Mmdel, M. L. Stitch, and C. H. Townes, 86, 629 (1954).
28
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
2.3.3.2.1.5. Millimeter-Wave Spectrometers. Since the absorption coefficient, a,varies as the third power of the frequency, a in the wavelength region of 1 to 10 mm is often high enough so that absorptions can be seen directly without the need for Stark-modulation or superheterodyne techniques. Furthermore, Stark modulation cells are quite lossy a t millimeter wavelengths so that instead one uses frequency modulation of the signal klystron. This source modulation was used in a millimeter-wave spectrometer by Rosenblum et al.6,1sin their very high precision determination of the carbon, oxygen, and hydrogen mass ratios. 2.3.3.2.2. MOLECULAR BEAMELECTRIC RESONANCE.19s2o The ultimate limit upon linewidth imposed by the Doppler broadening in the gas absorption cell spectrometers, described in the preceding section, is '
MICROWAVE RADIATION
0
A-FIELD
C-FIELD
0-FIELD
\
FLOPPED-, BEAM
HOT WIRE DETECTOR
>
FIG.5. Molecular beam electric resonance apparatus.
circumvented in the electric resonance molecular beam type spectrometer shown in Fig. 5. The beam of molecules traverses successive regions of oppositely directed inhomogeneous electric fields separated by an intermediate region (the c-field) which for the study of rotational transitions has no static field. The effective electric dipole moment, peff,of a molecule is, among other things, a function of the rotational state, J. In addition, the force on a molecule is proportional to p e f f ( ~ E / d 2 Thus, ). by a suitable arrangement of stops a particular rotational state is focused at the detector. By irradiating the molecules in the c-region with microwaves of the proper frequency, transitions are induced from the J to J 1 rotational state, The resultant change in the J for some of the molecules changes the magnitude of their deflection in the B-field so that they are not refocused at the detector, causing a decrease in beam intensity.
+
B. Rosenblum, A. H. Nethercot, Jr., Phya. Rev. 97, 84 (1955). H. K. Hughes, Phye. Rev. 72, 614 (1948). zo J. W. Trischka, Phys. Rev. 74, 718 (1948). l*
lo
2.3.
MASS OF NUCLEI AND INDIVIDUAL PARTICLES
29
Doppler broadening is absent as the molecules travel in essentially one direction relative to the microwave field. The linewidth is determined by the amount of time T , the molecules spend in the transition region 1. Typical linewidths that have been as is given by the relation Av * T achieved in mass measurements with this type of spectrometer range from 10 to 20 kc.21 By use of split RF field techniques,22linewidths of 1 kc are possible although no mass measurements have been made by this technique. 2.3.3.2.3. MASER-TYPESPECTROMETER.23A third general type Of microwave gas spectrometer is the gas-type maser (Fig. 6). While no mass measurements have been made with this device, it represents such an important conceptual and practical advance in the state of the art that it is worthy of mention here. I t is a combination of the above methods in that a molecular beam is used but a change in microwave
-
! U -d OUTPUT GUIDE
INPUT GUIDE
I,=_=L,_,_,_,---'----------_--_______________--__----J
GAS SOURCE
CAVITY
FOCUSER
FIG. 6. Gas maser spectrometer.
power absorbed by the beam is measured. The beam of molecules passes through a system of electrodes which produce a field gradient which focuses only those molecules in a particular energy state above the ground state into a microwave cavity. Microwave radiation of the proper frequency will stimulate emission to the ground state so that the output radiation is enhanced. A linewidth of 4 kc has been obtained for NHI at room t e m p e r a t ~ r e . ~ ~ , ~ ~ The maser-type spectrometer should also be capable of far greater sensitivity than the conventional ones.26 2.3.3.3. Correlation between Isotopic Shift of Rotational Frequency MOLECULES. The correlation and Mass Change. 2.3.3.3.1. DIATOMIC between frequency shift in the rotational spectrum and mass change is C. A. Lee, B. P. Fabricand, R. 0. Carlson, and I. I. Rabi, Phys. Rev. 91,1395 (1953) . N. F. Ramsey, Phys. Rev. 76, 996 (1949). 23 J. P. Gordon, H. Zeiger, and C. H . Townes, Phys. Rev. 95, 282 (1954). Z 4 J. P . Gordon, Phys. Rev. 99, 1253 (1955). M. W. P. Strandberg and H . Dreicer, Phys. Rev. 94, 1393 (1954). Z 8 C. H. Townes, Phys. Rev. Letters 6, 428 (1960). *I
22
30
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
most easily made for a diatomic molecule. Equation (2.3.3.2)is only a first approximation which assumes a rigid rotator. The effect of the vibration of the molecule is in a first approximation represented by v = 2J[B, -
(Y(V
+ 4)]
(2.3.3.4)
where Be = h/8r21, and I , is the equilibrium moment of inertia assuming the nuclei are stationary at their equilibrium separation, r,; v is the vibration quantum number and a is a correction which measures the rotation-vibration interaction; I , = pr.% where p is the reduced mass of the molecule. By measuring v for two different vibrational states, a can be found and hence Be. The equilibrium internuclear distance, r,, is determined to a very high approximation only by the electronic structure of a molecule, so that it is the same for different isotopic species. Thus by measuring the equilibrium B values first with masses M and mo and then with M and ml, where ml and mo are isotopes of the same atom, the relative masses of ml and mo will be given by (2.3.3.5)
Note that knowledge of re or constants such as h is not required. Assuming (ml - mo) is small, an uncertainty in M/mo of A produces an uncertainty in ml/mowhich is given by (2.3.3.6)
In practice this is usually negligible. In RbC1, for example, an uncertainty of 1 mmp in both Rb and C1 produces errors of only 1 part in lo7in the C1 mass ratio or Rb mass ratio. There are a set of approximations or uncertainties connected with Eq. (2.3.3.5)most of which can usually be very accurately corrected for or are negligible, which will now be briefly described. (a) Centrifugal distortion of the molecule due to rotation. This correction to Be is of the order of B,3/u82where ws is the vibrational frequency of the molecule. In all but the very lightest molecules this correction is usually negligible, When it is important, however, we can usually be determined with sufficient accuracy from optical band spectra to make this correction. 4sb (b) Anharmonicity of the potential function.27 Here corrections are also of the order of B,3/wz and are usually negligible except in lighter molecules, where they can also be corrected for as in *'J. 4. Duaham, Phys. Rev. 41, 721 (1932).
2.3.
MASS OF NUCLEI A N D INDIVIDUAL PARTICLES
31
(c) Contribution of molecular electrons to moment of inertia. Implicit in the writing of Eq. (2.3.3.5) is that the mass of an entire neutral atom, nucleus plus electrons is concentrated at a point. In reality, of course, the electrons have an extension in space about the nuclei and one must consider their extra contribution to the moment of inertia of the molecule due to this spatial extension. If the electrons are assumed to be almost spherically distributed about their respective nuclei and if indeed they rotated rigidly with the molecule, then one might expect the moment of inertia to be greater than the point mass assumption by approximately an amount equal to the moment of inertia of the electrons about their respective nuclei. If really present, this contribution to the moment VALENCE ELECTRONS NUCLEUS I
/’ ./
CLOSED
FIG.7. Illustration of the slipping of closed shell electrons in a rotating molecule.
of inertia would be quite large; but fortunately it is absent because the orientation of a closed shell of electrons remains fixed in space as the molecule r0tates.28,2~This slipping of the closed shells as the molecule rotates is illustrated in Fig. 7 and is akin t o the motion of a chair on a ferris wheel. On the other hand, the valence electrons may be visualized as rotating rigidly with the molecule and will give an extra contribution t o the moment of inertia, A single rigidly jotating valence electron at an average distance of approximately 1 A from its nucleus would give an error of less than one part per million in the C136/C137ratio in the alkali halides and is in general negligible in the heavier molecules. However, a similar valence electron in a lighter molecule such as CO results in an uncertainty of one part in lo6in the C12/C1aratio, which is one hundred times greater than the experimental error in this case. 2 8 G. C. Wick, Phys. Rev. 75, 51 (1948). 29See C. H. Townes and A. L. Schawlow, “Microwave Spectroscopy,” p. 213. McGraw-Ell, New York, 1955.
32
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
However, the nonslipping valence electrons contribute proportionately t o the magnetic moment of the molecule, so that by measuring the magnetic moment of the molecule, one can very accurately correct for this extra contribution, A I of the valence electrons t o the moment of inertia above the idealized situation. This connection between the two derives from nothing more than Larmor’s theorem for rotating charges and is given by4J-g (2.3.3.7) where B:‘Om is the idealized Be value assuming all the mass for the neutral atom concentrated at a point. Here M p is the proton mass; m, the electron mass; p2&,the nuclear magneton; pJ, the magnetic moment of the molecule in the state J . As magnetic moments of molecules are of the order of a nuclear magneton for lighter molecules, this fractional correction t o the B value will be of the order of a part in several thousand and corresponded t o a correction of one part in loEin the C13/C12ratio in CO, which was one hundred times greater than the experimental error.6 (d) “Wobble” stretching: Rosenblum et aL6 have pointed out that in a Z molecule, even though the projection of the electronic orbital angular momentum along the internuclear axis is zero, it may still have perpendicular components which precess about the internuclear axis causing it t o “wobble” and hence produce a type of centrifugal distortion. The effect of wobble stretching can be taken into account only by using one accurately known mass ratio t o compare with the microwave value of the same mass ratio. In this sense it represents a basic limit on the accuracy of mass determination by the microwave method. This correction in CO (which was ten times the experimental error) amounted t o about 1 part in lo6 in the C and 0 mass ratios and was applied to the C12/C1sand 018/01E mass ratio determination using the known C12/C14 ratio to calibrate for this correction. The very detailed examination of the afore-mentioned electronic corrections t o mass determinations in the molecule CO by Rosenblum et aLs has confirmed the validity of the molecular theory involved t o very high accuracy. This is illustrated in Table I1 by noting the consistency of their results for the Cla/C12ratio in the molecules C 0 l Eand CO1*and the good agreement of these results with other determinations. In the higher mass region these electronic corrections are far less important. This is borne out by R b mass ratios in Table I1 which have not been corrected for elect,ronic effects. The microwave result in RbCl which was obtained with a molecular beam apparatus is seen t o be consistent with the determinations in RbI and RbBr which were made in a
2.3.
33
MASS O F NUCLEI AND INDIVIDUAL PARTICLES
TABLE11. Illustrative Mass Determinations in Diatomic Molecules Mass ratio Molecule
From microwave spectra
Rbsa/Rb87 RbC136 RbI RbBr C13/C1* C016 COls
0.9770163 k 45" 0.9770177 f 45d 0.9770146 f 55d 1.08361283 f 27" 1.08361290 f 28"
From nuclear reactions ~~~
~~~~
From mass spectra ~~
~
0.9770191 f 20 1.08361308 f 20a 1,08361278 k 200
1.08361309
3*
See reference 5. T. T. Scolman, K. S. Quisenberry, and A. 0. Nier, Phys. Rev. 102, 1076 (1956). J. W. Trischka and R. Braunstein, Phys. Rev. 06, 968 (1954). See reference 17. a T. L. Collins, W. H. Johnson, Jr., and A. 0. Nier, Phys. Rev. 94, 398 (1954).
0
high temperature gas absorption cell, and all three in good agreement with the mass spectrographic result. 2.3.3.3.2. MOLECULES MORECOMPLEX THANDIATOMIC MOLECULES. In molecules other than diatomic, zero point vibrations prevent the determination of accurate mas6 ratios but nonetheless an accurate determination of atomic masses can be made in terms of mass difference ratios. In analogy to the representation of the rotation-vibration interaction for a diatomic molecule, the effective B value, Bo for a polyatomic molecule may be written as2-4
B O = Ba -
2
ai(Vi
+ di/2)
(2.3.3.8)
i
where Be = equilibrium B value, ai is the rotation-vibration interaction, vi is the vibrational quantum number, and di the degree of degeneracy of the ith vibrational mode. In order to determine Be, one must measure BOfor the same rotational transition for each vibrational mode in both the ground vibrational state, vi = 0, and the first excited state, vi = 1. This is, however, exceedingly difficult as the Boltzmann population of the excited vibrational states of the less abundant isotopic species will be too low so that to measure all the a's is in general beyond the reach of the sensitivity of present-day microwave techniques. In addition, the a's in a polyatomic molecule depend in a rather complex way upon anharmonic potential constants and mass, so that knowing their values for a more abundant isotopic species does not allow one to theoretically predict their values for the less abundant species as one can do in a simple diatomic molecule. By a straightforward application of the parallel axis theorem of mechanics, it can be shown that the following relationship obtains for
34
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
isotopic substitutions of an atom lying on the symmetry axis of a symmetric-top or linear molecule : 4 * 6
m l - mo - BLl)) - M1B!2)(BLo) M~B:~)(B:o’ -~12))‘ m2 - mo
(2.3.3.9)
Here mo,m land m2 are the isotopic masses, M1 and MO are the molecular masses with isotopes ml and m2, respectively, and the B,‘s are the equilibrium B values corresponding t o the three different isotopic species. While the equilibrium B values appear in Eq. (2.3.3.8), it can be shown that the error introduced by using instead the effective or measured B value, Bo (determined from the measured rotational frequency by vrot = 2BoJ), is small. For example, a detailed estimate* of the error introduced by this neglect of a in the (P3- S a 2 ) / ( S 3 *- S12) mass ratio as determined from the linear molecule OCS shows that it is less than 1 part in 15,000. This corresponds to an uncertainty of 0.03 mmp in the determination of S33mass taking the and V4masses as known, and is representative of the accuracy obtainable in the middle mass region. As a matter of fact the best evidence for the smallness of the error introduced by neglecting zero point vibrations is furnished by the very good agreement between mass difference ratios determined by the microwave method, as outlined, neglecting zero point vibrations, and other methods. Such comparisons have been tabulated in several p l a ~ e s For . ~ exam~ ~ ~ ~ ~ ple, in OCS the oxygen atom, being light, undergoes large displacements from its equilibrium position so that one might expect the effect of zero point vibrations to show up in this case; yet the microwave value of (0’’ - 016)/(018 - OI6) = 0.501042 f 8 6 is in good agreement with the most recent mass spectroscopic values of 0.501045 & 3 3 1 and 0.5010446 jz 9 . 3 2 In addition, it can be shown that the typical uncertainty in M1/Mz is also generally unimportant.6 Thus if one measures the rotational frequencies corresponding to three isotopic substitutions for an atom on the symmetry axis of a symmetric top molecule and takes the masses of two of the isotopes as known, one can accurately determine the mass of the third. Taking the masses of two of the isotopes as known can be regarded as a calibration of the isotope shift and the effect of zero point vibrations. Note that no information concerning the structural parameters of the molecule, such as bond angles and internuclear distances is required. 30 S.Geschwind, in “Nuclear Masses and Their Determination,,’ p. 163. Pergamon, New York, 1947. 31 H. Ewald, Z. Naturjorsch. 6a, 293 (1951). 31 T. T. Scolman, K. S.Quisenberry, and A. 0. Nier, Phys. Rev. 102, 107 (1956).
2.3.
MASS OF NUCLEI AND INDIVIDUAL
PARTICLES
35
Further increase of the experimental accuracy, would bring one into the region where electronic effects and zero point vibrations would undoubtedly become important. It would be very difficult to correct for these effects in polyatomic molecules as one is able to do for diatomic molecules and so one must regard the present state of the theory of the electronic structure of polyatomic molecules as setting the limit of accuracy attainable in mass determinations in polyatomic molecules.
2.3.4. Measurement of Mass with Cloud Chambers and Bubble Chambers*
Cloud chambers have proved very useful in the determination of the mass of the unstable nuclear particles produced in nuclear disintegrations, and with the rapid development of bubble chamber technique, there is every hope that mass measurements will be commonly made with those instruments. The mass of a charged particle detected in a cloud or bubble chamber may be determined in various ways. The most important of these are the following: 1. 2. 3. 4. 5.
Momentum-ionization Momentum-range (energy) Momentum-change in momentum Ionization-range Scattering-range
Momentum-ionization. If the momentum p = mopyc and the velocity @c are measured, the mass mo can be found, since y depends only on p. The momentum can be measured either prior to the entry of the particle into the chamber, or in the chamber, if the chamber is immersed in a magnetic field. Likewise the velocity may be determined either within the chamber by measurement of ionization, or outside the chamber by time-of-flight, cerenkov, or ionization techniques. Since these two measurements are independent, the error in the mass is obtained from
where A p and AI are the errors in p and I , and m ois taken to be a function of p and I . It is found that the error in ionization usually contributes
* Section 2.3.4 is by W.
6. Fretter.
36
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
more to the error in mass than does the error in momentum, when both are measured' in the same chamber. Under favorable circumstances the error in mass for an individual particle may be as small as 5%. I n the relativistic region, where the slope of the rise is rather slow, the error in mass determination is much larger, but it is still possible to distinguish mesons from nucleons in the momentum range of 5-20 Bevlc. Momentum-range. If the particle can be stopped and its range determined, and a range-energy relation is available, the mass can be found from a combination of the equations p = moPycand kinetic energy
= moc2(y -
1).
There are two general procedures used, depending on the energy of the particle: either the particle is stopped in the medium of the chamber, or in heavy material as in a multiplate chamber. In either case a separate measurement of the momentum is usually necessary, since momentum measurements are virtually impossible in a multiplate chamber and inaccurate in a chamber when a particle approaches the end of its range. In a large bubble chamber, however, momentum and range measurements are simultaneously possible. Where the range-energy relation is accurately established, and accurate momentum measurements can be made, the momentum-range method is the most accurate for mass measurements. The range straggling error on an individual measurement is only one or two per cent, and momentum measurements can usually be made to a per cent or so under good conditions. Depending on the region in which the measurement is done, the error in mass on an individual determination is also of the order of one or two per cent. Momentum-change in momentum. When it is inconvenient to bring a particle to rest, but momentum measurements may be made, the momentum loss of a particle in passing through a given thickness of material may be used together with the momentum itself, to determine the mass. This method was discussed by Wheeler and Ladenburg.* It is not very accurate because it involves differences in momentum measurements which are ordinarily fairly small, and is not often used. Ionization-range. If momentum measurements cannot be made, the mass can be determined if the velocity is found from the ionization, and the energy from the range of the particle. Because of the inaccuracies in the ionization measurements, and the limited region over which they can be made, this method has not been used very much, except for fairly rough measurements. In a large bubble chamber, where simultaneous range and velocity measurements should be possible, this should provide
* W. B. Fretter and E. W. Friesen, Rev. Sci. Innstr. 26, 703 (1965). * J. A. Wheeler and R. Ladenburg, Phys. Rev.
60, 754 (1941).
2.3.
MASS
OF NUCLEI A N D INDIVIDUAL
PARTICLES
37
a method for mass measurements in the absence of a magnetic field. If the medium is so heavy as to give large multiple scattering making an ordinary magnetic field useless, this method should give useful results, as i t does in emulsions. Scattering-range. Rarely applied to chamber conditions, this method gives mo-yp2cand moc2(r- 1) which can be combined to determine the mass. It has been applied to multiplate chambers3 and to high-pressure cloud chamber^.^
2.3.5. Determination of Mass of Nucleons in Emulsions* I n previous sections (2.1.1.3, 2.2.1.1, an d 2.2.1.2) the method of charge, momentum, and energy determination of particles have been discussed. I n all calculations, connected with these problems, the knowledge of the particle mass or the simultaneous mass determination was implicitly involved. Therefore we will discuss below only a few cases of mass determination which are interesting from a methodical standpoint. The discussion will be limited to particles of single charge and mainly treat particles ending in the emulsion. 2.3.5.1. Constant Sagitta M e t h ~ d . l J .I~n the case of singly charged, nonrelativistic particles (pat = 2E), ending in the emulsion, the rangeenergy relation together with the measured mean scattering angle give sufficient information for the determination of the particle mass. The range-energy relation can be written in the form
E
= aRnm1-n
(2.3.5.1)
where a is a constant and n the power factor which actually is not a constant, but varies slightly with particle ~ e l o c i t yThe . ~ mean scattering angle is given by6
(2.3.5.2)
K being the scattering constant and t the cell length. Using relation C. Peyrou, Nuovo cimento [9]11, Suppl. No. 2, 322 (1954). P. Baxter and F. R. Stannard, Proc. Phys. Soc. (London) A70, 19 (1957). 1 S. Biswas, E. C. George, and B. Peters, Proc. Indian Acad. Sci. S A Y418 (1953). 2 C. Dilworth, S. J. Goldsack, and L. Hirschberg, Nuovo cimento 191 11, 113 (1954). 3 R. G . Glasser, Phys. Rev. 98, 174 (1955). 4 See Section 1.7.5.2. 6 See Section 2.2.1.2. 3 4
~
* Section 2.3.5 is by
Marietta Blau.
38
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
(2.3.5.1) with the approximation that the product momentum times velocity is equal to twice the particle energy, (2.3.5.2) can be written in the form 8 = -tllZR-nmn-l K
2a
(2.3.5.3)
where m is the only unknown magnitude, which therefore can be calculated. The method of constant sagitta, which has been briefly mentioned in connection with other scattering methods, was developed independently by the laboratories of Brussels and Bombay and was presented in a joint communication at the Congress on Cosmic Rays at Bagnere de Bigore, July, 1953. The most common procedure in scattering measurements of fast particles is the use of constant cell lengths along the whole track or limited track segments within which the particle energy can be considered constant. The rate of energy loss per unit length, however, increases with decreasing residual range and toward the end of a track the sections of nearly constant energy become exceedingly small. I n this region, measurements made with cell lengths, appropriate for the more energetic part of the track, lead by necessity to results of low statistical accuracy. In the constant sagitta method variable cell lengths are chosen in such a way that the mean sagitta remains constant along the whole track. With the aid of the range-energy relation one can calculate for particles of given mass the cell length required such that at a given residual range the mean value of the second difference I) has a predetermined value A, (2.3.5.4)
The relation between t, R, and m for a given value A can be written: t3/2
-
ml-nRn.
(2.3.5.5)
The general procedure in constructing a “scattering scheme” is the following. One starts at a point of the track with the residual range Ro and chooses for the first cell a value t o ; the subsequent cells tl, t2, . . . tn can be easily found from a log-log plot, describing the relation (2.3.5.5) for a particle of given mass m. A particle of mass m2 # ml, measured with the same scattering scheme will again lead to a constant second difference D,, which, however, is different from D,. The ratio of the two second differences is related to the ratio of the masses by (2.3.5.6)
2.3.
MASS OF NUCLEI AND INDIVIDUAL PARTICLES
39
Equations (2.3.5.5) and (2.3.5.6) are not strictly correct for various reasons. (1) The power factor n is not constant, but varies with particle energy. (2) The relation p@c = 2E is not valid for higher energies. (3) The scattering constant varies slightly with particle velocity and cell length. (4) Corrections for “noise,” so far neglected, have to be introduced. The effects due to factors 1-3 can be accounted for in various ways. Dilworth et aL2 have shown that an adequate procedure consists in first calculating the scattering scheme with a single power factor (the authors use n = 0.58), assuming a constant value for the scattering constant and neglecting deviations due to the assumption ppc = 2E, and afterward using correction terms to adjust the calculated value D,,,‘to Bobs,the value of the mean second difference observed. Biswas et al.’ use values for the power factor n which were found in experiments with protons with energies up to 35 MeV, and, for higher energies, values derived from a n empirical formula; for the scattering constant theoretical values, depending on cell length and particle energy, were used. Glasser3 determines the constants b and n
ID1 = bR-n
(2.3.5.4a)
from measurements on proton tracks of 57 Mev from a cyclotron beam by fitting the experimental data to a power law and found a power factor n, = 0.607. Equation (2.3.5.4a) is derived from (2.3.5.4) and the measurements were started a t a point RO= to of the track, where to is the initial cell length: the following cell lengths ti are then related to the first by ti = t o ( R / t ~ ) 2 n ’ [compare 3 (2.3.5.5)]. According to Glasser’s experiments the mean absolute second differences as a function of particle mass are then given
ID1 = 0.053
e)””” (to)0.893.
(2.3.5.7)
where m, is the proton mass. The noise problem (point 4) is less complicated in the constant sagitta than in the constant cell method. The cells are smaller and therefore the troublesome stage noise is either completely absent or has a t least a constant value. The most important source of noise is the grain noise, arising from the fact that the grain centers are not completely aligned along the trajectories: the error is independent of cell length and of the order of half a grain diameter. The independence of the error from cell length is a great advantage of the constant sagitta method, because of the possibility of choosing cell lengths in such a way that the signal-tonoise ratio assumes an optimum value for the whole trajectory.
2.
40
DETERMINATION OF FUNDAMENTAL QUANTITIES
According to Biswas et al.’ the optimum conditions for the signal-tonoise ratio can be expressed by the relation A = 2.42
(2.3.5.8)
where A, the mean absolute value of the true second differences, is given by A = dDZ - p2, and a the mean value of the noise. Relation (2.3.5.8) suggests that the scattering scheme should be chosen in accordance with particle mass, since A is a function of mass [Eq. (2.3.5.6)]. RESIDUAL RANGE R in mm,SCALE
IT
-
500
200 -SCALE
II
i
I
100
I I-
(3
z w
-1
50-
-I -I W
0
2 0 - SCALE
10
-
I
20
50 100 200 RESIDUAL RANGE IN p ,SCALE
500
1000
I
FIQ.1. Plot of cell length versus residual range for protons and pions
Figure I, taken from the paper by Biswas et aLli is a plot of cell length versus residual range for protons and pions, chosen in such a way that the mean absolute second differences after noise subtraction assume the constant values of 1, and 1.6 p, respectively. Glasser’sa experiments indicate that the optimum signal-to-noise ratio is obtained for D = 2.4Z where D is the observed value due to scattering noise. He finds that the optimum initial cell t o for R = t o is 4.4 p for protons, 1.9 for pions, and 3.9 p for a particle of mass 1000 me, where m, is the electron mms.
+
41
2 . 3 . MASS OF NUCLEI AND INDIVIDUAL PARTICLES
The error in mass measurements is relatively large, even if optimum conditions are used since: am 1 aA - = -_. m 1-nA [See Eqs. (2.3.5.4) and (2.3.5.5).] In Table I (Glasser) the percentage statistical error in mass measurements for particles of 1000 m, measured a t optimum conditions is given for tracks of various lengths. The error becomes considerably less severe, of course, if many tracks are available. TABLE I. Statistical Error in Mass Measurements Available length,
(I0 1000 3000 6000 10,000 50,000
Number of cells
45 81 131 178 350
Errors on 1 track
(%I
Errors on 25 tracks (%)
45 33 26 23 14
8.9 6.7 5.3 4.5 3
Tables with scattering schemes for various particles were published by different authors. 2.3.5.2. Photometric Method. Mass measurements on singly charged particles, ending in the emulsion, can be made with any one of the ionization methods described in earlier sections. However, for dense tracks near the end of the range, the blob counting or mean gap length methods are not suitable. In this case the method of gap or blob length per unit path length should be used, or a photometric method based on opacity or profile measurements of the track. In the following the Lund photometric method, the so-called mean track width method, will be discussed. The general measuring arrangement and the measuring procedure has already been described in the section on charge determination. Details of the multiplier equipment, the electronic part, and the stabilized high-voltage supply of the measuring instrument are published by von Friesen and Kristiansson.' The objective used in the measurements is a K.S. X 100, and the dimensions of the slit referred to the objective plane are 2 . 4 X 30 1.1. The optimum slit dimensions depend on the density of the track, the track inclination, and the background conditions. Each measurement with the track H. Fay, K. Gottstein, and K. Hain, Nuovo cimento Suppl. [9]11, 234 (1954). S.von Friesen and K. Kristiansson, Arkiv Fysik 4, 505 (1952).
42
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
segment below the slit was followed by two background measurements to each side of the track; the three measurements are used for the determination of the true opacity, due only to grains activated by the particle along its path through the emulsion. The method of “mass measurement” is based upon the following considerations, The mean track width, or MTW(Rm), of a particle of mass m a t a point with residual range R is for singly charged particles a function of the particle velocity, v , and hence of the ratio R/m. (2.3.5.9)
Because MTW depends also on development and emulsion conditions in a complicated way, it was found impossible to obtain a simple analytic expression for f(R/m). found it expedient to write the function MTW (Rm) The in the form of a power series: MTW(Z2m) = MTWo
+
(2.3.5.10)
where A, B . . . in the expansion depend on emulsion and development condition, but are independent of particle mass; MTWo, the mean track width a t zero range, is normalized to the value 100. The number of terms in the expansion depends on the length of the track. The constants A, B . . . can be determined for a special emulsion stack and development conditions if particles of known mass, ending in the emulsion are available. The MTW of two particles with masses ml and mz a t the same residual range can be written in the form MTW,
=
100
+ A I R + BIRZ+ -
. and MTWZ = 100
+ AzR + BzR2 +
*
.
*
where (Al/Az) = (mz/ml) and (Bl/B2) = (mz/m1)2. It is advantageous to check this ratio on tracks of particles with known mass in order to ascertain the absence of systematic errors. The mass of an unknown particle is determined by the method of least squares with the condition, that Eq. (2.3.5.10)should fit the experimental results as well as possible, or that
0=
2
{MTWL- MTW(Rm)) aMTW(Rm) am
n 8
S.von Friesen and K. Kristiansson, Arkiv Fysik 8, 121 (1954). K. Kristiansson, Arkiu Fysik 8, 311 (1954).
(2.3.5.11)
2.3.
MASS OF NUCLEI AND INDIVIDUAL
PARTICLES
43
where MTW1, MTWz . . . MTWL represent measured values of the unknown track a t residual ranges R I , Rz, . . . RL. The mass of the unknown particle m, is related to the mass m of the known particle by (m/m,) = (RL/RL,) if R I , R2, . . . RL and R1,,R2, . . . RL, are the residual ranges where the ionization and hence the MTW of the two particles are equal. Particles of unknown mass should be compared with protons or other known particle tracks found a t nearly the same part of the emulsion. The value of the MTW of a track depends on the depth in the emulsion; dept,h corrections can be applied, but may introduce additional errors. Furthermore the inclination of tracks used for comparison should be the same, and the best results can be obtained only if the dip angle is small. The error in mass measurements depends on the total track length measured of the particle used for comparison, on the residual range of the unknown particle, on the mass ratio, and finally on the resolving power of the emulsion. The errors are relatively small if optimum conditions prevail, because the method is based on a great number of single measurements. Kristianssons estimates the error in mass to 5 3 % ’ if tracks of a t least 1 cm length are available. 2.3.5.3. M a s s Measurement of Particles Which Do Not End in the Emulsion. I n the case of particles not ending in the emulsion, the conditions are more complicated, because in addition to the mass the particle velocity must be determined. It is difficult to treat this problem in a general way, since it depends strongly on the velocity of the particle, length of track, dip angle, and emulsion conditions. For particle tracks near minimum ionization, where the grain density does not change perceptibly over an appreciable length in the emulsion, the only possible method in determining the mass consists in a combination of ionization and scattering measurements (constant cell method). In this case it is helpful to use sets of curves-scattering angle or p p versus blob density-for particles of various masses. The accuracy of the method depends nearly entirely on the error made in the scattering measurement; in general this error will be large unless the tracks are very long and flat and the emulsion is free from distortion. I n gray tracks with more than 1.5 minimum ionization, the gradient of the blob density along the track gives additional information on the particle mass, provided that the track length is sufficient. Johansson and Kristianssonlo have applied the photometric method to gray tracks which do not end in the emulsion. They approximate the MTW versus range relation by a straight line, and the slope of this line represents the rate lo
F. Johansson and K. Kristiansson, Arkiv Fysik 11, 467 (1956).
44
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
of change, d(MTW),/dR, which is a measure of the particle mass. They again use normalization procedures and comparison with tracks of particles of known mass and velocity. The MTW and d(MTW)/dR, the latter in units of MTW, of the unknown track, are calculated by the least square method. They find that identification is possible even without additional scattering measurements if very long and flat tracks are available; the best results were obtained with blob densities between 2-5 times minimum value, but even with tracks as light as 1.75 times, minimum ionization mass measurements can be made if the track length exceeds 10 mm. Yekutielli and Rosendorf" have recommended blob density measurements along the track, plotted as a function of the distance from the entrance of the track to its final disappearance. They discuss the problem in a mathematical way and give the solution of the problem in an analytical form.
2.4. Determination of Spin, Parity, and Nuclear Moments 2.4.1. Spectroscopic Methods 2.4.1.1. Optical and Ultraviolet Spectroscopy.* t 2.4.1.1.1. HISTORICAL BACKGROUND. Hyperfine structure in optical spectra was discovered in 1891 by A. A. Michelson using his celebrated interferometer. Fabry and Perot, and Lummer and Gehreke also brought their interferometers to bear, and by 1910 a considerable number of observations had been made. At length it became clear that two causes of this structure were present, one due to the different isotopes of an element, now usually called isotope shift, and the other due to some effect within a single isotope. I t was not until 1924 that Pauli' suggested that the hfs in the case of a single isotope was due to nuclear moments. Subsequent experiment rapidly disclosed that many nuclei possess spin, and a number of magnetic and electric quadrupole moments were 11
R. Rosendorf and C. Yekutielli, Nuovo eimento Suppt. [9] 12, 416 (1954).
t See also Vol. 3, Part 2; Vol. 4, A, Chapter 4.2. W. Pauli, Naiurwissenschaften 12, 741 (1924).
J. E. Mack, Revs. Modern Phys. 22, 64 (1950). G. Laukien, Handbuch der Physik, XXXVIII/l, 338 (1958). G. H. Fuller, 1960 Nuclear Data Tables, National Academy of Sciences, National Research Council Part IV, Table 4, 72 (1961). J
4
~
* Section 2.4.1.1is by Joseph G.
Hirschberg.
2.4.
DETERMINATION OF SPIN,
PARITY,
A N D NUCLEAR MOMENTS
45
In the case of elements with several isotopes the measurement of nuclear moments is complicated by the fact that the structure of each of the various isotopes is displaced by the isotope shift. Tin, for example, has ten stable isotopes, several of which have additional hfs caused by nuclear spin. With the advent, since the Second World War, of the availability of separated isotopes, the evaluation of isotope shifts and nuclear moments has been greatly facilitated. 2.4.1.1.2. ISOTOPE SHIFT.A short description of isotope shift is included here for two reasons. In any evaluation of nuclear moments of an element with several isotopes, the isotope shift structure must be disentangled from the structure due to the moments of the nucleus in order to measure the latter. Also, the isotope shifts themselves, in the case of heavy elements may directly shed light on the structure of the nucleus. The isotope shift can be divided into three categories: that for light elements, that for heavy elements, and that for intermediate elements. For light elements, the effect is due to the difference in mass of the nucleus for the different isotopes, In its simplest form this occurs in the spectrum of hydrogen. In one-electron spectra the effect of a finite nuclear mass is taken into consideration by the use of the reduced mass p :
Mm ' " W m
(2.4.1.1.1)
where M is the nuclear mass and m is the electron mass. The energy T , of a particular level will depend on M , the nuclear mass according to (2.4.1.1.2) where T , is the value which would obtain if M were infinite. The energy levels of a lighter isotope are shifted more toward the series limit than those of a heavier one. This simple situation obtains only with the two-body case; hydrogen and hydrogen-like spectra. When more than one electron is involved, corrections to the simple reduced mass solution must be made. In this case, detailed knowledge of the wave functions is necessary to apply the corrections, and the measured shifts do not agree well with calculation, except in the simplest cases. Beyond mass 50 or 60, the mass effect becomes very small, and is finally obscured by the nuclear volume isotope shift which may be thought of as being caused by the finite size of the nucleus. Since the nucleus is not an infinitely small point, but has a radius of the order of All3 X 10-13 cm, the
46
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
Coulomb field breaks down in its neighborhood. Orbital electrons whose wave functions have an appreciable value at the nucleus, such as configurations involving s and to a smaller extent p electrons, are most affected. This effect becomes important for 2 > 20 and increases toward heavier nuclei approximately as Z2.The effect on the energy levels is just opposite to the mass effect, the energy levels of the lighter isotopes lying lowest in the energy level diagram. For details in the evaluation of both shifts the reader is referred to references 5, 6, and especially 7. 2.4.1.1.3. EFFECTSOF NUCLEAR MOMENTSAND SPIN. Besides the isotope shift another cause of hyperfine structure was soon required, since such structure was often found to occur in nuclei with only a single isotope. As indicated above, this was ascribed in 1924 by Pauli to the existence of a nuclear spin, I, and to the interaction between the resulting nuclear moments and those of the optical electrons. The most important of the nuclear moments is the magnetic moment pr. The change in the energy of the atom due to the magnetic energy between this moment and the orbital electrons is given by: AW,
=
-
-prH(O)cos prH(0)
(2.4.1.1.3)
where H(O)is the average value of the magnetic field of the orbital electrons in the neighborhood of the nucleus; p~ is the magnitude of the nuclear magnetic moment; H(0) and pr are the vector field and nuclear moment respectively. This expression becomes : AW,
=
A I J cos(1,J)
(2.4.1.1.4)
when it is remembered that H(0) and J are antiparallel (due to the negative sign of the electronic charge), and pr is parallel to I (because the proton has a positive magneticlmoment). A = -prH(O)/IJ and is evaluated by a method considered below. We may form a vector diagram for the composition of two angular moments (see references 6 and 7), as in Fig. 1, where I, the spin of the nucleus, and J, the total angular momentum of the electrons, are combined to form F. We can apply the law of cosines and write cos(1,J) =
FZ - 1 2 - J 2 21J
(2.4.1.1.5)
5E. U. Condon and G . H. Shortley, “Atomic Spectra.” Cambridge Univ. Press, London and New York (1935). 6 H. E. White, “Introduction to Atomic Spectra.‘’ McGraw-Hill, New York, 1934. 7 H. Kopfermann, “Nuclear Moments.” Academic Press, New York, 1958.
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
47
Thus :
W,
=
A - (F2 - I 2 - J')
(2.4.1.1.6)
2
A rigorous solution on the quantum mechanics6 necessitates the following changes: F2, 12,and J 2 are replaced by F ( F l), 1(1 l), and J ( J 1) respectively. The magnitudes pr and H ( 0 ) are replaced by the z components of and H(0) if I and J are taken parallel to the z axis. Thus we obtain: A (2.4.1.1.7) W , = 2, F ( P 1) - 1 ( 1 1) - J ( J 1)
+
+
+
+
+
+
where F can assume the values
I+J,
I + J - l ,
I + J - 2 * . . ( I - J ) . (2.4.1.1.8)
This expression for the splitting AW, results in the characteristic hyperfine flag patterns, in which the same interval rule occurs as in
FIG. 1 . Vector diagram for combining I and J.
FIG.2. Hyperhe structure levels showing interval rule.
ordinary fine structure: The energy difference between two adjacent hfs levels, for instance one of F and one of F - 1, is just proportional to F. For example, if I = and J = 2 then F can have the values 4, $, +, 4 by Eq. (2.4.1.1.8)and the levels will appear as in Fig. 2. In many cases, the nuclear spin can be determined simply from the number of levels, or from the intervals, but in more difficult cases where some of the structure cannot be resolved, intensities must be determined. Detailed methods are discussed by Kopfermann.7 Sometimes it is noticed that the interval rule is not exactly followed. This effect is attributed to an electrostatic interaction with the orbital electrons by the nuclear quadrupole moment. The only other nuclear moment which normally can be measured by
2.
48
DETERMINATION OF FUNDAMENTAL QUANTITIES
means of hyperfine structure, is the electric quadrupole moment, Q. The additional energy due t o the quadrupole moment AWQ is given by: (2.4.1.1.9)
where e is the elect,ronic charge, the electric gradient a t the nucleus in the z direction and 0 is the angle between the direction of the quadrupole moment and the electric field gradient. The quadrupole moment may be thought of as a measure of the departure of the nucleus from spherical shape and has positive or negative sign according t o whether the nucleus is prolate or oblate, respectively. Casimirs has expressed the formula for AWQ as follows:
where
c = F ( F + 1) - 1(1+ 1) - J(J + 1)
and
B
= eQ@JJ(O)
where ipJJ(0)is the average gradient in the J direction of the electric field due t o the orbital electrons at the nucleus. We have now obtained expressions for the effect on the atomic energy levels of the magnetic dipole and electric quadrupole moments in terms of two constants A and B. If these formulas are combined we obtain for the total effect: AWF = AWm AWQ. (2.4.1.1.11)
+
As has been suggested above, much the largest effect is due t o th e magnetic moment ; the AWQ is often negligible and when present can be treated as a perturbation on AW, resulting in a departure from the intervals predicted. The hyperfine structure levels are affected by external magnetic fields in a similar way t o the ordinary fine-structure levels, resulting in Zeeman and Paschen-Back effects. A discussion of these will be found in Kopfermann’ and White,6 and is beyond the scope of this treatment. I n the study of spectra, transitions between levels are observed, not the levels themselves. This requires a consideration of the combination rules, telling us which levels combine with each other. The general rule is that values of a given F will combine so that AF = 0 or k 1. The intensities depend on I and J as well as F , and tables may be found in standard texts. 8
H. B. G. Casimir, Teyler’s Tweede Genootshap, Verhandel. (Haadem) (1936).
2.4.
DETERMINATION
OF SPIN, PARITY,
AND NUCLEAR MOMENTS
49
As an example of the transitions to be expected, consider a simple example: I = $, and J = 0 and 1. Here (see Fig. 3) there are just as many lines as F levels in the lower J state since there is only one hyperfine upper state. Other simple cases arise when one of the states arises from a configuration not split by the nuclear moments, the result then being similar t o the example.
1/2
FIG.3. Hyperfine structure transitions.
2.4.1.1.4. EVALUATION OF MOMENTS. The nuclear moments which have been determined from the study of hfs are the mechanical or spin moment I , the magnetic dipole moment p, and the electric quadrupole moment Q. As mentioned above, F , the total angular momentum, can assume J - 2 * - ( I - J ) , (or if J is the values I J , 1 J - 1, I larger than I , two I 1 levels) and for each value of F there is a hyperfine structure level. I can be determined if the spectrum is sufficiently well resolved by counting the maximum number of hfs components. If this cannot be done, for example because of insufficient resolution, the interval rule, intensities, or Paschen-Back effect may be used. By the use of Eqs. (2.4.1.1.7) and (2.4.1.1.10), the constants A and B can be deduced from the measured structure. As indicated above, A is related t o the nuclear magnetic dipole moment and B t o the electric quadrupole moment, and these latter could be determined exactly if the values of the mean magnitude field H ( 0 ) and the mean electric gradient in the J direction in the neighborhood of the nucleus were known. Goudsmitg has deduced formulae for A in the single electron case in terms of the nuclear g-factor gr, where pr = (e/2MP)hIgr, e is the electron charge, h is the Planck-Dirac constant, M , is the proton mass, c is the velocity of light, and I is the nuclear spin; for electrons with I > 0:
+
+ +
A. =
+
MPhcR,a2Z3 mn3(Z
9
-
+ + ) j ( j + 1) gz
S. Goudsmit, Phys. Rev. 48, 636 (1933).
(2.4.1.1.12)
50
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
and for s electrons
8 M,hcR,a2ZS A, = 3
mn3
BI
(2.4.1.1.13)
+,
Here R, is the Rydberg, a the fine structure constant, 1 and j the orbital and total electronic momentum quantum numbers respectively. For alkali-like and alkaline earth-like spectra similar formulas exist, semiempirical in nature, and indeed these calculations can be extrapolated to complex spectra with good accuracy.I Casimirs has obtained an expression for the B factor involving the quadrupole moment Q : 1 2j B = -@&(2.4.1.1.14) < l / r 3 > Rr(l,j) H. 2j+ 2 Unfortunately, the factor < l / r > , the average reciprocal of the magnitude of the electron radius vector, appears in this equation, but. this can be eliminated by consideration of the A factor in the derivation of which < l / r > also appears. The following relation7 can then be obtained:
Q
+
B PB' 41(1 1) P I m 1 F, A e2 j(2j - 1) pn M, I R,
= EOPO- -
(2.4.1.1.15)
where eo and P O are the electric and magnetic fundamental constants, 8.85 X 10-l2 farads per meter and 41 X lo-' henrys per meter respectively, p n is the nuclear magneton, pB the Bohr magneton, p r the nuclear magnetic moment and P, and R, are relativity corrections tabulated by Kopfermann. 'I 2.4.1.1.5. LIGHTSOURCES. The hfs splitting is smaller than the ordinary fine structure of atomic spectra by about the ratio of the nuclear magneton p n t o the electron magneton, or roughly %&. Thus, the evaluation of nuclear properties by optical spectroscopic methods ordinarily involves the measurement of very small energy differences, often of the order of thousandths of a wave number unit, To measure energy differences of the order of a few thousandths of a wave number unit, only a few light sources can be used. The principal causes of broadening that are encountered are: Doppler effect, Zeeman effect, and Stark broadening. Doppler broadening resulting from random thermal motion is given by:
where r D is the half-intensity breadth of the Doppler-broadened line, T is the absolute temperature of the source, and M the molecular weight
2.4.
DETERMINATION
OF SPIN, PARITY, AND NUCLEAR MOMENTS
61
of the radiating atom. From this we see that for narrow lines a cooled source is necessary; for light elements cooling by liquid air is commonly used. An atomic beam, where the effective temperature may be very small indeed, has been used t o obtain some of the narrowest spectral lines yet attained. The Zeeman effect is proportional t o the magnetic field strength, and in most sources is not a troublesome source of broadening. The Stark effect is of two types. A relatively large effect proportional t o the electric field occurs in hydrogen, and hydrogen-like atoms. In other cases, a much smaller quadratic effect obtains, proportional to the square of the field. Because of random intermolecular and interatomic electric fields, the linear Stark effect can contribute t o the broadening of spectral lines, the Stark broadening, r8,being approximately proportional to the density. For hydrogen and hydrogen-like spectra subject to the linear Stark effect, rdis especially large and the spectral lines are greatly broadened, especially at high pressures or if large externally applied electric fields are present. Because of the factors discussed above, we see that sources for high resolution must exhibit low temperature and low density, and should have as small magnetic and electric fields as possible. Other requirements, in some cases, are at least moderate intensity and sometimes high excitation. The accompanying tabulation summarizes the most commonly used sources with their advantages. ~~~~~~~~~
Type of source Arc Spark Vacuum spark Hollow cathode
~
Excitation energy Medium High Very high Medium
Electrodeless Low (radio frequency) Atomic beam in Medium emission Atomic beam in Resonance absorption lines only
Doppler temperature High High High May easily be cooled May be cooled May be very low May be lowest
Brightness
Pressure
Fields
High High High Medium
High High Low Low
Medium High High Very low
Medium
Very low
Low
Low
Lowest
Very low
Low
Lowest
Very low
From this tabulation we see that three sources appear to be of greatest use for the study of hfs: the hollow cathode, the electrodeless discharge tube, and the atomic beam. The hollow cathode (see Fig. 4), first used by Paschen, has been brought to a high state of development by many workers, especially
52
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
Schiiler, Arroe, and Mack. It combines simplicity, ease of cooling, and very low electric fields with relatively low pressures, & to 10 torr. The electrodeless tube, excited by radio frequency, has the great advantage t ha t no electrodes are necessary, so that the tube can simply be of glass or quartz. It can also be cooled, but not quite so easily, and has the complication that suitable radio-frequency excitation must be provided. It also operates at low pressure, in the neighborhood of torr. An example is the well-known Meggers lamp, containing Hg198.
FIG.4. Hollow cathode discharge tube.
FIG.5. Atomic beam.
For ultimate resolution, where the line breadth must be reduced as far as possible, the atomic beam (Fig. 5 ) is used. Here the effective temperature is dependent only on the component of the atomic velocity in the line of observation. If the average velocity of the atoms emerging from the oven at temperature To is Do, where mijo2/2 = kTo, then the component V8 in the line of sight is 2fio tan 8, so that T,, the effective temperature, is given by T , = 4To tan2 0. For a well-collimated beam, 0 may be as small Thus even for a n oven a t 1000"K, the as lo, so that tan2 0 = 3 X effective temperature in the beam can well be reduced t o the order of l0K! The atomic beam is, however, much more complicated than the other light sources mentioned, requiring an oven and some means, for example an electron gun, for exciting the spectrum (in the case of the
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
53
emission beam) : it is usually not especially bright. A discussion of atomic beams has been given by Meissner.10 2.4.1.1.6. SPECTROGRAPHIC APPARATUS.* As noted above, the problem of determining nuclear properties by measuring the hyperfine structure of atomic spectra entails high resolving power. Of the many possible high-resolution spectroscopic instruments, only a few can be selected for discussion here and we will confine our remarks to the two most commonly used-the diffraction grating and the Fabry-Perot interferometer. As developed by Rowland in the nineteenth century and especially by Harrison in recent years, the diffraction grating (plane and concave) is a practical instrument of high resolution, suitable from the vacuum ultraviolet t o the far infrared. The theoretical resolving power R of the grating is given by the simple relation : (2.4.1.1.17) R = A/Ah = nN where n is the order number of the spectrum, and N is the total number of grooves in the grating. With modern ruling techniques it has been possible t o produce highly efficient gratings 25 cm wide with upwards of 15,000 grooves per centimeter, or more than 375,000 grooves. This affords a resolving power of over a million in the third order; such diffraction gratings can be very useful in the measurement of hfs, especially in the ultraviolet where high orders can be employed. The Fabry-Perot interferometer, devised toward the close of the nineteenth century, has emerged during the last decade with the advent of mnltilayer coatings and the photomultiplier as a n instrument t o be used for highest resolution. The resolving power of the Fabry-Perot is virtually unlimited, since it is just proportional to the separation bet ween the plates. Separations of more than a meter have been achieved, corresponding t o resolution widths smaller than any presently-available spectral lines. The Fabry-Perot, operating as it does at extremely high order, ordinarily requires a predispersion monochromator to eliminate unwantrd orders. Despite this complicating factor, the use of the Fabry-Perot, both photographically and photoelectrically, has become very widespread. This interferometer, as is well known, consists of a pair of partially reflecting mirrors, accurately flat and parallel (see Fig. 6). The light passed by the Fabry-Perot is a series of ring-shaped fringes a t infinity, the intensity of which, for perfect plates, is given by the Airy forrnula:
I * See also lo
=
T,,,[l
+ F Z sin2 6/2]-'
Vol. I , Part 7.
K. W. Meissner, Revs. Modern Phys. 14, 68 (1942).
X IO
(2.4.1.1.18)
54
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
where T,,, is the maximum transmittance, T,,, = T 2 / ( 1- R)2, where T and R are the transmittance and reflectance of the partially reflecting mirrors; F is proportional to the sharpness of the fringes and is given F = 2 di?/(l- R ) ; 6 is the phase difference of adjacent rays, and 10is the incident intensity. When the absorption of the partially-reflecting films is very low, T,,, is very nearly unity, and as Jacquinot” pointed out, the Fabry-Perot is, under these conditions, the most luminous of spectrometers ; its transmittance is highest for a given spectral range. Multilayer films achieve extremely high reflection, upwards of 95 %, with very low absorption, so that the sharpness of the fringes is limited not by the reflectivity through the function F, but by the available flatness of the plates.
Fro. 6. The Fabry-Perot interferometer.
The traditional method of using the Fabry-Perot is photographic and fully described elsewhere.10p12The photographic method has a number of disadvantages, however-among them nonlinearity in both wavelength and intensity, the first being quadratic and the latter logarithmic. A photoelectric method, developed by Meissner,I3 Jacquinot, and others, eliminates the two difficulties mentioned, and in addition has a considerable advantage in sensitivity. Here a diaphragm is placed at the center of the fringe pattern, and the light which passes through falls on a photomultiplier or other suitable detector. It can be shown that 6 = (%/X)2pt cos 8, so that I can be made to vary by changing 1/X, p , t, or cos 8. In the photoelectric method p is usually changed by varying the gas pressure between the interferometer plates, although sometimes t and 8 are varied. As 6 is changed, by whatever method, the spectrum is spread out repeatedly, and a monochromatic line appears as in Fig. 7. The condition P. Jacquinot, J . Opt. Sot. Am. 44, 761 (1954). S. Tolansky, “High Resolution Spectroscopy.” Pitman, New York, 1947. 18 K. W. Meissner, J . Opt. SOC.Am. 31, 405 (1941); ibid. 32, 185 (1942).
11
12
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTG
55
for the maxima is evidently 6 = 2 m , whence
nX = 2pt cos
e
(2.4.1.1.19)
often quoted as the fundamental formula for the Fabry-Perot. The halfintensity breadth, A61/z, for the perfect (exactly plane plates of infinite extent) interferometer is given to a first approximation by Meissner as 2F/r or 0.64F. Figure 8, which shows the logarithm of the transmittance TO,where To = Z/Zo [see Eq. (2.4.1.1.18)lof the interferometer plotted against the logarithm of the phase angle 6, indicates the strong dependance of the half-intensity width on the reflectance of the mirrors. In practical interferometers, a flatness of about & wavelength (green) can be expected, which would allow a reflectance of about 97% to be used. A 63
I I
2
6-
I
'
-I
FIG.7. Schematic appearance of a monochromatic line in a photoelectric Fabry-Perot.
higher R than this would merely lead to loss of light; for poorer plates, correspondingly lower reflectances must be used. As can be seen from Eq. (2.4.1.1.19), the wavelength free spectral range, @A, between successive peaks (n changing by l ) , is just X2/2pt, or, in terms of wave number, 6iv = l / 2 ~ t . In general, the predispersion instrument must have a wavelength band pass less than or equal to 26i to prevent the confusion of orders alluded to earlier. 2.4.1.1.7. RESULTSFROM SPECTROSCOPIC OBSERVATIONS. Of the many possibilities, two examples are given of the phenomena and methods described above. The first is a measurement of the spin of U 2 3 5 by Blaise et al.14Figure 9 shows two orders of a tracing by a photoelectric FabryPerot interferometer of a uranium hollow-cathode discharge cooled by is separated from the liquid nitrogen. The very strong line due to UZ3* U236and U233structure by the isotope shift. The U236components are not l4
J. Blaise, S. Gerstenkorn, and M. Louvegnies, J . phys. radium 18, 318 (1957).
8
FIG.8. The Airy formula.
-1
235
i- 233-l
FIQ.9. Hyperfine structure of Ua36. 56
235 :-233-1
.
100
50
0
Tb159 I 3/2 FIG.10. Hyperfine structure of
TbI59.
2.
58
DETERMINATION OF FUNDAMENTAL QUANTITIES
completely resolved after the first three, a, b, and c, but by the use of the intensity rules, the spin, I, of 4 was confirmed. The other example,16 Fig. 10, very clearly shows the hfs flag pattern in Tb169.Two orders of a photoelectric Fabry-Perot trace are presented, clearly showing I = $ and exemplifying the excellent resolution and intensity measurement possible with this method. 2.4.1.1.8. SUMMARY. The optical spectroscopic method provided the first knowledge of nuclear moments, and still is the source for much of present-day knowledge. Although superseded in some cases by more accurate methods, such as molecular beams and nuclear resonance, it is still indispensable in many cases where the other methods are not applicable or are applicable only with difficulty. Two examples are: (i) shortlived or rare isotopes where the quantities required for a beam are out of the question; and (ii) elements whose electronic ground state has no magnetic moment such as the rare gases. Several lists of moments have been p u b l i ~ h e d . ~ - ~ ACKNO w LEDGMENTS It is a pleasure for the author to acknowledge the kind help of many in preparing this section, especially of Dr. Jean Blaise, Bellevue, France, who put original photographs of hfs at the author's disposal, and of Professor Julian Ellis Mack of the University of Wisconsin, who read the section and made several valuable comments.
2.4.1.2. The Investigation of Short-lived Radionuclei by Atomic Beam Methods.* 2.4.1.2.1. INTRODUCTION.Atomic and molecular beams pro-
vide one of the oldest methods of experimental physics. Their earliest applications were, as to be expected, to the kinetic theory of gases, because study of the effusion of gas molecules from a source was of fundamental importance in understanding the Maxwellian theory of gases. The technique was soon recognized, however, as especially valuable for the study of molecular, atomic, and, later, nuclear structure. I n a beam, the atomic or molecular system is in effect isolated from any other molecule or atom. Thus, in a study of atomic, molecular, or nuclear properties, all experimental complications arising from the effects of the solid, liquid, or gaseous state can be avoided. If the experiment utilizes a resonance method, the lower limit to the width of the resonance line depends only upon the radiation lifetime and the duration of 0bservations.t lr
J. Blaise and F. Tomkins, private communication (1961).
t For further details of the atomic beam method see also Vol. 4, A, Chapters and 2.2; Vol. 4, B, Sections 7.1.1, 7.2.1, and 9.1.2.
* Section 2.4.1.2 is by John C.
Hubbs and William A. Nierenberg.
1.3
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
59
Advantages of the reaction-free state become apparent from consideration of the numbers involved in a typical experiment with a nonradioactive beam. Such a beam consists of 1O1O or lo1' atoms per second at the detector in an apparatus whose length is of the order of 1 meter. The distribution of velocities is the Maxwellian distribution multiplied by the velocity v. The mean velocity may be 50,000 cm/sec, giving a lineal density of -loB atoms/cm; for a beam 0.01 cm wide and 1 cm high, this is a volume density of -lo8 atoms/cm3. The result is a much smaller collision rate than in a standard gas density of 10lg atoms/cm3. Furthermore, if the cross section for collision of the beam atoms with themselves is spherically symmetric, the probability that products of collisions will remain in the beam is only a little greater than the apparatus transmission. Since this is about or less, the effect of collisions is negligible. Scattering due to residual gas in the apparatus is scarcely significant: the pressure of the residual gas is 5 X lo-' mm of Hg, or less, which corresponds to a mean free path ten times the length of the apparatus. The molecular (or atomic) beam became an important device for modern physical research when 0. Stern and his collaborators used it for studying such diverse phenomena as molecular scattering and space quanti~ation.l-~I n these experiments he studied the deflection of the atomic system in very strong inhomogeneous magnetic f i e l d ~ . ~The J specific conformation of magnet pole tips required to produce this inhomogeneity still goes by the name Stern-Gerlach field. The next major step in the technique, based on an interesting theoretical observation by Breit and Rabi,6 was put into experimental practice by Rabi and his collaborators.~-9 This concerned the strong magnetic interaction between the currents produced by the electron angular momentum of the atom and the nuclear magnetic moment. The nuclear spin is rather strongly coupled to the electronic angular momentum, and the two systems would not be decoupled in a magnetic field of less than 1 L. Dunoyer, Compt. rend. acad. sci. 162, 594 (1911); 167, 1068 (1913); Le radium 8, 142 (1911); 10, 400 (1913). 9 W. Gerlach and 0. Stern, Ann. Physik [4] 74, 673 (1924); [4] 76, 163 (1925).
0. Stern, 2.Physik 7, 249 (1921). R. D. Frisch and 0. Stern, 2.Physik 86, 4 (1933). I. Estermann and 0. Stern, Z. Physik 86, 17 (1933); 86, 132 (1933); I. Estermann, 0. C. Simpson and 0. Stern, Phys. Rev. 62, 555 (1937). 'G. Breit and I. I. Rabi, Phys. Rev. 38, 2082 (1931). I. I. Rabi, Phys. Rev. 49, 324 (1936). *I. I. Rabi and V. W. Cohen, Phys. Rev. 48, 582 (1933); D. R. Hamilton, Phys. Rev. 66, 30 (1939). SS. Millman, I. I. Rabi, and J. R. Zacharias, Phys. Rev. 63, 384 (1938); N. A. Renzetti, Phys. Rev. 67, 753 (1940). 4
60
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
several hundred gauss.* I n a weak field, therefore, the two systems behave as one, with an angular momentum that is the vector sum of the two angular momenta, but with a magnetic moment that, to a very good approximation, is due to the electronic structure alone. Thus the behavior of the system as a whole in a n external field is primarily influenced by the spin of the nucleus, and not by its magnetic moment, except through the hyperfine coupling. This was the basis for the zero-moment method used by Rabi and his collaborators for the measurement of the spins, magnetic moments, and quadrupole moments of the stable isotopes of the alkalies, gallium, and indium. (The Breit-Rabi diagram is later explained in some detail.) The next step in the development of the subject was the discovery and application by Rabi of the nuclear radiofrequency resonance method to molecular beams. This was an advance of many orders of magnitude in the precision of measurement of nuclear moments, and the application of the method to HD and D2led to the discovery and measurement of the deuteron quadrupole moment. lo Immediately afterward, the method was applied to the radiofrequency spectroscopy of alkali atoms by Millman et al.,” and because of its enormous precision, virtually supplanted the zero-moment method. Zacharias measured the spin of K40, the rare radioactive isotope of potassium, by an ingenious application of the method, and thereby demonstrated its applicability to trace amounts of material in the presence of large numbers of background atoms. l2 The resonance method was extended to measurements
* Throughout this paper all magnetic moments are in units of
the Bohr magneton,
PO, about 0.927 X lO-*O erg/gauss. The vector magnetic moment of the electron
system is gJpoJ, where J is the vector angular momentum in units of h, and gJ is known as the gyromagnetic ratio of the system and depends upon the details of the e‘ectronic motion. For a single electron in an s state, we have gJ E - 2, and, in general, g J - 1. For a nucleus, however, the magnetic moment is gIp& where I is the vector nuclear mlm. As a n example, for a single s spin in units of h, and, what is important, gr electron, the magnetic field at the nucleus due t o the spin oi’ the electron is roughly 2p0/r3. If r is taken as ~ 0 . X 5 10-8 cm, this is equivalent to about lo6 gauss. A unit nuclear moment in this field of 106 gauss is, therefore, equivalent to about 5 X erg. When this is divided by Planck’s constant, one gets 108 cps or 100 Mc/ sec, if frequency is used as a unit of energy. This represents a rather strong coupling of the nuclear spin t o the electronic angular momentum, and, since the electron precession frequency in a magnetic field, if uncoupled, is gJpoH/h, or about 1.4 Mc/ sec/gauss, for gr = 1, it would take a field of several hundred gauss to completely decouple the two systems. l o J. M. B. Kellogg, I. I. Rabi, N. F. Ramsey, and J. R. Zacharias, Phys. Rev. 66, 728 (1939); H. G. Koski, T. E. Phipps, N. F. Ramsey, and H. B. Silsbee, Phys. Rev. 87, 395 (1952); N. J. Harrick, R. G. Barnes, P. J. Bray, and N. F . Ramsey, Phys. Rev. 90, 260 (1953). I I S . Millman, P. Kusch, and I. I. Rabi, Phys. Rev.66, 165 (1939). l2 J. R. Zacharias, Phys. Rev. 60, 168 (1941); 61, 270 (1942).
-
-
2.4.
DETERMINATION
OF SPIN,
PARITY,
AND NUCLEAR MOMENTS
61
of quadrupole interaction in nuclei, including what is known as “pure quadrupole resonance.”13 The Kusch-Foley measurements established the anomalous magnetic moment of the electron14 and set the stage for the rapid development of the renormalized-electron-field theory and Lamb’s experiment leading to the discovery of the “Lamb shift.”16 For a broad discussion of procedures and techniques of atomic beams, the reader should refer to several articles that cover all the above applications. l 6 * This discussion is primarily concerned with the applications to the measurement of the spins, magnetic moments, and quadrupole moments of unstable nuclei of short half-lives. These quantities are of primary importance in the study of nuclear structure, but until recently there has been very little done in the field, considering the large number of known isotopes. As a result, although much has been known about the nuclei t ha t lie along the “stabie line” of the Seer&chart and longlived isotopes that may be produced in abundance from these in reactors, very little was known about those isotopes that are neutron-deficient by two, three, four, or more neutrons. The primary reason is that these are cyclotron-produced isotopes and, in general, are not available in quantities greater than about 1013 atoms. By a n amalgamation of the techniques of nuclear physics with atomic beams, measurements have been successfully carried out on 1 O l o atoms and on species whose halflives are as short as 10 minutes. Within the last four years, the nuclear spins of about one hundred isotopes have been determined by this extension of the method of atomic beams. The half-lives of these isotopes have ranged from 10 minutes to 24,000 years. It should be emphasized that these measurements have been made with no loss of the great resolution available and thus have a very high reliability compared with other methods. The usefulness of the results can be appreciated from the fact that nuclear spins are now known for ten isotopes of Cs, ranging from a magic number of neutrons for 65Csi23’ to ten less than the magic number of neutrons for 5sCs:227. I n addition, long-lived isomeric states have been investigated. Much useful information has been obtained on Au and on Ag isotopes. Because these atoms are in an electronic state with J = $, it is not feasible to measure the quadrupole moments of these nuclei.
* See also the Vol. 4 references cited on page
58. W. A. Nierenherg and N. F. Ramsey, Phys. Rev. 72, 1075 (1947). 1 4 P. Kusch and H. H. Foley, Phys. Rev. 74, 250 (1948); 72, 1256 (1947). 16 W. E. Lamb, Jr., R. C. Retherford, S. Treibwasser, and E. S. Dayhoff, Phys. Rev. 73, 241 (1947); 79, 549 (1950); 81,222 (1951); 86, 259 (1952); 86, 1014 (1952); 89, 98 (1953); 89, 106 (1953). 16 J. G. King and J. R. Zacharias, Advances in Electronics 8 , l (1956); N. F. Ramsey, “Molecular Beams.” Oxford Univ. Press, London and New York, 1956. 13
62
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
Instead, the hyperfine-structure anomalies for these nuclei are observed. This term is also nuclear-shape-dependent. The advantage to be gained is that measurements can be made on a J = $ state, and a quadrupole (as well as an octupole) term can be measured. If this is possible, it will result in the knowledge of the quadrupole moments for eight or nine isotopes of the same element, and should be extremely useful in the interpretation of the unified nuclear model and similar theories. 2.4.1.2.2. GENERAL PRINCIPLE.* The fundamental method for the spin and magnetic moment measurements is based on space quantization. Because of its spin 1, the nucleus alone has a spatial degeneracy of 21 1. Because of the electronic angular momentum J , the electronic system has a degeneracy 2J 1 . In the absence of any external fields or interactions between the electrons and the nucleus, the total degeneracy is (21 1)(2J 1 ) . Because of the magnetic interaction between the nucleus and the electrons, the degeneracy between levels of the total angular momentum, F = I J, is removed. There are either 21 1 or 2J 1 of these hyperfine levels, whichever number is the smaller. F takes on the values I J, I J - 1 , . . . \I - JI,and each of these levels is 2F 1 degenerate. On the application of a weak magnetic field H , this degeneracy is removed, and the energy dependence of the levels is a constant plus a term proportional to H . The difference between any two energy levels is a possible transition frequency under the influence of an applied radiofrequency magnetic field, and this transition is observed, provided that it is allowed by the selection rules, and provided that the apparatus is designed to detect the particular transition involved. There are many different types of transitions, and a t least three different ways of designing the atomic beam magnets to observe them. As an example, the class of transitions that involve AF = _ + I is particularly suitable for precision measurements of hyperfine intervals. If no previous value of this separation is available, however, a tedious search is involved because of the high resolution of an atomic beam apparatus. A procedure that operates in stages and requires only a minimum amount of isotopes has been devised for determination of the various nuclear constants of interest. The quantities to be determined are, in order, the spin, the magnetic dipole term in the hyperfine interaction, the electric quadrupole term, the sign of each, and, if possible, the magnetic octupole term and the hyperfine-structure anomaly. From these terms, the values of the nuclear magnetic moment and electric quadrupole moment can be determined by ratios. The sequence as described requires monotonically increasing precision. As an example, the alkalies have a 2S1,2ground state; therefore, the quadrupole and octupole terms vanish identically, and
+
+
+
+
+
+
+
* See also Vol. 3, Part 6.
+
+
+
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
63
there is no hope of measuring these moments in this atomic state. Since the hyperfine-structure anomaly may be as high as several per cent, there is no point in measuring the dipole interaction with extreme accuracy for the purpose of determining the nuclear magnetic moment b y comparison with a known isotope. However, in order to determine the sign of the interaction] it is generally necessary to measure the hyperfine-structure constant to a greater accuracy to check the consistency of the assignment. As a n example of some aspects of this technique, we consider C S ' ~ ~ . The spin of this isotope is 2. Figure 1 is the plot of the energy levels versus the magnetic field. The energy is in units of a, the hyperfinestructure constant, and the abscissa is the dimensionless quantity,
+
X = (-gJ gr)poHha(I $1 +
(9.7
< 0 for electrons)
(2.4.1.2.1)
The Hamiltonian representing the interaction between nuclear spin and electronic angular momentum, and the interaction with the external applied field, is X
=
a1
- J - gJpoJzH - ____ grpoIpH h h ~
(2.4.1.2.2)
where X and a are in frequency units. The Land6 factor g.7 is known very accurately from atomic beam experiments on the stable Cs isotopes and is very nearly equal to -2; g I is small, mrnlm; a is approximately several kilomegacycles per second. Initially, in the weak field or Zeeman region, the levels vary linearly with H . The corresponding frequency differences between adjacent upper levels, F = 8, AF = 0, and AM = f1, are equal to one another and to approximately Q poH/h, or about 0.5 Mc/sec/gauss. The field is too weak to decouple the nucleus from the electrons, but it does cause a precession of the entire system due to the torque on the total dipole, which is why all the upper frequencies are the same for low field. For a free electron ( I = 0), this frequency is gJpoH/h or 2.8 Mc/sec/ gauss. The nucleus adds inertia to the system because of its angular momentum, but makes a negligible contribution to the torque; therefore the precession frequency is reduced. The exact expression for 2S1/2states to first order in the field for the upper levels is y o = - -
g.rpoH/h 21 1
+
2IgrpoH/h. 21 1
+
(2.4.1.2.3)
The last term is negligible for initial measurements in the determination of the spin. If a resonance can be observed a t low enough values of H , the spin is determined by the factor (21 l)-l. The advantage is the discreteness of the search. The resonance of the carrier or known isotope
+
64
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
2 .O
M
1.0
F.5
2
w -
AW
F
=% -1.c
-2.0
I
I
I
.
1.0
2.0
30 3.0
X
ENERGY LEVELS
J . 12
1.2,
FIQ.1. The Breit-Rabi diagram for I
=
2.
is used to calibrate the field, and the position of possible resonances for different spins can, in general, be discretely predicted and compared. Once a spin resonance is located and verified, an estimate can easily be made of the hyperfine interval Av, the difference in energy between the upper and lower F levels: A v = [(2I 1)/2]a. This is done by following the resonance to a field that will introduce a small deviation from the linearity expressed by Eq. (2.4.1.2.3).This term is, in general, quadratic, and is due to the incipient deeoupling of I and J by the external field,
+
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
65
and is 2Iv02 Av
(2.4.1.2.4)
V ~ V O + - -
to second order in the field. The shift 21v02/Av gives a rough estimate of Av, and this shift can be increased by increasing the frequency until Av is determined to any reasonable desired accuracy. Since this procedure is capable of determining Av to 0.1% or better if necessary, one must use the exact solutions for the Hamiltonian, including the terms in g r ; gr itself is estimated from Av by comparison with a known isotope and the relation” Avi/Avz = (211 l ) g r 1 / ( 2 I 2 l)grE. (2.4.1.2.5)
+
+
For alkalies and other 2S1/2 states, this formula is good to about 1%, which is more than enough for the small correction it affords. The deviation of this relation from equality is a measure of the finite distribution of the nuclear magnetism and is known as the hyperfine anomaly. However, the sign of gr is not determined, and the usual method is to check the constancy of the calculated A v versus H for assumed gr > 0 or gr < 0. If the resolution of the apparatus is sufficiently good, there is enough discrimination to tell the sign. So far, nothing has been said about the apparatus needed to observe this resonance. A particular arrangement of magnets in an atomic beam apparatus is used that results in what is inelegantly called a “flop-in)’ trajectory rather than the original “flop-out” design. Reference to Fig. 1 shows the rather interesting phenomenon that the particular level corresponding to the most negative value of M = - I - J (e.g., the level -$ in the diagram) varies exactly linearly with H and “crosses the diagram” in the Paschen-Back region. to take its place with the levels WLJ = The level immediately above, M = - I - J 1 (e.g., in the diagroup. The slope of gram) eventually winds up with the WLJ = these lines at a particular magnetic field is a measure of the force on an atom in that state in an inhomogeneous magnetic field. The two levels, M = - I - J and M = - I - J 1, therefore have equal but opposite forces exerted on them in the same inhomogeneous field, providing only that the field is sufficient to decouple the angular momenta completely as in the Paschen-Back region. Figure 2 is a schematic of an actual apparatus. The atoms leave the source and pass between the pole tips of two successive magnets that supply the field necessary to cause the decoupling and the inhomogeneity to deflect the atoms. These magnets are called the “A” and “B” magnets respectively, and are so arranged that the fields of each and the field gradients of each are in the same
-+
+
17E.Fermi,
Z . Physik 60, 320 (1930).
+ +&
-+
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
I
FIG.2. A cross section of an atomic beam apparatus.
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
67
direction. The apparatus then acts like a Stern-Gerlach system in the first instance. The atoms are deflected either to the left or to the right, depending upon the sign of mJ, and there is no signal a t the detector. Between the two deflecting magnets there is a uniform field, the “C” field, which is normally at just a few gauss for spin determinations. The resonance takes place in this field via a small perturbing rf field induced by a pair of wires, generally called a “hairpin,” running parallel to and near the beam. If the resonance condition is met, transitions are caused between the two states,
F=I+J -I - J
MF =
and M F =
-I - J
+ 1.
The result is that the moments of the atoms initially in these states may be reversed in sign in the B field with respect to the A field, and these atoms will go in a sigmoid path and be refocused onto the detector yielding a signal. The result is a positive signal compared to no signal, which gives far better statistics than the flop-out method in which the indication is a reduction in beam intensity. This is an essential feature in experiments involving trace amounts of material. I t is important to note that the matching and stability of the A and B fields can be quite crude without too much effect on the efficiency of the apparatus. The C field must be stable within a fraction of the line width, and since this may be as low as 20 kc/sec, this is often a stringent requirement. Since only two 1) are involved, the best fractional inlevels out of a total of 2 ( 2 1 tensity obtainable is ( 2 1 l ) - I , and because of various losses, about one-quarter of this is actually realized. The final identification of the radioisotope is usually made certain by a determination of the half-life of a collected resonance sample. Figure 3 shows the results of the decay of a Cs beam produced by a-particles on Xe. The undeflected-beam decay curve is composite, with a t least three components. Three of the spin buttons collected showed counts that were above background and each decayed with a half-life ~ ~ and , CsI3* characteristic of a known species. Thus the spins of C S ~CsI3l, were measured. Cs129 and Cs131 had been measured earlier after having been produced by different methods. The procedure for atomic and nuclear levels where both I and J are greater than one-half is more complicated because of the existence of the quadrupole term. However the spirit of the method is the same, but two different transitions must be observed for sufficient accuracy, this is
+
+
68
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
always possible because there are always two or more “flop-in” transitions in this case. Further accuracy gives a direct measurement of gr. 2.4.1.2.3. CONSIDERATIONS IN APPARATUS DESIGN. The design of an atomic beam apparatus for the study of radioactive species should clearly involve the optimization of the transmission which is the total number of atoms from the source that reach the detector. This is a different condition from the many that are usually employed, so a brief description is required here. The apparatus designed and constructed a t
-
0
5
I5
10
20
25
DAYS
FIQ.3. The decay of three resonances corresponding to three different Cs isotopes.
Berkeley will be discussed as an example. I t is more fully described in the thesis of R. A. Sunderland, and the calculations that follow are abstracted from more complete accounts in his thesis. The apparatus being considered is to be designed so that the largest possible fraction of the manufactured atoms is available a t the detector. The C field will, therefore, be made as short as possible, thus sacrificing ultimate precision, and dead spaces between the source, the magnets, and the detector will be held to a minimum. In order to obtain a first idea, it will be assumed that the C-field length and the dead spaces are negligible. The deflection
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
69
of an atom from a straight-line path is then (2.4.1.2.6)
where 6y is the deflection perpendicular to the axis of the apparatus, p is the effective moment of the atom, a H / a y is the perpendicular magnetic field gradient, 1 is the length of the deflecting field, M is the mass of the atom, and v is the velocity of the atom. For l 2 = 1000 cm2,
aH/ay
=
6000 gauss/cm
T = 1000"K, and p = 10Vo erg/gauss, the deflection is 0.3 cm. In what follows, numerical factors of the order of unity will be discarded. If T is the source temperature, Mu2 = ICT for a n average atom. The field inhomogeneities are developed by pole tips which are cylindrical arcs. If the radius of one of the pole tips is a, we can assume ( l / H ) ( a H / a Y ) = U/a> therefore
6y = PHp/kaT.
(2.4.1.2.7)
If the exit slit is of height h and width w, we may assume, dimensionally, that the source slit and width are proportional to h and w (and, in fact, h, w, and 6y must be proportional to a in a n analysis of this type). The total number of atoms that reach the detector is then (remembering that 1 a h a w 0~ 6y a a) (2.4.1.2.8)
Here p is the number of atoms per cm3 in the vapor phase in the oven. For an experiment with an unlimited supply of material, the maximum permissible p 00 l / w by the Knudsen condition. This is the condition that the slit width should not exceed the mean free path in the oven, so that the condition for pure molecular effusion can be maintained. I n this case, N varies as the length of the apparatus. The only limit to the increase in 1 is the scattering of the beam atoms by the residual gas and the expense. In practice, it has seldom been necessary to consider an apparatus of length greater than 2 meters, and most of this length is due to the C field needed for high precision. In the situation in which trace amounts of material are involved, the total amount of material is fixed and all of it is needed. This is the case a t hand. The proper condition is that phw equal a constant. Then we find
N
a
Hp/T.
(2.4.1.2.9)
70
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
This means that the efficiency of the apparatus is independent of its length. The apparatus can be made quite short until the dead spaces and the finite C field introduce other limiting factors. A convenient over-all length is about 75 cm. Of course, the half-life of the material also enters these considerations. As the apparatus is shortened, the rate of efflux is decreased and the carrier needed must be decreased. This can reach the point a t which the amount of carrier is too small to be successfully handled. The foregoing analysis was carried through on the assumption that the ratios of ail lengths were held constant. There are other considerations, however. Provision must be made for a collimating slit and for a stop wire. The latter is an obstacle that just blocks the direct beam from the detector. It prevents the fast, relatively undeflected tail of the Maxwellian distribution from reaching the detector. Furthermore, the ratio of the A to B magnet lengths need not be unity. I n fact, if the A magnet has all its dimensions reduced, it remains equivalent in focusing power and transmission, as the above analysis showed (including the necessary reduction in source dimensions). If carried to the limit, this means a factor of two in length for the total apparatus and, therefore, a factor of four in transmission. In practice, a factor of three can be obtained, A factor of this size is not trivial when viewed against cyclotron time and counting time. One pays for this improvement with a higher source pressure. For the alkalies, this represents a reasonable temperature rise. For low-vapor-pressure substances, such as U, Pu, Am, Np, Th, etc., the necessary rise in oven temperature is barely tolerable, and the improvement in intensity probably is not worth it. Under these conditions, the stop wire allows only about one-half the atoms through that undergo a change of effective moment of 1 Bohr magneton. The transmission of such an apparatus is about when a cosine distribution is assumed. Channeling the source never seems to produce the improvements expected from the simple theory and, further, is less effective for material of short half-life. The exit slit is 0.1 by 0.5 in., and the source is 0.002 by 0.080 in. The one remaining question of importance is the length and value of the rf field. A typical beam velocity is 50,000 cm/sec, and if the hairpin is 1 cm in length, the transit time is 1/50,000 sec, given a resonance band with 50,000 cps. At a field of 10 gauss, a nucleus of spin I = 0 in an alkali atom resonates at 28 Mc/sec, and this bandwidth will just resolve spins in the neighborhood of I = 50. Such resolution is far more than necessary, but usually the C field is not sufficiently uniform to attain ideal resolution, and resolutions of 200 kc/sec are more common. This is still more than adequate. The strength of the rf field is determined as follows.
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
71
The system precession a t resonance about the C-field direction can be viewed from a rotating coordinate system that has the same velocity and sense of rotation. In the rotating system, the angular-momentum vector appears stationary, and the applied magnetic field is taken as zero. Such is the essence of Larmor’s theorem. The rf field, which appeared as a rotating vector perpendicular to the C field in the laboratory system, is now stationary if the oscillator is tuned to resonance. In this coordinate system, the spin will precess about the perturbing field H’ with the Larmor frequency corresponding to H’. If the strength is just right, it will make just one-half revolution in the time the atom spends in the rf field. Denoting this time by At, the condition for a precession of ?r is (2.4.1.2.10)
For spin 0, At = 1/50,000 sec-I, this yields a field of 20 mgauss, a field that is very easily obtained up to quite high frequencies. For a monoenergetic beam, the resonance will decrease if this field is exceeded. I n practice, the Maxwellian velocity distribution will smooth out the oscillations of the resonance versus perturbing field. 2.4.1.2.4. P R O B L E M S INVOLVED I N P A R T I C L E DETECTION OF ATOMIC BEAMS.2.4.1.2.4.1.General Considerations. Only experimental methods that take advantage of particle detection can now give the high sensitivities required for the investigation of short-lived radioisotopes. Two essentially complementary techniques now being used meet this requirement; one is the detection of individual beam atoms by ionization, massspectrometric analysis, and subsequent detection with electron multipliers; the other is radioactive collection and detection. For the first scheme, the following limiting situation is typical: at least lo3 positive ions must be observed a t the electron multiplier to obtain a statistically significant sample of the beam; mass-spectrometer transmission is about 10%; and the fraction of beam atoms that is ionized ranges from a lower limit of (with the universal electron-bombardment detector first introduced by Wessel and Lewls) to unity for special elements, such as the alkalis and Group IIIb series, in which Langmuir-Taylor surface ionixationlg is part,icularly applicable. For radioactive collection and detection this situation is typical; a beam sample is required to have a decay rate in excess of 1 cpm and is observed in detectors with a n efficiency of 25%. Under these simple assumptions the crossover point, where one method becomes comparable to the other in sensitivity, occurs G. Wessel and H. Lew, Phys. Rev. 92, 641 (1953). I. Langmuir and K. H. Kingdon, Proc. Roy. SOC.A107, 61 (1925); J. B. Taylor, 2.Physik 67, 242 (1929); Phgs. Rev. 86, 375 (1930). ‘8
l9
72
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
for materials with half-lives between one week and 100 years, depending upon the ionization efficiency attainable. There is, however, another consideration, difficult to evaluate, but nonetheless important. Even when there is no common stable isotope having the same mass as a nuclide to be investigated by the ion method, there is inevitably some spillover of carrier, beam contamination, and apparatus background, especially oils, into the mass channel of interest. The effect is to make the ion method somewhat less sensitive than would first appear. The present cross-over, therefore, occurs somewhere in the neighborhood of 1- to 100-year activities. An excellent r6sum6 of the ion system is contained in an article by King and Zacharias16 and the publication of Lew,20therefore, this discussion is generally limited to research on materials with lives of one year or less. The central procedures of the radioactive method are the collection of beam material for a suitable integration time and the detection of the sample decay by a n appropriate counter. Let us first consider the conditions under which the material available may be most effectively utilized. The central fact is that if reliable counters are available, the relative uncertainty t in a beam sampling is (I, t =
+
c)l'2 ct"2
+
(2.4.1.2.11)
where c is the sample decay rate, b G is the observed decay rate, and t the counting time. It is assumed that the background b is well known. We dispense with the b term by noting that in most instances an efficient counting system with background of about 1 cpm or less can be found (more on this point later), and that 1 cpm is also about the lower limit of decay rate set by the time required t o count down the results of a run and by the investigator's patience. Thus any discussion of radioactive detection rather naturally separates into two categories; into the first class fall all materials with half-lives of several days or more, and into the second go isotopes with shorter lifetimes. For investigations in the first class, the rate of progression of research is strictly proportional to the rate a t which the beam exposures can be counted, since the exposures themselves can be taken a t essentially any rate, and since the investigator will have sufficient time to analyze the data and decide on a n optimum search procedure. The rate a t which exposures can be counted is, from Eq. (2.4.1.2.11), ne2c2 (2.4.1.2.12) d N / d t = -= nt2c c+b za
H. Lew, Phys. Rev. 74, 1550 (1948); 76, 1086 (1949).
2.4.
DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS
73
where n is the number of counters used, c is a prescribed relative uncertainty in the beam sampling, c is the net counting rate of beam samples, and b the counter background. The total activity available for the experiment will invariably be limited by availability, expense, or health precautions, and the counting rate c will be some appropriate apparatus constant times the total activity divided by the number N of exposures required to execute the experiment. Thus perhaps a more illustrative form of Eq. (2.4.1.2.12) is 1/r = 2Qne2/N2 (2.4.1.2.13) where r is the counting time required for the entire research project, and Q is proportional to the total activity. Note that there is a large premium connected with optimum search procedures, which reduce N, and with increase of the allowable relative uncertainty in the individual resonance points. In addition, the counter effect occurs as the first power of the number of counters. We now consider investigations falling into the second class, where one deals with materials in which the lifetime itself is a major factor in the control of exposure and counting procedures. Here the progress of the research is taken to be proportional to the number of resonance points that can be obtained per run, although other considerations, such as the fact that no extensive data analysis is possible during the course of the run, also play an important role. The maximum counting time per resonance point is T n / N , where N is now the number of resonance points taken during the run, T is a characteristic time which is essentially the half-life, and n is the number of counters. But, as before, the individual counting rates are & / N ; so for this case, assuming b << c, we find N = e(Qnr)1/2. (2.4.1.2.14) Although this result is formally like Eq. (2.4.1.2.13),the physical situation is quite different, for T is now fixed. First of all, under these circumstances, there is little to be gained from large numbers of counters, quite aside from the fact that, in the interest of reliability and uniformity, cach beam exposure should be counted in each counter, so that the dead time involved in switching samples would become high (particularly for half-lives of less than a few hours). Also interesting is the fact that N goes only as the square root of Qr, which essentially measures the total radiation h k e n by the investigator. Because the run generally extends over tbe ent,ire lifetime of the material and little time is left for calculations, the principal profit is in precalculation of situations that may be of interest during the course of the search procedure. We now consider a question pertinent to either class of investigationindeed, to any search procedure-namely, what method of resonance
74
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
coverage, i.e., spacing of resonance points, will lead to the fastest convergence from an initial uncertainty u in a parameter such as gJ or the hyperfine structure, whose effect on the resonance frequency is controllable (e.g., by the strength of the static magnetic field), to a final uncertainty u’. Assuming that the assigned uncertainty in each resonance center is the interval between resonance points, and that a t each stage the effect of the desired parameter on the resonance frequency is multiplied by a factor n, the total number N of points needed for the procedure will be approximately given by u/u’ =
n”n.
(2.4.1.2.15)
Minimizing N with respect to n, we find that n is ideally e = 2.7 with a flat maximum between n = 2 and n = 5. The effect of this consideration, in combination with the practical ones of field drift and time spent in adjusting the field, is to make an optimum procedure one in which the effect of the parameter is increased by a factor of 3 to 6 per stage, and in which resonance points are separated by intervals of 4 to 8 the line width. Once the material has been produced, chemistry must, in general, be performed to separate the product from the few grams of target material and to put the product in a suitable state for production of an atomic or molecular beam. The range of such “chemistry” procedures is limited only by the ingenuity and resources of the investigator. The methods are thus best illustrated by examples. The central theme of all such processes is that between 1Olo and loLe atoms must be transferred under conditions of high radiation level to the apparatus in which the beam is to be produced under controlled conditions. The natural procedure for n - y reactions is to bombard the pure element, which is then placed in bulk quantities into an appropriate source. Investigations using this procedure have principally been carried out by Goodman and WexleP and by Srnith.z2 For cyclotron bombardments or secondary reactions in piles involving a product that is not the same element as the parent, the opportunity for controlling the quantity of the carrier is a very considerable advantage, as is discussed below. Therefore, in the investigations of cesium and rubidiurn,23~24the target *l L. S. Goodman and 9. Wexler, Phys. Rev. 95, 570 (1954); 97, 242 (1955); 90, 192 (1955); 100, 1245 (1955); 100, 1796 (1955). z* J. Smith, Thesis, Harvard University, Cambridge, Massachusetts, 1951. ** J. P. Hobson, J. C. Hubbs, W. A. Nierenberg, and H. B. Silsbee, Phys. Rev. 96, 1450 (1954); 104, 101 (1956). 24 W. A. Nierenberg, J. C. Hubbs, H. A. Shugart, H. B. Silsbee, and P. 0. Strom Bull. Am. Phys. SOC.[2] 1,343 (1956); W. A. Nierenberg, H. A. Shugart, H. B. Silsbee, and R. J. Sunderland, Phys. Rev. 104, 1380 (1956).
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
75
is not a n alkali bromide but an alkaline earth bromide, and chemical separation of the target and the product is made, with a suitable quantity of carrier whose primary purpose is to prevent losses of the minute quantities of the radioisotope. The final product form is rubidium bromide or cesium iodide. The alkali halide is then reduced, in the oven, b y calcium a t a temperature of about 400°C. The chemistry of these two alkalis might have been done, however, in any one of the following ways.
‘I
It
.
I
II 1 ,
::
I s
I
I !I:
It
(I It
\
,GLASS _+HIGH
WATER IN-
TO METAL SEAL
VOLTAGE
EVAPORATOR
FIG.4. Detail of part of an evaporator for separating thallium from gold.
The halide nitrate could be produced in the chemistry process, the nitrate put in the oven, and then successively decomposed first to the oxide and then to the alkali metal a t high temperatures; or the azide of the alkali could be produced in the chemistry process and then decomposed directly to the metal in the oven. The gross target material, barium bromide or iodide plus carrier plus radiobromide o r iodide, might be placed in the oven and the entire load reduced by calcium a t a n appropriate temperature. I n investigation of the neutron-deficient thallium25 isotopes, it was found that chemistry procedures took entirely too long for the investigation of the 1- and 2-hour isotopes, therefore a different procedure was used. The thallium was evaporated (Fig. 4) from the gold target into a small cup with a n appropriate amount of carrier thallium. This material was then placed in the oven and a beam made of thallium metal. Again, t 6 G. 0. Brink, J. C. Hubbs, W. A. Nierenberg, and J. L. Worcester, Phys. Rev. 107, 189 (1957).
76
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
however, one alternatively might have put the gold target directly into the oven and evaporated out the thallium. In gallium investigationslZ6it was discovered that the activity coefficient of gallium in copper metal is so small that one is unable to separate gallium and copper by vapor pressures, even though the ratio for the free metals a t the melting point for copper is 103/1, and as a result, a long and involved chemistry process is required. 2.4.1.2.4.2. Oven Channeling. A possibility not heretofore mentioned is that the fraction of beam atoms that reach the detector, the effective transmission, may be materially increased by the use of long thin tubes or channels a t the source orifice to reduce the fraction of beam atoms issuing a t large angles to the beam direction. This technique has been extensively used by the MIT group in their investigations of the rare and long-lived alkalies and halides. l 6 They report typical forward effusion gains as large as several hundred from the channeling process. This technique is not of particular importance for the investigation of short-lived (less than 1 day) isotopes for the following reasons. First, in this case, one must effuse the entire activity in a period of an hour to a day. With a standard oven geometry, only a few hundred milligrams may be effused in this time. If, now, the source is channeled by a factor K , the total effusion rate is decreased by a factor K , and the source pressure must also be reduced by about a factor K . Therefore, the total quantity of material that can be effused per unit time is decreased by a factor K 2 . Absorption of the beam material by the walls of the source and inevitable sample contamination resulting from hurried chemistry set an effective lower limit of about 1 mg on the carrier; therefore, in this case, the maximum allowable channeling gain is about 20. But, from Eq. (2.4.1.2.14), the effect on the over-all research picture will be a factor of only 4 or 5. An effect this small does not justify the additional line-up problems and increased probability of failure of a run. On the other hand, for moderately long-lived materials (greater than 1 day), channeling is desirable, for the chemistry procedure can be made arbitrarily clean, ovens can be properly outgassed before use, and furthermore, the effect on the research goes as the first power of the channeling gain. [SeeEq. (2.4.1.2.13).] 2.4.1.2.4.3. Beam Collection. A general problem uniquely associated with the radioactive method is the collection of the beam on some surface in such a manner as to be useful later for counting. While this process need not necessarily be 100 % efficient, reproducibility is highly desirable. Bellamy and Smith, in their parent investigations of alkalies by the 18 J. C. Hubbs, W. A. Nierenberg, H. A. Shugart, and J. L. Worcester, Phys. Rev. 106, 1928 (1957); J. L. Worcester, J. C. Hubbs, and W. A. Nierenberg, Bull. Am. Phys. Soc. [2] 2, 310 (1957)$ J. L. Worcester, private communication.
2.4.
DETERMINATION O F SPIN, PARITY, AND NUCLEAR MOMENTS
77
radioactive method, have used a scheme where the beam is first ionized and then attracted t o a brass surface by a potential of several hundred volts.22It is found by Hobson and Hubbs that the retention by a brass surface for rubidium ions, but at ion currents lo2lower than that reported by Bellamy and Smith, is about 1% for ion energies between 10 and 1000 ev.27,28The retention rises a t either end of this range. It is known from other sources that the retention of aluminum for 3-kev ions in this mass region is essentially It is also found by Hobson and Hubbs that the retention of surfaces for thermal rubidium atoms is similarly complex. A tabulation of their data, some of which were taken only once, is given in Table I. The data were taken with a tracer technique, whereby TABLE I. Retentivity Coefficients 1. Brass 2. Silver
3. Platinum 4. Tungsten 5. Sulfur
0.10 f 0.01 0.07 f 0.01 0.24 f 0.02 = 0.21 f 0.03 = 1.0
a = a = a = LY
a
the counting rate of samples was compared with the integrated current to a surface ionization detector, from a beam whose ratio of carrier to active rubidium is known. Because of the well-known difficulties in the precision measurement of the latter ratio, the results are normalized to unity for sulfur. (The value obtained from the measured activity ratio was about 125% for the retentivity of sulfur.) On the other hand, Goodman and Wexler find large and reproducible retention for thermal cesium and indium atoms falling on a brass surface at liquid nitrogen temperature,21and Hamilton et al. find a similar situation for beams of copper and gold on surfaces of The Berkeley group finds that sulfur surfaces give reproducible and probably unit retention for beams of cesium, rubidium, potassium, thallium, gallium, indium, copper, silver, gold, and neptunium. They also find flamed platinum surfaces to be equally effective for beams of thallium, plutonium, uranium, thorium, americium, protactinium, and curium. Difficulties are found by this group in the collection of bismuth, iodine, and bromine, and by Cohen in the collection of astatine.31Lipworth et ~ 1 . 27 J. P. Hobson, Thesis, University of California, unpuhlished; J. C. Hubbs, Thesis, University of California, unpublished. 28 H. Lew and G. Wessel, Phys. Rev. 90, 1 (1953). 2gL.Davis, D.E. Nagle, and J. R. Zacharias, Phys. Rev. 76, 1068 (1949). 30 D. R. Hamilton, A. Ternonick, F. M. Pipkin, and J. B. Reynolds, Phys. Rev. 96, 1356 (1954). 31 V. W. Cohen, private communication. 31 E. Lipworth, private communication.
~ 2
78
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
have tried several surfaces, including platinum and sulfur, and find a silver surface to be the only acceptable one for the collection of iodine and bromine and to give retention that is less than unity and not highly reproducible, even under conditions where the silver surface is deposited by vacuum evaporation, removed from the evaporator, and stored under mechanical pump vacuum until ready for use. In summary, only sulfur is found to give uniformly acceptable retentivities for all the electropositive elements tried. This is presumably the result of a negative bonding energy for these materials and of the fact that sulfur forms in air a surface layer of acid which is rapidly pumped off in vacuum. The effective vapor pressure of solid sulfur a t room temperature is about lo-$ mm Hg; therefore, the lifetime of a surface atom is more than an hour. A corresponding “universal” surface for electronegative elements has not yet been found. 2.4.1.2.4.4. Beam Detection. The cross-sectional area of an atomic or molecular beam is typically 0.1 cm2 or less. This immediately allows the investigator who takes advantage of this fact to design counting systems having backgrounds far below the average levels encountered in nuclear physics research. While each nuclide presents unique opportunities for the design of low-background and high-counting-efficiency systems, the attention required by other phases of the research is such that we have standardized on three counter types (described below), which detect essentially any decay with reasonable efficiency and with background less than 5 cpm. Counter for X-rays, low-energy-y-rays, and 0-particles. The of Rbsl first prompted the development of high-efficiency, low-background counters to detect K X-rays (15 kev) emitted when the material K-captures or internally converts. The product of the development is equally useful when decay goes largely through @-emissionor when there is a prominent gamma (with energy below 150 kev) in the delay scheme. The counter and appropriate collecting surface are shown in Fig. 5. The counting head is a brass cup that fits over the face of the photomultiplier, and into which are cemented the y-window of 0.5-mil 2-5 aluminum, the crystal, 0.25 by 0.5 inch by 1 mm, and the quartz light window. Care is taken to prevent a film of the cement, Myva Wax, from covering either face of the scintillator. The y-window transmission is approximately 90 % for rubidium K X-rays; thin-film techniques might well be used to extend the useful lower limit of operation to 1 kev. Sodium iodide crystals are obtained in large blocks and split to size in a dry box with relative humidity less than 5% a t 30°C. Cleavages are 3 3 J. B. Hobson, J. C. Hubbs, W. A. Nierenberg, and H. B. Silsbee, Phys. Rev. 99, 612 (1955).
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
79
made by delivering a light blow to a razor blade held against the surface in the direction of a crystal plane, and acceptable fragments with no sign of fracture or discoloration are immediately sealed between the two windows to prevent surface decomposition. Photomultiplier tubes are selected from the large number of tubes based on the requirements of less than one dark-current pulse per minute
BH 11 I
1
EDGE
I
I I
SIDE
TOP
BOTTOM
FIQ.5. Drawing of the phototube assembly for the X-ray counters. The lower part of the drawing is the sketch of the sample carriers or buttons.
above a pulse height equivalent to 7-kev gamma rays in NaI. The common denominator of tubes so selected is a very large amplification for a given dynode voltage and presumably a high photocathode conversion efficiency. The relative dark-current rate is very nearly independent of dynode voltage over the normal region of operation of the tubes. Highvoltage curing of the photomultipliers leads to no improvement of the noise figure. Operation. Counting is performed with conventional preamplifiers and single-channel differential analyzers. The only stabilization used is that of the analyzer and a Sorenson regulator, which is used to supply
80
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
110 voltsac. The counter window is customarily set to accept 90% of the X-ray line, which for 15-kev X-rays results in a 100% window, is., from one-half to three-halves of the peak counting-position. Only a 50 % window is required to achieve the same results above 50 kev. Under such adjustments as this, small drifts in window width and height contribute a negligible change in counter efficiency and background, a factor of paramount importance for counting periods which sometimes are as long as three months. The counters are shut down only after a counter electronics failure. The counter head is shielded with a 2.541. thick lead cylinder. For window settings as given above, the counter background in the shield is one count per minute or less for all energy settings between 5 kev and 100 kev. Minimum background occurs in the neighborhood of 40 kev, where it is typically 0.35 cpm. Counter background outside the shield is approximately 5 cpm. Integral settings of the counter in the shield, with a threshold corresponding to 3 to 5 kev, result in approximately 8 cpm background for a situation in which all p mesons, low-energy gamma rays, and medium-energy electrons are detected with unit efficiency. The integral contribution of dark-current pulses to counter background is, for a typical unit, approximately lo4 exp(-EE/0.5 kev) cpm between 1 and 10 kev. The scintillation counter is ineffective for low-energy p- and &-emitters because of the relatively thick (7 mg/cm2) aluminum window. Alpha emitters are detected in flow proportional counters large enough that the a is stopped in the counting gas. Discrimination against pulses corresponding to less than 3 or 4 Mev lost in the gas effectively eliminates all background except that arising from large showers and from a’s from contaminants in the chamber walls. This latter contribution is minimized by coating the chamber walls with a heavy layer of common aquadag which has been found to contain a-emitters in much lower concentration than metals. Such chambers have backgrounds of three to ten counts per hour and a detection efficiency of about 50%. p emitters are detected in similar large-volume flow counters, or in special units with extremely small interior wall area (about 20 cmz). With large chambers it is again found that a layer of aquadag substantially reduces chamber background. Large units have been found to have a detection efficiency of 50% for p’s above 5 kev and a background of about 3 counts per minute. The special small-volume chambers built of stainless steel typically have a background of 1 count per minute with 50% detection efficiency for p’s above 30 kev. 2.4.1.2.5. ISOTOPE PRODUCTION AND PREPARATION. In this section, the problem of producing sufficient radioactive nuclei for detection and
2.4.
DETERMINATION
OF SPIN,
PARITY, AND NUCLEAR MOMENTS
81
measurement is analyzed, and some numerical examples are given. For illustration, the examples of Rbsl, Rbs2, RbS3,and RbS4are discussed. The stable isotopes of Rb are Rbss and RbS7,with spins of Q and Q respectively. Their nuclear magnetic moments are 1.35 and 2.75 in units erg/gauss. The most convenient of the nuclear magneton, 5.05 X way to produce the neutron-deficient isotopes is via a-bombardment of Br. Since Br has two stable isotopes in nearly equal abundance, B P and BrS1,there may be more than one way to produce the same isotope. If 50-Mev a-particles are available, reactions of the type (a,kn) are possible, where k = I, 2 , 3, or 4.Other reactions are possible, but the yield of
ALPHA
CALCULATED
LAB ENERGY
-
MEV
EXCITATION FUNCTIONS
Br(a, hn) Rb
FIG.6. The differential cross section for a-particles on bromine to produce rubidium.
radioisotope is so low that they may be neglected. Furthermore, we will see that the identification, by means of chemistry, vapor pressure, electronic structure, and characteristic X-rays, is sufficiently rigid to remove any possibility that the finally detected beam is not rubidium. Figure 6 is a production curve for a-particle reactions, estimated from the evaporation model. These curves and the initial energy of the a-particles can be used in a numerical integration to estimate the yield of the radioisotopes. The numbers used as a n example are based on an actual run a t the Berkeley 60-in. cyclotron. The overriding problem in working with the neutron-deficient isotopes of short half-life is one of intensity. It is important to obtain yields as great as possible from a given cyclotron bombardment. This means that the high-intensity beam should be uied. I n working with isotopes of Au
82
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
that can be made by a-particles on Ir, this is no problem because the heat developed on a conventional water-cooled target can never be enough to melt or vaporize the Ir or Au under reasonable circumstances (up to about 100 pa of beam at -45 Mev). However, for most substances, such as the bromides in this example, a beam of only a few microamperes a t this energy vaporizes and decomposes the salt in a conventional holder. Therefore, a special target assembly (Fig. 7) was designed to permit bombardment by currents up to 40 pa without destruction of the salt. For estimation of the number of atoms involved, reference is made to Fig. 6, which gives the differential cross section for each of the four reactions (a,kn) on bromine, for k = 1, 2, 3, 4. Knowing the density of BaBrz and interpolating a n appropriate range-energy relation for a-particles, one can reach the estimates given in Table 11. The yield for an TABLE11. Results of Br
(a&) Rb Reactions from Bombardment of Br with a Cyclotron Beam of 45-Mev a-Particles at 24 ha (Average)
Total effective production cros8 section (emz X 3.19 3.06 1.50 0.310 Number of atoms produced per minute (XlO'Z) 1 . 0 0.97 0.47 0 . 1 Equivalent number of disintegrations per minute after 5.75-hr bombardment ( X lo8) 570 420 0.94 0.49 Reaction not completely realized because energy available (45 MeV) takes in only half the ((rI4n)peak.
average cyclotron beam current of 24 pa and 45-Mev a-particles is also given in the table, as well as the number of disintegrations per minute immediately after a 5.75-hr bombardment. These numbers can be most useful if they are translated to counts per minute. The counters used are K X-ray counters (more fully described later). We use here only those numbers necessary for intensity estimates. First, we have the factors common to all four isotopes: (a) Counter solid-angle factor of 0.5. (b) Window efficiency-the effect of the finite resolution of the counter discriminator. This is about 0.85. (c) A fluorescent yield of 1/1.62. for K versus L captures. (d) A factor of Combined, these factors give a counting eficiency of 0.24. Now there are factors that are different for each isotope: Rb8': 0.87 for the electron capture versus @+ emission. RbS2:0.94 for the electron capture versus /3+ emission. In addition, there is a guessed factor of for competition in production with the metastable RbS2that does not decay to the ground state.
+
FIG.7. High-yield cyclotron target for production of the radio-alkalies from halide salts. The cross ribbing is for the more effective removal of the heat.
2.
84
DETERMINATION OF FUNDAMENTAL QUANTITIES
Rbs3: The low-energy y-rays of Kr*lm,the daughter, which augments the counting by a factor 1.4. Rbs4:A factor of 0.76 because of electron branching. The total predicted counting rate immediately after bombardment is Rb81: 1.2 X RbE2:0.5 X Rbs3: 3.0 X RbE4:0.9 X
10” cpm 10” cpm lo8 cpm lo8 cpm.
This can be compared with the results obtained from the analysis of a sample from the “straight-through” or undeflected beam. The ratio of the observed counting rates for the Rb83 Rb84 components t o
+
t
I I
Rbe’
i
Rb
Rbe2
f
Rbs3
5 b
8 8
8
e
I0
%
, .60
F I ~8.. The normalized peak resonance intensity of the Rb isotopes for a number of runs. The intensities do vary approximately as 2 / ( 2 I
+
+ 1).
predicted and 1.9 X Rb81 Rb82 is 2.5 X observed. A rough absolute check also can be undertaken. The sample used for beam analysis was 0.02 of the total beam determined by monitoring the beam carrier. The transmission of the apparatus is 3.3 X 10-6/?r,and the decay rate (extrapolated) for the sample was 13,700 cpm, giving 0.7 X 10” cpm total, or about one-half that predicted. This is very good agreement,,
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
85
considering the nature of the estimates. There is still another check on the intensities, and th at involves the intensities of the observed resonances. The spins of Rbsl, RbE2,Rbs3, and Rb84 are $, 5, Q, and 2, respectively ; their resonances were observed under many magnetic field and production conditions. If each of the counting rates a t the spin resonances is normalized to the particular component present in the beam and l), they do indeed roughly scatter about a plotted versus 1/(21 straight line through zero (Fig. 8). The theory predicts just such a n intensity dependence. 2.4.1.2.6. CONCLUSION. The experimental methods outlined here are being used to materially extend our knowledge of the ground-state properties of nuclei. Since the radioactive technique is essentially universal, the detection limitations formerly imposed on atomic and molecular beam investigations have essentially disappeared for a class of isotopes considerably larger than th at which encompasses the stable and verylong-lived ( > 10-yr) species. Perhaps the most significant contribution to knowledge of nuclear structure has come from the enumeration of ground-state spins and magnetic moments for a large number of isotopes of the same element; most theoretical treatments of the nuclear system are considerably simplified by the fact that the “core” remains the same across such a series. I n addition to this general feature of the research, results are obtained having particular and specific interest, such a s the spin of Cs127 and Cs129, which showed conclusively the breakdown of the shell model in this region,2‘ and the spin 0 of Gass that, besides being interesting in its own right, has had some import in parity experiment^.^^
+
2.4.1.3. Microwave Method.* t During the last decade, the microwave method of measuring nuclear spin and moments has offered a n additional tool for determining nuclear properties. The principal feature of this method lies in the observation of the hyperfine structure which may be present in the microwave spectra of atoms or molecules. The existence of a n observable hyperfine interaction is the necessary condition for the derivation of any nuclear information from microwave spectroscopy. 3 4 H. Frauenfelder, A. 0. Hanson, N. Levine, A. Rossi, and G. DePasqualli, Phys. Rev. 107, 910 (1957). t See also Vol. 2, Part 10; Vol. 3, Chapter 3.1; Vol. 4, A, Sections 4.1.2 and 4.2.3.
* Section 2.4.1.3 is by C.
K. Jen.
86
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
Since the energy splitting due to nuclear interactions is in general rather small, the high resolving power inherent in microwave and low-frequency techniques proves to be particularly effective in such investigations. There are essentially two areas where the microwave method has been used in the measurement of nuclear properties. One is the microwave spectroscopy in gases, and the other is the paramagnetic resonance in solids. The former, dealing principally with free molecules in electronically nonparamagnetic states, has led to a number of new determinations on nuclear spins and electric quadrupole moments and, to a lesser extent, on nuclear magnetic moments. The latter, dealing principally with paramagnetic ions in solids under the influence of external magnetic fields, has yielded another set of new determinations on nuclear spins and moments. Although the two approaches just mentioned differ somewhat in their detailed experimental execution, they are basically similar in applying microwave techniques to the observation of absorption spectral lines. 2.4.1.3.1. EXPERIMENTAL METHODSI N MICROWAVE SPECTROSCOPY. * 2.4.1.3.1.1. General Considerations for Microwave Spectroscopic Observation. a. General Description. When reduced to its minimum essentials, a spectroscopic system of the absorption type should contain a monochromatic source of radiation with a continuously variable frequency, a material whose absorption of radiation is under investigation and a detector of the radiation following absorption. These conditions are readily met by a microwave system. A monochromatic microwave source for the spectroscopic work is usually supplied by a klystron tube whose frequency can be varied continuously over a range. The wave propagates from the source through an absorption cell containing the material to be studied. The absorption cell may be a section of the waveguide which when containing a gaseous material can be blocked off at the two ends by sheets of material (usually mica) transparent to the microwave, or it may be a microwave resonant cavity which contains a small solid substance for the study of paramagnetic resonance. In any case, the outgoing wave is allowed to reach a detector for which a semiconductor (usually silicon) crystal is commonly used. Thus it is, in principle, possible to observe an absorption spectrum by noting the change of the crystal
* Most of the material presented in Section 2.4.1.3.1 can be found in the general references given below.laJb,lo la C. H. Townes and A. L. Schawlow, “Microwave Spectroscopy.” McGraw-Hill, New York, 1955. lb N. F. Ramsey, “Molecular Beams.” Oxford Univ. Press, London and New York, 1956. lo B. Bleaney and K. W. H. Stevens, Paramagnetic resonance. Repts. Progr. in Phya. 10, 108 (1953).
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
87
current with the frequency (or alternatively the magnetic field as in the case of paramagnetic resonance). The absorption of a material is usually described by the imaginary part (x”) of the complex quantity x denoting the susceptibility of the material as
x = XI - i f .
(2.4.1.3.1)
The susceptibility may be due to electric dipoles as in most cases in gaseous microwave spectroscopy, or it may be due to magnetic dipoles as in paramagnetic resonance. I n practice it is seldom necessary to consider the absorption due to both kinds of susceptibility at the same time. b. Absorption in a Gas-Filled Waveguide. It is customary to use the absorption coefficient a instead of XI‘ for the description of the gaseous absorption. The relationship between the two quantities is (2.4.1.3.2)
where X is the free space wavelength and X I ‘ is the susceptibility per unit volume. Since X I ’ is dimensionless, a has the dimension of cm-l. If we let y denote the absorption of an empty waveguide, then we have
pl
=
Poe-(~+Y)~
(2.4.1.3.3)
where 1 is the length of the waveguide absorption cell, PO and PI are, respectively, the powers a t the input and output ends of the cell. Since in general a << y, we have for the power change due to gaseous absorption (i.e., signal power) P, = alPoe-Y1. (2.4.1.3.4) Equation (2.4.1.3.4) indicates that P, reaches a maximum if the cell length is chosen to be 1 = y-1. But this optimum condition is somewhat altered when ultimate sensitivity is considered in relation to the noise power (considered in a later section). The absorption coefficient a for a spectral line is given by the Van Vleck-Weisskopf equation: (2.4.1.3.5)
where A = constant of a gas which varies inversely as the absolute temperature, n = number of molecules per unit volume, v o = resonant frequency of a spectral line, and Av = linewidth parameter. At the pressure range most commonly used mm Hg), the linewidth is almost all due to “pressure broadening” and is directly proportional to
88
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
pressure. Hence, the peak absorption is independent of pressure [when there is no “intensity saturation” which Eq. (2.4.1.3.5) does not consider]. c. Absorption in a Cavity Containing a Paramagnetic Sample. A cavity can be characterized by a resonant frequency and a quality factor Q. The fact that a cavity is equivalent to a waveguide of length (X/27r)& and is physically concentrated in a small volume makes it ideally suited to a paramagnetic resonance experiment, where a sample is usually a small crystal placed in a strong homogeneous magnetic field. Since the absorption by magnetic dipoles is the only point of interest in this connection, the sample should be placed in the region of strongest microwave magnetic field, which in a cavity is fortunately also the location of weakest electric field. In a resonance experiment, the microwave and steady magnetic fields should be mutually perpendicular. Assuming that the microwave magnetic field is represented by H I cos at, we have for the power absorbed by the sample
P,
$wx:’H~~
=
(2.4.1.3.6)
where xi‘ is the imaginary part of the magnetic susceptibility for the whole sample, having the dimension of a volume. The absorption by a n empty cavity can be similarly expressed as w
P “ --o-- Q
HI2V 87r
(2.4.1.3.7)
where &I, is the “unloaded” quality factor, and V is effective volume in the expression < J H 2 do>.. = H l 2 V . For maximum sensitivity, QO should be made twice as large as the “loaded” Q , leading to the result P, = P0/2,where Po is the input power. Using the above relations, we have for the power absorbed by the sample (2.4.1.3.8) The expression for xi’ of a paramagnetic substance is often rather similar to that in Eq. (2.4.1.3.5) and can be generally written as
xi’
= CN*J(u)
(2.4.1.3.9)
where C is a constant of the substance and varies inversely as the absolute temperature, N , is the number of paramagnetic’ions in the sample, and f(v) is the spectral form factor. At the center of the resonance line, f(v) = v o / A v , where Au is the half-width.
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
89
d. Noise Power. We shall not give an analysis of the various sources of the noise power except to mention that the noise generated at the crystal is often the largest among them. The crystal noise power can be described by the expression
P (crystal noise)
=
(k T + Y2) ~
Af
(2.4.1.3.10)
where k = Boltzmann’s constant, I’ = absolute temperature, C = constant of a crystal, I = crystal current, f = output frequency, Af = output frequency bandwidth. The first term in Eq. (2.4.1.3.10) denotes the thermal agitation or the “Johnson” noise, and the second term the specific noise of a crystal. Equation (2.4.1.3.10) suggests that, for minimization of the crystal noise, f should be as high and Af as low as possible. The lowest limit of the noise power is set by the thermal term kT At. It is a common practice to describe the over-all noise power as N k T Af, with N designated as a noise figure not only for the crystal, but for the whole system. e. Sensitivity of Detection. The sensitivity of detection in a microwave spectroscopic system can be expressed by the minimum detectable a in gaseous spectroscopy or x:’ in paramagnetic resonance. This can be done by equating the signal voltage to the effective thermal noise voltage. For a gas-filled waveguide, the result is (2.4.1.3.11)
where e = Naperian base, I, = 2/7 (“optimized” length of waveguide). Similarly, for the case of a cavity, the result is (2.4.1.3.12)
Equations (2.4.1.3.11) and (2.4.1.3.12) point to the disadvantage of observing a signal at a very low output frequency, because the noise figure N with reference to Eq. (2.4.1.3.10) would be an enormously large number. Thus a spectrograph operating on the principle of detecting dc or very low frequency signals is apt to be extremely insensitive. The situation is somewhat improved if the microwave frequency is periodically swept (e.g., by a saw-tooth voltage) at a chosen rate across the frequency of a spectral line so that a band-pass receiver at a high frequency can be used. But the method that really gives a major improvement in sensitivity lies in the use of a modulation frequency such that the inverse frequency noise contribution in Eq. (2.4.1.3.10) is greatly reduced and the bandwidth Af can be made very narrow.
90
2.
DETERMINATION OF FUNDAMENTAL QUANTlTIES
2.4.1.3.1.2. Methods of Modulation and Detection. a. Stark Modulation. The method of modulation by employing the Stark effect is most extensively used in microwave spectroscopy of gases.* The modulation frequencies used by various workers are in the range of 6-100 kc. Consider first that the microwave frequency is tuned to an absorption line. If we apply an electric field, the absorption at this frequency can decrease considerably and often to zero because of the displacements in frequency for the Stark components of the original line. In the case of a periodically varying electric field, the fluctuating absorption would give a signal a t the modulation frequency. The variation of the amplitude of this signal with a slowly swept microwave frequency would yield a spectral presentation of the absorption line. The use of a square-wave (so biased that the field is zero half of the time) is especially rewarding in that the resulting spectrum will not only show the zero-field line, but also the Stark components at various positions as determined by the constant field during half of the cycle. However, there are also cases where we rather want to avoid seeing the Stark components when, for instance, we observe the Zeeman splitting of a spectral line. This is exactly our situation in the study of nuclear magnetic moments. Under these circumstances, we can use, for example, a coaxial waveguide for the absorption cell in which the electric field is very inhomogeneous, and hence will cause the Stark components to smear out. Such an arrangement is actually used in the experimental layout shown in Fig. 1. b. Zeeman Modu1ation.t Zeeman or magnetic field modulation is extensively used in paramagnetic resonance experiments. The modulation frequencies used by various workers are in the range of 10-1000 cps. In a few cases the modulation frequency has been raised to 100 kc or higher. A small sinusoidally varying magnetic field is usually placed along the direction of a high steady magnetic field. If the amplitude of the modulation field is much smaller than the spectral line width, then a signal at the modulation frequency would appear after the detection of the microwaves with a magnitude proportional to the derivative of the absorption coefficient (i.e., da/dH). c. Detection of Modulation Signals. Where a modulation scheme is used, only the information carried by the modulation frequency component in the detector output needs to be preserved. This signal is usually fed in succession to a preamplifier, a narrow-band amplifier, and a phase detector. In the phase detector, the signal is mixed with a modulation frequency reference voltage to produce essentially a dc output which can be delivered to an oscilloscope or a recorder. If it happens that there
* See for example, Vol. 3, Section 2.1.4.
t See also Vol. 4, A, Section 4.2.3.3.
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
91
is more than one modulation frequency being used for special reasons, then each frequency component can be separately handled through its own channel as in the above description. The spectroscopic sensitivity of a relatively high frequency modulation system has been experimentally found to be considerably higher than that of a nearly dc system. But, even here, the noise figure N in typical setups is far greater than the ideal ( N = 1).Referring to Eq. (2.4.1.3.11), under the conditions of a good waveguide spectrometer, e.g., I,, lo3cm, T 300"K, A! 30 cps, P o 1 mw, the experimental value for amin is
-
-
MICA WINDOW ROTATABLE POLARIZER
1
-
-
/,,\ \ LONGITUDINAL FIELD COIL
TRANSVERSE FIELD ELECTROMAGNET STARK ELECTRODE
I
ROTATABLE lDETECTOR
AMPLIFIER
T147] OSClLLOSCOPE
1OOkC
PHASE
Fro. 1. Schematic diagram of a microwave Zeeman spectrograph.
roughly in the range lo-* to cm-l. The corresponding value for N is of the order of lo3. The application of field modulation method to paramagnetic resonance in solids calls for some special consideration. Here the modulation frequency for the magnetic field is, with a few exceptions, usually restricted below 1 kc because of the eddy current limitations of the cavity used as the absorption cell. I n order that full advantage can be taken for the lower noise figure a t higher frequencies, a superheterodyne scheme can be adopted with increased sensitivity. A local microwave oscillator is set a t a frequency differing by a constant amount, for example, 30 Mc/sec, from the signal microwave frequency. The difference frequency due to the mixing of the local oscillator output and the signal wave after absorption can be handled by an I.F. amplifier, and eventually by a phase-sensitive detector. If a field modulation is used, then the output contains the
92
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
signal at the modulation frequency which can be handled as discussed before. The sensitivity of a paramagnetic resonance spectrometer can be conveniently rated according to the minimum number of unpaired electron spins N , in Eq. (2.4.1.3.9) that the apparatus can detect when the sample is a t a given temperature. By the use of Eq. (2.4.1.3.12),it can be shown that the ultimate sensitivity of a cavity spectrometer is of the order of 1011 spins at room temperature. Practical spectrometers are considered to be good if their capability of detecting spins covers the range 1 0 1 2 to 1013. 2.4.1.3.1.3. High-Resolution Spectroscopic Methods.* High-resolution spectrometers are occasionally needed to resolve very closely spaced lines (hyperfine structure or Zeeman splitting) , or to enable more accurate determination of line frequencies. Increase in resolution is mostly accomplished by seeking ways to reduce the linewidth. When a modulation method is used for the sake of sensitivity, it also broadens the line in the same process. This means that when high resolution is important one must choose a sufficiently low modulation frequency and/or low degree of modulation at the expense of sensitivity. In gaseous spectroscopy, the linewidth can often be reduced by using lower pressures when pressure broadening is the dominating feature. For further reduction of linewidth, methods for decreasing Doppler broadening have been successfully worked out. But the systems used are often rather complex. For paramagnetic resonance in solids, the linewidth can be reduced by decreasing the concentration of the paramagnetic species, or sometimes by lowering the temperature. A rather different form of high-resolution spectroscopy has been introduced by FeherId with his double resonance method. Here, a resonance condition is first established for the electrons in a paramagnetic species. The resonance is actually between the hyperfine levels of two magnetic states of the electron. A second resonance is then introduced between one of the participating hyperfine levels and another hyperfine level of the same electronic state. The second resonance is essentially of the nuclear magnetic type, being in the range of radio frequencies. If the electronic resonance is initially intensity saturated with a marked reduction of signal, then the application of the nuclear resonance can cause a visible change in the electronic resonance. There is thus a natural selection of only such nuclear transitions that can affect a given electronic transition. This method has the advantage of combining the high resolution of a nuclear resonance experiment with the high sensitivity of an
* See also Vol. 4, A, Section 4.2.3.2. Id
G. Feher, C. S. Fuller, and E. A. Gere, Phgx. Rev. 107, 1462 (1957).
2.4.
DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS
93
electron resonance experiment. But, more important for our present subject, the method has been demonstrated to give a direct determination of the nuclear g factor from the separation between two lines in the radiofrequency spectrum. 2.4.1.3.1.4. Measurement of Frequency, Magnetic Field, and Spectral Intensity. 2.4.1.3.1.4.1. Frequency measurement.* Measurement of the signal oscillator frequency is mostly done with reference to an absolute frequency standard like the transmissions of the station WWV. In gaseous spectroscopy, the frequencies a t the centers of all the spectral lines are to be measured. However, for the measurements involving the hyperfine structure or Zeeman spectrum, only frequency differences between lines are essential. This less stringent requirement can often be met by a simpler operation. In paramagnetic resonance, the oscillator frequency is usually held at a fixed value while the resonance condition is fulfilled by the variation of the magnetic field. It is comparatively a simple matter to make an absolute measurement of a constant frequency. The accuracy of frequency determination depends upon the stability of the signal frequency and the spectral linewidth. Under typical conditions, it is possible to determine the line frequency within about one-tenth of the linewidth. Since a typical linewidth in gaseous spectroscopy is of the order of 1 Mc/sec, the accuracy of the line frequency determinations is usually within the range 0.01 to 0.1 Mc/sec. 2.4.1.3.1.4.2.Magnetic jield measurement.t The measurement of magnetic field is mostly done through the use of a proton resonance magnetometer. Under typical conditions, it is usually practicable to measure the field with an accuracy within 0.1 oersted for a field of several thousand oersteds. This accuracy is more than adequate for the Zeeman measurements in gaseous spectroscopy and is quite satisfactory for most of the paramagnetic resonance work. 2.4.1.3.1.4.3.Intensity measurement.$ Unless extreme care is taken, intensity measurement is subject to many types of variability. Fortunately, only relative intensities are of interest for all measurements concerning the hyperfine structure from which nuclear information is derived. The average accuracy of relative intensity determination is of the order of 1%. 2.4.1.3.1.5. Illustrative Microwave Spectrographs and Spectral Presentations. 2.4.1.3.1.5.1. Spectroscopy in gases. As an illustrative example, the schematic diagram of a microwave spectrographs specially designed
* See also Vol. 2, Chapter 9.2.
t See also Vol. 1, Chapter 9.3.
1See Vol. 3, Section 2.1.7.5. 5 The specific description given here refers to
a spectrograph of this type at the Applied Physics Laboratory of The Johns Hopkins University.
94
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
for measuring the Zeeman effect in gaseous spectroscopy is given in Fig. 1. The absorption cell is formed by a cylindrical waveguide 0.62 in. inside diameter and 53 in. long, with mica windows a t the two ends. A center conductor of 0.125 in. outside diameter serves as the Stark electrode, being supported by perforated mica discs in the waveguide. A zero-biased square-wave voltage a t 500 volts maximum and 100 kc is used for the Stark modulation. The absorption cell is placed in the gap of a 4-fOOt long electromagnet, which can supply a transverse magnetic field up to about 8000 oersteds. An auxiliary coil wound around the absorption cell can supply a longitudinal magnetic field up to about
24,019.6 Mc/ssc
FIG.2. Observed hyperfine spectrum of O I @ C W 3for J
= 1 + 2.
1000 oersteds. The longitudinal field is used only for the determination of the sign of the magnetic moment. For the latter operation, a circular polarization of the microwaves is produced by a “polarizer” (mica phase changer) and detected by an “analyzer” (rotable rectangular wave with crystal detector). Using a spectrograph as described in Fig. 1, we can observe the hyperfine spectrum of a molecule in the absence of a magnetic field. Such a spectrum for 016C12S33 is shown in Fig. 2, for the rotational transition J = 1 4 2. A very good fit between the observed pattern and the theoretical predictions can be obtained only when the following assumptions are made: I ( S 3 3 ) = +, en& = -29.1 Mc/sec [see (Eq. 2.4.1.3.13)l. When a magnetic field is applied, each line in the hyperfine spectrum is split in manner characteristic of the magnetic properties of the par-
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
95
ticipating states. The Zeeman pattern2 is the simplest for the transition F = + 4, where the magnetic splitting gives a doublet shown in Fig. 3. The nuclear g factor for S33can be calculated with the known value for the molecular g factor of 0cs3*( -0.026).
+
F=
v2
-bZ
I
t
24,019.6 Mc/sec
H = 3912 oc. -2
-I
0
I
2
Mc/sec
FIG. 3. Zeeman splitting (after Esbach et ~ 1 . 2 ) .
(r components)
for F =
-
8 4 (J
= 1 + 2) of O‘~C”Sa8
CAVITY AND SAMPLE
OSCl LLATOR
1
FIG.4. Schematic diagram of a microwave paramagnetic resonance spectrograph. 2.4.1.3.1.5.2. Paramagnetic resonance. A moderately simple paramagnetic resonance spectrograph is given schematically in Fig. 4. A field modulation of several hundred cycles per second can be used, even with a thick cavity wall. If a thin metallic wall for the cavity is used, 2 J. R. Esbach, R. E. Hillger, and M. W. P, Strandberg, Phys. Rev. 86, 532 (1952).
96
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
then the modulation frequency can be raised to the range of several hundred kilocycles with the attendant advantage of high sensitivity. The source modulation and its related circuitry are used to stabilize the klystron frequency to the sample cavity resonance frequency. This frequency should be kept at least several times higher than the field modulation frequency. There should be a provision, not indicated in the figure, for rotating the sample in the cavity if a single crystal is used. A sample spectrum of paramagnetic resonance is given in Fig. 5 which shows the magnetic hyperfine structure of Ce141 obtained by Kedzie et aLs with an irradiated fission product of U236 in a L a ~ M g ~ ( N 0 , ) ~ ~ - 2 4 H ~ 0 single crystal. * The eight-line spectrum is clearly associated with a
t---
1060 o e r s t e d s
t
3725 oersteds
FIQ.5. Hyperfine structure due to Ce"' in the paramagnetic resonance of Ce3+ ions in LazMgs(N03)~ 2 4 H ~ This 0 . spectrum was observed at 4.2"K and 9380 Mc/sec (after Kedzie et aZ.3).
nuclear spin of 4. The magnetic moment of Ce141is evaluated from the measured hyperfine coupling constant to be about 0.89 nm. BACKGROUND AND RESULTS OF MEASUREMENT 2.4.1.3.2. THEORETICAL FOR SPIN AND NUCLEARMOMENTS.2.4.1.3.2.1. Spins and Nuclear Moments by Microwave Spectroscopy in Gases. Microwave spectroscopy in gases has been largely concerned with the study of the rotational transitions in various molecules with negligible mutual interaction except for the effect of molecular collisions on line broadening. For the most part, the molecules studied do not possess a resultant electron spin or orbital angular momentum. This is the natural situation for almost all stable gaseous molecules in their ground electronic states with an extremely small number of exceptions. To be eligible for the present consideration of nuclear interaction, the molecule in question must have a t
* In their original work, the authors got several other types of spectral lines including those for Nd"7 from the same sample. We have reproduced only the part for Ce141for the benefit of a simple presentation. R. W. Kedzie, M. Abraham, and C. D. Jeffries, Phys. Rev. 108, 54 (1957).
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
97
least one nucleus with a spin. Generally speaking, both the electric and magnetic hyperfine interactions are possible under these conditions. It turns out, however, that magnetic hyperfine interactions are usually negligible in view of the fact that the magnetic fields, due to paired electrons, tend to cancel out almost completely. The few molecules which do show a detectable magnetic hyperfine structure have not yet figured significantly in giving new nuclear moments. On the contrary, the electric quadrupole hyperfine interaction in a molecule is often very prominent except for a nuclear spin of +, which has no quadrupole moment, or, in some cases, a hyperfine coupling much too small for detection. Assume now a molecule answering the above description with just one nuclear spin and also being, for simplicity, a symmetric top variety (which includes a linear molecule as a special case). The restriction to a single nuclear interaction and a simple molecular configuration is fortunately not too stringent because most of the pertinent information to be presented has been obtained in just this way. Also assume that no magnetic field is being applied externally for the moment. Then the nuclear electric quadrupole interaction energy is, by a standard formula (reference la, page 154)
where
+
+
+
c = F ( F 1) - 1(1 1) - J ( J 1) F = J + I , J + I - 1 , . . . [J-I1 eqQ = quadrupole coupling constant K = component of J on molecular symmetry axis in a symmetric top, being zero for a linear molecule in the ground vibrational state. The quadrupole coupling constant is the product of the proton charge e and the nuclear quadrupole moment Q and the second derivative of V , q = r32V/r3z2,evaluated along the molecular axis z for the potential V a t the nucleus due to all charges outside of the nucleus. The quantity WQ is the hyperfine energy for each state characterized by the quantum number F , with all the F states associated with the same rotational state characterized by J and K . The rotational energy for a symmetric top is W R= B J ( J 1) - ( A - B ) K 2 , where A and B are the usual rotational constants. Any given transition is a result of the relation
+
A(WR
+ WQ)
=
hv
where Y is the microwave frequency and h is Planck’s constant. The selection rules in F , J, and K are: AF = 0, 1; A J = 0 , +_ 1; AK = 0 .
+
98
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
The intensity distribution for the transitions is well known and will not be given here. It is clear from the above discussion that, with the knowledge of J and K ,the nuclear spin I can be determined by matching the pattern of a calculated hyperfine spectrum (including intensity distribution) based upon an assumed I with that of the observed spectrum. The adjustable scale factor for perfect matching gives directly the constant eqQ. As a result of such measurements over a large number of molecules, the known spin values of very many nuclei have been confirmed; but, more significantly, more than ten new nuclear spins have been established by the microwave spectroscopic method for the first time. Table I gives a listing TABLE11 Atom Mass number I
B
0 S
c1 Ge Se
I
10 11 17 33 35 36 73 75 77 79 125 129 131
dPN)
3
Q
$ +0.633(10)
8 2
i
+l.OO(lO) +1.32(8)
H
8 8 4
-1.018(15) 2.74(14) 2.56(12)
cma) Reference
+
1.80081 (49)] [+2.68852(4)1 [-1.89370(9)] [O.64342(13)]
+0,06(4) +0.0355(2) -0.005(2) -0,050 +O ,035 [ + I .28538(6)] 0.0172(4) [ -0.8791 4 (12)] -0.21 (10) 1.1 [+0.534058(14)] I < 2 x 10-31 0.7 -0.66 [+2.617266(12)] - 0 . 4 3 0 5 )
[
i3
&(lo-24
Ib lb lb lb,2,4 lb lb,5 lb 6 lb lb 7,8 lb,9 8
This table lists only those atoms whose nuclear spins Z were first determined by microwave spectroscopy in gases. The other quantities p and & were also first obtained from the same method, but, where they are improved by another method, the more accurate results are given in square brackets. The numbers in parentheses indicate experimental errors in the last figures of the associated values.
of the nuclei and their new spin values, together with other items to be d i s c ~ s s e d . ~ ~Notable ~ ~ ~ 4 -among ~ these new spins are the ones such as I(B1O) and I(C136)which ran counter to the then current predictions. It is just this type of information that is most helpful to the theories of nuclear structure. We come now to the other result of the hyperfine structure observation, J. R. Esbach, R. E. Hillger, and C. K. Jen, Phys. Rev. 80, 1106 (1950). C. Aamodt and P. C. Fletcher, Phys. Rev. 98, 1317 (1955). EL. C. Aamodt and P. C. Fletcher, Phys. Rev. 98, 1224 (1955). 7 P. C. Fletcher and E. Amble, Bull. Am. Phys. SOC.2, 30 (1957). * P. C. Fletcher and E. Amble, Phys. Rev. 110, 536 (1958). W. Gordy, 0. R. Gilliam, and R. Livingston, Phye. Reu. 76, 443 (1949). 6L.
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
99
namely, eqQ. Unlike the direct determination of nuclear spin, the nuclear quadrupole moment Q is as yet concealed in its product with the quantity q which needs to be ascertained. Of course, if the electronic wavefunction is accurately known, there should be no difficulty in calculating q. As this is practically not the case, some methods of approximation have to be used. The problem is greatly simplified by the observation of Townes and Schawlow (see reference la, page 232) that, for a heavy atom participating in a covalent bond in a molecule through a p orbital, the contribution to q is nearly the same as in an atom. In the case of an atom, q can be calculated readily from the fine structure separations or the magnetic hyperfine structure. However, measurements of eqQ for molecules containing the same atom show rather large variations, which can only mean that q varies from molecule to molecule on account of local conditions. The local conditions are construed to consist mainly of the ionic character of the bond and the degree of s or d hybridization. I n spite of some inherent uncertainties in this manner of analysis, it has been demonstrated that reasonable values for the nuclear quadrupole moments (with a specification of their signs) can be obtained. Such values for Q are listed in Table I only for those nuclei whose spins were first determined by the microwave method. There have been, in fact, many other nuclear quadrupole moments estimated from microwave measurements even though the corresponding nuclear spins were established by other methods earlier. We shall now take up the subject of nuclear magnetic moments in microwave spectra of gas molecules. I n the absence of an external magnetic field, nuclear magnetic moment can only reveal itself in a magnetic hyperfine interaction. However, except for an exceedingly few molecules, each of which has an electronic angular momentum, most molecules very rarely show any detectable magnetic hyperfine structure, which is rather difficult for interpretation in any case. Therefore, the information on nuclear magnetic moment has to be sought in the Zeeman splitting of the quadrupolar hyperfine lines previously discussed. Assume now we apply an external magnetic field H . For an electronically nonparamagnetic molecule, the only quantities which could be influenced by the magnetic field are the nuclear magnetic moment and the rotational moment of the molecule. These moments are usually of the same order of magnitude and are sufficiently small that the magnetic energy involved is often considerably smaller than the quadrupolar interaction for the fields commonly applied. Under these conditions, the first order magnetic energy (reference l a , page 290)
WH = -MpArH(ar.rg,
+ wgz)
(2.4.1.3.14)
2.
100
DETERMINATION O F FUNDAMENTAL QUANTITIES
where
M
magnetic quantum number associated with F nuclear magneton g J = rotational g factor = p(rotation)/J qr = nuclear g factor = ~ I / I (YJ = [F(F 1) J(J 1) - I ( I 1)1/2F(F 1) ffI = 1 - ( Y J . As a consequence of this situation, and subject to the well-known selection rule of AM = 0 for a ?r transition and AM = 1 for a u transition, there results a number of Zeeman components, with a defined intensity distribution, for each of the hyperfine lines. From the measured splittings for two or more hyperfine lines, it is possible to determine 1gJJ and 1911 and their relative sign, assuming that F , J , and I values are all known. The absolute sign of the g-factors can be experimentally determined by using circularly polarized microwaves. The aforementioned method has offered a few new determinations of nuclear magnetic moments before many of these moments were redetermined by some other methods (chiefly nuclear resonance) with much higher accuracy. In the interest of the present discussion, the first determinations of nuclear magnetic moments by the microwave spectroscopic method are listed in Table I, with the more accurate values by some other methods placed in square brackets. 2.4.1.3.2.2. Spins and Nuclear Moments by Paramagnetic Resonance. Paramagnetic resonance means, by usage, the electron spin magnetic resonance rather than any other paramagnetic resonance such as nuclear magnetic resonance. The presence of an unpaired electron spin, then, is an essential feature of this study. When a spinning electron, as a part of an atom, is placed in a magnetic field, a resonant absorption of radiation energy occurs when the radiation frequency is related to the magnetic field in a specific manner. It so happens that, with a convenient laboratory field of a few kilogauss, the resonant frequency falls directly in the microwave region. This is the only visible connection that paramagnetic resonance has with the microwaves. Paramagnetic resonance can be and has been done in gases, liquids, or solids. More productive work, however, has been done with the solid phase than any other, partly because most paramagnetic compounds are solids. The most extensively studied paramagnetic atoms or ions are those in the transition groups, the iron and the rare earth groups of elements. Paramagnetic resonance studies in recent years, mainly led by Bleaney at Oxford, have supplied the much needed experimental basis for many of the deductions and speculations in the quantum theory of paramagnetic solids. Among the significant contributions on the subject is a clear =
~ L N=
+ +
+
+
+
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
101
demonstration of the effect of the crystalline field on the splitting of the orbital degenerate energy levels, “quenching” of the orbital angular momentum, and magnetic anisotropy. Though somewhat disparaging to our present discussion, it must be said that the nuclear effects come in somewhat as by-products of the principal development in paramagnetic resonance in much the same way as they do in gaseous microwave spectroscopy. The nucleus of a paramagnetic ion can make its presence felt only through the channel of hyperfine interaction. Whenever the nucleus possesses a resultant angular momentum, and hence a nuclear magnetic moment, it can have a prominent magnetic hyperfine interaction with the electronic magnetic moment. Also, there may be a nuclear electric quadrupolar interaction, although generally very much smaller. The dominating role of the crystalline field in paramagnetic resonance of solids, and the variety of field symmetries involved, make it difficult to describe the situation in most general terms. For simplicity of presentation as well as to cover practically the most important case, let us assume that the crystalline field has an axial symmetry and the “spin Hamiltonian” takes the formga X = DSz2
+ P[gllHzSz + gI(HzSz + HuSg)l + AIzSz + B(1zSz(2.4.1.3.15) +
where the coordinate system is chosen with z along the symmetry axis and Sz,Su,S,= components of the electron spin S H,,H,,H, = components of magnetic field H 911 = electronic g factor parallel to symmetry axis gI = electronic g factor perpendicular to symmetry axis p = Bohr magneton and A , B, and D are constants. I n Eq. (2.4.1.3.15),it is assumed th a t the lowest orbital state is a singlet and the nuclear electric quadrupole energies are negligible. The complete solution of Eq. (2.4.1.3.15) is too complicated to be given here, but a simple description of the results can be made in sufficiently strong magnetic fields. If the field-independent terms are all negligible (i.e., A B D A 0), then there is just one resonance with the relation hv = gPH witheb
--
92 =
gI12COS*
8
+ gL2 sin2 8
((2.4.1.3.16)
where e = angle between magnetic field and the symmetry axis. If the term DSz2 is small but not negligible, there are 2 s equally spaced lines with sa
B. Bleaney and K. W. H. Stevens, Repts. ProgT. ,in Phps. 16,134 (1953). reference 9a, page 139.
@hSee
102
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
the intensity decreasing from the center. This spectrum is known as the fine structure, since it is completely independent of the nuclear effects. If, in addition, A and B are also not negligible, there is a hyperfine structure of 21 1 equally spaced lines for each fine-structure line present. All the hyperfine lines in one group are of equal intensity, since all the discrete nuclear orientations relative to the magnetic field are equally probable under normal temperature conditions. The spacing K between any two adjacent hyperfine lines is given assb
+
K2g2= A2g$ cos2 9
+ B2gL2sin2 0.
(2.4.1.3.17)
It is clear from the preceding discussion that the nuclear spin I can be directly determined from the number of hyperfine components. The nuclear magnetic moment can be easily determined if the moment of an isotopic nucleus is known from a previous direct measurement (e.g., nuclear magnetic resonance). Then the ratio of the nuclear magnetic moments between the two isotopes is simply equated to that of the total splittings of the two corresponding hyperfine spectra observed in the paramagnetic resonance experiment. When such information is not available, then one has to calculate A or B from the observed spacing between the hyperfine lines. Either A or B is directly proportional to the nuclear magnetic moment p with a proportionality factor which, for a non-s electron, depends principally upon the factor < F ~ >which , is the inverse cubic distance between the electron and the nucleus averaged over the spatial function of the electron. Without going into the detailed processes of evaluation, it suffices to say for our present purpose that the nuclear magnetic moments have been estimated for a number of nuclei from results of paramagnetic resonance. Table I1 tabulates the nuclear spins and magnetic moments, along with some rough values for the nuclear quadrupole moments, which represent new results first obtained by paramagnetic r e s o n a n ~ e . ~ ~ ~ l ~As , ~in ~ ~Table 0 - 1 9I, more prelo
M. M. Weiss, R. 1. Walter, 0. R. Gilliam, and V. W. Cohen, Bull. Am,. Phys.
SOC.2, 31 (1957).
Dobrowolski, R. V. Jones, and C. D. Jeffries, Phys. Rev. 104, 1378 (1956). R. V. Jones, W. Dobrowolski, and C. D. Jeffries, Phys. Rev. 102, 738 (1956). 13 W. Dobrowolski, R. V. Jones, and C. D. Jeffries, Phys. Rev. 101, 1001 (1956). l4 J. Owen and I. M. Ward, Phys. Rev. 102, 591 (1956). 16 M. Abraham, R. Kedsie, and C. D . Jeffries, Phys. Rev. 108, 58 (1957). 16 J. G . Park, PTOC. Roy. SOC.A246, 118 (1958). P. B. Dorain, C. A. Hutchison, Jr., and E. Wong, Phys. Rev. 105, 1307 (1957). B. Bleaney, P. M. Llewelyn, M. H. L. Pryce, and G . R. Hall, Phil. Mag. [7] 46, l1 W. l*
991 (1954). J. C. Hubbs, R. Marrus, W. A. Nierenberg, and J. L. Worcester, Phys. Rev. 109, 390 (1958).
2.4.
DETERMINATION
OF SPIN, PARITY,
AND NUCLEAR MOMENTS
103
cise values later derived from some other methods are placed in square brackets. Not listed in Table 11, but which can be found elsewhere, are the cases where paramagnetic resonance confirmed previous spin assignments but offered new values for nuclear magnetic and, sometimes, quadrupolar moments. TABLEI I e Atom Mass number I
-P
v Mn
co Mo Ru Ce Nd Sm
Eu DY
Er U P
PbN)
&(lo-24 cm*) Reference
-
32 49 50 53 56 57 60 95 97 99 101 141 143 145 147 147 149 152 154 161 163 167 233 235 239 24 1
1
5
-0.2523(3) 4.46(5)
6
4
4
4
5
[+3.34702(94)] 5.050(7) 3.855(7) 4.65(20) 3.800(7)
4
[ -0.93270(18)] [ -0.95229(10))
Q
= 1.09(3)
3 & 4 4 4 4
0.89(9) --1.0(2) -0.62(9) 0.56(6) -0.78(37) -0.65(23)
3
2.1 -0.37(4) +0.51(6) 0.50(12) 0.48-0.54 0.32-0.35 0.4(2) 1.4(6)
4 4 4 8 4
$
.’”” = 0.965(10) &I64
+
1.1(4) +1.3(4) 10.2(30) 3.3-3.5 3.9-4.1
[O ,021
Id 10 lb 11 12 lb 13 lb, 4 lb,14 Ib lb 3 Ib Ib 3 lb lb 15 15 16 16 lb 17 17 18,19 18
This table lists only those atoms whose nuclear spins Z were f i s t determined by paramagnetic resonance in solids. The other quantities p and Q were also first obtained from t h e same method, but, where they are improved by another method, the more accurate results are given in square brackets. The numbers in parentheses indicate experimental errors in the last figures of the associated values.
In concluding the subject on the microwave method for measuring spins and nuclear moments, it may be said that this method has offered a number of new assignments for the spins and many reliable values for the magnetic and quadrupolar moments of nuclei. This type of measurement has the advantage of requiring only small samples, a factor of great importance, particularly in connection with radioactive nuclei.
104
2.
DETERMINATION O F FUNDA4MENTAL QUANTITIES
2.4.1.4. Nuclear Magnetic and Quadrupole Resonance. 2.4.1.4.1.*
t
PRINCIPLES OF THE METHOD. 2.4.1.4.1.1. Introduction. Since their introduction by Purcell, et al.1 and by Bloch et a1.,2the techniques of nuclear magnetic resonance and nuclear induction have been successfully used in the precision determination of the magnetic moments of over 100 nuclei and the spins of about 25. The subsequent observation of nuclear quadrupole resonance absorption in solids by Dehmelta and by Pound4 has resulted in the measurement of nuclear electric quadrupole moments for some 20 nuclei. These experiments are all so closely interrelated that they will be collectively treated. The method is a modified extension of Rabi's principle of magnetic resonance (see Section 2.4.1.2) to nuclei contained in matter of normal density; here the resonance condition is observed by a macroscopic electromagnetic absorption or dispersion in contrast to the deflection and subsequent detection of free atoms. It should also be noted that the method is the nuclear analog of microwave paramagnetic resonance absorption (see Section 2.4.1.3). Consider a nucleus characterized by its angular momentum Ih and total magnetic moment p = yhI. Here p represents the component along I of the resultant magnetic moment of the individual nucleons, which are so tightly coupled by nuclear forces in comparison to the energies involved in magnetic resonance that the assumed proportionality between p and I is completely justified. The maximum possible components of I and t) in any given direction are called the spin I and the magnetic moment The proportionality factor y = p / I h is here called the gyromagnetic ratio; for protons, y E 2.67 X lo4 sec-l oersted-'. The interaction of the nucleus with a n applied dc magnetic field H will be given by the usual magnetic dipole term X M = - y . H = -yhI * H. The 21 1 equally spaced energy levels (see Fig. 1) are given by Em = -ykHom where I 2 rn 2 - I , if we assume H is directed along t See also Vol. 2, Chapter 9.7;Vol. 3, Chapter 4.1;Vol. 4, A, Chapter 4.4. E. M. Purcell, H. C. Torrey, and R. V. Pound, Phys. Rev. 69,37 (1946);N.Bloem-
+
bergen, E. M. Purcell and R. V. Pound, Phys. Rev. 73, 679 (1948). *F. Bloch, W. W. Hansen, and M. Packard, Phys. Rev. 89, 127 (1946); F. Bloch, Phys. Rev. 70, 460 (1946);70, 474 (1946). a H. G. Dehmelt, Am. J . Phys. 22, 110 (1954),which see for earlier references. 4R.V. Pound, Phys. Rev. 79, 685 (1950). 6 I.( is in Gaussian units, which are used throughout this section; the unit of magnetic field H i s the oersted. Sometimes nuclear magnetic moments are given in units of t h e nuclear magneton 10 = eh/2mc = 5.05 X erg/oersted. Usually experiments are performed in magnet air gaps in which H (oersteds) = B (gauss) to high precision; for this reason, the literature of magnetic resonance often specifies t h e magnetic field in units of t h e gauss.
* Sections 2.4.1.4.1 through 2.4.1.4.7.2 are by C. D. Jeffries. Section 2.4.1.4.7.3 is by G. Feher.
2.4.
DETERMINATION OF SPIN,
PARITY,
AND NUCLEAR MOMENTS
105
the z axis with magnitude Ha. Consider now a large number of such nuclei which are immersed in a lattice, i.e., a reservoir of thermal energy, such as the protons in a sample of water or a crystal of ice. Because of thermal agitation, there will be relaxation processes which tend to bring the system of proton spins into thermal equilibrium with the lattice.
m =+A 2
(b)
(a)
FIQ.1. (a) Energy levels of a magnetic dipole with spin I = in a magnetic field. (b) Additional splitting due to nuclear electrical quadrupole interaction with electric field gradient in a crystal.
We can describe the system by assigning Boltzmann population factors to the energy levels. Between an adjacent pair of levels there will exist the population ratio: N(m)/N(m
+ 1)
=
exp[-E,/kT
+ E,+l/lcT]
E 1 - rhHo/kT
1-
lo-'
(typical value a t room temperature). Now if electromagnetic radiation at the Larmor frequency wo = [Em- Em+J/h = ~ H radians/sec o is applied to the system, stimulated magnetic dipole transitions may occur according t o the selection rule Am = fl. However, there will be more stimulated absorption (Am = - 1) than stimulated emission (Am = 1) because of the slight excess population in the lower state of any pair of levels. Thus there will be a net resonant absorption of electromagnetic energy by the system, resulting in a slight heating of the lattice. For HD fields of a few thousand oersteds, Y O = w 8 / 2 r is of the order of 10 Mc/sec, and the resonance absorption can be detected by straightforward radio frequency techniques, e.g., b y wrapping a coil around the sample and observing by means of a bridge the loss factor of the coil for frequencies in the neighborhood of va. As in related phenomena, the absorption is accompanied by a dispersions which can be observed by adjusting the
+
R. d e L. Nronig, J . Opt. SOC.Am. 12, 547 (1926); H. A. Kramers, Atti del congr. J. Gorter and R. de L. Kronig, Physica 3,1009 (1936); A. R. Von Hipple, "Dielectrics and Waves." Wiley, New York, 1954. 6
inst. jk., Como 2, 515 (1927); C.
106
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
bridge to detect the change in reactance of the coil. Thus by observing the resonance frequency v o in a known magnetic field, one obtains in a relatively simple experimental arrangement the nuclear gyromagnetic ratio p / I h , from which the magnetic moment p may be determined if the spin I is known. Furthermore, as shown later, the spin may be determined from the amplitude of the resonance absorption. If I > 1, the nucleus may be further characterized by an electric quadrupole moment and there may be in addition to an interaction of the form XQ = Q * AE, which represents the dyadic product of the nuclear quadrupole moment operator Q with the electric field gradient tensor AE a t the nuclear site. The nuclear electric quadrupole moment, defined explicitly in reference 44, is a measure of the deviation from spherical symmetry of the nuclear charge distribution. Although the quadrupole interaction may be averaged to zero in the rapid tumbling of liquids, we find in single crystals that the magnetic energy levels are shifted unequally (see Fig. 1) by this additional term, so that resonance absorption between pairs of levels may be observed a t several different frequencies. I n the case of zero applied magnetic field, resonance absorption between pure quadrupole levels may be observed. The measurement of the resonance frequencies determines the product qQ of a principal value q of the electric field gradient tensor with the scalar quadrupole moment Q . Calculated values of q must be used to obtain Q from the data; consequently the precision in the determination of Q, in this or any other method, is about two or more orders of magnitudes lower than for p. However the precise ratio of the quadrupole moments for two isotopes can be easily measured. Furthermore the spin I is explicitly given by the number of resonance lines. From the broad field of nuclear magnetic and quadrupole resonance absorption, we discuss here only those aspects of direct interest to nuclear physics. For details we refer to more general and extensive treatments.’-’l 2.4.1.4.1.2. The Bloch Phenomenological Theory. The simple optical model of magnetic resonance absorption given above does not take into consideration the interactions of the spin system with itself or with the lattice. Detailed treatments of these relaxation processes have been N. F. Ramsey, “Nuclear Moments.” McGraw-Hill, New York, 1953. G. E. Pake, Solid State Phys. 2, 1 (1956). E. R. Andrew, “Nuclear Magnetic Resonance.” Cambridge Univ. Press, London and New York, 1955. lo A. Abragam, “Principles of Nuclear Magnetism.’’ Oxford Univ. Press, London and New York, 1961. 11 T. P. Das and E. L. Hahn, Nuclear quadrupole resonance spectroscopy. Solid Stale Phys. Suppl. 1 (1958).
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
107
given,lJ2-14 but from the viewpoint of nuclear physics it is sufficient to consider the original phenomenological treatment of Bloch in which these effects are represented by two empirical relaxation times.2 Consider n identical nuclei per unit volume of spin 1 and magnetic moment p under the influence of an applied field H with the components: H , = 2H1 COB wt, H , = 0, H , = H ; here H is a large and uniform field; HI is a small oscillating field applied by means of a coil around the sample to induce magnetic dipole transitions. Following Ehrenfest’s theorem one assumes that M, the resultant magnetization per unit volume of all the nuclei, obeys the classical equation of motion dM/dt = M = rM X H in the absence of relaxation effects. These are next taken into account as follows: owing to thermal processes in which the spin system exchanges energy with the lattice, it is assumed that the longitudinal component of the magnetization M , approaches its thermal equilibrium value Mo in a characteristic thermal relaxation time T1 according to the relation: k?,= - ( M z - Mo)/T1. Here M ais given by the familiar Curie formula
+
M~ = I--HO1 npz I 3kT
= (I -I- 1 ) l - nyh2 wo
3kT
(2.4.1.4.1)
where T is the absolute temperature of the sample and k is Boltzmann’s constant. It is also assumed that the transverse components M,, M , will vanish exponentially in a characteristic time T 2according to the relations M , = -M,/T2; a, = -M,/TZ. The time T, 5 TI and is called the transverse or total relaxation time, and is essentially the characteristic time for incoherence of M,, M , due to processes in which the total energy of the spin system does not change; more specifically, there may be mutual spin flips, or variations of the locally effective value of H throughout the sample due to the dipole fields of neighboring nuclei. By the Fourier theorem the resonance linewidth Aw 0 1/T2, unless there is appreciable inhomogeneous broadeningl4” due, e.g., to magnetic field nonuniformities. I n the following we assume that there is no appreciable inhomogeneous broadening, as is often the case for liquids but not for solids. The total effect of both applied fields and internal interactions is then approximately given by the phenomenological equations :
M,
=
kf,= Ma =
-MZ/TS - r M , H -MM,/T2 r(M,2HI cos w t - M,H) - ( M z - Mo)/TI - rM,2H1 cos wt,
+
R. K.Wangsness and F. Bloeh, Phys. Rev. 89, 728 (1953). F. Bloch, Phys. Rev. 102, 104 (1956); 106, 1206 (1957). l 4 A. G. Redfield, ZBM J . Research Develop. 1, 1 (1957). 148 A. M. Portis, Phys. Rev. 91, 1071 (1953). 12
(2.4.1.4.2)
108
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
In the Purcell technique’ the experimentally observed quantity is essentiaIIy a bridge unbalance voltage V , proportional to A$z. I n the Bloch technique2 of nuclear induction, an additional coil is wound around the sample along the y axis (see Fig. 3), and one observes the induced voltage V , a Mu. I n either technique, resonance is usually observed by noting the periodic change in V , or V , as the applied field H is periodically swept through the resonance value H O = wo/y by superposing on the dc field a small modulation field H , cos Qt. From the nature of Eqs. (2.4.1.4.2), it is seen that the exact shape of the resonance curves V,(H) and V,(H) will depend upon the rate a t which the resonance is swept through, i.e., upon whether or not the passage through the resonance line in a time T is adiabatic in the Ehrenfest sense (7 >> l/yHl), and also upon the ratios r/T1 and ~ / T zPerhaps . the simplest and most common case is that of adiabatic slow passage: resonance is swept through so slowly that M , continuously reaches its steady-state value under the influence of HI and the relaxation processes. This requires T >> T l , T z , l/yHl. Another interesting case is that of adiabatic fast passage in which T >> l/yH1, T << T1,Tz; this has the property of reversing the algebraic sign of M,, i.e., turning over the magnetization during the passage. Still another case of experimental interest is that of nonadiabatic passage (very small H I ) with T 5 Tz, which gives rise to transient oscillation or “wiggles.”1,’6These and other modulation effects have been treated, 16--20 but here we will be chiefly concerned with the adiabatic slow passage case for which Eqs. (2.4.1.4.2) yield these solutions, if H is considered to be the variable and AH = ( H o - H ) is the deviation from the resonance value :
M,
= u cos w t
- v sin w t
(2.4.1.4.3a) (2.4.1.4.3b)
M u = T (u sin w t + v cos w t )
(2.4.1.4.3~)
ly)2HiTz2 AH
(dispersion mode) + ( T z Y A W ’ + ( Y H I ) ~ T I ABO TZ (2.4.1.4.3d) lrlH1Tz M (absorption mode). u(H) = 1 + (Tzr AH)’ + (YHI)~TITZ (2.4.1.4.3e)
u(H) =
1
o
In Eq. (2.4.1.4.3b) the (-) sign is to be taken for positive y, the (+) sign B. A. Jacobsohn and R. K. Wangsness, Phys. Rev. 73, 946 (1948). E.E.Salpeter, Proc. Phys. Soc. (London) A68, 337 (1949). 17 K. Halbach, Helv. Phys. Acta 27, 259 (1954);29, 37 (1956). R. Karplus, Phys. Rev. 73, 1027 (1948). 19 B. Smaller, Phys. Rev. 83, 812 (1957). 50 J. Burgess and R. Brown, Rev. Sci. Znstr. 23, 334 (1952). 16
2.4.
DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS
109
for negative y. In the nuclear induction technique, e.g., there will be induced into the receiver coil along the y axis a voltage
V,
a
M,
=
T (u, cos w t -
ow
sin w t ) .
Because of incomplete decoupling with the transmitter coil along the z axis, there will also be induced into the receiver coil a leakage voltage V e = A sin w t B cos w t , which may generally be considerably larger than V,. By adjusting the phase of this leakage voltage by means of flux steering paddles and/or phase shifting networks, one can make either A / B << 1 or A / B >> 1. Thus the total receiver coil voltage V , V , can be made essentially proportional to either v or u;i.e., one may observe either the absorption or the dispersion (or any combination). Similar considerations hold for the Purcell single coil technique : by adjusting the balance of an rf bridge containing the coil as a n element, one can V e at nuclear resonance make the total bridge unbalance voltage V , proportional to either the imaginary or the real part of the complex nuclear susceptibility corresponding to the absorption v or the dispersion u,respectively. Note that whereas the algebraic sign of the nuclear induction signals are dependent upon the sign of the gyromagnetic ratio y, those of t>hesingle coil signals are not. This is due to the circumstance that two (crossed) coils are needed to establish the sense of a rotating field. The shape of experimentally observed homogeneously broadened resonance curves for protons in HzO under adiabatic slow passage conditions are shown in Figs. 2a and 2b. They follow closely the form expected from Eqs. (2.4.1.4.3): the absorption v has a maximum at resonance ( A H = 0) and a half-width at half-maximum H1,2 = l/yTz, provided the rf field H 1 is small enough so that ( ~ H I ) * T I
+
+
+
(yH1)2TlT, = 1 after which it decreases while the line width increases. This behavior is called saturation: the transition probability between two levels due to a sufficiently large H can considerably exceed the transition probability due to thermal relaxation processes, resulting in a population difference, and hence a signal, considerably less than that at thermal equilibrium. The dispersion u, on the other hand, does not display this saturation effect. Bot,h u and v have the same maximum value: ( M o / 2 ) . ( T Z / T I ) ~ / ~ . The magnitude of the received signal Vs, which is proportional to uu or W U , will thus have the optimum value:
110
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
Equation (2.4.1.4.4) shows clearly that to make the signals as large as possible the highest available static magnetic field (and hence frequency) should be used. It is possible in the case of liquids to shorten Ti by the addition of paramagnetic ions until TIc Tz.This gives optimum signal heights which, for equal numbers of nuclei, are proportional to p(I 1)
+
a
b
d
C
FIG.2. Oscilloscope traces of signals for protons in 1 cc of HpO as observed at I f o s 3300 oersteds with apparatus of Fig. 3. A 40-cps modulation field is used, sweeping through resonance twice in each cycle, the phase being shifted to separate the two traces. In (a) and (b) are shown the adiabatic-slow-passage dispersion and absorption modes, respectively, observed in a 1.0 molar aqueous solution of Fe(N03)3. In (c) and (d) are shown the nonadiabatic-passage dispersion and absorption modes, respectively, observed in a 0.01 2M aqueous solution of FeCla; note the characteristic “wiggles.”16
+
at constant frequency and to p s ( l 1)/Pa t constant field. Tables are availablezoEgiving these relative sensitivity figures for all nuclei for which p and I are known. Figures 2c and 2d show observed signals for protons in H 2 0 under nonadiabatic conditions. Such signals are frequently observed in prac*0*
Varian Associates, Palo Alto, California.
2.4.
DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS
111
tice; for the absorption mode, the resonance field H o lies on the line of symmetry of the figure.16 The phenomenological theory given above yields the so-called Lorentz line shapes of Eq. (2.4.1.4.3);these are usually observed for liquids under slow passage conditions. However, in viscous fluids or solids the assumed model is not valid, and the observed line shapes are often nearly Gaussian; we refer to Pake and Purcel12' for a general discussion of line shapes. 2.4.1.4.2. EXPERIMENTAL APPARATUS. Since the early experiments, there have been many changes and improvements in the apparatus used to observe nuclear magnetic resonance resulting in greatly improved sensitivity and resolution. For example, with present-day techniques, the resonance of the deuterons in 1 cc of ordinary water may be observed several times above the noise background; relative line widths as small as H 1 p / H o = lo-* have been measured for protons. The details of various experimental apparatus have been given adequately in the literature to which we refer. The two coil nuclear induction arrangement of Bloch et aL2 has been modified for various applications by ProctorlZ2Weaver,23 Redfield,24ArnoldlZ6and others. The single-coil bridge method of Purcell et a1.l has been adapted to a more readily tunable twin-tee bridge by Torrey,26 AndersonlZ7Collins,28and others. Another notable single-coil method is the regenerative detector of Pound and Knight,29Roberts130 and others, in which the absorption is detected by its reaction on an oscillator. Other related techniques have also been d e ~ c r i b e dI.n~ experi~ ments in which the nuclear magnetic resonance is split by quadrupole interaction, any of the preceding methods may be used. In zero magnetic field experiments, where there is no preferred axis of orientation, only the single-coil methods will work. Circuits developed particularly for this case have been described by Dehmeltla2W ~ n gL,i ~ i~n g s t o nand , ~ ~others. G. E. Pake and E. M. Purcell, Phys. Rev. 74, 1184 (1948). W. G . Proctor, Phys. Rev. 74, 35 (1950). 23 H. E . Weaver, Phys. Rev. 89, 923 (1953). p* A. G. Redfield, Rev. Sci. Znstr. 27, 230 (1956). 26 J. T. Arnold, Phys. Rev. 102, 136 (1956). * 6 H. Torrey, Phys. Rev. 76, 1059 (1949). $7 H. L. Anderson, Phys. Rev. 76, 1462 (1949). 2 8 R. L. Collins, Rev. Sci. Znstr. 28, 502 (1957). 29 R. V. Pound and W. D. Knight, Rev. Sci. Znstr. 21, 219 (1950). 30 A. Roberts, Rev. Sci. Znstr. 18, 845 (1947). 3 1 J. R. Zimmerman and D . Williams, Phys. Rev. 76, 350 (19491; H. S. Gutowsky, L. H. Meyers, and R. E. McClure, Rev. Sci. fnstr. 24, 644 (1953); H. W. Knoebel and E . L. Hahn, ibid. 22, 904 (1951). 82 H. G. Dehmelt and H. Kriiger, Z.Physik 129, 27 (1951). 33 T. C. Wang, Phys. Rev. 99, 566 (1955). 3 4 R. Livingston, Ann. N . Y . Acad. Sci. 66, 800 (1952). 21 22
112
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
Well-engineered apparatus of both the single- and double-coil types are now commercially a ~ a i l a b l e . ~ ~ ~ ~ ~ All techniques essentially measure the same thing: the components of the nuclear magnetization given by u and v in Eq. (2.4.1.4.3).The exception is that in the regeneration detectors only the absorption mode may be detected; also the range of operation of the applied rf field H I is somewhat restricted so that optimum sensitivity may not be achieved for very short or very long relaxation times. On the other hand, the regeneration detector is basically the simplest, since it does not involve a separate transmitter and receiver and is easily tunable with a single variable condenser; no frequency-sensitive balance conditions are involved as in the single-coil bridge method and in nuclear induction. These latter methods have the advantage, however, that they operate over wide ranges of H1 and may be preset to observe the dispersion under optimum signal conditions: (7H1)2T1Tz> 1. This feature is particularly useful in searching for weak signals. Current practice in most nuclear induction arrangements is to gang and track the transmitter and receiver so that single knob frequency control is available. Independent in-phase and out-of-phase leakage controls facilitate a rapid adjustment of the desired mode of operation. The sign of nuclear magnetic moments can be determined only with the nuclear induction arrangement. Permanent magnets may be used to provide a highly stable moderate field (-5 to 8 kilo-oersteds) with a minimum of upkeep. Greater flexibility is usually desired, however, and electronically regulated precision electromagnets of the Stanford type" have been widely used. They provide a continuously variable field from 0 to 15 kilo-oersteds with a stability of 1 in los or better. To prevent excessive broadening of the resonance lines by field nonuniformities within the sample, large diameter poles (-12 in.) and/or shimming by thin ferromagnetic materials or auxilliary coils is usually resorted to. For nuclear moment determinations, an inhomogeneity 6H = l / y T z 5 0.1 oersted over a typical sample volume of 1 cc is desirable. Otherwise it is difficult to shorten TI by the addition of paramagnetic ions to make TI = T z . An upper limit of the signal-to-noise ratio to be expected with a nuclear induction apparatus, for example, can be calculated approximately as follows: the voltage induced at resonance into a cylindrical receiver coil of N turns and diameter d by a sample completely filling the coil will be V , = Nn2d2i@,X 10-8 volt. If the sample does not comVarian Associates, Palo Alto, California. Perkin-Elmer Corporation, Norwalk, Connecticut. 37 Available from Varian Associates (reference 35) ; Pacific Electric Motors, Oakland, California; Harvey-Wells Corp., Framingham, Massachusetts. 86
36
2.4.
DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS
113
pletely fill the coil, the voltage will be correspondingly less because of incomplete flux linkage. For slow passage homogeneously broadened lines, the maximum value of A$, = (MOw/2) l/Tz/Tl. If the coil is resonated with a capacitor in a parallel resonant circuit of quality factor Q, the available rms signal voltage across this resonant circuit then will be
V , = QV,
=
+
[ n ~ 2 N d ~ & H o ~ 7 ~1)Ih2/6 (1 4
3 kT] d m X lo-'
volt. (2.4.1.4.5)
Taking the typical values d = 1 cm; N = 10 turns; Q = 100; H o = 10,000 oersteds; y = 4 X lo3rad/oersted for deuterons; I = 1; n = 6.6 X 1OZ2/cc for D20; T = 300°K; TI = T2, one finds V , = 2 X volt. T h e rms Johnson noise voltage across the same circuit will be Vn = 44kTRAv volts, where Av is the bandwidth of the receiver in cps, and R = QwL is the shunt resistance of the resonant circuit. For cylindrical coils of length = diameter d, L = Q.7N2dX lo-* henry. The signal-to-noise ratio then becomes
where the factor f 2 1 takes into account all additional noise, e.g., in the amplifier, and is here called the noise figure of the apparatus. It is possible to achieve f = 2 in practice. From Eq. (2.4.1.4.6)one finds that a n over-all working rule for the actual observable signal-to-noise ratio in a typical apparatus a t room temperature under optimum condi= ~13Tis:~ tions [TI = T z ; ( ~ H I ) ~ T
(2.4.1.4.7) where ~.l/~.lo is the nuclear magnetic moment in units of the nuclear magneton, Ho is the field in oersteds, n = number of nuclei/cc in sample, Av = apparatus bandwidth in cps. Similar considerations for the single coil method have been given b y Pound.' The general conclusion is that for all the methods, about 102O nuclei are required to give a visible signal 1, p PO, H lo4 oersteds. for the typical values I A block diagram for a nuclear induction spectrometer is shown in Fig. 3. Strong signals are simply displayed on a n oscilloscope by using a modulation field H , several times the linewidth, requiring a bandwidth Vv lo3 cps. For weak signals, a modulation field approximately 4 the linewidth is often used in conjunction with a narrow band (Av s 0.1 cps) lock-in amplifier to record the derivative of the absorption, as shown.
- - -
-
114
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
m
oscil lotor
frequency
FIG.3. Block diagram of nuclear induction spectrometer.
FIG.4. Recorded derivative of the dispersion signal of
0 1 7 in 1 ec of natural HzO; 10,000 oersteds; H I ir. 0.2 oersted; H,,, E O . 3 oersted; lock-in bandwidth AV = 0.1 cps. [H. E. Weaver, Phys. Rev. 89, 923 (1953).]
Ho
sz
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
115
This narrow-banding technique usually reduces the noise by a factor of 100. The derivative signal from 0'' in natural HzO recorded under optimum conditions is shown in Fig. 4. Figure 5 is a block diagram of an apparatus for observing pure quadrupole resonances.
FIG.5. Block diagram of pure quadrupole resonance spectrometer. The resonance is modulated by applying periodically a magnetic field which causes Zeeman splitting.
2.4.1.4.3. TYPICAL PROCEDURES IN NUCLEAR MAGNETIC RESONANCE. The linewidth = l / r T z is usually considerably narrower in nonviscous liquids than in solids because of the averaging out of the local dipole fields by the rapid tumbling motion in the former case. For this reason, and also because T I may be controlled by the addition of paramagnetic ions, liquid samples are usually used in the determination of nuclear moments by magnetic reaonaiice. The nuclei in question are often contained in ions or complexes in aqueous solution. One usually chooses the most soluble compounds to maximize n. If the nuclei have an appreciable quadrupole moment, this will usually prove to be effective in providing sufficient thermal relaxation so that TI = T z ; in this case no paramagnetic ions need to be added. In fact, the quadrupole relaxation is sometimes so effective that the relaxation times are too short and the
116
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
line widths consequently too broad for the resonance to be seen. I n these cases it is necessary to use a complex ion that is highly symmetrical so that the resultant electric field gradient is minimized. For example, the resonance of As16 can only be seen in the AsOT- or ASST- complexes. Similarly, the resonance of Zr91 has been observed only in the compound (NH4)zZrFa.When it is desired to make TI = Ta by the addition of paramagnetic ions (e.g., Fe++ or Cu++), one may roughly assume that T z is determined by the known magnetic field inhomogeneity, and then use the approximate formula:' l/T1 y2p;JVi X (10q/lcT) sec-I, where y = gyromagnetic ratio of nucleus in rad/oersted; peff= magnetic moment in units of the Bohr magneton of the paramagnetic ion and has the typical values 2-6; Ni = number of paramagnetic ions per cc; 1 = viscosity of liquid; k = Boltzmann's constant; and T = absolute temperature. The rare earth and transuranic elements are known to form only paramagnetic compounds and are not amenable to nuclear magnetic resonance studies in liquids because of the relatively huge (-lo6 gauss) and randomly oriented magnetic field existing a t the nuclear sites due to the hyperfine coupling with the unpaired electrons. Nuclear moments for these particular elements have been measured, e.g., by observing the hyperfine structure of the microwave paramagnetic resonance in single crystals (see Section 2.4.1.3). The other transition groups ( 3 4 4d, 5d) can fortunately be made to form diamagnetic complexes and are amenable to both nuclear magnetic resonance and paramagnetic resonance studies. I n some cases, one can also observe nuclear resonances in paramagnetic materials, as discussed in Section 2.4.1.4.7.2. 2.4.1.4.3.1. Magnetic Moment Measurements. The usual operation in a nuclear magnetic resonance experiment is to measure the ratio y/yp = p/2ppI, where y = p/Ih is the unknown gyromagnetic ratio and y p = 2pp/h is the proton gyromagnetic ratio. It is sometimes more convenient to measure y / y d , where Y d is the deuteron gyromagnetic ratio, and then use the precisely known ratio y d / y p . The unknown p in units of the nuclear magneton is then given by P/PO =
2I(Y/YP)
*
(PP/VO).
(2.4.1.4.8)
The proton magnetic moment in units of the nuclear magneton, pP/po, has been by comparing the magnetic resonance frequency to the cyclotron frequency of protons in the same magnetic field. The
** H. Sommer, H. A. Thomas, and J. A. Hipple, Phys. Rev. 82, 697 (1951). 99 F. Bloch and C. D. Jeffries, Phys. Rev. 80, 305 (1950); C. D. Jeffries, ibid. 81, 1040 (1951). '0 D. J. Collington, A. N. Dellis, J. H. Sanders, and K. C. Tuberfield, Phys. Rev. 99, 1622 (1955);J. H. Sanders, Nuovo cimento Suppl. [lo]6, 242 (1957). 'l K . R. Trigger, Bull. Am. Phys. SOC.[ 2 ] 1, 220 (1956).
2.4.
DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS
117
best value, as compiled by Cohen et ~ l . is, pp/pa ~ ~ = 2.79275 k 0.00003, including the diamagnetic correction. The ratio y / y p is usually measured as follows. A suitable sample is prepared containing both the unknown nuclei and the reference nuclei, H or D, as HzO or DzO, for example. If the signals are large, they can be quickly found on the oscilloscope and, holding the dc magnetic field constant, the ratio of the resonance frequencies w / w , = y/yp is measured by tuning the spectrometer to first one resonance and then the other, being careful in each case to adjust to a pure absorption mode so that the oscilloscope pattern is exactly symmetrical about its center point. A weak unknown signal is best found through searching a t constant frequency with a narrow band recording spectrometer by slowly varying the field H with a motor-driven current control. For optimum sensitivity, the dispersion mode is used under the following conditions: T1 = Tz; After the signal is located, the ratio (yH1)2T1Tz= 2; H , = u / w p = y / y p may be measured in constant field by sweeping through the two resonances consecutively with a slow motor drive on the frequency control, and putting frequency markers on the recorder tapes. If the lines are reasonably sharp (Hip -0.1 gauss) the ratio w / w , can be easily measured to 1 part in lo5 and with care to 1 in los, Such precision is not always warranted, because diamagnetic effects may cause the magnetic field at the unknown nucleus to differ slightly from that a t the reference nucleus, thus violating slightly the assumed equality w / w p = y / y p . Necessary corrections are discussed in Section 2.4.1.4.5.2. The sign of p is determined relative to p, by simply noting, with a nuclear induction spectrometer, whether the two signals have the same or opposite polarity when consecutively examined by varying the H field, leaving the spectrometer frequency and the leakage controls fixed. 2.4.1.4.3.2. S p i n Measurements in L i q u i d s . Under conditions of adiabatic slow passage and negligible saturation, we obtain from Eq. (2.4.1.4.3)for the signal voltage from the absorption mode at resonance V 8 = yH~Tzii?ow= H1wz[Tzy2n(1 1)1]. Now Tz = l/yHllz and hence [ y n ( I l)I] = V,Hl,z, which, being the product of the signal height and the half-width is proportional to the area A under the absorption curve if displayed as a function of the magnetic field. Thus, if we have two nuclei with identical resonance line shapes, and the two resonances are observed in turn by varying H while keeping W , H I , and the apparatus sensitivity all constant, we can write;
+
+
2 ):( ):( =
(11 (12
+ 1)Il + 1)Iz'
(2.4.1.4.9)
4 2 E. R. Cohen, J. W. M. Dumond, T. W. Layton, and J. S. Rollett, Revs. Modern Phys. 27, 363 (1955); E. R. Cohen, Nuovo cimento Suppl. [lo] 6 410 (1957).
118
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
Thus we can determine the spin I1 relative to I z by comparing the signal areas, knowing the gyromagnetic ratios and the relative nuclear concentrations in the sample. This signal area comparison can be made directly on the oscilloscope for strong signals, after making certain that both are observed under unsaturated adiabatic slow passage conditions. Weak signals are usually examined by a narrow band differentiating type of apparatus which records the derivative of the resonance lines. To consider this case and to allow for other than Lorentzian line shapes, we rewrite the undifferentiated signal for unsaturated adiabatic slow passage conditions as V , a nw21ylH1(I 1)1 - f ( H ) , where f ( H ) is a resonance line shape function with a maximum valuef(H = H o ) = l/Hlp and so normalized that f ( H ) dH = 1. Now the differentiated signal recorded by the lock-in apparatus is proportional to
+
lom
G(H)
=
(aVa/dH) . Hm
=
+ l)nf'(H).
w2H1HmlylI(I
The first moment MI of the differentiated signal will be M1 =
lo" G(H)
*
(H
- Ho) d H
= w2H1HmlylZ(Z
+ 1)n
as is seen from a partial integration and the normalization of f ( H ) . Thus for the differentiated signals of two nuclei observed in turn by varying H while keeping w , HI, and Hm constant, we find (2.4.1.4.10)
The condition (yH1)2T1T2<< 1, necessary for the validity of Eq. (2.4.1.4.10), must be experimentally checked by making for each resonance a saturation curve, i.e., a plot of V , versus H I , from which a n upper limit for a nonsaturating value of H I can be determined. Also the modulation field H , should be reasonably small, compared to H l l z , to insure a true derivative in each case. In the event that the resonance curve is displayed as a function of frequency instead of field, then the area A' = V,(v)dv so determined may be used in Eq. (2.4.1.4.9), if the right side is multiplied by yI/y2; similarly the first moment M1' = G(v) * ( v - VO) dv of the differentiated curve may be used in Eq. (2.4.1.4.10), if the right side is multiplied by (y1/y2)2. From Eq. (2.4.1.4.9),it is seen that the area ratio is proportional to I(Z 1) a 3, 15, 35, 63, 99, . . . , for odd nuclei, for example, so that increasingly greater accuracy of measurement is necessary to distinguish
lom lom
+
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
119
between higher spin values. Particular care must then be taken to insure that there is no spurious signal height distortion by modulation effects, etc. It is evident that if I I and Iz are both known for two isotopes, then the first-moment ratio determines the relative isotopic abundance. This is the basis of a simple nondestructive technique for measuring isotopic abundance ratios to within a few per cent.43 2.4.1.4.4. QUADRUPOLE RESONANCES. 2.4.1.4.4.1. QuadrupoZe Splitting of Magnetic Resonance. In crystalline samples one must consider the total interaction X = -7hI * H Q AE. One of the simplest cases is that in which the magnetic interaction term is much larger than the quadrupole term. Using first-order perturbation theory, the energy levels are found to be, for electric fields of axial symmetry,4
+ -
E,(e)
=
e2qQ + 41(2I - 1) [3m2 - I ( I + 1)][+ cos2 8 - +]
(2.4.1.4.11)
where 0 is the angle between H and the axis ( 2 ) of field symmetry; eq = -2d2V/&2, where V is the electrostatic potential a t the nucleus due to the surrounding electronic charges; Q = scalar nuclear electric quadrupole moment.44 For crystalline fields of cubic symmetry, q vanishes; for fields of lower than axial symmetry, an additional parameter 7 = (azV/azz)- (aZV/dyz) is introduced and Eq. (2.4.1.4.11)is modified." For a single crystal, the energy levels of Eq. (2.4.1.4.11)have the discrete pattern as shown in Fig. 1 for a hypothetical spin I = 3. In general, there are 21 possible transitions of the magnetic dipole type (Am = + 1 ) between the levels. These can be induced by the applied rf field H I , and such resonances are readily observed by the magnetic resonance techniques discussed above. In contrast to liquids, the nuclear dipole-dipole interaction between neighboring nuclei is not averaged out, with the result that usually Tz<< TI in diamagnetic crystals; for this reason, the expected signal-to-noise ratio is less. A differentiating type of spectrometer is usually used, and the lines are observed by varying either the field or the frequency. A sample of -1 gram is usually required. The crystal is so mounted that 0 may be varied. At 0 = arccos (l/d3), the lines become nearly superposed, since by Eq. (2.4.1.4.11) the quadB. E. Holder and M. P. Klein, Bull. Am. Phys. SOC.[I]29, 17 (1954). Q is defined [see, e.g., H. B. G. Casimir, Interaction between atomic nuclei and electrons. Teyler's Tweede Genootschap Haarlem 11, 36 (1936)]as Q = e-'Jpr2(3 cos28 - 1) dr integrated over the nuclear volume for the state m = I , where p is the 43
44
nuclear charge density at a volume element dr at position T from the origin; 8 is the angle r makes with the nuclear spin axis.
120
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
rupole term vanishes to first order; higher order terms omitted from Eq. (2.4.1.4.11) may prevent an exact superposition. For polycrystalline samples, one observes a much less discrete spectrum representing the spatially averaged values of Em(@. In addition to those induced by the rf magnetic field HI, transitions corresponding to Am = f 2 , & 1 could be induced by an applied rf electric field gradient. Sufficiently intense gradients cannot be produced electrically in the laboratory; they may be produced, however, by strong lattice vibrations induced by supersonic t e c h n i q ~ e s . ~ ~ - 4 ~ 2.4.1.4.4.2. Pure Quadrupole Resonance. When single crystals are not available, and also when the quadrupole interaction is larger than the magnetic interaction, it is convenient to examine the so-called pure quadrupole spectrum. In zero magnetic field the energy becomes
e2qQ Em = 41(2I - 1) [3m2 - 1(1
+ l)]
(2.4.1.4.12)
for axially symmetric crystalline fields. Between these pure quadrupole levels an applied rf field HI may still induce magnetic dipole transitions (Am = & 1) by acting on the nuclear magnetic moment. For half-integral spin there are I - such allowed transitions; for integral spin there are I transitions. In a nonaxial field the levels are shifted but the Iml degeneracy is not removed for half-integral spin; for integral spin, there is a splitting of the m levels, and additional lines are observed. Since the energy, Eq. (2.4.1.4.12), is not orientation-dependent, powdered samples are usually used, and the resonances are observed with r e g e n e r a t i ~ eor~ super-regenerati~e~~ ~ detectors as the frequency is slowly varied. The signal is modulated by either Zeeman splitting, or by frequency-modulating the oscillator. Resonances over very wide ranges (42-700 Mc) have been observed. 2.4.1.4.4.3.Determinations of Nuclear Spin and Quadrupole Moments. In both cases 2.4.1.4.4.1and 2.4.1.4.4.2above, by measuring the number of lines and the resonance frequency for each, one can obtain the spin I and the coupling constant e2qQ in single crystals or powdered samples. For two isotopes of the same element, q will be the same, so the ratios of the nuclear quadrupole moments will be exactly equal to the ratio of the coupling constants. A number of such ratios have been measured.3 The sign of Q cannot be determined by these methods. The general problem is to evaluate q so that Q can be determined from the measured
+
45 W. G. Proctor and W. H. Tahttila, Phys. Rev. 101, 1757 (1956); W. Proctor and W. A. Robinson, Phys. Rev. 104, 1344 (1956). 4 6 M. Menes and D. I. Bolef, Phys. Rev. 109, 218 (1958).
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
121
value of e2q&. Moderately accurate values of q have been estimated by the following procedure^.^' The value of q in the molecular solid is correlated with that for a free molecule;48 then, using the semiempirical considerations of Townes and D a i l e ~ ,the ~ ~ q for a free molecule is related to that for a free atom, which is then estimated by the established procedures of optical hyperfine structure,60including the corrections of Sternheimer61 and Koster.62 2.4.1.4.5. CORRECTIONS I N MAGNETIC RESONANCE. The actual magnetic field a t the nuclei may differ slightly from the applied magnetic field H O in which one observes magnetic resonance a t frequency wo. Several small corrections are necessary in calculating the gyromagnetic H 1 H 1 l . Here H1 is the ratio y and one usually writes w o / y = Ho magnetic shielding field a t the nuclei due to induced diamagnetic circulation of the electrons in the atom, ion, or molecule; and H” is the field at the nuclei due to the paramagnetism of the sample itself. 2.4.1.4.5.1. Diamagnetic Corrections. The field HI is proportional but opposite in sense to Ho and is written as H 1 = -uHo, where u is the magnetic shielding factor. Lamb63 calculated theoretically u for free atoms, and values have been tabulated by Lamb and by D i c k i n s ~ n . ~ ~ for hydrogen to u = for uranium. They range from u = 2 X Measured magnetic moments that have been multiplied by the correction factor (1 u ) are said to be diamagnetically corrected. If the resonances are observed in atoms or ions, the Lamb correction is probably adequate. However, many resonances are observed for nuclei contained in molecules, and Ramseys6 has calculated the additional corrections necessary. Unfortunately, they can be evaluated only for the protons in the H2 molecule; the result, including the Lamb term, is u = 2.66 X low6.By comparing the resonance frequency of protons in H2 to those in the HzO and in mineral oil, the shielding factors for the latter two are found and u = 2.60 X respectively. experimentally to be u = 2.82 x This shift in the resonance frequency for a given nucleus contained in different compounds is called the “chemical shift” and has been observed
+ +
+
47 See, for example, T. P. Das and E. L. Hahn, Nuclear quadrupole resonance spectroscopy. Solid State Phys. Suppl. 1, Chapters 7-9 (1958). 48 C. H. Townes and B. P. Dailey, J . Chem. Phys. 20, 35 (1952). 4 9 C. H. Townes and B. P. Dailey, J . Chem. Phys. 17, 782 (1949). 60 See, e.g., H. Kopfermann, “Kernmomente.” Akademische Verlagsges., Leipzig, 1940; reprinted by Edwards, Ann Arbor, Michigan. 6 1 R. M. Sternheimer, Phys. Rev. 84, 244 (1951); 86, 316 (19.52); B6, 736 (1954). 62 G. F. Koster, Phys. Rev. 86, 148 (1952). 53 W. E. Lamb, Phys. Rev. 60, 817 (1940). 54 W. C. Dickinson, Phys. Rev. 80, 563 (1950). 5 5 N. F. Ramsey, Phys. Rev. 77, 567 (1950); 78,699 (1950); 86,243 (1952).
122
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
for many nuclei.6s Although usually of the order of O.Ol%, a few cases have been found where the shifts are as large as l%.&' For this reason, it is important in measuring magnetic moments to observe the resonance in several different compounds, including free ions if possible; presumably the resonance which occurs at the lowest field (for fixed frequency) shows the least diamagnetic shielding. There are, of course, no diamagnetic corrections necessary in measuring the ratios of magnetic moments for two isotopes of the same element. Special high-resolution techniquesz6 have been developed to study these chemical shifts from the view point of structural chemistry.68 2.4.1.4.5.2. Paramagnetic Corrections. The field H" due to the magnetization M , of added paramagnetic ions has been experimentally measured by Dickinsons8for different ions and geometries of the sample. For a spherical sample, the depolarization field -(+r)M, cancels the Lorentz field (+r)M, and one might expect the field H" to be zero; experimentally, this is not found to be so, probably because of spinexchange interactions. Dickinson found that H" depends upon the sample geometry, the chemical compound containing the resonating nucleus, and the particular paramagnetic ion and its concentration. For example, a sample of HzO of roughly spherical proportions containing FeClz has a measured value H l l = 2cHo X oersted, when c is the molarity (moles/liter) of FeC12. For long cylindrical shapes, H l 1 may be negative. A general safety rule is, for ( H " / H ) < lo--*, it is necessary that c < 0.3 mole/liter. In powdered metal samples, Knightso has found that the paramagnetism of t,he conduction electrons contributes to H". This Knight shift in the resonance frequency may be as large as 1%; experimental and theoretical values have been tabulateds1 for many metals. 2.4.1.4.6. MEASUREMENT OF MAGNETIC FIELDS BY NUCLEAR MAGNETIC RESONANCE. Driscoll and Benders2 have measured the proton gyromagnetic ratio in absolute units by observing the resonance frequency w o in a magnetic induction field Bo,which was measured by a 68 A summary of data is given by H. E. Walchli, in "A Table of Nuclear Moment Data," ORNL-1469 and supplements. Oak Ridge National Laboratory, Oak Ridge, Tennessee. 67 W. G. Proctor and F. C. Yu, Phys. Rev. 81, 20 (1951). 68 W. A. Anderson, Phye. Rev. 102, 151 (1956); J. N. Schoolery, Discussions Faraday Sac. 18, 215 (1955). 60 W. C. Dickinson, Phys. Rat. 81, 717 (1951). So W. D. Knight, Phys. Rat. 78, 1269 (1949). 6 1 W. D. Knight, Solid State Phye. 2 , 93 (1956). (* R. L. Driscoll and P. L. Bender, Phys. Rev. Letters 1, 413 (1958); see also H. A. Thomas, J. A. Hipple, and R. L. DriscoU, Phys. Rev. 78, 787 (1950).
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
123
current balance. They find y p = wo/Bo= (2.67513 & 0.00002) X lo4rad/ sec/gauss, without diamagnetic correction. Correspondingly, this result, can be used to measure a magnetic field in absolute units to high precision; one simply observes, in the field to be measured, the proton resonance frequency v P of protons in liquid Hz0 containing, say, 0.02 mole per liter of FeClz. The sample geometry may vary between a sphere and a long cylinder. The applied field is then given directly by the formula
B (in gauss)
=
234.873 X v p (in Mc/sec).
(2.4.1.4.13)
For high fields, it is sometimes more convenient to observe the deuteron resonance frequency Vd in DzO (containing, say, 0.5 mole/liter of FeClz), for which the following formula applies:
B (in gauss)
=
1530.06 X
Vd
(in Mc/sec)
(2.4.1.4.14)
For precision measurements of fields of a few gauss, the devices of Pack. ~ particularly ~ ~ suitable. ard and Varians3 and Abragam et ~ 2 1 are 2.4.1.4.7. SPECIAL METHODS. We discuss very briefly special methods, which, in addition to the usual steady-state methods described above, have some applications to nuclear physics. 2.4.1.4.7.1. Spin Echoes. By applying the rf field HI in the form of pulses instead of continuously, Hahn64 induced several transient responses of spin systems which he called spin echoes. This method is particularly useful for measurement of the relaxation times TI and Tz, and may be applied to spin systems either in magnetic interaction with a dc field, or in pure quadrupole interaction.O6I n the latter case, unusually small quadrupole coupling constants have been measured by the use of signals from spin system A transient double resonance a t a relatively high resonance frequency are observed as a function of a second relatively low frequency applied to a system B, which is in weak interaction with system A . At the resonance frequency of B , the signal strength of A is observed to change. In effect, one has a high frequency, and hence high sensitivity, probe for weak interactions. This method would be particularly appropriate for measuring quadrupole coupling constants smaller than a few Mc/sec. M. Packard and R. Varian, Phys. Rev. 93, 941 (1954). A. Abragam, J. Combrisson, and I. Solomon, Compt. rend. m a d . sci. 246, 157 (1957). 84 E. L. Hahn, Phys. Rev. 80, 580 (1950). 66 M. Bloom and R. E. Norberg, Phys. Rev. 93,638 (1954);E.1 1. Hahn and B. Heraog, ibid. 93, 639 (1954);W. G. Proctor and W. H. Tanttila, ibid. 98, 1854 (1955); M. Bloom, E. L. Hahn, and B. Heraog, ibid. 97, 1699 (1955). 66 B. Heraog and E. L. Hahn, Phys. Rev. 103, 148 (1956). 67 D. Kaplan and E. L. Hahn, Bull. Am. Phys. SOC.[2]2, 384 (1957). 6s
83s
124
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
2.4.1.4.7.2. Dynamic Nuclear Orientation. This method is closely related in technique to microwave paramagnetic resonance (Section 2.4.1.3), and is a consequence of some generalizationses of a suggestion by O v e r h a u ~ e r It .~~ has been used to determine the spins and magnetic moments of radioactive nuclei70~71by observing the change in angular distribution of y-rays accompanying certain resonance conditions. The principle is to enhance the population of certain nuclear magnetic states by saturating the microwave magnetic resonance of an electron-nuclear hyperfine-coupled system, in analogy t o the earlier work of Pound.72The resulting nuclear orientation, dynamically produced by the applied microwave field and associated relaxation processes, is to be contrasted to that produced by static hyperfine interactions a t very low temperatures, as discussed in Section 2.4.2.3. With reference to Section 2.4.1.3.2,we give a simplified illustration of the method by considering a hyperfine system with nuclear spin I = 1 and electron spin S = in an external field H along the x axis, the pertinent interactions being given by
+
X = gPHS,
+ AIgS,+ B(I,S, + I$$,).
(2.4.1.4.15)
The physical system may be, e.g., a single crystal of zinc sulfate in which about 0.1% of the Zn++ ions have been replaced by (paramagnetic) Co++ ions. The energy levels g@HM AmM are shown in Fig. 6, where the states are labeled by the high-field quantum numbers ( M = S,, m = I,); usually A m M << gPHM. The term in B may be rewritten as iB(I+S- I-S+), where I+ = I= iI,, etc.; and it is seen that this slightly admixes state b with state f, and also state c with state e. The consequence of this is that a microwave magnetic field applied parallel to H can induce the so-called forbidden transitions AM = k 1, A m = T 1, labeled 4 in Fig. 6. Suppose the transition b ~f is saturated, i.e., the populations are made equal by applying sufficient microwave power. Then the relative
+
+
+
F. Bloch, Phys. Rev. 93, 944 (1954); A. Overhauser, ibid. 94, 768 (1954);J. Kor ringa, ibid. 94, 1388 (1954); C. Kittel, ibid. 96, 589 (1954);P.Brovetto and G . CiniNuovo cimento [9]11, 616 (1954);P. Brovetto and S. Ferroni, ibid. [9]12, 70 (1954), A. Abragam, Phys. Rev. 98, 1729 (1955);A. Abragam, Compt. Tend. acad. sci. 242, 1720 (1956);A. Abragam and J. Combrisson, Compt. rend. acad. sci. 243, 576 (1956); G. Feher, Phys. Rev. 103, 500 (1956);G.Feher and E. A. Gere, ibid. 103, 501 (1956); C. D. Jeffries, ibid. 106, 164 (1957); D. Pines, J. Bardeen, and C. P. Slichter, ibid. 106, 489 (1957). 6p A. Overhauser, Phys. Rev. 89, 689 (1954);92, 411 (1953). M. ilbraham, R. W. Kedaie, and C. D. Jeffries, P h p . Rev. 106, 165 (1957);Bull. Am. Phys. SOC.[2]2, 382 (1957). 7 l F. M. Pipkin and J. W. Culvahouse,Phys. Rev. 106,1102 (1957);109,1423 (1955). 72 R. V. Pound, Phys. Rev. 79, 685 (1950).
2.4.
DETERMINATION OF SPIN, PARITY,
AND NUCLEAR MOMENTS
relative populations, saturating
(M 1 m)
b-f
I
e-A a ? ' I f
f
125
re1ative populations, saturating a-f I
I
I
FIG. 6. Energy levels, transitions, and populations for a hyperfine system with S = and Z = 1. Forbidden transition 4 and allowed transitions a are induced by microwave fields parallel and perpendicular, respectively, to the dc field H .
4
populations of the other states will be as shown, if we assume that the usual strong paramagnetic relaxation processes maintain thermal equilibrium between a and f, between b and el and between c and d ; it is also assumed that a time-dependent term B'(t) in the hyperfine interaction provides a relaxation process to keep c and e in thermal equilibrium. The resulting nuclear alignment
P
-
=
(N,
+Nf +N, +
N d
- 2Nb - 2 N , ) / Z N
-
x
-hv/3lcT
where v is the microwave frequency and T is the absolute temperature. For v 101O cps and T N 1"K, P 0.1, and this alignment will produce a detectable anisotropy in the angular distribution of the y-rays emitted by the nuclei. I n a typical experimental arrangement, Fig. 7, one observes a signal proportional to the difference in the y-ray intensities emitted parallel and perpendicular to the dc field H , as the field is slowly varied, keeping the oscillator frequency v fixed to the cavity resonance frequency. An anisotropy is recorded at the field corresponding to the transition b tf f, and also a t the field corresponding to c t)e ; in general, 21 anisotropy signals are observed, and the nuclear spin is thus directly determined. The field spacing between signals is proportional to the coupling constant A , from which the nuclear magnetic moment is determined,
126
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
either through hfs calculation, or by direct comparison to that of a known stable isotope. It is also possible to obtain nuclear orientation by saturating the usual allowed transitions AM = k 1, Am = 0, labeled a in Fig. 6. For example, by saturating the transition a ++ f, the populations shown are obtained if one assumes the same relaxation processes as before. The resulting
(;v3
microwove oscillator
radioactive porarnognetic crystal in microwove cavity
\
% I iq.
~~
magnet pole
I ig h t f Pipe
L
A d iff e r e n- c
Fro. 7. Experimental arrangement for dynamic orientation of radioactive nuclei.
alignment is P = hv/6lcTl and in general one observes a y-ray anisotropy at 21 1 field values. In more general situations, the effects of relaxation processes between a and b, etc., and between a and el etc., must be taken into account. Feher6*has proposed and demonstrated that, independently of relaxation processes, if the populations of first a and f, and then a and b, are inverted by adiabatic fast passage (see Section 2.4.1.4.1.2),the resultant populations correspond to a nuclear orientation comparable in magnitude to those above.
+
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
127
Dynamic orientation experiments are usually done at 1- or 3-cm wavelengths a t the temperature of pumped liquid helium (-1.3"K). Conventional microwave resonance techniques are employed. Saturation powers between microwatts to milliwatts are required, and activities from microcuries to several millicuries have been used. Since the signal-to-noise ratio is determined by the number of decays rather than by the number of nuclei, this method is particularly appropriate for short-lived nuclei, in contrast to the method of direct paramagnetic resonance absorption, which is more suitable for long-lived and stable nuclei. The above discussion has been particularly appropriate for a, nucleus in relatively strong hyperfine coupling with its own paramagnetic ion, but similar ideas726 may be applied to the case of nuclei .inweak long-range dipole-dipole coupling with electron spins, for example, the protons in a piece of plastic containing free radicals. If the forbidden transitions involving a simultaneous flipping of an electron spin and a nuclear spin are saturated, an appreciable polarization of most of the nuclei in the sample may be achieved. For details we refer to a review article.72b
2.4.1.4.7.3. The Electron Nuclear Double Resonance (ENDOR) Technique.* I n this method,7aone deals with nuclear and electron paramagnetic resonances simultaneously (see Section 2.4.1.3). It is applicable to some paramagnetic substances in which the nucleus, in addition to the applied magnetic field, is subjected to the hyperfine field of the electron. The ENDOR method is illustrated in Fig. 8 for the simple case of a hyperfine system with I = +,S = +. The magnetic field is set to the proper value to observe the microwave transitions hv,, as indicated by the arrows in Fig. 8. The amplitude of the signal due to these transitions is proportional to the difference in population in levels in A and A'. If we partially saturate this resonance by increasing the microwave field, the population difference will be diminished and the signal reduced. By inducing now the transitions h v N or h v ~ (see , Fig. 8), the population difference in levels A and A' can be increased and thereby the electron spin resonance signal enhanced. Thus one monitors the electron spin resonance signal while nuclear transitions are being induced. Since the electronic moment is approximately three orders of magnitude A. Abragam and W. G. Proctor, Compt. rend. acad. sci. 246, 2253 (1958). D. Jeffries, Progr. i n Cryogenics 3, 129 (1961). 73 G. Feher, Phys. Rev. 103, 500 (1956).
72.8
nbC.
-
*Section 2.4.1.4.7.3 is by G. Feher.
128
2.
DETERMINATION O F FUNDAMENTAL Q U A N T I T I E S
larger than the nuclear moments, the sensitivity of the ENDOR technique greatly exceeds that of the ordinary nuclear resonance. It is also independent of the size of nuclear moments as long as they are large enough to give rise to a set of resolved ENDOR lines. For those two reasons, the method is particularly advantageous in determining moments of nuclei which have a small gyromagnetic ratio, or nuclei which are radioactive. The transition-inducing field at the microwave frequency v, is applied to the sample in a resonant microwave cavity, which may be thin-walled and slotted to allow the fields a t the radio frequency V N to be applied by coils wound around the cavity. m= ,
+ J-
?= I:-
m,- -I
ti-+
ms=+I
*
ml=
m5 = -I 2
mz= +'
A'
4.
FIG. 8. Energy levels of a system with Z = i,S = Arrows indicate the transitions involved in the ENDOR technique. [G. Feher, Phys. Rev. lOS, 500 (1956).]
By observing the two transitions hvrv and hvw, the nuclear are given in the high field region by the relation7 hvN
- hvN1
=
2grpH.
91
values
(2.4.1.4.16)
The method has been applied by Pipkin and Culvahouse71 to radioactive and by Feher et u Z . , ~ ~to P32. Pipkin and Culvahouse made use of the double resonance method to align the nuclei and used a y-counter to monitor the nuclear transitions. Halford et aZ.,76 have applied it to determine directly g1 for Nd143and Nd146;corrections similar to the chemical shift in nuclear resonance are required in this case.76a The ENDOR technique is also useful in the investigation of quadrupole interactions, as demonstrated in the work on F centers in KCl.V6 The quadrupole interactions of both the C136J7 and K39s41 were clearly reG. Feher, C. S. Fuller, and E. A. Gere, Phys. Rev. 107, 1462 (1957). D.Halford, C. A. Hutchison, Jr., and P. M. Llewellyn, Phys. Rev. 110,284(1958). 7ba €3. Bleaney, J . p h p . radium lS, 826 (1958). 76 G.Feher, Phys. Rev. 106, 1122 (1957). l4
76
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
129
solved. In this case the quadrupole interaction of the chlorines is only approximately 40 kc/sec, which indicates the relatively high resolution attainable by this method. It should be noted that the method is not applicable to all paramagnetic substances, but only to those that can be saturated and exhibit an inhomogeneously broadened77 line. If those two conditions are satisfied, transitions whose frequencies are much smaller than the electron spin resonance line width may be observed.73
2.4.2. Indirect Methods
2.4.2.1. Angular Correlation,* t 2.4.2.1.1. INTRODUCTION AND LITERAThe importance of angular correlation techniques for determining nuclear moments and parities stems mainly from two facts: a straightforward theoretical basis exists, and the experimental realization is relatively simple and inexpensive. These attractive features have instigated an enormous amount of theoretical and experimental work. The present discussion does not cover all of the major aspects, but is intended only as an outline of basic principles and a guide to the literature. For further studies and for all details, additional papers and reviews should be consulted. A first, naive introduction into the theory and a discussion of many experiments can be found in the reviews of Deutsch’ and F r a ~ e n f e l d e r . ~ . ~ A more complete and involved theory is given by Biedenharn and Rose.4 The influence of extranuclear fields has been discussed extensively by Steffen.s A recent survey, by Devons and Goldfarb,6 deals not only with cascade processes, but also with nuclear reactions and with the application to mesons and hyperons. An introduction into the modern theory TURE.
’7
A. M. Portis, Phys. Rev. 91, 1071 (1953).
t We
use the term angular correlation as comprising directional correZation and polarization correlation. In directional correlation only the directions of the radiations are observed; in polarization correlation one determines also the polarization of at least one radiation. M. Deutsch, Repts. Progr. i n Phys. 14, 196 (1950). 2 H. Frauenfelder, Ann. Revs. Nuclear En‘. 2 , 129 (1953). 3 H. Frauenfelder, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 531. Interscience, New York, 1955. L. C. Biedenharn and M. E. Rose, Revs. Modem Phys. 26,729 (1953). R. M. Steffen, Advances in Phys. 4, 293 (1955). (I S. Devons and L. J. B. Goldfarb, in “Handbuch der Physik-Encyclopedia of Physics” (S.Flugge, ed.), Vol. 42, p. 362. Springer, Berlin, 1957.
’
* Section 2.4.2.1 is by Hans Frauenfelder.
130
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
is provided by Biedenharn.’ All of these reviews contain extensive bibliographies. The information provided by angular correlation experiments alone very rarely is sufficient to determine all nuclear properties involved. I n most cases, angular correlation is only one tool out of many and other approaches (cf. Sections 2.4.1, 2.4.2.2, and 2.4.2.3) must be used in conjunction with it to get a complete picture.* 2.4.2.1.2. THEPHYSICAL PICTURE. The measurement of nuclear spins by angular correlation methods depends on the fact that the probability of emission of a particle or quantum by a nucleus is a function of the angle between the nuclear spin axis and the direction of emission. This function is determined by the spin of the decaying nucleus, the type of emitted radiation, and the angular momentum carried away by it. Observation of the radiation pattern with respect to the nuclear spin axis hence amounts to a measurement of the nuclear spin, provided the type of radiation and its angular momentum are known. The desired radiation pattern can be observed if the nuclei are partially or completely aligned, i.e., if their spins are parallel. Under ordinary circumstances, however, nuclei are randomly oriented in space and the resulting radiation pattern is isotropic. Three different methods can be used to produce an ensemble of nuclei which are not randomly oriented. (1) If the decaying nuclei are placed a t a very low temperature in an extremely strong magnetic field (electric field gradient) they are polarized (aligned). The directional distribution of the emitted radiation with respect to the direction of the applied field can then be measured and yields the desired information. (See Section 2.4.2.3.) (2) Nuclei can be produced in a preferential orientation by a spin dependent nuclear reaction. The directional distribution of the secondary radiation with respect to the incoming beam can then be used to determine the nuclear spin. (See Section 2.7.4.) (3) One can select only those nuclei whose spins happen to lie in a preferred direction. Such a selection can be executed if the nuclei decay through successive emission of two radiations R1 and Rz. The observation of R1in a fixed direction selects an ensemble of nuclei which has a nonisotropic distribution of spin orientations. The succeeding radiation Rz then shows a definite angular correlation with respect to R1. The observation of two directions, a8 discussed above, allows in general only the determination of the nuclear spins and the multipole
* For certain aspects of the problems discussed in this section see also Vol. 4, A, Chapters 3.5 and 4.5. ’I L. C. Biedenharn, in “Nuclear Spectroscopy” (F. Ajzenberg-Selove, ed.), Part B. Academic Press, New York, 1960.
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
131
orders of the emitted gamma rays, but not of their parities. In order to find the parities, one must, in addition to the directions, also measure the linear polarizations of the emitted radiations. (Experiments involving conversion electrons are an exception. There the directional correlation alone is already parity sensitive.) The idea underlying the determination of electric and magnetic nuclear moments can now be explained quite easily. Assume a certain nuclear orientation to be established by one of the methods discussed above. The nucleus remains in this excited state for a time T and then decays. The particle emitted in the decay is used to obtain information about the nuclear orientation. This information is correct only if the nucleus is unperturbed during its stay in the excited state. Any perturbation, magnetic or electric, will change the radiation pattern. This change can be used to extract information about the electromagnetic moments of excited nuclear states. If one, for instance, applies a magnetic field to the nucleus, it will precess during the time T.The corresponding rotation of the radiation pattern depends on the time T , the applied field H , and the nuclear g factor. It can thus be used to find the nuclear g factor. If, in addition, the nuclear spin is known, the nuclear magnetic dipole moment follows immediately. Similarly, the change of the correlation pattern in an applied electric field gradient yields the quadrupole coupling. The theoretical description of the correlation process can be patterned in such a way that it follows the physical picture given above rather closely :6-g Assume for instance a gamma-gamma cascade. The initial state is then described by a density matrix.'O Using first-order perturbation theory, one calculates the density matrix describing the intermediate nuclear state, which results after the emission of the first gamma ray. Again using first-order perturbation theory for the emission of the second gamma ray, one calcuIates the various interesting properties. The theory of angular correlation will not be discussed any further; all required formulas and results will be borrowed without derivation from the various theoretical papers cited above. 2.4.2.1.3. DETERMINATION OF SPINS.The schematic picture of a directional correlation measurement is shown in Fig. 1. A nuclear decay occurs between states a, b, c, . . . with spins I,, I*,I,, . . . through the successive emission of particles (or photons) R1, R z , . . . . (It is assumed here that the spins and parities of the states a, b, c, . . . are well defined.) The assignment of spins to the nuclear states and of angular momenta to the radiations then involves four steps. F. Coester and J. M. Jauch, Helu. Phys. Acta 26, 3 (1953). U. Fano, Phys. Rev. 90,577 (1953). l o U. Fano, Revs. Modern Phys. 29, 74 (1957). 8
132
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
I. The calculation of the various possible theoretical correlation functions W,,,,,(e). We denote here with W(e) dQ the relative probability that the radiation Rz is emitted into the solid angle dfl a t a n angle 0 with respect to R1 [Fig. l(b)]. For details see step I below. 11. The measurement of the coincidence rate C(6) as a function of the angle 6 included by the axes of two counters a t the source. See Fig. l(c) and step I1 below.
Source C
IC
a
b
C
FIQ.1. (a) Nuclear cascade. (b) W(0)dil is the relative probability that the radiation Rz is emitted into the solid angle dil a t an angle e with respect to R1. (c) The two counters 1 and 2 subtend an angle 29 at the source. From the coincidence rate as a function of angle C @ ) , one obtains after suitable correction the correIation function W(0).
111. Correction of C(8) in order to find W,,,(6). At each angle 19 between the counters, coincidences between radiations including a range of angles e are recorded. Corrections are hence needed to find w,,,(e) from the measured curve. These corrections are discussed in Section 2.4.2.1.7. IV. Comparison of W,,,(e) and Wthear(0).Once the experimental correlation function has been determined, i t has t o be compared with the various possible theoretical functions in order to determine the most likely assignment. I n most cases, a unique assignment is possible only if other methods (e.g. internal conversion, measurement of ground state spin, lifetime, etc.) contribute. Even so, the determination of a unique scheme more often resembles a jig-saw puzzle than a well-defined physics problem. I n the following, we discuss the first two steps in more detail. I. T h e theoretical directional correlation function. The most convenient form in which to express the correlation function W(6)is (2.4.2.1.1)
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
133
The functions P k are Legendre polynomials. (Sometimes, the much less convenient expansion in terms of cask 0 is used. Formulas for the conversion from one expansion into the other can be found in the papers by Steffen5 and Lloyd.") The coefficients A k contain all the physical information. Usually, one chooses Ao = 1, so that the correlation function, integrated over all angles, is unity: JV(0) dQ = 1. Three general rules for the coefficients Ak hold for all types of particles:
k integer, even. 0
I Ic, A k
I min(21b,2L1,2L2). =
Ak(a,b)Ak(b,c).
(2.4.2.1.2) (2.4.2.1.3) (2.4.2.1.4)
Rule (2.4.2.1.2), expressing the fact that only even Legendre polynomials appear, holds as long as one measures only the directions and linear polarizations of the radiations. The observation of the circular (longitudinal) polarization, however, introduces also odd integers (cf. Chapter 2.5). Rule (2.4.2.1.3) states that the highest term ,k in the expansion (2.4.2.1.1) is smaller than, or equal to, the smallest of the three numbers 21b, 2L1, 2L2, where LI and L2 are the angular momenta carried away by R1 and R2. This rule implies isotropic correlation if the angular momentum of either of the radiations or of the intermediate state equal 0 or +. Equation (2.4.2.1.4) expresses the fact that the coefficients Ak for a cascade can be broken up into two factors, each factor depending on only one transition of the cascade. The second factor Ak(b,c), for instance, is entirely determined by the properties of the levels b and c and of the radiation Rz. Numerical values for the factors Ak(a,b) for given properties of the levels a and b and of the radiation R have been calculated and tabulated for most cases of interest. Relevant references are collected in Table I. Consider now a simple example, the directional correlation between two successive gamma rays in the cascade shown in Fig. 2(a) ( I , = 2, I b = 2, I , = 0 ) . The second gamma ray in this cascade must be pure quadrupole, and from the references given in Table I, we find the corresponding directional correlation factors A2(b,c) = -0.5976, A4(b,c) = -1.069. If we assume the first transition to be pure also, for instance dipole radiation (L1 = l), we find immediately A2(a,b) = -0.4183, Ar(a,b) = 0 , and hence A2 = 0.2500 and A4 = 0. These values have to be compared with experiment. Generally, however, the first transition in a 2 -+ 2 3 0 cascade will be mixed, ie., carry away more than one single value of angular momentum. Angular momentum selection rules allow L1 = 1, 2, 3, or 4. As a rule, 11
S. P. Lloyd, Phys. Rev. 8S, 716 (1951).
134
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
TABLEI. References for Directional Correlation of Various Types of Radiations [This table is intended for quick reference only; no credit is given for original work. Compare Biedenharn” and Gibbs and Wayb also.] Radiation Pure ga.mma rays Mixed gamma rays More than two gamma rays Conversion electrons Alpha particles Heavy particles with spin Beta particles
Theory
Tables
a, c, 4 e a, c, d, e a, c, e
c, d, f c, d, f, u c, f, h c, i c, f j, k c, 1
c, e
a, c, e a, e c , el 1
0 L. C. Biedenharn, in “Nuclear Spectroscopy” (F. Ajzenberg-Selove, ed.), Part B. Academic Press, New York, 1960. * R . C. Gibbs and K. Way, “A Directory to Nuclear Data Tabulations,” U.S. Atomic Energy Commission, 1958. L. C. Biedenharn and M. E. Rose, Revs. Modern Phys. 26, 729 (1953). H. Frauenfelder, i n “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 531. Interscience, New York, 1955. S. Devons and L. J. B. Goldfarb, i n (‘Handbuch der Physik-Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 42, p. 362. Springer, Berlin, 1957. M. Ferenta and N. Rosenzweig, AEC Report ANL-5324 (1955). 0 R. G. Arns and M. L. Wiedenbeck, Phys. Rev. 111, 1631 (1958). J. M. Kennedy, B. J. Sears, and W. T. Sharp, AECL-106, CRT-569 (1954). ’ M. E. Rose, L. C. Biedenharn, and G. B. Arfken, Phys. Rev. 86, 5 (1952). j W. T. Sharp, J. M. Kennedy, B. J. Sears, and M. B. Hoyle, AECL-97, CRTO-556 (1953). A. J. Ferguson and A. R. Rutledge, AECL-420, CRP-615 (1957). I. Hauser, Nuovo cirnento Suppl. [lo] 6 , 182 (1957).
a
b
FIG.2. (a) 2 -+ 2 0 cascade, occurring frequently in even-even nuclei. (b) The first gamma ray 71 is usually mixed in this case; the two dominant angular momenta being L I = 1, LI‘ = 2.
2.4.
DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS
135
only the two lowest multipoles, L1 = 1 and L1'= 2, occur with measurable intensity [cf. Fig. 2(b)]. The coefficients Az(a,b) and Ad(a,b), and hence also A , and Ad, now become continuous functions of the mixing ratio 6, which in this case indicates the ratio of amplitude of the quadrupole to that of the dipole contribution. (6 is defined as the ratio of reduced matrix element for the emission of L' to that of L r a d i a t i ~ n , ~ ~ ~ - ~ * ~
FIG.3. Parametric plot1%of the directional correlation coefficients A2 and A P for the cascade 2 + 2 -+ 0. 6 ia the mixing ratio (quadrupole to dipole radiation) in the first transition. The curve is labeled with the values of the parameter 6' = [S/(l - ISI)]. Two experimental points are shown.
the ratio of the total intensity of the L' pole to that of the L pole is equal to az.) The most convenient representation of the dependence of A2 and A4 on 6 is a parametric pZot:12 one draws the curve A z = A z ( 6 ) , A4 = A4(8) in an A2 - A4 plane. [The functions Az(6) and A4(6) are easily calculated from the formulas and coefficients given by Frauenfelder,3 Biedenharn and Rose14Ferentz and Rosenzweig,13 and Arns and Wiedenbe~k.1~1 A parametric plot for the cascade 2 -+2 -+ 0 is shown in Fig. 3. The C. F. Coleman, Nuclear Phys. 6, 495 (1958). M. Ferenta and N . Rosenaweig, AEC Report ANL-5324 (1955). 14 R. G. Arns and M. L. Wiedenbeck, Phys. Rev. 111, 1631 (1958). 12 1s
136
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
experimental points with coordinates and must lie on this curve and thus yield values for sign and magnitude of the mixing ratio 6. 11. The experimental determination of C(8). The first problem is to decide when two radiations are genetically related, i.e., have been emitted by the same nucleus. It can be solved by observing the decays in a cloud chamber or a photographic emulsion (coincidence in space), but more often a coincidence in time is used. Two radiations from a source are assumed to originate from the same decay if they are observed within a very short time interval (resolving time). A correction for “accidental coincidences” is then applied. (See Section 2.4.2.1.7.)
FIQ.4. Slow-fast coincidence setup for directional correlation measurements.
A typical experimental setup for gamma rays is shown in Fig. 4 (slowfast circuit). The gamma rays y1 and yz are counted in the scintillation counters 1 and 2 (cf. Section 1.4.1). The outputs of these counters are divided into a OW^' and a “fast” channel. The slow channels, using linear amplifiers, preserve the pulse heights and determine the energies of y1 and y z . I n the fast channels, only the very first part of each pulse is used. Wide band amplifiers “blow up” the pulse; the very first part of the leading edge is then shaped into a very short pulse, usually with delay lines. The widths of these pulses essentially determine the resolving time of the circuit. The fast coincidence circuit responds only if the shaped pulses from both counters arrive within the “resolving time.” The output of the two slow channels and of the fast coincidence circuit are fed into a slow coincidence circuit. An output signal results only if the two gamma rays possess proper energies and if the two pulses are simultaneous (i.e.,
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
137
arrive within the resolving time). The simple arrangement shown in Fig. 4 is adequate for many y-y directional correlation experiments. Improvements and changes, however, can be made in various directions. For work requiring extremely short resolving times, the ordinary photomultipliers can be replaced by types not requiring additional amplification. The NaI counters are replaced by other detectors if radiations other than y rays are observed (cf. Chapters 1.2-1.5, 1.8, 2.2, and papers by Frauenfelder3 and Devons and GoldfarbO). If one radiation is in coincidence with many others or with a continuous spectrum, as for instance in a beta-gamma coincidence experiment, then the use of multichannel pulse-height analyzers permit,s the simultaneous recording of the directional correlation function a t various energies.en15 Data gathering, quite often the limiting factor in correlation work, can be speeded up by the use of more than two counters.16 Detailed discussions of the electronics involved in directional correlation experiments can be found in papers by Bell,16v17 Kandiah and Chaplin, l 8 de Benedetti and Findley,lg and in the present series.20 2.4.2.1.4. DETERMINATION OF PARITIES; LINEARPOLARIZATION OF GAMMARAYS. The parity of a nuclear state with respect to that of another state (usually the ground state) can be found by determining the parity of a transition linking the two. As in the case of the spin determination, the decaying nucleus is first partially or completely oriented by one of the three methods discussed under Section 2.4.2.1.1.The parity of a t,ransition can then be inferred from the observation with respect to the direction of orientation of (a) the directional distribution of conversion electrons, or (b) the directional distribution and linear polarization of gamma rays. The first approach is better suited to low energy transitions (say, E < 0.3 Mev), where the conversion coefficient is appreciable; the second is more favorable for high-energy gamma rays (say, E > 0.2 Mev). 2.4.2.1.4.1.Conversion Electrons. The directional correlation involving conversion electrons is similar to that of gamma rays treated in Section 2.4.2.1.3.The coefficients Ah(a,b),however, depend now also on the parity of the conversion electrons; i.e., for a given multipole order L, they differ ‘6 T. R. Gerholm, in “Proceedings of the Rehovoth Conference on Nuclear Structure’’ (H. J. Lipkin, ed.), p. 483. Interscience, New York, 1958. 18 R. E. Bell, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), Chapter 18. Interscience, New York, 1955. l 7 R. E. Bell, Ann. Revs. Nuclear Sci.4, 93 (1954). K. Kandiah and G. B. B. Chaplin, Progr. in NucEear Phys. 6, 89 (1956). 19 S. de Benedetti and R . W. Findley, in “Handbuch der Physik-Encyclopedia of Physics” (S. FlCgge, ed.), Vol. 45, p. 222. Springer, Berlin, 1958. 2 0 S P Vol. ~ 2 of this series.
138
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
for electric and magnetic radiation. Numerical values for the coefficients A,(a,b) can be found in the references given in Table I. The experimental realization requires special care: the source must be very thin and uniform in order to minimize scattering of the conversion electrons, scattering between source and counter has to be avoided, K electrons must be selected and recorded in order to permit comparison with the theoretical calculations, and extranuclear effects must be avoided or taken into account. Details regarding all these points can be found in the reviews,ass in the references given there, and in the papers by Gimmi, Heer, and Scherrer,21Coburn, Kane, and FrankeljZ2and Snyder and FrankeLZs 2.4.2.1.4.2. Linear Polarization of Gamma Rays. Electric and magnetic radiation of the same multipole order L have opposite parity; their electric and magnetic vectors are related through the transformation E 4 H, H 4 -E. This transformation leaves the Poynting vector and hence the directional distribution of the radiation unaltered. A measurement of the directional correlation thus yields L, but gives no information about the parity. With L known, the observation of the direction of E (i.e., of the linear polarization of the gamma ray) furnishes the parity, however. I n angular distribution experiments, one can directly detect the linear polarization of emerging y rays, without a coincidence setup. I n cascade decays, one must observe the linear polarization of y rays in coincidence with a preceding or following radiation. Relevant formulas for such polarization-direction correlations can be found in papers by Biedenharn and RoseI4Devons and Goldfarb16and Biedenharn.' Basically, such a measurement is similar to that treated in Section 2.4.2.1.3 and hence the discussion here is restricted to the new aspect, namely the detection of the linear polarization of y rays. Before treating methods for measuring the linear polarization, a few remarks concerning the description of the state of polarization of photons are in order. The complete description in terms of Stokes parameters can be found in various r e f e r e n ~ e s ~ J ~(cf. , ~ 4Section 2.5.2). For the following discussion, an outline of selected pure cases is sufficient. Circularly polarized y rays have their spin ( l h ) parallel (right c.P.) or antiparallel (left c.P.) to their direction of motion, Linear polarization cannot be described so simply in terms of the spin of photons. Classically, a beam of y rays with complete linear polarization corresponds to electromagnetic radiation in which the electric vector E lies always in the direction of polarization. Polarimeter, i.e., detectors sensitive to the state of polarization of 2'
F. Gimmi, E. Heer, and P. Schemer, Helv. Phys. Acta 2B, 147 (1956).
** H. C. Coburn, J. V. Kane, and S. Frankel, Phys. Rev. 106, 1293 (1957). 2a 24
E. S. Snyder and S. Frankel, Phys. Rev. 106, 755 (1957). w. H. McMaster, Am. J . Phy8. 22, 351 (1954).
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
139
gamma rays, are always based on some kind of scattering. Methods to determine the degree of circular polarization are generally based on the Compton effect2S*26 for low-energy y rays ( < 6 Mev), and on the nuclear p h o t ~ e f f e c t ~ ’ for - ~ ~higher energies ( E > 3 Mev). In addition to these two major methods, some additional polarization sensitive effects can probably be used-the atomic photoeffect30 and pair p r o d ~ c t i o n . ~ ~ As yet, no detectors based on these effects have been built. The Compton p ~ l a r i m e t e r , ~treated ~ - ~ ~ here in some detail as an example, is based on the Klein-Nishina f o r m ~ l afor ~ the ~ * differential ~~ cross section du/dQ for Compton scattering
@
dn
-
[
k !f 2 ko2 ko
+ ko-k - 2 sin2 6
0082
1.
3
(2.4.2.1.5)
Here, ko is the initial energy of the y ray and k the energy of the scattered quantum k0 (2.4.2.1.6) k = 1 (ko/mc2)(1 - cos 6)‘
+
6 is the scattering angle, 3 the angle between the direction of polarization of the incident photon and the plane of scattering, ro = e2/mc2 the classical radius of the electron. Equation (2.4.2.1.5)shows that du/dQ depends on 4 and that the observation of Compton scattering hence can be used for the determination of the direction and degree of linear polarization. Figure 5 shows the geometry used in a n ideal experiment; Fig. 6 displays t8heintensity of scattered radiation, for fixed values of ICO and 6, as a function of the angle 4. The idealized polarimeter shown in Fig. 5 operates by recording the number of Compton scattered photons as a function of the angle 3. The minimum of the corresponding curve (Fig. 6) indicates the direction A. Wightman, Phys. Rev. 74, 1813 (1948). N. R. Steenberg, Can. J . Phys. 31, 204 (1953). 2’ D. H. Wilkinson, Phil. Mag. [7] 43, 659 (1952). z8 L. W. Fagg and S . S. Hanna, Phys. Rev. 92, 372 (1953). 2’) I. S. Hughes and D. Sinclair, Proc. Phys. SOC.(London) A89, 125 (1956). F. L. Hereford and J. P. Keuper, Phys. Rev. 90, 1043 (1953). s1 M. M. May, Phys. Rev. 84, 265 (1951). F. Metzger and M. Deutsch, Phys. Rev. 78, 551 (1950). s3 J. J. Kraushaar and M. Goldhaber, Phys. Rev. 89, 1081 (1953). * 4 D. R. Hamilton, A. Lemonick, and F. M. Pipkin, Phys. Rev. 92, 1191 (1953). s6 G. R. Bishop and J. P. Perez y Jorba, Phys. Rev. 98,89 (1955). 9 B C. F. Coleman, Phil. Mag. [S] 1, 166 (1956). 3’ P. H. Stelson and F. K. McGowan, Phys. Rev. 106, 1346 (1957); 109,901 (1958). 3* W. Heitler, “Quantum Theory of Radiation,” Oxford Univ. Press, London and New York, 1954. 2s
z8
140
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
Polarization Vector scatterer
FIQ.5. Geometry for a polarimeter based on the Compton scattering. Polarization Vector Counter
FIQ.6. Polardiagram showing the cross section o!u/ds2 for Compton scattering as a function of the angle ip between the direction of polarization of the incident gamma ray and the plane of scattering. The incident photons with energy ko are completely polarized and enter the scatterer perpendicular to the plane of the figure. The scattering angle 6 is fked at 80". The intensity distribution is shown for two incident energies, 0.5 Mev and 3 MeV.
of polarization. To determine the degree of polarization, one assumes that the incident y-ray beam of arbitrary linear polarization consists of two completely polarized and mutually orthogonal beams, with intensities I,,and II. The polarization vector of 111lies in the direction of the scattering minimum (as shown in Fig. s), that of II a t right angle to it. The
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
141
degree of polarization of the incident beam can then be described by (2.4.2.1.7) The measured numbers of scattered y rays in the two extreme points are given by N,i, = I , ,&(a = 0") 11 dU(* = 90") (2.4.2.1.8) N,,, = I , ,dU(* = 90") 11 &(a = 0").
+ +
From (2.4.2.1.7)and (2.4.2.1.8),the degree of polarization P follows as (2.4.2.1.9) where R, called the asymmetry ratio of the polarimeter, is defined by du(@ =
R = &(a
=
90") 0")
(2.4.2.1.10)
For the idealized polarimeter of Fig. 5, R can be calculated from the Klein-Nishina formula, Eq. (2.4.2.1.5). An actual polarimeter can be characterized by five quantities: asymmetry ratio, efficiency, solid angle, energy resolution, and signal-to-noise ratio. The asymmetry ratio R is defined by a n obvious generalization of Eq. (2.4.2.1.10).For intensity reasons, scatterer and counter subtend large solid angles. The angles 6 and a hence possess considerable variation (e.g., A 6 = A0 = 50"), and for each energy ko, R must be either calculated by integration, or it must be determined experimentally. (See Section 2.4.2.1.7 and the papers by Metzger and D e u t s ~ h Kraushaar ,~~ and G ~ l d h a b e r Hamilton ,~~ et ~ l . Bishop , ~ ~ and Perez y J ~ r b aColeman,36 ,~~ and Stelson and M~Gowan.~') A good polarimeter should of course combine a small energy resolution with the largest possible values of the other characteristic quantities. The simplest Compton polarimeter consists of a y-ray collimator (lead or tungsten), a scatterer (e.g., lead or copper), and a counter.3g The collimator shields the counter from direct radiation. The mean scattering angle 6 is so chosen that the asymmetry ratio R becomes a maximum. The major disadvantage of such a polarimeter is a small signal-to-noise ratio and a bad energy resolution. Considerable improvement is achieved by using a scintillation counter as scatterer and measuring coincidences between signals in the' scatterer and the counter (now called analyzer) .32-40 The signal-to-noise ratio for S. de Benedetti, in "Beta- and Gamma-Ray Spectroscopy" (K. Siegbahn, ed.),
3%
p. 674. Interscience, New York, 1955. 40 J. I. Hoover, W. R. Faust, and C. F. Dohne, Phys. Rev. 86, 58 (1952).
142
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
coincidences is much larger than that for single counts. The energy resolution is still bad, since the spread A6 in the scattering angle 6 causes a corresponding unavoidable spread of pulse height in the scatterer and the analyzer. The spread of the energy can be seen by calculating (klko) from Eq. (2.4.2.1.6) for the various angles 6. Energy resolution and signal-to-noise ratio can, however, be further improved.36The scintillation crystal of the analyzer is made sufficiently large so that the scattered y rays are nearly always completely absorbed when being counted. The output pulses from the scatterer and the analyzer are again fed into a coincidence circuit, but they are also added. A true event is recorded only if a coincidence occurs and if simultaneously the sum pulse possesses a pulse height corresponding to the initial energy ko. 2.4.2.1.5. DETERMINATION OF GYROMAGNETIC RATIOS.The basic idea underlying the determination of gyromagnetic ratios has been sketched in Section 2.4.2.1.2.Detailed treatment of the theoretical and experimental aspects can be found in the papers by Frauenfelder,* Biedenharn and Rose,4Steffeq6 Devons and Goldfarb,* Gerholm,I6Alder et u Z . , ~ ~Abragam and Aeppli et Krohn and R a b ~ ySteffen , ~ ~ and Z0be1,~~ and Bodenstedt et al.46aThe present discussion is restricted to an outline of the experimental approach. The gyromagnetic ratio of an excited nuclear state can be measured if the following conditions are fulfilled. sec and sec. (a) The lifetime T of the state lies between about (b) The state is either the intermediate state in a nuclear cascade or it can be produced through a nuclear reaction (e.g., Coulomb excitation), (c) The state decays by emission of a radiation (usually a y ray) which displays a nonisotropic directional distribution with respect to either the direction of the incoming beam or the direction of emission of the first radiation. (d) The directional distribution or correlation is not appreciably disturbed by quadrupole interaction. [Quadrupole interaction can always be present, since condition (c) requires that the intermediate state possess a spin of at least 1.1 Assume now that a cascade, yI-yz, has been found which fulfills the first three conditions. The next step consists in preparing a source in 41 K. Alder, H. Albers-Schonberg, E. Heer, and T. B. Novey, Helv. Phys. Actu 26, 761 (1953). 48 A. Abragam and R. V. Pound, Phys. Rev. 93, 943 (1953). 4) H.Aeppli, H. Albers-Schonberg, H. Frauenfelder, and P. Schemer, Helv. Phys. Acta 26, 339 (1952). 4 4 V. E. E o h n and S. Raboy, Phys. Rev. 97, 1017 (1955). 45 R. M.Steffen and W. Zobel, Phys. Rev. 103, 126 (1956). 46s E. Bodenstedt, H. J. Kroner, G. Strube, C. Giinther, J. Radeloff, and E. Gerdau, 2.Physik 163, 1 (1961).
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
143
which the nuclei experience little or no quadrupole interaction. This problem can often be solved by preparing a nonviscous liquid ~ o u r c e , ~ , ~ . ~ , 21,42,44,46-49 by using the substance in the melted p h a ~ e ,by ~~ embedding ~,~ the radioactive atoms in a cubic crystal3 or in a n oriented single crystal.60 Next, the directional correlation function W ( 0 ) , Eq. (2.4.2.1.1) is determined (Section 2.4.2.1.3).Assume that the measured coefficients Az, Ah, together with other data yield an unambiguous value for the spin I of the intermediate nuclear state. Now, the gyromagnetic ratio can be found by applying to the source a magnetic field H which is perpendicular to the plane of the two counters and the source [Fig. 7(a)J.Each of the two counters shall be sensitive to only one y ray of the cascade. During the time t , which elapses between the emission of y 1 and that of yz, the nucleus precesses in the field H by an angle W H t , where W H is the Larmor frequency (2.4.2.1.11)
the nuclear magneton. Instead of the correlation W ( 0 ) ,one observes W(O - W H t ) , averaged over the range of time t which the coincidence circuit accepts. Suppose that it registers only if yz is emitted within the time interval T I to Tz after the emission of y 1 [Fig. 7(b)]. The measured correlation function is then given by p o is
w(e,H)
-
JTT’exp(-t/T)W(B - W H t , H
=
0) dt.
(2.4.2.1.12)
In practice, the time T Iand Tz are determined by the resolving time T , of the coincidence circuit and a delay Ta inserted into the channel which counts y l (Fig. 7). The choice of Ta and T,depends on the lifetime of the nuclear state under investigation. The discussion of the following three cases shows all the important aspects of the method. Assume for simplicity that the highest term in Eq. (2.4.2.1.1) is given by lc, = 2. Moreover, instead of using Eq. (2.4.2.1.1),it is more convenients1to employ the equivalent expansion in terms of cos n0: w(e) = 1 B~ cos 20 (2.4.2.1.13) B , = 3A2/(4 Ad.
+
+
Three important special cases can now be treated easily. H. Albers-Schonberg, E. Heer, and P . Schemer, Helv. Phys. Acta 27, 637 (1954). R. M. Steffen, Phys. Rev. 103, 116 (1956). 48 V. E. Krohn, T. B. Novey, and S. Raboy, Phys. Rev. 106, 234 (1957). 4s G. Goldring and R. P. Scharenberg, Phys. Rev. 110,701 (1958). SOH. Albers-Schonberg, E. Heer, T. B. Novey, and P. Schemer, Helv. Phys. Acta 27, 547 (1954). 6 1 K. Alder, Helv.Phys. Acta 26, 235 (1952). 4fi
47
144
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
Magnetic Field H
b 1
Td
_I Coincidences
Resolving TimeTr
Decoys in this intervol ore accepted by the coincidence circuit
Emission ot
'1
3;
I_ Delay Td in
Channel I
Td T2
_I
(b) FIG.7. (a) Setup for the determination of the gyromagnetic ratio of the intermediate state in a y-7 cascade. The magnetic field H is applied perpendicular t o the plane containing the source and the two counters. (b) N ( t ) dt denotes the probability of emission of ya in the time interval t to t dl after emission of 7 , . T indicates the lifetime of the intermediate nuclear state. The coincidence circuit accepts only decays which occur in the interval T Ito Tz.T Iand TPare fixed by the delay Td in channel 1 and the resolving time T,: TI= Td - T,,T2 = T d T,.
+
+
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
145
I. Td = 0, T , >> r , each counter sensitive to one y ray only. This case applies particularly to nuclei with very short lifetimes (T = 10-lo sec). Present technique then does not permit using coincidence circuits with shorter resolving times and the assumptions made represent actual experiments quite well. The selection of y 1 in channel 1 and yz in channel 2 is performed with a slow-fast circuit (Fig. 4). The effect of the magnetic field on the correlation pattern under the assumptions I can be calculated by inserting Eq. (2.4.2.1.13) into Eq. (2.4.2.1.12) and setting T1 = 0, T 2 = 0 0 . The result
+
WI(@,H)= 1 [B,/d1 f (2WHT)2]cos2(8 - AO) (2.4.2.1.14) where A0 = & arc tan 2 ~ ~ clearly 7 , shows an attenuation [by a factor l/dl ( ~ O R Tand ) ~ ] a rotation (by an angle A@) of the correlation pattern. For small angles, AO is approximately equal to the classical angle of precession W H T . The magnitude of the g factor can be determined from the attenuation or from the rotation. The sign of g follows from the direction of rotation. 11. Td = 0, T , >> r , each counter accepts both y rays. Such a situation occurs if both y rays possess very nearly equal energy. The corresponding effect on the correlation pattern can be seen by calculating the function WII(O,H)= +(Wl(O,H) Wl(O, - H ) j , since exchange of y1 and y z corresponds to a transformation H -+ -H. Hence
+
+
As expected, only an attenuation occurs, but no rotation. Only the magnitude, but not the sign of g can be found. 111. Td > T,, T , << T . In this case, coincidences occur only if y 1 enters channel 1 and yz channel 2 [cf. Fig. 7(b)]. From Eq. (2.4.2.1.12),one finds directly WIII(@,H) = - WHTd, H = 0). (2.4.2.1.16) The original correlation function is hence only rotated, but not attenuated. The g factor must be found from the rotation. Case I11 can be applied to relatively long-lived states ( ~ : 1 0 -sec ~ to lop7see). The preceding outline displays the most important features of the directional correlation method for determining nuclear g factors. A number of additional remarks are in order, however. (1) The assumption kmaX= 2, used in I and I1 above, can easily be removed. Relevant formulas can be found in papers by Biedenharn and Rose,4 Steffen,6 Devons and Goldfarb,s Alder et uZ.,~' Abragam and Pound,42 Alder,61and Nierenberg.62
w(@
62
W. A. Nierenberg, Ann. Revs. Nuclear Sci. 7, 349 (1957).
146
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
(2) Cases intermediate between the ones treated above can also be handled without too much complication. Calculations for correlation functions with arbitrary values of T d , T,, and r can be found in papers by Alder et aZ.41 and Steffen and Z ~ b e l . ~ ~ (3) The limits for the applicability of the angular correlation method, sec are given by the following considerarain= sec and rmX = tions. The presently available maximum field strengths, H,, 5 lo6 gauss, determine rain.For shorter lifetimes, the resulting rotation of the correlation pattern is too small to be observed. The upper limit, rmX,is fixed by the circumstance that an optimum measurement of the attenuation or rotation requires a resolving time T, which is comparable to r. The number of accidental coincidences, however, is proportional to T , and outweighs the true coincidences for r > sec. (4) The experiments discussed above all yield the nuclear gyromagnetic ratio. If the spin I of the intermediate state is also known, then the magnetic moment follows CC = glro. (2.4.2.1.17) (5) Assumption (d) above, demanding the absence of quadrupole interactions, is often not fulfilled. Nevertheless, the true g factor can be found, either by applying corrections to the g factor inferred from attenuation, or by using the g factor given by the r o t a t i ~ n . ~ ~ , ~ ~ . ~ ~ (6) Experimental details can be found in papers by Gerholm,16Aeppli et al.,43Krohn and R a b ~ ySteffen , ~ ~ and Z0be1,~~ Krohn et Goldring and S ~ h a r e n b e r g and , ~ ~ Albers-Schonberg et U Z . ~ O It is obvious that the strong magnetic field present a t the source necessitates, in addition to the usual requirements for anguIar correlation experiments, light pipes between scintillation crystals and photomultipliers and very good magnetic shielding of the multipliers. (7) I n the discussion of cases I and 111, it is assumed that the complete correlation function is measured as a function of the applied field. Actually, it is advantageous to use three (or more) counters in fixed positions, e.g., one at 0', one a t 135', and one a t 225'. Coincidences 0'-135' and 0'-225' are measured. The ratio R = C(0'-135')/C(0'-225') as a function of the applied field H then is a measure for the rotation of the pattern and allows a very accurate determination of W H . (See Krohn and Raboy44 and Steffen and Z0be1.~~) (8) Further improvement results if more than three counters are used and if coincidences between all pairs of counters are recorded. Such an arrangement greatly improves statistical precision.16B6a (9) Even in strong magnetic fields, and with states of very long life6)
T. R. Gerholm, T. Lindqvist, and H. de Waard, Nuclear Instr. 1, 107 (1957).
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
147
time, the nuclei perform only a small number of revolutions. Magnetic resonance experiment^^^ hence seem not feasible but it may be possible to use the method of “stroboscopic coincidence^."^^ (10) Gyromagnetic ratios can also be determined if the excited state is reached by a nuclear.reaction. The magnetic field is applied to the target and the corresponding rotation of the directional distribution of the de-excitation radiation is observed. No coincidences are involved and all nuclei, regardless of the actual time 2 they spend in the excited state, are detected. The approach thus corresponds to case I above. Usually, two counters are used in fixed positions and the ratio R [see remark (7)]as a function of the applied field yields the g factor. (See the papers by Goldring and S ~ h a r e n b e r gLehmann ,~~ et u Z . , ~ ~TreacylK6Sugimoto and Mizob u ~ h iPhilips , ~ ~ and Jones,68and S ~ g i m o t o . ~ ~ ) 2.4.2.1.6. DETERMINATION OF ELECTRIC QUADRUPOLE MOMENTS. The measurement of gyromagnetic ratios, as discussed in Section 2.4.2.1.5, is quite straightforward. The application of similar ideas to the determination of electric quadrupole moments is much more intricate and has yielded far fewer results. The discussion here is hence very brief. Two facts render experiments on quadrupole moments difficult. (a) The interaction between an electric quadrupole moment and an inhomogeneous electric field depends quadratically on the magnetic quantum number m, which describes the orientation of the nucleus with respect to the extranuclear field. Classically, this m degeneracy means that the nuclei precess with equal probability in both directions. No rotation of the correlation pattern occurs and all the information must be gathered from the attenuation. This approach is less precise and also does not yield the sign of the quadrupole moment. (b) No artificial field gradients can be produced which are large enough to perturb an angular correlation, hence the natural field gradients in solids and liquids must be used for the study of the interaction. Since the radioactive atoms are usually foreign to the lattice in which they are embedded, and since they also recoil when undergoing a decay, it is extremely difficult to estimate reliably the electric field gradient present a t the nucleus. Results of most measurements are expressed in terms of the quadrupole coupling, i.e., the product Q * VE, where Q is the nuclear quadrupole moment and V E the electric field gradient. The separation into the two factors is almost always extremely difficult. V. L. Telegdi, unpublished. P. Lehmann, A. LBvBque, and R. Pick, Phys. Rev. 104, 411 (1956). 68 P. B. Treacy, NuckaT Phys. 2, 239 (1956). 67 K. Sugimoto and A, Mizobuchi, Phys. Rev. 103, 739 (1956). 68 W. R. Philips and G. A. Jones, Phit. Mag. [S] 1, 576 (1956). 69 K. Sugimoto, J . Phys. SOC. Japan 15, 240 (1958). 64
66
148
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
Information on the determination of the quadrupole coupling has been described in the following papers: F r a ~ e n f e l d e r ,Steffen16 ~ Devons and Albers-Schonberg et aZ.,46 Goldfarb,6 Alder et aZ.,41Abragam and Steffen,47 Krohn et al.,48 Albers-Schonberg et a l l s o Lehmann and Miller,6oand Harris.6Oa 2.4.2.1.7. EXPERIMENTAL CORRECTIONS AND TESTS.Three experimental phases will be outlined briefly. Tests must establish that the equipment works properly; a procedure for getting reliable data has to be worked out in each experiment; the raw data must be corrected for all deviations of the actual setup from the ideal one. 2.4.2.1.7.1. Tests. The invention of tests forms a n essential part of the preparation for any experiment. During the setting-up of the equipment, ideas for tests occur naturally. The following list is hence not a complete guide, but it is intended only as a n example. In a directional correlation experiment, the following tests are valuable. (1) For each channel, one determines stability, energy resolution, energy calibration, linearity of response, and efficiency with radioactive isotopes of well-known radiation characteristics. (2) The performance of the counting systems as a function of source strength is observed. (3) The solid angle Q of the counters is determined, either by calculation61or by experiment.62 (4) The resolving time T r of the coincidence circuit is investigated as a function of source strength and energy of incident radiation. T , can be found either by measuring the coincidences from a “prompt” cascade (i.e., a cascade with a very short lifetime of the intermediate state) with a delay Td > Tr in one channel, or by using two independent sources and separating the counters. ( 5 ) Scattering from one counter into the other, or from material close to source or counters, is checked. (6) The over-all performance of the directional correlation system is tested by measuring V(6)for well-known cascades (Nieo,Pd106). Most of these tests apply also to polarization-direction correlations. I n addition, however, the asymmetry ratio R of the polarimeter must be determined by one or more of the following experiments. (7) R can be found by measuring the linear polarization of gamma rays which were scattered by a n angle of about 90°.32--34,40 (8) The polarization-direction correlation of a well-known cascade (NiSo,Pdlo6)can serve to fix R.33 P. Lehmann and J. Miller, J . phys. radium 17, 526 (1956). S. M.Harris, Nuclear Phys. 11, 387 (1959). M.E.Rose, Phys. Rev. 91, 610 (1953). 82 J. S. Lawson, Jr., and H. Frauenfelder, Phys. Rev. 91, 649 (1953).
2.4.
DETERMINATION OF SPIN, PARITY, .4ND NUCLEAR MOMENTS
149
(9) The linear polarization of y rays following Coulomb excitation in even-even nuclei is well known and hence can also be used to find R.37 (10) The linear polarization of annihilation quanta yields a value for R a t the energy 0.51 MeV. This experiment, however, requires two polarimeter. 39 2.4.2.1.7.2.Procedure. It is simplest to discuss a y-y directional correlation. Other measurements are obvious generalizations of this basic experiment. Before a run, and at intervals during runs, resolving time and back, are determined. N a n d C denote single counts ground counts [ N i 0 ( 8 ) Co(9)] and coincidences, respectively, per unit time. 8 denotes the angle subtended a t the source by the counters (Fig. 1). A source of proper strength is prepared. The maximum source strength is determined by the ratio of accidental to true coincidences. For accurate measurements, this ratio should be smaller than about 0.2;under unfavorable conditions, one may be forced to make it unity or even higher. The source is then centered to about 1 %; i.e., the counting rate N ( 9 ) of the movable counter as a function of the angle 6 should be constant to about 1 %. After the source is centered, the coincidences and the single counts are recorded for various angles. The angle should be changed frequently, e.g., about every 20 min. For short-lived radioisotopes, more frequent changes are advisable. The choice of angles depends on the form of the correlation function. I n general, it is best, to use a few angles only (e.g., go", 120", . . . 270") and to obtain a t each angle a large number of coincidences. For extended experiments, automatic equipment for changing the angles and recording the data is very convenient. From the measured data, one obtains the "true" values Ct(S) and N t ( 8 ) by subtracting background, accidental coincidences, and contributions due to disturbing radiation^;^^ e.g. :
Cf(fi)= C m e a s (9) - Co(8) - Carc(9) @(9) = 2TrNr;1""(9)Ny""(8).
- Cd(9) (2.4.2.1.18) (2.4.2.1.19)
The contribution Cd(8),due to other coincident y rays present in the source must be found in a separate measurement, with the energy selection adjusted in such a way th at the various contributions can be singled out. The true number of single counts and coincidences can now be written as : Nit(&) = Mpifli~i (2.4.2.1.20) Ct(8) = ~ P , P , Q l E l Q 2 ~ 2 E , ~ ( i ) . (2.4.2.1.21) 63
N. Levine, H. Frauenfelder, and A. Rossi, Z. Physik 161, 241 (1958).
150
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
M is the number of nuclear disintegrations per unit time, pi the probability per disintegration that the radiation selected in counter i is emitted, Qi the solid angle in units of 4n, ei the efficiency of channel i, and ec the efficiency of the coincidence circuit. K (8) denotes the directional correlation function, as measured with the equipment under discussion. Comparison of Eq. (2.4.2.1.21) with Eq. (2.4.2.1.19) shows that the accidental coincidences increase with the square of the source strength, the true coincidences only linearly. As mentioned above, an optimum source strength exists which can be calculated from Eqs. (2.4.2.1.19)(2.4.2.1.21). Decreasing the resolving time T , permits the use of correspondingly stronger sources. Equations (2.4.2.1.19)-(2.4.2.1.21) also show that the ratio of true to accidental coincidences is unaffected by the solid angle. Very large solid angles hence are often advantageous. (Optimum solid angles are discussed in Devons and Goldfarb.e) For the further evaluation, the coincidence ratio C f ( 8 ) / N 1 ' ( 8is ) formed. From Eqs. (2.4.2.1.20) and (2.4.2.1.21), one then finds for the desired uncorrected experimental correlation function K ( 8 ): (2.4.2.1.22)
If one uses movable counters, it is seen from this equation that one best divides through by the counting rate in the niovable counter, i.e., that one identifies counter 1 with the movable one. The ratio K ( 8 ) then becomes independent of the solid angle of the movable counter. Furthermore, it is independent of the decay of the source, of the branching probability pZ and of the efficiency of the movable counter. Small errors in the centering of the source are corrected in first order. The numbers K ( 8 ) are now fitted by
+ A;~'~(COS8) +
K ( 8 ) = Ka(1
*
*
.
+ A~maPkmar(COS 8))
(2.4.2.1.23)
using the method of least square^.^^^^^-^' It usually suffices to take = 4. Details concerning the determination of the coefficients A k ' and their errors are given in RoselB1 Breitenbergerl64 and Price;6s applications of the method are treated in Klema and McGowanBsand Breitenberger.67 2.4.2.1.7.3. Corrections. K ( 8 ) corresponds to the correlation function W(0)only under the assumption of centered point sources, point detectors,
,k
E. Breitenberger, Proc. Phys. Soc. (London) A69, 489 (1956). 86
P.C.Price, PhiE. Mug.[7]46,237 (1954);Proc. Cumbridge Phil. Soc. 60,491 (1954).
s8E. D.Klema and F. K. McGowan, Phys. Rev. 91, 616 (19531. E. Breitenberger, Proc. Phys. SOC.(London) A69, 453 (1956).
2.4.
DETERMINATION
OF SPIN, PARITY,
AND NUCLEAR MOMENTS
151
TABLE11. Corrections in Angular Correlation Experiments [The references are not complete; in some cases, only a few out of a large number of papers are given as examples.] Correction for
References
Perturbing influence of extra nuclear fields Finite size of counters Finite sire of source Decentering of source Scattering in the source Count er-to-counter scattering Disturbing radiations
a, b, c, d b, eel, r b, c, e, f, i, k
m
f,
c, n c, e, 0 j , P, q
A. Abragam and R. V . Pound, Phys. Rev. 92, 943 (1953). Devons and L. J. B. Goldfarb, in “Handbuch der Physik-Encyclopedia of Physics” (5.Flugge, ed.), Vol. 42, p. 362. Springer, Berlin, 1957. H. Frauenfelder, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 531. Interscience, New York, 1955. R. M. Steffen, Advances in Phys. 4, 293 (1955). M. Walter, 0. Huber, and W. Ziinti, Helv. Phys. Acta 23, 697 (1950). f S. Frankel, Phys. Rev. 83, 673 (1951). 0 M. E. Rose, Phys. Rev. 91, 610 (1953). E. Breitenberger, Proc. Phys. Soc. (London) A66, 846 (1953). i A. M. Feingold and S. Frankel, Phys. Rev. 97, 1025 (1955). i E. L. Church and J. J. Kraushaar, Phys. Rev. 88,419 (1952). J. S. Lawson, Jr., and H. Frauenfelder, Phys. Rev. 91, 649 (1953). E. D. Klema and F. K. McGowan, Phys. Rev. 92, 1469 (1953). E. Breitenberger, Phil. Mug.171 46, 497 (1954). E. Breitenberger, Proc. Phys. SOC.(London) A67, 1108 (1954). H. Frauenfelder, Ann. Revs. Nuclear Sci. 2, 129 (1953). p E. D. Klema, Phys. Rev. 100, 66 (1955). q N. Levine, H. Frauenfelder, and A. Rossi, 2.Physik 161, 241 (1958). H. I. West, Jr., Univ. of California Report UCRL 5451 (1959). a
* S.
no scattering, no disturbing radiations, and no extranuclear effects. In order to compare experimental and theoretical results, K(6) must be corrected for all deviations from such an ideal arrangement. All of the necessary corrections have been discussed in the literature ; relevant references are collected in Table 11. After applying the corrections, the final form of the experimental correlation function becomes P P ( 6 )=
K O (1 + AYP~(cos6) + . . .
+ A ~ s x P k , , ( ~6)). ~~ (2.4.2.1.24)
One now makes the assumption 6 = 0 and compares theoretical coefficients A k with AYp in order to extract the desired physical information.
152
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
2.4.2.2. Conversion Coefficients.* 2.4.2.2.1. INTRODUCTION; THEOCONSIDERATIONS. The ratio of the probabilities for emitting Y-shell conversion electrons (see Section 2.2.3.1; Y = K , LI * * Mv * Ltotal *) to that of nuclear gamma rays is called the Y-shell conversion coefficient ay. This quantity, as well as the ratio of conversion coefficients in different shells depend on the character of the transition under consideration, i.e., on whether the parities of the initial and the final state are the same or not, and on the distribution of the angular momenta of the emitted gamma rays. As discussed in Section 2.4.2.1, the last datum is connected with the difference between the spins of the initial and the final levels. Information about spins and parities of nuclear levels can, therefore, be obtained by comparing measured conversion coefficients, or ratios of conversion coefficients for the same transition in different shells, with the theoretically computed values. We will therefore first discuss some results of conversion theory and then some experimental procedures for measuring conversion coefficients. Conversion coefficients also depend on the transition energy and on the nuclear charge. Since these quantities are mostly known with sufficient precision, their influence will not be discussed here (except for remarking that conversion coefficients mostly increase strongly with increasing Z and with decreasing E ) . Moreover, only theoretical results for pure multipole transitions will be considered; the probabilities for emitting gamma rays and those for emitting conversion electrons of different multipole orders have to be added incoherently. Conversion coefficients depend on the nuclear structure, both in a static as well as in a dynamic way.' The static effect arises through the action of the average nuclear charge distribution on the atomic electrons. Practically, it is dependent on the nuclear radius only. The dynamic effect originates in the circumstance that nuclear matrix elements due t o penetration of the electron wave function inside the nucleus allow ejection of conversion electrons but no emission of nuclear gamma rays. These matrix elements are normally much smaller than the gamma ray matrix elements which then determine the conversion coefficients (almost independent of nuclear structure). But in cases where the gamma matrix elements become very small, either by I-forbiddenness* or through violation of asymptotic
RETICAL
- -
--
1 L. A. Sliv and M. Listengarten, Zhur. Ekuptl. i Teoret. F i z . 29, 29 (1952); T. A. Green and M. E. Rose, Phys. Rev. 110, 105 (1958); E. L. Church and J. Weneser, Ann. Rev. Nuclear Sci. 10, 193 (1960). 2 E. L. Church and J. Weneser, Phys. Rev. 104, 1382 (1956).
-
* Section 2.4.2.2 is by A. H. Wapstra.
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
153
selection rules13the penetration matrix elements become prominent, thus giving rise t o relatively large conversion coefficients. Evidently, large deviations can occur only for delayed transitions. K selection rules apply to both kinds of matrix elements, so that transitions delayed by K forbiddenness alone should have normal conversion coefficients.3 I n only slightly delayed transitions, interference between the two kinds of matrix elements may cause conversion coefficients t o be even somewhat smaller than those for undelayed transition^.^ The influence of the nuclear size effect on K, LI, and ,511 conversion coefficients is approximately the same; that on LIIIconversion coefficients is very much less, even in a case6 where LI and LII conversion coefficients are large by a factor 20. Theoretical conversion coefficients for the K , LI, LII, and LIII shells have been computed by RoselBtaking into account the influence of the static correction for the finite nuclear size and of the screening of the electric field of the nucleus by the electron shells around it. The same book gives M-conversion coefficients uncorrected for the last two effects. As a result, these conversion Coefficients are somewhat large (e.g., about 40% for E2 and 2 = 80); the ratios of the conversion in the M subshells are more nearly correct. Sliv and Band’ published K and L conversion coefficients computed from a model in which the nuclear currents are located on the surface of the nucleus, thus including the static correction and part of the dynamic correction. Their results differ from those of Rose by less than 5% in almost all cases of physical importance. Data permitting the calculation of conversion coefficients for any nuclear model have been published by Green and R0se.l The precision in measuring conversion coefficients is mostly limited due t o the difficulties in measuring gamma ray intensities. Ratios of Conversion coefficients for one gamma transition, which are identical with ratios of conversion line intensities, can be determined much more accurately. Since these ratios too depend on multipolarity and character, they can also be used to get information about these data. The K to L S. G. Nilsson and J. 0. Rasmussen, Nuclear Phys. 6, 617 (1958). L. S. Kisslinger, Bull. Am. Phys. SOC.2,358 (1957); A. H. Wapstra and C. J. Nijgh, Nuclear Phys. 1, 245 (1955); K. 0. Nielsen, 0. B. Nielsen, and M. A. Waggoner, ibid. 3
4
2, 476 (1956). 5 F. Asaro, F. S. Stephens, J. M. Hollander, and I. Perlman, Phys. Reu. 117,492 (1960). 6 M. E. Rose, “Internal Conversion Coefficients.’’ Interscience, New York, 1958. ‘L. A. Sliv and I. M. Band, “Coefficients of Internal Conversion.’’ Acad. 8ci. U.S.S.R., Moscow, 1956 and 1958; translated as Reports 57-ICC-K1 and 59-ICC-L, Department of Physics, University of Illinois, Urbana.
2.
164
DETERMINATION OF FUNDAMENTAL QUANTITIES
conversion line ratios can be measured most easily, but are often not very sensitive to the multipolarity (see Fig. 1). The L subshell ratios give much more information (see Fig. 2), but they can only be measured with instruments of high resolving power. IOC
2-
K / L ratios
Gamma Energy
FIQ.1. Ratio of K and L conversion lines for different pure multipole transitions, for Z = 79. At other nuclear charges, the same K / L ratios appear at approximately the same values for E / Z z .
2.4.2.2.2. DETERMINATION OF BETA AND GAMMA RAY INTENSITIES, The most natural way for obtaining conversion coefficients is to measure electron intensities in a beta ray spectrometer and gamma rays in a gamma ray spectrometer. Measurements of relative electron intensities in a magnetic or electric beta ray spectrometer with moderate precision offers no difficulties, since in a well designed instrument of this type the efficiency is nearly independent of the energy (see Section 2.2.1.1). It is, however, necessary t o take several precautionss if high precision is desired. 8 See, e.g., A. H. Wapstra, “Report on the Israel Conference on Nuclear Physics.” North-Holland Publ., Amsterdam, 1957.
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
155
2-
I
0.2
1.0E
,
I
0.4
0.3
.
I
I
I
I
l
0.5 0.7 Gamma Energy
l
I
I MeV
2
3-
#/-
0.2
I
I
0.3
0.4
t
1
I
I
I
l
0.7 Gamma Energy 0.5
l
I MeV
I
1
2
2-
=--_-_ ---------------
0.01
I
I
I
I
I
I
-------_
MI I
I
l
l
I
156
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
For measuring absolute intensities, beta ray spectrometers can be calibrated using sources with simple spectra, the strength of which has been determined using 4~ counters or coincidence techniques. In cases with few, well separated electron lines, electron intensities may be measured in proportional or scintillation counter spectrometers, or with the new solid state detectors (see Sections 1.3.1, 1.4.1, and 1.8.1). Efficiencies of gamma spectrometers vary with energy in a way which, in many cases, cannot be computed with the necessary accuracy. These instruments have, therefore, t o be calibrated even for measurements of ratios of intensities of gamma ray lines with different energies. The most accurate measurements can be obtained, in principle, with crystal diffraction spectrometers.9 Such instrhments demand very strong sources. The more popular scintillation spectrometer needs much weaker sources, but has a much lower resolving power. Even for sources with few, well separated gamma lines, intensity measurements in a scintillation spectrometer with a precision of 5 % or better are mostly hard to make due to the difficulty of separating the lines from the Compton continua, caused by lines of higher energies in an unambiguous way. Secondary electrons emitted by a thin “converter” placed at the source position of a beta ray spectrometer and irradiated by the gamma ray source can be measured as much sharper lines than those found in scintillation spectrometer (see Section 2.2.3.1.2). By choosing a converter of material with high atomic number, the ratio of the intensities of the photo lines and the Compton continua can, moreover, in principle be made larger than is possible in scintillation spectra. Uranium converters are most suitable, but for many purposes lead converters, which can be obtained more easily, are sufficient. Sources for this type of measurement should be about a hundred times stronger than those used in scintillation spectrometry. In the computation of gamma ray intensities from the measured photopeak areas, one needs to know the photo cross sections as a function of the direction at which the photo electron is emitted. Habitually, the influence of this function is expressed as the product of the total photo cross section T Y and an angular distribution factor fY, so that the photopeak area A Y (if plotted as number of electrons per momentum interval versus momentum) becomes A y = 1 7 T y f y db Q the quantity d being the thickness of the converter and b a dimensional factor, and Q being the solid angle of acceptance of the spectrometer used. Values for the total cross sections can be derived from the tables of White 9
See Vol. 5, A, Section 2.2.3.2.
2.4.
DETERMINATION O F S P I N , PARITY, .4ND NUCLEAR MOMENTS
157
Grodstein or from older ones by Davisson.’O The recent work of Hultberg and his collaborators on the computation of the angular distribution factor fK is extremely important for the use of this method in intensity measurements. Yet stronger sources are necessary if one wants t o measure Compton lines,12 since then the gamma rays have to be collimated before they strike the converter, which in this case is preferably of low 2 material. The last method has the advantage that it produces only one line for every gamma ray. 2.4.2.2.3. DETERMINATION O F CONVERSION COEFFICIENTS; SPECIAL METHODS. Accurate measurements of conversion coefficients can be made in special cases, e.g., a single y transition preceded by a single /3- transition. The conversion coefficients can then be found by measurement of the intensities of the conversion lines and the beta continuum in a beta spectrometer: 1/aK
=
/3/K
- (K +L + M
+
. . .)/K
(2.4.2.2.1)
The symbols K , L , M , . . . stand for the intensities of the K,L,M, . . . conversion electrons. The intensity /3 of the beta continuum is mostly obtained by extrapolating the high energy part of the spectrum t o lower energies with help of the theoretical spectrum shape, since the measured spectrum is always distorted at low energies by counter window absorption and by the occurrence of scattered electrons. This extrapolation can, in allowed spectra, be made most easily with help of a Fermi plot. Forbidden spectra in the so-called “unique” class have also well-known shapes.I3 The other forbidden beta spectra have shapes depending on the accidental values of some nuclear matrix elements which are unknown. This fact introduces an additional uncertainty in the measurement of the lo G. White Grodstein, Natl. Bur. Standards (U.S.) Circ. No. 683 (1957); C. M. Davisson, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 857. Interscience, New York, 1955. 1 1 S. Hultberg, Nuclear Instr. & Methods 10, 24 (1960); S. Hultberg, D. Horen, and J. Hollander, Nuclear P h p . 28, 471 (1961); W. F. Frey, J. H. Hamilton, and S. Hultberg, Arkiv Fysik 21, 383 (1962). 12 M. Mladjenovic and A. Hedgran, Arkiv Fysik 8, 49 (1954); B. S. Dzhelepov et al., Nuclear Phys. 2, 408 (1956). 13 N. Dismuke, M. E. Rose, C. L. Perry, and P. R. Bell, U.S. Atomic Energy Comm. Rept. 0.R.N.L.-1222 (1952); M. E. Rose, C. L. Perry, and M. M. Dismuke, U.S. Atomic Energy Comm. Rept. 0.R.N.L.-1459 (1953); K. Siegbahn, ed., “Beta- and Gamma-Ray Spectroscopy,” cf. p. 875. Interscience, New York, 1955; A. H. Wapstra, G. J. Nijgh, and R. van Lieshout, “Nuclear Spectroscopy Tables.” Interscience, New York, 1959.
158
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
intensity of the continuous beta spectrum, though in the most frequent case of first forbidden transitions, the deviations are mostly small.14 The influence of this uncertainty on the conversion coefficients can in principle be avoided by measuring the electron spectrum in a beta ray spectrometer in coincidence with K X-rays due to internal conversion; the X-rays may be detected by a scintillation counter. In the special case discussed above we now find 1 / a ~=
TL,-K/TLSK
- (K
+L + M +
*
*
.)/K
(2.4.2.2.2)
in which npg indicates the number of coincidences per nuclear electron selected by the beta spectrometer (of course corrected for backgrounds and accidental coincidences), and n,-K the same per K conversion electron. Equations (2.4.2.2.1) and (2.4.2.2.2) are rather similar; the difference is that, using Eq. (2.4.2.2.2),n g K can be measured at places in the beta spectrum where the influence of disturbing effects and backgrounds is lowest. In the same simple cases, the K conversion coefficient may be obtained by comparing the gamma ray intensity with that of the K X-rays caused by the conversion process: a~ =
(E,/EK)~K/TWK
where X K and y represent the photopeak arcas in the scintillation spectrum, W K is the fluorescent yield and E, and EK are the respective efficiencies for detecting the gamma and the X-rays in the photopeak of the scintillation spectra. Sources used in this method should be thin enough to avoid absorption or fluorescent excitation of X-rays. The ratio E J E K can be computed accurately if the gamma energy is so low that the mean free path for absorption is smali compared with the dimensions of the counting crystal. I n this case, this method is capable of high precision. A somewhat more involved coincidence method was applied by Petterson et aZ.16on Aulg8.They measured coincidences between the 412 kev gamma rays and the nuclear electrons in Aul9s, and between 134 kev gamma rays and the L conversion line of the 165 kev transition in Hglg7 (decay schemes see Fig, 3). In this case,
in which n are again the number of coincidences per detected gamma ray; N ~ ~and Z KNp are the count rates of nuclear and conversion electrons, at the top position of the conversion line and at the point where the Py coinl4
l6
A. H. Wapstra, Nuclear Phya. 9, 519 (1958). B. G.Petterson, J. E, Thun, and T,R. Gerholm, NucZeur Phys. 24,243 (1961).
2.4.
DETERMINATION
OF SPIN, PARITY, A N D NUCLEAR MOMENTS
159
cidences were taken respectively. The factor c is the L conversion probability of the 165 kev line except for a 4 % correction for the difference between the line shapes of the 412K line and the complex 134L line. Another met,hod was used by Lewin et a1.16 in a study of TIzoo.They compared coincidences between the group of 1200 kev lines (see Fig. 3) preceding the 368 kev transition and the K conversion line of the last TLOO
ffl
Au 198
7
E @ I97
h
' .
O+
O+ ~ 1 9 8
YZ+
HqKX>
p" ~1203
FIQ. 3. Decay schemes of Hg197, Aulgg, Tlzo'J, and HgZoa,the middle two partly simplified. The values give transition energies in kev.
transition with coincidences between the 412 kev gamma rays and the nuclear electrons in Aulg8.Then, l / a ~= c(nsr41z/n,-,izoo)(n,/ns>
-
(K
+L + M + .
*
.)/K.
Here, ne is the ratio of top counting rate and area of a monoenergetic electron line, and ng is the ratio of the counting rate of nuclear electrons in AulS8a t the point where nar is measured and the area of the continuous beta spectrum. The quantities ng, and ne-,are the numbers of coincidences per recorded gamma ray; the factor c, nearly unity, corrects for the dependence of the coincidence and counting efficiencies on th e electron energy, and for the fact that the actual decay schemes are somewhat more involved than depicted in Fig. 3. I n complicated cases, some of the above methods cannot be applied t o yield accurate results. Then, the method of comparison of external and internal conversion lines may become very useful. If these lines are measured in the same spectrometer, both are proportional t o the same solid angle, which therefore does not occur in the final result allowing higher precision. l6
W. H. G. Lewin, B. van Nooijen, and A. H. Wapstra, Nuclear Phys. 27, 681
(1961).
160
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
If no strong enough sources are available to allow external conversion measurements, one still has not necessarily to know the absolute efficiencies of electron and gamma spectrometers. Often, a conversion coefficient of one of the transitions in the nuclide studied may be known; it can then be used to calibrate the electron and gamma counters relative to one another. Or, they may be relatively calibrated by measuring a standard nuclide in both. In nuclides decaying by negaton decay alone, the total amount of K X-radiation should be equal to the number of K conversion electrons multiplied by the fluorescent yield.17 Then, the ratio of K Auger electrons to that of K X-rays is known with considerable accuracy. And in nuclides emitting positrons, two annihilation quanta of 511 kev are emitted for every positon. Thus, one of the last three possibilities may be used in calibrating beta and gamma spectrometers relative to one another in order to determine conversion coefficients. In cases where the above methods fail, one may adopt a theoretical conversion coefficient for one of the transitions studied. Thus, the transition between the first excited 2+ state and the ground state in even-even nuclei is necessarily a pure E2 transition; and its conversion coefficient is often found to agree well with theory (though a few real exceptions may exist la). 2.4.2.2.4. EXAMPLE OF USEOF CONVERSION DATA.As a rather typical example for the use of conversion coefficient measurements we consider the case of HgZo3. This nuclide decays as depicted in Fig. 3 to a 279.12 kev excited level in TlZo3 decaying to the ground state by a single transition; the conversion coefficients can therefore be measured by the special methods outlined above. Accurate meas~rernents’~ yielded the results given in Table I; all experimental data have errors of only a few per cent. Theoretical values for the conversion coefficients of pure multipole transitions of a 279.12 kev gamma transition in TI are also given in Table I. Since parity is a good quantum number for electromagnetic transitions, only those multipole transitions can occur mixed in which the change in parity is the same (the upper and lower parts in Table I, respectively). The conversion coefficient for a mixture of two multipolarities is a! = aa1 (1 - a)a!,
+
1’ A. H. Wapstra, G. J. Nijgh, and R. van Lieshout, “Nuele&rSpectroscopy TabIes.” Interscience, New York, 1959. 18 J. F. W. Jansen, S. Hultberg, P. F. A. Goudsmit, and A. H. Wapstra, Nucleaf Phys. (to be published). 1 9 G. J. Nijgh, A. H. Wapstra, L. Th. M. Omstein, N. Salomons-Grobben, J. R. Huizenga, and 0. Almh, Nuclear Phys. 9,528 (1958), and other work discussed there.
2.4.
DETERMINATION O F SPIN, PARITY, AND NUCLEAR MOMENTS
161
and LYZ being the theoretical conversion coefficients for these multipolarities and a the percentage gamma rays of the first type (a is therefore the same for conversion in different shells). Comparison of the measured K / L ratio (the most easily obtained datum) with the theoretical values shows that the transition is a combiLYI
TABLEI. Conversion Coefficients ( x 104) and Ratios for the 279 kev Transitions in T I 2 0 3 (Theoretical data for transitions in which the parity does not change are printed above the experimental data printed in bold-face type, the other below. All theoretical data are obtained from Rose's tables6 except those printed in italics which are due to Sliv.') Conversion ratio
fif 1
41 10 3920
E2 M3
Exp. El M2 E3
764 770 41200 1620 278 14650 2060
646 684 479
586
55
616
64
108
618
110
19700 480 46 3890 4530
12700 242 34 3170 499
243 271 2400 166 7 434 2890
5 5 127 137 4550 82 6 291 1135
6.37 6.76 1.60 1.49
2.10 3.37 6.08 3.76 0.45
0.907
0.086
0.900
0,093
0.226 0.212 0.647
0.508 0.623 0.122
0.604 0.732 0.814 0.110
0.326 0.145 0.111 0.639
0.007 0.007 0.266 0.265 0.231 0.171 0.123 0.075 0.251
TABLE 11. K Conversion Coefficients and L Subshell Ratios Giving the Correct K / L Ratio for the 279 kev Transition in T l * ' J 3
+ + M2 + E3, Rose
E2, Rose MI E2, Sliv Experimenta1 MI
OIK
Lr / L
LllIL -
LIIIIL
0.179
0.479 0.517 0.604 0.732
0.351 0.332
0.170 0.151
0.326 0.173
0.171 0.095
0.196
0.162 1.33
+
+
nation of either M 1 E2, or M 2 M3. The theoretical K conversion coefficients and the LI :LII :L I I Iratios for the mixtures yielding the correct K / L ratio are given in Table 11. Comparison with the experimental values E 2 mixture. The ground state has a spin points very strong t,o a M 1 and positive parity; our result therefore proves that the 279 kev level in TI2O3has a spin and positive parity. The theoretical values in Table I1 do not agree completely with the experimental result, probably due to the dynamic finite size effect discussed in the beginning of this section. According t o an estimate by Kiss-
+
+
162
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
linger,20the influence of this correction would decrease the theoretical value of LYK by about 10%. Unfortunately, this arbitrariness in the theoretical conversion coefficients for pure multipoles hampers an accurate determination of the multipole mixing ratios a from conversion coefficients alone: the values for a as computed from Rose’s theoretical values and the experimental K , LI, LII, and LIII conversion coefficients are 0.256 0.012, 0.280 k 0.013, 0.46 f 0.03, and 0.37 k 0.03, respectively. One would be inclined to reason that the LIII conversion coefficients are least affected by the dynamic finite size effects since the LrII electron wave function has a minimum at the nucleus. The value for a as obtained above from the LIII result agrees indeed fairly well with the result a = 0.31 5 0.025 found by McGowan and StelsonZ1from angular distribution measurements. The theoretical results of Sliv lead to a value a = 0.42 f 0.03, in somewhat worse agreement with McGowan and Stelson’s result.
2.4.2.3. Nuclear Orientation.* 2.4.2.3.1. INTRODUCTION. A nucleus of total angular momentum I is oriented when the ( 2 1 1) magnetic substates have unequal occupational probabilities. Such a nucleus defines a certain spatial anisotropy and, when subsequent emission or absorption processes occur, this anisotropy is carried over into the final state as a consequence of the conservation laws involved. The detailed theories of these effects are closely related to those for angular correlations, and both can be treated by the same very general and refined
+
L. S. Kisslinger, Bull. Am. Phys. SOC.2, 358 (1958). F. K. McGowan and P. H. Stelson, Phys. Rev. 107, 1647 (1957);and private communication. 1 L. C. Biedenharn and M. E. Rose, Revs. Modern Phys. 26, 729 (1953). F. Coester and J. M. Jauch, Helv. Phys. Acta 40, 3 (1953). a S. Devons and L. J. B. Goldfarb, i n “Handbuch der Physik-Encyclopedia of Physics” (S. Flugge, ed.), Vol. 43, p. 362. Springer, Berlin, 1957. M. E. Rose, “Elementary Theory of Angular Momentum.” Wiley, New York, 1957. 5 U. Fano and G. Raoah, “Irreducible Tensorial Sets.” Academic Press, New York, 1959. * M. Jacob and G. C.Wick, Ann. Phys. (N.Y.) 7,404 (1959). 7 L. J. B. Goldfarb, i n “Nuclear Reactions” (P. M. Endt and M. Demeur, eds.). North-Holland Publ., Amsterdam, 1959. 8 L. C. Biedenharn, in “Nuclear Spectroscopy” (F. Ajzenberg-Selove, ed.), Part B, Chapter V.C. Academic Press, New York, 1960. 9 S. R. de Groot and H. A. Tolhoek, in “Beta and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 613. Interscience, New York, 1955. $0
$1
-
* Section 2.4.2.3is by E. Ambler.
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
163
Indeed, it is appropriate to regard nuclear orientation and angular correlation as special cases of the most general observation, where a complete set of observations of the angular distribution and polarization of particles and radiation is made. Of course, for any particular purpose, e.g., for extracting given quantities of physical interest, many of these observations would be irrelevant or redundant, and one would have to select those which are appropriate. In particular, one would need to determine whether it is a n advantage to know the orientation of the initial nucleus. At this point we should note that there are two types of nuclear orientation: nuclear polarization, where there is a net nuclear moment; and nuclear alignment, where orientations parallel and antiparallel to a given axis are preferred over other orientations although no net moment exists. The experiments that have been carried out to date involve mainly the orientation of radioactive nuclei and the observation of the angular distribution of the decay products, although work in other fields, e.g., that of nuclear reactions, has begun. It should be remembered, however, that the method is an indirect one, and quantities such as spins and parities are seldom determined in a straightforward and unambiguous way starting from a completely unknown situation. The experimental method separates naturally into the following parts: (I) the production of oriented nuclei; (2) the determination of the degree of nuclear orientation produced; (3) the measurement of the relevant angular distributions, including in some cases polarization or correlation measurements; (4) the computation of theoretical curves and their subsequent comparison with experiments.
As a general rule, the measurement of spins and parities is possible simply because, for a given degree of nuclear orientation, angular' distributions for certain spin sequences can be fit to the data and others cannot. The measurement of nuclear moments, on the other hand, is possible when a decay scheme is known so that the degree of nuclear orientation, which is directly related to a nuclear moment, can be determined. The method is entirely reliable for the measurement of spins provided sufficient initial information concerning the decay scheme is available. The same can hardly be claimed for the measurement of nuclear moments, where the sensitivity is seldom such as to give accuracies better than lo%, and where agreement with more direct determinations is often not very good. The method has the advantage that it can be carried out with radioactive nuclei of which only a small quantity is available.
2.
164
DETERMINATION O F FUNDAMENTAL QUANTITIES
TABLE I. Experiments Using Oriental Nucleia Nucleus
Ref.
So46 Cr61 Mn 62 Mn MnKZ MnKZ Mn 6zm MnK4 Mn6* MnK4 MnK6 Mn66 Mn 68 Mn 68
1 2 3 4
co
C066
co
co CO67 co co co Cob8 C068
co
c o 80 coeo
co
ED
COSO CoEO
co co 80 go
Co60 As16 In114m
In116 SblZ2 Sblz2 I131
Cela7 ce1a7m
Cela9 Cela9 CeI4l Cel4I
Method S6
5 6
53 53 53 s3 53
7
53
8 9 7
53 53 s3 53
10 11
12 13
14 13 15,16 17 18 19-21 22 23 24 6 17 25
20,26 22 27 28 29-31 1 24,32,33 34 35 36,37 35 38,39 40 41 41 42 43 43 44
52
53 53 s3
s3 82 53 52 s2 s2 s2
53
s3 53 s3 s2 52 53 53 s5 S6 53 D5 S6
s1 S6
D 4 & D5 54 52 s2 52 52 s2 s2
Substance
Observation
Iron alloy “Brine holes” in CMN CMN CMN CMN CMN CMN CMN CMN CMN Mn (ND4)2(S04)2.6D~0 NFS CMN CMN CMN CMN CU(NHdz(S04)~.6HzO CMN CU(NHJ z (SO,) 2.6H20 CU(NHdz(S04)~6HzO CU(NHAz(S04)2.6H20 NFS CMN CMN CMN CMN CU(NH~Z(SO~)Z.~HZ~ CU(NH,)z(SOdz.6HzO CMN CMN Co metal Iron alloy CMN As in silicon Iron alloy Metal Iron alloy Sb in silicon Co (Zn)p-t,p-i NES NES CES CMN CMN CES
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
165
TABLE I (Continued) Nucleus
Ref.
Method
Substance
45 46 47 13 48 49 43 50 51 52 53 54 55,59 56 56 57 58 59 60 35 61 61 61,62
s2 & s 3 s2 & 53 s2 s3 s2 S2,S3 s2 52 s2 & S3 s3 Sl S2,S3 S2,S3 s2 s2 s3 52 s3 52 S6 s4
NES NES CES CMN N ES NES CMN NES NES SES Metal NES NES NES NES HES NES NES YES Iron alloy URN URN URN
54 S2$4
Observation
a Column one refers to the nucleus oriented, column two to the original publication, column three to the method used (cf. Table 11), column four to the substance used, and column five to the observations made. In the fourth column we have used the following abbreviations: CMN = cerous magnesium nitrate, NFS = nickel fluosilicate, Co (Zn)p-t,p-i = cobalt (zinc) p-toluenesulfonate, p-iodobenzenesulfonate, NES, CES, SES, HES, YES = neodymium, cerium, samarium, holmium, ytterbium ethyl sulfates respectively, and URN = uranyl rubidium nitrate. I n the fifth column the angular distribution of alpha, beta, and gamma rays is denoted by w(a,q), w(p,q), and w(k,v) respectively; w(k,v)& and w(k,q).5 refer to plane polarization and circular polarization measurements of gamma rays respectively; an refers to a measurement of slow neutron capture cross section; w(p,k,v) refers to beta-gamma angular correlation; w(kl,k2,q)to gamma-gamma angular correlation, and w(f,q) to the angular distribution of fission fragments.
REFERENCES TO TABLE I 1. A. V. Kogan, V. D. Kul'kov, L. P. Nikitin, N. M. Reinov, I. A. Sokolov, and
M. F. Stel'makh, Zhur. Eksptl. i Teoret. Fiz. 99, 47 (1960); Soviet Phys. JETP 12, 34 (1961). 2. M. Kaplan and D. A. Shirley, Phys. Rev. Letters 6, 361 (1961). 3. W. J. Huiskamp, M. J. Steenland, A. R. Miedema, H. A. Tolhoek, and C. J. Gorter, Physica 22, 587 (1956).
166
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
REFERENCES TO TABLE I (Continued) 4. W. J. Huiskamp, A. N. Diddens, J. C. Severiens, A. R. Miedema, and M. J. Steenland, Physica 23, 605 (1957). 5. E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys. Rev. 110,787 (1958). . . 6. H. Postma, W. J. Huiskamp, A. R. Miedema, H. A. Tolhoek, and C. J. Gorter, Physim 24, 157 (1958); H. Postma, Thesis, University of Leiden, Netherlands,
unpublished. 7. R. W. Bauer, M. Deutsch, G. 8. Mutchler, and D. G. Simons, Phys. Rev. 120, 946 (1960). 8. M. A. Grace, C. E. Johnson, N. Kurti, H. R. Lemmer, and F. N. H. Robinson, Phil. Mag.[71 46,1192 (1954). 9. G. R. Bishop, J. M. Daniels, H. Durand, C. E. Johnson, and J. Perez y Jorba, Phil. Mag.L71 46,1197 (1954). 10. S. Bernstein, L. D. Roberts, C. P. Stanford, J. W. T. Dabbs, and T. E. Stephenson, Phys. Rev. 94, 1243 (1954); J. W. T. Dabbs and L. D. Roberts, ibid. 96, 970 (1954). 11. P. Dagley, M. A. Grace, M. Gregory, J. S. Hill, and C. V. Sowter, reported by N. Kurti, Physica Suppl. S164 (September, 1958). 12. R. W. Bauer and M. Deutsch, Phys. Rev. 117, 519 (1960). 13. H. Postma and W. J. Huiskamp, PTOC.7th Intern. Conf. Low Temp. Phys. p . 183 (1961). 14. R. W. Bauer and M . Deutsch, Nudear Phys. 16, 264 (1960). 15. L. J. Gallaher, C. Whittle, J. A. Beun, A. N. Diddens, C. J. Gorter, and M. J. Steenland, Physica 21, 117 (1955). 16. 0. J. Poppema, J. E. Siekman, and R. van Wageningen, Physica 21, 223 (1955). 17. E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys. Rev. 108,503 (1957). 18. G. R. Bishop, M. A. Grace, C. E. Johnson, A. C. Knipper, H. R. Lemmer, J. Perez y Jorba, and R. G. Scurlock, Phil. Mag. [7] 46, 951 (1951). 19. J. M. Daniels, M. A. Grace, H. Halban, N. Kurti, and F. N. H. Robinson, Phil. Mag.[7] 43, 1297 (1952). 20. J. C. Wheatley, D. F. Griffing, and R. D. Hill, Phys. Rev. 99,334 (1955). 21. D. F. Griffig and J. C. Wheatley, Phys. Rev. 104, 389 (1956). 22. G. R. Bishop, J. M. Daniels, G. Goldschmidt, H. Halban, N. Kurti, and F. N. H. Robinson, Phys. Rev. 88, 1432 (1952). 23. P. Dagley, M. A. Grace, J. S. Hill, and C. V. Sowter, Phil. Mag. [S] 5, 489 (1958). 24. E. Ambler, R. W. Hayward, D. D. Hoppes, R. P. Hudson, and C. 6. Wu, Phys. Rev. 106, 1361 (1957). 25. E. Ambler, M. A. Grace, H. Halban, N. Kurti, H. Durand, C. E. Johnson, and H. R. Lemmer, Phil. Mag. [7] 44, 216 (1953). 26. B. Bleaney, J. M. Daniek, M. A. Grace, H. Halbaa, N. Kurti, F. N. H. Robinson, and F. E. Simon, Proc. Roy. Soc. A221, 170 (1954). 27. P. S. Jastram, R. C. Sapp, and J. G. Daunt, Phye. Rev. 101, 1381 (1956). 28. J. C. Wheatley, W. J. Huiskamp, A. N. Diddens, M. J. Steenland, and H. A. Tolhoek, Physica 21, 841 (1955). 29. M. A. Grace, C. E. Johnson, N. Kurti, R. G. Scurlock, and R. T. Taylor, in “ConfBrence de physique des basses temphratures,” p. 263. Centre National de la Recherche Scientifique and UNESCO, Paris, 1956; BuZl. Am. Phys. Soc. 2, 136 (1957).
2.4.
DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS
167
REFERENCES TO TABLEI (Continued) 30. G. R. Khutsishvili, Zhur. Eksptl. i Teoret. Fiz. 29, 894 (1955); Soviet Phys. J E T P 2, 744 (1956). 31. J. M . Daniels and M. A. R. LeBlanc, Can. J . Phys. 37, 1321 (1959). 32. C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys. Rev. 106, 1413 (1957). 33. C. E . Johnson, M. A. Grace, R. G. Scurlock, and C. V . Sowter, Phil. Mag. [8] 2, 1050 (1957). 34. F. M. Pipkin and J. W. Culvahouse, Phys. Rev. 109, 1423 (1958). 35. B. N. Samoilov, V. V. Sklyarevskii, and E. P. Stepanov, Zhur. Eksptl. i Teoret. Fiz. 38, 359 (1960); Soviet Phys. J E T P 11, 261 (1960). 36. J. W. T. Dabbs, L. D. Roberts, and S. Bernstein, Phys. Rev. 98, 1512 (1955). 37. A. Stolovy, Phys. Rev. 118, 211 (1960). 38. F. M. Pipkin, Phys. Rev. 112, 935 (1958). 39. G. E. Bradley, F. M. Pipkin, and R. E. Simpson, Phys. Rev. 123, 1824 (1961). 40. C. E. Johnson, J. F. Schooley, and D. A . Shirley, Phys. Rev. 120, 1777 (1960). 41. J. N. Haag, C . E. Johnson, D, A. Shirley, and D. H. Templeton, Phys. Rev. 1% 591 (1961). 42. M. A. Grace, C. E. Johnson, R. G. Scurlock, and R. T. Taylor, Phil. Mag.,to be published. 43. E. Ambler, R. P. Hudson, and G . M . Temmer, Phys. Rev. 101, 196 (1956). 44. C. F. M. Cacho, M. A. Grace, C. E. Johnson, A. C. Knipper, R. G. Scurlock, and R. T. Taylor, Phil. Mag. 171 46, 1287 (1955). 45. D. D. Hoppes, E. Ambler, R. W, Hayward, and R. S. Kaeser, Phys. Rev. Letters 6, 115 (1961). 46. D. D. Hoppes, Natl. Bur. Standards (US.)Rept. 93 (unpublished). 47. M. A. Grace, C. E. Johnson, R. G. Scurlock, and R. T. Taylor, Phil. Mag. [81 3, 456 (1958). 48. C.A. Lovejoy, J. 0. Rasmussen, and D. A . Shirley, Phys. Rev. 123, 954 (1961). 49. D. A. Shirley, J. F. Schooley, and J. 0. Rasmussen, Phys. Rev. 121,558 (1961). 50. G. R. Bishop, M. A. Grace, C. E. Johnson, H. R. Lemmer, and J. Perez y Jorba, Phil. Mag. [Sl 2, 534 (1957). 51. G. A. Westenbarger and D. A. Shirley, Phys. Rev. 123, 1812 (1961). 52. L. D. Roberts, S. Bernstein, J. W. T. Dabbs, and C. P. Stanford, Phys. Rev. 95, 105 (1954). 53. A. Stolovy, Bull. Am. Phys. SOC.[II] 6 (3), 275 (1961). 54. C. A. Lovejoy:andLD. A . Shirley, Proc. 7th Intern. Cmf. Low Temp. Phys. p. 164 (1961). 55. C. E. Johnson, J. F. Schooley, and D, A. Shirley, Phys. Rev. 120, 2108 (1960). 56. Q. 0. Navarro and D. A . Shirley, Phys. Rev. 123, 186 (1961). 57. H. Postma, H. Marshak, V. L. Sailor, F. J. Shore, and C. A . Reynolds, Bull. Ant. Phys. SOC.[I11 6 (3), 275 (1961). 58. H. Postma, A. R. Miedema, M. C. Eversdijk Smulders, Physiea 26,671 (1959). 59. H. Postma, M. C. Eversdijk Smulders, and W . J. Huiskamp, Physica 27, 245 (1961). 60. M. A. Grace, C . E. Johnson, G. R. Scurlock, and R.:T. Taylor, Phil. Mag. [81 2, 1079 (1957). 61. J. W. T. Dabbs, L. D. Roberts, and G. W. Parker, Physica Suppl. 569 (September 1958); Proe. 7th Intern. Cmf. Low Temp. Phys. p. 174 (1961). 62. S. H. Hanauer, J. W. T. Dabbs, L. D. Roberts, and G. W. Parker, ORNL Report 2919 (unpublished).
168
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
It is clear that the analysis of the data of each experiment will be very different in detail, and that the results of such analysis may be complicated. It would be neither possible nor appropriate, therefore, to redescribe all the experiments that have been done to date. We have tried to list them concisely in Table I, however, so that the reader may refer to the original papers for important and special details. The aim of the greater part of the article is to give an account of the general physical principles, some of the special techniques, and some of the mathematical methods of the theory that are fast becoming developed to the point where the experimentalist might be expected to use them. The reader will find useful a number of review articles.1°-17 2.4.2.3.2. THEPRODUCTION O F O R I E N T E D NUCLEI-STATICMETHODS. We may divide into three groups the methods proposed for orienting nuclei, and call them the static, dynamic, and transient methods. We have listed these methods together with certain information about them in Table I1 (a), (b), and (c) and have numbered them S1, S2, etc., for easy reference. The basic ideas of the static methods are as follows. The degeneracy of the ( 2 1 1) spatial orientations of the nucleus can be removed either by a coupling of the nuclear magnetic moment to a magnetic field or by a coupling of the nuclear quadrupole moment t o an electric field gradient. If the nuclei are then cooled to a temperature, T , such that kT ( k is Boltzmann’s constant) is of the order of the hyperfine splitting, the lower levels, which correspond to preferred orientations of the nuclear spin, become preferentially populated. The use of external electric fields is not possible since the magnitudes required exceed the dielectric strength of the substances to which they must be applied. The use of a n external
+
10 R. J. Blin-Stoyle and M. A. Grace, i n “Handbuch der Physik-Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 42, p. 555. Springer, Berlin, 1957. 1lR. J. Blin-Stoyle, M. A. Grace, and H. Halban, i n “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 600. Interscience, New York, 1955. 12 M. J. Steenland and H. A. Tolhoek, i n “Progress in Low Temperature Physics” (C. J. Gorter, ed.), Vol. 11, p. 293. North-Holland Publ., Amsterdam, 1957. 13E. Ambler, in “Progress in Cryogenics” (K. Mendelssohn, ed.), Vol. 2, p. 235. Heywood, London, 1960. 14 C. D. Jeffries, in “Progress in Cryogenics” (K. Mendelssohn, ed.), Vol. 3, p. 129. Heywood, London, 1961. 14& G. R. Khutsishvili, U s p e k h i Piz. Nauk 71, 9 (1960); Soviet P h y s . - U s p e k h i 3, 285 (1960). R. P. Hudson, in “Progress in Cryogenics” (K. Mendelssohn, ed.), Vol. 3, p. 99. Heywood, London, 1961. 16 N. Kurti, Nuouo cimenfo Suppl. [lo] 6, 1101 (1957). 17 W. J. Huiskamp and H. A. Tolhoek, in “Progress in Low Temperature Physics” (C. J. Gorter, ed.), Vol. 111, p. 333. North-Holland Publ., Amsterdam, 1961.
2.4.
DETERMINATION
OF SPIN,
PARITY,
AND NUCLEAR MOMENTS
169
magnetic field, on the other hand, provides a most direct and general way of orientating nuclei. Although it has been used successfully to a certain extent, rapid development of this so-called “brute-force” method has been hindered largely by the slow technical development of reliable steady high magnetic fields. With the continued development of high power solenoids, and the further possibility of high field superconducting solenoids, however, this situation is likely to change, and since the method has such a wide applicability we shall discuss it in some detail. We shall then discuss the other methods that have been proposed, which rely upon the presence in certain compounds of high internal fields, i.e., large hyperfine splitting. Some of the methods in this last group have met with considerable success. 2.4.2.3.2.1. External Field Polarization-El. The feasibility of the method in principle,’* and the appreciation of the practical problems in detai1,lg have long been known. More recent and extensive studies have allowed the method to be used in a significant although limited way.20-22 If an assembly of nuclei of magnetic moment p is coupled to an external field H a t temperature T , the nuclear polarization is given by: fl
= &(X)
=
21
+
1 ___ coth 21
(T + 21
1
1
x) - 2f coth
($x )
(2.4.2.3.1)
where &(x) is the Brillouin function and x = p H / k T . The value of f l is the same as the ratio of the nuclear magnetization to that a t saturation, and this quantity has been tabulated already in connection with electronspin m a g n e t i ~ m . 2 ~If~we 2 ~ adopt the figure of f l = 0.2 as being a reasonably large nuclear polarization for carrying out experiments, we find that for nuclear magnetic moments normally encountered, we require H I T = 106-107 gauss/”K. Now, as we shall discuss later, a value of T = 0.05”K can be obtained fairly easily, whereas a value of T = 0.01”K should be possible; therefore, magnetic fields in the range 104-106 gauss are suitable. As an illustration, we take what we believe to be a reasonable figure, viz., H / T = 4 X lo6 gauss/”K (e.g., 100 kilogauss a t 0.025”K) F. E. Simon, Conrpt. rend. ccungr. sur le magnt%isme, Strasbourg, 1939,p. 1 (1939). N. Kurti, “Les phknombnes cryomagn&iques,” p. 29. Coll&gede France. Paris, 1948. 20 J. W. T. Dabbs, L. D. Roberts, and S. Bernstein, ORNL Report, central files number 55-5-126(un-published). 2 1 J. W. T. Dabbs, L. D. Roberts, and S. Bernstein, Phys. Rev. 98, 1512 (1955). 22 A. Stolovy, Phys. Rev. 118, 211 (1960). 2s J. R. Hull and R. A. Hull, J. Chem. Phys. 9, 465 (1941). 24 L.P. Schmid and J. S. Smart, LLTables of Some Thermodynamic Functions which Occur in the Theory of Magnetism.” Navard Report 3640, U.S. Naval Ordnance Laboratory, White Oak, Maryland. 1s 19
TABLE 11. List of Methods Proposed for Orienting Nuclei (a). Static Methods of Orienting Nuclei Method
No.
Ref.
External field polarization
s1
1-3
Magnetic hfs alignment
52
4
w
l
0
Magnetic hfs polarization Electric hfs alignment
S3
5,6
s4
7
Ferromagnetics and anti-ferromagnetics
55
8-10
Dilute solid solutions in above
S6
10-13
Criteria
-
&upling p"H.1 HIT lohi07 gauss per degree K Coupling AS,Z,or
+
B(SJ, S J d S = 4 and A >> B or B >> A S>&andA,B#O Z > $ As above, but S = $ and A = B included P#O
Large enough hfs; coupling for alignment as in S2 and for polarization as in 53 As above
Applicability
All nuclei with pn. Long nuclear relaxation times may limit method to metals and alloys Paramagnetic compounds, e.g.9 Ce2Mg,(N03)1~24H20 Nd(C,H,SO,)a.SH,O Rare earth metals (See
Special Requirements Adiabatic demagnetization and contact cooling. Poor thermal conductivity is a severe problem Adiabatic demagnetization, usually
55) As above
Covalently bonded complexes, e.g., UOZ in (U02)Rb(N03)r High density of relevant nuclei is a very important advantage for work on nuclear reactions Whenever mechanism (e.g. diamagnetic core polarization in iron alloys) producing large hfs is operative
As above, but external field
102-103 gauss needed Temperatures < 1"K, usually
As above
As above
(b). Dynamic Methods of Orienting Nuclei Method Optical pumping
No.
Ref.
Criteria
D1
1
Resonance with low-field Zeeman splitting; F = I S, is oriented by transitions induced by a polarized light beam As D1 but orientation is passed o n t o other atoms in vapor by collisions. Large cross section for spin-flip exchange with atoms of added vapor. Depolarization at walls must be small Z relaxes via A S . I coupling; saturate electron spin resonance
Applicability
+
Optical pumping with ‘spin transfer’
D2
2
rf and microwave pumping in metals
D3
3
Extension t o nonmetals and semiconductors
D4
4-6
Use of ‘forbidden’ transitions
D5
Atoms in vapor
Metals with unpaired s-electrons
Nuclear relaxation via Paramagnetic centers with resolved hfs AS . I; or A S J , B(SzZz &ZV) with H along z-axis hfa with some anisotropy _ _ in order to give reasonable transition probabilities with rf and external fields parallel. Relaxation transitions, which produce orientation in D3, are ‘driven’ by rf field. Differential effect with inhomogeneously broadened line
+
7
Atoms in vapor, e.g., alkali metals, mercury. Possible extension t o solids
+
Special Requirements Vapor at ordinary temperaturea. When radioactive species used, large background due t o wall absorption is troublesome A s above
T
‘V 1°K preferably; relatively high rf powers needed, which can be a disadvantage T ‘V 1°K preferably
As above
TABLE I1 (Continued) (c). Transient Methods of Orienting Nuclei Method
No.
Ref.
Double resonance
T1
1
hfs must be resolved, but Paramagnetic centers with detailed form not so resolved hfs important. Necessary to make adiabatic rapid passage through electron spin resonance and nuclear spin resonance successively. Nuclear spin relaxation must not be too short
Three other transient effects
T2
2
Z and S relax via A S . I coupling, i.e., main relaxation process is that where S and Z flip together
c. -3 t 9
T3
T4
3
Criteria
Hyper6ne coupling is mainly of the type AS * I
Applicability
Special Requirements
T
‘v
1°K preferably
Only necessary to turn on external field Saturate two electron resonances in turn where Z does not flip After thermal equilibrium, decrease external field from Paschen-Back to Zeeman region (a) so that change is at all times adiabatic, and (b) so that the time taken is << electron spin-lattice relaxation time
REFERENCES TO TABLEII(a) C. J. Gorter, Physik. Z. 36, 923 (1934). N. Kurti and F. E. Simon, Proc. Roy. SOC.A149, 152 (1935). F. E. Simon, Compt. rend. sur le magndtisme, Strasbourg p, 1 (1935). B. Bleaney, Phil. Mag. (71 42, 441 (1951). C. J. Gorter, Physica 14,504 (1948). M. E. Rose, Phys. Rev. 76, 213 (1949). R. V. Pound, Phys. Rev. 76, 1410 (1949). 8. J. G. Daunt, Proc. Intern. Conf. Low Temp. Phys., Ozfordp. 157 (1951). 9. C. J. Gorter, Proc. Zntern. Conf.Low Temp. Phys., O z f o r d p . 158 (1951). 10. W. Marshall, Phys. Rev. 110, 1280 (1958). 11. B. N. Samoilov, V. V. Sklyarevskii, and E. P. Stepanov, Zhur. Eksptl. i Teoret. Fiz. 38, 359 (1960); Soviet Phys. J E T P 11, 261 1. 2. 3. 4. 5. 6. 7.
2
(1960). 12. D. A. Coodings and V. Heine, Phys. Rev. Letters 6, 370 (1960). 13. R. E. Watson and A. J. Freeman, Phys. Rev. 123, 2027 (1961).
REFERENCES TO TABLEII(b) A. Kastler, Proc. Phys. SOC.(London)A67, 853 (1954); J. Opt. Soc. Am. 47, 460 (1957). H. G. Dehmelt, Phys. Rev. 106, 1487 (1957); 109, 381 (1958). A. W. Overhauser, Phys. Rev. 92, 411 (1953). J. Korringa, Phys. Rev. 94, 1388 (1954). F. Bloch, Phys. Rev. 93, 944 (1954). A. Abragam, Phys. Rev. 98, 1729 (1955). 7. C. D. Jeffries, Phys. Rev. 106, 164 (1957).
1. 2. 3. 4. 5. 6.
REFERENCES TO TABLEII(c) 1. G. Feher, Phys. Rev.
103, 500 (1956). 2. D. Pines, J. Bardeen, and C. P. Slichter, Phys. Rev. 106, 489 (1957). 3. A. Abragam, Compt. rend. 242, 1720 (1956).
2.
174
DETERMINATION OF FUKDAMENTAL QUANTITIES
and plot f l against nuclear moment in Fig. 1. We also indicate the actual polarization that could be obtained in a number of favorable but useful cases. Since the polarization goes practically linearly with x over a significant range, the following high temperature approximations are useful : (2.4.2.3.2) f2
+ 3)
1 (21 - 1)(21 * I ( I 1)
+
= fl2
1
(2-4.2.3.3)
where f l is the polarization and fz the alignment (cf. Section 2.4.2.3.5). The quantity in the square bracket varies from 0.25 for I = 1 to 0.4 for 1 = m, so that the alignment, being proportional to flz, would be very small in this range.
fI
50%
40% 30 % 2 0 Yo I0
0
0
I
2
3
4
5
6
Pn FIG.1. Nuclear polarization, f l , in a magnetic field of 100 kilogauss and at a temperature of 0.025"K, for various values of nuclear spin, plotted as a function of nuclear moment, p a , in nuclear magnetons. Values corresponding to a few specific isotopes are indicated. The dotted line on hydrogen isotopes applies to the room temperature ortho-para ratios for moIecular hydrogen.
The main obstacles in the way of the full utilization of this method have been practical ones associated with the simultaneous production of very high steady magnetic fields and very low temperatures. I n order to
2.4.
DETERMINATION O F SPIN, PARITY, AND NUCLEAR MOMENTS
175
cool substances to very low temperatures, they must be brought into contact with a paramagnetic salt which has itself been cooled by the process of adiabatic demagnetization. This may be done by taking many silver or copper wires in parallel, embedding one set of ends in the paramagnetic salt, and attaching the other ends to the sample. With such an arrangement the main resistance to heat flow is usually found to be surface contact resistance, e.g., between the wires and the salt. In addition to this practical difficulty, there is a more fundamental one associated with a very long nuclear spin-lattice relaxation time found in insulating substances at low temperatures. This should not present any serious difficulty in the case of metals, however, and in the case of insulators it is often possible to devise ways of shortening the relaxation time, e.g., by including a trace of paramagnetic impurity. When we are interested in cooling relatively small numbers of nuclei, e.g., in experiments on radioactivity, the amount of heat to be transported is quite small, so that cooling can be effected fairly quickly. If we are interested in cooling a large number of nuclei, on the other hand, e.g., for experiments on nuclear reactions, the cooling may be rather slow. The rate of cooling may be computed quite simply since the algebraic expressions for all the relevant quantities, such as the entropy, S , and polarization, fl, of the nuclei in an external magnetic field are known and have been tabulated for different values of the parameter 5 . 2 3 , 2 4 Thus, for example, in the “high” temperature range (cf. Section 2.4.2.3.8) considered above, the reduction in entropy per mole, over the value of R ln(21 1) for random orientation ( R is the gas constant), is given for a change in temperature dT, by:
+
dS where
p
= =
p d T / T 3 erg deg-I 0.037pn2H2(1 l)/Ierg deg, and
+
(2.4.2.3.4) (2.4.2.3.5)
is given in nuclear magnetons, H in kilogauss. The heat flow Q is given by Q = T dS/dt. (2.4.2.3.6)
pn
In a given experimental arrangement the heat flow between the nuclear stage and the electronic stage (a demagnetized paramagnetic salt at temperature, I’E,say) will be governed by a number of thermal resistances R1, Rz, . . . etc., in series. Thermal resistance will arise through finite nuclear-spin relaxation times, the thermal resistance of the metallic link between the stages, the contact between the link and the salt and the finite thermal diffusivity of the paramagnetic salt. In addition to the heat of magnetization to be extracted from the nuclei, there will be spurious heat leaks such as are due to vibrations, residual gas conduction,
2.
176
DETERMINATION O F FUNDAMENTAL QUANTITIES
eddy current heating, and so on. These factors have been discussed in some detai1,20*26 and ways of coping with the difficulties that arise have been discussed. Perhaps the most significant factor is the thermal resistance of the boundary26 between the paramagnetic salt and the wires connected to the nuclear stage. Good bonds are necessary a t the interface, therefore, and a number of ways for making them have been discussed.26-28A good working rule is t o take the heat flow across an area A em2 to be given by Q = lo3* A(T3 - T E ~ergs/sec ) (2.4.2.3.7) where T is the temperature of the nuclear stage, and to assume that a very large mass of paramagnetic salt is used so that TE remains sensibly constant. If a more detailed analysis is required, the necessary data may be found in the references already quoted. For the present, we prefer to follow a simple procedure in order to obtain an idea of the practical limits of the method. Thus, for example, suppose we wish to polarize the nuclear spins in one mole of Li7, having at our disposal H = 100 kilogauss, and assuming T E = 0.01”K. Take A = 2 X lo4 em2. Using Eqs. (2.4.2.3.4-7), we can compute the temperature and nuclear polarization of the sample as a function of t, the time after the cooling starts, i.e., after demagnetization of the upper stage. The result is shown in Fig. 2; it will be seen that a significant nuclear polarization is built up after about an hour. The heat content of the nuclear spins in an external field is quite large a t these low temperatures, so that it is possible to do experiments on the nuclei in which a significant amount of heat is developed. By way of illustration we have shown the projected behavior of the system under “cyclical” operation, and we see, for example, that with a heat dissipation of 30 pw an average nuclear polarization of over 20% can be held for 10 minutes. Thus experiments with incident particles that produce small heat dissipation in the source, e.g., photons and neutrons, could be done very easily, whereas experiments with incident particles such as protons, though very possible, may prove to be laborious. 2.4.2.3.2.2.Magnetic hfs Alignment2g--S.2, and Magnetic hfs Polarization30J--XS. These methods rely upon the fact that in certain paramag26
N. Kurti, F. N. H. Robinson, Sir Francis Simon, and D. A. Spohr, Nature 178
450 (1956). 28 21
See, e.g., W. A. Little, Can. J . Phys. 57,334 (1959), and references quoted therein. A. R. Miedema, H. Postma, N. J. Van der Vlugt, and M. J. Steenland, Physica
26, 509 (1959). 28 A. C. Anderson, G . L. Salinger, and J. C. Wheattley, Rev. Sci. Instr. 52,1110 (1961). 28 B. Bleaney, Phil. Mag. [7] 42, 441 (1951). 30 C. J. Gorter, Physica 14, 504 (1948). 31 M. E. Rose, Phys. Rev. 76, 213 (1949).
2.4.
DETERMINATION
OF SPIN,
PARITY, AND NUCLEAR MOMENTS
177
netic salts there are large magnetic fields (-lo6 gauss) a t the nuclei of the paramagnetic ions due to the unpaired electrons, and at temperatures of the order of 0.01"K the nuclear magnetic moments become oriented with respect to these fields. When the electron moments are oriented a t these very low temperatures, therefore, nuclear orientation will automatically follow. In the case of magnetic hfs polarization the
T "K
fI
.I 2
.6 0
.I 0
.50
.O 0
.40
.O 6
.3 0
.O 4
.2 0
.o 2
.I0 0
0 -
0
60
I20 TIME - mins
180
240
FIG.2. Temperature, T,and nuclear polarization, fl, plotted as a function of time for a specimen of 1 mole of Li7 cooled by contact with a paramagnetic salt at a temperature, TE,of 0.01"K. The heat flow, 4, is supposed to be governed by contact resistance .
electron moments are polarized by the application of an external field, and since the electron magnetism is easily saturated a t low temperatures, a field of a few hundred gauss suffices. In the case of the magnetic hfs alignment method, use is made of the interaction between the paramagnetic ion and the electric field arising from its neighbors in the crystalline lattice, and in a rough way one may say that the electric field aligns the electron moment. This is an oversimplified picture, however, as is the way of describing the interaction between the electron and nuclear spins in terms of a magnetic field. The behavior of paramagnetic ions in a crystal
2.
178
DETERMINATION OF FUNDAMENTAL QUANTITIES
lattice is complex, although the spin Hamiltoniana2provides a very concise way of describing this behavior a t low temperatures. The use of the spin Hamiltonian for describing the behavior of paramagnetic ions in the solid state is based upon the fact that the energy levels of the free ion are perturbed by the electric fields in the crystal in such a way that there is a low-lying group of levels more or less wellseparated in energy from the rest. At very low temperatures only these levels are populated, and their behavior, for most purposes, can be considered independently from the rest. They are described collectively by a fictitious spin operator S , the value of which is obtained by setting (25 1) equal to the number of electronic levels in the group. These levels, which are not necessarily all degenerate, often consist of a complicated mixture of orbital and spin eigenstates of the free ion. Their magnetic properties may be quite anisotropic, a fact that is taken into account by combining S with an anisotropic g-factor. Usually the environment of the ion is such that there is symmetry around a given axis, and this is chosen to be the z-axis in the spin Hamiltonian. It happens frequently, although it is not invariably the case, that all ions of a given kind are equivalent in the crystal, i.e., they have the same axis of symmetry and are described by the same spin Hamiltonian. The z-axis then usually coincides with some crystal axis that may be identified easily from the natural shape of the single crystals. For a single axis of symmetry the spin Hamiltonian 52 is
+
where /3 is the Bohr magneton, fin the nuclear magneton, and gn the nuclear g-factor. D represents an electronic splitting originating solely from the action of the crystal electric field. A and B represent the magnetic hyperfine splitting and are proportional to the nuclear moment of a given isotope. P represents the electric hyperfine splitting and is proportional to the product of the nuclear spectroscopic quadrupole moment and the electric field gradient a t the nucleus, The terms Az and A, are present in the cases of ions with an even number of electrons, e.g., Pr, Ho, Tb. I n these cases the energy levels are not necessarily degenerate in the sense of Kramers’ Theorem,89 although in many substances, owing to the particular crystal symmetry, this may happen. I n particular for some cases A. Abragam and M. H. L. Pryce, PTOC.Roy. SOC.A206, 135 (1951). H. A. Kramers, Koninkl. Ned. Akad. Wetenschap., PTOC. 93,953 (1930); Physica 1, 182 (1934). See also M. J. Klein, Ant. J . Phys. 90, 65 (1952). 52
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
179
the ground state may be an “accidental” doublet. This doublet may then be split by very slight distortions of the crystal field from its assumed symmetry, and the terms A, and Au are included to describe this effect. The effects they describe are not usually sharp, so that mean values about a statistical distribution are assigned to A%and Au. For the cases mentioned above, a Gaussian distribution has been found to describe adequately experimental data on paramagnetic resonance.34 Note that for electric interactions the magnetic substates + M referring to the electron spin, and the magnetic substates f m referring to the nuclear spin, are degenerate, i.e., electric interaction alone can never produce polarization, only alignment. To produce a nuclear polarization an external magnetic field must be applied which couples either directly to the nuclear spin (method Sl) as represented by the last term in Eq. (2.4.2.3.8), or indirectly through the electron spin as represented by the terms in A and B (method 53). It should be noted that for a given paramagnetic compound, different isotopes of the same element have spin Hamiltonians that differ only in the hyperfine coupling constants, and that A and B are proportional to p / I , and P to &/1(21 - 1) [cf. Eq. (2.4.2.3.11)l. This is an essential point in the method usually adopted for the determination of nuclear moments. For a further discussion of the spin Hamiltonian, as well as for numerical values of the constants in particular cases, the reader is referred to a number of review articles.36-38 As classified in Table 11, the methods of nuclear orientation can then be discussed by assuming that certain terms in the spin Hamiltonian are dominant.29-39Thus, for example, for method S1 we need consider only the last term in Eq. (2.4.2.3.8). For method 52, two cases are important, the case of S = 4 where the D term automatically vanishes, and the case of S > +. For S = 4 there are two extreme cases. When A >> B the energy levels are as shown in Fig. 3a, with the nuclear magnetic substates I , = + I lowest. I n this case the nuclear spin aligns parallel and antiparallel to the z-axis. When B >> A the energy levels are as shown in Fig. 3b, the nuclear magnetic (for I half-integral) or I , = 0 (for I integral) lie substates I , = lowest, and nuclear spin “aligns,” or rather precesses, in a plane perpendicular to the z-axis. When S > p there are again two cases, viz., D negative and D positive. We have shown the energy levels in the former
*+
s4 J.
M. Baker and B. Bleaney, Proc. Roy. Soc. A245, 156 (1958). B. Bleaney and K. W. H. Stevens, Repfs. Progr. in Phys. 16, 108 (1953). K. D. Bowers and J. Owen, Repts. Progr. in Phys. 18, 304 (1955). J . w. Orton, Repts. Progr. in Phys. 22, 204 (1959). M. H. L. Pryce, Nuovo cimento Suppl. [lo] 6, 817 (1957). 39 A. Simon, M. E. Rose, and J. M. Jauch, Phys. Rev. 84, 1155 (1951).
36
180
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES rn
m f 1/2
=T
( i 3 / 2 ,i1/2) f f 3/2
if51/2 /2
f 5/2
i 5/2
f 3/2 f 1/2
f 1/2 f3/2
f 5/2
f 3/2
L2P
+1/2
L-
(C)
FIG. 3. Energy levels for various values of the parameters A , B, D and P in Eq. 2.4.2.3.8;in (a) D = 0, P = 0, and A >>B, in (b) D = 0, P = 0, and B >> A , in (c) P = 0, D >> A , B and positive, in (d) D = 0, A = 0, B = 0 and P positive. The energy levels are given assuming I = $, with S = 6in (a) and (b),and S = Q in ( c ) . The quantity m labels the eigenstates of I..
case in Fig. 3c, and have assumed for simplicity that A 2 B and D >> A . The case of D negative corresponds to the case where the electron spin levels 8,= + S are lowest, and is more favorable for producing nuclear orientation than the case of D positive. We have listed in Table I11 the paramagnetic salts most frequently used
2.4.
181
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
TABLE 111. List of Paramagnetic Salts Used for Nuclear Orientation by Methods 52 and 538
~~
8
Mn55 (X)(5) (Y) C O (X)(5) ~ ~ (Y) M+++
1.997 1.997 4.108 7.29
f g4 -_I
S
911
4+ 3i
CeI4l(6) Pr141 NdI4'(6)
4
1.997 - 80 1.997 -215 4.385 2.338 -1.84 2.72
0.25 1.55 0.45
+
D
91 ___-
90 90 85 283
- 90 - 90 103 5 1
A
B
P
A
126 237
-
-
-
-
-
-
(-20) 770 (-40)
-
-
-
-
-
-
Ethylsulfates: M + + + ( C ~ H S S O ~ ) ~ . ~ H ~ O Cooling substances: Ce(C2HSS04)3.9H20; gll(7) = 3.80, 91 = 0.22, T-T*correl.(8) Nd(C2HoS04)3.9HzO;911 = 3.535, g 1 = 2.072, T-T* correL(9)
I Ce(l0) ~r141(11) Nd14a ~m147(12) Sm147 ~b169(11 Dy167 Ho l E 616) ( Yb17s(17)
IS/
-
911
i i
D 91 50.4
3.80 $ 1.525 3 3.535 2.072 4 0.432 0.604 4 0.596 8 i 17.72 < 0 . 3 (#)i 10.76(14: 0 4 15.36 0 C4)+ 3 . 4
-
+
+
-
-
A
I
B l
755 380 165 60 2090 691(15 3340 02
A
-
(288) ( 3) -
3870 -
650 -
P
-
.
Cooling substance: (Ni 15%, Zn 85%) SiFr6HzO; S = 1, g = 2.29, D = -1400, T-T* correL(l8)
_ -s
4$
co59
4
911
2.000 5.82
_
91 _
_
D
_
2.000 -134 3.44 -
~
-
A
91 184
A
P
A
47
-
-
A
P
A
B
- +150(19) -1654(21) +178 +301 862
-
-
-
a For the cooling substances references are given t o g-factors and t o T-T* correlation data. For other ions, constants in spin Hamiltonian are given for one isotope;
182
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
those for other isotopes differ only in the values of A , B[ a r / Z ] and P [ a Q / I ( 2 I - l)]. Except where explicit references are given (in parentheses), the data are taken from references 35-37 of text. The units for A , B, P, and A are lo-' cm-'. It should be noted that the constants have not usually been measured directly in the cooling substances, but rather in an komorphous diamagnetic salt (see references 35-37), and in some cases differences have been noted. REFERENCES A N D FOOTNOTES TO TABLE I11 1. M+++, M++, and Mf refer to trivalent, divalent and monovalent ions respectively. 2. A. H. Cooke, H. J. Duffus, and W. P. Wolf, Phil. Mag. [7] 44, 623 (1953). 3. R. P. Hudson, R. 5. Kaeser, and H. E. Radford, Proc. 7th Intern. Conf. Low Temp. Phys. p. 100 (1961). 4. J. M. Daniels and F. N. H. Robinson, Phil. Mag. [71 44, 630 (1953). 5. There are two sets of nonequivalent ions in unit cell; the ratio of X: Y appear to be about 1 6 : l [B. Bleaney, K. D. Bowers, and R. S. Trenam, Proc. Roy. Soc. A228, 157 (1955)l. 6. R. W. Kedzie, M. Abraham, and C. D. Jeffries,Phys. Rev. 108,54 (1957). 7. A. H. Cooke, 5. Whitley, and W. P. Wolf, Proc. Phys. Soc. (London) B68, 415 (1955). (See also reference 14 of text for discussion.) 8. C. E. Johnson and H. Meyer, Proc. Roy. Soc. A263, 199 (1959). 9. H. Meyer, Phil. Mag. [8] 2, 521 (1957). 10. No direct measurement of hyperfine splitting; see reference 41 of Table I. 11. J. M. Baker and B. Bleaney, Proc. Roy. Soc. A246, 156 (1958). 12. H. J. Stapleton, C. D. Jeffries, and D. A. Shirley, Phys. Rev. 124, 1455 (1961.) 13. Not directly determined, but see references 48 and 49 of Table I. 14. A. H. Cooke, D. T. Edmonds, F. R. McKim, and W. P. Wolf, Proc. Roy. SOC. A262, 246 (1959). 15. See reference 56 of Table I. 16. J. M. Baker and B. Bleaney, Proc. Phys. Soc. (London) A68, 1090 (1955). (See also reference 11.) 17. See reference 60 of Table I. 18. See reference 14 of text. 19. See reference 61 of Table I. 20. M. H. L. Pryce, Phys. Rev. Letters 3, 375 (1959). 21. See reference 62 of Table I. 9
for nuclear orientation by methods S2 and 53, together with the appropriate spin Hamiltonian constants. It will be noticed that almost all the substances used are those met with in work on magnetic cooling, and the nuclei to be oriented (usually radioactive isotopes) are incorporated substitutionally. This method, using a crystal of mixed constitution, works particularly well since the nuclei are in excellent thermal contact with the coolant. It is important to note, however, that spin Hamiltonian constants are usually obtained from paramagnetic resonance measurements made on a salt considerably diluted with isomorphous diamagnetic ions, and it is not invariably the case that they are the same as for the magnetically concentrated salt.
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
183
To produce nuclear polarization we must, of course, apply an external magnetic field large enough to cause appreciable polarization of the electron moment. At the temperatures reached by magnetic cooling a few hundred gauss will usually suffice, provided the direction is suitably chosen with respect to the crystal field axes. Thus if we have a large D term in Eq. (2.4.2.3.8),a very anisotropic g-factor with 911 >> gl, or a very anisotropic hfs with A >> B , a magnetic field applied perpendicular to the z-axis will produce only a second order Zeeman effect. I n this case the spins remain oriented with respect to the z-axis, which is determined by the crystal field symmetry and not the external magnetic field. It is clear that the desired effect can be achieved by applying the magnetic field along the z-axis. There is an obvious advantage, therefore, in using single crystals with one kind of magnetic ion in unit cell instead of either polycrystalline specimens or single crystals with ions with different magnetic axes, e.g., the alums. The methods described in Section 2.4.2.3.8may be used to obtain the following “high” temperature approximations for the polarization and alignment : fl
= g,PHzAS(S
fz = -P(I
+ 1)(1 + 1)/9k2T2
+ 1)(21 + 3)(21 - 1)/451kT
(2.4.2.3.9)
+ (A2 - B2)S(S+ 1 ) ( 1 + 1)(21 - 1)(21 + 3)/2701k2T2 + P2(1 + 1)(21 + 5)(21+ 3)(21 - 1)(21 - 3)/18901k2T2. (2.4.2.3.10) 2.4.2.3.3. Electric hfs Alignrnent4O-S4. The coefficient P in Eq. (2.4.2.3.8)is given by41 3eQ d2V P= (2.4.2.3.11) 41(21 - 1)
where e is the absolute value of the electronic charge and Q the nuclear electric quadrupole moment. V is the electrostatic potential a t the nucleus, and therefore d2V/dz2 is the electric field gradient along the symmetry axis. Now we require P to be a t least 10V ev to obtain an appreciable nuclear alignment, and since values of Q may be of the order of one barn in favorable cases, we can see that electric field gradients of the order of 1Ol6 esu are required. Such intense field gradients are found locally in certain compounds, where the electron cloud surrounding the nucleus is very asymmetrical, for example, of the type encountered in certain covalent bonds. A notable series of compounds where this R. V. Pound, Phys. Rau. 76, 1410 (1949). See, e.g., A. R. Edmonds, “Angular Momentum in Quantum Mechanics,” p. 115. Princeton Univ. Press, Princeton, New Jersey, 1957. 40
41
184
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
situation obtains is the actinyl double nitrate^,^^^^^ where the hyperfine splittings are very large and where alignment can be achieved without having recourse to magnetic cooling. I n order to give a n idea of the pattern of the energy levels when the hfs is due to quadrupole splitting only, we have drawn the case of P negative in Fig. 3d. It will be noted from Eq. (2.4.2.3.10) that whereas with magnetic hfs alignment f 2 a 1/T2 a t high temperatures, in quadrupole alignment fz cc 1/T. 2.4.2.3.2.4. The Use of Ferromagnetics and Anti-ferromagnetics. Several of the transition and rare earth metals and alloys, as well as numerous intermetallic compounds, are ferromagnetic, and some are known to have a sufficiently large hfs so th at nuclear orientation becomes practicable (see Table I). The hfs is produced through magnetic coupling, although with transition metals, unlike the situation that obtains in paramagnetic salts, the electrons that give rise to the magnetic moment are not tightly bound. It is also a remarkable fact44that normally diamagnetic ions, when dissolved in small quantities in a ferromagnetic material such as iron, show sufficiently large hfs to make nuclear orientation possible. The mechanism that brings this about seems to be due to a polarization of the inner electron shells, which, in the case of S-shells is particularly potent in producing hfs. The method promises to have very wide application, especially for the investigation of radioactive nuclei that are normally nonmagnetic in the soIid state. Before going on to discuss the dynamic and transient methods for nuclear orientation, we shall now discuss some points of technique which pertain especially to the static methods. TECHNIQUES.2.4.2.3.3.1. Specimen 2.4.2.3.3. SOMEEXPERIMENTAL Preparation.* Most of the salts shown in Table I11 may be obtained as single crystals b y growing them from an aqueous solution. Small seed crystals are obtained, which are placed one in each of a number of small beakers of a few cm3 capacity. The beakers are filled with saturated solution which is allowed to evaporate very slowly. As the seed crystal grows, a layer of faults often occurs a t the place where the original surface of the seed was located. This can be avoided very often by warming the solution slightly (about +"C) before pouring it over the seed. The seed tends to dissolve slightly and if the warming is overdone may dissolve
* See also Vol. VI, A, Part 41
2.
J. C. Eisenstein and M. H. L. Pryce, Proc. Roy. SOC.A229, 20 (1955); A238, 31
(1956). 43 B. Bleaney, Discussions Faraday Sac. No. 19 (1955). 44 B. N. Samoilov, V. V. Sklyarevskii, and E. P. Stepanov, Zhur. Eksptl. i Teoret. Fit. 38, 359 (1960); So& Phys. J E T P 11, 261 (1960).
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
185
completely! A slight dissolving, however, seems to be of value in avoiding faults at the surface of the seed. It is preferable to start with the highest grade of materials available, and sometimes a number of further purifications by recrystallization are beneficial. The rate of growth must be kept very slow and regular (one or two cm3 per week) in order to obtain clear single crystals, and even then all the crystals obtained are unlikely to be entirely free from occlusions of solution. To achieve the best results the ambient temperature should be controlled to better than i-1”C,and it is preferable to control the humidity also. The beaker should be covered to avoid dust and other contaminants entering the solution, which might cause fresh seeds to form. In practice it is not always possible to prevent the formation of new seeds, although these can be kept from interfering with the growth of the main crystal by a suitable arrangement, e.g., by suspending the main crystal in the middle of the solution away from the newly formed seeds which generally fall to the bottom of the beaker. It is always possible to stop t,he growth temporarily and to filter the solution, but there is a chance of new faults appearing when growth is restarted. Special care must be exercised in heating some solutions, particularly the ethyl sulfates since there is a tendency for the ethyl sulfate to break down into acid and alcohol. This is particularly serious because further deterioration is rapid. In growing ethyl sulfates the acidity of the solution should be checked a t regular intervals. It is also possible to grow the crystals from alcohol solutions, and often an advantage to grow them a t temperatures below normal room temperatures. The anhydrous uranyl rubidium nitrate crystals may be grown satisfactorily from solution in concentrated nitric acid. Controlled evaporation of the solvent may be effected by slowly passing dry nitrogen gas over the solution. If it is necessary to include radioactive nuclei in the crystal for gammaray measurements, the activity may be placed in the solution. Allowance should be made for the fact that in this case, as in the general case of growing crystals of mixed composition, the relative concentration of ions in the solution may not be the same as that in the crystal. When beta radioactivity is to be investigated, however, or when only a small amount of activity is available or when the half-life is very short, the activity can be mixed with a drop of solution which is then placed upon the surface of an inert crystal and allowed to crystallize. Similar remarks apply to specimens which are alpha radioactive although, because of the greatly differing scattering properties of alpha particles and electrons in matter, the need for a thin source is not so critical. The obtaining of an “ideal” beta-particle source under the circumstances is of course not possible. The preparation of a “good” beta-particle source is a t best somewhat of an
186
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
art, although it is possible in most cases to check the quality by comparing beta and gamma counting rates. In the case of the preparation of metal specimens, a number of alternative methods of preparation have been employed. It is possible to irradiate metal single crystals in a pile, for example, and subsequently anneal them.44s46 It is also possible to plate the activity on the surface and then heat treat so as to allow a slight diffusion to take place.44 2.4.2.3.3.2. Magnetic Cooling. Although there are a few useful experiments that can be done at about 1°K (the practical low temperature limit which can be reached by causing liquid He4 to boil under reduced pressure), and a t about 0.25”K (the practical low temperature limit reached by similarly using liquid He9, in most cases temperatures of about 0.01”K are required and the method of magnetic cooling is used. The principle of the method of magnetic cooling is to apply a magnetic field H O(typically in the range 10-30 kilogauss) to a suitable paramagnetic salt a t a low temperature T O (usually about 1°K). After thermal equilibrium has been established the salt is isolated thermally, usually by pumping away exchange gas, and the magnetic field is removed. The temperature then falls to a value T, as the field falls to a value H such that T, = -To *H. (2.4.2.3.12) Ho When the external field becomes very small, Eq. (2.4.2.3.12) ceases to hold and the temperature becomes dependent upon internal rather than external constraining forces, such as crystal fields and hfs. When we approach a cooperative region, as is often the case in practice, the relation is more complicated, and in fact at temperatures below a Nee1 point, the application of an external field may actually cause a slight cooling of the sample. For a detailed discussion of these and other points we refer the reader to more detailed treatments of the subject of magnetic co0ling,4~-~~ and also to Section 2.4.2.3.8. We shall refer here only to a few points of technique that are somewhat special to nuclear orientation. 41 M. A. Grace, C. E. Johnson, N. Kurti, R. G. 8curlock, and R. T. Taylor, in “Conference de physique des basses temptktures,” p 263. Centre National de la Recherche Scientifique and UNESCO, Paris, 1956. 48 C. G. B. Garrett, “Magnetic Cooling.” Harvard Univ. Press, Cambridge, Massachusetts, 1954. 47 E. Ambler and R. P. Hudson, Repts. P r o p . in Phys. 18, 251 (1955). 48 D. de Klerk and M. J. Steenland, in “Progress in Low Temperature Physics” (C. J. Gorter, ed.), Vol. 1, p. 273. North-Holland Publ., Amsterdam, 1955. 49 D. de Klerk, in “Handbuch der Physik-Encyclopedia of Physics” (S. Flugge, ed.), Vol. 15, p. 38. Springer, Berlin, 1956.
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
187
In experiments with radioactive nuclei the activity is included in a host crystal essentially as an impurity. The type of paramagnetic ion in the host crystal can then be chosen to produce efficient cooling. Among the considerations that affect this choice is the condition that the coolant ion have little or no hfs. The reason for this can best be illustrated by considering the entropies involved. For a given type of ion the maximum entropy of the electron and nuclear spin systems is R ln[(2S 1)(2Z l)]. In order to produce considerable orientation, it is necessary to remove as much of this entropy as possible from the nuclear spin system. By magnetizing at 1”K, however, the maximum amount of entropy that can be l), i.e., the electron removed from the system as a whole is R ln(2S spins are completely ordered, whereas the nuclear spins are completely disordered. Since the demagnetization process is isentropic this is also the final entropy, although it is now shared between the two systems. For I appreciably greater than S, it is clear that the final state will not be a highly ordered one, i.e., the nuclear orientation will be small. When a second type of ion, of the kind mentioned above, is incorporated in small concentration, however, the entropy balance is very favorable for producing a large nuclear orientation. The best example of this is the case of cerous magnesium nitrate,60*61 which may be cooled by demagnetization to a temperature of 0.003”K1partly because it is a very dilute salt magnetically and partly because there is no hfs. This salt has the further advantage that it is a double salt into which both trivalent rare earth ions and divalent transition metal ions may be incorporated. Another point to be noted about magnetic cooling is that it is a “single shot” process, i.e., after demagnetization the specimen begins to warm up again, and measurements are taken during this warming period. Since the heat capacities of the salts are often very small, the question of thermal insulation becomes most important. Particular attention must be paid to technique in this connection, and sources of heat influx such as arise from radiation, and more particularly from vibration of the sample and conduction by residual exchange gas desorbing from the walls of the cryostat, must be minimized. For a carefully designed apparatus, a heat influx of 10 ergs/min into a sample weighing a few grams would be considered very satisfactory. For details of specimen support inside the cryostat, the use of exchange gas, thermal “guard-rings,” etc., the reader is referred to references 10-22, 25, and 4 6 4 9 . When a radioactive substance is being investigated, the heating due to absorption of some of the energy released during decay may become noticeable. For the more commonly used
+
+
+
J. M. Daniels and F. N. H. Robinson, Phil. Mag. [7] 44, 630 (1953). R. P. Hudson, R. S. Kaeser, and H. E. Radford, Proc. 7th Intern. Conf. Low Temp. Phys. p. 100 (1961). Kt
188
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
isotopes with beta-gamma cascades, a useful average working figure is 1 erg/min per microcurie of activity. The amount of activity required is usually in the range 1-100 microcurie. Thus even with cooling substances such as cerous magnesium nitrate, warm up times of one hour can be achieved, although since much larger effects are observed at the lowest temperature, not all of this time is equally useful. I n fact, the more usual procedure is to take data only at the lowest temperatures, and then deliberately to warm up the sample and take the “warm” data needed for normalization purposes. Besides requiring less stringent conditions for counter stability, this method overcomes inaccuracies due to inhomogenous heating in the sample, which arises on account of the rather low thermal diffusivity of the salts a t low temperatures. In fact, for the most accurate work, the taking of “cold” data is often confined to the first few minutes after demagnetization. There is a further point to note when experiments are being performed with polarized nuclei, I n this case, there is an external field present which affects both the specific heat, C , and the temperature of the sample. Now we have c = [b CH2J/T2 (2.4.2.3.13)
+
where c is the Curie constant and b is the “specific heat constant.” For work on radioactivity, or on nuclear reactions where the incident particles are uncharged, no serious trouble due to small heat capacities should be encountered. For charged particle reactions, however, the situation is likely to be very different, and a change of technique would be needed. One could consider producing continuous refrigeration by the use of He3 as ref1igerant~2-~4or by the development of ways of producing cyclical magnetic as well as resorting to experiments with a rather poor duty cycle, as illustrated for a somewhat extreme case in Fig. 2. In the case considered above for producing nuclear polarization, the residual external magnetic field limits the extent of the cooling and hence the polarization. An alternative procedure is to use two different specimens separated sufficiently in space but connected by a thermal link, the first specimen being cooled directly by demagnetization and the second by contact with the first, in the manner outlined in the previous section. This method has the further advantage that the substances in the V. P. Peshkov and K. N. Zinov’era, Repts. Progress in Phys. 22, 504 (1959). W. Taconis, in “Progress in LOW Temperature Physics” (C. J. Gorter, ed.), Vol. 111, p. 153. North-Holland Publ., Amsterdam, 1961. 54 E. Ambler and R. B. Dove, Rev. Sci. ZnstT. 32, 737 (1961). 56 C. V. Heer, C. B. Barnes, and J. G . Daunt, Rev. Sci. Instr. 26, 1088 (1954). 52
6s K.
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
189
second stage need not necessarily be a paramagnetic salt. There are a number of points of technique involved, as we have already indicated, the most important one being that of obtaining good thermal contact between the thermal link and the salt. I n certain cases, demagnetization to a nonzero field can be achieved most advantageously without sacrificing the degree of cooling by using single crystals which are magnetically very anisotropic and by essentially rotating the sample with respect to the magnetic field.66Equation (2.4.2.3.12)is then modified to read: (2.4.2.3.14) where go and g are the g-factors in the directions of H o and H respectively. The advantage of using this method when g o >> g , as is the case with cerous magnesium nitrate, is evident. When the g-factor along H is not small, i t is necessary to allow for saturation effects. This is the case, for example, with neodymium ethyl sulfate where very often a polarizing field is applied along the z-axis while susceptibility measurements are made perpendicular. I n this case a useful relation is i
T*
=
( T H *tanh y)/y
(2.4.2.3.15)
where y = gllpH/2kT, TH* is the measured magnetic temperature, and T* is the one to be substituted in the T-T* correlation (see below). Finally, we should point out th at when we are in the range of cooperative effects in paramagnetic salts, changes in temperature may be complicated but, can be found by a procedure based upon the thermodynamic relati~n~’*~~*~~ (2.4.2.3.16) where M is the magnetic moment of the sample. I n order t o determine the degree of nuclear orientation (see Section 2.4.2.3.5),it is usually necessary to know the temperature of the specimen. When working with liquid He4, 68 or liquid He3, 69 temperatures can be obtained simply from the vapor pressure, although corrections for thermomolecular pressure differences may be significant.6oI n the magnetic cooling 56A.
H. Cooke, S. Whitley, and W. P. Wolf, Proc. Phys. SOC.(London) B68, 415
(1955).
W. F. Giauque, Phys. Rev. 92, 1339 (1953). F. G. Brickwedde, H. van Dijk, M. Durieux, J. R. Clement, and J. K. Logan, J. Research Natl. Bur. Standards A64 1 (1960). 59 S. G. Sydoriak and T. R. Roberts, Phys. Rev. 106, 175 (1957). 60 T. R. Roberts and S. G. Sydoriak, Phys. Rev. 102, 304 (1956). 57 68
190
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
region the absolute temperature is usually correlated with the susceptibility of the paramagnetic salt and published as a T-To correlation. The procedure is to measure T*, compute T o , and then use the published data to find T46-61[cf. refs. in Table 1111. The following relation defines T*:
c/T* = M / H ,
(2.4.2.3.17)
where c is the Curie constant and M the magnetic moment caused by a small external measuring field, He. T* is thus related only to measurable quantities, although it will be different for differently shaped specimens. The symbol To refers to a spherically shaped sample, and it is customary to relate all T* values to To through the relationship
+ /3
(2.4.2.3.18)
(n,- n)c
(2.4.2.3.19)
T o = T* where
/3
=
where n, is the demagnetizing coefficient for a sphere (= 47r/3), and n that for the specimen used. The following points are worth noting: at higher temperatures, T o 3 T; a t lower temperatures the determination of the T-Tb correlation is a lengthy, and also somewhat inaccurate, procedure; for magnetically anisotropic substances, the T-To correlation will depend upon the direction of measurement with respect to the crystallographic axes; and finally the value of T o measured in the presence of a magnetic field, is not necessarily equal to that which would be measured in zero field at the same temperature, since saturation effects can occur at low fields in the magnetic cooling range. Once the T-TD correlation is known, secondary thermometers, such as carbon thermometers-or, more particularly, y-ray anisotropies of radioactive nuclei, can be calibrated and used. This latter method, the “nuclear thermometer” as it has come to be called, has many advantages. It can be used, for example, to give the temperature of a very specific region (e.g., for work on beta decay of the surface layer of a paramagnetic salt). It is also convenient to use in experiments on radioactivity, since the “thermometric” isotope can be simply included in the crystal and the gamma ray intensity, and hence the temperature, recorded simultaneously with the other data. 2.4.2.3.4. THE PRODUCTION OF ORIENTED NUCLEI-DYNAMICAND TRANSIENT METHODS. The relevant properties of substances used in this class of experiment can usually be described by a spin Hamiltonian. Preferential nuclear population distributions are not caused primarily by lowering temperature; however, they are caused by driving transitions between different energy levels by means of oscillating fields a t a fre-
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
191
quency chosen to satisfy some resonance condition. The oscillating field is usually an electromagnetic field (although the use of acoustic vibrations for the purpose has been proposed) and the frequency ranges from the optical region down through the microwave to the rf region. The basic idea is to induce transitions a t rates comparable to the natural relaxation rates of the system. Saturation effects then occur, and the system is driven away from thermal equilibrium into a state of dynamic equilibrium where certain levels, not necessarily the lowest in energy, are preferentially populated. The new situation can then be thought of as being maintained by a pumping action, where the applied field excites the system in one way and the internal processes relax the system in another way. The optical method, which was the first of its kind to be suggested,61 is in many ways different from the others, For example, it is carried out a t or near room temperature and resonance is performed on free atoms in the vapor phase. The orientation is with respect to the total angular momentum of the atom, F = J I, where J refers to the electronic angular momentum of the atom. Relaxation occurs through spontaneous emission, rather than by taking place through thermal processes as in other methods, although often unwanted relaxation mechanisms such as wall collisions are present and prove to be troublesome. Relaxation effects may be induced also by collision with impurity atoms, a phenomenon that can in fact be used to orient the impurity nuclei themselves.s2 In general, for work with radioactive nuclei, these methods have the disadvantage that the activity tends to stick to the walls of the containing vessel, and since nuclei in such a position are not oriented, they merely give unwanted background counts. Other proposals which have been made refer generally to solid state systems, and t o a large extent stem from a suggestion by O v e r h a u ~ e r . ~ ~ His suggestion was based upon a theoretical study of the relaxation mechanisms for nuclear spins and the spins of conduction electrons in metals. Given hyperfine coupling of the type AS * I, it was shown that upon saturation of the electron spin resonance signal, there accompanied the disappearance of the electron polarization an enhanced nuclear polarization of approximately equal magnitude (in angular momentum units) to that which originally existed among the electron spins. The method is not confined to metals, but it is applicable to a much wider variety of substances; in fact, to most paramagnetic s u b ~ t a n c e s . ~ ~
+
61 A. Xastler, Proc. Phys. SOC.(London) A67, 853 (1954); J . Opt . Soc. Am. 47, 460 (1957). ' 62 H. G . Dehmelt, Phys. Rev. 106, 1487 (1957); 109, 381 (1958). a* A . W . Overhauser, Phys. Rev. 92, 411 (1953). 6' A. Abragam, Phys. Rev. 98, 1729 (1955).
192
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
I n paramagnetic salts we can study the various dynamic effects by simply adding to Eq. (2.4.2.3.8) the appropriate term ho * S cos wt for the applied rf field of amplitude ho, and a random time perturbation X ( t ) for the thermal relaxation effects,l4 The detailed b e h a v i ~ r ’of~the ,~~ system will depend upon matrix elements of these perturbing terms between the various energy levels of the unperturbed spin Hamiltonian. The form of X ( t ) may be quite complicatede6but in some cases, e.g., dielectrics at low temperatures, it gives rise to only one kind of relaxation process, e.g., where only the electron spin is flipped by interaction with the lattice waves (phonons) with no change in direction of the nuclear spin. These processes by themselves do not affect the nuclear polarization directly, and the required effects are brought about by second order processes where the phonons modulate the hyperfine coupling. Consider the simplest case of S = +, I = +, with coupling AS I, and with relaxation processes as described above. I n thermal equilibrium in an external field the energy levels and populations are as shown in Fig. 4a, where it is assumed for convenience that x << 1. The nuclear polarization f l ‘v 0. In the Overhauser method we would apply an rf field transverse to the steady field in order to saturate the two allowed electron spin transitions, as shown in Fig. 4b. Under the combined action of these rf fields and the relaxation processes (remembering that the only relaxation for I is through the term AS * I) the population of the I++ > level will become enhanced at the expense of the - - > level, and a nuclear polarization f l = +s is obtained. In the case that only one of the two allowed transitions is saturated the polarization will be x/2. I n the case of a metal, where the electrons taking part in the processes belong to a highly degenerate Fermi-Dirac system, the details are somewhat more involved but the basic idea is the same. Indeed it has been showne4 that the method should work to a varying degree for a wide variety of substances possessing different types of hyperfine coupling. A variation of these ideas which has proved to be most successful was proposed by Jeffries. The method relies on the fact that, as shown in Fig. 4c, all the energy levels are not necessarily exact eigenstates of S , and I,, and a so-called weak “forbidden” line can be excited. The idea is to saturate this line with an external field and thereby drive the transition which, in the previous method, is caused by second order thermal relaxation processes. In the steady state, shown in the figure, the same degree of polarization but of opposite sign is obtained. In discussing the dynamic methods it has been assumed that the different hyperfine components of a resonance line are resolved. It has been 86 See, e.g., C. H. Townes, ed., “Quantum Electronics.” Columbia Univ. Press,
I
New York, 1960.
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
193
shown recently that this criterion is by no means necessary, and that in fact polarization effects can occur in surrounding nuclei, e.g., protons in the water molecules of crystallization, or in plastics where the protons can be coupled to F-centers. This occurs in spite of the fact that the nuclei are only weakly coupled by dipolar forces to the electron spins. It
I-
x
--
1+I -
I - 2 x I++>
Main relaxation relaxation processes
I i ASJ,
”forbidden”
Saturate
t hrf
S to lotticc
I+X
I -x
ItX ---.-
Itx I
a Thermal Equlllbrlum 1, = o
1t2x
I-->
--I
I b. Overhauser Effect f,.+X
c. Jeffries Effect f,=-X
FIG.4.Energy levels and relative populations of an electron and nuclear system with S = and Z = & in an external field H , and coupled through the hyperfine interaction, are shown. The quantity z = p H / k T is assumed to be <
+
is necessary, however, in order for the method to work at all, that the width of the resonance absorption line should be caused by a mechanism giving rise to inhomogeneous broadening. There are further conditions imposed upon the line width and shape, arising out of the fact that allowed and forbidden transitions will occur at the same time (i,e., different centers will have slightly different resonance frequencies at any given time) and we have to rely on a differential effect. The effect turns out to be greatest in the region of greatest slope of absorption line, and zero when frequency is set a t the center of the line. The method is of special interest
194
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
since nuclei can be oriented which do not necessarily belong to paramagnetic ions, e.g., protons. For a list of references to the numerous experiments carried out to investigate and develop this method the reader is referred e1~ewhere.l~ We shall not discuss the other ingenious methods (listed in Table 11) that have been proposed for producing oriented nuclei since they do not appear to have yet reached the stage where they can be exploited in a practical way for experiments of the type we are considering. 2.4.2.3.5. THE DEQREE OF NUCLEAR ORIENTATION.2.4.2.3.5.1. Description and Notations. In the expressions for the angular effects associated with oriented nuclei, the degree of nuclear orientation enters most simply and naturally in terms of the statistical These provide an alternative statistical description to the density matrix, and they are preferred for present applications since they represent the multipole moments of the orientation. The statistical tensors are defined, in the notation of references 5 and 3, respectively, by the relation [
]g)= p k * ( d , ” ’ I ’ ) (-)r’-m’
=
- m’lkq>
(2.4.2.3.20)
mm’
where < ImI’m’lkq> is a vector addition coefficient, and the last term is an element of the density matrix, pa, for the nucleus in the physical state [a>. The symbols a,a’represent those quantum numbers that refer to quantities other than nuclear spin. The statistical tensors, or rather their complex conjugates, are also the expectation values, , of a set of tensor operators T,k..3973&69 We shall need to consider only systems with sharp angular moment (I’ = I ) and with axial symmetry ( q = 0; m = m’). In this case we may also drop a, a’ and obtain the following simple relations. The only nonzero elements of the density matrix are = am (2.4.2.3.21) where am is the occupational probability of the mth nuclear magnetic substate, and [ ]hk)
=
1(
-)r-m<
Iml - mlkO>
a,.
(2.4.2.3.22)
m
6e A.
Simon, Phys. Rev. 92, 1050 (1953).
A. Simon and T. A. Welton, Phys. Rtc. 90, 1036 (1953);93, 1435 (1954). 68 M.E.Rose, Phys. Rev. 108, 362 (1957);G.F.Koster and H. Statz, ibid. 113,445
67
(1959). 89 E. Ambler, J. C. Eisenstein, and J. F. Schooley, J . Math. Phys. (January, 1962), 3, 118 (1962);ibid. 3, 760 (1962).
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
195
Other notations that appear in the literature and are identical to that defined in Eq. (2.4.2.3.22) are fk(I)’OJ1 and Rk.72Other notations that which are related as follows: differ somewhat are Bk73and fk(I),70~74
Rk
=
Bk/l^ =
(y)
Ik[(2h
+ 1)(21 - k)!/(21 + k f 1)!]”2fk(I) (2.4.2.3.23)
2 = (21 + 1)1‘2.
where
The first four nuclear orientation parameters are given explicitly by (2.4.2.3.24a) (2.4.2.3.24b) f3
f4
[ m3am- ( 3 P + 31 - 1) ma,/5] = I-4[ 2 m4am - (612 + 6 I - 5) 1m2am/7
=
I-3
m
m
m
m
+ 3(1 - 1)(1)(I + 1 ) ( I + 2)/35].
(2.4.2.3.24d)
For nuclear alignment, as opposed to nuclear polarization, the odd orientation parameters, f l , f3, etc., are zero. It should be noted that in many cases we are dealing with a nuclear spin coupled to an electron spin, as displayed in Eq. (2.4.2.3.8). The two systems in physical states a and s, say, are c ~ r r e l a t e d ,so~ ~we must consider their joint density matrix pas. We then have
= TrE(
}
(2.4.2.3.25)
where the quantum numbers M refer to the eigenstates of Sz, and Trl stands for the trace taken over the electron states. I t will be noted also from Eq. (2.4.2.3.8) that it is not possible in general to diagonalize the H. A. Tolhoek and J. A. M. COX, Physica 19, 101 (1953). M. Morita and R. S. Morita, Phys. Rev. 109, 2048 (1958). 7* R. B. Curtk and R. R. Lewis, Phys. Rev. 107, 1381 (1957). 7 3 T. P. Gray and G. R. Satchler, Proc. Phys. SOC.(London) A68, 349 (1955). 74 5. R. de Groot and H. A. Tolhoek, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 613. Interscience, New York, 1955. ‘&See,e.g., U. Fano, Revs. Modern Phys. 29, 74 (1957). 70
71
196
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
energy and I , simultaneously. In computing the a,a it is then preferable to proceed by solving the secular determinant, as we illustrate later. In addition to describing the initial nuclear orientation, the statistical tensors can be used to describe the orientation of any other particle or radiation involved. I n general, for a particle with spin s, only tensor moments up to rank 2s can be nonzero, Thus for particles of spin +,only the tensor of rank 1 exists, and the orientation is completely specified by the polarization. The photon is in a special category. For present purposes it may be regarded simply as a particle of spin 1 with zero rest mass and only two independent polarization states. The polarization states may be chosen in a number of ways, e.g., two orthogonal states of plane polarization, or the two states of circular polarization, i.e., angular momentum projections along the wave vector Ic of + 1 (optically left circular polarization) or - 1 (optically right circular polarization). I n the former case we choose directions of plane polarization, Al and A2, for the vector potential such that (A1,A2,k) forms a right-handed set of axes.7ev77The polarization of any photon can then be written
+
A = aiAi a z A 2 where a1 and a2 are complex numbers such that la112
+
(2.4.2.3.27a)
1
la212 =
(2.4.2.3.26)
and A1 and Az are unit vectors. We now write
-
la212
=
41
aza1*
=
62
i(a1az" - aza1*) =
53
la112 a1az*
+
(2.4.2.3.27b) (2.4.2.3.27~) (2.4.2.3.27d)
so that El measures the difference in intensity of radiation polarized along A1 over that polarized along A2, t2measures a similar quantity for axes rotated through n/4 in a direction A1 + A2, and E3 measures the intensity of optically left-circularly polarized radiation over optically right-circularly polarized radiation. We have for the density matrix
1 P = t 2
+2
El
P = 77
31
- its 2
+2 i t 8
which can be written 76
52
1
- 61 2
+t
'
4.
H. A. Tolhoek, Revs. Modern Phys. 28, 277 (1956). F. W. Lipps and H. A. Tolhoek, Physica 20, 85, 395 (1954).
(2.4.2.3.28)
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
197
For further discussion of the quantities El, t2, t3, which are closely related to the Stokes’ parameters, we refer the reader e l~ e w h e re .~ O-~ ~ If we choose the more appropriate spherical representation for vector quantities
eo = e, (2.4.2.3.29)
and write the following equation to describe the photon polarization
A
+ LYA-
=
the density matrix is given b y 1 P =
+
53
-(El
2
-(51
(2.4.2.3.30)
+itz)
2
- it21
12
2
(2.4.2.3.31) F3
Finally, the statistical tensors are given by
Roo
=
1/43,
Rlo = &/d2, R2o
=
1/46,
Rzkz =
i.5)/2 (2.4.2.3.32a)
-(I1
with
R1*1
=
Rz*l
=
0.
(2.4.2.3.32b)
I n the description of polarization so far, we have considered a single particle in a pure state. The concepts need to be extended in order to describe a beam of radiation which may be unpolarized, partially polarized, or completely polarized. This may be done, for example, simply by taking the appropriate incoherent average over the large number of particles in the beam.7s-79 For photons, for example, a beam traveling along a direction described b y polar angles (0,4) with intensity W(O,4) can be described with the aid of Stokes’ parameters and written as a 4 X 1 matrix:
w e , + ) . (I,w.
w(e,4)- ( i , P t 1 , P ~ 2 , P t 3 )
(2.4.2.3.33)
The magnitude P measures the magnitude of the polarization and ranges from zero for a n unpolarized beam to 1 for a completely polarized beam The components of describe the type of polarization, a s described above. 78 7t1
U. Fano, J. O p t . SOC.Am. 39, 859 (1949). W. H. McMaster, Revs. Modern Phys. 33, 8 (1961).
198
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
2.4.2.3.5.2. Determination. We shall defer until the next section any discussion on the measurement of the degree of polarization of a beam. Here we discuss the determination of the degree of nuclear orientation of the initial state. For the static methods, if we know the Hamiltonian of the system and the temperature of the sample, we have the formal expression : = Tr(T,kp)/Tr p (2.4.2.3.34)
where, for thermal equilibrium
The trace is taken over all states, i.e., if X is the spin Hamiltonian over both electronic and nuclear states. It is useful to have expansions in powers of 1/T, which are obtained by the methods described in Section 2.4.2.3.8and have been given explicitly for a number of cases in Section 2.4.2.3.2. At the lowest temperatures, however, these expressions are not accurate enough, so that it is necessary to solve the eigenvalue problem and compute the am's.Fortunately this is often quite simple since the secular determinant factors and closed . ~ ~ ~ parameters ~ ~ ~ ~ ~ ~ ~ expressions can be obtained for the u , ' ~ Orientation for a few simple cases have been tabulated for a number of values of The secular determinant corresponding to the most frequently met with spin Hamiltonian, viz., A=, Av, and D all zero, and H along the z-axis, in the representation that makes M and m diagonal, factors into two linear equations and 2 1 quadratic equations. The latter are of the form: m,
m
-t
= 0.
- 1, +t
a22
(2.4.2.3.36)
-W
The eigenvalues are given by ‘w = ;[(a11
+ a d k ((a11-
+ 4 a l ~ a ~ ~ )(2.4.2.3.37)
a22)2
1’2]
and the transformation between energy and angular momentum eigenstates is IWr> = a r , m l m , -& > 4- a,,,-llm - 1, ++> (2.4.2.3.38) 80 81
N. R.Steenberg, Proc. Phys. Soc. (London) A66, 399 (1953). T.P. Gray and 0. R. Satchler, Pmc. Phys. Soc. (London) A08, 349 (1955).
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
with a,,, and
ar,m-l
199
given by (a11 -
wr)ar,m
+ a i 2 ( ~ ~ , ~ -01 =
(2.4.2.3.39)
and the normalization condition
We then obtain am by means of the equation
When the terms in D, As and A,,, or H , and H,, are present the secular determinant may not factor so simply. Perturbation methods can be employed in many cases, however, while in the most complicated cases the dimensionality of the problem [i.e., (2s 1)(21 l ) ] is small enough to permit rapid solution with the aid of an electronic computer. 2.4.2.3.5.3. Perturbing Effects. Two effects will be considered that can alter the nuclear orientation from the values computed by the above methods, viz., the effect of interionic couplings and the effect of perturbations acting on intermediate states. Strictly speaking, Eq. (2.4.2.3.8) applies only to substances of high magnetic dilution, where the interaction between the paramagnetic ions themselves can be neglected. This coupling must certainly be taken into account in concentrated salts, e.g., in computing specific heat, and the question arises whether the coupling affects the nuclear orientation. The type of interionic coupling predominant in the salts we are considering is of the dipole-dipole type. This can be included formally by summing Eq. (2.4.2.3.8)over all ions, and adding the interaction
+
+
where the prime denotes summation over all pairs. The components of the magnetic moment p are (g.pSz,gypSy,gzpS,). It has been ~ h ~ ~ nbY~ ~ ~ ~ ~ applying the methods described in Section 2.4.2.3.8that these interactions do not appear in the leading terms of the orientation. Thus, for example, they do not appear in the expression for fz until the term in l/T4. There are a few cases found experimentally, however, where the interactions do have a significant effect on the orientation, viz., the cases of the 8z
N. R. Steenberg, Phys. Rev. 93, 678 (1954). Can. J . Phys. SB, 1133 (1957).
J. M. Daniels,
200
2.
DETERMINATION O F FUNDAMENTAL Q U A N T I T I E S
divalent ions Co++ 84 and Mn++ in cerous magnesium nitrate. It has been suggested that the effect is the result of antiferromagnetic ordering,86 and it has also been conjectured how a strong interaction between pairs of ions might be i m p ~ r t a n t . ~An ’ explanation is still lacking, however, although from the more practical point of view i t is worth noting that the disturbing effects can be overcome completely by applying an external In the special case of neomagnetic field of a few hundred dymium ethyl sulfate, interionic interactions are also important, but here the crystal structure is such that often a good approximation is obtained by considering only the two nearest neighbors.88 Reorientation effects in an intermediate state of a decay sequence have been investigated a great deal in connection with angular correlation s t u d i e ~ , ~and ~ - to ~ ~a lesser extent specifically for oriented n~clei.~~*94,96 The effects are divided into two groups, dynamic and static, depending on whether or not the interaction is described by a Hamiltonian depending explicitly on time. Static perturbations comprise effects due to external magnetic fields, hyperfine splitting, and so on. Dynamic perturbations comprise effects due to spin-lattice relaxation, nuclear recoil (e.g., as in a-decay), and rearrangement of electrons among the atomic orbits (e.g., following p-decay and, particularly, K-capture). The static effects can be estimated quite accurately, whereas only qualitative arguments are made concerning the dynamic effects. Generally it appears that reorientation effects could be important for mean nuclear lifetimes longer than about lo-” sec. The experimental evidence on the subject, which is meager, seems to indicate that for cascades initiated by p-decay, effects if they exist at all are small, but for K-capture they may be large. Thus in practically the same substance, there are no effects observableg6 a t all for CoflO, 84 E. Ambler, M. A. Grace, H. Halban, N. Kurti, H. Durand, C. E. Johnson, H. R. Lemmer, and F. N. H. Robinson, Phil. Mag. 171 44, 216 (1953). 85 M.A. Grace, C. E. Johnson, N. Kurti, H. R. Lemmer, and F. N. H. Robinson, Phil. Mag. [7]46, 1192 (1954). 86 M.W.Levi, R. C. Sapp, and J. W. Culvahouse, Phys. Rev. 121, 538 (1961). ST E. Ambler, R. P. Hudson, and G. M. Temmer, Phys. Rev. 101, 196 (1956). 88 C. E. Johnson, J. F. Schooley, and D. A. Shirley, Phys. Rev. 120, 2108 (1960). ED A. Abragam and R. V. Pound, Phys. Rev. 92, 943 (1953). V. Gillet, Nuclear Phys. 20, 561 (1960). 91 H. Frauenfelder, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 531. Interscience, New York, 1955. 92 R. M. Steffen, Phil. Mag. Suppl. [8]4, 293 (1955). 93 E. Heer and T. B. Novey, Solid State Phys. 9, 199 (1959). 94 N. R.Steenberg, Phys. Rev. 96, 982 (1954). s6 H. A. Tolhoek, C. D. Hartogh, and S. R. de Groot, J . phys. radium 16,615 (1955). S6 H.R. Lemmer and M. A. Grace, Proc. Phys. SOC.(London) A67, 1051 (1954).
2.4.
DETERMlNATION O F SPIN, PARITY, A N D NUCLEAR MOMENTS
201
whereas the 137 kev level in FeS7(lifetime -5 X sec) seems to be perturbed ~ t r o n g l yfollowing ~ ~ , ~ ~ K-capture in Cofi7.For further discussion on dynamic perturbations we refer the reader to references 91-93, There is one essential point of difference between static perturbations as they affect angular correlation as compared to nuclear orientation. In the former case one tries to choose substances where the effects of external fields at the nucleus are small, whereas in the latter case they must par excellence be large. This does not mean that attenuating effects should be larger in the latter case; on the contrary, they may be less. As is well known from angular correlation theory, an external field when suitably oriented, i.e., along the direction of the first radiation, serves to preserve or restore the angular correlation. Similarly, if the symmetry of the perturbing field is the same as the (assumed) axially symmetric field that produced orientation, angular distributions from oriented nuclei may be unaffected. The same conclusion would hold for angular correlations with oriented nuclei, provided the first radiation is observed along the axis of symmetry, but would not necessarily hold if the first radiation were observed in a perpendicular direction. This comes about firstly because the nuclear orientation remains axially symmetric, and secondly -and partly as a result of this-because the spin flipping terms are small. Thus if we suppose that the interaction in the intermediate state is given by a spin Hamiltonian, X', with constants g', A', B', etc., the terms that give rise to reorientation are off-diagonal, e.g., B', and if these are zero there is no reorientation a t all. 2.4.2.3.6. MEASUREMENTS OF ANGULAR DISTRIBUTIONS AND POLARIZATIONS. The majority of experiments have been carried out below 1°K and have involved magnetic cooling. The apparatus used has been developed using techniques already established in this field, with appropriate modifications for the different types of particle to be counted. Measurements have been made so far on the angular distribution of gamma rays,98,99 the plane polarization of gamma rays, l o O , l o l the circular polarthe angular distribution of beta parization of gamma ray~,~02,103 97 G. R. Bishop, M. A. Grace, C. E. Johnson, A. C. Knipper, H. R. Lemmer, J. Perez y Jorba, and R. G. Scurlock, Phil. Mag. [8]46,951 (1955). 98 B. Bleaney, J. M. Daniels, M. A. Grace, H. Halban, N. Kurti, F. N. H. Robinson, and F. E.Simon, Proc. Roy. SOC.A221, 170 (1954). s9 0. J. Poppema, M. J. Steenland, J. A. Beun, and C. J. Gorter, Physica 21,233 (1955). loo G. R. Bishop and J. Perez y Jorba, Phys. Rev. 98,89 (1955). lo* R. W. Bauer and M. Deutsch, Nuclear Phys. 16,264 (1960). lea J. C. Wheatley, W. J. Huiskamp, A. N. Diddens, M. J. Steenland, and H. A. Tolhoek, Physica 21, 841 (1955). 1°3 1%.W. Bauer and M. Deutsch, Phys. Rev. 117, 519 (1960).
202
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
t i c l e ~ , ’ ~the 4 ~ ~angular ~~ distribution of alpha particles, 106,107 and the angular distribution of fission fragments.los In addition to the original references given above to descriptions of apparatus, we also refer the reader to references 10-14. Experiments have been made also on the angular distribution of gamma rays using dynamic methods of nuclear orientation. 14,108 We cannot hope to describe all the above apparatus in detail, so we shall discuss those aspects of the design that are particular to the nuclear orientation method and are not normally encountered in nuclear spectroscopy on the one hand and magnetic cooling on the other. These stem largely from the fact that one has to deal with thick sources placed inside a cryostat. When one is dealing with penetrating particles, e.g., gamma rays of energy greater than about 10 kev, there is no need for any special technique. Conventional NaI (Tl) scintillators may be placed outside the cryostat, and only small corrections are required for absorption and scattering within the source and in the walls of the cryostat. The use of a multichannel a n a l y ~ e for r ~recording ~ ~ ~ ~data ~ ~ is useful, among other things, for assessing the amount of scattering, as well as for recording data for several gamma ray energies simultaneously. For short range particles, problems arise from the fact that the source cannot be made close to ideal, i.e., “massless,” and from the inability of such particles to penetrate the walls of the cryostat. The procedure adopted with alpha and beta particle sources is to take an inactive crystal weighing a few grams and place on one well-defined surface a drop of concentrated solution of the salt containing the activity. It is found that when the solution is allowed to crystallize it does so as a continuation of the base single crystal. I n the case of alpha particle sources the base crystals of uranyl rubidium nitrate are normally employed (it should be noted that there is in these samples a background heating due to alphaparticle activity of about 5 ergs per minute per gram). I n the case of beta-particle sources it is found that suitable sources can be made by confining the activity to a surface layer about 10011 thick. Although there is some multiple scattering and considerable backscattering preslo4 C. 8. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys. Rev. 106, 1413 (1957). IosH.Postma, W. J. Huiakamp, A. R. Miedema, M. J. Steenland, H. A. Tolhoek, and C. J . Gorter, Physica 24, 157 (1958). lo8J. W. T.Dabbs, L. D. Roberts, and G. W. Parker, Physica Suppl. S69 (September, 1958). lo’s.H. Hanauer, J. W. T. Dabbs, L. D. Roberts, and G. W. Parker, ORNL Report 2919 (unpublished). 108 F. M. Pipkin, Phys. Rev. 112, 935 (1958). loO D. A. Shirley, J. F. Schooley, and J. 0. Rasmussen, Phgs. Rev. 121, 558 (1961).
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
203
ent, so that substantial corrections have to be applied, a check can generally be made on the consistency of this procedure. This involves comparing the value of f~ deduced from beta-particle asymmetries, the value of fi deduced from gamma-ray anisotropies, and the temperature as determined by a measurement of T* for a salt whose thermal and magnetic properties are known and for an isotope whose decay scheme is known. Some discussionllOolllhas been given of the more important corrections to be applied when using beta-particle sources. Of the electrons emitted in the backward direction, between and are returned to the forward hemisphere. Although there is some degradation in energy, this is not large for beta particles, so that it is hard to make an energy discrimination. The procedure adopted is to reject the very lowest part of the beta-particle spectrum since the corrections are too large to be allowed for accurately. The remaining part of the spectrum is then compared with a “reconstructed11spectrum formed by taking the spectrum of an ideally thin source and computing the thick source behavior, using experimental data on backscatteringl12,ll3of material of the same average Z as the crystal. This is also compared to the spectrum taken with a betaray spectrometer, first of the thin source alone, and then with the same source placed on a crystal of the paramagnetic salt. A correction must be applied also for Compton electrons generated in the surroundings. Since this is directly related to the geometry involved, it must be determined in relation to the particular apparatus used. It may be evaluated by making spectral measurements with and without an absorber placed over the source, just thick enough to stop the beta particles. Although in the particular case of positrons it is possible to measure the asymmetry by measuring the annihilation radiation with scintillators placed outside the cryostat,lo6 it is generally preferable to place the counters inside the cryostat. lo4 This immediately brings up the question of the behavior of light pipes, phototubes, scintillators, and other detectors, such as surface barrier counters a t low temperatures. The more usual photomultiplier tubes cease to operate a t low temperatures, principally because the photocathode does not remain electrically conducting as the temperature is lowered. Special types may be obtained, however, with photocathodes that can be operated down to liquid helium temperatures and which have remarkably low noise. An alternative
+
IlaR. W. Hayward and D. D . Hoppes, I R E Trans. on Nuclear Sci. NS-6, No. 3 (December, 1958). 111 D. D. Hoppes, Natl. Bur. Standards Tech. Note No. 99 (1961), unpublished. 114 W. Bothe, 2.Naturforsch. 49, 542 (1949). 113 H. H. Seliger, Phys. Rev. 88, 408 (1952).
204
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
arrangementlo4to that of operating a photomultiplier a t low temperature is to use a lucite light pipe, 50-100 cm long, between the scintillator in the cryostat and the photomultiplier a t room temperature. Since lucite is both a poor thermal conductor and opaque to infrared radiation, fairly large diameter light pipes can be used to extract light pulses with good efficiency, without introducing prohibitively large amounts of heat into the cryostat. As a scintillator for counting beta particles, anthracene works quite well when operated at low temperatures, although rapid cooling or warming will cause cracking and consequent loss of resolution. Under good conditions an overall resolution of about 15% can be obtained with the 624 kev conversion electrons from Cs’37. Anthracene crystals also work satisfactorily a t low temperatures as alpha-particle detectors, although the simplest arrangement in this case, and also for fission fragments, is to use surface barrier counters. We refer the reader elsewhere for further details.1l4~~~~ Up to the present time, measurement of the polarization of particles emitted from oriented nuclei has been confined to gamma radiation, for which detectors based upon Compton scattering have been used. The sensitivity of such a detector to polarization is obtained from the KleinNishina formula, and can be handled very simply by means of Stokes’ parameter^.^^,^^ Thus, in the notation of Eq. (2.4.2.3.33),if the incident and scattered photon beams are given by No(l,P0fo)and N,(l,P&) respectively, then (2.4.2.3.43)
where the matrix T is given by Eq. (2.4.2.3.44). T
= 2
(A)*(i)’ mac
1
+
x
OOS~
+ (ko - k ) ( l - cos x) sin2 x
sin2 x
1
0 -(1 - c o a x ) ( k c o s x + k o ) ’ S
COB
COB*
x
0
(1 - c o s x ) ( n a X n ) . ( k X S )
+ .
0 ‘0
2
+
x
-(1 - COB x)(ko coa x k) S (1 - 008 x)(n X no) (ko X S) (1 - cos x)(ko X n) S 2 coa x (ku k ) ( l COB X)COB x
. . -
1
.
(2.4.2.3,44)
- con x)(k X no) . S + In these expressions, ko and k are the momenta (in units of moc) of the incident and scattered photon respectively, and noand n are unit vectors (1
F. J. Walter, J. W. T. Dabbs, and L. D. Roberts, Rev. Sci. Instr. 81, 766 (1960). J. W. T. Dabbs and F. J. Walter, eds., “Semiconductor Nuclear Particle Detectom.” Publ. No. 871, National Academy of Sciences-National Research Council, Washington, D.C., 1961. 1~
116
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
205
along these directions respectively. Th e magnitudes k o and k are connected by the well-known relation
l / k - l / k o = 1 - cos x (2.4.2.3.45) where x is the scattering angle. S is a unit vector along the initial direction of the spin of the scattering electron, mo is the rest mass of the electron, c is the velocity of light, and T is in units of cm2/electron. The choice of axes is such that, with the notation of Section 2.4.2.3.5
A1” = A 1 = (noX n) (2.4.2.3.46) where A” and A are the vector potentials of incident and scattered photon, respectively. The axes to which the ko-photon is referred (z’y’z’, with ko along 02’) will be related to those of the nuclear orientation (zyz, with n along 02) by a right-handed rotation through angle 0 about Oy. Maximum . aligned nuclei the sensitivity is obtained by taking e equal to ~ / 2 For beam can be described by the intensities, N,. and Nu<,of components polarized along x’ and y’ respectively. If the beam then undergoes Compton scattering through angle x , with a n azimuthal angle # about Oz’, the scattered intensity is given b y N Z i Nu, N = +[lOOO]T*R[ N,. - Nut (2.4.2.3.47)
+
:
rl where
0
0 sin 2#
]
01 (2.4.2.3.48)
0 With the choice of axes made, the direction # = 0 represents scattering perpendicular to the nuclear alignment axis, and # = ?r/2 parallel to it. If respectively, we have the scattered intensities are N I and (2.4.2.3.49) where the plane polarization P ~ =I ( N , ) - N,t)/(N,t (2.4.2.3.27b)and (2.4.2.3.33)], and & ( x , k ~= ) sin2 x / [ l
+ N U j ) ,[cf. Eqs.
+ cos2 x + (ko - k)(1 - cos x ) ] .
(2.4.2.3.50)
against x for different values of The value of Q ( x , k @ )can be pIotted116,117 Ice, and the optimum value of x can be chosen for any given photon energy. 116 117
H. Schopper, Nuclear Znstr. 3, 158 (1958). L. Grodsins, Progr. in Nuclear Phys. 7 , 163 (1959).
206
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
Thus, for example, for photons of energy 1 MeV, the optimum value of x is about 76". At such large scattering angles the cross sections are small, so that it is desirable to use strong sources and as large a scattering volume as is consistent with good geometry and the avoidance of double scattering. The counters N I and NI,must be well shielded from direct radiation. Spurious counts are further minimized by using a scintillator as scatterer (usually a plastic scintillator) and setting the counts caused by the recoil electrons in coincidence with N I and "1. The method of linear addition of the pulses has been used to discriminate further.lol The Compton scattering from magnetized iron is usually used to detect circular polarization. The scattered intensity in the general case is easily obtained from Eq. (2.4.2.3.43).For the special case when S, n, ko and k are coplanar we have
Ni
-
+
+
[l cos2 x . (ko - k ) ( l - cos x)] - sin2 xP51 f (1 - cos x ) [ ( ~ o k)cos 8 cos x - k sin B sin xJfP& (2.4.2.3.51)
+
whcre N+ and N - refer to S and r] parallel and antiparallel respectively. Note that the angle between k and s is 18 x l , and that with our definition P& is positive when A is in n - ko plane. Note also that S and the magnetization are antiparallel, and f is the average polarization for electron in the scatterer (for iron, f = A). The figure of merit, M , exclusive of corrections, e.g., for solid angles, is defined by the relation
+
M
=
(N+ - N - ) / ( N +
+ iv-).
It is not very large; thus, for example, for completely polarized radiation with P = 1 and t~ = 0, the value of M for photon energies around 1 Mev is a little over 3%. WITH THEORY. Formulas for the angular distri2.4.2.3.7. COMPARISON bution and polarization for various types of radioactive decay of oriented nuclei have been given by many authors. We quote the more frequently used formulas here explicitly. We denote by I, the spin of the initial nucleus, and by Ib the spin of the final nucleus. The symbol f is used for (21 1)lI2, and n for a unit vector along the axis of nuclear orientation about which we always assume cylindrical symmetry. 2.4.2.3.7.1. Alpha-particle Emission. Since the alpha particle has spin zero, no polarization analysis is possible and the only relevant formula is that for the angular distribution. The relative probability of emission along a direction p is given by4J18J19
+
(2.4.2.3.52) < L O L ' O I ~ O > B ~ P ~ el. (COS P. J. Brussard and H.A. Tolhoek, Phyaica 24, 233, 263 (1958). 119 N. R. Steenberg and R. C. Sharma, Can. J . Phys. 88, 290 (1960).
118
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
207
The quantities of U L , U L J are the transition matrix elements for orbital angular momenta L and L’ respectively. Since there is no reason to suppose t>hat they will be real, the argument of the complex number is combined with the Coulomb phase shift in the quantities (YL and a L I . The angle 0 is that between p and n. We note that the vector-coupling coefficient vanishes unless ( L L’ k ) is even, and 1 I, - l b l < L < I, 4- I b ; that parity conservation allows L to take only even (odd) values when the initial and final nuclear states have the same (opposite) parity; and that except when I b = 0, more than one L value is possible, and we may get interferences between different matrix elements. I n most cases the values of aL decrease fairly rapidly with increasing L, so that usually only two aL values need be considered. The quantity in the curly brackets is the 6-j It should be remembered, however, that if the alpha-particle counters do not resolve alpha particle fine structure, a suitably weighted average of expressions such as Eq. (2.4.2.3.52)must be taken, and since the contributions came from different final states, they combine incoherently. For further details the reader should consult references 106 and 107. 2.4.2.3.7.2. Gamma Emission. The relative probability of emission of gamma rays along a direction k with polarization t is given by
+ +
W(k,t,n) =
1MLtML*BkFk(LL’;IbIa){ 1
LL’
Pk(C0s
0)
k,even
where
(2.4.2.3.54)
Fk(LL‘ ;I b I a )
=
Fk(L‘L;Ib I a )
= (-)lo+lb--l(-)L+L’&€~
- llkO>
I’
( L L‘
ak
(2.4.2.3.55)
Ib]
+
and r L 7 is 1 (or - I) when the parity of the radiation L’ is even (or odd). It will be noted that the only terms with odd k values, e.g., those depending upon nuclear polarization, involve a measurement of ,&,, i.e., the circular polarization. A measurement of sense of the circular polarization, therefore, serves to measure the sign of B1 and hence the direction of the nuclear spin relative to an external magnetic field. Since the direction of the magnetic field at the nucleus can also be deduced (note that 120 M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, “The 3-j and 6-j Symbols.” Technology Press (Mass. Inst. Technol.), Cambridge, Massachusetts, 1959.
208
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
when there is hyperfine coupling the directions of the external field and hyperfine field may be opposed), this measurement gives the sign of the nuclear magnetic moment. The parity of the radiation only enters explicitly in the last term, which involves a measurement of the plane polarization of the photon. Since the amplitudes, M L , MLt, etc., of the various multipoles decrease rapidly with increasing L, it is useful to give the explicit formula for a mixed 2L-pole and 2L+1-poletransition. Assuming we have aligned nuclei so that we can drop the second term in Eq. (2.4.2.3.53), we have
W(k,t,n)
Bk(Fk’Pk(cos 0)
=
+ F:’Pk2(cos 0),5}
(2.4.2.3.56)
k,even
where Fk’
=
+ 62Fk(L-k 1, L + 1; IbI,) + 2bF(L, L + 1; + 6’1 (2.4.2.3.57) & [fk(LL)Fk(LLlbI,) - 6’fk(L + 1, L + 1)Fk(L + 1, L + 1; - 26fk(L, L + 1)Fk(L, L + 1; IbI,)]/[l + 6’1 (2.4.2.3.58)
[Fk(LLIbI,)
IbIa)]/[l
and
F:’
=
IbIa)
where the upper sign is taken when L is electric radiation and the lower sign when it is magnetic radiation. Equations (2.4.2.3.57) and (2.4.2.3.58) L be determined uniquely show that the “mixing” ratio 6 = M ~ + I / M can by measuring the angular distribution and plane polarization. The values of the M L are usually taken to be real. This is always possible, provided interactions are invariant under time reversal, by a proper choice of phases for the angular momentum eigenfunctions. This is done in this case by fixing the phase of the vector potential and hence the multipole operators. The effect of spin and spin changes are reflected in the coefficients Fk(LL’Id,). Since these quantities have been t a b ~ l a t e d , ~it~is~ -a ~ ~ ~ fairly simple matter to construct theoretical angular distributions using trial spin assignments to compare with the experimental curves. Finally, since we are seldom dealing with a n oriented isomeric state, we must consider the case that the gamma emitting state may be fed from an initially oriented state with spin lo,through a cascade l o - + I1+ 121
Iz+ .
*
*
+ In-, + I,
I , + It,. (2.4.2.3.59)
M. Ferentz and N. Rosenzweig, Argonne National Laboratory Report ANL-
5324.
K. Alder, B. Stech, and A. Winther, Phys. Rev. 107, 728 (1957). A. H. Wapstra, G. J. Nijgh, and R. van Lieshout, “Nuclear Spectroscopy Tables.” Interscience, New York, 1959. 122
113
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
209
The orientation parameter of the initial level, Bko, is related to that referred to in previous formulas, Bk., say, by the relation Bk.
=
BkoUk,Uk2’
’
’
where we have one “depolariaation coefficient,” transition, and where u k , refers to the part
uk, for
each intermediate
LqL.’
Is-1
(2.4.2.3.60)
u k ,
(2.4.2.3.61)
I,
of the cascade. If this transition is mixed, as shown in Eq. (2.4.2.3.61), so that the total angular momentum carried away is L,, L8‘,Lit, etc., with amplitudes a, a’, u” etc., we have uk,
+ a”Uk:(L,’) +
= [a2Uk,(Ls)
’
* ] / [ af 2 a” f
’
*
’1
(2.4.2.3.62)
where
When we are considering a beta-transition in this connection we note that the angular momentum carried away by the beta-neutrino pair is usually small. When the nuclear spins involved are quite large the depolarization coefficients are close to unity and the angular distribution is not greatly affected. When the nuclear spins involved are quite small, however, e.g., in the case of CoS8( I = 2), the effect may be considerable. The effect can then be used to obtain the transition probabilitie~~124.126 a2, a’2, etc., through the coefficients uk in a similar, though less sensitive, way to that used to obtain the 62 from the Fk. 2.4.2.3.7.3. Beta-particle Emission. The relative probability for emission of electron with momentum p is’22q126-12*
The parameters bk’(LL’) have been defined slightly differently by several authors, and also contain apparent differences due to different normaliza124
Iz6
P. Dagley, M. A. Grace, J. S.Hill, and C . V. Sowter, Phil. Mag.[8]3,489 (1958). F. M. Pipkin and J. W. Culvahouse, Phys. Rev. 109, 1423 (1958).
A. M. Bincer, Phys. Rev. 112, 244 (1958). A. Morita and R. S. Morita, Phys. Rev. 109, 2048 (1958). For a more complete list of references to phenomena on beta radiation, see Table I of N. Kurti, Nuovo cimento Suppl. [lo] 0, 1101 (1957). 126
127
210
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
tions of lepton wave functions. I n Eq. (2.4.2.3.64)the parameters bk’(LL’) are given by
_”; 0”) ifaLx’(
-)“‘bk(LL’)
bk’(LL’) = 16d‘(Z,E)
(2.4.2.3.65)
where the ba(LL’) are those explicitly given in reference 126 for electrons; by a simple substitution those for positrons can also be obtained. The normalization is such that the absolute transition rate, N ( p ) d p , for electrons with momentum p to p d p , is obtained by multiplying the term k = 0 in Eq. (2.4.2.3.64) by (l/2r3)p2q2 d p where p is the neutrino momentum. The parameters bk(LL’) contain, as well as reduced nuclear matrix elements, reduced lepton matrix elements. The latter are evaluated, with certain approximations, in reference 126 with the normalization of 1 in a box of unit volume. If other evaluations are required, note must be taken not only of normalization, but also of conventions used in tabulating Coulomb phase shifts, Ag. Thus the Coulomb wavefunctions given by Bhalla and must be multiplied by ( 2 ~ 9 - l in ’ ~ order to be used in the bk(LL’)of reference 126. The prescription is also given for obtaining the factors depending upon different choices of Coulomb phase shifts A,. The quantities L and L’ are the multipole orders of the beta radiation, i.e., the angular momentum carried off by the beta-neutrino pair. We note that when the neutrino is not observed ( L L‘ k) is even. The multipole order must satisfy the condition A(LIaIt,),and k must satisfy the conditions A(kLL’),A(kIJa). For allowed beta decay the relevant multipole operators are C V ( ~ )( ,L = 0) and C A ( ~ () L, = 1); for first forbidden beta decay they are CAY&and CA(-ii. 2) with L = 0, Cv(ir), CA(d X r), and Cv(yso) with L = 1, and CA(iBiJ with L = 2. The phases of the operators are chosen so that the reduced nuclear matrix elements are explicitly real with proper choice of phase for the angular momentum eigenfunctions. The coupling constants Cv and C A are included to show the connection of the operator to either the vector or axial-vector part of the interaction. The expression for the beta-gamma correlation function from oriented nuclei can be found in reference 130. IN POWERS OF 1/T. It is generally useful, and in 2.4.2.3.8. EXPANSION some cases sufficient, to be able to obtain various properties such as the orientation parameters in a “high temperature” approximation. It is possible to obtain formulas for any of the statistical tensors as a power
+
+ +
lZQC. P. Bhalla and M. E. Rose, Oak Ridge National Laboratory Reports ORNL2954 and ORNL-3207. Iso M. Morita and R. S. Morita, Phys. Rev. 110, 461 (1958).
2.4.
211
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
series in 1 / T , by using a method originally due to Van Vleck.I3l The average value of any operator, 0, is given by:
0
=
(Tr Op)/(Tr
(2.4.2.3.66)
p)
where Tr stands for the trace, and p is the density matrix for the system. For thermodynamic equilibrium p =
exp( - X / k T ]
= 1- X/kT
+ X2/2!k2T2 - X3/3!k3T3+ - -
where X is the Hamiltonian.
...O = [ < O >
- /kT -
*
+
/2!k2T2 .][1 - < X > / k T <X2>/2!k2T2 -
+
. *I-' *
where
=
(Tr OXn)/Tr (1).
Thus
O
=
[
-
/kT
+ /2!k2T2 -
+ { - < X 2 > / 2 ! k 2 T 2+ < X > 2 / k Z T 2 )+
( a
*
a )
*
+
*
.][1
+ <X>/kT
. . .]. (2.4.2.3.67)
This expression simplifies further since for all tensor operators, except for those of rank zero, the trace is identically zero. The trace of the spin Hamiltonian, moreover, is generally made zero by adding constant terms, so that Eq. (2.4.2.3.67) finally becomes, setting 0 equal to T,",
F,"
=
[- < T , " X > / k T ]
+ [ { 3 < T , " X > <X'>
+ [/2!k2T2] -
< T q k X 3 > ) / 3 ! k 3 T+] 3
*
.
(2.4.2.3.68)
The advantage of the method lies in the fact that the trace of the matrix is invariant under canonical transformation. The traces contained in Eg. (2.4.2.3.68)may be evaluated in any representation, therefore, and it is unnecessary to solve the eigenvalue problem. The advantage is greatest when we have a problem of large dimensionality, e.g., when we wish to investigate the effect of interionic couplings discussed below, but the method has proved extremely useful in many other cases. The appropriate form of the Hamiltonian to be taken is that given by Eq. (2.4.2.3.8),and the operators in which we are interested are the T,k of Section 2.4.2.3.5. The T," are listed explicitly in reference 69. An important point to remember is that the different parts of 0 and X do not necessarily commute, so that care must be taken in multiplying out the terms in Eq. (2.4.2.3.68). One is finally left with a sum of traces of products of angular momentum operators raised to a certain power, e.g., I z p ~ ~ ~ I numerical z7, values of which have been given in tables.6e 131
J. H. Van Vleck, J . Chem. Phys. 6, 320 (1937).
212
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
As an example of the use of Eq. (2.4.2.3.68),we compute the nuclear polarization and nuclear alignment to order 1/T2 for the spin Hamiltonian of Eq. (2.4.2.3.8),neglecting the last term. The appropriate relations are, from Section 2.4.2.3.5, =
fl
R1[(21
+ 2)(21 + 1)(21)]1'2/2 4
3I
=
/I
and f z = R2[(21
+ 3)(21 + 2)(21 + 1)(21)(21 - 1)]"2/6 4 5 I 2 = < I z 2 - Z(I + 1)/3>/12.
Since in taking the trace we must sum over both electronic and nuclear levels. we have T r 1 = (25 1)(21 1 ) .
+
+
The nuclear polarization contains no term in 1/T (since we have neglected the direct influence of the external field), and for the term in 1/T2 we need
=
+
+ 1)/9
2g$3HzA Tr(SZ2Iz2)/Tr 1 = 2g1@Z,A5(5 1)I(I
whence we obtain Eq. (2.4.2.3.9). In the alignment a term arises in 1/T through the quadrupole coupling, and we need <[fx2
- I(I
+ 1)/3]X>
=
PI(I
+ 1)(2I + 3)(21 - 1)/45
from which we obtain the first term of Eq. (2.4.2.3.10). For the term in 1/T2 we obtain the following nonvanishing terms
+
<[I,* - I(I 1)/3]X2> <[I,' - f(f f 1)/3][A2Sz21z2 B2(Sz21z2 Sy21y2) p'(1.z' - I(I 1)/3)2]> = ( A 2 - B2)S(S l)Z(I 1)(21 - 1)(21 3)/135 P 2 1 ( I 1)(21 5)(21 3)(21 - 1)(21 - 3)/945
+
=
+ +
+
+
+ + +
+
+
+
whence we get the remainder of Eq. (2.4.2.3.9). It will be recognized that the number of terms in the expansion Eq. (2.4.2.3.68) increases considerably as we go to higher powers in 1/T. There is also the disadvantage that, since all terms are treated on an equal footing, it may be necessary to consider a prohibitively large number of terms. Some advantage can often be gained by a specific choice of representation, 131 in particular an interaction r e p r e s e n t a t i ~ n . ~ ~ Some of the remarks made in Section 2.4.2.3.3 may be made more quantitative with this method. Standard expressionsse for the molar
2.4.
DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS
213
entropy S and the molar specific heat C are
S
R[ln(2S
=
+ 1)(21 + 1) - < X 2 > / 2 ! k 2 T 2+
and
. .]
c = T as/aT.
Let us consider a paramagnetic salt in a n external field H a t angle 8 to the z-axis in the zz-plane, and for simplicity take the Hamiltonian Eq. (2.4.2.3.8)without the terms in A=, Ay,D , and gn. Since we are to be concerned with the properties of the salt as a whole we must allow couplings between the ions. We therefore consider a n assembly of N ions coupled through the dipolar coupling given in Eq. (2.4.2.3.42). The complete Hamiltonian is X =
2
{ gllPH cos 8
i
+ glpH sin e + A is,iI, + B(iS, iI,+
+ P ( i I z 2- 1(1 + 1)/3)} +
2'
((g112P2 is,
isz+ g12p2 is,js, +
ij
- 3(g1\P' S z Z c j
g12p2
r?. $1
iI,)
is, is,)
+ glP 'Srhj+ g i P iS,gij)(gllPjSxzij + glBfSzx;j + glP jS,yij) rjj
where the first summation is over all N spins, and the second (primed) summation is over all pairs i and j . iS denotes the spin of the ith ion, After expanding X2 and evaluating the nonvanishing traces, we make the y;j = r:. - z:j and obtain the expression substitution x:j
+
, - 2k2T2 - (gl12pzHzcos2 8 gL2p2H2sin2 8)(S(S 1)/3) (A2 2B2)S(S 1)1(1 1)/9 P21(1+ 1)(21 - 1 ) ( 2 1 + 31/45 1
+
[
+
+
+
+
+
+ terms in higher powers of 1/T
+
1.
(2.4.2.3.69)
This expression has been given p r e v i o u ~ l y , 'together ~~ with the explicit evaluation of the lattice sums for a number of salts. It will be seen th a t lS*
J. M. Daniels, PTOC.Phys. SOC.(London) A66, 673 (1953).
214
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
the first term depending on l/Tz in the brackets represents the entropy reduction due to the external field; the second, that due to the magnetic hyperfine splitting; the third, that due to the electric hyperfine splitting; and the last, that due to the mutual magnetic coupling between the ions. It can be seen from Eq. (2.4.2.3.69) that in magnetic cooling when the external field falls to such a value that the term in H2 is no longer very large compared with the others, we should expect deviations from formula (12) of Section 2.4.2.3.3. It will be noted that Eq. (2.4.2.3.69)is only useful for fairly small values of HIT. For large enough values of field, however, we may neglect the internal forces, in which case the expression for the entropy becomes quite simple, viz., S = R[ln(25
where y
=
+ 1)(21 + 1) - Y&(?J)~
goSH/kT with g the g-factor along H, i.e., g2
= g1I2 cos2 0
+ gL2 sin2 8.
Since the entropy is then a function of only gH/T, the derivation of Eqs. (2.4.2.3.12) and (2.4.2.3.14) becomes clear. Finally we can see that Eq. (2.4.2.3.13)follows from Eq. (2.4.2.3.68).
2.5. Determination of the Polarization of Electrons and Photons"
2.5.1. Introduction The present chapter is restricted to the discussion of circularly polarized photons and of longitudinally and transversally polarized electrons. A more detailed description of these topics can be found in reviews by Grodzins,' Fagg and Hanna,2 and Page.s The determination of the linear polarization of photons is dealt with in Section 2.4.2.1.4.The polarization of nucleons is treated in reviews by Wolfen~tein,~ Faissner,6 and Squires,6 among others. 1
L. Grodains, Progr. in Nuclear Phys. 7 , 163 (1959).
* L. W. Fagg and S.S.Hanna, Revs. Modern Phys. 31, 711 (1959). aL. A. Page, Revs. Modern Phye. 31, 759 (1959); Ann. Rev. Nuclear Sci. 12, 43 (1962). L. Wolfenstein, Ann. Rev. Nuclear Sci. 6, 43 (1956). 6 H. Faissner, Ergeb. ezakt. Naturw. 32, 182 (1959). 6 E . J. Squires, Progr. in Nuclear Phys. 8, 47 (1960).
* Chapter 2.5 is by
H. Frauenfelder a n d A. Rossi.
2.5.
POLARIZATION OF ELECTRONS AND PHOTONS
215
All experiments with polarized electron and photon beams are based on the interaction of polarized particles with external fields or with matter. In the present chapter, results from the relevant theories are used without derivation. The references given in Chapter 1.1, Vol. 5 A, and in the present chapter should be consulted for details. The term “polarization” and related concepts will be defined in Section 2.5.2. Here it suffices to say that, in general, a particle with spin S possesses 2s 1 possible orientations of its intrinsic angular momentum with respect to a given direction. A particle beam is called unpolarized if all of these states are equally populated for any choice of quantization axis. The electron, a particle with spin +,possesses two polarization states. If the direction of quantization is chosefi along the direction of motion, i.e., if spin and momentum are parallel or antiparallel, one talks about longitudinal polarization or helicity. Transverse elet tron polarization corresponds to a choice of polarization direction a t right angles to the electron momentum. Longitudinal and transverse polarization are, of course, only two convenient cases and any other direction may be chosen as the reference axis. The understanding of the polarization states of photons is more difficult. Photons carry an intrinsic angular momentum of lh, i.e., S = 1 . ’ ~Instead ~ of the expected three polarization states, a photon, and indeed any particle with zero rest mass, possesses only These two polarization states can, for instance, be chosen to be perpendicular to the momentum or to be parallel to it. The first choice corresponds to the linear polarization, the second to circular polarization. A photon in a right (left) circularly polarized state has its spin along (opposite) to its momentum. These two states are actually the only spin eigenstates of a free photon; a measurement of the component S , of the spin along the momentum will yield the values +1 or -1, but never 0. The linear polarization does not correspond to an eigenstate of S,.11,12 Because both electrons and photons possess two states of polarization, the formal description of the polarization is very similar. More important, the longitudinal polarization (helicity) is, to a large degree, conserved in the various transformations involving photons and electrons. Thus,
+
7 W. Heitler, “Quantum Theory of Radiation,” Appendix 1. Oxford Univ. Press, London and New York, 1954. 8 J. M. Jauch and F. Rohrlich, “Theory of Photons and Electrons,” Section 2.8. Addison-Wesley, Reading, Massachusetts, 1955. 9 M. Fierz, Helu. Phys. Acla 13, 95 (1940). lo E. Wigner, Revs. Modern Phys. 29, 255 (1957). 11 J. Hamilton, “The Theory of Elementary Particles,” Gection I, 14. Oxford Univ. Press, London and New York, 1959. I* R. P. Feynman, “Theory of Fundamental Processes,” Chapter 19. W. A. Benjamin, New York, 1961.
216
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
lohgitudinally polarized electrons give rise to circularly polarized photons in bremsstrahlung, and circularly polarized photons produce longitudinally polarized electrons in pair production and in the photoeffect. This close connection is useful in understanding the over-all picture and very important in experimental investigations, where sometimes the polarization of photons is easier to determine than that of electrons or vice versa. Experiments involving the polarization of electrons and photons are important in two respects. First of all, they yield information about the basic properties of electrons and photons. Secondly, polarization measurements can be a tool in a number of fields. For many years, the first aspect dominated because it required considerable time and effort to show that photons possess the properties of “massless particles with spin 1” and that free electrons display the polarization phenomena predicted by the Dirac theory for spin particle^.^^^'^ Since the satisfactory solution of these basic questions, polarization measurements have been used, for instance, to determine the g factor of free electrons16J6and to investigate properties of nuclear levels.1TJ8Of major importance is the application to problems connected with parity noncon~ervation.’~-~~ This importance is due to the fact that the spin S is a pseudovector and that experimentally observable terms like S p (where p is a momentum) indicate whether parity is conserved or not. With the laws of beta decay established, such measurements can be used to investigate nuclear decays, assign parities and spins, and measure ratios of nuclear matrix elements. Polarization experiments will remain a valuable and useful tool not only in nuclear spectroscopy, but also in other fields.
+
-
2.5.2. Description of Polarized Beams*
2.5.2.1. The Polarization of Electrons. 2.5.2.1.1. PRELIMINARY REElectrons possess two polarization states; for any given direction
MARKS.
l 3 N. F. Mott and H. S. W. Massey, “Theory of Atomic Collisions.” Oxford Univ. Press, London and New York, 1949. l 4 H. A. Tolhoek, Revs. Modern Phys. 28, 277 (1956). W. H. Louisell, R. W. Pidd, and H. R. Crane, Phys. Rev. 94, 7 (1954). l8 A. A. Schupp, R. W. Pidd, and H. R. Crane, Phys. Rev. 121, 1 (1961). l7 J. C. Wheatley, H. J. Huiskamp, A. N. Diddens, M. J. Steenland, and H. A. Tolhoek, Physica 21, 841 (1955). l* G. Trumpy, Nuclear Phys. 2, 664 (1956). lo T. D. Lee and C. N. Yang, Phys. Rev. 104,254 (1956). 20 C. S. Wu, Revs. Modern Phys. 31, 783 (1959). 2 1 E. J. Konopinski, Ann. Rev. Nuclear Sci. 9, 99 (1959). 2* J. H. D. Jensen, in “Lectures in Theoretical Physics” (W. E. Brittin and R. W. Downs, eds.), Vol. 11. Interscience, New York, 1960. 2 3 C. S. Wu, in “Theoretical Physics in the Twentieth Century” (M. Fierz and V. F. Weiaskopf, eds.). Interscience, New York, 1960. * See also Vol. 4AJ Chapter 3.5.
2.5.
POLARIZATION OF ELECTRONS AND PHOTONS
217
z, the spin can be either ‘(up” or “down.” Assume that one has defined z by an experimental arrangement and that one has determined the probability N ( + ) for finding electrons with spin “up” and N ( - ) for finding electrons with spin “down.” One then defines a polarizution:
(2.5.1)
Except for the special case 1P.I = 1, such an experiment does not yield all the information that can be extracted from a given beam. Assume for instance an electron beam that is completely polarized in a direction x, where x is perpendicular to y and z. P, and P, then will be zero and a measurement of P, and P, alone gives very little information. For a complete description of the polarization of a beam, the measurement described above is performed along three mutually orthogonal directions z, y, and z. The degree of polarization of such a beam is then given by (2.5.2) P = d P Z Z P,2 P,2.
+
+
P can vary between 0 and 1. The basic ideas involved in the experimental determination of electron polarization are simple and many experiments have been devised and executed without the use of the formal description in terms of density matrices and Stokes parameters.a Nevertheless it is often helpful, particularly when dealing with relativistic electrons, to have a complete formal description of polarized beams. A brief outline of such a description will be given here and in the following sections. Extensive treatments of the polarizations of spin 4 particles can be found in references 14, 22, and 24-29. 2.5.2.1.2. ELECTRONS AT REST.The description of the polarization of ~ ~ ~ ~ ~the ,~~ individual electrons in their rest frame is s t r a i g h t f o r ~ a r d . Using Pauli matrices 0 -i 61 = 6 2 = (i () 63 = (2.5.3)
(; ;)
)
(; ”)
H. P. Stapp, Phys. Rev. 103, 425 (1956). L. Michel, Nuovo cimento 14, Suppl., 95 (1959). 26 M. E. Rose, “Relativistic Electron Theory.” Wiley, New York, 1961. 27 D. M. Fradkin and R. H. Good, Jr., Revs. Modern Phys. 33, 343 (1961). 28 R. P. Feynman, “Quantum Electrodynamics,” p. 60. W. A. Benjamin, New York, 1961. 29 W. S. C. Williams, “Introduction to Elementary Particles,” Chapter 8. Academic Press, New York and London, 1961. a 0 W. Pauli, Die allgemeinen Prinzipien der Wellenmechanik, in “Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 5, Part 1, p. 109. Springer, Berlin, 1958. 8 1 E. Merzbaeher. “Quantum Mechanics,” Chapter 13. Wiley, New York, 1961. 24
26
218
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
one defines a polarization vector P as the expectation value of the spin vector d = (UI,U~,U~)in the normalized state $:
Pk
=
($‘*,bk$)
k
P=
.
= 1, 2,
3
(2.5.4)
or (2.5.5)
P transforms like a vector and it is legitimate to say that P points in the direction of the electron spin. Equation (2.5.1) follows from Eq. (2.5.4) fork = 3 = z. The component of the polarization in an arbitrary direction, defined by the unit vector A, is given by the expectation value of the operator dab: P, = < d * d > . (2.5.6)
If the angle between P and ii is given by 8, then the probabilities that a measurement of P, yields the value + l , respectively - 1, are given by prob (+1) prob (-1)
= =
cos2(8/2) sin2(8/2).
(2.5.7)
Equations (2.5.4) to (2.5.7) describe fully polarized electrons, as can be seen easily by calculating the degree of polarization P , Eq. (2.5.2), for Eq. (2.5.4) or (2.5.6): I’ = 1. (2.5.8) This result, a t first sight puzzling, is due to the fact that only pure spin states have been considered so far, i.e., states that can be described by a wave function $. In the general case, where the electrons can no longer be described by a single wave function, the degree of polarization can be between 0 and 1. Such a situation is called a mixed state. To find the resulting polarization, one must average the polarization of the various pure states: (2.5.9) (2.5.10) Equation (2.5.7) is replaced by prob (fl) = P cos2(8/2) prob (- 1) = P sin2(e/2).
(2.5.11)
Mixed states cannot be described by wave functions, since they consist of an incoherent superposition of pure states. The most convenient tool to
2.5.
POLARIZATION OF ELECTRONS A N D PHOTONS
219
handle such systems is the density matrix which will be briefly described in Section 2.5.2.3. 2.5.2.1.3. NONRELATIVISTIC ELECTRONS. The step from electrons a t rest to nonrelativistically moving particles leads to some useful definitions. Assume that the electron has a momentum p and that @ is the unit vector in the direction of the momentum. One then defines the longitudinal pofarization or helicity P, by
P,
=
(2.5.12)
P, = +1 means that the polarization vector is parallel to the momentum p; P, = - 1 indicates antiparallel orientation. In general, the polarization vector will be neither parallel nor antiparallel to the momentum and the absolute value of P , will be less than 1. One can also measure the expectation value of the polarization in a direction i perpendicular to p, i.e., the transverse polarization Pt :
P,
= .
(2.5.13)
In such a measurement, an azimuthal angle is implied. Usually, the conditions of an experiment dictate the choice of the direction i. Often i is selected in such a way that Pt is maximized. 2.5.2.1.4. RELATIVISTIC ELECTRONS. The situation for relativistic electrons is more complicated, because in the Dirac theory the spin and the momentum of a particle are coupled. The approach used for the nonrelativistic electrons therefore does not work. The description of the polarization of relativistic particles is nevertheless very important. This importance has actually increased in the last few years, because the high-energy accelerators not only produce relativistic electrons, but also relativistic muons, nucleons, and hyperons. The simplest description of the polarization of a relativistic electron was originally suggested by Darwin3P:one defines the polarization as the expectation value of the spin operator d in the rest frame of the electron. To find the polarization in any other coordinate system, one uses a Lorentz t r a n ~ f o r m a t i o n . ' ~ ~ ~ ~ - ~ ~ The polarization can, however, also be defined directly by selecting an appropriate operator. The various ways in which such operators can be introduced, their justification, their properties, and the relationships C. G. Darwin, Proc. Roy. SOC.Al'dO, 621 (1928). H. A. Tolhoek and S. R. deGroot, Physica 17, 1, 17, 81 (1951). 34 G. Ascoli, 2.Physik 180, 407 (1958). *& T. K. Khoe and L. C. Teng, Argonne National Laboratory Report ANLAD 69, 1962. Unpublished. 3*
*3
220
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
among different points of view have been the subject of a number of papers. 2 2,24-27,86-46 One of the most convenient generalizations of d has been introduced by S t e ~ h For . ~ ~electrons, the corresponding operator can be written as22,27,44,46 where
z = (d ' PIP + P x
(bJx
is the 4 X 4 generalization of Eq. (2.5.3) and
(:
P)
(2.5.14)
is the 4 X 4 Dirac matrix
:1)*
The operator B commutes with the Hamiltonian and with the momentum operator. In the nonrelativistic limit B is identical with the spin operator d.
From Eq. (2.5.14), using Dirac spinors, one can see that the expectation value of the helicity is the same in the laboratory system as in the rest system of the electron : (2.5.15) The nonrelativistic operator d 6 for the transverse polarization is replaced by pd 6. The matrix fi introduces a factor mc2/W, where W is the total energy of the electron. The transverse polarization, measured in the laboratory system, is hence smaller than the transverse polarization measured in the rest system: mc2 P,(lab) = <&> = = -Pt(rest). (2.5.16)
-
W
2.5.2.2. The Polarization of Photons. Photons also possess only two polarization states and the formal description of the photons polarization seL. Michel and A. S. Wightman, Phys. Rev. 98, 1190 (1955). B. Stech, 2. Physik 144,214 (1956). C.Bouchiat and L. Michel, Nuclear Phys. 6, 416 (1958). J. Werle, Nuclear Phys. 6, 1 (1958). 40 C. Fronsdal and H. UberalI, Phys. Rev. 111,580 (1958). R. H. Good, Jr., and M . E. Rose, Nuouo cimento 13, 1182 (1959). 4-2 M. Froissart and R. Stora, Nuclear Znstr. 7,297 (1960). 4 * F. Calogero, Nuovo cimento 20, 280 (1961). 44 M.E.Rose and R. H. Good, Jr., Nuouo cimento 22, 565 (1961). 45 D.M.Fradkin and R. H. Good, Jr., Nuovo cimento 22, 643 (1961).
*'
2.5.
POLARIZATION
OF ELECTRONS AND PHOTONS
22 1
is hence very similar to the one for electrons. One main difference exists, however. The spin of the electron can point in a n arbitrary direction with respect to its momentum. The photon spin, however, lies always parallel or antiparallel t o the m ~ m e n t u m . A~ photon . ~ ~ ~ in ~ ~a ~plane-polarized ~ state is not in an eigenstate of the photon spin operator; its state is described by a linear superposition of the two eigenstates of the spin operator. Formally, one can describe the complete polarization state of a photon moving along the + z axis b y the expectation value of the Pauli spin vector Eq. (2.5.3): P(photon) = . (2.5.17) The physical interpretation of the spin vector is different, however, for electrons and photons. For electrons, P is a vector in physical space. P(photon), on the other hand, is a vector in a "polarization space" (PoincarB space)8 and its components are defined as follows: P I is the plane polarization with respect to two arbitrary orthogonal axes; Pz is the plane polarization with respect to two axes oriented a t 45" to the right of the previous set; and P o is the circular polarization (helicity, longitudinal polarization). 2.5.2.3. Density Matrix and Stokes Parameters. T h e complete description of an electron or photon beam, propagating in a given direcTABLE I. Stokes Parameters for Photon and Electron Beams Stokes parameter
I PI
Pz P8
Photons
Electrons
Intensity Plane polarization Plane polarization a t 45" to PI Helicity (circular polarization)
Intensity Transverse polarization Transverse polarization a t 45' to PI Helicity (longitudinal polarization)
tion, requires four experimentally determinable quantities : the intensity 1 and three components of the polarization vector P . (Methods for measuring beam intensities are treated in Chapter 2.8.) A particular set of these four quantities, known as Stokes parameters, is given in Table 1.8,46-61
U. Fano, J. Opt. Soc. Am. 39, 859 (1949). D. L. Falkoff and J. E. McDonald, J . O p t . SOC.Am. 41, 862 (1951). 48 W. H. McMaster, Am. J. Phys. 22, 351 (1954). U. Fano, Revs. Modern Phys. 29, 74 (1957). 6o G. Passatore, Nuovo cimento 6, 850 (1957). 61 W. H. McMaster, Revs. Modern Phys. 33, 8 (1961). 411 47
222
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
Most experimental arrangements are designed without explicit reference to the Stokes parameters. Their value and convenience becomes apparent, however, in the theoretical treatment of the experiments. Hence, some of the properties of Stokes parameters are listed in the following : (1) The Stokes parameters can be written in the form of a column vector : (2.5.18)
(2) If several independent beams are superposed incoherently, the Stokes parameters of the resulting beam are the sum of the Stokes parameters of the individual beams. (3) From the Stokes parameters, the density matrix: p30s31948849 describing the polarization state can be constructed immediately: p =
+(I
+P
*
d)
(2.5.19)
where d is the Pauli spin vector (Eq. 2.5.3). (4) A polarization semitive detector can also be described by a set of Stokes parameters. Assume that the detector passes only particles or photons characterized by the Stokes parameters (1,Q). The set (1,Q) is then used to represent the detector. (5) The probability W of detecting with a detector (1,Q) a particle out of a beam characterized by (1,P) is given by (2.5.20) (6) If a particle undergoes a polarization-sensitive interaction, then the Stokes parameters of the scattered beam (I,P) are given by (2.5.21)
The interaction matrix 3 has been worked out explicitly for a number of cases.46-61 2.5.2.4. Sign Conventions. A particle or photon is called “righthanded” when its spin and its momentum are parallel; it is called “lefthanded” when they are antiparallel. From Eq. (2.5.12) it then follows that a right-handed particle possesses positive helicity, and a lefthanded particle negative helicity. A right-handed particle corresponds to a right-handed screw.
2.5.
POLARIZATION O F ELECTRONS AND PHOTONS
223
This definition is opposite to the one used in classical optics, where observers face the source of light. The magnetic moment of the electron is opposite to its spin; an electron with positive helicity thus has a magnetic moment which trails its momentum. The sign of the transverse polarization occurring in nuclear interactions of particles with spin is fixed by the Basel Convention: Positive polarization is taken in the direction of the vector product
+
ki X ko, where ki and koare the wave vectors of the incoming and outgoing particles respectively. [SUN Document, Physics Today 16, No. 6, 23 (1962).] 2.5.3. Motion of Electrons in Electromagnetic Fields* 2.5.3.1. Preliminary Remarks. A knowledge of the behavior of spin particles in electromagnetic fields is crucial for designing and evaluating experiments involving their polarization. I n homogeneous fields, the trajectory of a particle beam is independent of its polarization. The problem can therefore be separated into two parts, the motion of “spinless” particles in external electromagnetic fields and the influence of these fields on the polarization. The trajectories of particles in external electromagnetic fields have been calculated for many cases of interest. Discussions of this problem and further references can be found in references 52-54. The behavior of the polarization in electromagnetic fields is of major interest for the applications discussed in the present chapter. Some important cases and a list of references are given in Section 2.5.3.2. In Section 2.5.3.3, experimental arrangements for the transformation from longitudinal to transverse polarization are treated. 2.5.3.2. Theoretical Results. The motion of polarized particles in electromagnetic fields has been treated in a number of papers.14~26,21~33-36. In discussing a particular problem, one has to decide whether
+
42,43966--62
6zK. T. Bainbridge, in “Experimental Nuclear Physics” (E. SegrB, ed.), Voi. I, Part V. Wiley, New York, 1953. 63 0. Chamberlain, Ann. Rev. Nuclear Sn’. 10, 161 (1960). 6 4 M. S. Livingston and J. P. Blewett, “Particle Accelerators,” Chapter 5. McGrawHill, New York, 1962. b6 H. Mendlowitz and K. M. Case, Phys. Rev. 97, 33 (1955); 100, 1551 (1955). 6E K. M. Case, Phys. Rev. 106, 173 (1957). s7 L. M. Carassi, Nuovo cimento 6, 955 (1957); 7, 524 (1958). s8 H. Mendlowitz, Am. J . Phys. 26, 17 (1958). 69 V. Bargmann, L. Michel, and V. L. Telegdi, Phys. Rev. Letters 2 , 435 (1959). V. S. Popov, Soviet Physics J E T P 11, 1141 (1960). R I R H. . Good, Jr., Phys. Rev. 126, 2112 (1962). E2 H. J. Meister, 2. Physik 166, 468 (1962). * See also Vol. 4A, Part 3.
224
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
the treatment must be relativistic, whether one has to take the anomalous magnetic moments into account, and whether the fields can be considered to be homogeneous. These three criteria will become more apparent in the following. In Sections 2.5.3.2.1 to 2.5.3.2.5, typical situations will be treated. The notation is as follows: y = [ l - (v/c)”-”~
(2.5.22)
E and B denote the electric and magnetic field strengths; fi and are unit vectors in the direction of the momentum and the transverse polarization, respectively; v is the particle velocity. The rest mass and the charge are m and e; for an electron, e < 0. The dimensionless g factor is defined as the ratio of the magnetic moment I to the spin S:
I = SPOS
(2.5.23)
p o = eh/2mc.
(2.5.24)
with For positive Dirac particles, g and PO are both positive; the spin and the magnetic moment are parallel to each other. For negative Dirac particles, where the magnetic moment points opposite to the spin, Eqs. (2.5.23) and (2.5.24) can lead to some confusion. The negative sign can be introduced in g or in pa. In the definition used here, e and hence po are negative for the electron; g is positive, The g factor has a normal and an anomalous part, g = 2(1
+ a).
(2.5.25)
Numerical values for the anomalous part a are given in Table 11. TABLE 11. Anomalous Parts of the Particle Electron Muon Proton a
a = (g - 2)/2
(1160.9 f 2 . 4 ) X (1162 f 5 ) X 1.79276 0.00003
g
Factor Reference a b
Schupp, Pidd, and Crane.I6 Charpak et al.83,84
4 3 G. Charpak, F. J. M. Farley, R. L. Garwin, T. Muller, J. C. Sens, V. L. Telegdi, and A. Zichichi, Phys. Rev. Letters 8, 128 (1961). 04 G. Charpak, F. J. M. Farley, R. L. Garwin, T. Muller, J. C. Sens, and A. Zichichi, Phys. Letters 1, 16 (1962).
2.5.
POLARIZATION OF ELECTRONS A N D PHOTONS
225
The values in Table I1 and the equations in the following sections together determine whether the anomalous magnetic moment must be taken into account in the design and the evaluation of a given experiment. 2.5.3.2.1. HOMOGENEOUS EXTERNAL FIELDS.I n order to describe the behavior of the polarization in external electromagnetic fields, one introduces the angle # between the polarization vector P and the momentum unit vector 0 : (2.5.26) (P * p)/P = cos #.
d is called the polarization angle. I n an external electromagnetic field, 4 will, in general, be a function of time. The laboratory time rate of change, d#/dt, in a homogeneous field is given by59,36
1""
[
5 + i . (fi X B ) ( g - 2 )
fi=-=e " (9 - 2 ) dt 2mc ( u / c ) Y
Equation (2.5.27) is valid as long as the distances over which the fields vary are large compared to the dimensions of the wave packet describing the particle. Inhomogeneous fields are treated in reference 61. Equation (2.5.27) can be generalized to include the effect of external electromagnetic fields on particles with electric dipole rn0ments.b~t6~ The resulting equations have been used in the evaluation of experiments to determine the electric dipole moment of muons and electr0ns.6~-6* 2.5.3.2.2. LONGITUDINAL FIELDS.Assume that both the electric and the magnetic fields are parallel to the momentum:
EX$=O
BXP=O.
The longitudinal polarization (Eq. 2.5.15) remains constant. The transverse polarization (Eq. 2.5.16) is affected in two ways. I n a longitudinal magnetic field, the maximum transverse polarization also remains constant (# = const) but the transverse polarization processes around the field direction with an angular frequency WB
=
geB/2myc
=
gpoB/hy.
(2.5.28)
This rotation has been used for a direct determination of the g factor of the free e l e c t r ~ n . ' When ~ using lens spectrometers in experiments BB
R. L. Garwin and L. M. Lederman, Nuovo cimento 11, 776 (1959). D. F. Nelson, A. A . Schupp, R. W. Pidd, and H. R. Crane, Phys. Rev. Letters 2,
492 (1959). 67 G. CharpRk, L. M. Lederman, J. C. Sens, and A. Zichichi, Nuovo cimento 17,288 (1960). 68 G. Charpak, F. J. M. Farley, R. L. Garwin, T. Muller, J. C. Sens, and A. Zichichi, Nuovo cimento 22, 1043 (1961).
226
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
designed to determine a transverse polarization, one must either compensate or correct for the resulting rotation of the polarization vector (see Section 2.5.5). In a longitudinal electric field, the transverse polarization remains unchanged in the rest system of the particle. The electric field, however, changes the total energy W of the particle. According t o Eq. (2.5.16), the transverse polarization observed in the laboratory system will then change also. 2.5.3.2.3. TRANSVERSE ELECTRIC FIELD. Assume an electric field which is always perpendicular to the momentum:
E*f=E
B=O
E*fi=O.
The trajectory in such a field will be circular and the laboratory angular frequency of the particle will be given by: WE =
(2.5.29)
eE/myv.
The momentum will also rotate with a frequency angle 4 changes according to d$/di
= QE = & E [ ( g
- 2)y
WE.
- g/rl.
The polarization (2.5.30)
If 60is the polarization angle of a particle before entering the electric field, and if the momentum is turned by an angle A$ = w ~ by t the field, then the polarization angle becomes 4’ = $ J ~ A 4 , where
+
A4
=
4A$[(g - 2 ) ~ g/~l.
(2.5.31)
Equation (2.5.31) yields for the angle A#* required to transform longitudinal into transverse polarization (A4 = --.rr/2) : (2.5.32)
In the nonrelativistic limit, y = 1, the situation becomes very simple. From Eq. (2.5.30) it follows that QE(nonre1) = - W E . Seen from the laboratory, the polarization vector keeps its direction in space; the polarization direction is not affected by the electric field. The angIe A$* in Eq. (2.5.32) becomes a/2. 2.5.3.2.4. TRANSVERSE MAGNETIC FIELD. I n a transverse magnetic field, E=O B*QXfi=B the time rate of change (Eq. 2.5.27) of the polarization angle becomes
n,
eB 2mc
= - (g
- 2).
(2.5.33)
2.5.
POLARIZATION
OF ELECTRONS
AND PHOTONS
227
The orbital angular frequency of a charged particle in this field is given by the Larmor frequency: (2.5.34) (2.5.35) Equations (2.5.33) and (2.5.35) show that a particle with g = 2 will precess in a transverse magnetic field without change of the polarization angle 4. A particle with g # 2 , on the other hand, will suffer a change of the polarization angle and this change can be used to determine the anomalous part a = &(g - 2 ) directly. Such experiments have been performed with free electrons'6 and with free muon^.^^,^^ For high-energy polarized beams, the property expressed by Eq. (2.5.33) c8n be used to transform the polarization angle. 2.5.3.2.5. CROSSED FIELDS. Consider transverse crossed fields,
E.B=O
B*IX@=B
(2.5.36)
adjusted in such a way that the particle trajectory is straight:
E
= -V
X B/c.
(2.5.37)
Inserting Eqe. (2.5.36) and (2.5.37) into Eq. (2.5.27) yields QEB
=
W L ~ / ~ Y .
(2.5.38)
Arrangements other than the ones treated in Sections 2.5.3.2.2 to 2.5.3.2.5 can be discussed along similar lines, always starting from Eq. (2.5.27). 2.5.3.3. Polarization Transformers. In some experiments it is desirable to change the polarization of a beam. One example is the determination of the longitudinal polarization of beta particles emitted in nuclear beta decay. If Mott scattering is used to measure the polarization, the longitudinal polarization must first be changed to a transverse one, since Mott scattering is only sensitive to the transverse polarization. A second example is the determination of the anomalous part of the g factor of the electron or the muon. The transformation of the polarization state of a particle beam can be achieved in a number of ways, all based on applications of Eq. (2.5.27). In the following sections, the most important polarization transformers will be discussed. One general remark is in order. Most of the spin transformers act also as energy, momentum, or mass filters. It is therefore possible to use them simultaneously for the selection of a certain energy or momentum band and for the polarization transformation.
228
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
Nevertheless it is often desirable to precede the transformer in the experimental set-up by an energy selector in order to prevent particles with undesired energies from reaching the detector after scattering from the walls of the polarization transformer. 2.5.3.3.1. CYLINDRICAL AND SPHERICAL CONDENSERS. It is clear from Section 2.5.3.2.3 that a radial electric field acts as a spin transformer. Equation (2.5.31) gives the change A$ in the polarization angle $I if the momentum is turned by an angle A$. Equation (2.5.32) describes the angle Afi* through which the momentum has to be turned in order to transform a longitudinal polarization in to a transverse one. A radial electric field can be realized with cylindrical and spherical condensers. Both types have been used extensively as velocity selectors6* and as polarization transformers. l r 3 References and data for designing velocity selectors are given in reference 52. The spherical condenser has been treated in references 69-73. Among the many publications in which the use of radial electric fields as polarization transformers is described, the following offer the most information for the experimental design and evaluation : cylindrical c o n d e n ~ e r sspherical ,~~ condensers.76-7s Figure 1 shows a setup in the cylindrical g e ~ m e t , r y Figure . ~ ~ 2 shows the schematic arrangement for a polarization transformer using a spherical ~ondenser.7~ The deflection voltage which must be applied to the plates of the condensers is given by62 (2.5.39)
for the cylindrical case, and by (2.5.40) M. Purcell, Phys. Rev. 04, 818 (1938). E. Persico and C. Geoffrian, Rev. Sci. Znstr. 21, 945 (1950). 71 F. T. Rogers, Rev. Sci. Znsfr. 22, 723 (1951). 72 N. Ashby, Nuclear Instr. 3, 90 (1956). 7aR. H. Ritchie, J. S. Cheka, and R. D. Birkhoff, Nuclear Znslr. 6, 157 (1960); 69E.
70
8, 313 (1960). 74 J. S. Greenberg, D. P. Malone, R. L. Gluckstern, and V. W. Hughes, Phys. Rev. 120, 1393 (1960). 76 H. Bienleh, K. Guthner, H. von Issendorff, and H. Wegener, Nuclear Instr. 4,
79 (1959). 76 A. Ladage, Ph.D. Thesis, University of Hamburg, 1961 (unpublished). 77 R. 0. Avakyan, G . L. Bayatyan, M. E. Vishnevskii, and E. V. Pushkin, Soviet Phys. JETP 14, 491 (1962). 78 A. R. Brosi, A. I. Galonsky, B. H. Ketelle, and H. B. Willard, Nuclear Phys. 33, 353 (1962).
2.5.
POLARIZATION OF ELECTRONS AND PHOTONS
229
Grounded
At. C o l l i m a t o r Plbstic Scintillator
S , ,S,
S7 -Collimating
Slits
FIG.1. Spin transformation with a cylindrical condenser. The figure also shows the arrangement to determine the transverse polarization with Mott scattering. The d i g r a m is not to scale. (From Greenberg, Malone, Gluckstern, and Hughes, reference 74.)
for the spherical case. Rz and R1 are the radii of the outer and the inner plates, respectively, m is the rest mass and v the velocity of the particle, and y is given by Eq. (2.5.22) (Gaussian units). Some typical design parameters and properties for two systems are given in Table 111. TABLE 111. Design Parameters for Electric Condensera Type
Cylindrical
Spherical
Reference Mean bending radius Gap width Deflection angle Azimuthal acceptance angle Maximum voltage Transmission Energy resolution
74 20 cm l.8cm 90"
78 25.4cm 1.Ocm 137" 45" 50 kv 5 x 10-4 45%
5
50 k v x 10-5 5%
230
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
The following additional remarks apply to the design and the evaluation of experiments involving cylindrical and spherical condensers : (1) It is difficult to use voltages higher than 50 kv without sparking, regardless of gap width. The systems, hence, must be so designed that the highest energy to be investigated can be passed without exceeding this limit. (2) It is preferable to apply the voltage symmetrically to the two plates so that the mean path is a t ground potential.
FIG. 2. Spin transformation with a spherical condenser. The indicated value for the polarization angle 6 applies to electrons of 616 kev. (From Brosi, Galonsky, Ketelle, and Willard, reference 78.)
(3) The transformation from longitudinal to transverse polarization in any given set-up is complete only for the one energy which satisfies Eq. (2.5.32). However, for energies within a very wide band, the deviations from a complete transformation are small and a corresponding correction is easy to apply. (4) The walls of the system should be lined with a material with a low backscattering coefficient (polyethylene) in order to reduce the background. (5) Corrections which must be applied to the data in order to find the final polarization angle are discussed in references 74-78. (6) As can be seen from TabIe 111, the spherical condenser has a t least one order of magnitude better transmission than the cylindrical con-
2.5.
POLARIZATION
OF ELECTRONS AND PHOTONS
23 1
denser. It is, however, much more difficult to construct and to adjust. The choice between the two hence depends on the particular problem to be solved. (7) As pointed out above, it is desirable to preselect the energy of the particles entering the polarization transformer, for instance with a magnetic spectrometer. Unfortunately it is impossible to match a magnetic spectrometer to a spherical analyzer without a considerable loss in intensity. One thus has two alternatives, either a preselection with a corresponding loss in intensity, or no preselection and a higher background and increased scattering. 2.5.3.3.2. CROSSEDFIELDS (WIEN FILTER).Consider crossed electric and magnetic fields, adjusted according to Eq. (2.5.37) :
E / B = lE[/lB[
= V/C.
(2.5.37’)
Particles with the velocity v then travel undeflected through this system. Their polarization vector, according to Eqs. (2.5.38) and (2.5.34), is turned by an angle At$ = egBL/2mcvy2 (2.5.41) where L is the length of the system. Such crossed fields have been used as polarization transformers in a number of investigation^.'^-*^ An example of a setup is shown in Fig. 3. To give some idea about the orders of magnitude involved in the design of a Wien filter for low-energy electrons, consider a n apparatus of length L = 30 cm and assume electrons of 200 kev energy ( v / c = 0.695). From Eq. (2.5.37) it follows that E (in kv/cm)/B (in oe) = 0.21. If a complete transformation from a longitudinal to a transverse polarization is to be obtained, A 4 in Eq. (2.5.41) is r / 2 and one finds for the required magnetic field strength B = 120 oe. The electric field strength then becomes E = 25 kv/cm. Crossed fields have a number of advantages over other types of polarization transformers: (1) The entire arrangement is axially symmetric. This symmetry reduces the appearance of experimental asymmetry effects. 78 P. E. Cavanagh, J. F. Turner, C. F. Coleman, G. A. Gard, and B. W. Ridley, Phil. Mag. 21, 1105 (1957). A. I. Alikhanov, G. P. Eliseiev, V. A. Lyubimov, and B. V. Ershler, Nuclear Phys. 6,588 (1958). P. E. Spivak and L. A. Mikaelyan, Nuclear Phys. 20, 475 (1960). 82 R. Sosnowski, Z. Wilhelmi, and J. Wojtkowske, Nuclear Phys. 26, 280 (1961).
232
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
(2) The magnetic and the electric field can be varied in such a way that the transformation of the polarization is complete, i.e., A 4 = n/2, for a wide range of energies. (3) If one reverses B and E simultaneously, the geometry of the apparatus remains unchanged but A 4 changes its sign. This property allows a simple elimination of asymmetry effects due to different counter efficiencies and other instrumental asymmetries.
O&O 2&- J
Ir
I
\
\
cm
I,
‘._I
IL-.---A
-
FIG. 3. Spin transformation with a crossed field. The upper part shows the horizontal, the lower the vertical cross section. The electron energy is preselected with a magnetic spectrometer (2); 1 indicates the source; 3 denotes the magnet pole and 4 the electrostatic plates of the crossed field system. Magnetic shields (5) are used to define the length of the crossed fields. The transverse polarization of the electrons leaving the transformer is determined with Mott scattering; 6 indicates the scattering foil and 7 the counter telescopes. (From Spivak and Mikaelian, reference 81.)
(4) Since the change in polarization can be chosen to be go”, small errors in spin rotation due to an error in the determination of the effective length L are small. ( 5 ) Crossed fields can be used as mass filters, even for particles with very high energy.s3@It may be possible to use the combined properties of mass filter and polarization transformer. One major disadvantage of a Wien filter is the fact that gamma rays from the source can pass straight through the system without encountering an absorber. This disadvantage can be removed, with some loss of intensity, by inserting a magnetic spectrometer (lens spectr~rneter’~ or transverse sector fields1)between source and polarization transformer. 83
A. Yokosawa, Argonne National Laboratory Report ANLAD-70, 1962. Unpub-
lished.
2.5.
POLARIZATION OF ELECTRONS AND PHOTONS
233
2.5.3.3.3. COULOMB SCATTERING. Coulomb scattering in materials of low atomic number 2 can also be used to transform the polarization state of electron^.^^.^^ The Coulomb field is essentially a radial electric field. For low 2, the spin-orbit coupling is small and all the arguments presented in Section 2.5.3.3.1 apply. Details about experiments can be found in references 86-90. A particularly elegant m e t h ~ d consists ~ ~ s ~ ~in using a semicircular aluminum foil as the polarization transformer. With properly placed source and second scattering foil, electrons scattered anywhere in the foil are changed from a longitudinal to a transverse polarization state. The semicircular scatterer results in an appreciable intensity gain over a flat scatterer. The main advantage of this type of polarization transformer is its simplicity; the main disadvantages are the necessity for a correction for depolarization and the fact that Coulomb scattering does not act as an energy selector. 2.5.4. Polarization Transfer* 2.5.4.1. interactions between Electrons and Photons. It was pointed out in Section 2.5.1 that the helicity is to a large extent conserved in the various transformations involving photons and electrons. This “persistence of helicity” forms the basis of some methods to determine the longitudinal polarization. It can, for example, be more convenient in a certain experiment to measure the helicity of photons than of electrons. In this case one forces the longitudinally polarized electrons to produce bremsstrahlung ; the bremsstrahlung is circularly polarized and one detects this circular polarization. In the present section, the various processes of interest for experimental applications will be outlined. Two of the processes of interest are shown in Fig. 4, both involving electrons of momentum p and energy E (including the rest energy), and photons of momentum k and energy E?. Only the observed particles are shown in Fig. 4.Figure 4a hence can represent bremsstrahlung or positron annihilation. Figure 4b can stand for pair production or photoeffect. To describe the particles and the interaction processes, it is best to use the Stokes parameters (Table I) and the interaction matrix 3, Eq. (2.5.21). L. J. Tassie, Phys. Rev. 107, 1452 (1957). F. Gursey, Phys. Rev. 107, 1734 (1957). 86 A. De-Shalit, S. Kuperman, H. J. Lipkin, and T. Rothem, Phys. Rev. 107, 1459
S4
86
(1957).
J. Heintze, 2. Physik 160, 134 (1958). A. I. Alikhanov, G . P. Eliseiev, and V. A. Liubimov, Nuclear Phys. 7, 655 (1958). 89 W. Bahring and J. Heintze, 2. Physik 163, 237 (1958). 9 0 S. Cuperman, Nuclear Phys. 28, 84 (1961). * See also Vol. 4R, Chapter 7.6.
87
88
234
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
Explicit representations of the interaction matrix for various processes can be found in references 46-51. (Before using the Stokes parameters or interaction matrices given in a paper, it is advisable to check the definition of the components of the Stokes parameters since it is not the same in all papers.)
X
t
FIG. 4. Interaction between electrons and photons. (a) An incident electron creates one (or more) photons. (b) An incident photon gives rise to one (or more) electrons. Only one emerging particle is shown. The interaction takes place in the zz plane.
If the incident electron in Fig. 4a has a longitudinal polarization P, and the outgoing observed photon a circular polarization Pa, one defines a helicity transfer egiciency L by the relation Pt(photon)
=
LP,(electron).
(2.5.42)
L is a function of the parameters describing the process under consideration. Examples are discussed later. If the incident electron is transversely polarized, the outgoing photon can be circularly polarized; the corresponding transfer efficiency T is given by Pt(photon) = TPt(electron).
(2.5.43)
Similar transfer efficiencies can be defined for processes represented by Fig. 4b, for instance P,(electron) = L’Ps(photon).
(2.5.44)
In general, the calculation of the helicity transfer efficiencies L and L’ requires a detailed investigation of the processes under consideration. However, for incident particles of high energy (>>mcz)>,a general connection holds: I n a Coulomb field, almost complete momentum transfer implies that helicity is also transferred nearly completely, i.e., L -+ 1,
2.5.
POLARIZATION OF ELECTRONS AND PHOTONS
235
L’ -+ 1. This result can be interpreted as a consequence of conservation ~ ~the , ~following ~ sections, specific examof spin angular m o m e n t ~ m .In ples will be given. 2.5.4.2. Bremsstrahlung.* Of particular interest for experimental applications is bremsstrahlung. Regardless of the polarization state of the incident electron beam, bremsstrahlung is, in general, polari~ed.~3-~* If the incident electron beam is unpolarized, the bremsstrahlung is partially linearly p o l a r i ~ e d . ~At ~ -electron ~~ energies of a few Mev or less, the polarization vector (vector of the electric field) is perpendicular to the plane of interaction (plane of photon emission) for low photon energies. At the high energy tip of the bremsstrahl spectrum, the polarization vector lies in the plane of interaction. A typical plot of the linear polarization versus photon energy is shown in Fig. 5 . At high electron energies, the polarization vector is always perpendicular to the plane of interaction and the amount of polarization decreases with increasing photon energy (Fig. 6). The high energy tip of the bremsstrahl spectrum of high energy electrons shows very little or no linear polarization. The polarization of the bremsstrahlung due to polarized electrons has been treated in many papers.61~91~92~95~g7-107 The following facts emerge. ( 1 ) The linear polarization of the bremsstrahlung is independent of the polarization of the incident electrons if the spin of the final electrons is not observed. ( 2 ) L o n g i € u d i ~ a lpolarized l~ electrons give rise to circulady poIarized photons. The sign of the helicity remains unchanged; right handed U. Fano, K. W. McVoy, and J. R. Albers, Phys. Rev. 116, 1159 (1959). R. H. Pratt, Phys. Rev. 123, 1508 (1961). 98 A. Sommerfeld, “Atornbau und Spektrallinien,” Vol. 2, Chapter 7. Vieweg, Braunschweig, Germany, 1939. 9 4 H. A. Bethe and E. E. Salpeter, Quantum Mechanics ofthe One- and Two-Electron Atoms, Section IV, c. Springer, Berlin; Academic Press, New York, 1957. 96 R. L. Gluckstern, M. H. Hull, Jr., and G. Breit, Phys. Rev. 90, 1026 (1953). 96 R. L. Gluckstern and M. H. Hull, Jr., Phys. Rev. 90, 1030 (1953). 97 C. Fronsdahl and H. Uberall, Phys. Rev. 111, 580 (1958). 9s C. Fronsdahl and H. Uberall, Nuovo cimento 8, 163 (1958). 99 H. Olsen and L. C. Maximon, Phys. Rev. 114, 887 (1959). 1 0 0 1 . B. Zel’dovich, Doklady Akad. Nauk S.S.S.R. 83, 63 (1952). 101 K. W. McVoy, Phys. Rev. 106, 828 (1957); 111, 1333 (1958). 102 G. L. Visotskii, A. A. Kresnin, L. N. Rozentsveig, Zhur. Eksptl. i Teoret. Fiz. 82, 1078 (1957); Soviet Phys. J E T P 6, 883 (1957). 103 A. Claesson, Arkiv Fysik 12, 569 (1957). 104 G. Bobel, Nuovo cimento 6, 1241 (1957). 106 B. K. Kerimov, I. M. Nadzhafov, Izvest. Akad. Nauk S.S.S.R. ser. Fiz. 22, 886; Columbia Tech. Transl. 22, 879 (1958). H. Banerjee, Phys. Rev. 111, 532 (1958). 107 S.Sarkar, Nuovo &mento 21, 410 (1961). * See also Vol. 4A, Section 3.5.3. 91
92
236
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
electrons give rise to right-handed photons. The transfer eflciency L, i.e., the ratio of the photon helicity to the electron helicity [see Eq. (2.5.42)], depends on the energy of the electron, on the fractional energy E,/(E - mc2) of the photon, on the photon emission angle 8, and on the charge Z of the nucleus in whose field the bremsstrahl process occurs.
E ~ (E / - M C 2~
FIG. 5. Polarization effects in bremsstrahlung from electrons with initial energy = 2.5 MeV. The transfer efficiencies L (Eq. 2.5.42) aod T [(Eq. 2.5.43)]and the linear polarization Plin are plotted vs. the fractional photon energy E , / ( E - mcz). The curves are for 2 = 26 (iron) and for a photon emission angle e = 10'. (From Fronsdal and tZberall, reference 97.)
E
L reaches a maximum near the high energy tip of the bremsstrahl specthis maximum is unity for all trum. For high energy electrons ( E >> m2), elements. Typical curves for L are shown in Figs. 5 and 6. (3) Transversely polarized electrons also give rise to circularly polarized photons. The transfer efficiency T [Eq. (2.5.43)] is considerably smaller than unity. Two typical curves for T are given in Fig. 5 and Fig. 6. (4) For arbitrary polarization, characterized by a degree of polarization P [Eq. (2.5.2)] and a polarization angle 4 [Eq. (2.5.26)], the circular polarization P 3 of the bremsstrahlung is given by combining (2.5.42) and (2.5.43) as Pa(photon) = P ( L cos 4 T sin 4). (2.5.45)
+
2.5.
POLARIZATION OF ELECTRONS AND PHOTONS
237
The helicity of the bremsstrahlung is thus proportional to the degree of polarization of the electrons; unpolarized electrons cannot emit circularly polarized bremsstrahlung. (5) The intensity of the bremsstrahlung from transversely polarized electrons exhibits an azimuthal asymmetry about the direction of motion of the incident electron beam.108
0
0.2
0.4
0.6
0.0
I .o
Ey/ E FIG.6. Polarization effects in bremsstrahlung from electrons with initial energy E = 50 MeV. The transfer efficiencies L (Eq.2.5.42) and T (Eq.2.5.43) and the linear polarization Pli, are plotted vs. the fractional photon energy E,/(E - mez) = E,/E. The curves are for Z = 82 (lead) and for a photon emission angle e = 0.23". (From 01sen and Masimon, reference 99.)
(6) For longitudinally polarized electrons of very high energy, finite nuclear size effects can become noticeable in the transfer efficiency L.Io9 2.5.4.3. Pair Production. Pair production can be considered to be the inverse process of bremsstrahlung ; the physical observables of interest can be obtained simply from the ones calculated for bremsstrahlung (see references 8, page 161; references 51, 101). For experimental applications, the following results are important :61,99, 101,110,111
W. R. Johnson and J. D. Roeics, to be published. B. K. Kerimov and F. S. Sadykhov, Zhur. Eksptl. i Teoret. Fiz. 40,553 (1961); Soviet Phys. J E T P lS, 387 (1961). 110 K. W. McVoy and F. J. Dyson, Phys. Rev. 106, 1360 (1957). 111 I. G.Ivanter, Z h r . Ekspft. i Teoret. Fiz. S6, 1093 (1959); Soviet Phys. J E T P 9, 777 (1959). 108
109
238
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
(1) A photon beam with helicity PI produces an electron-positron beam that is longitudinally polarized. (2) The faster particle of the pair is always polarized in the same sense as the circularly polarized photon. (3) The longitudinal polarization P, of the faster particle of the pair is proportional to PI. At the upper end of the spectrum, where this particle takes all the available energy, the helicity transfer efficiency L' = P,/Pa reaches a maximum. (4) For high energy photons ( E , >> mc2), the maximum transfer efficiency for any element becomes unity. (5) I n a shower, created by a longitudinally polarized high energy electron, the initial helicity remains conserved to a very large extent."O This follows from the remarks here and in Section 2.5.4.2:in both bremsstrahlung and pair production, the helicity is conserved a t high energies. This preservation of helicity may be used to determine the helicity of the particle originating the shower. 2.5.4.4. Photoeffect.* The photoelectric effect is closely related to bremsstrahlung. In first order, it is the inverse of the high-energy limit of bremsstrahlung a t which the incident electron radiates all of its kinetic energy and is left with zero momentum.112 The calculation of the relevant cross sections how^:^^^^^^^^^*^^-^^^ (1) I n the high energy limit, photoelectrons produced by circularly polarized gamma rays are longitudinally polarized, regardless of the angle of emission. The helicity transfer efficiency P,(electron)/P3(photon) is unity. (2) At lower energy, circularly polarized photons also produce polarized electrons. However, the transfer efficiency is less than unity and the polarization vector of the electron is in general not parallel to the electron momentum. Curves and equations to find t'he magnitude and direction of the polarization vector are given in references 114-117. 2.5.4.5. Positron Anni hi I a tion-in- FIi g ht. The annihilation-in-fligh t of positrons is the basis for some experimental methods to determine the positron polarization by measuring the polarization of the annihilation quanta. Polarized positrons, when annihilating in flight, transfer a certain amount of their polarizarion to the resulting quanta:
t
K. W. McVoy and U. Fano, Phys. Rev. 116, 1168 (1959). llJK. W.McVoy, Phys. Rev. 108, 365 (1957). 114 H.Olsen, Kql. Norske Videnskab. Selskabs. Forh. 51, Nos. 11, Ila (1958). 116 H. Banerjee, Numo eimento 11, 220 (1959). U.Fano, K. W. McVoy, and J. R. Albers, Phys. Rev. 116, 1147 (1959). 11' B. Nagel, Arkiv Fysik 18, 1 (1960). 118 H.Kolbenstvedt and H. Olsen, Nuovo cimento 22, 610 (1961). * See also Vol. 4B, Chapter 8.2. t See also Vol. 4A, Section 2.1.9. Ila
2.5.
POLARIZATION OF ELECTRONS AND PHOTONS
239
(1) In single quantum annihilation of longitudinally polarized positrons, the resulting single gamma ray is circdarly polarized in the same sense as the positron. The helicity transfer efficiency increases with positron energy and becomes unity for high energies.llaJ16Single quantum annihilation is forbidden for free electron-positron pairs. However, if a third body takes up momentum, the process can take place. Usually, single quantum emission will only account for a small fraction of the observed annihilation quanta. 119 (2) The two quantum annihilation of longitudinally polarized positrons in an unpolarized target gives rise to circularly polarized photons.s.61J20-122 The higher energy gamma ray, emitted in the direction of the positron momentum, carries the larger helicity, with the same handedness as the positron. The helicity transfer efficiency is unity a t high energies; it is about 0.5 a t a positron energy of 100 kev. (3) Transversely polarized positrons also give rise to circularly polarized annihilation quanta.lZ3 (4) The annihilation quanta from transversely polarized positrons show a very small left-right asymmetry.124This asymmetry is, however, too small to be experimentally useful a t the present time. (5) The photon emission left-right asymmetry can be quite large in single quantum annihilation in high 2 atoms. 2.5.5. Detection of Electron Polarization* An experimental setup to determine the polarization of an electron beam will, in general, consist of a momentum selector and a polarizationmeasuring device. Information about momentum selectors can be found in Chapter 2.2 and in references 52, 125, and 126. The most important methods used to measure the polarization are listed in Table IV and discussed in the following sections. 2.5.5 1. Electron-Electron Scattering.t 2.5.5.1.1. THEORETICAL BACKGROUND. If two colliding (negative) electrons have parallel polarizations, the differential cross section for scattering is smaller than if they have 119
L. Sodickson, W. Bowman, J. Stephenson, and R. Wehatein, Phys. Rev. 124,
1851 (1961).
A. Page, Phys. Rev. 106, 394 (1957). W. R. Theis, 2.Physilc 160, 198 (1958). ‘22 W. H. McMaster, Nuovo cimento 17, 395 (1960). I2*L.A. Page, Phys. Rev. 109, 2215 (1958). 1 2 4 s . C. Miller and R. M. Wilcox, Phys. Rev. 126, 629 (1962). 120L. 121
l2s M. Deutsch and 0. Kofoed-Hansen, in “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 111, Part XI. Wiley, New York, 1959. 128 K. Siegbahn, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), Chapter 111. North Holland, Amsterdam, 1955. (Second ed. in preparation.) * See also Vol. 4A, Chapter 3.5. t See also Vol. 4B, Section 6.1.3.
TARLE IV. Methods to Determine the Polarization of Electrons Method
Used mainly for: PolarizaCharge tion Energy
Section
~
electronelectron scattering
~~
P, P,
>0.3Mev >0.3Mev
2.5.5.1 2.5.5.1
Remarks
~
~
Depends on the spin dependence of electron scattering. Electrons are scattered from thin magnetized iron foils. Effect is small (
Mott scattering
e-
Pt
0.03-1 Mev
2 . 5 . 5 . 2 Depends on the spin+rbit interaction in scattering from heavy nuclei. The required thin scattering foils introduce difficulties. Theory of scattering not complete at low energies. Effects can be large (up to factor 2) but corrections can be very large also.
Bremsstrahlung
e+,e-
P,
>0.2Mev
2.5.5.3
Longitudinally polarized electrons produce circularly polarized bremsstrahlung. Circular polarization is determined by methods given in Section 2.5.6. Effects are small ( < l o % ) ; strong sources are required.
EC
Annihilation in flight (Helicity transfer)
e+
P,
>0.2Mev
2.5.5.3
The helicity of the high energy annihilation quanta is determined.
Annihilation in flight or a t rest (cross section)
ef
P,
2 0 . 0 3 MeV
2.5.5.4
The annihilation of positrons in magnetized iron depends on the direction of polarization.
Annihilation a t rest in magnetic fields
e+
PP
2.5.5.5
The energy term d - B permits the determination of the direction of d with respect to the applied field B.
2.5.
POLARIZATION OF ELECTRONS AND PHOTONS
24 1
antiparallel polarizations. This fact can be predicted by a naive application of the Pauli principle: If the electrons are in the same spin state, they cannot be in the same space state and hence cannot collide. The dependence on polarization is strongest for “head-on collisions,’’i.e., scattering by 90” in the c.m. system. For such collisions, the parallel polarization cross section vanishes entirely a t nonrelativistic energies and the polarization dependence remains strong at all energies. It weakens as the scattering angle departs from 90’ and vanishes a t 0” and 180”. Positronelect,ron collisions have the same polarization dependence as electronelectron collisions in the extreme relativistic limit, but the polarization dependence vanishes nonrelativistically. The polarization dependence of electron-electron (Mdler), electronpositron (Bhabha), and electron-muon collisions has been investigated theoretically by many author^.^^^'*^-'^^ These calculations do not contain radiative corrections. Such corrections are very small a t energies of interest in experiments on ordinary beta decay. In very high energy scattering experiments, they may have to be taken into a c c ~ u n t . ~ ~ ~ ~ ~ The electron-electron cross sections for parallel and antiparallel spin directions are given in Section 1.1.5.3. Of particular interest is the ratio of parallel to antiparallel differential cross section. As shown in Section 1.1.5.3, this ratio depends on the fractional energy transfer w (see Eq. 1.1.83); for a given electron energy, it is smallest for w = 6, i.e., for a head-on collision (see Fig. 9 in Chapter 1.1). The ratio up/u. for w = is shown in Fig. 7 for electron-electron and positron-electron scattering. (The designation parallel and antiparallel can lead to ambiguities. Here, parallel means that the two spins are parallel. In some theoretical papers, parallel means that each electron has its spin parallel to its momentum.) 2.5.5.1.2. THE EXPEMMENTAL METHOD.To use the polarization dependence of the Mgller cross section in measuring electron polarization, A. M. Bincer, Phys. Rev. 107, 1434 (1957). A. A. Kresnin and L. N. Rosentsveig, J. Exptl. Theoret. Phys. ( U . S . S . R . )32, 353 (1957); Soviet Phys. J E T P 6, 288 (1957). I29 G. W. Ford and C. J. Mullin, Phys. Rev. 108, 477 (1957). lS0J. M. C. Scott, Phil. Mug. [8] 2, 1472 (1957). A. I. Mukhtarov and I. S. Perov, Izvest. Akad. Nauk. S.S.S.R. 22, 883 (1958); Columbia Tech. Transl. 22, 876 (1958). I32 R. Raczka and A. Racska, Bull. acad. polon. sci. 6, 463 (1958). 133 K. Bockmann, G. Gamer, and W. R. Theis, 2. Physik 160, 201 (1958). la4 K. Nagy and I. Farkas, Nuovo ciniento 7, 570 (1958). Is6 P. Stehle, Phys. Rev. 110, 1458 (1958). S. Sarkar, Nuovo cimento 24, 139 (1962). l3’ G. Furlan and G . Peressutti, Nuovo cimento 15, 817 (1960); 19, 830 (1961). la* Y. S. Tsai, Phys. Rev. 120, 269 (1960). lZ7
128
242
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
electrons must be scattered by target electrons of known polarization, and some aspect of the scattering process must be measured. The troubles inherent in absolute measurements of cross sections can be avoided by measuring a quantity that depends on the ratio of cross sections for two polarization states of the target electrons. One such quantity can be developed as follows. Let Pi be the polarization vector of the electrons to be measured, while Pois the polarization 1.0
l
'
l
-
l
.
1
2
-
ELECTRON-POSITRON (Bhabha)
0
0.5
1.0
ENERGY (Mcv)
1.5
2 .o
FIQ.7. Cross section ratio (parallel to antiparallel spins) for Mprller and Bhabha scattering as a function of the incident electron energy in MeV. The curves are valid for symmetric scattering, where both emerging electrons have the same energy. The energy transfer in this case is w = i. vector of the target electrons. Let u+ be the scattering cross section with both electrons completely polarized in the direction of Pi and Po,while u- is the cross section with complete polarization but with the direction of POreversed. The cross section for scattering of electrons with uncorrelated polarization will be (u+ u-)/2. The magnitude of P is the fraction of particles that are fully polarized. Therefore a fraction PiPo of the collisions takes place between particles that are fully polarized, and have cross section u+ or a- depending on the direction of Po. The rest of the collisions have cross section (a+ u-)/2. With Poin one direction, the counting rate will be proportional to
+
+
(2.5.46)
2.5.
POLARIZATION OF ELECTRONS AND PHOTONS
243
With the direction of P o reversed, it will be proportional to (2.5.47) From these equations, one obtains (2.5.48) The polarization dependence of the cross section is often expressed in terms of u-/u+, and experimental results are given in terms of 8 , the relative change in counting rate on polarization reversal : 6=2
c+- c- = 2PtPo 1 - u-/u+* c++ c1 + .-/+
(2.5.49)
TARGET ELECTRONS. I n order to use the relation 2.5.5.1.3. POLARIZED Eq. (2.5.49), one must have target electrons of known polarization whose direction of polarization can be reversed. Such electrons exist in a magnetized ferromagnetic foil. About two of twenty-six orbital electrons in iron are aligned at saturation. The fraction f of electrons aligned can be obtained by measuring the magnetization of the foil. It is related to the magnetization through f = MJNP (2.5.50) where M, is the magnetic moment per unit volume in the material due to electron spin, N is the number of electrons per unit volume, and p is the Bohr magneton. M a is not equal to the total magnetic moment per unit volume in the material: in addition to Ma there is a small but significant orbital c o n t r i b u t i ~ n . ‘The ~ ~ relative magnitudes of spin and orbital contributions can be obtained from a measurement of the gyromagnetic ratio for the foil material by an Einstein-de Haas or Barnett experiment. M, is related to the total magnetization M by M,=2-
g’
-1 9’
M
(2.5.51)
where g‘ is the magnetomechanical f a ~ t 0 r . l ~ ~ Values of the magnetomechanical factor g’ (which is sometimes also called the gyromagnetic ratio) are collected in reference 140. The data in reference 140 also allow reasonable estimates for alloys. C. Kittel, “Introduction to Solid State Physica,” 2nd ed., pp. 408-414. Wiley, New York, 1956. 140 G . G. Scott, Revs. Modern Phys. 54, 102 (1962).
244
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
2.5.5.1.4. THE DETECTION OF ELECTRON-ELECTRON SCATTERING. The cross section for Mgller scattering from all electrons of an atom is smaller than the Mott cross section for scattering from the nucleus by a factor of about the atomic number of the scatterer. Therefore, Mgller-scattered electrons must be detected among a much larger number of Mott-scattered ones. Fortunately, there are differences between Mott-scattered and Mller-scattered electrons. The Mgiller-scattered ones come from collisions between particles of equal mass. In such collisions, as seen in the lab system, the incident electron gives up much of its kinetic energy to the target electron, and both emerge from the collision. This allows Mgller collisions to be distinguished from the Mott background by counting the two electrons in coincidence, Use of coincidence counting is helped by the fact that Mgller scattering can be treated as a two-body process as long as the bombarding energy is much greater than the binding energy of the target electrons. If the incident momentum is known, there is a fixed relation between the momenta of the two outgoing electrons. The counters can be placed so that if one electron reaches one counter, the other electron will almost certainly reach the other. This relation can be obscured by multiple scattering in the foil. Energy conservation in the collision can be used to discriminate against spurious coincidences by requiring the energies of the two electrons to add up to the incident energy. 2.5.5.1.5. EXPERIMENTAL ARRANQEMENT. The scattering method described above has been used to determine the helicity of electron^,'^^-^^^ of p o s i t r ~ n s , ' ~and ~ J ~of~ muons.147 A very simple experimental arrangement is shown in Fig. 10 of Chapter 1.1. An improved system is represented schematically in Fig. 8. A magnetized ferromagnetic foil provides polarized target electrons and coincidence counting is used to detect Mgller collisions. The angle 0 is the I 4 l H. Frauenfelder, A. 0. Hanson, N. Levine, A. Rossi, and G. DePasquali, Phys. Rev. 107, 643 (1957). 14* N. Bencaer-Koller, A. Schwarzschild, J. B. Vise, and C. S. Wu, Phys. Rev. 109, 85 (1958). 143 J. S. Geiger, G. T. Ewan, R. L. Graham, and R. D. MacKensie, Phys. Rev. 112, 1684 (1958). 144 J. D. Ullman, H. Frauenfelder, H. J. Lipkin, and A. Rossi, Phys. Rev. 122, 536 (1961). 146 D. M. Harmsen and K. Holm, 2.Physik 166, 227 (1962). * 4 6 J. C. Hopkins, J . B. Gerhart, F. H. Schmidt, and J. E. Stroth, Phys. Rev. 121, 1185 (1961). 147 G. Backenstoss, B. D. Hyams, G. Knop, P. C. Marin, and U. Stierlin, Phys. Rev. Letters 6, 415 (1961).
2.5.
POLARIZATION
OF ELECTRONS A N D PHOTONS
245
laboratory scattering angle corresponding to a 90" c.m. scattering angle. A beta monochromator selects electrons in the desired momentturn band. The Mdler-scattered electrons a t 90" c.m. scattering angle possess half the selected energy. Pulse-height analysis in the two counters thus distinguishes the Mplller-scattered electrons from those which have undergone Coulomb scattering. Energy selection and coincidence arrangement together form a very effective means for singling out the desired events. I n principle, the arrangement shown in Fig. 8 can be used to detect longitudinal as well as transverse polarization. The cross section ratio u+/u- is not very different from one for transverse polarization, h ~ w e v e r . ~
SOURCE Magnetized
MONOCHROMATOR
in direction of arrow
COUNTERS
FIG.8. Basic arrangement for the measurement of electron and positron helicity by means of Moller and Bhabha scattering.
In practice the method is therefore always used for the determination of longitudinal polarization. In this case the target electrons should be polarized in the direction of the incident electron beam. Unfortunately, extremely high fields are necessary to magnetize a thin foil in a direction normal to its surface. Therefore, it is necessary to magnetize the foil along its surface and incline i t a t an angle to the electron beam, as shown in Fig. 8. If the fraction of electrons aligned in the foil is f, and the angle of inclination of the foil to the electron beam is a, the target electrons will have longitudinal polarization f cos LY when looked at in the c.m. system. The relative change 6 of the coincidence counting rate then follows from Eq. (2.5.49) if one sets P O = f cos a and uJu+ = u p / u a :
(2.5.52)
The ratio up/ua is given in Chapter 1.1, Eq. (1.1.86). AND PROCEDURE. One realization of the arrange2.5.5.1.6. EQUIPMENT ment in Fig. 8 is shown in Fig. 9. A detailed drawing of a similar instrument is reproduced in reference 145. Additional experimental information
246
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
LEAD STOP,
SOURCE MQNOC;HAOMATOR COILS
n
HELMHOLTZ
\I
COILS
FIG.9. Experimental realiaation of the arrangement shown schematically in Fig. 8. (After Ullman, Frauenfelder, Lipkin, and Rossi, reference 144.)
can be found in references 141-147. A few remarks about equipment and experimental procedure follow here. (1) The angle 28 between the two counters must be adapted to the energy selected by the spectrometer. If the total energy of the incoming electron is W , and if both scattered electrons have equal energy (w = t), then sin2 e =
2mc2
w + 3mc2
(2.5.53)
At small energies, the two electrons emerge a t an angle 20 = 90"; a t higher energies, scattering is peaked forward and 8 decreases. (2) The design of the spectrometer is important. It is best to use one with small spherical aberration (intermediate image spectrometer, see Section 2.2.1.1). Otherwise only a small fraction of the electrons striking the scattering foil will reach the counters. (3) The scattering foil must be selected so that the multiple scattering in the foil is not too disturbing. The fraction of electrons aligned should be as large as possible and the external magnetic field necessary to polarize should be aa small as possible. The first requirement is satisfied if the foil
2.5.
POLARIZATION OF ELECTRONS AND PHOTONS
247
thickness is selected such that the root-mean-square scattering angle for the incident electrons equals the counter aperture.144To meet the other two conditions one usually selects Deltamax or Supermendur foils. Superm e n d ~ r with l ~ ~ a composition of 49% Fe, 49% Co, and 2 % V, has a fraction f of electrons aligned which varies between 5 and 9 % and can be kept near saturation with a holding field of 3 oe. Deltamax possesses an f around 5 % and requires a holding field of about 4 oe. These values apply to foils of thickness 2-8 mg/cm2. The determination off is straightforward, but somewhat difficult; details are discussed in references 143 and 144. Equations (2.5.50) and (2.5.51) are used to find f from the measured magnetization M . (4) Scintillation counters are usually selected as detectors. The thickness of the (plastic) scintillators is chosen to be about the range of the electrons to be counted. ( 5 ) The coincidence circuit should have a resolving time which is as short as possible in order to minimize accidental coincidences. A conventional slow-fast system permits a good pulse-height selection and a resolving time of the order of a few nanoseconds. (6) In order to determine 6, Eq. (2.5.49), the foil is magnetized in one direction and coincidences are recorded for a fixed period of time. The magnetization of the target foil is then reversed and the corresponding coincidence rate is measured. (7) The coincidence rates so determined must be corrected for accidental coincidences and for spurious coincidences due to background, to beta-gamma cascades, and to annihilation radiation.144 (8) Asymmetries in the equipment can be tested by replacing the ferromagnetic scattering foil with a nonmagnetic foil of about equal thickness (A1 or Cu). (9) Counter-to-counter scattering must be checked and eliminated. (10) The effect of reversing the magnetic field at the scattering foil on the counters and on the electron trajectories must be minimized. 2.5.5.1.7. CORRECTIONS. The polarization Pi, determined from 6 by Eq. (2.5.52), must be corrected for the following effects: 2.5.5.1.7.1. Finite Apertures and Finite Energy R e ~ o l u t i o n . ~The ~~J~~ relations (Eq. 2.5.52) are true for a single energy, scattering angle, and angle between the trajectory of electrons and the axis of the foil. I n an actual experiment, a fairly wide range of all three of these quantities has to be accepted to get a reasonable counting rate. The problem of averaging the expression on the right of Eq. (2.5.52) over all the accepted events is, in general, very complicated. 14*
H. L. B. Gould and D. H. Wenny, Elec. Eng. 76, 208 (1957).
248
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
The problem is simplified if the accepted energy range is reasonably narrow. Except a t low energy, where the Mprller scattering method is not at its best anyway, both up/ua and Pi vary slowly with energy. Pi can then be treated as a constant, and up/uacan be calculated over the mean accepted energy, The averaging of cos a over electron trajectories can be separated from the averaging of R, Eq. (2.5.52)) over scattering angles as long as the accepted distribution of scattering angles does not vary appreciably over the range of variation of a. This can be assured by making the counter faces large enough. The averaging of cos a over electron trajectories is a straightforward geometry problem. Averaging R over scattering angles is more complicated. First, one must know the distribution in scattering angle of the counted events. Controlling this distribution is helped by the fact that in Meiller scattering there is a definite reIation between the scattering angle and the energy of an electron after the collision. This correlation is important, because the energy will hardly be changed by multiple scattering that radically changes the direction of motion. It is possible to limit the accepted range of scattering angles by limiting the accepted range of energies. If the counters are large enough to intercept all the electrons allowed by the energy selection, multiple scattering will have little effect on the accepted distribution. The distribution of accepted events as a function of electron energy after scattering can be measured by sweeping a narrow channel across the energy range accepted by one counter. This procedure gives the distribution of Mplller-scattered electrons reaching the counter. The resulting curve, when multiplied by the energy-sensitivity curve of the counter as used in the polarization measurement, yields the accepted distribution of events. Once this distribution is known, the averaging process can be carried out by numerical integration. The averaging process is discussed in references 144 and 145. 2.5.5.1.7.2. D e p o E ~ r i z a t i o n Electrons . ~ ~ ~ ~ ~ ~can ~ be depolarized by multiple and single scattering in the source, and by multiple scattering in the foil and any material between the source and the foil. It is generally true that depolarization corrections are less important in experiments involving Mprller scattering than in those using Mott scattering (Section 2.5.5.2)) for the following reasons. The energies involved in MPler-scattering experiments are usually larger than those involved in Mott scattering. The depolarization in the source is hence smaller. Depolarization in the scattering foil is less important in Mprller scattering because the energy and angle conditions which an event must satisfy in order to be counted are stringent. Details of the corrections for depolarization in the source
2.5.
POLARIZATION OF ELECTRONS AND PHOTONS
249
and in the absorber are given in references 144 and 145 and in Section 2.5.5.2.6. 2.5.5.2. Mott Scattering.* 2.5.5.2.1. THEORETICAL BACKGROUND. The classical way to study electron polarization is by means of Mo tt scattering, i.e,, by the scattering of the electron in the Coulomb field of a nucleus. l3v14 The spin-orbit interaction leads to a difference in scattering a t a fixed scattering angle between electrons with spin up and electrons with spin down. This spin dependence is the basis of the method discussed in the present section. Mott scattering is best suited for the analysis of the transverse polarization of negative electrons of energies between about 30 and a few Lob.
System
Rest System of the Electron
FIG. 10. Mott scattering. In the lab system, the incident electron with spin d perpendicular to the plane of scattering is deflected by an angle e to the right. In the electron rest system, the nucleus orbits partially around the electron and creates a magnetic field B. I n the situation shown in the figure, this magnetic field and the spin d are antiparallel. The magnetic moment and the field B hence are parallel, the spin+rbit interaction adds to the Coulomb attraction, the scattering to the right is enhanced.
hundred kev. If the energy is lower than about 30 kev, the experimental problems of preparing thin enough scattering foils become formidable. If the electron energy is much higher than, say, 600 kev, the asymmetry due to the spin-orbit interaction becomes very small. The method is also not well suited for positrons. The Coulomb repulsion keeps the positron away from the nucleus; hence the spin-orbit interaction and the asymmetry in scattering remains small a t all energies. The basic physical idea underlying Mott scattering can be described b y reference to Fig. 10. Assume that an electron is scattered in a plane perpendicular to its spin. The existence of a spin-orbit interaction is then easiest to see in the rest system of the electron, where the nucleus moves around the electron. This motion gives rise to a magnetic field at the electron. If the magnetic field and the magnetic moment of the electron are parallel, the attractive potential is increased. I n the case of Fig. 10, this happens for electrons scattered to the right and one expects increased * See also Vol. 4A, Section 3.5.1.
250
2. DETERMINATION
OF FUNDAMENTAL QUANTITIES
scattering to this side. Hence one has the following rule, useful in discussing experimental arrangements : Electrons with spin up, i.e., with the magnetic moment pointing down, approaching a nucleus of positive cha,rge straight ahead, are predominantly scattered to the right.
(2.5.54)
2.5.5.2.2.THEASYMMETRY FUNCTION 8(e): THEORETICAL RESULTS."~'4 Mott scattering is discussed in Section 1.1.5.3. The two quantities important for experimental work are (du/dQ), the differential cross section for an unpolarized beam, and 8(8,WJZ), the asymmetry function. 8(e,W,Z)is the polarization produced if an unpolarized electron beam of energy W is scattered by a thin target of atomic number 2; 13 is the scattering angle. In the following, S(e,W,Z) will be written S(e). To describe the Mott-scattering process of a polarized beam, one can introduce spherical coordinates, with the z-axis along the momentum of the incident electron. The transverse polarization P of the incident electron shall lie along the azimuth Q O and the scattering shall be described by a polar angle 0 and an azimuthal angle Q. The cross section for Mott scattering then can be written as
Assume that the scattering occurs in the plane perpendicular to the polarization vector, so that Q - (DO = 3a/2 = -a/2 for scattering to the left and Q - Q O = +a/2 for scattering to the right. "Left" is thus defined by a unit vector such that
e
L=PXp
(2.5.56)
where $ and are unit vectors in the direction of the polarization and the momentum of the incident electrons, respectively. (This definition of left and right agrees with the one shown in Fig. 10.) The ratio of intensities of electrons scattered to the left to electrons scattered to the right then becomes
Hence one finds for the transverse polarization P
p
1L-R =--.
SL+R
(2.5.58)
2.5.
POLARIZATION
OF ELECTRONS AND PHOTONS
25 1
The generalization to the case where the polarization vector is not perpendicular to the scattering plane is trivial. The most complete recent calculation of du/dQ and s(0)has been performed for electrons with p = v/c = 0.2, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, for scattering on mercury (2 = 80), cadmium (2 = 48), and aluminum (2 = 13).149Partial results are shown in Figs. 11 and 12 of Chapter 1.1. Additional values for gold (2 = 79) and aluminum for v/c = 0.49 and 0.59 are given in reference 150. A number of additional remarks may be useful: (1) Figure 11 in Chapter 1.1 shows that S(e) for mercury is negative for most values of v / c and e. For positive P, Eq. (2.5.58) shows that scattering then occurs predominantly to the right, in agreement with rule (2.5.54). (2) In all calculations, the transverse polarization P is defined in the rest system of the electron. (3) The calculation149has been performed using unscreened Coulomb functions. A number of calculations using screened functions exist,151J62 and these indicate that screening is indeed important below about 150 kev. Unfortunately, these calculations are not accurate and extensive enough to be of much use in experiments. Experimental methods hence must be resorted to at the present time for estimates of the correct asymmetry function s(e)at low electron energies (see Section 2.5.5.2.3). (4) The calculations of s(e)in references 149 and 150 assume point nuclei. The influence of a finite nuclear size on s(0)has also been investigated153;a t the energies used in most experiments, the corresponding correction will be less than 0.1 %. (5) The contribution to s ( 0 ) due to inelastic electron scattering is less than 0.5% for gold scatterers16*and it can hence be neglected in most experiments. (6) The scattering of polarized muons from extended nuclei has also been c a l ~ u l a t e d . ~The ~ ~ resulting ~ ~ ~ 6 asymmetry may be useful to determine the polarization of muon beams. 2.5.5.2.3. THEASYMMETRY FUNCTION s(6): EXPERIMENTAL RESULTS. In the absence of a completely satisfactory calculation of s(0) for lowN . Sherman, Phys. Rev. 103, 1601 (1956). N.Sherman and D. F. Nelson, Phys. Rev. 114, 1541 (1959). J. Bartlett and T. Welton, Phys. Rev. 69, 281 (1941). 16* C.B. 0. Mohr and L. J. Tassie, Proc. Phys. Soe. (London) 67, 711 (1954). l K 3B. K.Kerimov and V. M. Arutyunyan, J . Exptl. Thewet. Phys. (U.S.S.R.) S8, 1798 (1960);Soviet Phys. J E T P 11, 1294 (1960). lK4 G. Felsner and M. E. Rose, Nuovo cimento 20, 509 (1961). l K 6J. Franklin and B. Margolk, Phys. Rev. 109, 525 (1958). lK6G.H.Rawitscher, Phys. Rev. 112, 1274 (1958). 14*
lKo
252
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
energy electrons, the existing experimental data give some useful information about the influence of screening. Two methods have been used to find s(e). The more straight-forward approach is to use double scattering of e l e c t r o n ~ . * ~ In~ ~the 4 first scattering, an unpolarized electron beam becomes partly polarized; the second scattering acts as an analyzer. The asymmetry in the double-scattering experiment yields the product S(0,) X S(0,). I n a second method, one assumes that electrons emitted in allowed beta transitions possess a polarization P , = v / c . The longitudinal polarization is first transformed to a transverse one (Section 2.5.3.3) and the asymmetry of the transversely polarized beam is then determined. With the assumption P = v / c , S(0) then follows from Eq. (2.5.57). The early experiments, restricted to double scattering, are summarized in references 1, 13, and 14. Recent investigations have used double A doubles ~ a t t e r i n g * ~ ~and - l ~polarized l electrons from beta decay.162*163 scattering experiment a t an electron energy of 1.5 kev has been performed with a mercury atomic beam.ls4 The most recent data are plotted in Fig. 11, which shows the ratio S,,,(0)/Sthe0,(O) as a function of the electron energy. Sexp(0) is the asymmetry function as determined by one of the two methods outlined above. &haor(e) is the value calculated without s ~ r e e n i n g . ' * ~ ~ ' ~ ~ Figure 11 shows that screening is very important a t energies below 150 kev. Even above 150 kev, there may still be some deviations from the "unscreened" value. Figure 11 also demonstrates the need for more and more accurate experiments and for improved theoretical calculations. At the present time, the data in Fig. 11 are useful for estimates of the correct asymmetry function S ( 6 ) for electron energies between 40 and 200 kev. 2.5.5.2.4. DETERMINATION OF THE TRANSVERSE POLARIZATION OF ELECTRONS. Mott scattering is best suited for the measurement of the W. G. Pettus, Phys. Rev. 109, 1458 (1958). D. F. Nelson and R. W. Pidd, Phys. Rev. 114,728 (1959). P. E. Spivak, L. A. Mikaelyan, I. E. Kutikov, and V. F. Apalin, Nuclear Phys. 23,
lS8
lS9
169 (1961). P. E. Spivak, L. A. Mikaelyan, I. E. Kutikov, V. F. Apalin, I. I. Lukashevich, and G . V. Smrinov, J . Exptl. Theoret. Phys. (U.S.S.R.)41, 1064 (1961); Soviet Phys. J E T P 14, 759 (1962). V. A. Apalin, I. Y. Kutikov, I. I. Lukashevich, L. A. Mikaelyan, G . V. Smirnov, and P. Y. Spivak, Nuclear Phys. 31, 657 (1962). 162 H. Bienlein, G. Felsner, K. Guthner, H. von Issendorff, and H. Wegener, Z. Physik 164,376 (1959). H. Bienlein, G. Felsner, R. Fleischmann, K. Giithner, H. von Issendorff, and H. Wegener, 2.Physik 166, 101 (1959). '64 H. Deichsel, 2.Physik 164, 156 (1961).
2.5.
253
POLARIZATION OF ELECTRONS A N D PHOTONS
transverse polarization of (negative) electrons with energies between about 40 and a few hundred kev. The basic setup for such a measurement is very simple. The beam of electrons strikes the scattering foil. A small fraction of the electrons (10-4-10-6) suffers a large angle scattering and enters one of the two symmetrically arranged counters C L or CR. The ratio L / R of the intensities in the two counters is a direct measure for the transverse polarization [Eq. (2.5.58)]. In theory this procedure is straightforward and simple. I n practice there exists a n enormous number of difficulties and the experimental ratio
"1" 1 I
REFERENCES
0.71 ,
0.60
,
,
,
100
,
,
,
,
200 ELECTRON ENERGY (kew I
,
,
300
FIG. 11. Ratio of the experimentally determined asymmetry function 5.,,(120") to the function St~,,,(120"), calculated without screening. All values apply for a gold scatterer (Z = 79).
( L / R ) e x pmust be corrected for many deviations from a n ideal setup before a reliable value for the polarization P can be obtained. I n the present section, some of the important points in designing a system for such an experiment will be sketched; the corrections which must be p discussed in Section 2.5.5.2.6. applied to ( L / R ) e x are Electrons can only be transversely polarized if a direction, different from that of the momentum, is given by some physical process or quantity. I n double scattering, the normal to the scattering plane in the first scattering constitutes such a quantity. I n radioactive decay, electrons can be transversely polarized if measured in coincidence with a preceding or following radiation. An example is the transverse polariza-
254
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
tion of conversion electrons following beta decay.16b169The theory predicts that the conversion electron will, in general, be emitted polarized, with the polarization vector parallel to the beta electron momentum. Thus, conversion electrons emitted at right angles to the beta particles are transversely polarized. I n order to select these electrons one must record coincidences between a beta counter and two counters which record the left-right asymmetry after scattering. Because this type of
FIQ. 12. Schematic representation of a setup to measure the transverse polarization of conversion electrons following beta decay. (From Blake, Bobone, Frauenfelder, and Lipkin, reference 172.)
experiment shows all the basic aspects of a transverse polarization measurement, a typical setup will be discussed here. The transverse polarization of conversion electrons following beta decay has been determined in a number of experiment~.17~-~7~ A typical setup is shown in Fig. lZ.172 The beta particles emitted by the source S are detected in the counters 0 and 00. The subsequent conversion electrons are focused by a two-lens magnetic spectrometer upon the gold foil T. 166
H. Frauenfelder, J. D. Jackson, and H. W. Wyld, Phys. Rev. 110,451 (1958).
186
€3. V. Geshkenbein, Nuovo cimento 10, 365 (1958).
V. B. Berestechij and A. P. Rudik, Nuovo cimento 10, 375 (1958). R. L. Becker and M. E. Rose, Nuovo cirnento 18, 1182 (1959). 169 H. R. Lewis and J. R. Albers, 2. Physik 168, 155 (1960). 170 J. E. Alberghini and R. M. Steffen, Nuclear Phys. 14, 199 (1959). 171M. E. Vishnevskii, V. A. Lyubimov, E. F. Tretihkov, and G . I. Grishuk, J. Ezptl. Theoret. Phya. (U.S.S.R.) 38, 1424 (1960); Soviet Phys. J E T P 11, 1029 (1960). 178 B. Blake, R. Bobone, H. Frauenfelder, and H. J. Lipkin, Nuovo cimento 26, 942 (1962). 167
168
2.5.
POLARIZATION OF ELECTRONS AND PHOTONS
255
The currents in the two spectrometer lenses are of equal magnitude but opposite sign in order to eliminate a rotation of the conversion electron spin. The transverse polarization of the conversion electrons is then determined by Mott scattering from the gold foil T a t an angle of 120". The counts L and R in the two detectors CL and CR, respectively, are recorded in coincidence with the events in the beta detectors p and Po. The coincidences with the detector Po, which counts beta particles emitted parallel to the spectrometer axis, serve to determine the instrumental asymmetry. The source in all experiments should be as thin as possible and as uniform as possible, in order to minimize depolarixation and backscattering. The energy or momentum selector and the slit systems should be completely symmetric and should not contribute spuriously scattered electrons. Lining the entire system with polystyrene and constructing the scattering chamber as large as possible reduces the amount of unwanted scattered e l e c t r o n ~ . ~ ~ ~ 7 ~ A thin gold foil is usually employed as the target. I n principle, uranium foils would be better, because the asymmetry function S(0) becomes larger with increasing 2. However, gold is the heaviest material which can be worked easily into thin uniform foils. The position of the scattering foil is crucial. In early scattering experiments, foils were placed a t 45" to the beam and the intensities L and R were then determined. It turned out, however, that such an arrangement is extremely inconvenient. Multiply scattered electrons may mask the polarization effect nearly c0mp1etely.l~The foils are now usually placed perpendicular to the beam and the scattered electrons are observed a t angles around 120" (Figs. 2, 3, and 12) or around 70" (Fig. 1). I n the first case, the intensity of the Mott-scattered electrons is small, but the asymmetry function S(l2OO) is large. In the second case, the intensity is much larger, but S(70") is quite small. The choice between these two solutions depends on the particular problem. Generally, the larger angle is to be preferred. Two methods allow the determination of some instrumental asymThe first, more common one, is to replace the gold scatterer by an aluminum foil. S(0) is small for A1 and the two counters CL and C R hence should give the same counting rates. In the second method, one uses the fact that even for gold S ( 0 ) is very small a t scattering angles around 30" (see Fig. 11, Chapter 1.1, Vol. 5 A). Two additional counters at small scattering angles are employed to get the data necessary for the corrections of the instrumental asymmetries.l* Unfortunately, these measurements eliminate only some of the instrumental asymmetries. Even small misalignments of the electron beam with respect to the symmetry axis of the system can introduce large errors. These errors and the
256
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
methods to correct for them are discussed in detail in references 74, 75, and 78. 2.5.5.2.5. DETERMINATION OF THE LONGITUDINAL POLARIZATION OF ELECTRONS. The basic idea involved in the measurement of the longitudinal polarization of electrons by Mott scattering is now clear. First, the longitudinal polarization is transformed to a transverse one (see Section 2.5.3.3). The polarization transformer usually acts also as a momentum selector. Second, the transverse polarization is then determined as outlined in Section 2.5.5.2.4.Three basic systems are shown in Figs. 1-3. References are also given in Section 2.5.3.3. One particular idea deserves to be mentioned here. The difficulty of the only approximately known asymmetry factor s(0) a t low electron energies can be partly avoided by accelerating the electrons to a fixed energy. Relative measurements can then be made easily and absolute ones are ~ i m p l i f i e d . ~ ~ 2.5.5.2.6. CORRECTIONS. I n order to find the polarization, the raw data L and R must be corrected for depolarization in the source, the target, and the polarization transformer, the background must be taken into account, and geometric factors resulting from the finite sizes of the source, the spectrometer apertures, the target, and the detectors must be determined. These corrections are treated in detail in references 1, 74-76, 78, 79-81, 87-90. Here a few remarks concerning the most important corrections are made. 2.5.5.2.6.1. Plural and Multiple Scattering in the Target. This contribution is the most important source of errors in Mott scattering.14 Plural scattering consists of two scattering events of about equal angle; multiple scattering consists of a succession of small-angle scattering. Both effects contribute to the counting rate in the detectors CL and CR. In either case, the individual scattering event will involve angles of less than, say, 60'. Figure 11, Chapter 1.1, Vol. 5 A, then shows that s(e < 60') is nearly zero. The contribution from plural and multiple scattering thus does not show an asymmetry, and it must be subtracted from the total counting rate in order to find the true asymmetry L / R . The contribution from multiple scattering can be calculated reasonably well173(see also the discussion and the references in Section 1.1.6). The foil can be made thin enough so that the contribution from multiple scattering is very small. It is much more difficult to get rid of plural scattering or to correct for it by calculation. Foils thin enough to eliminate the contribution due to double scattering yield only very low counting rates. Moreover, the ever-present wavyness of very thin foils considerably enhances the contribution of double scattering over the amount expected from theoretical estimates. H. Wegener, 2.Phyaik 161, 252 (1958).
2.5. POLARIZATION OF
ELECTRONS AND PHOTONS
257
The best method to correct for the contribution from plural and multiple scattering in the target consists in determining the intensities L and R as functions of the target thickness and then extrapolating to zero thickness. For thin targets (thickness t < 1 mg/cm2, even at electron s e d One . energies of 600 kev), a linear extrapolation is ~ plots either L / R or ( L - R ) / ( L R ) versus the foil thickness and extrapolates to zero thickness.
+
1.6
fl
LEAST-SQUARES STRAIGHT-LINE FIT TO THESE POINTS
FIG.13. Experimental correction for plural and multiple scattering in the target. extrapolated to zero target thickness, yields [PS(O)]-'l*.The kinetic energy of the electrons is 616 kev, the scattering angle 0 is 135". (From Brosi, Galonsky, Ketelle, and Willard, reference 78.)
An improved approach, which fits the data for a much larger range of target thicknesses, is described in reference 78: one plots the function
[PS(e>]-"~ =
d(J5 + R)/(L- R)
(2.5.59)
versus the target thickness. The extrapolation to zero thickness then yields the desired value of P . Figure 13 shows an example of such an extrapolation. 2.5.5.2.6.2. Depolarization. It was pointed out in Section 2.5.3.3.3 that the polarization vector of electrons remains approximately fixed with respect to the laboratory system during multiple scattering. More accurately, if the momentum is turned by an angle A# by multiple scattering in low-2 material, then the polarization vector turns only
~
2.
258
DETERMINATION OF FUNDAMENTAL QUANTITIES
through an angle A 4 , given by (2.5.60)
where v is the velocity of the electron. This effect leads to an energydependent depolarization of an electron beam. Electrons initially moving in the direction towards the analyzer, i.e., the target or the polarization transformer, suffer no appreciable depolarization. Electrons which initially moved in other directions and are scattered into the acceptance angle of the analyzer have their polarization vector pointing in different directions. Electrons originally moving backwards and being scattered forwards may even have polarization vectors pointing in the opposite direction from the vectors of the “true electrons.” The scattered electrons hence decrease the degree of longitudinal polarization of the electron beam. The multiple scattering of polarized electrons has been studied by various authors.129,174-1” Detailed formulas can be found in these references. In an actuaI experimental setup, depolarization can occur in the source, in the source-backing material, in all material between the source and the target foil, in the walls, and in the polarization transformer. The magnitude of the depolarization is usually estimated with the help of theoretical calculations. Moreover, the calculations are best checked by additional experiments, for instance varying the source or the sourcebacking thickness. Most important, theoretical estimates are used to find values for the maximum thickness of the various components; the system is then designed in such a way that all corrections remain small. In the souwe, depolarization occurs either by a large-angle single scattering or by small-angle multiple scattering, These two effects must be considered separately. Calculations have been performed for longit ~ d i n a l l y ”and ~ t r a n s v e r ~ e l y ’polarized ~~ electrons. It turns out that the two contributions, multiple and single scattering, contribute terms of the same order of magnitude. Under identical conditions, transversely polarized electrons are much less depolarized than longitudinally polarized ones. E. Rose and H. A. Bethe, Phys. Rev. 66, 277 (1939). B. Muhlschlegel and H. Koppe, 2. Physik 160, 474 (1958). I. N. Toptygin, J . Ezptl. Theoret. Phys. (U.S.S.R.) 80, 488 (1959). G. Passatore, Nuovo cimento 18, 532 (1960). B. Miihlschlegel, 2.Physik 166, 69 (1959). B. Blake and B. Miihlschlegel, 2.Physik 167, 584 (1962).
114 M.
ll6 ln
2.5.
POLARIZATION OF ELECTRONS AND PHOTONS
259
Figure 14 shows an experimental investigation of the influence of source thickness and backing thickness on the measured value of the polarization. The depolarization due to scattering from the walls, from the foil and source holders, and from the plates of the polarization transformer or the momentum selector is much more difficult to study. Theoretical calculations are nearly impossible. Experimentally, one can find some information by changing the various parts of a setup in a controllable manner.’8
P v/c
--
co60 Source 2 rng /cm2
0
Co60 Source
Thickness = I rng/crn2
0.8 I.6 2.4 3.2 Thickness of Ni Backing (mg/crn2)
4.0
FIQ.14. The effect of the thickness of the source backing and the source on the meaaured polarization. Electron energy 194 kev. (From Greenberg, Malone, Gluckstern, and Hughes, reference 74.)
2.5.5.2.6.3.Finite-Angle Corrections. The source, the polarization analyzer, the target, and the counters all subtend finite solid angles. Moreover, the momentum selector will also possess a finite momentum resolution. Corrections must be applied for all these effects.The solid angles will also introduce a spread in the angle & between the polarization vector and the scattering plane. Equation (2.5.57) thus must be used in the form in which the term sin(+ - &) is still retained. The function sin(+ - cb0) then must be averaged over the accepted angles.78 2.5.5.3. Methods Depending on Helicity Transfer. In the present section, experiments are discussed in which a longitudinally polarized electron gives rise to one (or more) circularly polarized gamma rays, and the circular polarization of the gamma rays is measured. The discussion can be very brief because the polarization transfer is already treated in Section 2.5.4 and the determination of the circular polarization of gamma rays is dealt with in Section 2.5.6. 2.5.5.3.1. EXTERNAL BREMSSTRAHLUNG. The basic idea underlying the determination of the electron helicity by using the bremsstrahlung is
2.
260
DETERMINATION OF FUNDAMENTAL QUANTITIES
extremely simple. Electrons are allowed to emit bremsstrahlung in a radiator. The polarization of the emitted radiation is then measured in a polarimeter. 180-18’ An experimental arrangement is shown schematically in Fig. 15. Before discussing Fig. 15, some differences between internal and external bremsstrahlung must be mentioned. The external bremsstrahlung is produced
FIG.15. Arrangement for measuring the helicity of internal and external bremsstrahlung. The main part of the external bremsstrahhng originates in the radiator of high atomic number 2. The absorber (low Z ) prevents electrons from entering the polarimeter and producing spurious bremsstrahlung.
by an electron which leaves the nucleus and then radiates in the field of another nucleus. The intensity of the external brernsstrahlung is approximately proportional to the atomic number 2 of the material in which the radiation occurs (see Section 1.1.5.1). The internal bremsstrahlung, which originates in the atom of the decaying nucleus, is much weaker than the external bremsstrahlung and its intensity is approximately independent of the atomic number of the source.18* In the arrangement shown in Fig. 15, the source will emit internal M. Goldhaber, L. Grodzins, and A. W. Sunyar, Phys. Rev. 106, 826 (1957). F. Boehm and A. H. Wapstra, Phys. Rev. 109, 456 (1958). 18* H. Schopper and S. Galster, Nuclear Phys. 6, 125 (1958). 188 E. G. Beltrametti and S. Vitale, Nuovo cimento Q, 289 (1958). 184 U. Amaldi, Jr., M. Bernardini, P. Brovetto, and S. Ferroni, Nuovo cimento 11,
180
181 ’
415 (1959). 18)
A. Bisi and L. Zappa, Phys. Rev. Letters 1, 332 (1958); Nuclear Phys. 10, 331
(1959). 186
G . Culligan, 5. G. F. Frank, and J. R. Holt, Proc. Phys. Isoc. (London) 78, 169
(1959). 187
P. Lipnik, J. P. Deutsch, L. Grenacs, and P. C. Macq, Nuclear Phys. SO, 312
(1962). 188 C. S. Wu, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 649. North Holland, Amsterdam, 1955.
2.5.
POLARIZATION OF ELECTRONS AND PHOTONS
26 1
bremsstrahlung and beta particles. The internal bremsstrahlung quanta are circularly polarized. 189-191 The beta particles give rise to circularly polarized external bremsstrahlung quanta in the radiator. The experimental setup must be so designed that two conditions are fulfilled: (1) No electrons should give rise to bremsstrahlung originating in the polarimeter
FIG.16. Total intensity of internal and external bremsstrahlung, after subtraction of the background, a8 a function of the atomic number 2 of the radiator. The int.ensity a t 2 = 0 is due to internal bremsstrahlung alone. Points are for photon energies between 0.6 and 1 Mev (x), a.nd between 1 and 1.4 Mev ( 0 ) . (After Schopper and Galster, reference 182.)
which could falsify the results. This condition is enforced by the absorber, which consists of a very low 2 substance, such as water, polyethylene, or paraffin; (2) A separation between internal and external bremsstrahlung must be possible. It is achieved by measuring the bremsstrahlung intensity as a function of the atomic number, 2, of the radiator. 182,188 The result of such an experiment is shown in Fig. 16. The main problem consists in finding the correct correspondence between the longitudinal polarization of the original electron and the G. W. Ford, Phys. Rev. 107, 321 (1957). R. E. Cutkosky, Phys. Rev. 107, 330 (1957). lv1 A. Pytte, Phys. Rev. 107, 1681 (1957). I89 I90
262
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
circular polarization of the bremsstrahlung. The helicity transfer has already been discussed in Section 2.5.4. I n addition, however, the following problems arise: (1) The depolarization in the source, the radiator, and the absorber must be calculated. Backscattering must be taken into account. (2) The energy distribution of the beta rays entering the radiator must be known. (3) The angular and the energy distribution of the bremsstrahlung must be found. The arrangement shown in Fig. 15 is not restricted to electrons from radioactive sources. Electrons emitted by short-lived reaction products can also be used. In this case, where the experiment is done at an accelerator, a pulsed beam and a gated detector can be employed to reduce the background. lE7 2.5.5.3.2. ANNIHILATION-IN-FLIGHT. As pointed out in Section 2.5.4.5, the higher energy gamma ray from positron annihilation-in-flight carries most of the helicity of the original positron. This property has been used to determine the positron helicity :lg2--1g4 positrons impinge on a converter, in which they annihilate in flight. The highest energy annihilation quanta, emitted in a relatively narrow cone forward, are then analyzed with a Compton polarimeter (Section 2.5.6). It is advantageous to use a spectrometer to select the energy of the incoming p o s i t r o n ~ . ~ ~ The 2 , ~ 9converter ~ can be made out of lucite. I n order to reduce the background, it is possible to use a plastic scintillator as converter and then measure coincidences between the analyzed gamma rays and the converter.lg4 2.5.5.4. Positron Annihilation with Polarized electron^.^^^ The spin dependence of positron annihilation has been used in two different ways to determine the helicity of positrons. Both methods are experimentally quite difficult and require a detailed knowledge of the depolarization during the slowing-down of positrons. As a result, very few experiments have been performed and i t seems unlikely that either method will be used often in the future. 2.5.5.4.1. ANNIHILATION-IN-FLIGHT IN A POLARIZED TARGET. Assume that a longitudinally polarized positron beam strikes a target which contains electrons polarized parallel or antiparallel to the momentum of the incoming positrons. The situation is thus superficially the same as in the 191 M. Deutsch, B. Gittelman, R. W. Bauer, L. Grodzins, and A. W. Sunyar, Phye. Rev. 107, 1733 (1957). 19a F. Boehm, T. B. Novey, C. A. Barnes, and B. Stech, Phys. Rev. 108,1497 (1957). lD4J. B. Gerhart, F. H. Schmidt, H. Bichsel, and J. C. Hopkins, Phye. Rev. 114,
1095 (1959).
2.5.
POLARIZATION
OF ELECTRONS AND PHOTONS
263
Bhabha-scattering experiments (Section 2.5.5.1). However, the magnetized target is chosen to be so thick that most of the incident positrons are stopped, and instead of observing the scattered electrons, one determines the number of annihilation quanta of a given energy. The gamma rays of 0.511-Mev energy are due to annihilation-at-rest. The gamma rays of higher energy are produced by annihilation-in-flight. Annihilation-inflight is strongly spin-dependent. 1,61,120--122 In the exact forward direction, angular momentum can only be conserved if the spins are antiparallel. In other directions, the spin dependence is a function of the positron energy. At low energies, annihilation with opposed spins is predominant; a t high energies, annihilation in the triplet state is more probable. To use this spin dependence for a determination of the positron helicity, positrons from a very strong source are focused by magnetic lenses onto a magnetic annihilator foil.l g 6The gamma-ray intensity for various gammaray energies is then observed for the two directions of magnetization. From the observed intensity difference, the polarization of the positrons can be calculated. The main difficulties of this method are: (1) the counting rate is small even for very strong sources and the background is large; (2) the observed intensity difference is small (1-3%) even for completely polarized positrons; (3) the depolarization in the thick annihilator can be large.129,1g6 2.5.5.4.2. ANNIHILATION OF SLOW POSITRONS. The annihilation of slow positrons in magnetized ferromagnets can be used for the determination of the helicity of the positron^.^^^^^^^ This method depends on the fact that at zero positron energy, two-quanta annihilation can occur only in the singlet state, i.e., for antiparallel spins. In iron, about two of the d-electrons can be polarized but the s-electrons cannot. Annihilation of slow positrons occurs mainly with s- and d-electrons. In order to favor the observation of the annihilation quanta from d-electrons, one uses the fact that the d-electrons are faster than the s-electrons. The angular distribution between the two annihilation quanta will hence be wider for annihilation occurring with d-electrons than for those occurring with s-electrons. The intensity of the wings of the angular distribution, determined as a function of the direction of magnetization of the annihilator, thus gives information about annihilation with d-electrons. The effects upon revers196s.
Franltel, P. G . Hansen, 0. Nathan, and G . M. Temmer, Phys. Rev. 108, 1099
(1957). 197
S. M. Neamtan, Phys. Rev. 110, 173 (1958). S. 5. Hanna and R . S. Preston, Phys. Rev. 106, 1363 (1957); 108, 160 (1957);
109, 716 (1958). lQ*
R. S. Preston and S. 5. Hanna, Phys. Rev. 110, 1406 (1958).
264
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
ing the magnetization can be quite large (-lo%), but the accurate calculation of the polarization of the incident positrons is extremely difficult. 2.5.5.5. Positron Decay in Magnetic Fields. The formation and the decay of positronium in a magnetic field can be used to determine the polarization of p o ~ i t r o n s . ~ , ~The ~ ~ 9theory 8 underlying this method is described in references 1, 3, and 199. The unavoidable depolarization effects which occur in thermalizing positrons make this method unsuitable for very precise experiments. 2.5.6. Detection of Photon Polarization
2.5.6.1. Introduction. In principle there exist a number of possibilities to detect the circular polarization of gamma rays200: (1) Gamma rays can interact with polarized matter; (2) The helicity of gamma rays can be transferred to secondary particles, for instance, electrons, and the polarization of these particles can be measured ; (3) The Mossbauer effect can be used to study the intensity of the various components of a nuclear Zeeman pattern.
The possibilities (1) and (2) are discussed in detail in reference 200. In the present section, only those methods are described that have been successfully applied to experiments, namely Compton scattering and Mossbauer effect. A t the present time, Compton scattering from polarized ferromagnets is the best method t o determine the circular polarization of gamma rays in the energy range from about 0.2 t o a few MeV. The method suffers from the same limitations that all experiments with polarized ferromagnets do: only about two out of twenty-six electrons can be polarized and the maximum effect to be expected from any polarimeter is thus of the order of 10%. The method using the Mossbauer effect can be applied only to the small class of gamma rays that can be emitted and absorbed without recoil and that can be split into individual Zeeman components by a magnetic field. If these conditions are fulfilled, the helicity can be determined to a much higher degree of accuracy than with the Compton effect. General discussion of the methods used to determine photon helicities is contained in references 1, 2, 3, 14, and 200. 2.5.6.2. Compton Polarimeter. 2.5.6.2.1. THEORETICAL BACKGROUND. The interaction of unpolarized gamma rays with matter is treated in l@Q
L. A. Page and M. Heinberg, Phys. Rev. 106, 1220 (1957). Inslr. 8, 158 (1958).
zoo H.Schopper, Nuclear
2.5.
POLARIZATION OF ELECTRONS AND PHOTONS
265
Section 1.1.7; the Compton effect, in particular, is dealt with in Section 1.1.7.2. The differential cross section for Compton scattering is described by the famous Klein-Nishina formula, Eq. (1.1.118). For the application to the determination of the polarization of gamma rays, the form of the Klein-Nishina formula shown in Eq. (1.1.118) is unsuitable since i t applies to unpolarized gamma rays and unpolarized electrons. The scattering of a gamma ray with circular polarization P3 from a n electron with spin d is given by the following expressions : 1 ~ 2 ~ 1 4 ~ 4 6 ~ 4 9 ~ 6 1 ~ 2 0 0 - 2 0 ~ (2.5.61) 9 0 @3
=
+
+
1 cosz 6 (Ico - k)(l -(I - cos 0)(ko cos 0
=
- cos 0)
+ k)
*
d.
(2.5.62) (2.5.63)
Here, T O = e2/mc2is the classical electron radius, ko the initial and k the final photon momentum. a0 denotes the ordinary (KIein-Nishina)
FIG. 17. Compton scattering. The incoming gamma ray with momentum ko is scattered by an electron with spin d; k is the scattered gamma ray. A reversal of the spin of the electron, d + -d, corresponds to a change $ + $ T.
+
Compton cross section; @ 3 is that part of the cross section th a t depends on the helicity Pa. 0 is the scattering angle; f denotes the fraction of polarized electrons. Introducing the angle between ko and d, and 4 between the (kod) and the (kok)planes (Fig. 17), one can rewrite Eq. (2.5.63) as
+
=
- (1 - cos 0)[(ko
+ k)cos 6 cos + + k sin 0 sin + cos 41.
(2.5.63')
Equations (2.5.61) to (2.5.63') show that Compton scattering can be a tool for determining the circular polarization Pp of gamma rays. (The Zo1 *OZ
W. Franz, Ann. Phys. 33, 689 (1938). F. W.Lipps and H. A, Tolhoek, Physicu 20, 85, 395 (1954).
266
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
dependence of the Compton cross section on the linear polarization, and the corresponding application to measurements of the linear polarization, are discussed in Section 2.4.2.1.4and in reference 2.) Circular polarization measurements using the Compton effect can be performed in two different ways. One either observes the attenuation of a gamma ray in a polarized absorber as a function of the direction of magnetization of the absorber (transmission method), or one determines the intensity of the scattering from polarized matter as a function of the magnetization (scattering method), For any given problem, one must decide which method will yield the largest effect, the highest intensity, and the best signal-to-noise ratio. This decision will depend on the gamma-ray energy, the intensity of the available sources, the background radiation, and on whether the gamma rays are observed in coincidence with other radiations. In each case, two directions of magnetization exist which can be denoted with and -. The change of a counting rate N on reversal of the polarizing magnetic field is observed. The relative change 6 in the counting rate N is given by
+
(2.5.64) where N+ and N - are the counting ratee with the magnetic field in one or the other direction respectively. Which direction is defined as depends on the arrangement. The factor 2 is convention and it is sometimes omitted. In Fig. 18, 6 is plotted versus the energy of the incoming photons for four different arrangements. Each curve is calculated under optimum conditions. In actual experiments, the effect to be expected is usually smaller. The problem of production of sources of circularly polarized gamma rays deserves a remark. The fact that electrons emitted in nuclear beta decay are longitudinally polarized is intimately connected with parity nonconservation in weak interactions. Gamma rays, however, are created in electromagnetic interactions and these always conserve parity. Gamma rays hence can only be circularly polarized if they are emitted by a nuclear system which is polarized. Such a polarization can be the result of the application of low temperatures and very high fields, or of the observation of a preceding or following radiation in coincidence with the gamma ray under investigation. I n both cases, extranuclear effects can destroy part of the circular polarization of the gamma ray. (Similar effects occur in angular correlation experiments; they are briefly discussed in Section 2.4.2.1. References to detailed investigations are also given there, par-
+
2.5.
267
POLARIZATION OF ELECTRONS AND PHOTONS I
I
I
I
I
C. BACKWARD SCATTERING
15
1
b. FORWARD SCATTERING
10
8
IN% 0 . TRANSMISSION
5
d. BEARD- ROSE 0
I
I
I
1
I
I
2
3
4
5
K O IN UNITS OF
4
MC2
FIG. 18. Relative change 8 in counting rate N as a function of the incident photon energy ko (in units of mcz) for various experimental arrangements. Curve a applies to transmission with optimum length. Curve b refers to forward scattering under optimum angle. Curve cis for backward scattering, and d for the Beard-Rose method. All curves are calculated under the assumption of saturated iron and completely polarized incident photons.
ticularly in Table 11.) A strong influence of extranuclear fields on the circular polarization of gamma rays has indeed been observed.203 2.5.6.2.2. THETRANSMISSION METHOD.The simplest way to measure the circular polarization of a gamma ray consists in determining the transmission through a polarized absorber, as shown schematically in Fig. 19. In this experiment, the total Compton cross section is used. By integrating the differential cross section (Eq. 2.5.61), one obtains the total cross section for Compton effect in the form Cr
101
=
6 0 +.fP363
F. Boehm and J. Rogers, Nudear Phys. 33, 118 (1962)
(2.5.65)
268
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
where the polarization sensitive part
US
is given by
The upper sign corresponds to the photon spin parallel to the electron spin and hence antiparallel to the magnetization.
SOURCE
SCINTILLATION CRYSTAL MAGNET
,
PHOTOMULTIPLIER I
FIG.19. Measurement of the helicity of gamma rays by transmission. The magnet is alternatively magnetized parallel and antiparallel to the direction of the gamma ray. The corresponding change in counting rate is proportional to the gamma-ray helicity .
The relative change 6 in counting rate (Eq. 2.5.64) on reversing the sign of the magnetization is then given by 6 = 2 tanh(NLfZu3-P3).
(2.5.67)
N is the number of iron atoms per cm3,L is the length of the iron absorber, and Zf is the number of oriented electrons per iron atom [Zf = 2.06 at saturation; see Section 2.5.5.1.3, particularly Eqs. (2.5.50) and (2.5.51)J. The sign in Eq. (2.5.64) applies if the electron spin is parallel to the direction of the incident photons. Equation (2.5.66) and Fig. 18 show that a change in sign of 6 occurs at 0.65 MeV. Around this energy, the transmission method is obviously not usable. At higher energies, the transmission is greatest when the photon spin is parallel to the electron spin. The relative change 6 increases with increasing length of the magnetized absorber; but the counting rate decreases. The optimum length Lo of the absorber can be calculated easily,200it is given by
+
NLOT, = 2
(2.5.68)
where T~ is the total absorption coefficient of the gamma ray. The curve a in Fig. 18 has been calculated for this length LO.
2.5.
POLARIZATION OF ELECTRONS AND PHOTONS
269
A number of experiments have been performed with the transmission method. 18,180,187,204--207 The disadvantages of the transmission method are its small counting rate and the small effect (see Fig. 18). Its advantages are the basic simplicity and the fact that energy discrimination is simpler than in the scattering methods. In the latter experiments, the gamma-ray lines are degraded in energy and widened ; in the transmission arrangement, the gamma-ray counter can be set on the energy of the incident gamma ray. One additional difficulty in the transmission method arises in the determination of the effective length L, because the ends of the iron absorber are not fully magnetized. A toroidal analyzer has been used to avoid this diffi~ulty,20~.207 but the results from these experiments indicate that some unsolved problems remain. 2.5.6.2.3. FORWARD SCATTERING. The majority of all experiments has A typical arrangebeen performed with forward scattering.17~1s1~184,208-212 ment is shown in Fig. 20. The gamma rays are scattered from magnetized iron and the counting rate is determined as a function of the direction of magnetization. The entire setup is cylindrically symmetric in order to increase the counting rate. A central absorber prevents the detection of unscattered gamma rays. The effect t o be expected from such an arrangement is given by Eqs. (2.5.61) to (2.5.63):
(2.5.69)
N+ is the number of scattered photons if the electron spin is approximately parallel to the incoming gamma ray (0 _< $ < a/2) and N - is the corresponding counting rate with the electron spin approximately antiparallel to ko. As shown in Fig. 17, reversal of the electron spin, d + -d, corresponds to a change $ --f $ T . “Approximately antiparallel” therefore is represented by angles $ such th at T I $ < 3 ~ / 2 ,and the corresponding cross section as- is given by
+
93-
+3
(T
5
$
< 3~/2).
(2.5.70)
a04 S. B. Gunst and L. A. Page, Phys. Rev. 02, 970 (1953). zosA. Lundby, A . P. Patro, and J. P. Stroot, Nuovo cimento 6, 745 (1957); 7, 891 (1958). *On L. A. Page, B. G. Peterson, and T. Lindqviat, Phys. Rev. 112, 893 (1958). an’ I. Marklund and L. A. Page, Nudear Phys. 9, 88 (1958/59). *08 H. Schopper, PhiE. Mag. 181 2, 710 (1957). R. M. Steffen,Phys. Rev. 116, 980 (1959). zla H. Appel, 2. Physik 166, 580 (1959). D. Bloom, L. G. Mann, and J. A. Miskel, Phys. Rev. 136, 2021 (1962). *laH. J. Behrend and D. Budnick, 2. Physik 168, 155 (1962).
ANALYZER MAGNET
8- DETECTOR
ANTHRACENE
MAGNETlC
MAGNETIC SHIELD
Y- DETECTOR
FIQ.20. Apparatus to determine the circular polarization of gamma rays by Compton scattering from magnetized iron. The gamma rays are detected in coincidence with the preceding beta particles. (From Steffen, reference 209.)
70'
I
I
I
60'
mE
50'
40'
30'
I
I
4
2 k0
I
6
I
FIQ. 21. Optimum scattering angle Om for forward scattering as a function of the incident photon energy k~ (in units of mc*). The angle fi between the incoming photon and the electron spin is optimized according to Eq. (2.5.71). (From Sohopper, reference 200.) 270
2.5.
POLARIZATION OF ELECTRONS AND PHOTONS
27 1
The difference in sign in the definitions of 6 (Eq. 2.5.64) and 6' (Eq. 2.5.69) is due to the fact that 6 is defined for transmission and 6' for scattering.200 In order to design a n optimum instrument, one must compute @s-/@o for various scattering angles and determine its maximum for various gamma-ray energies. The optimum angle $, between the direction of the incoming photon ko and the electron spin is given byzoo (2.5.71) where 8 is the scattering angle. The optimum scattering angle em must be calculated numerically. Figure 21 shows the optimum scattering angle for t,b = $m as a function of energy of the incident photon. The corresponding change 6' in the counting rate is given by curve b in Fig. 18. To determine the sign of the gamma ray helicity, the following rule, which follows from Eqs. (2.5.69) and (2.5.63), can be useful: More photons are scattered in the forward direction if the electron spin and the photon spin are antiparallel. Right-handed photons thus are scattered more strongly in the forward direction if the magnetization in the scatterer points away from the gamma-ray source.
I n the experimental realization of the ideas outlined above, a number of factors must be taken into account: (1) The magnetic scatterer should contain as high a fraction f of polarized electrons as possible. Usually, Armco iron is chosen as material. Hyperco alloy has been used for improved efficiency.203 The calculation off from the saturation magnetization is discussed in Sections 2.5.5.1.3 and 2.5.5.1.6. (2) The gamma rays observed in the counter (see Fig. 20) should come only from the saturated part of the magnetic scatterer, but not from other material. To satisfy this condition as completely as possible, lead collimators are introduced and the magnetic core is chosen to be so thick that scattering from the magnet coils is negligible. A second possibility of avoiding scattering from the coils, and at the same time minimizing multiple scattering in the magnetic material, is the use of a large air gap.211 (3) I n the cylindrical geometry shown in Fig. 20, only very few gamma rays are scattered at the optimum angles. It is possible to shape the magnetic scatterer in such a way that most gamma rays are scattered at about the optimum angle 0,,,.200~203~213 (4) The influence of the magnetic field on the gamma ray counter must *I*
€3. Behopper and S. Galster, Nuclear Phys. 6, 125 (1958).
272
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
be eliminated. Long light pipes and magnetic shielding can reduce changes in the counting rate on switching the field to less than 0.1 %. (5) In every system there exist drifts. The influence of such drifts can be minimized if the magnetic field is changed at very short intervals of time. One such system, with a cycle of less than 10 see, is described in reference 21 1. (6) The raw data from a polarimeter must be corrected for background, for multiple and plural scattering in the magnetic scatterer,200and for deviations from an ideal geometry. Calculations of the effects of the finite size of source, scatterer, and absorber can be found in references 184, 200, 209, and 210.
‘;;lo
SCATTERER
DIRECTION OF MAGNETIZATION
COLLIMATOR
FIG.22. Principle of the method of Beard and Rose for the determination of photon helicity. The iron scatterer is magnetized alternately in the two directions indicated. Two counters are used to speed up data collection and to reduce experimental asymmetries.
2.5.6.2.4. BACKWARD SCATTERING. Figure 18 shows that, especially for small energies, backscattering yields the highest changes in relative counting rate. Nevertheless, experiments in the backscattering geometry are hard to perform, because the cross section is small and the energy of the backscattered quanta is nearly independent of the incident energy. The selection of the desired photons is hence difficult and the background is high. Details about the determination of the photon helicity using backscattering can be found in references 214 and 215. 2.5.6.2.5. THE BEARD-ROSE METHOD. The geometry shown in Fig. 22 permits the simultaneous observation of the scattered gamma rays in two direction^.^^^^^^^ Data gathering hencecan be faster than inother methods. *I4
M. Bernardini, P. Brovetto, S. DeBenedetti, and S. Ferroni, Nuooo cimento 7,
416 (1958).
R. M. Steffen, Phys. Rev. 118, 763 (1960). D. B. Beard and M. E. Rose, Phys. Rev. 108, 164 (1957). 217 E. C . Beltrametti and S. Vitale, Nuovo cimento 9, 289 (1958). z16
218
2.5.
POLARIZATION OF ELECTRONS AND PHOTONS
273
Curve d of Fig. 18 shows th at this approach is best suited to gamma-ray energies around 1 MeV. 2.5.6.3. Determination of the Helicity of Gamma Rays by Mossbauer Effect. 2.5.6.3.1. THEMOSSBAUEREFFECT.^^*-^^^ A free nucleus N * of mass M that emits a gamma ray of energy E will recoil with an energy R given by R = E2/2Mc2 (2.5.73) and the gamma ray line will, in general, show Doppler broadening. If the decaying nucleus is embedded in a solid, a fraction f of all decays will show a remarkable behavior,21* namely : (1) the emitted gamma ray line will show the natural line width r ; and ( 2 ) the gamma ray will not show any measurable energy loss due to recoil.
The natural line width
r is given by rr = h
(2.5.74)
where T is the mean life of the excited nuclear state. The fraction f of recoilless emissions is determined by
f
=
exp(- /X2)
(2.5.75)
where < x 2 > is the mean square deviation of the nucleus from its equilibrium position in the lattice, and 2nX = 2nhc/E is the gamma ray wavelength. If the solid can be described by a Debye model, and is characterized by a Debye temperature 0, then
f
=
exp
[- a 6R
1
+
2):(
L””s]) (2.5.76)
where T is the ambient temperature of the solid. In practice, only lowenergy gamma rays ( E 5 100 kev) show an appreciable fraction of recoillessly emitted gamma rays. I n order to observe these gamma rays, a material containing the nucleus N in its ground state is used as an absorber. The gamma ray beam contains a fractionf of gamma rays with the natural line width. A fraction 2 1 8 R. L. Mossbauer, Z . Physik 161, 124 (1958); Naturwissenschajten 46, 538 (1958); 2.Naturforsch. 14a, 211 (1959). 219 R. L. Mossbauer, Ann. Rev. Nuclear Sci. 12, 123, 1962. 220 H. Frauenfelder, “The Miissbauer Effect.” W. A. Benjamin, New York, 1962. 221 A. Schoen and D. M. J. Compton, eds., “The Mosshauer Effect, Proceedings of the Second Mossbauer Conference.” Wiley, New York, 1962.
274
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
f’ of these gamma rays are then absorbed without recoil, and this excess absorption can be observed. If the source is moved towards the absorber with a velocity v, the emitted gamma ray will receive an additional Doppler energy AE: AE = (v/c)E. (2.5.77) By giving the source various velocities v and hence various additional amounts of energy, the absorption curve can be used to trace out the natural line shape, or, more accurately, the overlap of the natural lines of source and absorber. If the source or the absorber is placed in a strong magnetic field of magnitude B,either artificially created or produced by extranuclear fields in solids, the gamma-ray line splits into Zeeman components. Assume the very simplest case where the excited state N* has spin 1 and the ground state N spin 0. The emitted line then splits into three Zeeman components. These components can be resolved if
g*poB > 2 r
(2.5.78)
where g* is the g factor of the state N* and p o is the nuclear magneton. If the gamma ray is observed in the direction of the magnetic field, only two lines will be seen and these two components are completely circularly polarized (longitudinal Zeeman effect). 2.5.6.3.2. THEOBSERVATION OF THE H E L I C I T Y . The ~ ~ ~observation J~~ of the circular polarization is now straightforward. The gamma-ray beam emitted in the direction of the magnetic field is observed with an absorber possessing an unsplit absorption line. The absorption spectrum will show two lines, and the intensity of these two lines is proportional to the strength of the right and left circularly polarized component respectively. Alternately, the absorber can be placed in a strong magnetic field and it can then be used to find the helicity of an unsplit gamma ray. A detailed description of the theory underlying this method and of some experiments is contained in reference 222. It is essential to point out, however, that this method can only be applied to gamma rays that show an appreciable Miissbauer effect and that satisfy the condition expressed by Eq. (2.5.78).
**sH. Frauenfelder, D. E. Nagle, R. D. Taylor, D. R. F. Coohran, and W. M. Visscher, Phys. Rev. 126, 1065 (1962).
2.6. Determination of Life-Time 2.6.1. Long life-Time* 2.6.1.1. Radioactive Decay. For a (large) number N of identical radioactive nuclei, the decay rate is proportional to the number of the existing nuclei : - -dN =
AN.
dt
(2.6.1.1)
The constant h is called the decay constant. Integration of Eq. (2.6.1.1) leads to N = Noe-At (2.6.1.2) as the number of the nuclei still existing a t the time t, if this number was Noat t = 0. N and d N / d t are to be understood as statistical mean values. The mean life is found to be 7 =
i*N t
dt
- -.
(2.6.1.3)
h*Ndt A more common figure than 7 is the half-life T , defined as the time when N = N0/2. One has T = 7 log 2 = 7 * 0.693.
If there are two ways of decay, e.g., alpha and beta decay, the partial half-life TI is given by TI =
e2 T
(2.6.1.4)
P1
+
with p l / ( p l p z ) as the branching ratio for the decay via branch 1. An analogous expression holds for n ways of decay. For a constant production rate d N / d t , the number of active nuclei N increases as (2.6.1.5) N = N,(1 - e--ht) where N , is the saturation activity and X the decay constant defined in Eq. (2.6.1.1). I n the case of decay chains, Eq. (2.6.1.2) and Eq. (2.6.1.5) must be combined, N, being replaced by N of the mother substance. Figure 1 shows, as an example, the amount of RaA, RaB, and RaC (Po218, Pb214,and Bi214)in a source which contained only RaA a t the time t = 0. We speak about radioactive equilibrium if the ratio between
-
* Section 2.6.1 is by H.
Daniel and W. Gentner. 276
276
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
the activities does not change. The equilibrium is called secular if the decrease in time is negligible. Then the number of nuclei show the same ratio as the half-lives while the activities are equal:
N1:Nz: . . . N ,
=
T1:Tz:
*
*
*
T,.
(2 .6.1.6)
I n the case of a nonsecular equilibrium, the number of nuclei as well as the activities of the daughter substances N 2 through N , are larger than the values obtained from Eq. (2.6.1.6).
t-
FIG.1. Amounts of RaA, RaB, and RaC in a source which contained RaA only at the time t = 0.
The laws of radioactive decay have frequently been checked by experiment, especially in the beginning of radioactive research. Excellent review is given by Meyer and Schweidler,’ Rutherford el a1.,2and B ~ t h e . ~ The theory of radioactive decay is treated here and by Rasetti4 and Segr&.6In particular, Bothe3 and SegrP deal with the statistical fluctuSt. Meyer and E. Schweidler, “Radioaktivitiit,” 2nd ed. Teubner, Berlin, 1927. E. Rutherford, J. Chadwick, and C. D. Ellis, “Radiations from Radioactive Substances.” Cambridge Univ. Press, London and New York, 1930. * W. B o t h , in “Handbuch der Physik” (H. Geiger and K. Scheel, eds.), 2nd ed., Vol. XXII, Part 1. Springer, Berlin, 1933. F. Rasetti, “Elements of Nuclear Physics.” Prentice-Hall, New York, 1936. E. SegrB, in “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 111, pp. 1-53. Wiley, New York, 1959. 2
2.6.
DETERMINATION OF LIFE-TIME
277
ations in radioactive decay. The decay constants X have finally been found, in every case, to be, within very narrow limits, practically unchangeable numbers characteristic for the individual activity. Small changes, however, have definitely been observed. Leininger et a1.6 measured the difference in the decay constants between Be metal and BeFz to be X(Be) - X(BeF2) = (0.84 5 0.10)10-3X(Be); Kraushaar et ~ 1 . ~(0.74 7 5 0.05)10-3; and Bouchez et U Z . , ~ (1.2 5 0.1)10-3. Similar results were found for the difference between X(Be0) and X(BeF2).6s7The radioactive isotope Be7 decays by electron capture, and the small effects on the decay constant can be explained by the change of the electron density at the nucleus. In the case of Tcggm which decays by a strongly converted isomeric transition, Bainbridge et d 9found the following differences : and
X(KTc04) - X(TczS7) = (2.7 5 0.1) 10-3X(Tc&) X(TC)- X(TczS7) = (0.31 0.12)10-3X(T~zS7).
Again the effect is caused by a change of the electron density a t the nucleus. Byers and Stumplo measured the effect of low temperature and superconductivity on the decay constant of metallic Tcggm. They obtained X(4.2'K superconductive) - X(293'K) = (6.4
k 0.4)10-4X(2930K).
The effect was greatly reduced in the absence of superconductivity: X(4.2'K
normal) - X(293"K)
=
(1.3 f 0.4)10-4X(2930K).
Porter and McMillan" treated theoretically the effect of compression on the decay rate of metallic Tcggm. The question of a cosmological variation of the decay constants cannot definitely be answered a t present. The reader is referred to the review article of Dicke.I2 Before measuring unknown half-lives, it is useful to estimate the expected order of magnitude. In any type of radioactive decay, the decay rate increases with increasing energy available. Besides, the decay rate depends on spin and parity change of the transition, and on other factors such as the more or less overlapping of the initial and final state wave functions. Instead of discussing these questions in detail, we want to give some help for rapid orientation. R. F. Leininger, E. SegrB, and C. Wiegand, Phys. Rev. 81, 280 (1951). J. J. Kraushaar, E. D. Wilson, and K. T. Bainbridge, Phys. Rev. 90, 610 (1953). 8R. Bouchez, J. Tobailem, J. Robert, R . Muxart, R. Mellet, P. Daudel, and R . Daudel, J. phys. radium 17, 363 (1956). K. T. Bainhridge, M. Goldhaber, and E. D. Wilson, Phys. Rev. 90,430 (1953). lo D. H. Byers and R. Stump, Phys. Rev. 112, 77 (1958). l 1 R. A. Porter and W. G . McMillan, Phys. Rev. 117, 795 (1960). 12 R. H. Dicke, Revs. Modern Phys. 29, 355 (1957). 7
278
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
Figure 2 shows the relationship between half-life and energy for alpha transitions between the ground states of even-even nuclei.1s For these transitions the half-lives can be calculated most accurately. Transitions between other nuclei are often strongly hindered (by up to more than a factor of thousand). Extensive information is contained in the review article of Perlman and Rasmussen.13 For beta transitions, with
FIQ.2. Plot of logarithms of partial alpha half-lives for ground state transitions versus the inverse square root of the effective total alpha-decay energy (Qea = alphaparticle energy recoil energy electron screening correction). Points used in the analysis are indicated by open circles, and points not used, by full circles. Nuclei with 126 or fewer neutrons are not shown on this plot. The last digit in the mass number of the alpha emitter is given beside each point.
+
+
the decay energy, spins, and parities known, one calculates the order of magnitude for the half-life from the expected ft-value. A method for rapid determination of ft-values is given in references 14 and 15. A rough figure for the expected gamma transition probability is obtained from the Weisskopf formula.16~” Here again, large deviations are known : enhancements by more than two orders of magnitude, and hindrance I. Perlman and J. D. Rasmussen, in “Handbuch der Physik-Encyclopedia of Physics” (S. Fliigge, ed.), Vol. XLII, pp. 109-204. Springer, Berlin, 1957. 14 S. A. Moszkowski, Phys. Rev. 88, 35 (1951). D. Strominger, J. M. Hollander, and G. T. Seaborg, Revs. Modern Phys. SO, 585 (1958). la V. F. Weisskopf, Phys. Rev. 83, 1073 (1951). 17 S. A. Moszkowski, in “Beta and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), pp. 373-395. North-Holland, Amsterdam, 1955.
2.6.
DETERMINATION O F LIFE-TIME
279
factors of up to 10’. Competing processes, especially internal conversion, speed up the total decay according to Eq. (2.6.1.4). 2.6.1.2. Survey on the Methods. There are a number of ways of measuring long half-lives. The most important ones are: (a) direct observation of the decay; (b) determination of activity and number of radioactive nuclei; and (c) determination of daughter product content, collected during a known time interval. These three methods will be treated below in some detail. The decay of the free neutron and double beta decay will be treated separately. Besides the methods (a) through (c), there are methods that are applicable in special cases, which shall not be discussed, e.g., half-life determination via the mole ratios for members of radioactive decay chains (cf. Section 2.6.1.1). Direct observation of the decay is useful with shorter half-lives and is applicable up to the order of ten years half-life. This method becomes more involved and less accurate with increasing half-life, a t least, for half-lives longer than about a month. Variations of the efficiency of detection probably are the main source of error. Zero methods are advantageous. Variations within the source, such as loss of activity or change in the self-absorption, can be avoided by careful source preparation. The activity can be followed either by counting methods or by integral methods with an ionization chamber or calorimeter. Counting methods are more suitable for shorter half-lives and less suitable for longer half-lives, as compared with integral methods. Ionization chambers or calorimeters are useful for half-lives longer than about an hour. Ionization chambers need lower activities than do the calorimeters and are therefore more common in use. Half-lives longer than about ten years are usually measured with the specific-activity method. This method is applicable down to about one year half-life. Errors arise mostly from uncertainty in the determination of the specific activity as well as from determination of the number of radioactive nuclei in the sample. In many cases dilution is necessary and gives rise to additional errors. Which of the three single processes contributes most to the over-all error depends on the individual activity. For example, determination of the number of radioactive nuclei is easy and no dilution is necessary in the case of Rb87, whereas measuring the specific activity is difficult. I n the case of C14,however, all three processes are not easy; determination of the specific activity is most difficult. For determining the specific activity 4r counters are preferred, particularly in the case of beta-active nuclei. The content of radioactive nuclei in the source is frequently measured with the help of a mass spectrometer. The dilution process is done mostly in the liquid phase. I n a number of cases, the gaseous phase is chosen.
2.
280
DETERMINATION OF FUNDAMENTAL QUANTITIES
For measuring very long half-lives the collection method (c) is preferred. More important is the inverse method: with the half-life known, one determines the daughter product and hence the age of a sample. The suitable age determination method is often uniquely determined by the sample itself. I n many cases, however, different methods are applicable, thus providing a check or indicating different types of ages. 2.6.1.3. Direct Observation of the Decay. I n this method the decay rate -dN/dt is measured as a function of the time t. According to Eqs.
I
-
\ \
I
I
I
I
I
I
\
I
I 1 \ 1
I
I
I
I
I
t (arbitrary units)
1
I
I
-
FIG.3. Counting rate from a source containing two activities with half-lives T I and Tz, respectively. A constant counter background is assumed.
(2.6.1.1) and (2.6.1.2) the logarithm of the decay rate, when plotted against the time, is a straight line if there is only one activity present. Figure 3 shows the decay of a source with two different activities, with a background assumed. I n practice it is advantageous to follow the decay as long as possible, e.g., for detecting radioactive contaminations. The classical instrument for measuring decay curves is the ionization chamber.* Here the current which arises between two electrodes as a consequence of the ionization and a voltage difference is measured directly. All the charge-transporting ions in the gas between the electrodes * See also Vol. 2, Chapter 4.1.
2.6.
DETERMINATION OF LIFE-TIME
28 1
are directly produced; the chamber is operated in such a way that ion multipIication does not take place. Gas pressure and voltage are such that one works in the region of saturation (no recombination). For more accurate measurements the differential ionization chamber, introduced by Rutherford,'* is preferred. This instrument consists of two single chambers, as identical as possible, operated with equal voltages of different sign. One chamber is irradiated by an activity of known half-life and, the other is irradiated by the activity as a rule, long half-life (e.g., RazZn); with unknown half-life (for measuring small differences, as in the Be7 and Tcggm experiments discussed in Section 2.6.1.1, one takes the same isotope
BRASS
SOURCE HOLDER
POLYSTYRENE LUCITE
4
B O O CERAMIC OR CRUSHED Bc METAL COVEREP WITH WAX
FIG.4. Schematic diagram of balanced ionization chambers.
for both chambers). The source strengths are chosen in such a way that a t t = 0 both chambers show about the same current, or the chamber with the shorter-lived activity a little more. The ionization current difference is taken as a function of time. Figure 4 shows schematically the differential ionization chamber of Segr& and Wiegand.l9 Here the sources are inside the chambers. The instrument is operated with argon having a small excess pressure (1356 mm Hg). The voltages are provided by batteries. The galvanometer, measuring the difference current after amplification by an electrometer circuit, is operated by the rate-of-drift method. Bainbridge et al.9 describe a high-pressure differential ionization chamber welded of steel. The sources are also inside the chambers, if possible at a position where the ionization current is maximum so that small source displacements have only a negligible effect. TobailemZ0gives a description of a differential ionization chamber filled with air at atmospheric pressure. The sources 18 E. Rutherford, Sitzber. Akad. Wiss. Wien, Math.-natunu. Kl., Abt. IIa 120, 303 (1911). 10 E.Segrh and C. E. Wiegand, Plays. Rev. 76, 39 (1949). *o J. Tabailem, Ann. phys. (12110, 783 (1955).
282
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
are outside the chambers. Robertzl reports a differential ionization chamber which can be operated a t 4 atm and 1600 volts per chamber. In all cases the chambers are connected by a tube for pressure balance, The chambers are shielded against each other with lead (5-10 cm Pb). The induced currents are very smalI and therefore need special means for measuring. Robert21reports for each chamber with the source a t the center a current of 7.5 x l O - l 4 amp per microcurie Ra including daughter products, with a filter of 2 mm lead. Practically, one measures the charge of a well insulating condenser (capacitance of the order of 5000 ppf) coupled,-e.g., to a vacuum tube electrometer. Care must be taken to insure a high input impedance of the system (1012-1014ohms) and a low zero point drift of the galvanometer. Tobailem20describes a setup which approximately compensates for the discharge of the heater battery for the electrometer tube. Mme. Curiez2describes a method which makes use of a piezoelectric crystal and a balance. In order to compensate for differences between the two chambers the sources may be exchanged: first, source 1 is measured in chamber 1 and source 2 in chamber 2; then source 1 in chamber 2 and source 2 in chamber 1 ; then again source 1 in chamber 1 and source 2 in chamber 2, etc. Exact source positioning is important with this method. This may lead to difficulties in the case of setups with the sources outside the chambers.2l If the difference current vanishes at time t = 0, the current i as a function of 2 is given by
i = IW[l- exp(-AX,t)]
(2.6.1.7)
I , = current a t t = 00, A, = decay constant of unknown activity. TobailemzOalso treats the following subjects : decay of standard not negligible; different background in the two chambers; effect of a source on the other chamber; and a method for defining t = 0. The general operation of ionization chambers is treated by S t a ~ b . ~ ~ The ionization chamber makes use of the ionization produced by the radioactive radiation. Instead, use can also be made of the calorimetric effect for determining d N / d t , particularly in the case of strong sources. The produced heat may either be measured directly or be compensated for by the Joule Thompson or the PeItier effect. There are objections against the use of the Peltier effect, because of the strong ionization present.a1 Differential calorimeters have been proved successful. Accord-
-
*‘J. Robert, Ann. phye. [13]4,89 (1959). 22 M.Curie, “RadioactivitB.” Hermann, Paris, 1935. H. H. Staub, in “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. I. Wiley, New York, 1953.
2.6.
DETERMINATION
OF LIFE-TIME
283
ing to Angstromz4 and von Schweidler et a1.,2Kone takes two single calorimeters as identical as possible, one of them heated by the radioactive source under investigation and the other heated by an electric current. Comparison is made by a thermocouple. The current is adjusted so as to give vanishing temperature difference. Measurements are best started after approximate temperature equilibrium has been reached. Robertz1describes a microcalorimeter he has recently used for precise half-life determinations (Fig. 5 ) . The radioactive source is inside a small calorimeter made of gold or lead, which hangs on silk strings within
FIG.5. Microcalorimeter (schematically). S source, C calorimeter, 5°Cthermocouple, E red copper vessel, Ch chimney, B water bath.
a copper vessel; good thermal insulation is necessary for a high sensitivity. The copper vessel is surrounded by a large water bath. A thermocouple compares the temperatures of calorimeter and copper vessel. The water temperature is measured by a precision mercury thermometer divided into hundredths or thousandths of a degree. The water temperature is kept at the calorimeter temperature while measuring, by adding warm water. The water temperature is taken as a function of time. The ionization chamber and the calorimeter are preferred because of their great stability. However, for short half-lives (a few minutes or less), they are too slow. Here (and up to several years half-life) counting methods are suitable. Moreover, counters usually are more handy. Proportional counters are more stable than Geiger counters. I n particular, age effects are much smaller. Another disadvantage of Geiger counters is the low counting rates permissible. Therefore, statistical errors are large K. Angstrom, Physik. 2. 6, 685 (1905). E. von Schweidler and V. F. Hess, Sitzber. Akad. Wiss. Wien, Math.-naturw. KZ.. Abt. I I a 117, 879 (1908); St. Meyer and V. F. Hess, ibid. 121, 603 (1912). 24
26
284
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
in the case of short half-lives. Hence Geiger counters will be useful only for survey measurements, e.g., for identifying the activity. Scintillation counters withstand high counting rates but are usually not stable enough for precise work. A severe source of error is the dependence of the photomultiplier gain on the counting rate.26 This effect makes a high discrimination level unrecommendable. Instead, the use of a suitable filter is indicated for eliminating soft unwanted radiation. I n order to reduce gain variations a stabilizer may be ~ s e d . ~ ’ , ~ ~ A combination of magnetic spectrometer and scintillation counter has proved to be very useful (see, e.g., reference 29). The magnetic spectrometer is set at the maximum of the intensity distribution of a beta spectrum while the scintillation counter threshold is set about halfway between the line and zero. I n this way even large variations of the spectrometer setting, the photomultiplier or amplifier gain, or the discrimination threshold do not seriously change the counting rate. Of course, the resolving power of the magnetic spectrometer may also be used for discriminating against unwanted radiation. Use of a good mechanical clock or a quartz timer is recommended for all counting measurements. The power line frequency is not always constant enough to serve as a time standard. I n the case of very short half-lives, of the order of a second, the time is fixed by a quartz timer which automatically switches a scaler plus printer or two scalers with printers alternatively. Another method consists in operating two scalers with lighting tubes, one of them being driven by a quartz clock and the other by the radiation counter. The scalers which are not switched are photographed a t about equal time intervals. Another method uses the constant sweep speed of a cathode ray oscilloscope as time axis, the counts being indicated by vertical deflection pulses. The scope screen is again photographed. Other electrical means for measuring decay curves have been described.a0 After registration of a pulse, the counting setup, in most arrangements, will not be ready to register any pulse coming within a fixed time interval, the dead time TO.When T ohas elapsed the setup will be ready for the next pulse. Under these conditions the measured counting rate n has to be corrected to give the “true” counting rate no according to no = n / ( l
- nTo).
(2.6.1.8)
H. Jung, Ph. Panussi, and J. Jlnecke, Nuclear Znstr. 9, 121 (1960). B. Astrom, Arkiv Fyaik 12, 237 (1957). ** H. de Waard, Nucleonics 13, No. 7, 36 (1955). z9 H. Daniel, Z . Naturforsch. laa, 363 (1957); H. Daniel and U. Schmidt-Rohr, Nuclear Phys. 7, 516 (1958). 80 A. J. Bureau and C. L. Hammer, Phya. Rev. 106, 1006 (1957).
2.6.
DETERMINATION OF
LIFE-TIME
285
The conditions under which Eq. (2.6.1.8) holds are not always fulfilled. A counting setup may have several dead times, e.g., the dead time of the first scaling circuit and the dead time of the mechanical counter. For more . ~ ~ complicated cases, the reader is referred to the paper of J o ~ t There are several experimental methods for determining the dead time correction. A very direct method consists in taking the decay of a strong pure source. One can also work with several sources of known strength constant in time. However, it is not advisable to vary the counting rate by varying the distance between source and detector and to depend on the validity of the 1/74 law. This law does not hold accurately enough, because of absorption and backscattering from the walls. The dead time can also be measured electronically with a double pulse generator. It is good t o apply two independent methods in parallel. The counter background is measured before and after the measurement of the sample. The decay curve itself should be followed as long as possible, e.g., to notice radioactive contaminations. Weak long-lived contaminations can be corrected for fairly easily if one has measured long enough. If the decay curve falls off too rapidly a t the beginning, this may indicate a short-lived activity or may be caused by a counting-ratedependent gain. When higher precision is wanted a least square fit of the decay curve must be made. Standard least square methods are sufficient. P e i e r l ~ ~ ~ treats counting statistics including background problems and data evaluation. A graphical method, in the course of which the remaining number of pulses is plotted, is given by KuiEer et al.33 I n the methods described above, the counting rate is taken as a function of the time. In some cases it is practical t o measure directly the number of remaining nuclei N as a function of time. For this purpose, the mole ratio between the radioactive isotope and an isotope of the same element is determined with a mass spectrometer. In this way Wiles et aL3* measured the half-life of CsI37 with C S ~ ~ ' / C Sand ' ~ ~T, h ~ d the e ~half-life ~ of Kr*5 with Krs5/Kr*6. Another method, which makes use of the alpha radioactivity of Rn222, the gamma radioactivity of some daughter products, and of the very different ionizing effect of the two radiations, has been applied by Bothe36 in order to measure the RnZz2half-life with high precision. 31
R. Jost, Hetv. Phys. Acta 20, 173 (1947).
52
R. Peierls, PTOC.Roy. SOC.A149, 467(1935).
33
34
I. KuSEer, M. V. Mihailoid, and E. C. Park, Phil. Mag. [ S ] 2,998 (1957). D. R. Wiles, B. Smith, R. Horseley, and H. G . Thode, Can. J . Phys. 31, 419
(1953). 3 6 R .K. Wanless and H. G. Thode, Can. J . Phys. 31, 517 (1953). 36 W. Bothe, 2. Phyaik 16, 266 (1923).
286
2.
DETERMINATION OF FUNDAMENTAL QUANTITIEE.
2.6.1.4. Specific Activity Method. The specific activity method is favored in cases where the half-life is too long to measure directly the decrease of activity. Instead, one measures the specific activity and the number of radioactive nuclei per weight unit. If
A =
decay rate weight unit
=
specific activity
and @=
number of active nuclei weight unit
then it follows from Eq. (2.6.1.1) X = A/g.
(2.6.1.9)
For the direct decay rate measurement, counting methods are applied. However, the ionization chamber and calorimeter are also used. With these methods, the energies of the radiations must be known whereas with counting methods approximate knowledge of the decay scheme is often sufficient. The calorimetric method has the advantage of being free of self-absorption effects and the disadvantage of needing strong sources. The heat production rate .J is given by
J
= 5.93 X
10-TB watt
with C = source strength in curies and 2 = mean energy per disintegration in MeV. For counting alpha rays, counters with a defined small solid angle are suitable because of the small directional scattering of alpha rays. Figure 6 shows a proportional counter due to Hurst and Source holder and collimating baffles are a rigid unit. In this way constancy of the geometry factor, i.e., the reciprocal of the fractional solid angle, is guaranteed. This geometry factor amounts to 716.8 & 0.8, including a ball correction of 2.5%. The source is mounted on a platinum disk in the middle of the source-holding plate. Small dislocations on the plate are not critical. The counter is usually operated as a proportional counter filled with 60 mm Hg of methane. The degree of evacuation in the region between source and counter, necessary to ensure that all alpha particles emitted into the fractional solid angle reach the counting volume, can be calculated. It is advantageous to make all alpha particles reaching the counter travel through the entire counter, for in this way pulses of about the same height result. s7R.Hurst and G. R. Hall, Analyst 77, 790 (1962).
2.6.
DETERMINATION
OF LIFE-TIME
287
To Amplifier
t
FIG.6. Alpha proportional counter with defined small solid angle.
Counters with a small solid angle are not suitable for low specific a large proportional activities. Macfarlane and K ~ h m a ndescribe ~~ counter 011 the walls of which (1200 cmz) the active layer is deposited (27r geometry). The pulses are registered with a multichannel analyzer. With shielding and anticoincidence counters, the background between 1 and 3 Mev alpha energy amounts to only 9 pulses per hour. Four n counters are also suitable for alpha rays. I n a n y case, the sources must be thin enough to avoid self-absorption losses. The source backing, however, can be made thick. With thick backing a correction must be 38
R. D. Macfarlane and T. P. Kohman, Phys. Rev. 121, 1758 (1961).
288
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
applied which, in the case of 2a geometry, has been c a l ~ u l a t e d The .~~ detection probability is ( F = without correction) :
+
F ( B ) = +[l
+ 0.201@(B)].
@ ( B )is a tabulated function depending on material and range.39 With Pt as backing and U as source, 0.2013(B) = 0.016. I n the case of nonnegligible source thickness, the self-absorption must be corrected This correction depends also on the discrimination bias. For beta rays counters with defined small solid angle are also used. The method is recommended by its simplicity. The accuracy, however, is limited by a number of necessary corrections which are not very well known: Within the source the beta particles lose energy, are scattered, and are absorbed. The scattering leads to increased intensities in directions not too far away from the normal direction.40 Backscattering also increases the intensity.41The phenomenon of backscattering, which because of its complexity is hard to treat theoretically, has been subject to many experiments. The results are not in good agreement, especially with respect to the difference between electrons and positrons. The various measurements are not always comparable, however, because the results depend on the geometry chosen. Qualitatively, it can be said that the backscattering strongly increases with increasing atomic number for backings of equal surface density. 41-44 Therefore 8 thin film made from a low-atomic-number material is taken as source support. This film should be conductive in order to avoid charging the source, (e.g., Al, formvar, or collodion with an evaporated Au layer on the back). The most accurate absolute measurements of beta activities are performed in 4?r geometry, either with a gaseous source where the active gas is added to the counting gas or is the counting gas itself, or with a thin solid source. For gaseous sources, cf. Section 2.6.1.8. Figure 7 shows a 4n counter due to Pate and Yaffe.4KThe counter consists of two hemispherical brass cathodes 7 cm in diameter and two ring-shaped anodes of 1 mil tungsten wire. The source is supported by a plastic film of 5-10 pg/cm2, evaporated with 2 pg/cm2 gold. Methane of 39 B. B. Rossi and H. H. Staub, “Ionization Chambers and Counters.” McGrawHill, New York, 1949. 40 D. Fehrentz and H. Daniel, Nuclear fnstr. 10, 185 (1961). 4 1 W. Paul and H. Steinwedel, in “Beta and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), pp. 1-23. North-Holland, Amsterdam] 1955. 42 W. Bothe, 2. Naturjorsch. 4a, 542 (1949). 43 G . Marguin, Ann. phys. 1131 2, 318 (1957). 4 4 H. H. Seliger, Phys. Rev. 78, 491 (1950); 88, 408 (1952). 46E. D. Pate and L. Yaffe, Can. J . Chem. 33, 610 (1955).
2.6.
DETERMINATION
289
OF LIFE-TIME
atmospheric pressure is taken as counting gas; it is dried by silica gel. The counter is operated in the proportional region. The slope of the ' per 100 volts. The detection probability is unity plateau is less than 0.1% within 0.1 %. The finite thickness of the supporting film makes a correction necessary. Pate and Yaffe46discuss the various methods according to their own 0040' COPPER ANODE LEAD
TEFLON INSULATOR
,/
GA> n n
INLET (2001' )I* TUNGSTEN
AND MOUNT
GAS OUTLET
FIG.7. 4~ counting chamber.
experiments. (a) With the sandwich method as described b y Hawkings et a1.,47 the source is first measured in 47r geometry, then on the front side covered with a film identical to the backing and again measured in 4n geometry. The observed reduction in the counting rate is taken as correction for the first measurement. The result should be independent - ~ ~ a correction of the film thickness but is noL48 (b) Mann and S e l i g e ~give formula the result of which depends on two not well known quantities, the fractional backscattering factor and the fractional absorption factor. B. D. Pate and L. Yaffe, Can. J. Chem. 33, 929 (1955). R. C. Hawkings, W. F. Merrit, and J. H. Craven, i n Proceedings of Symposium on Maintenance of Standards, Natl. Phys. Lab., May, 1961. (Puhl. by H. M. Stationary Office, London, 1952.) 48 1).B. Smith, Brit. Atomic Energy Rept. A.E.R.E. I/R 1210 (1953). 4B W. B. Xlann and H. H. Seliger, J. Research Natl. BUT.Standards 60, 197 (1953). 46
47
290
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
(c) With the absorption curve the counting rate is taken as a function of the backing thickness for the same source. This curve, which is not a straight line, is extrapolated to zero. The error of the disintegration rate is of the order of 0.2% even for beta emitters of low maximum energy.42The preparation of thin films is described in references 50-52. Particular difficulties often arise from self-absorption within the source, sometimes even if the specific activity of the sample is sufficient to prepare a source of a low average thickness. If possible, sources should be prepared by evaporation in vacuo because this method gives the most uniform sources. Electrodeposition frequently gives good results, too. I n some cases drying by evaporation of colloidal suspensions is applicable. Evaporation of salt solutions, on the other hand, results in sources of a crystalline structure with an effective thickness generally much larger than the calculated average. I n order to keep the crystals small, one may try special drying methods such as freeze drying or the application of insulin.63Source preparation is treated in the handbook of Siegbahn.K4 The source quality is not easy to check. Autoradiographies help in detecting inhomogeneities. Inspection with a magnifying glass or a microscope is advisable in any case. The average source thickness can be determined with a balance or with an alpha gauge. Estimating the source thickness from activity, area, and specific activity of the material frequently leads to values too small. Pate and Yaffe66compare various methods for source preparation. In order to correct for self-absorption in homogeneous sources, the self-absorption curve is taken. This procedure is tedious because each point needs a source of its own, the preparation method being always the same. The self-absorption curve is fitted by the theoretical curve of Gora and Hickey6sor extrapolated directly to zero. The error is of the order of 1%. Bayhurst and Prestwood67give a method for transforming, within an error of 3%, results obtained with thick sources into values for negligible source thickness. Heintze and Fischbecks8 describe a setup for measuring absolute beta activities in 2 r geometry for sources of saturation B. D. Pate and L. Yaffe, Can. J. Chem. 33, 15 (1955). E. Huster and W. Rausch, Nuclear Instr. 3, 213 (1958). 61 H. Mahl, in “Technische Kunstgriffe bei physikalischen Untersuchungen,” 12th ed. (H. Ebert, ed.), pp. 115-122. Vieweg, Braunschweig, Germany, 1959. 68 L. M. Langer, Rev. Sci. Instr. 20, 216 (1949). 6 4 H. Slatis, in “Beta and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), pp. 259-272. North-Holland, Amsterdam, 1955. 66 B. D. Pate and L. Yaffe, Can. J. Chem. 34, 265 (1956). 66 E. K. Gora and F. C. Hickey, Anal. Chem. 26, 1158 (1954). 61 B. P. Bsyhurst and R. J. Prestwood, Nucleonics 17, No. 3, 82 (1959). SS.1. Heintze and H. Fischbeck, 2. Physik 147, 277 (1957). *L
2.6.
DETERMINATION OF LIFE-TIME
29 1
thickness which has an error of 5%. In the two cases the decisive step for eliminating the effect of the spectral shape is the use of the mean beta energy (instead of the maximum energy). Vincentb9gives a description of setup (spherical 4s proportional counter) and method (variation of gas pressure) for measuring absolutely electron capture activities. Instead of proportional or Geiger counters, use can also be made of scintillation counters for 47r counting. Nevertheless, difficulties arise from the higher background. The scintillator may be liquid or solid.60-62 If the activity is in solution or incorporated into the crystal, no difficulties with source thickness or backscattering are encountered. If there is a second electron in cascade a correction must be applied if the solid angle is less than 47,or if the electron emission is delayed, as in the case of CsLS7. In some cases specific activities can be determined with the help of coincidence methods. Consider a nucleus emitting one beta particle and one gamma quantum per disintegration only. The beta counter shall accept beta rays only, and the gamma counter gamma rays only. The efficiency of the coincidence circuit shall be unity. Then one has, as can easily be derived, for the source strength - d N / d t (disintegrations per unit time) - d N / d t = npn,/n,, (2.6.1.10) independent of the beta and gamma detection probabilities; np, ny, and neodenote the beta, gamma, and coincidence counting rates, respectively. Of course, the method is applicable also in the case of alpha-gamma coincidences and gamma-gamma coincidences, and can be extended to more complicated decay schemes (see reference 63). In practice there are many sources of error. The beta counter usually also accepts gamma radiation. The method frequently in use to measure the gamma sensitivity of the beta counter, by putting an absorber in front of the counter, leads in most cases to values which are too high, because the secondary electrons from the absorber are also counted. Coincidence circuits with a short resolving time often have efficiencies less than unity. Gamma quanta may be scattered from one counter into the other. To reduce this effect the counters are put a t right angles to the source. In this way the effects of anisotropic directional correlations are also eliminated. Because of the finite resolving time, there are also accidental coinciD. H. Vincent, Nukleonik 1, 332 (1959). C. F. G. Delaney and I. R. McAulay, Sci. Proc. Roy. Dublin Sac. A l , 1 (1959). 6 1 K. F. Flynn and L. E. Glendenin, Phys. Rev. 116, 744 (1959). 62 G. M. Lewis, Phil. Mag. [7] 43, 1070 (1952). G . Wolf, Nukleonik 2, 255 (1961).
69
292
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
dences. I n a twofold coincidence circuit the rate of accidental coincidences is given by n, = 2 m p , where 27 = (double) resolving time; r is determined by adding a delay into one of the pulse channels, by irradiating the two counters with independent sources, or with sources like which do not give true coincidences. I n the case of a modern fast-slow coincidence circuit, determining the accidental coincidences is not so simple. Mayer-Kuckuk and N i e r h a d 4 describe a method which consists in measuring all possible twofold coincidences (with elimination of the third coincidence requirement). For extremely long-lived activities no direct counting methods are applied. The photographic emulsion is a detector which allows accumulation of data for a long time. I n this way Riezler and co-workersR5investigated the alpha decay of a number of nuclei. I n many cases it is not possible to determine directly the number of radioactive nuclei in a source. Sometimes the composition of the source material must be found out first. If there is enough material available the number g of radioactive nuclei per weight unit can be determined with a balance.* One can also add a known amount of inactive material to the unknown amount of active material and, in this way, prepare a standard solution. The mole ratio between active and inactive nuclei can be determined mass spectroscopically. An aliquot is taken from this standard solution and its decay rate measured. When preparing aliquots, care must be taken that the activity is really shared according to the volume ratio. I n carrier-free solutions the activity tends to stay on the walls. It is therefore advantageous to add as much carrier as is compatible with the activity measurement. The minimum amount corresponds to a layer a few atoms thick on all surfaces which come in touch with the activity during the whole procedure. All surfaces should be rinsed before use. The conditions, e.g., the pH value, should be chosen in such a way that the radioactive ions really stay in solution. As an example of a mass spectrometric analysis (CI3*source produced in a reactor), the work of Boyd et ~ 1 . should 6 ~ be cited, and as an example of a decay rate determination including source preparation, etc., the work of Bartholomew et U Z . , ~ ’ performed with the same source.
* Sometimes this number is too small for direct weighing. T. Mayer-Kuckuk and R. Nierhaus, Nuclear Instr. 8,76 (1960). W. Porschen and W. Riezler, 2. Naturforsch. l l a , 143 (1956); W. Riezler and G. Kauw, ibid. laa, 665 (1957); 13a, 904 (1958); 14a, 196 (1959). 66 A. W. Boyd, F. Brown, and M. Lounsbury, Can. J. Phya. 33,35 (1955). 67R. M. Bartholoqen;, A. W. Boyd, F. Brown, R. C. Hawkings, M. Lounsbury, and W. F. Mqrrit, Cdn. J. Phys. SS,43 (1955). 64
6s
I
2.6.
DETERMINATION OF LIFE-TIME
293
2.6.1.5. Decay of the Free Neutron. The beta decay of the free neutron was presumed by Chadwick and Goldhaber,'j8 and established by Snell and Miller'j9 and by Robson.'O The free neutron is the simplest beta-unstable nucleus. Quantities usually not calculable can reliably be calculated in this case. One of the quantities most important for beta decay theory is the half-life of the free neutron. However, special experimental arrangements are necessary, because of the impossibility of keeping neutrons a t a given place. The high background resulting from neutron capture disturbs the measurements. All neutron half-life measurements done u p to now are of the specificactivity type with direct observation of the decay. There is a proposal, however, to collect the decay product hydrogen in a n evacuated vessel.71 Figure 8 shows the setup of Sosnovsky et al.,72which has led to the most accurate value T = 11.7 k 0.3 min. A well collimated neutron beam from a reactor runs from left to right through the vacuum chamber 1 (pressure 1-2 X 10-6 mm Hg) in which there is a high voltage electrode 6 (20 kv). Neutrons of the beam decaying off the baffle 2 are registered if the recoil protons are emitted in the direction of the proton counter 5 . Before reaching the counter the protons fall through the potential of 20 kv and are simultaneously focused. A proportional counter with thin window serves as a proton detector (collodion 0.07 p , grid supported). The decay rate of the neutrons is measured with a constant-power reactor. I n practice, difference measurements (reactor on-reactor off; accelerating voltage on-accelerating voltage off) are necessary for eliminating the background. The transmission of the system and the source volume were carefully calculated with regard to the fact that the neutron density is not constant over the beam cross section. As the recoil protons enter the accelerating region with a small divergence only, the method is almost independent of the proton energy spectrum which depends strongly on the beta decay interaction and is not known well enough. Besides the decay rate, the number of neutrons in the source volume is needed. This quantity is measured by activating N a or Au targets. R ~ b s o n ?puts ~ two magnetic lens spectrometers back to back and leads a neutron beam from a reactor through the spectrometers perpenJ. Chadwick and M. Goldhaber, Proc. Roy. SOC.A161, 479 (1935). A. H. Snell and L. C. Miller, Phys. Rev. 74, 1217 (1948). J. M. Robson, Phys. Rev. 78, 311 (1950). 71 B. T. Feld, in "Experimental Nuclear Physics" (E. SegrB, ed.), Vol. 11, p. 218. Wiley, New York, 1953. 7 1 A. N . Sosnovsky, P. E. Spivak, Yu. A. Prokofiev, I. E. Kutikov, and Yu. P. Dohrinin, Nuclear Phys. 10, 395 (1959); Zhur. Eksptl. i Teoret. Fiz. 36, 1012 (1959); 36, 1059 (1958). 7 a J. M. Robson, Phys. Rev. 88, 349 (1951). 88 69
FIG.8. Diagram of the apparatus for measuring the neutron half-life. KEY:(1) body of apparatus; (2) diaphragm; (3) diaphragm with grid; (4) spherical grid; (5) proportional counter; (6) high voltage electrode; (7) flange of dzusion pump; (8) monitors; (9) reactor shield.
2.6.
DETERMINATION OF LIFE-TIME
295
dicular to their axis. One of the spectrometers detects the electrostatically accelerated recoil protons, with an ion-sensitive multiplier tube as detector; the other spectrometer detects the electrons. The two spectrometers are operated in coincidence; the finite time of flight of the protons is compensated for by a delay line. Snell et ~ 1 . ’also ~ take coincidences between electrons and recoil protons, but without magnetic spectrometers. The electrons are detected by two proportional counters in coincidence, and the protons by an ion-sensitive multiplier tube. D’Ange10’~ works with a diffusion chamber for detecting the electrons. The recoil protons cannot be identified. In order to reduce the background] the neutron beam is cleaned of gamma radiation by reflexion on a nickel mirror, and the chamber is operated with an oxygen-helium mixture. 2.6.1.6. Double Beta Decay. Double beta decay is a process of second order in the beta interaction. As this interaction is weak, extremely long half-lives are expected. Reliable experimental results for the occurrence of double beta decay do not exist a t present. I n double beta decay the nuclear charge is changed by two units. Instead of positron emission, electron capture is possible. Accordingly, four cases are to be distinguished if one asks for the role of the charged particles only: Emission of two electrons] emission of two posit,rons, emission of a positron and capture of an electron, and capture of two electrons. Each of the four processes may occur without neutrino emission or with emission of two neutrinos (in the case of double 8- decay more precisely : antineutrinos). I n the neutrinoless case, the socalled Majorana case, the energy sum of the two emitted electrons is always constant and equal to the Q-value, whereas in the case of neutrino emission, the socalled Dirac case, the energy sum is a continuous distribution between zero and the Q-value. This difference has important consequences for the experimental technique. The expected half-lives are much shorter in the Majorana case than in the Dirac case (of the order of 10l6 years and of the order of loz3years, respectively). For the connection with the regular beta decay, and generally for more information, the reader is referred to the excellent article by Primakoff and R ~ s e n . ’ ~ Experiments for detecting double beta decay can essentially be done in three different ways: (a) by counting the two electrons in coincidence; (b) by observing the two electrons with a cloud chamber or with a photographic emulsion; and (c) by determining the amount of daughter substance grown up in a mother substance of known age. Method (a) is the A. H. Snell, F. Pleasonton, and R. V. McCord, Phys. Rev. 78, 310 (1950). N. D’Angelo, Phys. Rev. 114, 285 (1959). 76 H. Primakoff and S. P. Rosen, Repts. Progr. in Phys. 17, 121 (1969).
296
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
most direct. In principle, method (a) and method (b) give information concerning not only the half-life but also the energy spectrum. A counting experiment of the type (a) was performed by Dobrokhotov et ~ 1 on. Ca48. ~ The ~ source consisted of 423 mg Ca4*,enriched to 76%; the inactive sample for background determination consisted of enriched Ca44. The two samples were alternatively put between two scintillation counters with a diameter of 50 mm. The solid angle subtended by the source at each counter was nearly 2 ~ A. 70 liter liquid scintillation counter viewed by 21 photomultiplier tubes served as an anticoincidence counter. Coincident pulses were displayed on a scope screen and registered with a camera. The whole arrangement was located in a lead shield at a depth of 65 meters water equivalent. Two different runs were made in different seasons, of 430 hours and of 300 hours, respectively. The over-all result was a lower limit of 7 X 10l8yearsfor the double beta decay half-life of Ca48.Other counting experiments were performed by Cowan et U Z . , ~ ~ A w s c h a l ~ mMcCarthy,so ,~~ Detoeuf and Moch,81Fulbright,82Pearce and D a r b ~Kalkstein ,~~ and LibbylE4and Fireman.86Berthelot et aLE6searched for double K capture. As an example of an experiment of the type (b) we sketch the work of Winter.87 Winter used a cloud chamber of 24 cm in diameter. The foils under investigation were stretched across the chamber. Stereoscopic pictures were taken at a random expansion rate of 2/min, in a magnetic field of 790 gauss. Pictures with two electrons, according to a first rough inspection, coming from the same place were stereoscopically projected back for further inspection. The last criterion was the energy balance. The following results were obtained: Cdlle, Tb-8- > 1 X 1017 years; > 6 X 10l6 years; Mo'OO, To-8- > 3 X 1017years; and MoQ2, Cd1O6,TB+B+ TB+B+> 4 X 10l8years. The values for Tp+p+are partial half-lives because of competing electron capture. Other experiments of this type were 77
E. I. Dobrokhotov, V. R. Laearenko, and S. Yu. Lukyanov, Zhur. Eksptl. i Teoret.
Fiz. 36, 76 (1959); English translation. Soviet Phys. J E T P 36, 54 (1959). 78
C. L. Cowan, Jr., F. B. Harrison, L. M. Langer, and F. Reines, Nuovo cimento
[lo] 3, 649 (1956).
M. Awschaloni, Phys. Rev. 101, 1041 (1956). A. McCarthy, Phys. Rev. 87, 1234 (1955); DO, 853 (1953). 8 1 J.-F. Detoeuf and R. Moch, J. phys. radium 16, 897 (1955); Compt. rend. acad. sci. 941,393 (1955). 8* H. W. Fulbright, Physica 18, 1026 (1955). 8 3 R. M. Pearce and E. K. Darby, Phys. Rev. 86, 1049 (1952). S4 M. I. Kalkstein and W. F. Libby, Phys. Rev. 86, 368 (1952). SsE. L. Fireman, Phys. Rev. 76, 323 (1949). 8 6 A. Berthelot, R. Chaminade, C. Levi, and L. Papineau, Compt. rend. acad. sci. 236, 1769 (1953). 87R.G. Winter, Phys. Rev. 88, 88 (1955). 79
805.
2.6.
DETERMINATION
OF LIFE-TIME
297
performed by Fremlin and Walters,*8 Fireman and S c h ~ a r z e r ,and ~~ Lawson. Inghram and R e y n o l d ~performed ~ ~ ~ ~ ~an experiment of the type (c). They investigated mineral samples of BizTe3which, according to geologic information, were 1500 k 500 million years old, and searched for Xe'3n from double beta decay of Te13n.The whole Xe content of the samples and the mass spectroscopical composition of the Xe were determined. In the second i n v e ~ t i g a t i o n 2.4 , ~ ~X 61113 S.T.P. argon plus 2.,, X lo-' cm3 S.T.P. xenon were extracted out of 371 gm Bi2Te3with 124 gm T e by heating in vacuo. A large excess of XelZg, Xe130, and Xe131was found as compared with atmospheric xenon. The excess of Xe129 and Xe131can be accounted for by (n,r) reactions with TeI28 and TeI3O and additional (T = 1.7 X 107 years). There was uranium XeIz9from the decay of in the neighborhood which can have provided the neutrons. A similar explanat,ion is excluded for Xe13"because 1 1 2 9 is unstable. The author^^^^^^ ascribe the XeL30excess to double beta decay of Te130 and obtain a halflife of 1.4 X loz1years. This investigation was later redone by Hayden and Inghram.93If all the Xe13nfound by these authors in their mineral sample is ascribed to double beta of TeI3O, a half-life of 3.3 X lo2' years is found. A similar experiment was performed by Levine et aLg4 The question of the stability of the free proton and of the nucleon bound in a beta-stable nucleus has also been investigated experimentaIly.96~*6 The free proton, e.g., might disintegrate into two positrons and an electron. Therefore the experimental arrangement to detect the decay rate by a counting method is not too different from the arrangements used in double beta decay work. The best results were obtained b y the CERN groupg6for the lower limits of the half-lives: 1.5-2.8 X years for bound nucleons and 2 .2 4 .7 x 1024 years for protons in hydrogen, the variations being due to postulates made about the decay mode. 2.6.1.7. Determination of Branching Ratios. I n many cases the partial half-lives are interesting (cf. Section 2.6.1.1). We do not want to treat 88 J. H. Fremlin and M. C. Walters, Proc. Phys. Soe. (London) A66, 911 (1952). E. L. Fireman and D. Schwarzer, Phys. Rev. 86, 451 (1952). J. S. Lawson, Jr., Phys. Rev. 81, 299 (1951). 9 l M. G. Inghram and J. H. Reynolds, Phys. Rev. 76, 1265 (1949). 92 M. G. Inghram and J. H. Reynolds, Phys. Rev. 78, 822 (1950). 93 R. J. Hayden and M. G. Inghram, i n "Mass Spectroscopy in Physics Research." Natl. Bur. Standards (U.S.) Circ. No. 622 (1953). 9 4 C. A. Levine, A. Ghiorso, and G. T. Seaborg, Phys. Rev. 77, 296 (1950). 96F. Reines, C. L. Cowan, Jr., and M. Goldhaber, Phys. Rev. 96, 1157 (1954); F. Reines, C. L. Cowan, Jr., and H. W. Kruse, ibid. 109,609 (1957); G. N. Flerov, D. S. Klochov, V. S. Skobkin, and V. V. Terentiev, Soviet Phys. Doklady S, 78 (1958). 96 G. K. Backenstoss, H. Frauenfelder, B. D. Hyams, L. J. Koester, Jr., and P. C. Marin, Nuovo cimento [lo] 16, 749 (1960). 89
90
2.
298
DETERMINATION OF FUNDAMENTAL QUANTITIES
the determination of partial half-lives of competing but equal processes, e.g., the decomposition of a beta continuum into its components, but restrict ourselves to the branching-ratio determination in the case of different processes, e.g., alpha and beta decay. Determination of branching ratios is particularly easy if the measurements themselves can be done with alpha particles only. This happens often with radioactive decay chains.07 In this way Schupp et aLg8determined the branching ratio ./(a p ) of Bi2'2 by comparison of the alpha lines of and Po212with a CsI scintillation counter. The error was small because there were two alpha groups with energies and intensities not too different but which could well be resolved, and because the angular scattering and backscattering of alpha particles is small in any case. Eastwood el U Z . ~ measured ~ the very small branching ratio a/(a p) of Bk248by separating chemically the daughter of beta decay Cf24aimmediately before the measurement of the alpha activity and following the increase of the over-all alpha activity. Extrapolation to the time of separation gave the alpha activity of Bk249.The specific beta activity was determined with a 47r counter (Section 2.6.1.4). Aliquots were used for both alpha and beta counting. For measuring electron-positron branching ratios, use can be made of a magnetic spectrometer, which is able to distinguish between electrons and positrons. Most "flat" spectrometers, such as the double focusing or the 180" spectrometer, do this; also, lens spectrometers which have a helical baffle. Weak positron branches are detected with the help of the annihilation radiation. In contrast to nuclear gamma radiation, radiation from annihilation in rest shows a strong directional correlation (angle between the two quanta, 180") which can be used to distinguish between the two. The branching ratio K / p + of K capture to positron emission can often be measured by comparing the intensities of K X-ray and Auger line on the one hand and positron radiation on the other. Difficulties arise in the case of light elements, because of the low K X-ray energy. Scobie and Lewis'o' measured the K / p t ratio of C1l with a wall-less proportional counter.101 Such a counter is also suitable for measuring the ratio between K and L capture.'O' 2.6.1.8. Age Determinations. Age determinations are the inverse of the problem of measuring half-lives: With the half-life known, one asks for the "age," i.e., the time which was needed to produce a certain experimentally determined amount of daughter product, or to reduce the
+
+
97
H. Daniel, Ergeb. exakt. Naturw. 82, 118 (1959). Schupp, H. Daniel, G. W. Eakins, and E. N. Jensen, Phys. Rev. 120, 189
98G.
(1960). 98 T. A. Eastwood, J. P. Butler, M. J. Cabell, H. G. Jackson, R. P. Schumann, F. M. Rourke, and T. L. Collins, Phys. Rev. 107, 1635 (1957). 100 J. Scobie and G . M. Lewis, Phil. Mag. [8] 2, 1089 (1967). 101 R. W. P. Drever, A. Moljk, and S. C. Curran, Nuclear Indr. 1,41 (1957).
2.6.
DETERMINATION OF LIFE-TIME
299
initial activity by a certain experimentally determined factor. Age determinations are of great importance for the study of the history of the earth. According to Eq. (2.6.1.2) the age t is given by (2.6.1.11)
If there is only one daughter product consisting of D nuclei, Eq. (2.6.1.11) can be rewritten t = -log 1
x
(I
+);
(2.6.1.12)
P = N being the number of parent nuclei still existing a t present, i.e., at the time t. In the case of the potassium-argon method with its two daughter products Ca40and A40 one has instead of Eq. (2.6.1.12)
( ;,g:)
t=Xlog 1 + - - ,
(2.6.1.13)
A,/X is the branching ratio for electron capture to A40, A, being the partial decay constant for this branch. A40/K40 means the mole ratio between A40and K40at present. Equations (2.6.1.11) through (2.6.1.13) hold only if there was no loss of material. Such a loss is particularly likely if the daughter product is a noble gas. If necessary, a correction must be applied.lo2 In many cases not the whole amount of daughter substance contained in a sample has been produced by radioactive decay of the mother substance. Help is given by the mass-spectrometric analysis. For example, Pb206is the final product of UZ38decay but is also contained in primordial lead. From the total PbZo6 amount determined in a Uz98-Pbzo6age determination, the fraction corresponding to the primordial lead must be subtracted. This fraction is found with the help of the PbZo4,PbZo7,and Pb208abundances. It is assumed here that these isotopes are of primordial origin. This is not quite right: Pb207is also produced by the decay of U236(abundance 0.72%) and its daughter products. If the sample also contains thorium it will contain as well the final product of the ThZS2 decay, PbZo8. Of the various age determination methods, only two shall be treated here, the potassium-argon method for dating large ages and the radiocarbon method for dating small ages. Table I lists the more common methods for age determination. 102 H. Fechtig, W. Gentner, and J. Zahringer, Geochim. el Cosmochim. Acta 19, 70 (1960); E. G. Wrage, ibid. 28, 61 (1962);H.Fechtig, W. Gentner, and S. Kalbitzer, ibid. 86, 297 (1961).
300
2.
DETERMINATION. OF FUNDAMENTAL QUANTITIES
TABLE I. Methods for Age Determinations" Parent
End product
U23B
6rB7 Pb*"a
K40
A40
U236
PbZ07
c136 C'4 H3
NI4 He 3
RbB7
2'112
(5.3 (4.51 (1.25 (7.14 (3.1 5568 12.28
+ Ca40
A36
of parent
Determined age
f O.2)1O1Oyears f 0.02)108 years f 0.05)108 years f 0.11)108 years f 0.3)106 years
Formation age Formation age Formation age Formation age Exposure age
f 30years f 0.03 years
Exposure age
0 The half-life values quoted are mean values computed from the information The stated errors are standard available in the literature (except in the case of 0"). deviations only, computed from the deviations from the mean (except CI4).Formation age, under ideal conditions, means the time elapsed since the last separation of parent and end product. Exposure age (for meteorites) means the time the sample (or its surface) has been exposed t o cosmic rays. The CI4method determines the time that the organic material has been dead.
Figure 9 shows a potassium-argon setup in use at the Heidelberg laboratory; in practice there is a second melting and purifying part which is also connected with the mass spectrometer. Before being placed in the melting oven, the sample is mechanically crushed to grains of 0.6 to 1 mm in diameter. The argon determination proceeds in three steps: extraction, purification of the extracted gases, and isotopic analysis. The chamber of the static high-vacuum mass spectrometer is made of glass. It can be heated by a sinkable oven. Transportation of the argon into the purifying part is done by adsorption on charcoal. The potassium content of the sample is measured by isotopic dilution with the same setup, or with a spectrophotometer, or by activation analysis. copillary
oven
-=+=7r - - - - - - - - - - -1
cal. gas
Ca
t-
melting
--i
Cu-CuO
purifying
Ca
1-
I
ion source
-
mass spectrometer
I
i-
FIG.9. Potassium-argon setup. rf radio frequency, M o Mo-crucible, T P Toepler pump.
2.6.
DETERMINATION OF LIFE-TIME
30 1
I n radiocarbon dating, the low activity still present in the sample is to be measured [Eq. (2.6.1.11)]. This is usually done with the help of a proportional counter but Geiger counters and, less frequently, scintillation counters are also used. In order to reduce the background, the counter is shielded b y absorbing material and anti-coincidence counters.
El
75 O h PARAFFIN 2 5 O/o BORIC ACID
GEIGER COUNTER
FIG.10. Proportional counter for radiocarbon dating. A vertical section of the whole shield perpendicular t o the axis of the proportional counter. Wire: stainless steel, 0.05 mm in diameter, sensitive length 300 mm. Inner diameter of the counter: 45 mm. Background: 0.98 cpm a t 20°C and 3 a t m Co2. Modern carbon: 9.0 cpm a t 20°C and 3 atm COZ. Maximum age: 38,000 years a t 3 atm COZwith t h e usual criterion of net counting rate = 4u, where u is the statistical deviation a t 48 hours of counting.
Figure 10 shows a counter arrangement. I n most cases, the C14 is added as a gas to the counting gas or forms the counting gas itself, notwithstanding the fact that Libby, who introduced and developed radiocarbon dating, worked with a solid source covering the walls of a Geiger counter. An explicit description of a C14 dating station using the COz proportionally counting technique is given by Olsson. lo3 Delaney and McAulayeodescribe a liquid-scintillator setup for radiocarbon dating. A general outline of the method, together with his Geiger counter technique, is given by Libby.104 The techniques used for sample preparation 1 03 104
I. Olsson, Arkiv Fysik 13, 37 (1958). W. F. Libby, “Radiocarbon Dating,” 2nd ed., Univ. Chicago Press, Chicago, 1955.
302
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
vary depending on the method chosen for determining the C14 activity but, as a general ruie, all the measurements are performed in comparison to a standard. As standard, wood about a hundred years old is preferred because it is easy to date with the help of the tree rings and there is no danger of a contamination. Such contaminations are found in very recent organic material as a consequence of industrial coal consumption (contamination with inactive carbon) and atomic bomb tests (contamination with C14). By determining the C14 content in comparison with a standard, all the uncertainties arising from wall effects in the counter, from counting losses because of the discrimination bias, etc. are eliminated. A serious source of error consists in contaminating the sample with recent organic carbon, An uncertainty arises also from possible variations, locally as well as in time, of the C14 content of the atmosphere.lo6 TABLE11. Some Important Half-Lives (Other Than Those Given in Table I)" n 014
Na2* NaZ4 Pal Srao Iiai
11.7 71.0 2.59 14.98 14.29 26.3 8.09
f 0 . 3 min rt 0 . 2 s e c f 0 . 0 3 years f 0 . 0 2 hours f 0 . 0 6 days 1.7years -fr 0 . 0 2 dayys
*
CS"'
PbS10 Bia1O((RaE) PbPl3
Rn"$ Rez2% Th232
28.5 f 1 . 2 years 21.3 f 0 . 8 years 5.01 f 0.01 days 10.64 f 0.01 hours 3.8242 f 0.0007 days 1607 f 11 years (1.41 f O.O1)lO1Oyears
For the computation cf. Table I footnote 8. Exceptions are here n and Na22.
For routine measurements, samples containing about 2-3 gm C are usually taken. It is, however, possible to work with smaller amounts, e.g., of the order of 0.5 gm C. Extended information on age determinations is found in references 104 and 106. When the C14 half-life itself is measured, corrections for the counter efficiency are to be applied. I n the case of a gaseous activity, the end effect is eliminated by taking two counters of the same diameter but of different length, and the wall effect can be found similarly with two counters of the same length but of different diameter or by a computation.104 Table I1 lists some important half-lives. 106 E. H. Willis, H. Tauber, and K. 0. Miinnich, Am. J . Sci. Radiocarbon. Suppl. 2, 141 (1960). 106L. T. Aldrich and G. W. Wetherill, Ann. Rev. Nucl. Sci. 8, 257 (1958); L. H. Ahrens, Repts. P r o p . in Phys. 19, 80 (1956); T. P . Kohman and N. Saito, Ann. Rev. Nuclear Sci. 4, 401 (1954); A. A. Smales and L. R. Wagner, eds., "Methods in Geochemistry." Interscience, New York, 1960; H. E. Guess, Ann. Rev. Nuclear Sci. 8,243 (1958).
2.6.
DETERMINATION OF LIFE-TIME
303
2.6.2. Short Lifetimes* 2.6.2.1. Introduction. An excited state of a nucleus which is not subject to strong external perturbations will in general decay according to an exponential law N = Noe-‘/r where T is the mean life of the state. The mean life is a n unambiguous property of the state and can be measured even when not all modes of decay are known or detectable. If there is more than one mode of decay, we can write 1/r =
x
=
xi a
where the A, are the rates of decay by each of the modes. The measured mean life is the same, regardless of the mode detected, provided the observed decay is exponential. In this section we deal with the meassec and primarily urement of short lifetimes, i.e., generally 7 < T < sec. Most of the states with lifetimes in this range, and still long enough t o be measured, decay by electromagnetic processes, i.e., gamma-ray emission or internal conversion and pair conversion. There are a few beta decays with lifetimes in the millisecond range among the light nuclei, e.g. BX2.Shorter beta-decay periods cannot be expected because the decay energies required would in general imply instability against nucleon emission. Alpha-particle decay of heavy nuclei can be expected to occur with periods in the range of interest. In fact, the lifesec) and ThC’ (3.0 X lO-’sec) were the first time of RaC’ (1.64 X very short lifetimes measured. Most direct measurements of r proceed by measuring the individual lifetimes T of a large number of nuclei and determining the mean value from their exponential distribution. Sometimes the mean of the measured values is determined without observing the exponential distribution itself. Such a direct measurement of the mean is the most obvious procedure, and in principle the most efficient, b ut in many cases it increases the magnitude of systematic errors. In some cases the measuring device itself takes the average and the individual values of T are not observed but only r. The lifetime T of a n individual nuclear state is the interval between its “birth” or formation and its “death” or decay. The second of these is always marked by the emission of some radiation which can be detected. The formation may be marked by emission or absorption of a particle which can also be timed or detected. Sometimes this radiation is not the one immediately leading to the state in question but one which precedes the formation by a negligible
* Sect,ion 2.6.2 is by M. Oeutsch.
304
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
time interval. This situation is fairly common because usually the states of interest lie quite low in the nuclear level scheme. For example the 0.50 Mev state in F17may be formed in the reaction Ol6(p,y)Fl7through a cascade of y-rays. The mean life of this state has been measured : (1) by detecting the 1.5 Mev gamma ray leading to the formation of the 0.5 Mev level and the 0.5 Mev gamma ray emitted in its decay and measuring the time delay between them; ( 2 ) by timing the bombarding proton and detecting the decay gamma ray. The hold-up in the gamma ray cascade is negligible compared with the 4 X 10-lo sec mean life of the 0.5 Mev state. Nuclear states formed after radioactive beta decay are frequently reached by a cascade of gamma rays through very short-lived states. For example the first excited state of deformed nuclei in the rare-earth region generally has an excitation energy below 0.1 Mev and a mean life of the order of sec. When it is formed by beta decay to a highly excited state followed by several gamma rays, the mean life is usually measured by detecting and timing the beta ray and a conversion electron from the decay transition, disregarding the delay due to the intervening gamma transitions. Of course i t must be proved in each case that this procedure is justified. Sometimes the detected radiation is not of nuclear but of secondary atomic origin. For example K X-rays following orbital electron capture may be used to time the formation of a state. The lifetime of the atomic levels, of the order of lO-'4 sec, may be neglected. Sometimes the 0.51 Mev annihilation radiation is detected to time the emission of positrons. This procedure must be used with caution since the mean time for annihilation ranges from about 2 X sec in some metals to about sec in air. Another possible source of error may arise when the radiations forming and depopulating the excited state are observed in a definite angular relationship with each other, as is frequently the case. The two radiations usually show an anisotropic angular correlation if the nuclear orientation is preserved during the intervening lifetime (cf. Section 2.4.2). The ambient molecular fields in the source material cause nuclear reorientation with a relaxation time which may be between sec and several minutes, depending on the nuclei and surroundings. The angular correlation of the radiations changes from its initial anisotropy to isotropy in a, time of the order of the nuclear spin relaxation time. If this time is comparable with the nuclear lifetime, the efficiency for decay varies significantly, distorting the apparent decay rate. Similar distortion can also occur in some circumstances' due to nuclear resonance absorption in the source or surrounding material. 1
F. J. Lynch, R. E. Holland, and M. Hamermesh, Phys. Rev. 120, 513 (1960).
2.6.
DETERMINATION OF LIFE-TIME
305
All of these interfering effects connected with the nuclear processes appear only rarely and under special circumstances. They usually cause only minor errors compared with those introduced by the technical difficulties of timing and detection. Indirect measurements of specific decay rates frequently also permit a determination of T, e.g., by measuring the cross section of an inverse process which is closely related to the decay. The most important example of this is Coulomb excitation of states which decay by y-ray emission and internal conversion. The same is true for determination of the width of the state which is connected with the mean life by the relation
r
=
These and similar indirect means of deducing lifetimes lie outside our subject. Any direct measurement of the lifetime T depends basically on a clock which starts at the moment of formation and stops a t the moment of decay. The phase or reading of the clock is examined to find T. Sometimes two clocks are involved, one starting at formation, the other a t decay. The phase difference between the two clocks is then measured. The rate of some clocks, such 8s oscillators, may be known precisely. Other clocks depending on individual atomic processes may be subject to statistical fluctuations. In practice even “cl&ssical”clocks have statistical uncertainties. A great variety of processes have been used as clocks. Most of them can be classified among a few general types. One distinction is provided by the manner in which the clock is started and stopped. Broadly speaking there are two types of clock: those attached to the nucleus itself and those under control of the experimenter. Of the first type we consider primarily the position or velocity of the nucleus after formation of the excited state. These can be used as clocks when there is sufficient information concerning nuclear recoil in the formation process and if the change of the recoil velocity due to interaction with the surrounding medium is sufficiently well understood. The second type of clock is usually an electronic device although there are a few examples of mechanical clocks. The starting and reading of the clock, especially an electronic clock, may be accomplished in two principal manners. Either a signal is derived from the process of formation or decay -a “birth” or “death” announcement-or else the clock is used to control the physical process by which the state is formed or its decay detected. We may call the first type the detector method, the second the shutter method. The greatest volume of results has been obtained by means of electronic clocks, using the detector method. When both the
306
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
formation and the decay are detected it is usuaIly called a coincidence method. When formation is gated by an electronic shutter we usually speak of a pulsed-beam experiment. A1though the distinction between the two procedures is not always perfectly sharp, the characteristics of the various schemes are sufficiently distinct to make this division useful for discussion. The classification of the main methods of lifetime measurement according to this scheme is shown in Table I.* TABLE I. Classification of Timing Methods “External” clocks controlled by the experimenter
_____-
Mechanical clocks
Electronic clocks
Detection of for- “Delayed coincidence” methods mation, and Time-to-pulse height converters Oscilloscope trace photography decay Chronotron Gating of forma- Pulsed accelerator beame t ion ,detect ion of decay Gating of forma- Electromagnetic deflection of decay particles with pulsed tion and deaccelerator or with coincicay dence detection
Measurement of recoil distance with coincidence detection Recoil distance measurement using Doppler shift t o detect stopping Recoil distance measurement, using collimators Mechanical shutters, rotating wheels, etc.
“Atomic” clocks depending on rate of extranuclear processes Doppler shift from recoil nuclei in various media Competition of nuclear decay with electronic transitions: Double-vacancy X-ray lines Positron “lines” from pair formation in ionized atoms
2.6.2.2. Electronic Timing. 2.6.2.2.1. DETECTORS FOR ELECTRONIC TIMING. The detector used with an electronic clock must produce a signal
capable of turning the clock on or off, maintaining the timing of the arrival of a ray from the source with a precision which is as least comparable with the lifetime to be measured. A constant time delay between the event and the signal does not cause any inconvenience. In principle it is also not necessary that the signal be very “fast,” i.e., that it contain very high frequency components, since the phase of a slow signal can still be determined with good precision. It is only necessary that this phase bear a fixed relation to the arrival time, not varying from event to event. In practice a fast signal is usually desirable to assure stable operation. * See also Vol. 2, Section 10.6.2 on atomic clocks, and various other chapters on electronic clocks, oscilloscopes, and other devices.
2.6.
DETERMINATION OF LIFE-TIME
307
Virtually all detectors depend on the collection of charge produced, directly or indirectly, by the passage of an ionizing particle through matter. Since this charge usually represents only a small number of electrons, fluctuations in this number are important and better time resolution can sometimes be obtained with a detector in which a larger amount of charge is collected relatively slowly. Apart from its timing characteristics, a detector must have good sensitivity and, in many applications, good selectivity for the relevant radiations. Other important considerations may be the ability to function a t high counting rates, suitable size, and reasonable simplicity. By far the most widely used detectors for lifetime measurements are scintillation counters employing photomultipliers. The most important, properties of these detectors are described in Chapter 1.4 of this volume.* The following is a brief summary of these. The scintillator transforms part of the energy loss of an ionizing particle into light; the photocathode converts part of the photons into electrons; the electron multiplier collects and accelerates the photoelectrons and amplifies the resulting current by repeated secondary emission. In each of these steps, except the photoelectric conversion, there occurs a time delay, subject to statistical fluctuations. Therefore timing information is lost. The statistical fluctuations are of two kinds: time fluctuations and efficiency fluctuations. The latter indirectly cause timing errors because of the manner in which the information is presented to the clock. The following are the main sources of time fluctuations: (1) Duration of the scintillation. After excitation, the scintillator begins to emit light without measurable delay, with an intensity which, initially, decays approximately according to a n exponential law. The period is of the order of a few nanoseconds for the fast organic phosphors, about 300 nsec for activated NaI, and about 30 nsec for pure N a I a t liquid nitrogen temperature. (2) Light collection. The time required for the scintillation light to reach the photocathode depends on the distance traversed. Along the line between the source of nuclear radiation and the photocathode the time At spread introduced in the distance d is (2.6.2.1)
where V L is the light velocity in the phosphor and V R the velocity of the radiation. At right angles to this line the spread is d/vL. With the usual detector dimensions and a reasonable geometric arrangement the spread sec. This may be increased by is rarely more than about 2 X multiple reflections in the phosphor and photomultiplier. * Ser also Vol. 2, Chapter 11.1.
308
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
(3) Transit time. In most of the widely used end-window photomultipliers the average time between the formation of a photoelectron and the arrival of the electron avalanche at the anode is about 50 nsec. Only the fluctuations about this mean transit time cause loss of timing information. Such fluctuations arise from differences in path length and, to a lesser degree, in electron velocity. By far the most important fluctuations occur in the region between the cathode and the first multiplying dynode because of the relatively long distance and weak electric field. Typical variations of transit time as a function of the point of origin of the photoelectron are about 3 nsec. This is reduced by as much as a factor ten when only the central portion of the photocathode is used. Ionizing particles, especially heavy ions, produce a sufficient number of secondary electrons from a solid surface to permit detection in an electron multiplier without the intervention of a phosphor and photocathode. Such devices eliminate most of the time fluctuations described. They have been used in time-of-flight applications but not in lifetime measurements. Gas ionization detectors were used universally before scintillation detectors became available. They are hardly useful for the measurement of lifetimes below about 100 nsec. Parallel-plate chambers yield a current pulse which starts immediately upon passage of the ionizing radiation. The current is however only of the order of 10-lo to lo-* amp and timing information is lost in the amplifier noise. Proportional counters and G-M counters avoid this difficulty. The geometry of such counters is necessarily such that transit time fluctuations in the migration of the primary electrons to the multiplying region are of the order of 0.1 psec. Parallel-plate spark counters are free of this difficulty and can, indeed, be used for nanosecond timing. Their construction and operation are rather awkward and they suffer from a long recovery time. Solid semiconductor ionization detectors are analogous to the parallelplate ionization chambers. Because of their smaller dimensions they yield currents a t least one hundred times larger than gas chambers. The noise problem has, nevertheless, still prevented their application to very fast timing. Devices of this type using collision multiplication may become useful in the future. 2.6.2.2.2. ELECTRONIC CLOCKS.The electronic device which most nearly resembles the usual idea of a clock consists of a gated oscillator* which is turned on by the formation signal and turned off by the decay (Fig. la). The number of oscillator periods can be counted by a scaling circuit which is examined after each event. Modern high-speed scalers could, in principle, allow this method to be used for periods of the order * On oscillators in general, see Vol. 2, Chapter 6.3.
2.6.
DETERMINATION
OF LIFE-TIME
309
of sec. It has been used, in this form, only for times in the microsecond range or longerl2ssprimarily in neutron time-of-flight measurements. When the resolution obtained by counting the nearest full cycle is insufficient, the phase of a single cycle may be examined. Most electronic
/---
c
T
-,‘
b
I
/
on
.. \
off
.’..-,
,
I
-T-
C
4 off
on
d
-T-
On
off
FIQ.1. Modes of operation of various electronic clocks.
clock schemes can be interpreted in this manner.4 Other discussions, from different points of view, have been given, e.g., by Bay,6 Baldinger and Franzen16 and deBenedetti and F i n d l e ~ .In ~ the simplest scheme, an R. A. Swanson, Rev. Sci. Znstr. 31, 149 (1960). * R . A. Reiter, T. A. Romanowski, R. B. Sutton, and B . G. Chidley, Phys. Rev. Letters 6, 22 (1960). E . Gatti and V. Svelto, Nuclear Znstr. 4 , 189 (1959). 6 2 . Bay, Nucleonics 14, No. 4, 56 (1956); Z. Bay, V. P. Henri, and H. Kanner, Phys. Rev. 100, 1197 (1955). E. Baldinger and W. Franzen, Advances in Electronics and Electron Physics 8, 289 (1956). S . deBenedctti and R. W. Findley, i n “Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 45. Springer, Berlin, 1958.
‘
310
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
“on” signal starts the oscillator which is turned off, before completing even one quarter cycle, by an “off” signal (Fig. Ib). The oscillator phase may be measured by the voltage reached. If the oscillator period is sufficiently long compared with the time interval measured, this output signal is nearly linear with T . For infinite period the oscillator becomes a constant current generator (Pig. Ic). This is the idealized mode of operation of most coincidence circuits and time-to-height converters. The distribution in output pulse heights can be conveniently sorted by a pulse-height analyzer. The triggered sweep circuit of a fast oscilloscope is such a constant current generator. If the sweep is started by the “on” signal, the “off” signal may be applied to the vertical plates or to the intensifier. Its position on the screen measures the time interval T.8s9This scheme is of value primarily when other characteristics of the signals are to be recorded together with T , e.g. the pulse height, and when the total number of events is relatively small. In practice, the clock scheme is usually somewhat modified. If the signal is derived from a detector which starts the clock unconditionally, the output will present a large number of meaningless long time intervals ocurring a t the rate of single counts instead of the rate of relevant time intervals. This is sometimes acceptable with shutter schemes in which the turn-on signal is provided by an oscillator, but not in coincidence schemes in which the signals occur a t random in time. Therefore, a gating signal is provided which allows the clock to be turned on only when an “off’) signal occurs within the interesting time interval. This can be achieved, e.g., by providing a relatively slow coincidence circuit as a gate and delaying both fast signals to the clock until the gate has time to operate. For example, in the oscilloscope presentation described above, the sweep trigger is derived from a gate circuit and both the “on” and “off” signals are displayed after a suitable delay. The distance between them is then used to measure T. By far the most widely used method to select only intervals in the desired range is the “overlap” method (Fig. Id). Here both the formation and decay signals must have a sharply defined duration. The clock current flows only during the interval when both signals are present. Therefore, the clock indicates the interval between the onset of the second pulse and the end of the first, i.e., the overlap time of the two pulses. The two pulses of well-defined duration are usually achieved by delay-line shaping of a step-function signal. We note also that the clock is now symmetric in the sense that either signal may be considered as 8 9
R. Hofstadter and J. A. McIntyre, Phys. Rev. 78, 24 (1950). K. G. Malmfors, J. Kjellman, and A. Nilsson, Nuclear Instr. I, 186 (1957).
2.6.
DETERMIXATION OF LIFE-TIME
311
the “on” or “off” signal. The symmetric arrangement is unable to distinguish intervals T from - T. In many experimental situations the nature of the detectors assures that in an event of interest the earlier signal always appears a t the same input. Accidental intervals may, however, appear in either order. When this constitutes a disadvantage, the same pair of signals may be presented to a second, similar circuit with a relative time delay such that only events occurring in the desired order will produce timing signals in both c i r c ~ i t s . ’ ~ Figure 2 illustrates an asymmetric clock arrangement of the type used in accelerator-chopper experiments’O in the manner illustrated in Fig. lc. The constant current generator is the tube V-4. It is turned on by a negative signal a t input 1 and turned off a t input 2. Similar circuits have been built with transist>orsreplacing the vacuum tubes.” Symmetric “overlap” circuits are frequently divided into two types,6 nonlinear addition or parallel-switch circuits and multiplication or series-switch circuits. The first type is basically derived from the Rossi coincidence circuit. The two pulses are added and the added signal is presented to a discriminator-integrat,or circuit. Many combinations of circuit elements have been used for this operntion.6,6A widely used scheme is addition in the common plate circuit of a pair of pentodes followed by a series12 or parallel13 diode discriminator or a pentode dis~ r i m i n a t o r .The ’ ~ pentodes may be replaced by transistors or by currentbiased diodes.16 Addition can also be performed a t the common cathode or common emitter.16 The commonest form of series switch is represented by the type 6BN6 beam switching tube in which current can pass to a target anode only when b0t.h of two control grids receive a positive pulse. Series switches can also be made of two transistors in series. Figure 3 illustrates a circuit due t’o Green and Bell.17 The two gate pulses are formed by shorted cables a t the two pentode plates and present>edto the series-switch circuit labeled “converter” which is the basic clock c,ircuit. Through a pair of taps on the cables, added pulses also appear at. the discriminator diode labeled “coincidence.” This sccond
+
10
L. E. Beghian, G. H. R. Kegel, and R. P. Scharenberg, Rev. Sci. fn,str. 29, 753
( 1 958).
G . Culligan and N . H. Lipman, Rev. Sci. Instr. 31, 1309 (1960). R. E:. Bell, R. L. Graham, and H. E. Petch, Cuan. J . I’liys. 30, 35 (1952). I 3 R. Garwin, Rev. Sci. Znstr. 24, 618 (1953). 1 4 S. Gorodetzky, R. Richert, R. Manquenouille, and A . Knipper, Nuclear fnstr. 7, 50 (1960). 15 S. deBenedetti and H. J. Richings, Rev. Sci. Znstr. 23, 37 (1952). 16 P. C. Simms, Rev. Sci. Znstr. 32, 894 (1961). l i R. E. Green and R. E. Bell, N,uclea,r Instr. 8, 126 (1958). 11
l2
2.6.
DETERMINATION OF LIFE-TIME
313
+
clock circuit serves to distinguish T from - T intervals as discussed above. Figure 4 shows the complete arrangement with a slow triplecoincidence gate circuit which permits the time interval to be recorded only when both signals have appropriate pulse heights as selected by the
R
Fro. 3. An overlap-type series-switch time-to-height converter with an auxiliary nonlinear-adding coincidence circuit to distinguish intervals +T from - T (from ref. 17).
circuits labeled A D A and A D s and when the second clock signal is present in ADc. Measurement of the accumulated change is equivalent to a phase measurement only if the amplitude of the oscillation (Fig. lb) is known. This can be achieved only if the start (and stop) signals are completely
314
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
I
I
1
I 7
I
I
I
I
- 1
\I/
i T I AA I- -I q J . I
I
Triple
I
Gate
Gating pulse
I
I
Kicksorter
I
Coinc.
I
I
FIG.4. A complete time interval sortcr. The 6BN6 converter and the fast coincidence circuit are shown in Fig. 3. A D A and A D B are single-channel pulse-height selectors (from ref. 17).
uniform. When detectors are used to produce these signals this is never true.6v1sThe (re1at)ivelysmall) loss in timing precision is accepted because of the simplicity of the procedure. A very elegant scheme for measuring the true phase of a gated or shock-excited oscillator has been introduced by Gatti (Fig. 5).18,1eThe “on” signal starts an oscillator with frequency v 1 or is known to occur a t a definite phase of this oscillator. The “off” signal does not block this oscillator as in the schemes described before but instead starts Av. The two oscillator anot,her oscillator with a frequency v 2 = v1
+
I*
C. Cottini and E. Gatti, Nuovo ciniento 4, 156, 1550 (1956).
19
R.L. Chase and W. A. Higginbotham, Rev. Sci. Inslr. 28, 448 (1957).
't
316
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
signals are added. The phase of the resulting beat signal with frequency v 2 - vl, is the same as the phase difference between the oscillators. The phase of the much slower beat signal can be measured, for example, by determining the time of the first zero crossing, that is the first half period. This then measures the initial time interval T modulo l / v l . This scheme is applicable particularly in the case of lifetime measurements with pulsed accelerators (shutter scheme) where the frequency v1 can be derived directly from the oscillator of the accelerator. The phase of this continuous wave train is more easily fixed than that of a signal derived from a particle detector. Most recent measurements utilize electronic clock schemes of the types described with a multichannel recording device so that the entire time interval spectrum of interest is recorded simultaneously. Many older measurements utilize a circuit only as null indicators, i.e., as coincidence circuits of short revolving time. Schemes of this type are called delayed-coincidence measurements. They usually use a circuit of the overlap type followed by a discriminator which accepts a signal only when the output pulse has nearly its maximum possible value, e.g., T is nearly zero. The time interval spectrum is then obtained by delaying one of the signals by accurately known amounts T’ and measuring the frequency of coincidences as a function of T’. The device used to introduce the delayed T’ is effectively the clock for this type of measurement. For T‘ larger than sec, active circuits such as univibrators are sometimes used. For shorter times almost invariably measured lengths of coaxial cables provide the delay. The disadvantage of this sequential method of obtaining data for a decay curve is obvious. One advantage of the method is the fact that the coincidence circuit null-indicating clock functions in exactly the same manner for all intervals T , so that nonlinearity of response does not affect the precision of the experiment. A rather elaborate and very successful scheme is represented by the so-called chronotronZ0which does not quite fit our classification. In this circuit a series of overlap or nonlinear addition circuits are placed along a pair of delay lines such that the two signals arrive a t the various circuits with different relative delays. The circuit giving the largest signal is the one a t which the two signals arrived most nearly simultaneously. If all the pulse heights from the various meeting points are recorded, e.g., on an oscilloscope, it is possible to interpolate between the possible discrete time intervals.21Another modification which extends the number of measur2o I. A. D. Lewis and F. H. Wells, “Millimicrosecond Pulse Techniques,” p. 269. McGraw-Hill, New York, 1954. W. Grismore and W. C. Parkinson, Rev. Sci. Znstr. 28, 245 (1957).
2.6.
DETERMINATION O F LIFE-TIME
317
able time intervals is the “vernier chronotron”22 in which the pulses circulate repeatedly through the two delay Iines. The manner in which the signal from the detector is used to turn the clock on or off is decisive for the resolution of the devices. The full timing information is contained in the entire pulse derived from the detector however deteriorated its frequency response, although the later part contains usually less information than the initial part. Ideally, then, the clock should examine the entire pulse and analyze it in such a manner that all of the information is used with appropriate statistical weight. Of the methods in use at present, only photography of oscilloscope traces could lend itself to such a procedure, since it preserves the information long enough to permit examination and evaluation by an electronic computer. The method has not yet been attempted in the measurement of lifetimes of nuclear states. Most of the electronic devices of the time-to-height converter or coincidence overlap types used with scintillation detectors utilize only one or two pieces of information. The most widely used scheme, which may be termed the limiter method, utilizes the time a t which the pulse reaches a certain level. In practice this is achieved by the use of a “limiter” circuit which reaches its maximum current when the signal reaches the desired IeveI. The most widespread limiter circuit is a conducting pentode which is cut off by the signal from the photomultiplier. Other limiters employ transistorsz3 which may be turned on to saturation or turned off. Recently, negative resistance elements, especially tunnel diode@ and avalanche transistors, have come into use for this purpose. Post and Schiff26 have analyzed the rise of an idealized charge pulse which would be derived from a photomultiplier without transit time or gain fluctuations. The charge collected a t the anode would be an irregular staircase, each step corresponding to the arrival of the charge due to a single photoelectron at the cathode. For pulses yielding, on the average, R photoelectrons, the time at which the level corresponding to the Q’th photoelectron is reached, is given by (2.6.2.2)
where r p is the period of the phosphor decay, assumed to be exponential.
** H. W.
Lefevre and J. T. Russell, I R E Trans. on Nuclear Sn’. “3,146 (1958). R. M. Sugarman a n d W . A. Higginbotham, Intern. Conf. Instr. for High Energy Phys. Berkefey I960 p. 54. Interscience, New York, 1961. a4 G . C . Bret and E. F. Shrader, Nuclear Znstr. 13, 177 (1961). *I R. F. Post and L. I. Schiff, Phys. Rev. 80, 1113 (1950). *a
318
2.
DETERMINhTION OF FUNDAMENTAL QUANTITIES
The standard deviation of this time, At, is given by
where R >> 1, R >> Q. It would, therefore, appear in this approximation that it is always advantageous to set the limit8erlevel to correspond to the lowest practical value of collected charge, ideally to the first photoelectron. The same article gives similar formulas for other phosphor decay laws and for less restrictive conditions. Equation (2.6.2.3) shows that the time resolution is an equally strong function of the light yield and photoelectric efficiency as of the decay time of the scintillator. The statistical fluctuations in light yield and photoelectric yield are included in Eq. (2.6.2.3) but not the pulse height spread due to different particle energies and varying efficiency of light collection from different parts of the scintillator. The signal time spread introduced by these effectsz6is
It is, therefore, customary to determine the height of each pulsc as a second parameter besides the time of t,he initial rise. I n the great majority of actual experiments this is done b y seIecting in a separate discriminator circuit only pulses falling within a narrow pulse height range and accepting through a gating arrangement only the corresponding time pulse (slow-fast scheme) (Fig. 4). This procedure is satisfactory if a large fraction of the relevant pulses falls into a narrow pulse height window. This is frequently true, for example, when counting internal conversion electrons or a-rays. It is also true for low-energy 7-rays counted with a NaI scintillator. In some other cases, for example with a continuous p spectrum or with a complex y-ray spectrum, the loss in counting rate may be too severe and it may be desirable to measure the time spectrum for a wide range of pulse heights. One procedure is to record for each pulse the measured time interval and the pulse height (or both heights if two detectors in coincidence are used). This can be done in a suitable digital memory device. The scheme involves rather formidable circuitry even with digital storage on magnetic tape.27 It has been used in connection with neutron time-of-flight experiments but, to the author’s knowledge, not for lifetime measurements. An alternative scheme is represented by analog rather than digital storage of the 26 27
E. Bashandy, Nuclear Znstr. 6, 289 (1960). C. C. Rockwood and N. G. Straws, Rev. Sci. Znsfr. S2, 1211 (1961).
2.6.
DETERMINATION OF LIFE-TIME
319
information, and this can be done in oscilloscope trace photography. In many cases it is sufficient to reduce the additional time uncertainty due to pulse height spreads only in first approximation. This can be done by a simple analog method. The time interval spectrum is presented as a pulse spectrum by a time-to-height converter. By adding to (or subtracting from) this converted pulse another pulse proportional to the total pulse height, a linear dependence of the signal time on pulse height can be compensated. Sometimes i t is possible to reduce the pulse height spread b y appropriate design of the detector. For example, a fairly uniform pulse height can he obtained even with a continuous beta-ray spectrum by using a scintillator much thinner than the range of the relevant beta rays. Gatti4s28and otherse have pointed out that, in fact, the transit time spreads and other factors make the representation of the signal as a staircase quite unrealistic. I n the case of scintillation detectors, these factors make the pulse start more slowly than corresponds to the exponential decay of the scintillator. It then becomes advantageous to derive the signal from a larger number of initial photoelectrons than would be indicated by the formula of Post and Schiff. I n all electronic clocks which may be considered gated oscillators, the effective turn-on or turn-off time (“machine time”) is determined by the centroid of the signal pulse used. This is also true for the linearly charging time-to-height converters. For example, when a limiter is used i t is the centroid of th a t part of the current which derives the limiter to saturation. When the entire scintillation pulse is used the standard deviation of this machine time is At = T = / R ~assuming /~ the entire spread to be due to the statistical fluctuations of the scintillations and the photoelectric yield. This spread is larger roughly by a factor R1/2than th at of the first photoelectron in the idealized case of Post and Schiff. On the other hand, the time of the centroid is independent of pulse height or number of photoelectrons. Thus, when the dominant factor in the resolution time is the variation in pulse height, the best procedure is to use the centroid of the entire pulse t o yield the machine time. The radio-frequency vernier circuit of Cottini and Gatti18 is specially suited to this. In many cases it has been found preferable to use a method which makes use of the fact that the mode or peak of the current pulse from a fast organic scintillator is not very far from the mean or centroid. The current pulse is differentiated and the time of zero crossing is used as clock signal. When the dominant factor in the resolution time is the decay time of the phosphor, the first photoelectron remains the best signal for the clock. This situation exists for small but uniform pulses detected with a 2*
S. Colombo, E. Gatti, and M. Pignatello, Nuovo cimenlo 6, 1738 (1957).
320
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
relatively slow scintillator such as NaI (Tl). I n most practical cases the three factors of scintillator decay, transit time spread, and pulse height variation make comparable contributions to the resolution time. A detailed analysis shows that, when only the initial part of the pulse is used, the variance of its centroid shows a flat minimum for a number of accepted photoelectrons, which depends on the specific conditions of the experiment but is in most practical cases of the order 0.1-0.2 of the total number R for fast organic scintillators and much smaller for NaI (TI), provided the pulse height does not vary over a large factor from one pulse to the other. The problems associated with other types of detectors are similar to those discussed above. In gas ionization counters the dominant factor is the transit time fluctuations. Therefore, small dimensions and strong electric fields in all parts of the counter are required for short response time unless the particle of interest always traverses the same small region of the sensitive volume.* 2.6.2.2.3. ANALYSISOF COINCIDENCE MEASUREMENTS. The deduction of the mean life r from the data obtained with a device which has a time resolution comparable with r requires an analysis similar to other measurements with devices of finite resolutions. We consider first the case of a constant source, i.e., coincidence experiments or devices gating the detector rather than the source. Figure 6 shows the counting rate versus clock time for three different resolution curves. The counting rate for sufficiently long time delays always approaches an exponential with logarithmic slope X = 1/r provided the slope of the resolution curve itself is steeper than A. The mean life can be calculated29 from the exponential part of the curve by the expression I
r = Ni(ti
- f)(log Ni - log Ni)’
(2.6.2.5)
i
The standard error in this value is (2.6.2.6)
It follows from this that a t least under these idealized conditions it is not the width of the resolution curve but its “steepness” which determines the shortest lifetimes which can be measured. If this condition is not
* Various chapters of Vol. 2 give further information on electronic problems connected with the type of measurements described in this section. 39R. H. Bacon, Am. J . Phys. 21, 428 (1953).
2.6.
DETERMINATION OF LIFE-TIME
321
satisfied or if it is satisfied only for times so long that the number of counts is prohibitively small, the mean life can still be measured moderately well under favorable conditions. It follows from the definition of the mean life that the mean shift of the solid curves in Fig. 6, with respect to the dotted curves, is exactly T . This “centroid shift” or first moment a
I,I
,____+--I
b
FIG.6. Time interval distributions due t o an exponential decay (solid curves) for three different resolution curves (dotted lines).
method6 is very attractive and has been widely applied to measure very short lifetimes. In the expression for the centroid given by
t = -i -
2 Ni
(2.6.2.7)
i
Ni is the observed number of counts in the time interval ti f At where 2 At is the separation of successive points on the time distribution curve.
322
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
The standard error in t is given by (2.6.2.S)
I t is applicable only when all counts in the solid curves are due to a sequence of radiation with the mean life 1/X, i.e., in the absence of a background of “prompt” events. Also, points with long time delays, subject to uncertainties due to accidental coincidences and low counting rates, enter with large statistical weight. The latter difficulty is avoided in a method of analysis proposed by Newton.30 The most serious limitation of the centroid shift method is the requirement of a very precise knowledge of the shape of the time resolution curve (or “prompt” curve) including the position of “0” time under the conditions of the experiment. This requirement may be partly relaxed by the use of higher moments of the two curves.b~31 A number of ingenious methods have been developed to calibrate the time resolution curve. The difficulty of this problem arises primarily from the fact that the position and shape of this curve depend on the pulse heights of the process in the “on” and “off” channels. In addition many circuits show a dependence of their behavior on counting rate and, used a t the limit of their resolution, show drifts which may occur during the time between calibration and measurement. The simplest method, frequently very adequate, is to observe the time spectrum with a source of radiation known to exhibit no nuclear time delays and producing pulses of the height encountered in the experiment. This pulse height may be selected by appropriate pulse height selector circuits. In some cases i t is possible to use a so-called “self-comparison” method. l 2 This method reverses the roles of the detectors used for the “on” and “off” signals in the lifetime measurement and compares the resulting curves with each other rather than with the “prompt” curve. It is applicable primarily when both radiations are selected in a magnetic spectrometer.26It suffers from the same limitations as the centroid shift method. Figure 7 shows an experimental result which is typical of the best measurements obtainable a t present with coincidence methods. The apparatus used was similar to that illustrated in Figs. 3 and 4, with the addition of a circuit to compensate for pulse height variation. Conditions in this case32 are very favorable because of the relatively high energy Newton, Phys. Rev. 78, 490 (1950). R. S. Weaver and R. E. Bell, Nuclear Znstr. 9, 149 (1960). 32 R. E. Bell and M. H. JPlrgeiisen, Nuclear Phys. 12, 413 (1959).
30T.D. 81
2.6.
323
DETERMINATION OF LIFE-TIME
of the gamma rays involved. The figure shows the centroid shift and the property of the normalized resolution curve to intersect the decay curve a t the point corresponding to the mean life. The decay curve has a horizontal slope a t this p ~ i n t . ~ , ~ ~ A very elegant modification of the “self-comparison” method has . scheme ~ ~ is applicable whenever recently been used b y Simms et ~ 1 This
DELAY FROM CENTER OF PROMPT CURVE
( I d l o s a c UNITS)
FIG. 7. Time distribution of counts due t o gamma rays of abcut 1 Mev energy following t h e 0.536 Mev gamma ray in the decay of Tb16e.The “prompt” curve was obtained with a Coeosource (from ref. 32).
the cascade of radiations passing through the state of interest is also accompanied by another radiation which can be detected in triple coincidence with the “on” and “off” signals. A strong calibrating source giving “prompt” radiations of the appropriate energies and of considerably greater intensity than the cascade of interest is placed in the apparatus at the same time as the source investigated. The triple coincidence signal is used to steer the clock readings from the delayed cascades to one set of channels of the time analyzer. Those readings without triple coincidence are overwhelmingly due to the calibrating source and are 33
P. C. Simms, N. Benczer-Koller, C. S. Wu, Phys. Rev. 121,
1169 (1961).
324
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
steered to another set of channels. Thus, calibration and measurement are performed simultaneously in the same apparatus. The two sources are actually physically mixed to assure identical geometrical conditions. Different cascades from the same nuclide may even be used. The problem of choosing truly identical pulse height spectra for the two sources remains, of course, in general, the same. The calibration of the clock for coincidence experiments, although straightforward in principle, presents a surprising amount of difficulty, and is the limiting factor in the precision of otherwise favorable cases. The usual time standard is the signal velocity in measured lengths of coaxial cable introduced into one of the detector signals. In practice, variations of the order of several per cent may occur, especially in cables with air as dielectric. Cables deteriorate with time and with rough handling. Additional uncertainties are introduced by connectors and by attenuation and dispersion of the signal in the cable and by connectors and terminations, Unless special precautions are taken, lifetimes based on such calibrations are uncertain by about 5 %. A more reliable standard is provided by the velocity of light (7-rays) in air. If a source of coincident ?-rays located between the detectors giving the time signals is displaced by a distance 2 along the line joining the detectors, the time interval spectrum is shifted by an amount 22/c. Detector separations much larger than detector size are usually required for suitable precision. Coincidences between the annihilation quanta emitted from a positron source are very useful for this procedure because they yield conveniently high coincidence rates even for large counter distances. For greatest precision, compensation must be made for changing counting rates if the source is displaced. It is frequently most convenient to calibrate a set of coaxial cable delay lines in terms of the displacement of a positron source, and then in turn use these cables to calibrate the time scale for radiations of different energy and for different counter arrangements. If oscilloscope display of the pulses is used as a clock, an oscillator may serve as calibration. The shortest lifetimes which can be measured observing decay curves with present coincidence methods are of the order lo-’” sec. The various methods for determining the mean delay of the entire sample of events discussed in the previous section can reduce this lower limit by about a factor 3 under favorable conditions, although a t the risk of systematic errors. Limits to lifetimes can be set as low as 5 X lo-’* sec by the centroid-shift method since a null result is much less affected by background effects due to the “prompt” radiations. Discussion of the time resolution of various experimental devices is somewhat impeded by the confusion-mostly semantic-attached to the
2.6.
325
DETERMINATION OF LIFE-TIME
term “resolving time,” which is used in a variety of ways to measure two quantities which are not very closely connected: (1) the range of clock time intervals corresponding to a single clock reading or channel, i.e., the clock channel width; we shall call this the coincidence resolving time 7 , ; (2) the range of true time intervals between two events which will correspond to a single clock channel; this we call the resolution time 7x. Although the two descriptions seem very similar, they describe two entirely different quantities. The point may best be illustrated by considering a time analyzer, e.g., a multichannel time-to-height converter with symmetric inputs. Each output channel registers all cases in which the time interval between the “start” and “stop” signals is between t and t r,. This width 7 cis usually completely under the control of the experimenter and may be made arbitrarily short. The limits of 7, are completely sharp since nothing is said about the time of occurrence of the events which caused the clock signal to occur. If the signals in the two channels occur completely a t random in time a t rates N 1 and Nz sec-’, and if N1 << 1/7, and Nz << l / r c , then the rate a t which “accidental coincidence” counts occur in the channel with coincidence resolving time rc is N , = 7,N1N2. If the circuit scheme is symmetric, each channel corresponds also to the interval between - T and - ( T 7,),so that the accidental coincidence rate becomes N , = 27cN1N2. In an actual experiment the signals never occur completely a t random. In calculating N , the rates N 1 and N 2 must be reduced by the rates of genuinely correlated events. Also, most circuits record only the first “off” signal after each “on” signal. Thus, the rate of accidental coincidences depends on T . The exact manner in which these corrections must be applied depends on the details of the apparatus and may not be easy to determine. The same is true for accidental coincidences arising from other parts of the equipment. For example, when pulse height selection is used to gate the clock, closely spaced pulses may cause a gating pulse to occur for wrong events. An estimate of the rate of occurrence of such ‘(accidental” counts also requires detailed consideration of the functioning of the circuit, and attempts at general evaluations are usually inapplicable in actual cases. In practice, it is almost always necessary to restrict counting to rates a t which these second-order corrections are negligible. We now turn to the resolution time 7T.Pairs of events separated exactly by the interval T’ will cause the clock to register a range of “machine time” intervals T with a probability distribution clue to the various sources of time spread in the detector, etc. The convolution of this probability with the channel width rc gives the probability (T’,T) that a count registered in the channel recording the clock time interval between T and T 70corresponds to a true interval T‘. The width of this probabil-
+
+
+
326
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
ity distribution P is the resolution time T,.. It measures the precision with which the arrangement can determine the true time interval T between events. Obviously there is no advantage in choosing a channel width rCmuch narrower than 7,. This discussion is equally valid when a single time channel is used (delayed-coincidence method). Apart from the inherent timing limitations of the detectors, the precision of the coincidence methods is limited by the unavoidable background of accidental coincidences. This is essentially a “signal-to-noise” problem of measurement. Consider a decay scheme in which a certain faction a of the disintegration proceeds through the state with lifetime 7. Let the detector efficiency for the formation and decay radiations, including solid angle, pulse height selection, etc., be el and e2 respectively. Let the total efficiency for detecting other, irrevelant radiation be el/& and ez/& under the same conditions. S1 and Sz represent the selectivity of the detectors. Then the number of “true” time intervals between 0 and r counted per second for a source of strength N o sec-’ is:
Nt
=
cuNoele2(l - l/e).
(2.6.2.9)
On the other hand the number of accidental intervals is:
The ratio of the two R = Nt/N,
=
SlSZ + aS1)(1 +
0.63~~ NOT (1
~
(2.6.2.11 )
Thus for a fixed true-to-chance ratio R, (2.6.2.12)
We see that the efficiency and selectivity of the detectors enters almost, equally into the counting rate obtainable with a given signal-to-noise ratio R. Thus, when the cascade of interest is of low intensity and if there are no other limitations on available source strength, higher counting rates are obtained with the use of highly selective low-efficiency detectors such as p r a y spectrometers or triple-coincidence schemes. When all the decays proceed through the state of interest or if the detectors are perfectly selective, the obtainable counting rate N t = 0.4e1e2/R7. For mean lives of the order sec very high counting rates can obvi-
2.6.
DE T E R M I NAT I ON OF LIFE-TIME
327
ously be obt'ained. For example, for typical values =
and
10-9
el =
e2
R
103
=
=
10V
t8he counting rate in the first mean life is 40 sec-'. The corresponding source strength is about 20 microcuries. On the other hand, for r = lop4sec even R = 4 can be obtained only with a counting rate of O.l/scc. The corresponding source strength is lo3 disintegrations/sec. Detector efficiencies may, of course, be made higher, but in practice sec is probably close to the longest lifetime that can be measured by this method except under exceptionally favorable conditions. SHUTTER METHODS.Electronic shutter meth2.6.2.2.4. ELECTRONIC ods have been used primarily with charged particle beams from accelerators. Thus, the actual formation of the state is controlled by the clock rather than the time of detection. In most schemes the turn-on time is synchronized with the shutter, the turn-off signal being derived from a detector. The simplest and most widely used electronic shutter consists of electrostatic deflection of the charged-particle beam. 34--38* In its simplest version it is illust,rated in Fig. 8. A radio-frequency field usually of the order of 10 Mc and amplitude of the order of 10 kv/cm is applied at the deflecting plates. The beam can then strike the target only as a short pulse during a small range of rf phase. The duration of this pulse is determined b y the duty cycle, i.e., the range of oscillator phase accepted by the target slit, and by the properties of the incident beam, primarily its size and spread in energy and angle. For a perfectly monoenergetic beam the shortest possible time resolution At is of the order
At
=
( L Ax AO) l I 2 V
(2.6.2.13)
where v, Ax, AO are, respectively, the velocity, width, and divergence of the beam and L is the distance between deflector and target. If the beam energy is not completely homogeneous there is an addiL .general discussion of the factors tiond time spread of the order of A V ~ A 34 L. Cranberg and J. S. Levin, Phys. Rev. 103, 342 (1956); W. Weber, C. W. Johnstone, and L. Cranberg, Rev. Sci. Instr. 27, 166 (1956). 3 6 C. M. Turncr and S. 11. Bloom, Rev. Sci. Instr. 29, 480 (1958). sR M. Birlc, Q . Goldring, and Y. Wolfson, Phys. Rev. 116, 730 (1959). 37 A. T. Q. Ferguson, M. A. Grace, and J. 0. Newton, Nuclear Phys. 17, 1 (1960). A. W. Schardt, Phys. Rev. 122, 1871 (1961). * Hce also Vol. 2, Section 11.2.2 (discussion of electronic deflection in oscilloscopes) and Vol. 3, Part 5 (mass spectroscopy).
Deflection
Deflected ion beam
1
Li/
I Trajectory of undef lected beam
Limiting slit \
'
P 1 l aI t o r
FIG.8. Simple radio-frequencydeflecting shutter for particles from an electrostatic accelerator (from ref. 35).
2.6.
DETERMINATION OF LIFE-TIME
329
determining the time resolution of this method has been given by Fowler and Since, in general, the clock only measures the time interval from the last preceding beam pulse, the period of the chopping oscillator must be long compared with the lifetime to be measured. On the other hand, the duration of the beam pulse should be shorter than the lifetime. Thus, a relatively small fraction of the possible beam intensity can be utilized. Several methods have been proposed to circumvent this limitation. If the beam pulsing is performed at the ion source of the accelerator, the peak current is usually larger than can be obtained with steady-state ion injection, but the time resolution is not as good as can be obtained with deflection of the accelerated beam, because of transit time spread in the accelerator. A combination of ion source modulation and beam Considerable improvement in duty cycle deflection is therefore can also be obtained by using an isochronous target, arranged so th a t the different path lengths L for particles passing through the deflector a t different angles compensate for the different times of entry into the deflector. The simplest procedure is to incline the target, but more sophisticated versions have been proposed. A more effective device for increasing the available intensity consists of a magnet placed between deflector and target in such a way th a t the path length is changed in the desired manner. A scheme of this type, introduced by M ~ b l e y , ~ ~is- ~illustrated Z in Fig. 9. The gain in accepted radio-frequency phase is of the order of s/Ax for a deflection through 45". Radio-frequency accelerators such as cyclotrons and linear accelerators produce pulsed beams by the nature of their accelerating process. A thorough discussion of this process in fixed-frequency cyclotrons has been given by C ~ h e nThe . ~ ~duration of the pulses is, approximately, At
=
):(
(s) ($
(2.6.2.14)
where w is the cyclotron frequency, V othe dee voltage amplitude, d the gap between the dees, e the charge, and M the mass of the accelerated ions. K is a number smaller than unity, related to the spread in phase at injection. Resolution times of the order of sec have been obtained by this method which has been used primarily in neutron time-of-flight T. K. Fowler and W. M. Good, Nuclear Znstr. 7 , 245 (1960). 40R. C. Mobley, Phys. Rev. 88, 360 (1952). 39
41 42
R. E. Holland, Nuclear Znstr. 12, 103 (1961). L. Cranberg, R. A. Fernald, F. S. Hahn, and F. S. Shrader, Nuclear Znstr. 12,
335 (1961). 4sB. I. Cohen, Rev. Sci. Znslr. 24, 589 (1953).
330
2.
DETERMINATION OF FUKDAMENTAL QUANTITIES
experiments rather than lifetime determination^.^^^^^ In some of these experiments the natural bunching is combined with a deflection shutter to permit larger time intervals between beam Most beam shutter experiments seem to produce pulses of about 1 nanosecond duration, roughly gaussian in shape. Blaugrund and Goldring47~~~ have developed a shutter technique using a microwave cavity operating a t 2500 Mc as deflector instead of the usual
G MAGNET
N BEAM DEFLECTOR
T- o
REFERENCE PULSE+
FIG.9. The beam bunching scheme of Mobley (from ref. 40).
condenser plates. With this scheme and a highIy monoenergetic beam from an electrostatic accelerator, pulses of 3 X 10-l1 see duration were produced. Tn order to utilize this time resolution the turn-off signal must also be derived from a shutter rather than from a detector. A very ingenious shutter scheme was used by these authors (cf. Fig. 10). The decay radiation in their experiments consisted of monoenergetic conversion electrons emitted following production of the excited state b y Coulomb excitation. These electJrons were focused by a magnetic spectromet,er onto a detector. Between the beam target and the spectrometer 4 4 C. 0. Muehlhause, S. D. Bloom, H. E. Wegner, and G. N. Glasoe, Phys. Rev. 103, 720 (1956). 46 J. E. Draper, Rev. Sci. Znstr. 29, 137 (1958). 46 R. Grismore and W. C. Parkinson, Rev. Sci. Inslr. 28, 245 (1957). 4 7 A. E. Blaugrund, Y. Dar, and G. Goldring, Phys. Rev. 120, 1328 (1960). 4 8 G. Goldring, Nuclear Znstr. 11, 29 (1960).
2.6.
331
DETERMINATION OF LIFE-TIME
they were accelerated in a microwave field of the same frequency as the beam shutter. Only those electrons which pass through the cavity at the appropriate phase of the microwave field (e.g., when its mean value in the cavity is zero) will have the correct energy to be focused in the spectrometer. Thus the device acts as a shutter open only during a small phase
L
1
t
SCALER FOR X - R A Y S
4
SCALER FOR COINCIDENCES
t
SCALER FOR ELECTRONS
FIG.10. The microwave two-shutter scheme of ref. 47: (1) @-rayspectrometer, (2) target, (3) electron counter, (5) energy-modulating cavity for @-rays, (6) beamdeflecting cavity, (15) beam shutter slit. The components 7-13 are parts of the microwave system.
of the microwave cycle. The delay-time distribution is measured by observing the counting rate as a function of the relative phase of the two shutter cavities which are driven from the same source. Figure 11shows a typical experimental result. The curve exhibits the exponential build-up (during the beam pulse) and subsequent decay of the short-lived state, superimposed on a constant background. This device has been used suc-
332
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
ceesfully for lifetimes between 5 and 20 nsec. A detailed analysis has been given by G01dring.~' Blaugrund49 has applied the same type of electron microwave shutter to the study of states produced by radioactive decay rather than by ion bombardment. In this case two electron shutters are used with two magnetic spectrometers, focusing, respectively, conversion electrons due
1
0
I
I
0.2
I
0.4
I
I
a6
a8
1.0
1.2
x
T
TIME DELAY-
FIG.11. Determination of the 9 X lo-" sec mean life of a state in apparatus shown in Fig. 10.
Tm169 with
the
to the preceding and decay transitions. The main limitation of this method is the low detection efficiency. This efficiency is the product of the microwave duty cycles, the transmissions of the two spectrometers, and the two internal conversion coefficients. Johansson and AlvagerS0 have used a two-electron shutter scheme with the more conventional deflecting arrangement analogous to that shown in Fig. 8. The deflecting plates are placed a t the exit slits of the two magnetic spectrometers, This postdeflection scheme has the advantage over preacceleration that it is also applicable to continuous electron spectra. In principle, it also permits multichannel operation employing 49 50
A. E. Blaugrund, Physica 26, 185 (1959). B. Johansson and T. Alviiger, Arkiv Fysik 17, 163 (1960); T. Alviiger, ibid. 495.
2.6.
DETERMINATION OF LIFE-TIME
333
several detectors. Its disadvantage is primarily the more difficult electron optics involved and the greater importance of transit time spread. The lower frequency of the oscillator (146 Mc) makes it applicable to longer lifetimes. It has been used successfully for lifetimes of about 2 nsec. The limitation on the measurement of relatively long lifetimes imposed on the coincidence method by accidental coincidences does not apply to the see and sec shutter method. About ten mean lives between have been measured by this method, using a Van de Graaff g e n e r a t ~ r , ~ ~ J l a proton linear ac~elerator,6~ and a betatron.63 2.6.2.3. Nuclear Recoil Methods. If the excited nucleus, immediately after formation, is moving in a simple manner and with sufficient velocity, this fact may be used to measure the mean life. One can: (1) Measure the mean flight distance traversed between formation and decay; (2) Reduce the velocity at a known rate and measure the mean velocity at the time of decay. Methods of simple mechanical transport, such as rotating wheels, are limited to lifetimes greater than about 10-3 sec. Shorter times are measured by making use of the recoil imparted to the nucleus in a nuclear reaction. The earliest attempt at a measurement of this type was made by J a c o b ~ e n The . ~ ~ flight-distance method has been applied successfully to several reactions by T h i r i ~ nand ~ ~by Devons and collaborators.6a-69 The initial recoil veIocity of a nucleus with mass number A , absorbing or emitting a proton or neutron with energy E (MeV), is: E1/2
v,/c =
-'
21.7A
(2.6.2.15)
Typically, v, is of the order of lo6or lo9 cm/sec. Since one may reasonably expect to localize the point of origin of the decay radiation within cm, this method is applicable to lifetimes longer than about 2 X about 5 X 10-l2 sec. A. W. Schardt and A. Goodman, Phys. Rev. 123, 893 (1961). S. D. Softky, Phys. Rev. 98, 736 (1955). 68 S. H. Vegors and P. Axel, Phys. Rev. 101, 1067 (1956). 6 4 J. C. Jacobsen, Phil. Mag. [6] 47, 23 (1924). 66 J. Thirion and V. Telegdi, Phys. Rev. 92, 1253 (1953); J. Thirion, C . A. Barnes, and C. C. Lauritsen, Phys. Rev. 94, 1076 (1954). 58 S. Devons, J. Hereward, and G. R. Lindsay, Nature 164, 586 (1949). 67 S. Devons, G. Goldring, and G. R. Lindsay, Proc. Phys. SOC.(London) 67, 137 (1954). 68s. Devons, G. Manning, D. StP. Bunbury, Proc. Phys. Soc. (London) 68, 18 (1955). 69s. Devons, D. StP. Bunbury, G. Manning, and J. H. Towle, Proc. Phys. SOC. (London) 69, 165 (1956). 61 62
Proton b e a m from
/ magnetic analyzer
I
t
I
.020”
/
,Tungsten I
I
Detector W W rp
\Mild steel back plate
‘ Hardened steel shaft Bronze nut
FIG.12. A flight-distance arrangement used to measure the lo-” sec mean life of the 6.14Mev state in Or6.Gamma rays from recoil nuclei moving in a backward direction are detected if the recoil distance extends beyond the edge of the tungsten collimator before the decay. From D. Kohler and H. H. Hilton, Phys. Reu. 110, 1094 (1948).
2.6.
335
DETERMINATION OF LIFE-TIME
Recoil Target Chamber (top view)
0 . 5 mil Ni and
collimator
Brass recoil
Scale = 1"
I
article counter (side view)
FIG.13. Apparatus to measure recoil distance in the reaction Alz7(d,p)Alz8* (from ref. 60). The particle counter detecte protons in coincidence with the 32 kev -prays from the f i s t excited state (7 = 3 X 10-9 see).
An experimental arrangement of this type is shown in Fig. 12. Collimation is usually easier if the decay radiation consists of electrons rather than gamma rays.67Severiens and Hannaso have used a echeme which avoids the use of collimation but uses, instead, the variation of solid angle with flight distance before decay (Fig. 13). As the recoil catcher is moved closer to the target, it intercepts nuclei which would otherwise O0 J. C. Severiens and S. S. Hanna, Phys. Rev. 104, 1612 (1956).
336
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
have survived to decay closer to the counter. Reasonable sensitivity can be obtained with this arrangement only if the dimensions of the apparatus, including those of the counter, are of the order of v r t . Except under special circumstances, such as bombardment with very heavy ions, it is limited to the range of 10-9-10-7 sec. A typical result obtained by direct flight-distance measurement is shown in Fig. 14. I
FIG.14. Normalized counting rate as a function of target position for the 720 kev -prays from Beg(d,n)B'"*.Also shown is the counting rate from the target covered with a nickel foil. The lifetime deduced from this curve is 7 X 10-'Osec (from ref. 55).
Lifetimes shorter by several orders of magnitude can be measured if the stopping of the recoil nuclei is indicated by a Doppler shift measurement on the radiation rather than by geometric limitation of the detector. This method has been discussed in some detail by Devons et uZ.6*,61,62 Near the threshold of a sufficiently endoergic reaction the recoil nuclei have substantially uniform recoil velocity v,, carrying the forward momentum of the bombarding particles. If the recoil nuclei are stopped after traveling a distance x in vacuum, the Doppler shift for gamma rays observed a t an angle 0 with respect to the recoil direction will be, on the average, A E / E ~= (vr/c)cos e(1 - e - z ' v ~ ~ ) . (2.6.2.16) Thus the mean life can be determined by observing the shift in the mean y-ray energy as x is varied. This method has been used successfully, especially by Devons and c o - w o r k e r ~ . ~Typical ~J~ energy shifts are of the order of 1% ' or less. With sufficient care the resolution of a scintillation spectrometer is sufficient to observe the shift. A magnetic spectrometer is preferable. The relevant distance z is very small, usually in the 81 82
L. G . Elliott and R. E. Bell, Phys. Rev. 74, 1865 (1948). G. A. Jones and D. H. Wilkinson, Phil. Mag. 171 43, 958 (1952).
2.6.
DETERMINATION OF LIFE-TIME
337
range 10-3-10-6 cm, and it is frequently more convenient to use a lowdensity material rather than vacuum to stop the recoil ions. If the range of the recoil ions in the stopping material is R, the mean Doppler shift becomes (2.6.2.17) Therefore, r can be determined from Doppler shift measurements in various materials, provided the range R is known. Range measurements for heavy ions in this energy range are rather scarce. A survey of theory and experiments has been made by Harvey,g3especially for very heavy recoil ions. The older literature is discussed by Thomas and Lauritsenls4 particularly for light nuclei which are more relevant for experiments of this type. Bohr’s theorys6 of the process indicates that, very roughly, the range should be proportional to the initial velocity if the primary contribution to the energy loss arises from ionization and excitation, i.e., R = av,..A typical value of the constant is a = 4 X sec for Li7 recoils in copper.69 The absorber-shift method is applicable in the range between lo-” see and sec. Faster decays show the full Doppler shift even in the densest materials; longer lived nuclei come t o rest before decaying in all solids. The flight-distance method and the absorber-shift method are complementary since the former can usually set a lower limit and the latter an upper limit on 7. The Doppler shift method has also been applied to nuclei recoiling from radioactive (Y d e ~ a y , ~for ~ ,the ~ ’ 40 kev state of T1208(T = 4 X 10-l2 sec). The initial recoil velocity, v, = 3.3 X lo7 cm/sec, is due to the a decay of Bi2I2.The recoil direction was fixed in these experiments by observing the decay radiation in coincidence with a particles emitted in the opposite direction. Internal conversion electrons rather than y-rays were observed since the energy shift for L-conversion electrons is, in this case, almost four times as large as the y-ray Doppler shift, because the electron velocity is significantly less than c. The observed energy shift was about 0.38%. A magnetic spectrometer was used to observe this shift. The apparatus of Burde and Cohenss is illustrated in Fig. 15. The distance traveled by the recoil nuclei before striking the stopping foil is fixed by interchangeable silver spacers of thickness varying between 2 and 7 X 10-6cm. B. G. Harvey, Ann. Rev. Nuclear Sci. 10, 235 (1960). R. G. Thomas and T. Lauritsen, Phys. Rev. 88,969 (1952). 66 N. Bohr, Kgl. Danske Videnskab. Selskab Mat.-Fys. Medd. 28, No. 8 (1948). 66 J. Burde and S. G. Coben, Phys. Rev. 104, 1093 (1956). 67 J. G. Siekman and H. dewaard, Nuclear Phys. 8, 402 (1958). 83 64
338
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
In the case of exoergic reactions, or of endoergic ones well above threshold, the recoil velocities have a wide spread in magnitude and direction. A particular recoil direction may be selected by observing the gamma rays in coincidence with a reaction product particle.69~60
Stopping film Spacer Source
Support for stopping film
Source support A scintillator
Light pipe
Fro. 15. Apparatus for measuring the recoil distance of the 40 kev state of TlZ08 = 1.4 X sec) after the decay of Bi*ls. The energy of the conversion electrons is measured by a magnetic spectrometer located above the assembly shown in the figure (from ref. 66). (7
Alternatively, all y-rays may be accepted and the lifetime deduced from the combined shift and broadening of the y-ray line, since the recoil velocity distribution is usually well known from other studies of the reaction. The earliest rough estimates of very short lifetimesa1OB8 were obtained by this method. It has limited precision since in most cases the broadening is only of the order of 1 % of the gamma-ray energy and has generally yielded only upper limits for the lifetimes. (8
V. K. Rasmussen, C. C. Lauritsen, and T. Lauritsen, Phys. Rev. 76, 199 (1949).
2.7.
339
DETERMINATION OF NUCLEAR REACTIONS
2.7. Determination of Nuclear Reactions 2.7.1. Determination of the Q Value for Nuclear Reactions* The determination of the reaction energy, or Q value, of a nuclear reaction depends upon measurements of masses, momenta, and kinetic a- Y b, where X energies. Typically, in a reaction of the type X is the target nucleus, a the bombarding particle, Y the residual nucleus, and b the emitted particle, the rest masses of X , a, and b are known from previous measurements (see Chapter 2.3). The kinetic energy of the bombarding particle a and the kinetic energy of b a t a given angle of observation may be measured. We shall see that this is then sufficient to determine the mass of the residual nucleus Y and the Q value of the reaction. In the calculation of Q values, mass and energy are treated interchangeably. The relation between energy and mass is given by Einstein’s formula E = Mc2, where E is the energy corresponding to a mass M and c is the velocity of light. A mass of 1 gram corresponds to an energy of 8.98758 x 102O ergs ( c = 2.997929 X 1O1O cm/sec). It is more enlightening however to consider the energy equivalent of a mass of nuclear magnitude. The mass taken is the atomic mass unit, the amu (one gram). The energy equivalent of one amu is amu = 1.65979 x 1.49175 x 10-3 erg. Since one million electron volts (Mev) = 1.60206 X 10-l2 erg
+
5 0.010 Mev,lO based on the 1 amu = 931.441 f 0.010 MeV,” based on the 1 amu = 931.141
+
16 scale.
016
=
C12
= 12 scale.
t For general references on the material in this section, see references 1-9. R. L). Evans, “The Atomic Nucleus.” McGraw-Hill, New York, 1955. D. Halliday, “Introductory Nuclear Physics,’’ 2nd ed. Wiley, New York, 1955. 3 M. S. Livingston and H. A. Bethe, Revs. Modern Phys. 9, 277 (1937). 4 I. Kaplan, “Nuclear Physics.” Addison-Wesley, Reading, Massachusetts, 1955. 6 K. T. Bainbridge, i n “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. I. Wiley, New York, 1953. 8 L. Landau and E. Lifshitz, “The Classical Theory of Fields.” Addison-Wesley, Reading, Massachusetts, 1951. 7 J. Blaton, Kgl. Danske Videnskab Selskab Mat.-fys. Medd. 24 (20), 1 (1950). 8 A . S. Langsdorf, Jr., J. E. Monahan, and W. A. Reardon, Argonne National Laboratory Report A m - 5 2 1 9 (1954). 9 J. L. Fowler and J. E. Brolley, Jr., Revs. Modern Phys. 28, 103 (1956). 10 J. W. M. DuMond and E. R. Cohen, Reus. Modern Phys. 26, 691 (1953); E. R. Cohen and J. W. M. DuMond, in “Handbuch der Physik-Encyclopedia of Physics” (S. Flugge, ed.), Vol. 35. Springer, Berlin, 1957. 11 F. Everling, L. A. Konig, J. H. E. Mattauch, and H. A. Wapstra, Nuclear Phys. 16, 342 (1960). 1
2
* Sections 2.7.1 and 2.7.2 are by
Fay Ajzenberg-Selove.
340
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
The CI2 = 12 scale is now the internationally accepted scale, and the masses1l used in this section and in Section 2.7.2 are based on this scale. All masses of atomic nuclei are given in amu, and kinetic energies are usually measured in MeV, kev (10-3 Mev), ev (10-6 Mev), and Bev* (lo3 Mev).
2.7.1.1. The Meaning of the Terms "Q Value" and "Threshold Energy." When a nucleus X is bombarded by a, one of the possible reactions leads to production of a nucleus Y and the emission of b. Such a reaction may be represented by
x+a+
Y +b
+Q
where Q, the reaction energy for the X(a,b)Y reaction, is the difference between the rest mass energy of the initial nuclear system ( X a) and the rest mass energy of the final nuclear system ( Y b). If the rest mass energy of the initial system is greater than that of the final system, Q is positive, the reaction is said to be exoergic, and, assuming for the moment that X and a have no kinetic energy, kinetic energy of magnitude Q (from conservation of energy) is transferred to Y and b. From conservation of momentum, the amount of kinetic energy carried by Y and by b is inversely proportional to the masses of Y and b, in the center-of-mass system (the C system).
+
+
+
+
Example: Consider the reaction N14 n + C14 p. Assume that the kinetic energy of the initial system (N14 n) is zero; the total kinetic energy given to C14 and n derives entirely then from the Q of the reaction which is 0.627 MeV. The residual nucleus, C14, acquires a kinetic energy of approximately (A) (0.627) Mev [more accurately (m,/m(c14+,))* Q]and the proton acquires a kinetic energy of approximately (++) (0.627) Mev in the C system. The kinetic energies in the laboratory system are more difficult to compute (see Section 2.7.1.2). The assumption that the kinetic energy of the initial system is zero is unrealistic except for capture of thermal neutrons. In general the incident particle will have an appreciable kinetic energy.
+
If the rest mass energy of the initial system is less than that of the final system, Q is negative, the reaction is said to be endoergic, and the reaction cannot take place unless a sufficient amount of kinetic energy is supplied to the initial system ( X a ) . From conservation of energy, the amount of energy required in the C system to allow the reaction to take place is numerically equal to the Q value. I n the laboratory system
+
* Gev is sometimes used instead of
Bev.
2.7.
DETERMINATION
OF NUCLEAR REACTIONS
34 1
(the L system), the corresponding energy is called the threshold energy, Ethresh.
(see Section 2.7.1.4 for the derivation of this expression).
+
+
p + Be7 n has a Q value of -1.643 MeV. If the reaction is to take place, the system Li7 p must be given a kinetic energy in the C system of 1.643 MeV. This is done by accelerating the protons to a laboratory energy Ethresh., which would give 1.643 Mev [total kinetic energy] in the C system: E t h r e s h . N +(1.643) MeV. Example: The reaction Li7
+
2.7.1.2. The Calculation of Q Values. ( A ) If the masses of all the nuclei and particles involved in a reaction are known, then the Q value can be obtained by subtracting the nuclear mass of the final system from the nuclearI2 mass of the initial system, and converting the mass difference to energy units by using the conversion factor given earlier. For instance, the Q of the reaction X(a,b)Yis [(Mx4-M,) - ( M Y4- Ma)] X 931.441 Mev where the M’s are the nuclear masses, in amu, of the various components of the reaction. It is, however, not convenient to express Q in terms of nuclear masses, since mass spectroscopic measurements which have led to our knowledge of the masses of most nuclei are measurements of atomic masses. In a nuclear reaction of the type X ( a , b ) Y , it can easily be shown that atomic or nuclear masses can be used interchangeably since the number of electrons is the same in the initial and final systems: If Z x , Z,, Zy, 2 6 are, respectively, the atomic numbers of X , a, Y , and b, then ZX 2, = Z Y Za, from conservation of charge. If we then both add and subtract (ZX z b ) from the Q equation above, we obtain
+
Q
=
[ { (Mx
+ +
+ ZxmJ + (Ma + Zame)1 - ( ( M Y+ ZymJ + + Zbme)) + AB,] X 931.441 Mev (Mb
where meis the mass of an electron and AB, (a very small quantity which can be disregarded*) is the difference in the binding energyof the electrons in the atoms of the initial and final systems. This equation is equivalent to writing Q = [(mx m,) - (my mb)] X 931.441 Mev (2.7.1.1)
+
+
*For very heavy nuclei, B. may be hundreds of kev, and so in some reactions ( e g , fission) ABe niay be an appreciable fraction of a Mev. However, even in this case, A B , is small compared to the present mass uncertainties. lP A. B. Brown, C. W. Snyder, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 83, 159 (1951) [Appendix].
342
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
where the m’s are atomic masses. Thus the Q value is the energy equivalent of either the nuclear or the atomic mass difference between the initial and the final systems. It is often convenient in calculations of this kind to use “atomic mass excesses” instead of masses. The atomic mass excess F of an atom is defined as the difference between the atomic mass of the atom and its mass number A : F X = mx - A X where A X is the total number of nucleons in nucleus X . Tables of atomic mass excesses already converted to energy units are availablell; for instance: mB,s = 9.012186 amu; F
=
9.012186 - 9 = 12.186 X
amu = 11.350 Mev
mGll= 12.000000 amu; F = 12.000000 - 12 = 0 amu = 0 Mev amu = 15.994915 amu; F = 15.994915 - 16 = -5.085 X = -4.736 MeV.
mole
In this last case, the atomic mass excess should properly be called an atomic mass defect, but it is not. Atomic mass excesses can be used t o compute directly the Q value of a reaction since the total number of nucleons in an ordinary reaction is A , = Ay Ab. Therefore the Q value in Mev of conserved: A x reaction X(a,b)Y is the difference of the atomic mass excesses in Mev of the initial ( X a) and final ( Y b) systems. ( B ) In the general case, the masses of all the nuclei involved in a nuclear reaction are not known. In that case, it is necessary to measure kinetic energies (or momenta) to determine the Q value. The expression which relates the masses, the kinetic energies (in the L system) and Q is called the Q equation, and, in the nonrelativistic form, it may be derived as follows. (See Fig. 1.) Let E, be the kinetic energy (+m,va2)of particle a. The kinetic energies of X , Y , and b are 0, EY,and Eb; 6 is the angle of emission of Y and 6 of b in the L system. From conservation of energy (and mass),
+ +
+
+
En 4-Q
=
EY 4- Eb.
(2.7.1.2)
From conservation of linear momentum
+ pb cos e and py sin 9 pb sin 8. dmc, = 42rnyEy 9 + 42nlbEb 8.
pa = py cos 4 That is,
=
COS
2/2myEy sin 9 = l/2mb~bsin e.
COS
(2.7.1.3) (2.7.1.4)
Equations (2.7.1.2), (2.7.1.3), and (2.7.1.4) can be solved simultaneously to eliminate any two parameters. In general the kinetic energy of the
2.7.
DETERMINATION OF NUCLEAR REACTIONS
343
incident particle E, is known and Eb can be measured a t a given angle of emission 8. Therefore one generally wishes to eliminate EY and 9. I n this case the Q equation can be written
Usually the mass numbers A for a, Y , and b can be used instead of the atomic masses. However, if a n accuracy of several figures in Q is desired then the atomic masses must be used. Since frequently one of the masses is not known (generally my) one has to substitute for nay either A r or y-axis
t
BEFORE COLLISION
< --__-_
Go+--@ ma
AFTER COLLISION
mX
(target nuckur
al rrrtf
*
x -axis LABORATORY SYSTEM Fro. 1. Representation of laboratory system for the determination of the Q value.
an approximate value of my. Inspection of Eq. (2.7.1.5) shows that it does not matter whet,her mass units, energy units, or pure numbers (i.e., A's) are used for the m's. (C) The study of nuclear reactions has recently become a field of high precision. Q values have been measured to accuracies of the order of 0.1 % (-1 kev) with the aid of extremely monoenergetic beams of incident particles and very precise measurements of momenta and energies, in particular, by the groups a t Massachusetts Institute of Technology, California Institute of Technology, Wisconsin, Rice, and Oak Ridge. This means that esoteric sources of errors must be considered: for instance Langsdorf et aL8 considered the thermal motion of the target nucleus, and the mass uncertainty due to the variation in the number of electrons taking part in the collision and in the recoil processes. Both of these effects are of the order of tens of electron volts13 (that is, the error l a J. B. Marion and B. Allen, A Tabulation of Neutron Energies from Various Charged Particle Reactions. Shell Development Co. (August, 1955). Unpublished.
344
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
in E b from these causes is negligible compared to the uncertainties from angular resolution, target thickness, and beam spread effects). This requirement for high precision also means that even in energy regions which one would not ordinarily consider relativistic, i.e., E/mc2 5 0.01, the relativistic form of the Q equation must be used. To obtain this form, we will have to consider again Fig. 1 . If the correct relativistic forms of the momenta are introduced, p2
=
$ + 2mE
and if the equations are again solved simultaneously, then we obtain the relativistic Q equation:
This equation reduces to the nonrelativistic Q-value formula Eq. (2.7.1.5) if E/mc2 << 1 for all particles. (See also Brown et aZ.12) Figure 2 shows typical variations of Eb versus 0 as a function of E, for an exoergic reaction: H2(t,n)He4, Q = 17.588 Mev (here b is the neutron). The curves were plotted from the tables given in Fowler and Brolleye which were computed using the relativistic Q equation. The figure serves to illustrate several points: if E, 'v 0, then the energy of the center of mass is N 0 and the kinetic energy of particle b should be approximately the same in all directions [this can be checked by substituting E, = 0 in Eqs. (2.7.1.5) or (2.7.1.6) which become independent of el. This is illustrated by the El = 0.1 Mev curve on Fig. 2. As E, is increased, the ratio of the energies of particles b emitted in the forward and backward directions increases [Eb(Oo)/Eb( 180") increases as E, increases]. Eb is single-valued for a given fl.and E, (we have taken mb < my). (D)Regardless of whether b (or Y ) particles have single values of Eb (or EY) at a given 0 and E,, these particles have discrete energies. This uniqueness property of particle energies a t a given angle and E, is a direct consequence of the laws of conservation of energy and momentum for a two-body reaction (i.e., a reaction leading to two fragments). I n any multiple breakup (a reaction or decay leading to more than two particles) the energies of the particles involved will not be discrete. An example of multiple breakup is beta decay in which a nucleus X with atomic number 2 decays into a nucleus Y with atomic number 2 1
+
2.7.
DETERMINATION OF NUCLEAR REACTIONS
345
emitting a beta particle p- and a neutrino. The mass difference mx - my is transformed into kinetic energy which is distributed between Y,p-, and Y. The distribution of N ( @ )with EB (number of electrons having an energy ENas a function of Ep) has been studied in great detail. p can have EB ranging from 0 to approximately (mx - my)c2 (the latter case corresponds to E y = 0 and E , = 0). Other types of multiple breakups have not been studied as thoroughly (see, for instance, Berlin and Owen'4).
40h H'+ t4He4+n 0 =17.588MEV
u 60" 120°
OO"
180"
€+"(L)
FIG.2. Typical variations of Eb versus 0 as a function of E,, for an exoergic reaction.
(E) As we have seen, the problems of determining Q values in twobody reactions or in multiple breakups leading to charged fragments are fairly simple in principle. However work in high-energy physics and cosmic rays often involves reactions in which (1) the masses of some of the incident and emitted fragments are unknown, and (2) three or more particles are emitted, with one or more of these uncharged. Methods of computing Q values under these circumstances are treated for instance by ThomsonI6 (see also Brueckner and Thompson,1e Armenteros et aZ.,17 Podolanski and Armenteros,l* and Leighton et aZ.19). T. H. Berlin and G. E. Owen, N. Y. 0.-2052 (1957);Nucl. Phys. 6,669 (1958). 1sR. W. Thompson, in "Progress in Cosmic Ray Physics" (J. G. Wilson, ed.), Vol. 3. Interscience, New York, 1956. 16 K. A. Brueckner and R. W. Thompson, Phys. Rev. 87,390 (1952). 17R. Armenteros, K. H. Barker, C. C. Butler, and A. Cachon, Phil. Mag. [7] 42, 1113 (1951). 18 J. Podolanski and R. Armenteros, Phil. Mag. [7] 46, 13 (1954). 10 R. 3. Leighton, S. D. Wanlaas, and C. D. Anderson, Phys. Rev. 89, 148 (1953). 14
346
2.
DETERMINATION OF FUNDAMENTAL QUANTITIFS
Consider, for example, the reaction bo--* p
+ r- + Q.
One may want to know, on observing a V event (two tracks originating a t one point) in a bubble chamber or a Wilson cloud chamber, whether the event corresponds to the decay above, or whether it is due to another process. This means that the two particles must be identified as a proton and a r--meson, that one must be sure that a third, neutral, non-ionizing (and therefore invisible) particle was not emitted, and that the Q of the reaction is the correct one for this process. All this involves measurements of momenta. The difficulty is that the majority of Ao are so energetic that the proton and the *--meson are traveling a t relativistic speeds and
FIG 3. Vector diagram of a AO-decay into two charged fragments with momenta P is the momentum of the Ao. [From R. W. Thompson, in “Progress in Cosmic Ray Physics” (J. G. Wilson, ed.), Vol. 3. Interscience, New York, 1956; reproduced by kind permission of the author.] p , and p - .
that measurements of momenta can then be very difficult. Then the &-surface representation of Thompson becomes very useful. (See Fig. 3.) In this representation, a family of ellipses are drawn, each ellipse corresponding to a different value of Q. The axes are the Manchester a-parameter and p , where a = [(sin28- - sin2 t9+)/sin2e] and p , is the transverse component of the momentum of either of the emitted fragments. 8- and e+ are the angles which the positive (i.e., proton) and negative (i.e., *--meson) particles make with the incident neutral particle, and e is the angle between the two fragments (all in the L system). Since the incident direction of the neutral particle (Ao) is not known, the decay is assumed to be a two-body decay in which case p,+ = pv- which uniquely determines the direction of the no.For each event, there is then one value of cr corresponding to one value of p,. These are plotted on a diagram of the type shown in Fig. 4. If all the points are nested on a given Q curve, then all the events are two-body decays with that particular Q value. If for any a there is more than one value of p,, or if for a p, there is more than one value of a (outside of error), then there is not a unique twobody decay mode for the system. 2.7.1.3. The Calculation of Threshold Energy. The threshold energy for au endoergic reaction is the amount of energy in the L system neces-
2.7.
t
$+
P
+
DETERMlNATfON OF NUCLEAR REACTIONS
7r
347
Q = 100
+O(Mev.)
150-
100 -
R -
9
50-
1 1 ’ -
n
I
t0.3
I
I 4-0.4
I
+0.8
+0.5
+0.9
+o.6
FIG.4. Q curves for A’J-decaywith various values for Q from 10 to 100 MeV. [From R. W. Thompson, in “Progress in Cosmic Ray Physics” (J. G. Wilson, ed.), Vol. 3. Interscience, New York, 1956; reproduced by kind permission of the author.]
sary to make this reaction energetically possible. I n the C system this energy is - Q (since Q is negative for an endoergic reaction, - Q is a positive quantity). I n a two-body process where a stationary target nucleus of mass mx is bombarded by a particle with mass ma, the threshold is related to the Q value by the equation energy Ethresh.
(2.7.1.7) Equation (2.7.1.7) may be derived as follows. Let V , be the velocity of a in the L system, V,’ be the velocity of a in the C system, Vx’be the velocity of the target nucleus X in the C system. In the C system, from conservation of momentum maVal = mxVx’.
(2.7.1.1’)
In the L system, the relative velocity of a and X is V , since the velocity of X in the L system is 0 (it is a t rest). The relative velocity of a and X in the C system is V,’ VX’(the plus sign is due to the opposite directions
+
348
2.
DETERYINATION OF FUNDAMENTAL QUANTITIES
of travel of a and X in the C system). But, the relative velocities must be the same in the two systems and, therefore:
v, = v,‘ From Eq. (2.7.1.1’)
+
(2.7.1.2’)
VX’.
(2.7.1.3’)
Substituting (2.7.1.3’) in (2.7.1.2’), we obtain (2.7.1.4’)
and (2.7.1.5’)
Ec, the total kinetic energy of the system ( X Ec = +m,Vt2
+ a) in the C system is
+ +mxV:.
(2.7.1.6’)
Substituting (2.7.1.4’) and (2.7.1.5‘) in (2.7.1.6’), we obtain (2.7.1.7’)
+
where +maVa2is EL,the total kinetic energy of the system ( X a) (actually just the kinetic energy of a) in the L system. At threshold, and Eq. (2.7.1.7’) becomes Eq. (2.7.1.7). when EC = -&, EL = Ethresh. In the relativistic case, Eq. (2.7.1.7) becomes (2.7.1.8)
Both Eqs. (2.7.1.7) and (2.7.1.8) give the minimum value of E, which can give a real value of Eb-at 0”. In order to observe particles b at angles greater than O”, a value of E, larger than this minimum threshold is necessary. For instance the value of E , above which neutrons are emitted a t e 2 90” is given (in the relativistic form) by Ethd.(9O0) = - Q
mx - mb
-
2(mx
1-
- mt,)c2
(2.7.1.9)
Figure 5, plotted from the tables of Fowler and Brolley,g which were computed using the relativistic &-value formula, illustrates the variation
2.7.
DETERMINATION OF NUCLEAR REACTIONS
349
of Eb with 0 as a function of E,. The reaction is H3(p,n)He3 with a Q = -0.764 MeV, with the neutron as b, the emitted particle. If E, N 0, no reaction occurs. As E, becomes greater than the threshold energy, there is a small energy region where, in the forward direction in the L system, there may be particles b of two energies (double values of Eb). In the C system there is only one value of Eb for a given value of E,, but 24
20
16
3 12
5 c .c
w
8
4 E,=l.l
-
Mev
m
U
-
0"
I20°
60'
180'
+3"(L1
FIQ.5. Variation of Eb with e as a function of E . (using relativistic Q value formula).
if V C(the velocity of the center of mass) is greater than Va (in the C system), then the particles which moved forward and backward in the C system will appear with two different energies in the L system in the forward direction. There can be no b partides emitted with 8 2 90" in the L system. This behavior is illustrated by the inset on Fig. 5 : if the curve corresponding to E p [ ( E p= Ea)]= 1.06 Mev is considered, one 7 0.01 Mev are both present. observes that a t 0", neutrons of ~ 0 . 1 and At higher bombarding energies, the ratio of Ea(O")/Eb(180") increases with increasing E, (see Fig. 5 ) .
350
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
2.7.1.4. The Q Values of Nuclear Interactions. I n elastic scattering [X(a,a)X],Q = 0 and ma = m b ( = m ) . Equation (2.7.1.6) then becomes
o = (" + mY)E,'
- ( m y
my
-
my
") E, (".' + EA2 +
- E?)
2md
where Ea is the kinetic energy of the incident particle and E,' is that of the particle scattered at angle e in the L system. In the nonrelativistic approach, Eq. (2.7.1.5) becomes
if 0 = 0 , E, = E,,'. In inelastic scattering the Q is the negative of the excitation energy of the nuclear state to which the scattering takes place: X(a,a')X* where X * represents the nucleus X excited to a state of higher energy than the ground state, and u' is the inelastically scattered particle. Inelastic scattering interactions are always endoergic. The magnitudes of Q values of nuclear reactions depend on the type of reaction. In general, two-particle reactions induced by deuterons, tritons, and Hea particles are exoergic or at least not extremely endoergic. This is due to the fact that a nucleon is in general bound more tightly to the target nucleus than in one of these particles, which fact is reflected in the large atomic mass excesses of deuterons, tritons, and He3 particles (respectively, 13.135, 14.949, and 14.930 Mev). For the same reason, reactions leading to d, t , and Hea particles are seldom exoergic and are usually fairly endoergic. Reactions initiated by a-particles, neutrons, and protons are usually endoergic, f with the exception of capture processes [i.e., Beg(n,?)Bel0, Li6(a,r)Blo, C18((p,y)N14]which are always exoergic. Disintegrations induced by y-rays are always endoergic: the Q value is equal in magnitude to the binding energy of the particle being removed from the nucleus: the order of magnitude of photodisintegration Q values is - 8 Mev [however, some y-induced reactions are less endoergic, i.e., Ha(y,n)H1 with Q = -2.225 Mev and Beg(y,n)Be8 with Q = -1.665 MeV; these are due to the anomalously low binding energies of a neutron in H2 and Beg]. The "Q" of any spontaneous decay process is inherently positive since a nuclear state could not decay unless its mass were greater than that t The atomic mass excesses of a-particles, neutrons, and protons are respectively
-
2.425, 8.071, and 7.289 MeV.
2.7. DETERMINATION OF NUCLEAR REACTIONS
351
+
of the final state. For a-decay Q = mx - (my ma) where X and Y are the parent and the daughter states. In computing the Q's involved in beta decay and K capture, one may well ask if it is legitimate to use atomic rather than nuclear masses. Let us consider this question briefly (see also Evans'). 1. Negatron (P-) decay: Process of type ZX+ z+lY Pv. Let MX and MY be the nuclear masses of ZX and z+lY, me be the electron mass, mx and my be the atomic masses of X and Y . The neutrino can be assumed to have zero mass (m,5 1 kev). Then
+ +
Q = ( M x- ( M Y
+ m e ) }- c2.
Let us add and subtract Zmec2in this equation, then
Q
=
+
{(MX Zme) -
( M y
+ Zme + m e ) }. c2
but this is equivalent to writing
Q
= (mx
- my)c2.
+ +
Thus, e.g., if we consider the Cl4 @--decay, C144 "4 8- v, the Q of the reaction, which is the total kinetic energy shared by the N14, @-, and v in the C system, is (mclr - m N I 4 )= 0,156 Mev. 2. Positron (@+)decay: Process of type Z X -+ z-1Y P+ v
+ +
Q = { M x- (MY+ me)) *c'
(the mass of a negatron and a positron are the same). Let us add and subtract Zm,c2 to this equation:
Q
=
+
+
{(Mx Zme) - (MY (2 -
+
1)me
+ 2me))
*
c2
therefore Q = { mx - (my am,)} * c2. That is, atomic masses can still be used to compute the Q of the decay, but an amount 2m,c2 must be subtracted from the energy difference of the parent and daughter states in order to find the total available kinetic energy. For instance let us consider the N13 decay: " 3 4 C13 @+ v. The mass difference of N13 and CL3is 2.221 MeV. The energy equivalent of 2me is 1.02 MeV. Therefore the kinetic energy available to C13, /3+, and v is 1.19 MeV. 3. K capture: Process of type Z X e- -+ z - ~ Y v (capture of orbital electron by the nucleus Z X ) . Q = ( ( M x me) - M Y ) * c2, and in the same way as for (1) and ( 2 ) [but adding and subtracting (2 - l)mec2]we find that we can use atomic masses in K-capture processes also. In experiments involving y - ~ a y s ,which have no rest mass, only their energy hv need be taken into account. Any reaction leading to annihila-
+ +
+
+
+
352
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
+
tion of matter is exoergic: for instance ee + 4 27 is exoergic with a Q of 1.02 Mev ( = 2 m 2 ) . Any reaction in which matter is created is endoergic (this follows from conservation of energy). An example of this type of process is p p--, pp p p where p- is an antiproton.
+
+ + +
2.7.2. Determination of Nuclear Energy levels from Reaction Energies The stationary energy levels may be pictured as “representing arrangements of the nuclear constituents which are particularly preferred over other possible arrangements and which may exist for times many orders of magnitude greater than the time* required for a nucleon to traverse the effective dimensions of the nucleus.”’ Nuclear energy levels or stationary states are characterized by certain quantities: among these are total angular momentum, J (see Chapter 2.4),parity, K (see Section 2.4.1), isotopic spin T, lifetime (see Chapter 2.5), magnetic and quadrupole moments, and excitation energy.” The configuration of neutrons and protons in a nucleus which gives rise to the state of lowest possible energy (or mass) for this nucleus is called the ground-state configuration, and the nuclear state to which it corresponds is called the ground state. For instance, on the independentparticle model, the ground-state configuration for ,Bl0 is ( l ~ ~ ~ ~ ) ~ ( l p for the five protons [lp:,, means that there are 3 protons in a state with I = l(p) and with spin and orbital momentum parallel (+)I, and ( 1 ~ 1 1 2 ) ~ ( 1 ~ ~ for 1 ~the ) ~five neutrons. There is evidence that the spins of neutrons or of protons cancel in pairs. I n the case of B’O, the possible angular momenta of the ground-state configuration may be estimated by considering the possible combinations of the angular momenta of the single odd proton and the single odd neutron. -The -interaction of these two nucleons with j , = j , = Q leads to J ( = j, j,) = 3, 2, 1, 0. The state of lowest excitation, or ground state, in this particular case, is the one which corresponds to J = j , j , = 3. The states in B”J with J = 2, 1, and 0, corresponding to the ground state configuration given above, will have higher energies. There may, of course, be other J = 3 states in Bl0; however, the J = 3 will then result from the interaction of a proton and a neutron in states other than 1p3p: for instance, a proton * Assuming a nuclear diameter of 10-12 cm and a nucleon speed of l O Q cm/sec, the
+
+
time required to cross the nucleus is 10-21 sec. 1 T. Lauritsen, Ann. Rev. Nuclear Sci. 1, 67 (1952). 1s F. Ajaenberg-Selove, ed., “Nuclear Spectroscopy,” Vols. A and B. Academic Press, New York, 1960.
2.7.
DETERMINATION
353
OF NUCLEAR REACTIONS
in a fds,z state coupling with a neutron in a lslIz state can also give = 3, but the resulting state will have a higher excitation energy than the ground state. The difference in energy of the excited state from that of the ground state is called the excitation energy; the ground state of a nucleus thus has zero excitation energy: E, = 0. All excited states and some ground states are unstable. Excited states decay by emitting a y-ray, or a nuclear fragment, or by emission or capture of an electron. A ground state may decay in all these ways except by emitting a y-ray. The mean length of time that a state exists is called its lifetime (see Chapter 2.5). When the lifetime of a state is extremely short (<10-I6 sec) it is difficult to measure it directly. Instead the width of the state is measured. By Heisenberg’s uncertainty principle (At * AE ‘V h) the nuclear width is related to the lifetime of the state* (see, for instance, references 2-5 for further discussion). Neutrons and protons may be regarded as different states of the same particle, called a nucleon.* A generalized description can be used in which a nucleon is identified as either a proton or a neutron by use of a new variable, a charge coordinate variable, called the isotopic spin. I n analogy to the case for mechanical spin, in which a particle system of spin (in units of h ) has two possible values for the z component of angular momentum, a quantum number called the “third component” or “z component,” T,, of isotopic spin is introduced, defined to be -$ for a proton N - Z where N is the and for a neutr0n.i For a nucleus, T, = 2 number of neutrons and Z is the number of protons in the nucleus. The analogy between isotopic spin and mechanical spin also extends to the concept of the i(total’7spin. The total isotopic spin is indicated by the quantum number T . The 2T 1 different possible values of T , associated with a given value of T represent isobars with different values for the difference in neutron-proton numbers, N - 2. For a given TE,T can have the values n ITz]where n = 0, 1, 2, . . . .
J
+
++
~
+
+
* For instance, if the mean lifetime of a nuclear state is 10-19 sec, its width r will ] where T is in seconds. be -7 kev. r = 16.58 X I O - ~ ~ / TMeV, t An alternate definition which is also commonly used defines T, = for a proton and -2j for a neutron. T,is sometimes called the isotopic (or isobaric) spin number. J. M. Blatt and V. F. Weisskopf, “Theoretical Nuclear Physics.” Wiley, New York, 1952. H. A. Bethe and P. Morrison, “Elementary NucIear Theory,” 2nd ed. Wiley, New York, 1956. A. M. Lane, The reduced widths of nuclear energy levels. Atomic Energy Research Establishment T/R 1289 (1954). 6 M. H. MacFarlane and J. B. French, Revs. Modern Phys. 32, 567 (1960); see also J. B. French in Ref. la.
+
354
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
Data on nuclear energy levels are available from many sources.* 2.7.2.1. Determination of Excitation Energies. Excitation energies of nuclear states can be determined in a variety of ways: (1) from nuclear reaction Q values, (2)from the y-decay of nuclear states, (3) from radioactive decay, (4) from neutron threshold measurements, ( 5 ) from the measurement of yields of nuclear reactions and radiative capture. Let us consider briefly each one of these methods and, as an example, let us see how information from each of these sources fits together to give the excitation pattern of a nucleus. (1) Nuclear reaction Q values. Let us assume that a target nucleus X is bombarded by particles a yielding a residual nucleus Y and emitting particles b. The Q of the reaction depends upon the mass of the residual nucleus which in turn depends upon the excitation of Y . If the reaction leads to Y in its ground state the Q value will be the largest possible for the X(a,b)Y reaction [since my will be smallest, the difference ((mx ma) - (my mt,)}(= Q) will be largest]. Let us call this particular reaction energy Qo, and let us call Q1and Q2 the reaction energies corresponding to the excitation of Y to the first and second excited states. Then the excitation energies of these two excited states will be, respectively, E,(1) = QO - Q1 and E,(2) = QO - Q Z = (Q1 - Q2) E,(l). What will usually be observed experimentally is the energy or momentum spectrum of particles b at one or more values of 0 (the laboratory angle of emission of b ) and at one or more values of E, (the kinetic energy
+
+
+
* The following list is just a small fraction of the available review literature: Ajsenberg-Selove and Lauritsen,6.1 Burcham,8.0 Dzhelepov,lo Endt and Van der Leun,” Way et aZ.,laJ8J4Strominger et al.16 * F. Ajzenberg and T. Lauritsen, Nuclear Phys. 11, 1 (1959). 1 F. Ajsenberg-Selove and T. Lauritsen, Ann. Rev. Nucl. Science 10, 409 (1960). W. E. Burcham, Progr. i n Nuclear Phys. 2 (1952). W . E. Burcham, i n “Handbuch der Physik-Encyclopedia of Physics” (S. Flugge, ed.), Vol. 40. Springer, Berlin, 1957. loB. S. Dehelepov, Zzvest. Akad. Nauk S.S.S.R., Ser. Fiz. 17, 391 (1953); 18, 523 (1954); B. S. Dzhelepov and L. K. Peker, “Decay Schemes of Radioactive Nuclei,” Academy of Sciences of the U.S.S.R.,Moscow, 1958. 11P.M. Endt and C. Van der Leun, Nuclear Phys. 34, 1 (1962). l*K. Way, R. W. King, C. L. McGinnis, and R. Van Lieshout, Nuclear level schemes: A = 40 - A = 92, T.I. D.-5300 (1955). 18 I (. Way, N. B. Gove, C. L. McGinnis, and R, Nakasima, in “Energy Levels of Nuclei,” Landolt-Bornstein Tables, Vol. 1. Springer, Berlin, 1961. 14 K. Way, et al., Nuclear data cards. Natl. Acad. 8ci.-Natl. Research Council, Washington, D.C.; and yearly cumulations in December 31 issues of Nuclear Sci. Abstr. l 6 D. Strominger, J. M. Hollander, and G. T. Seaborg, Revs. Modern Phys. 30, 585 (1958); J. Scheer, in “Energy Levels of Nuclei,” Landolt-Bornstein Tables, Vol. 1. Springer, Berlin, 1961.
2.7.
DETERMINATION OF NUCLEAR REACTIONS
355
of the bombarding particles). Figure 1 shows a spectrum obtained by Bockelman et d . 1 6 which is typical of the high resolution work of Buechner's group at M. I. T.l7 The reactions studied were the elastic and inelastic scattering of protons and deuterons from B'O: BlO(p,p')B'O and Bl0(d,d')B1O(E,'v 7 Mev and B = 90'). The ordinate is the number of protons and deuterons scattered from the BI0target per 50 pcoul of charge, * The abscissa is the momentum of the protons and deuterons in kilogauss-cm which can readily be converted to kinetic energy since the masses and charges of the scattered particles are known. For the momentum region shown, ten peaks are observed in the proton spectrum and four in the deuteron spectrum. Four of the proton peaks are due to protons elastically or inelastically scattered from B" (the target was not pure B1O but rather a highly enriched mixture of BO ' (-95%) and Bl1) and from C12 (there is almost always some carbon contamination of targets from various sources in the vacuum system). The most intense proton and deuteron groups appearing at the highest momenta observed should be due to the elastic scattering, for which Q = 0 and for which the final state reached is the ground state. From Eqs. (2.7.1.5) or (2.7.1.6) in Section 2.7.1, one can calculate Rb(Eb = E, or Ed)a t the angle observed, go", and therefore the momenta. The groups appear at the calculated momenta. The other groups ascribed to the BIo reactions correspond to inelastic scattering with increasingly negative Q as the momenta of the groups decrease. To obtain the Q value corresponding to the given group, one converts the momentum of the group into kinetic energy, and substitutes the value of the kinetic energy in the Q-value equation. Then (for inelastic scattering) E,(1) = -Q1, E,(2) = - Q2, etc. The excitation energies which were obtained are shown in Pig. 2: Bl0 was excited by protons to states a t 0, 0.72, 1.74, 2.15, 3.58, and 4.77 MeV, and by deuterons to states a t 0, 0.72, 2.15, and 3.58 MeV. The example chosen in this section involved scattering, but any nuclear reaction would be treated in the same way: find the Eb of the various peaks a t a given B and E,, compute the Q values corresponding t o the various Eb, and from the Q values derive the excitation energies of the nuclear states. There are certain complications which have to be considered however. For instance, the identity of the emitted particle must be established. Then one must be certain which peaks are due to the reaction of interest,
* The charge of 1 proton or of 1 deuteron is 1.60206 t- 0.00003 X lO-'3 bcoul; therefore 50 ficoul = 3.12 X 10'4 protons or deuterons hitting the target. 16 C. K. Bockelman, C. P. Browne, W. W. Buechner, and A. Sperduto, Phys. Rev. 82, 665 (1953). 17 W. W. Buechner, Progr. in Nuclear Phys. 6 (1956).
PROTON GROUPS B'O(P.flB'O
Ep=6.92Mev
90'
1
4.77
-
I 215
EXGITATION ENERGY IN Elo IN Mev-
I.
2
0 W
H p
IN KILOGAUSS-CENTIMETERS
FIG. 1. Scattered proton and deuteron groups from Bl0. Groups arising from Bl0 are labeled with the correspondingexcitation energy. Other groups are identified with the contributing nucleus. The abscissa is linear in p, not in H p . For further discussion, see text. [From C. K. Bockelman, C. P. Browne, W. W. Buechner, and A. Sperduto, Phys. Rev. 92, 665 (1953); reproduced by kind permission of Dr. Bockelman.]
2.7.
DETERMINATION OF NUCLEAR REACTIONS
357
and which to impurities in the target. One way of finding out is by varying E, or 0, or both, and seeing how Eb vanes with these parameters. The variation of EI,with E, and B depends on the mass of the interacting target nucleus. However, since the difference in the variation of the EL1s for nuclei differing by only a few mass units is small, it is necessary to carry out very precise measurements of energies or of momenta. A third point is that the resolving power of the detection equipment must be sufficiently high that states that are closely spaced can be resolved. Otherwise the excitation energies which are quoted are essentially meaningless where such close spacing exists. A fourth point is that in order to calculate the Q value correctly, enough must be known about the factors contributing to the width of the groups (target thickness and uniformity, spread in incident energy, etc.) that it is meaningful to state that a certain point on a peak corresponds to the mean energy of the particle group. (2) Gamma decay of nuclear states. When an excited nuclear state decays, it may do so in a variety of ways, depending upon its excitation energy and quantum-mechanical character. If the excitation energy of the state is less than the binding energy of a proton, deuteron, triton, He3, or n-particle in the nucleus, that is if the nuclear state is bound, then it may decay only by y or p emission, with the former usually much more probable. If a state is unbound it may decay by y-emission or by emitting one of the particles with respect to which it is unbound; the probability of particle emission is generally dominant when it is possible at all. In either case the y-decay need not proceed directly to the ground state but may proceed to an excited state of lower energy which in turn decays in some fashion. A decay which proceeds in such steps is called a cascade. Whether a cascade or a direct ground-state transition occurs depends on the energy differences and on the J , T , and T differences between the parent excited state, the intermediate state(s), and the ground state. Let us assume that we have bombarded a Be9 target with 1.2-Mev deuterons and that we have measured the energies and intensities of all y-rays which appear: E, = 0.414 (14), 0.472 (15),0.717 (56), 1.022 (13), 1.433 (8.7), 2.152 (2.9)) 2.871 (4.6), 3.380 (6.2), and 3.604 (1.5) Mev where the numbers in parentheses are the relative intensities of the y-lines.'* The first problem is to determine in which residual nuclei these transitions take place. With 1.2-Mev deuterons, the possible reactions6 are Beg(d,p)Be1"(to the ground and first excited states of Be'"), Beg(d,n)B'" (to the levels with E , 5 5 Mev), Beg(d,a)LiT (to the levels with E, 5 7 Mev), Beg(d,d)BeQ(to the ground state of Be9) and Beg(d,t)Beg (to the 1s V. K. Rasmussen, W . F. Hornyak, and T.Lauritsen, Phya. Rev. 76, 581 (1949).
358
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
ground and first excited states of Bes). [The Beg(d,Hea)Li8reaction is extremely endoergic.] The reactions leading to the ground states do not yield y-ray lines (since a ground state cannot decay by y-emission), nor does the Beg(d,a)Li7 reaction to the states higher than the first excited state at 477 kev (these states are unbound and are known to decay by particle emission), nor does the Beg(d,t)Bes reaction to the first excited state which is known to decay by a-emission. This leaves the Bes(d,p)Bel0 reaction to the 3.37-Mev state (which accounts for the 3.38-Mev y-ray), the Beg(d,a)Li7reaction to the 477-kev state (thus the 0.472-Mev y-line is ascribed to this reaction) and the Beg(d,n)BIOreaction which accounts for the remainder of the observed y-rays assuming that there were no impurities in the target. Somehow all the observed y-ray transitions must be fitted into a consistent energy level picture of BIO. If particle groups have been observed (see Section 2.7.2.1.1), then it becomes relatively simple to fit the y-rays as cascade or direct transitions. (However, in this example, the 1.433-Mev y-ray could be due to a transition between the 3.58- and 2.14-Mev states, or between those a t 2.14 and 0.72 MeV, or between both.) If particle groups have not been observed with sufficient accuracy, which is often the case, then it becomes necessary to determine which y-rays are in coincidence. (If a 1.4-Mev y-ray is in coincidence with a 0.7-Mev y-ray, then it may be inferred that there are two states 2.1 Mev apart, between which the decay proceeds, a t least part of the time, via a state 0.7 Mev different in energy from one of the two.) Even if particle groups have been established, it is extremely useful to observe the y-decay of nuclear states since the relative transition probabilities, derived from the relative intensities, help to determine the J , T, and T of the states. In some cases also the excitation energy may be more accurately known from a measurement of E, than from a low resolution particle energy measurement. However, the E, may not determine directly the excitation energy (E, # I?,): if a nucleus radiates (emits a y-ray) before it has significantly decelerated in the target material, the average E , may be larger than the E, by as much as several per cent. Unless the lifetime of the radiating state is known, one does not know whether to apply the Doppler correction which takes this effect into account (see Thomas and Lauritsenlg). This may then lead to an appreciable uncertainty in E,. On the other hand if E, is known very accurately, from particle measurements for example, the comparison of E, with E , will give information on the lifetime of the state. For further discussions of the problems of y-ray spectroscopy, the reader is referred to Section 2.2.3 of Vol. 5 A, and Siegbahn’s excellent book.20 R. G. Thomas and T.Lauritsen, Phys. Rev. 88, 969 (1952). K. Siegbahn, cd., “Beta- and Gamma-Ray Spectroscopy.” Interscience, New York, 1955. l8
Po
2.7.
DETERMINATION
OF NUCLEAR REACTIONS
359
Our best knowledge of the low-lying states of nuclei comes from the preceding two methods. The low-lying states are of special interest since they are presumably due to fairly simple configurations which may be amenable to study and comparison with theories of nuclear structure and nuclear forces. (See, for instance, Inglis,21 Mayer and Jensen,22 Elliott and Lane,z3 Alder el aZ.,24 Heydenburg and Temmer,26 and reference la.) ( 3 ) Radioactive decay. If a nuclear state decays by 0- or p+ emission, information on the location and character of states of the daughter nucleus may be obtained. As we have seen (Section 2.7.1.4), in 0- decay Q = (mx- m y )* c2 where mx and my are the atomic masses of the parent and daughter states, and in @+decay
Q
=
(mx- m y ) c2 - 1.02 Mev
If we can measure the maximum kinetic energy (the “endpoint”) of the @-particles,* we can determine the mass differences between parent and daughter states and thus determine the location of the daughter relative to the parent state. For instance, in the b+ decay of C1o[C1o-+ Bl0 p+ 4 the positrons are observed27 to have maximum energies of 2.1 and 1.1Mev (kO.1Mev) with relative intensities of 98.4 and 1.65%. These maximum energies are, by definition, the Q values corresponding to transitions to states in the daughter nucleus B10. Let us consider the most intense transition with Ep+(max) = 2.1 MeV. The mass difference between the ground state of CIO and the B’O state to which the transition 1.02 = 3.1 Mev. However, the mass difference takes place is 2.1 between the Cl0 and Bl0 ground states is known to be 3.62 MeV. Therefore, the intense transition is to the B10 state at 0.72 MeV. In the same way one can calculate that the other transition is to the state a t 1.74 MeV. The fact that the ground-state transition is not observed indicates that
+ +
+
* The difficulties in dctermining the endpoint energies in a complex spectrum (one involving more than one group of 8-rays) can be considerable, particularly if some of the groups are of low intensities: see, for instance, A. C. G. Mitchell in Siegbahn20 and C. S. Wu.28 See also Evans.28 D. R. Inglis, Revs. Modern Phys. 26, 390 (1953). 21 M. G. Mayer and J. H. D. Jensen, “Elementary Theory of Nuclear Shell Structure.’’ Wiley, New York, 1955. 23 J. P. Elliott and A. M. Lane, “Handbuch der Physik-Encylopedia of Physics” (S. Flugge, ed.), Vol. 39. Springer, Berlin, 1957. 2 4 K. Alder, A. Bohr, T. Huus, B. Mottelson, and A. Winther, Revs. Modern Phys. 28, 432 (1956). 26 N. P . Heydenburg and G. M. Temmer, Ann. Rev. Nuclear Sci. 6 , 77 (1956). Z6 C. S. Wu, Revs. Modern Phys. 22, 386 (1950). 2’ R. Sherr and J . B. Gerhart, Phys. Rev. 91, 909 (1953). a* R. D. Evans, “The Atomic Nucleus.” McGraw-Hill, New York, 1955.
360
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
the spin (angular momentum) difference between t,he Cl0 and Bl0 ground states is large: AJ is in fact known to be 3 which makes the transition highly unlikely. If one did not already know the value of the mass difference of the two ground states, one might ascribe the intense transition to the decay to the ground state of BIO and from this derive an incorrect representation of the low-lying levels of Bl0. For this reason the y-rays from the decay of the daughter states reached in &decay are usually studied. In particular, measurements of P-y coincidences give unique answers to the level schemes: if one observes that /3+ with EB+(max)= 2.1 MeV are in coincidence with 0.72-Mev y-rays, one knows immediately that the transition was to an excited state, 0.72 Mev above some other state. Thus p-measurements often do not yield easily the excitation energies of nuclear states. It is rather the y-measurements which locate the states. However, the transition probabilities are an excellent source of information on the J and K of nuclear states. In K capture, the y-rays from the decay of excited states of the daughter nucleus may lead to information on level structure. In a-decay, the mass difference between parent and daughter state Q = E,[1 (m,/mx)] where E , is the kinetic energy of the a-particle in the laboratory system (usually determined by measuring the momentum of the a-particle in a magnetic field), m, is the atomic mass of an a-particle and mx is the atomic mass of the daughter state. The transition probabilities here again indicate the J,K difference between the two states involved in the transition: emission of a-particles with low orbital angular momentum, I , is favored (see Evans,28 for instance). (4) Neutron threshold measurements. One of the methods for measuring the energies of excited states involves an ingenious threshold technique, which in the last few years has yielded much interesting information. If Be9 is bombarded with deuterons, one of the possible reactions is Beg(d,n)BIOwith a ground-state Q value of 4.363 MeV. At any deuteron energy, there will be monoenergetic groups of neutrons corresponding to the transitions to various states of B10. The most energetic neutrons will correspond to the ground-state transition, the least energetic to the transition to the highest excited state which can be reached at the particular value of the deuteron energy and a t the particular angle of observation. Let us assume that there is a state of Bl0 a t 5.93 MeV. The Q of the Be9(d,n)B10reaction to that state is -1.57 Mev (4.363 5.93 Mev). The threshold energy for reaching this state is then [(mBeg rnd)/mBs*]( -&) which is 1.92 MeV. Therefore, if we bombard Be9 with deuterons having an energy, for instance, of approximately 1.93 to 1.98 MeV, we should detect more slow neutrons (corresponding to transitions to the 5.93-Mev state) than just before the 1.92-Mev threshold.
+
+
2.7.
36 1
DETERMINATION OF NUCLEAR REACTIONS
t
FIG.2. Simplified energy level diagram of Bl0. T h e excitation energies of t h e levels are indicated in Mev relative to the ground state. y-ray transitions are shown by arrows leading t o the levels t o which the transitions take place. The binding energies of a proton and an a particle in BIO are shown on the diagram: (Beg p ) - BIO = 6.587 Mev and (Li6 a) - Bl0 = 4.463 MeV. (All t h e numbers on the figure are energies in Mev.) Note that theE,shown on the diagram above (Be9 p ) are in the laboratory system and that they have t o be converted t o t h e center-of-mass system t o yield (9/10) (0.993)]. While the numericalvalues the excitation energies [i.e., 7.48 = 6.587 given for the particle energies are for the laboratory system, the plot is in terms of the corresponding energies in the center-of-mass system. When deuterons of 1.2-Mev kinetic energy ( L system) bombard Beg, transitions are observed t o the first four excited states of B'O by observation of t h e neutron groups t o these states and their subsequent 7-decay. More highly excited states of BIO have, for instance, been observed by measuring neutron thresholds in the Beg(d,n)BIO reaction. The observed curve is shown on the diagram with the L system energies of the incident deuterons displayed. Here again the excitation energies are obtained by adding t o the Q of the reaction the center of mass equivalent of t h e laboratory energy. The mass difference Clo - B*o is 3.62 MeV. The maximum energy of the positrons t o the first excited state of B1o is 1.92 Mev (the mass difference between Cl0 and BIO* is 3.62 - 0.72 = 2.94 MeV; therefore EB+(to the 0.72-Mev state) = 2.94 - 1.02 = 1.92 Mev).
+
+
+
+
362
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
The ratio of slow to fast neutrons increases a t approximately the threshold energy. Details of the method are given, for instance, in a paper by Bonner and Butler.29 A typicaI curve is shown in Fig. 2. This curve taken from the work of Bonner and Cooknoshows thresholds (at the upper left of the figure) corresponding to levels in Bl0 at 4.78, 5.11, 5.17, 5.93, 6.06, 6.16, 6.43, and 6.57 MeV. [On the figure, the deuteron energies are kinetic energies in the laboratory system: to find E, add (11/9)Ed to the Q value of the Be9(d,n)Bl0reaction.] The usefulness of the method lies in the fact that it can lead to the precise determination of excitation energies in regions of nuclei which would be difficult to explore with high resolution in charged particle reactions. The limitations of the method are the restriction to levels with excitation energies greater than the Q of the reaction used, and the restriction to states whose formation involves small changes in orbital angular momentum I (neutron thresholds which would involve large 1 transfers to the neutrons are usually extremely weak near threshold; the threshold cannot be easily located, and may not be detected a t all). Wide levels are also often difficult to measure by this method. (5) Yield measurements. The method which has yielded the determination of the excitation energies of more levels than any other method involves the measurement of the yields of nuclear reactions. Let us consider again a reaction of the type X(a,b)Y. If the yield of particles b is measured at a given t9 as a function of E,, then there may be observed values of E, for which the yield of particles b is much different than the yields for neighbouring values of E,. If these anomalies are maxima* in the yield curve, they are called resonances, the particular values of Ea for which the maxima occur being called resonant energies. The resonances correspond to energy levels in the compound nucleus ( X a). If we call S the compound nucleus formed when a hits X , then the excitation energy of the level in nucleus S corresponding to a resonance a t E, (kinetic energy of a in the laboratory system) is equal to the sum of the binding energy of particle a in nucleus S, and its kinetic energy in the center-of-mass system:
+
Ex(S) = ((mx
+ ma) - ms] c2 + [mx/(mx + ma)lEa. *
When Be8 is bombarded with protons of varying kinetic energies, observations of the yields of emitted particles and of y-rays leads to the location of states of Bl0 above 6.587 Mev (the binding energy of a proton in
* Broad maxima may not be due to energy levels in the compound nucleus. See, for instance, Blatt and Weisskopf.2 40 3O
T. W. Bonner and J. W. Butler, Phys. Rev. 88, 1091 (1951). T.W.Bonner and C. F. Cook, Phys. Rev. 96, 122 (1954).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
363
B’O). For instance, in the range E, = 0.8 to 1.2 MeV, the yield of yradiation with E, > 5.4 Mev reveals two resonances a t E, = 0.993 and 1.085 MeV. These correspond to levels a t 7.48 and 7.56 MeV. The determination of the correct resonance energy depends primarily on the correct determination of the incident energy. This in turn depends on absolute voltage calibrations for electrostatic generators, such as the ones carried out by Herb et aL3I (see G o v ~ ~ ~ ) . It is obviously impossible to discuss here the complexities of compound nucleus formation : the reader is referred to Wigner,a3Blatt and Weisskopf,2 and Bethe and M ~ r r i s o nFor . ~ a discussion of typical data analysis problems, see, for instance, Christy34 and Gove,a2 and for reviews of data see Bar~cha11,~~ Hughes et aZ.,36 and Jarmie and Seagrave.37 2.7.2.2. The Meaning of Isobaric States. One of the most illuminating ways of present energy level information is by means of isobar diagrams. These are level diagrams of all the isobars of a given mass number in which the relative positions of the ground state masses are corrected for “non-nuclear” effects. The mass of a nucleus is in part determined by nuclear effects, such as the arrangement of the neutrons and protons within the nucleus, and is in part due to non-nuclear effects such as the Coulomb energy contribution and the effect of the unequal masses of neutrons and protons. For instance, the atomic mass difference between 6B12and &!12 is measured to be 13.369 MeV. This number is, however, not a true gauge of the mass difference between these two nuclei due to inherently nuclear causes: (1) BIZ has one neutron more than C12. I n order to correct for the resulting increase in mass which is irrelevant from the nuclear point of view, we would have to subtract 0.783 Mev (the atomic mass difference between a neutron and a proton) from the mass of B12. Thus the B12 - C12 mass difference adjusted for this first effect would be 783 kev smaller. (2) GI2 has one proton more than B12. Therefore its inherent Coulomb energy is higher than that of B12. The Coulomb energy depends on the distribution of charge in the two nuclei. The simplest calculation involves the assumption that charge is uniformly distributed in nuclei. In that case, the Coulomb energy difference between R. G. Herb, S. C. Snowdon, and 0. Sala, Phys. Rev. 76, 246 (1949). H. E. Gove, in “Nuclear Reactions” (P. hi. Endt and M. Deumeur, eds.), Vol. I. North-Holland Publ., Amsterdam, 1959. 33 E. P. Wigner, Am. J . Phys. 23, 371 (1955). 1 4 R. F, Christy, Physicu 22, 1009 (1956). 36 H. H. Barschall, Am. J . Phys. 18, 535 (1950). 36 D. J. Hughes and R. B. Schwartr, Neutron cross sections. BNL-325, seconcl edition (1958); D. J. Hughes, B. A. Magurno, and M. K. Brussel, ibid. Suppl. 1 (1960). 37 N . Jarmie and J. D. Seagrave, Charged particle cross sections. LA-2014 (1957) ; D. B. Smith, N. Jarmie, and J. D. Seagrave, LA-2424 (1960). 3’
31
364
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
two nuclei with atomic numbers Z and 2’ and mass number A is 3 e2 5r
- - [Z(Z
- 1) - Z‘(Z’ - l ) ]
where e is the electronic charge and T is the nuclear radius. This reduces to
[ Z(Z - 1) A-
Z’(2’ - 1)
1/9
] (0.596) Mev
(2.7.2.1)
where r is taken to be 1.45 X 10-13A1’3cm. From Eq. (2.7.2.1), the C12 - B12 Coulomb energy difference is 2.60 MeV. Since the Coulomb energy of CI2 is the larger of the two, the mass difference B12 - C12 will be increased by 2.60 Mev when the Coulomb energy difference is taken into account. Thus the atomic mass difference BI2 - C12 from which the two non-nuclear effects have been removed as above is (13.369 - 0.783
+ 2.60) Mev = 15.19 MeV.
This number reflects the energy difference due to the different arrangements of the nuclear constituents in the B12 and C12ground states. The isobaric correction (the Coulomb energy difference less the n-p mass difference), which in the case of B12 - C12 is 2.60 - 0.783 = 1.82 MeV, is always subtracted from the mass of the isobar with the larger 2. Figure 3 shows the isobaric diagram for A = 12. The ground-state masses of B12, C12, and NI2 have been adjusted in the manner just discussed: the ground state of B12 is plotted 15.19 Mev above the ground state of C12and the energy level diagram of BlZthen starts a t 15.19 Mev relative to the ground state of C12.* The first thing that may be noticed is the striking agreement between the intrinsic mass differences B12 - C12 and N12 - C12 which we have just calculated (crudely) to be, respectively, 15.19 and 15.11 MeV. This shows that the intrinsic masses of B12 and N12 are closely the same: thus two systems, in their lowest mode of excitation, one made up of 7 neutrons and 5 protons (BIZ) and the other of 5 neutrons and 7 protons (N12), have the same mass if we do not consider the electrical energy provided by the charge of the extra two protons in N12 and the mass-energy provided by the added mass of the extra two neutrons in BIZ.The difference, from the nuclear point of view, between B12 and N12 lies in the additional p-p bonds in N12 and the additional n-n bonds in BIZ. The equivalence of the masses of the two nuclei implies that the interactions
* The isobaric correction for Nl2 - C1*is 2.34 MeV. Since the ,N’* - ,C1?mass difference is 17.45 MeV, the intrinsic Nla - C12mass difference is 15.11 MeV, and the N1* energy level diagram starts 15.11 Mev above the ground state of Cia.
2.7.
DETERMINATION OF NUCLEAR REACTIONS
365
between neutrons and neutrons and between protons and protons are very closely the same. This hypothesis is reinforced by the similarity in the level structures of B12 and N12. The close similarity in level structures of mirror (or Wigner) nuclei (isobars whose 2 and N can be interchanged) has been well verified (see, for instance, Lauritsen and Ajzenberg-Selove,r Inglis,21 and B ~ r c h a r n , ~ ~and 9 ) is a proof of the charge symmetry of nuclear forces (n- = p - p ) for the relatively low-energy interactions encountered in nuclei.
FIG.3. Isobaric diagram of the A = 12 tria
See text for discussion.
The isotopic spin numbers T , of B12, C12, and N12 are, respectively, + 1 , 0, and -1. In the excitation region for A = 12 shown in Fig. 3, levels of B12 and N12 may only carry an isotopic spin T of 1, while levels in C12 may have either T = 0 or T = 1. In C12, levels with T = 0 may occur throughout the region shown. Levels with T = 1 are observed to occur for E, 2 15.11 MeV. The levels shown a t 15.11, 16.10, 16.57, and 17.22 Mev have been determined to be T = 1 states. It should be noted that their energies relative to the ground state of CI2 are in excellent agreement with the relative energies of the ground and first few excited states of B12and N12 (for a given energy the analogous levels in the three nuclei are said to be part of an isotopic spin multiplet). The fact that the T = 1 states in an isobaric triad such as BIZ,C12, and N12 are in good agreement with respect to mass (see Wilkinson3*) and character (where known) is a strong argument for the charge independence of nuclear 3* D. H. Wilkinson, Phil. Mug. [S]1, 1031 (1956).
2.
366
DETERMINATION OF FUNDAMENTAL QUANTITIES
forces: the equivalence not only of n-n and p-p bonds (as shown in mirror nuclei) but also of n-p bonds (Ci2has an n-p pair outside the 5 proton, 5 neutron common core, whereas N12 and B12 have, respectively, a p-p and an n-n pair). In addition to the T = 1 levels, C12, a T, = 0 nucleus, has a large number of T = 0 states. N12 and B12 can have no T = 0 levels, since T must be a t least as large as IT.1, which is equal to 1 for these two nuclei. While charge independence is generally believed to hold in nuclear forces, the substantiating evidence for it from level structure is less reliable and less plentiful than the evidence for charge symmetry. Additional work is required to determine precise excitation energies in nuclei and, particularly, to determine the character of the states: J , T, T , and reduced widths. Detailed comparisons require also knowledge of charge distributions in nuclei. The coulomb correction based on the uniform model is only a first approximation. More careful calculations have, for instance, been made by Carlson and Talmi.3e I n addition, there is no reason to believe that the Coulomb corrections to the energies of excited states should be the same as those to ground states (for instance because of different radii for excited and for unexcited nuclei). A more rigorous treatment of the material in this section is given, for instance, by Blatt and Weisskopf,2 Bethe and M ~ r r i s o n ,B~ u r ~ h a m , ~ ~ and MacDonald.'*
2.7.3. Total Interaction Cross Sections* 2.7.3.1. Methods and Results for Monoenergetic Fast Neutrons. One of the simplest and most straightforward of all nuclear cross sections to measure, a t least in principle, is the total interaction cross section. At the same time, the accuracy of total cross-section measurements can be made quite high. The basic reason for this simplicity and relatively high accuracy is that no absolute measurement of a particle flux is required. Instead, using transmission-type experiments, only the number of nuclei per cm2 in a target sample and relative flux measurements are necessary. In general the cross section for any nuclear process is defined to be the number of events of the appropriate type occurring per unit time per 18 do
B. C. Carlson and I. Talmi, Phys. Rev. 96, 436 (1954). W. E. Burcham, Progr. in Nuclear Phys. 4 (1955).
__
* Sections 2.7.3-2.7.7
are by Louis Rosen and Dan W. Miller.
2.7.
DETERMINATION OF NUCLEAR REACTIONS
367
nucIeus per unit incident flux. In most experiments designed to measure cross sections for a specific nuclear process, the experiments are complicated by (a) the necessity of measuring only events of the appropriate type, (b) the necessity of determining absolute numbers of these events occurring per unit time per nucleus in the target, (c) the necessity of measuring the absolute flux of incident particles accurately, and (d) the requirement implicit in the definition that the flux remain essentially constant over all of the nuclei included in the target sample. The experimental problem is greatly simplified when the cross sections for all types of interactions are lumped together into the total cross section, for then it is only necessary to measure the rate of flux decrease in the sample regardless of the specific processes responsible for removing the particles from the beam. This is most easily accomplished by a simple transmission experiment, in which the particle flux a t the detector is compared with and without a target sample interposed between the particle source and the detector. The number of interactions per cm2per second which occur in a layer “dx” of the sample along the beam direction, i.e., the decrease in flux ( - d l ) in the layer, is given by the definition as -dI = Inat dz where n is the number of target nuclei per cm3 and ut is the total cross section. A simple integration shows that the ratio of the flux a t the detector with the sample interposed (I) to the flux a t the detector with the sample removed ( l o ) is given by:
T
I = lo
e-nto,
(2.7.3.1)
Here t is the total thickness of the sample and T the “transmission” of the sample. This familiar formula shows an exponential decrease of the exit flux with increase of sample thickness. It is important to note that Eq. (2.7.3.1) is valid only if two assumptions inherent in its derivation are complied with: (a) every interaction occurring between an incident particle and a target nucleus is assumed to remove the incident particle from the experimentally observed exit flux, and (b) the energy spread of the incident particle beam is assumed small compared to energy intervals in which the total cross section varies appreciably. Coneerning the first assumption, some possible types of interactions do not remove the incident particle from the beam. For example, the slowing of an a-particle in matter is primarily due to repeated interactions with orbital electrons, each of which produces energy loss of the alpha but does not remove it from the beam. The
368
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
over-all flux of alphas in a sample material then obviously does not decrease exponentially. I n the case of neutrons interacting with nuclei in a sample, assumption (a) will in general be fulfilled provided precautions are taken by the experimenter to minimize the possibility of forward-scattered neutrons being counted as if they had not experienced an interaction, and to correct for those that are so misinterpreted by any detector of finite size. Assumption (b) is also required to insure exponential absorption. If the cross section varies rapidly over the energy region included in the energy spread of the incident beam, each energy increment of the incident-particle spectrum will be absorbed exponentially, but with a different value of ut. The absorption of the entire flux will then no longer be a simple exponential. This is discussed further in Section 2.7.3.1.5. Rewriting Eq. (2.7.3.1), one obtains the familiar expression used to determine the total cross section from a transmission type experiment: ut = (l/nt)ln(l/T).
(2.7.3.2)
A discussion of the accurate experimental determination of the quantities T , the corrected transmission of the sample, and nt, the number of nuclei per cm2 of the sample, is given in Sections 2.7.3.1.3-2.7.3.1.5. Although total nuclear interaction cross sections can be determined for incident charged particles, these measurements have usually been performed at energies higher than the range covered in the present discussion. We will, therefore, confine our considerations to the determination of neutron total cross sections only. 2.7.3.1.1. BASICEXPERIMENTAL ARRANGEMENT. As already implied, the three basic components required in a simple transmission experiment for determining neutron total cross sections are (a) neutron source, (b) sample in which the neutron interactions occur, and (c) neutron detector. The sample is located between the source and detector so that it completely shadows the detector from the source. These three components, particularly the source and detector, vary markedly according to the neutron energy at which the measurements are desired. The basic arrangement, however, still is the same. A typical experimental arrangement, used by Barschall and collaborators at the University of Wisconsin for total cross-section measurements in the 1-3 Mev region, is shown in Fig. 1.' The neutron sources employed involve either charged-particle accelerators or nuclear reactors. These are discussed with reference to their use in determining neutron total cross sections in Sections 2.7.3.1.2 and R. E. Fields, R.L. Becker, and R.K. Adair, Phys. Rev. 94, 389 (1954).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
369
2.7.3.3.1. Sample size, position, and composition depend on various factors which are discussed in Sections 2.7.3.1.3 and 2.7.3.1.5. Neutron detectors are considered with regard to their application to total crosssection measurements in Sections 2.7.3.1.4 and 2.7.8.3.1. ALUMINUM TUBE 0.005" WALLTHICKNESS
-
7
/
- SAM.PLE,$DIA.
PROTON BEAM
NEUTRON SOURCE TUNGSTEN DISC
LIMITING APERTURES, PROTON BEAM.
1
HYDROGEN RECOIL COUNTER -l
0.15" DIA.
I* DIA.
FIG. 1. Typical good-geometry transmission geometry' for determining neutron total cross sections.
2.7.3.1.2. MONOENERGETIC SOURCES. * t Although neutron sources are discussed in detail in Section 3.2.2.4, it might be well to discuss briefly t,he sources used in various energy ranges and some of their experimental difficulties, particularly with regard to the corrections to the cross-section measurements. 1 kev-Id0 kev. The standard nuclear reaction employed for electrostaticgenerator measurements of the total cross section in the range from 1-120 kev is the Li'(p,n)Be7 reaction. This reaction is endoergic, with a threshold proton energy of 1.881 As a consequence of the centerof-mass motion a t threshold, two groups of neutrons are observed in the forward direction just above threshold. As the proton energy is raised, the laboratory energies of these two groups split further apart until the lower energy group disappears from the forward direction and reappears in the backward direction. At this point, the energy of the high-energy neutron group in the forward direction is 120 kev. It is not feasible t,o
* Refer to Sections 2.8.2.1 and 3.2.1.4, Vol. 5 B.
t See Section 2.7.7 for more recent supplementary remarks. 2 K. W. Jones, R. A. Douglas, M. T. McEllistrem, and H. T. Richards, Phys. Rev. 94, 947 (1954).
370
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
use the Li7(p,n) reaction in the forward direction for neutron energies below 120 kev because of the two neutron groups present. Instead, the usual procedure is t o employ neutrons emitted in a backward direction, usually at a laboratory angle of 120’. Here there will be a single neutron
HARCOAL TRAP
FIG. 2. Typical experimental arrangements for “back-angle” total cross-section measurements using Li?(p,n) neutrons.
group produced, starting a t zero neutron energy at a proton energy appropriate to the angle employed, and increasing in energy as the proton energy is raised. Figure 2 shows a representative experimental arrangement used by Gibbons8 for this type of measurement. Other (p,n) reactions can be employed using heavier target nuclei, J. H. Gibbons, Phya. Rev. 102, 1574 (1956).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
37 1
which result in smaller center-of-mass motion effects. As a result, the forward group of neutrons becomes monoenergetic at neutron energies around 1 kev, which provides certain advantages over the Li7(p,n) r e a ~ t i o n . ~However, -~ because of fluctuations in yield, etc., these reactions have not yet been widely used as neutron sources. 120 kev-600 kev. Above neutron energies of 120 kev, the Li7(p,n) reaction is normally employed using transmission experiments set, up in the forward direction relative to the proton beam. By raising the bombarding proton energy, the neutron energy in the forward direction also increases in a unique fashion determined by the kinematics of the nuclear reaction. However, the presence of an excited state at 430 kev in the residual nucleus Be7* causes a low-energy group of neutrons to appear above proton energies of 2.38 Mev.? As a consequence, above neutron energies of about 600 kev (for the primary group), one must either use anenergysensitive neutron counter to bias against the low-energy group, or correct for the contribution of the low-energy group to the transmission measurements. 0.6 Mev-4 MeV. The endoergic HS(p,n)He3 reaction is generally used for the production of monoenergetic neutrons in the energy range from 0.6-4 MeV. This reaction has a threshold a t a bombarding proton energy of 1.019 MeV, and yields monoenergetic forward neutrons from E , = 288 kev4 up to 5 Mev or more, since no excited states of He3 are observable. 4 Mev-9 MeV. I n this energy region, the standard reaction is the exoergic H2(d,n)He3reaction, with a Q value of +3.265 MeV.* This reaction is beset with certain experimental difficulties characteristic of deuteron-induced neutron sources, due primarily to the extraneous production of neutrons from the beam d i r e c t i ~ nThese .~ will be discussed briefly later in this section and in Section 2.7.3.1.5. Additional reactions useful in this energy region are the Bee(cu,n)C12, C13(cu,n)01B,and N14(d,n)O16reactions.sp10The ( c u p ) reactions are not 4 J. B. Marion and Betty Allen, A tabulation of neutron energies from various charged-particle reactions. Shell Development Company Report, August, 1955.Houston, Texas. R. M. Brugger, T. W. Bonner, and J. B. Marion, Phys. Rev. 100, 84 (1955);J. H. Gibbons, R. L. Macklin, and H. W. Schmitt, ibid. 100, 167 (1955). T. W.Bonner, Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1966 4,92 (1956). V. R.Johnson, M. J. Laubenstein, and H. T. Richards, Phys. Rev. 77, 413 (1950). D.M.Van Patter and W. Whaling, Revs. Modern Phys. 26, 402 (1954). OA. 0. Hanson, R. F. Taschek, and J. H. Williams, Revs. Modern Phys. 21, 635 (1949). l o R . L. Becker and H. H. Barschall, Phys. Rev. 102, 1384 (1956).
372
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
beset with the background problem characteristic of the deuteroninduced reactions. However, because of yield difficulties, the C13(a,n) reaction is of practical use only in the neutron-energy range from 4.4 to 5.6 MeV, and the BeQ(a,n)reaction from 7.0 to 8.6 Mev.l0 12 MeV-27 MeV. The highly exoergic HS(d,n)He4reaction [or its alternate H2(t,n)He4]with a Q = +17.578 Mev is used to produce the highest energy neutrons available using electrostatic generator techniques at the present time." The H3(d,n)He4 reaction has a high yield, and is often used with Cockcroft-Walton accelerators to produce 14-Mev neutrons. Neutron energies as low as 12 Mev can be obtained from this reaction by going to a backward angle relative to the beam direction. The H3(d,n) reaction is also beset with background difficulties due to neutron production by the deuteron beam on defining slits, foils, etc. Further, at deuteron energies above 4 MeV, low-energy neutrons are observed'* which presumably are caused by deuteron breakup or by the reaction H3(d,2n)HeS.For this reason there is a t present no satisfactory source of monoenergetic neutrons in the energy range from about 8 Mev to about 12 Mev.13 General source considerations. The most important targets described above are Li', H2, and H3 targets. The lithium targets are normally prepared by evaporation of lithium in a vacuum onto thin tantalum or wolfram backings, while the deuterium and tritium targets may either be in the form of gas cells or absorbed in thin zirconium or other metal layers. It is important that the neutron flux per incident charged particle remain constant during a particular set of transmission and background measurements, so that suitable precautions must be taken to prevent excessive beam heating of the target, wandering of the beam to nonuniform parts of the target, etc. From the standpoint of corrections applied to the cross-section measurements, every effort must be made to minimize extraneous neutron-producing materials in the neutron sources, particularly those employing an incident deuteron beam. Thus the gas cells, for example, usually employ thin nickel beam windows with liquid air traps in the vacuum system to inhibit carbon buildup. The target assembly in all cases is lightly constructed with a minimum amount of neutron-scattering material present. Another important source consideration is the location of the source relative to external neutron-scattering material. Thus it has become standard to locate the experimental area in large rooms with false floors to minimize the scattering of neutrons from the walls and floor into the neutron detector. Nevertheless, in 1*
J. L. Fowler and J. E. Brolley, Jr., Revs. Modern Phys. 28, 103 (1956). R. L. Henkel, J. E. Perry, Jr., and R. K. Smith, Phys. Rev. 99, 1050 (1955). L. Cranberg, A. H. Armstrong, and R. L. Henkel, Phys. Rev. 104, 1639 (1956).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
373
general a very worthwhile precaution is to measure the energy distribution of the source neutrons. Tables have been compiled of the neutron energies available from various, reactions for various incident particle energies and laboratory observation angles.4*11.14 2.7.3.1.3. SAMPLE REQUIREMENTS. Considerable care must be exercised in the choice of scattering samples for transmission total cross-section measurements. l 6 Some aspects of this choice are considered below from the standpoint of their importance to the cross-section determination. The diameter of the sample is chosen just large enough to shadow the active volume of the neutron detector. It is important that the sample not be any larger than necessary, however, because of the increased correction required for inscattering into the detector. This correction can be very important at high-neutron energies, and it is not easy to determine. Therefore, one attempts to minimize the correction by keeping the diameter of the sample and the diameter of the active volume of the detector as small as possible. Typical sample diameters vary from 2.5 to 5 cm. The location of the sample and detector depend upon such factors as the energy resolution required, the neutron flux available, the type of sample (e.g., solid, or gas), inscattering effects, etc. Generally the sample is located midway between the neutron source and the center of the active volume of the detector, since this location minimizes the inscattering correction (see Section 2.7.3.1.5). Suspension of the sample by a thin wire or rod support helps to reduce the background of neutrons scattered into the detector from objects external to the sample. Typical source-to-detector distances vary from 25 to 150 cm. Alignment of the source-sample-detector system is also quite important to prevent a systematic error in the cross-section determination. Any misalignment may cause a portion of the detector to “see” the neutron source directly, without being shadowed by the sample, resulting in a higher apparent transmission. A variety of alignment methods are used; e.g., a point source of light placed a t the source position will cast a shadow of the sample on the detector (e.g., ref. 16), or the crosshairs of a reversible telescope of the same diameter as the sample placed in the sample position can be centered on source and detector visually (e.g., ref. 17). The thickness of the sample along the beam direction is chosen as a compromise. In order to obtain good statistical accuracy readily, a low 14 A. S. Langsdorf, Jr., J. E. Monahan, and W. A. Reardon, Argonne National Laboratory Report ANL-5219 (January, 1954). 16 H. H. Barschall, Revs. Modern Phys. 24, 120 (1952). 16 G. Freier, M. Fulk, E. E. Lampi, and J. H. Williams, Phys. Rev. 78, 508 (1950). Jerry P. Conner, Phys. Rev. 109, 1268 (1958).
374
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
transmission and a thick sample are desired. However, in order to minimize corrections due to inscattering and “hardening” effects (see Section 2.7.3.1.5), a high transmission and a thin sample are required. Optimum values of the transmission run in the region of 50-70%, and sample thicknesses are normally determined by this requirement. In regions of rapidly varying transmission due to compound-nucleus resonance structure, several samples of different thicknesses are normally required. The composition of the sample is usually dictated by chemical considerations. If the element to be studied is available in a pure, solid, chemically nonreactive form, target preparation simply amounts to machining to proper dimensions. Whenever possible, this elemental form should be chosen. If the solid is subject to oxidation or absorbs moisture from the air, the common procedure is to seal it in a thin-walled metal can and compare its transmission with that of an identical empty can. Gases may be investigated either by the use of very high-pressure gas cells (e.g., refs. 18 and 19) or by using liquefied gases in a Dewar (e.g., ref. 20). I n many cases it is more convenient to observe the relative transmissions of a sample made from a compound involving the gas and a sample with the same number of atoms per cm2 of the other part of the compound (e.g., Be0 and Be, ref. 21). Serious difficulties can arise from impurities in samples, and considerable care must be exercised to prevent their presence if possible. Powders, for example, are often prone to absorb moisture, which can strongly affect the measurements. In determining the cross section from Eq. (2.7.3.2), it is necessary to know the number of nuclei per cm2 (“nt”) in the sample accurately. This in turn requires a precise determination of the length and the density of the sample. The accuracy of the length measurement is seldom of concern except in the case of very thin samples or cross-section measurements of very high accuracy, such as the determination of the neutron-proton cross section. However, the density measurement is always of considerable importance due to the possibility of sample variations and inhomogeneities. The density of each sample employed should normally be checked, particularly if it is made from materials like graphite, etc. Hafner et a1.,22discuss such effects in considerable detail. S. Bashkin, F. P.. Mooring, and B. Petree, Phys. Rev. 82, 378 (1951). J. D. Seagrave and R. L. Henkel, Phys. Rev. 98, 666 (1955). P. Hillman, R. H. Stahl, and N. F. Rameey, Phgs. Rev. 96, 115 (1954). SIB. Russell, D. Sachs, A. Wnttenberg, and R. Fields, Phys. Rev. 71, 508 (1947); R. K. Adair, H. H. Barschall, C. K. Bockelman, and 0. Sala, ibid. 76, 1124 (1949). **E.M. Hafner, W. F. Hornyak, C. E. Falk, G. Snow, and T. Coor, Phys. Rev. 89, 204 (1953).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
375
REQUIREMENTS. As suggested earlier, the require2.7.3.1.4. DETECTOR ments imposed upon neutron detectors by the desire to obtain total cross sections accurately depend upon the neutron-energy range involved. Basically, the detector should have a high counting efficiency for neutrons of the desired energy, but be relatively insensitive to y-rays and neutrons of lower energies. Although neutron detectors are discussed in detail in Vol. 5A, a brief summary of types of detectors generally employed for total cross-section measurements in various energy ranges is given below. For transmission measurements in the kev region, it is generally necessary to employ a counter especially designed for this application. I n the back-angle measurements using a Li7(p,n) source, for example, the yield drops greatly below 15 kev with a simultaneous increase in the relative background. For these measurements, special directional B F 3 counters with good efficiency and reduced background have been devel0ped.~,23I n the energy region from 20 kev to a few hundred kev, BFs gas counters have long been used because of their relatively high counting efficiency (see, e.g., refs. 24 and 25). These counters are usually proportional counters filled with enriched B10F3, surrounded by paraffin to increase the fast-neutron sensitivity, and further surrounded by a boroncarbide paraffin mixture and covered with cadmium to reduce the slowneutron sensitivity. Nevertheless, such counters operate with moderately high background counting rates which must be carefully measured. More recently, a number of varieties of scintillation detectors have been developed for this energy range. A summary of their properties and references is given by Muehlhause.2s Gas recoil counters are frequently used in the energy range from about 50 kev to several MeV. These may be operated as ionization chambers or proportional counters using hydrogen or helium a t various pressures up to many atmospheres, depending upon background conditions and neutron energy (see, e.g., refs. 24, 27, and 28). These have the advantage *a C. T. Hibdon, A. J. Langsdorf, Jr., and R. E. Holland, Phys. Rev. 86, 595 (1952); Carl T. Hibdon, ibid. 108, 414 (1957). 24 B. B. Rossi and H. H. Staub, “Ionization Chambers and Counters.” McGrawHill, New York, 1949. 2 S L . W. Seagondollar and H. H. Barschall, Phys. Rev. 72,439 (1947); R. E. Peterson, H. H. Barschall, and C. K. Bockelman, ibid. 79, 593 (1950); T. W. Bonner and C. F. Cook, ibid. 96, 122 (1954). *8 C. 0.Muehlhause, NucZeonics 14, No. 4, 38 (1956). *7H. H. Barschall and M. H. Kanner, Phys. Rev. 68, 590 (1940); C. H. Johnson, H. B. Willard, and J. K. Bair, ibid. 96, 985 (1954). ZSC. K. Bockelman, D. W. Miller, R. K. Adair, and H. H. Barschall, Phys. Rev.
84, 69 (1951).
376
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
over BF3 counters in that energy discrimination is made possible by accepting only pulses greater than a certain bias, corresponding to forward recoils from the group of primary neutrons. Under these conditions the background from scattered neutrons can be made negligible (less than 1 %) without difficulty. Effective discrimination against y rays is also possible with this type of counter. A number of types of scintillation counters have been developed for the neutron-energy range above about 1 MeV. Organic scintillators are efficient and easy to preparelZ9and detector designs have been developed which are insensitive to y-ray~.~O The Rice designa1and various modificat i o n ~incorporate ’ ~ ~ ~ ~ both y-ray insensitivity and neutron energy discrimination. This design consists of spheres or layers of organic scintillator (therefore containing considerable hydrogen) separated by a transparent nonscintillating material (e.g., glass or hydrogen-free fluid). The relative dimensions of the scintillating and inert materials are chosen so that recoil protons dissipate most of their energy in a single scintillator. Further, electrons due to y-rays give smaller pulses than the recoil protons in a single scintillator and stop before getting an opportunity to pass through more than one scintillator. I n this way the background due to y-rays as well as lowenergy scattered neutrons can be largely removed by suitable pulse-height discrimination. 2.7.3.1.5. CORRECTIONS. The procedure normally followed in a transmission experiment is to determine the flux a t the detector with sample in (1’)and the flux at the detector with sample out (lo’)for a sample with a known number of nuclei/cm2. However, a number of corrections must be applied to this raw transmission (T’ ,= I’/lo’) in order to determine the corrected transmission ( T ) for use in Eq. (2.7.3.2). Room-scattered neutron background. This correction must be applied to the raw transmission since part of the neutrons counted, both with sample in and sample out, generally will not come directly from the source but from scattering into the counter from floors, walls, etc. It is highly desirable to minimize this correction by the use of biased energysensitive neutron detectors (such as recoil or scintillation counters, see Section 2.7.3.1.4)or direction-sensitive detectors (see, e.g., refs. 3 and 23). To determine the room-scattered neutron background in total crosssection measurements, the most common procedure is to insert a shadow bar in place of the sample. The material and dimensions of this shadow C. E. Falk, H. L. Poss, and L. C. L. Yuan, Phys. Rev. 83, 176 (1951). W. F. Hornyak, Rev. Sci. Instr. 23, 264 (1952). 81 J H. McCrary, H. L. Taylor, and T. W. Bonner, Phys. Rev. B4, 808A (1954); H. L. Taylor, 0. Lonsjo, and T. W. Bonner, ibid. 100, 174 (1955). 3a H. S. Hans and S. C. Snowdon, Phys. Rev. 108,1028 (1957). 29
$0
2.7.
DETERMINATION OF NUCLEAR REACTIONS
377
bar are chosen to give a negligibly small transmission. For experiments in the range below about 2 MeV, a shadow cone made of paraffin or Lucite is commonly used, whereas at higher neutron energies shadow bars of iron, tantalum, copper, etc. are employed (see, e.g., refs. 33 and 34). Ot,her methods are sometimes employed to determine whether the shadow bar itself cuts out background neutrons. l 6 s a 4 Let the observed flux a t the detector with sample in be 1’, with sample and with a perfect shadow bar in place be PIo‘. Then, if extraout be lo’, neous neutrons from the beam direction can be neglected, the corrected flux (without background) a t the detector with the sample in should be
z
=
I’ - pro’
and with the sample out should be I0
=
la‘ - , N o ’ .
The corrected transmission ( T = 1/13 is then easily shown to be given in terms of the uncorrected transmission (T’ = I’/Io’) by
T
=
(T’ - P)/(l - 8).
(2.7.3.3)
Extraneous neutron background jrom beam direction. Extraneous neutrons from the beam direction pose a fairly difficult correction to evaluate. These neutrons arise from (a) legitimate primary neutrons produced a t the target a t large angles but scattered forward by targetholding equipment and material in the immediate vicinity of the source, (b) neutrons produced (particularly when a deuteron beam is employed) when the beam strikes collimating slits and diaphragms, entrance foils of gas cells, etc. and (c) neutrons produced by deuteron breakup in the source itself. I n all three cases, an ordinary shadow-bar measurement does not give an indication of this type of background because the background comes from the source direction. I n case (a), the effect is minimized by constructing all parts of the target system from as thin material as possible. Some estimate of the maximum magnitude of the effect can be obtained by looking for low-energy neutrons in the direct beam with an energy-sensitive neutron detector, or by considering the angular distributions of neutrons produced in the source, the source geometry, the cross sections of the materials making up the target system, etc.22 To determine the background neutrons, case (b), coming from deuterons striking collimators, gas cell windows, etc., a common procedure is to a* 34
J. H. Coon, E. R. Graves, and H. H. Barschall, Phys. Rev. 88, 562 (1952). H. L. Poss, E. 0.Salant, G. A. Snow, and L. C. L. Yuan, Phys. Rev. 87, 11 (1953).
378
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
remove the actual neutron-producing target and observe the neutrons remaining in the flux. This is particularly convenient when deuterium or tritium gas cells are used, because the background rate can be determined by replacing the deuterium or tritium with hydrogen or vacuum depending upon the test (see, e.g., refs. 13,22, and 35). Care must be used to guarantee that all of the deuterium or tritium has been carefully flushed out before making this measurement. In case (c), deuteron breakup in the source results in a typical continuum of neutrons due to a . spectrum three-body breakup process, e.g., of the type H Z ( d , n p ) H 2 This of low-energy neutrons prevents the H2(d,n) and H3(d,n) reactions from being monoenergetic above deuteron energies of about 4 Mev.’2,18 The effect can be eliminated in total cross-section measurements only by the use of time-of-flight or energy-sensitive detectors capable of biasing out the low-energy continuum. In the back-angle Li7(p,n) total cross-section measurements in the 1-120 kev region, a serious source of extraneous neutrons from the source direction is due to primary neutrons backscattered by the target backing. This is best taken into account by actual measurement of strong isolated resonances with samples of various thicknesses. Correction for finite energy resolution and nonexponential absorption in the thick samples then allows a fairly accurate estimate of this b a~k g ro u n d .~ Inscattering correction. For any transmission geometry with sample and counter of finite size, it is necessary to include a correction for neutrons scattered forward in the sample and counted by the detector as if no interaction had occurred. As already pointed out, this correction can be minimized by using as small a detector as can be tolerated from counting rate considerations, with a sample just large enough to shadow the detector. Such an experiment is referred to as a “good geometry” transmission measurement. The top part of Fig. 3 shows such a geometry, and the bottom part an enlarged diagram of the sample for use in the derivation below. If we assume (a) that the detector is sufficiently energy-sensitive that it will only count elastically scattered neutrons in the forward direction and (b) that the over-all length of the scattering sample t is small compared to both the source-to-sample distance L1 and sample-to-detector distance Lsl then the correction to be applied may easily be derived30 as follows, using Fig. 3. Let n = number of scattering nuclei/cm3, L = L1 + Lz = the source-to-detector distance, Q = number of neutrons produced in the source in the sample direction per second per unit solid angle, D = diameter of cylindrical scattering sample of length t , 86 86
R. B. Day and R. L. Henkel, Phys. Rev. 92,368 (1953). This derivation is similar to that given in ref. 17.
2.7.
DETERMINATION OF NUCLEAR REACTIONS
379
Fro. 3. Geometry for determining inscattering corrections.
cross-sectional area of the detector, u,(Oo) = the differential elastic-scattering cross section for neutrons in the detector direction, ut = true total cross section, and utf = total cross section calculated without applying inscattering corrections. The number of neutrons scattered in the sample between x and x dx which strike the detector is given by, A d =
+
where the factors in brackets represent the flux a t the depth x in the sample, the number of nuclei between z and z dz, the differential cross section for elastic scattering in the forward direction times the solid angle of the detector subtended a t the sample, and the probability that neutrons scattered a t will penetrate the remainder of the sample. The flux at the detector due to all neutrons singly scattered by the sample into the detector is obtained by dividing by A d and integrating from x = 0 to x = 1, which yields
+
380
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
Let us define the observed transmission of the sample by
T
=
To
+ TI + Tz +
.
*
where To is the true transmission excluding all scattered neutrons, T1 is the ratio of the flux of neutrons singly inscattered by the sample into the detector to the flux of neutrons without the scattering sample in place, etc. Then T 0 --e -nett
(2.7.3.4)
and I n general, by Eq. (2.7.3.2) 1
1 T - -1 AT -. nt T
u =-ln-
nt
An
and
=
For the present case,
or the apparent relative decrease in the total cross section from its true value due to the singly inscattered flux is
(2.7.3.5) From this we can obtain the true cross section ut from the apparent cross section at' by: (I:
=
all
1 - (Au/u*)'
(2.7.3.6)
-
By substituting L z = L L 1 in Eq. (2.7.3.5), differentiating with respect to L,, and equating to zero, one finds that the minimum inscattering correction occurs when L1 = Lz = L / 2 . For this case, which is most frequently used in practice, Eq. (2.7.3.5)becomes
(2.7.3.7)
If the detector is sensitive also to inelastically scattered neutrons in the forward direction, ud(Oo) must be replaced in Eqs. (2.7.3.5) and (2.7.3.7) by the differential cross section in the forward direction for elastic and inelastic scattering down to neutron energies equal to the bias setting of the detector.
2.7.
DETERMINATION OF NUCLEAR REACTIONS
38 1
If a sample is used whose length t is not negligible compared to L1 and L2, then in the symmetric case (L1 = Lz = L/2) if rD2/4 <
For the case t << L, Eq. (2.7.3.8) reduces to Eq. (2.7.3.7). Equations (2.7.3.5)through (2.7.3.8) apply only to single inscattering, which is the largest correction to be expected. The practical difficulty in determining the corrections given in Eqs. (2.7.3.5), (2.7.3.7), and (2.7.3.8) is caused by the necessity of determining u,(Oo), the differential elastic-scattering cross section for forward scattering. For neutron energies in the range of a few hundred kev, it is usually sufficiently accurate to assume isotropic scattering. I n this case the ratio (ue/ac)(Awd) is simply the solid angle subtended by the detector at the sample (Awd) divided by 4a, since in this energy region elastic scattering predominates and at = h a , . For higher neutron energies, however, it is necessary either to measure the angular distribution of the elastically scattered neutrons or to use a theoretical expression for u,(Oo). Since the former choice normally is a difficult experiment in itself, the latter procedure has usually been adopted. Only recently have sufficiently accurate angular distribution measurements, described in Section 2.7.6.1, become available to check the theoretical expectations. The most common method for estimating ue(Oo) at higher energies is based upon the diffraction scattering of neutrons. For small angles, this results in the familiar f ~ r m u l a , ~ ' (2.7.3.9) where R is the nuclear radius, k the wave number of the incident neutrons, 13 the scattering angle, and J 1 a Bessel function of the first kind. For 6 = 0" this becomes Ue(O0) =
k2R4
-.4
(2.7.3.10)
A diffraction theory based on a continuum mode138 yields an expression in more common current usage :
+
ae(OO)= ( k R 41c2
(2.7.3.11)
G. Plaezek and H. A. Bethe, Phys. Rev. 67, 1075A (1940). s * B . T. Feld, H. Feshbach, M. L. Goldberger, H. Goldstein, and V. F. Weisskopf, Final report of the fast neutron data project. NYO-636 (January, 1951). Unpublished. 37
382
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
The expected forward scattering clearly becomes more pronounced with higher energies and heavier nuclei. A recent survey of angular distributions of elastically scattered neutrons a t 14 MevS9seems to give rather good agreement with Eq. (2.7.3.11) for light elements and for lead, assuming R = 1.33 A1/3X 10-13 cm. However, Eq. (2.7.3.11) seems to predict too small values of ~ ~ ( 0for ”) intermediate A nuclei, where the so-called “Wick’s limit”40 a,(O”) =
(2)’
(2.7.3.12)
appears to give better agreement.gg Multiple inscattering. Under experimental conditions such that singleinscattering corrections are not large, the corrections due to multiple inscattering are normally negligible. An estimate of the effect can be obtained by calculating the increase in transmission due to the effect of neutrons doubly scattered in the sample into the detector. As before, the flux a t the detector of doubly scattered neutrons may be calculated from
where the factors in brackets represent the primary flux at the depth 21 in the sample, the number of nuclei between XI and 21 XI, the differential cross section for elastic scattering a t the angle 0 times the appropriate solid-angle increment, the probability that neutrons scattered a t 21 at the angle 6 will arrive a t x2 before suffering a second collision, the number of nuclei/cm2 in the thickness dxl,the cross section for a second elastic scattering at the angle 6 (now straight forward) times the solid angle subtended by the detector, and the probability that neutrons second-scattered forward a t 22 will penetrate the remainder of the sample. This can be evaluated a t fairly high energies by noting that then most of the scattering is forward, and the correction requires evaluation only for small 6. This approximation is especially good when the detector is biased to count only elastic or nearly elastic neutrons. Replacing cos 6 by unity, and integrating only from 0 = 0 to 6 = ern (representing the maximum angle of the forward lobe of the angular distribution), one obtains the ratio of the increase of transmission due to second scattering
+
J. H. Coon, R. W. Davis, H. E. Felthauser, and D. B. Nicodemus, Phye. Rev. 111, 250 (1958). 40
G. C. Wick, Atti accad. Ital. 13, 1203 (1943).
2.7.
383
DETERMINATION OF NUCLEAR REACTIONS
(Tz) to that due to first scattering [ T I ,from Eq. (2.7.3.4)] t o be (2.7.3.13) The upper limit for this result, valid for high energies where 8, is small, is obtained by replacing uez((e)by ue2(00),yielding’’ (2.7.3.14) A reasonable value of 8, is l / l ~ R . ~ 8 Multiple inscattering involving more than two collisions is best calculated by Monte Carlo methods. An experimental check on the calculated corrections for single and multiple inscattering may be made by using samples of various lengths and diameters, and plotting the computed cross section against the sample length and diameter. A comparison of the cross section determined from the slope of the transmission versus sample-length curve and that determined by extrapolating the cross section versus diameter curve to zero diameter then serves as a check on the validity of the calculated inscattering corrections.zz Nonexponential attenuation. A basic assumption made in Section 2.7.3.1 was that the energy spread of the incident neutrons was small compared to energy intervals in which the total cross section varied appreciably. If this condition is not met in an experiment, then the decrease of the primary beam flux is no longer exponential. Further, the exit flux from the sample will have been filtered in such a way that neutrons of certain energies will have been removed more strongly than others. For example, if the total cross section is a monotonically decreasing function over the neutron spectrum employed, the exit neutrons will be “hardened” in the same sense that the average energy of X-rays normally increases with passage through an absorber. If a strong scattering resonance occurs in the spectrum, neutrons of that energy will be largely removedfrom the exit flux. An accurate measurement of an energy-dependent total cross section can best be made by employing a neutron-energy spread which is small compared to energy changes over which the cross section varies appreciably. The energy spread of the neutrons employed in fast-neutron experiments is determined by three fact0rs:~1(a) the spread in energy of the charged-particle beam which produces the neutrons at the target, (b) the thickness of the target, and (c) the angle subtended by the detector 41H. H. Barschall, C. K. Bockelman, R. E. Peterson, and R. K. Adair. Phys. Rev. 70, 1146 (1949).
384
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
a t the target. For charged-particle beams produced in electrostatic generators, factor (a) can usually be made arbitrarily small by use of precision electrostatic or magnetic analyzers (see, e.g., ref. 42), except for the limitation of beam intensity. Factor (b) can also be made small by using very thin evaporated targets, for example, but with a corresponding reduction in neutron yield. Factor ( c ) is determined by the variation of the energy of the emitted neutrons with angle of emission. For neutron sources using light nuclei, the finite solid angle subtended by the detector a t the neutron source can introduce considerable energy spread in the detected neutrons. The limiting spread can be calculated purely from the kinematics of the reaction, but in practice a variation of detector efficiency over its surface makes the exact shape of the counted spectrum difficult to determine. For neutron sources in the light nuclei, the effect can be minimized by removing the detector further from the source. However, this not only decreases the counting rate but also increases the relative importance of room-scattered background. An alternative method is to use a neutron source which employs a heavier target n u c l e ~ sWith .~ the exception of this last alternative, the methods for minimizing the energy spreads due to factors (a), (b), and (c) all have the disadvantage of reducing the primary neutron flux a t the detector with a consequent increase in running time to gain suitable statistics. This intensity disadvantage can be overcome by an elegant method using energy modulation of the accelerator and neutron time-of-flight t e c h n i q ~ e s . ~ ~ If it is not feasible to reduce the neutron energy spread to a value smaller than the energy region over which a marked variation in the cross section occurs, one is forced to make a measurement of the cross section “averaged” over the energy region determined by the neutron energy resolution. If the fraction of incident neutrons with energies dE is f ( E ) dE, so that $f(E) dE = 1, then the genbetween E and E eral result for the expected transmission of a sample with thickness t is44*46
+
T
=
Jj(E)e-ntcr(E)dE.
(2.7.3.15)
I n the event that ut = constant, Eq. (2.7.3.15) reduces a t once to Eq. (2.7.3.1) of Section 2.7.3.1. If ut is a function of energy, then for very thin samples for which ntot << 1 (2.7.3.16) R. E. Warren, J. L. Powell, and R. G . Herb, Rev. Sci.Znstr. 18, 559 (1947). Cranberg, W. P. Aiello, R. K. Beauchamp, H. J. Lang, and J. S. Levin, Rev. Sci. Zn~lr.28, 84 (1957); L. Cranberg, R. K. Beauchamp, and J. S. Levin, ibid. 28, 89 44
4aL.
(1957). 44 46
R. G. Thomas, Phys. Rev. 98, 77 (1955). 8. E. Darden, Phys. Rev. 99, 748 (1955).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
Eq. (2.7.3.15) becomes where
T
‘V
e-7JtBi
385 (2.7.3.17) (2.7.3.18)
The “average” cross section is normally calculated from Eq. (2.7.3.17). However, this will not be the same as the average cross section defined by Eq. (2.7.3.18) unless the sample employed in the experiment is thin enough for condition (2.7.3.16) to be satisfied.46 Such a measurement must then be made with very thin samples and long runs in order to insure adequate counting statistics. 2.7.3.1.6. REPRESENTATIVE RESULTS. The results of total cross-section measurements with fast neutrons have given a variety of useful information on nuclear structure. However, it is not the purpose of this section to discuss tthe significance and interpretation of these results. Instead, a few representative figures will show the character of the results of measurements of the type described in Sections 2.7.3.1.1 to 2.7.3.1.5. A detailed compilation of nuclear total cross sections is available for further referen~e.4~ In the kev region, Fig. 4 shows a typical result (from Gibbons3) of the back-angle Li7(p,n) neutron total cross-section measurements described earlier. The sharp peaks represent the familiar resonant behavior of the total cross section due to compound-nucleus formation. These resonances can be observed throughout the periodic table47in the kev-energy range, but usually overlap too strongly to be observed in the heavy elements above 100 kev. Figure 5 shows a compilation of measurements (from Hughes and Harvey46) of the very fundamental total cross section of hydrogen for fast neutrons in the range from 1 to 100 MeV. The individual points represent very careful measurements performed at the appropriate energies by various workers. 1 2 0 . 2 2 - 3 4 , 4 * These references are particularly valuable because of the detailed and careful consideration given to the corrections required in total cross-section measurements. 4 6 D. J. Hughes and John A. Harvey, Neutron cross sections. Brookhaven National Laboratory Report BNL-325 (1955). 17H. W. Newson, J. H. Gibbons, H. Marshak, R. M. Williamson, R. A. Mobley, A. L. Toller, and R. Block, Phys. Rev. 102, 1580 (1956); H.Marshak and H. W. Newson, ibid. 106, 110 (1957). 48 C. L. Storrs and D. H. Frisch, Phys. Rev. 96, 1252 (1954); R. B. Day, R. L. Mills, J. E. Perry, Jr., and F. Scherb, ibid. 98, 279A (1955); R. Sherr, ibid. 68, 240 (1945); A. E, Taylor and E. Wood, Phil. Mag. [7] 44,95 (1953); L. J. Cook, E. M. McMillan, J. M. Peterson, and D. C. Sewell, Phys. Rev. 76, 7 (1949); J. DeJuren and N . Knable, ibid. 77, 606 (1950); R. H. Hildebrand and C. E. Leith, ibid. 80, 842 (1950).
386
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
En
(k*v)
FIG.4. A typical total cross-section measurement in the kev region.8
4
'1 0.1 I)
1.0
BNL- oc 42 Wis 24 A MI78 v Hrv2 A Cal 1.2.3 Hrv3 . Har-ac 2 L A - a c 1.7 x Hrvl o
2
4
6
810
2
4
6
8
100
E (MeV)
FIG.5. The total cross section for the scattering of neutrons by protons, as measured by various authors.'6
" 0
02
04
0.6
0.8
1.0
1.2
14
1.6
1.8
2.0
N E U T R O N ENERGY
2.2
2.4
2.6
28
3.0
3.2
3.4
(MEV)
FIG.6. A representative total cross-section measurement in the range from 0.05 to 3.2 M e ~ . 1 ~ . ~ ~ ~ * 9
388
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
A typical total cross-section measurement performed with good resolution on a light element (carbon) is given in Fig. 6.re,2s,49 The resonance at 2.08 Mev is not completely resolved, although a neutron-energy spread of 10 kev was employed. It is in a region of such sharp resonances that the “hardening” effects discussed in the last section become serious. Figure 7 shows a three-dimensional plot of observed total cross sections divided by r R 2 (where R is the nuclear radius) as a function of mass
FIG.7, A three-dimensional plot of ut/lrR2averaged over resonances versus neutron energy and mass number.61s6*
number A and neutron energy. Most of the data represent results from authors60s61 in Barschall’s group at Wisconsin, summarized by the threedimensional diagram shown.61.62 The surface represents the observed cross sections averaged over resonance structure. The accuracy of the results of total cross-section measurements can, in general, be made quite high because no absolute flux measurement is required, as previously mentioned. The highest accuracy has been achieved in measurements of the neutron-proton cross section, where * errors of less than M % are 2.7.3.2. Total Cross-Section Measurements Using Polyenergetic Neutron S0urces.6~ Nereson and Darded4 have developed a method by *Errors of 0.18% at E,, = 0.4926 Mev were reported by Charles Engelke, Bull. Am. Phys. SOC.(21 6, 64 (1981). 4s D.W. Miller, Phys. Rev. 78, 806 (1950). SOD. W. Miller, R. K. Adair, C. K. Bockelman, and 8. E. Darden, Phys. Rev. 88, 83 (1962); M. Walt, R. L. Becker, A. Okazaki, and R. E . Fields, ibid. 89, 1271 (1953). 6 1 A. Okazaki, S. E. Darden, and R. B. Walton, Phys. Rev. 93,461 (1954). 6 1 H. H. Barschall, Phys. Rev. 86, 431 (1952). 68 L. Rosen, Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1966 4, 97 (1956). 64
N. Nereson and 8.Darden, Phys. Rev. 89, 775 (1953); ibid. 94, 1678 (1954).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
389
means of which one may make use of a continuous spectrum to measure total cross sections a s a function of neutron energy. Basically, the method involves the isolation of narrow energy bands from a continuous spectrum through the use of a detector which operates as a neutron spectrometer. Once isolated, each energy band may then be used as the “neutron source” for a standard good-geometry transmission experiment. Figure 8 shows a perspective cutaway view of the detector, which is operated directly in a collimated beam of neutrons. A thin hydrogenous
FIQ.8. Cutaway of ionization chamber64 developed for total cross-section measurements using a polyenergetic neutron source.
layer at the front of the detector serves to transfer kinetic energy from neutrons to protons. After collimation, the recoil protons pass first through a small proportional counter and then into a parallel plate, Frisch grid, ionization chamber which is operated as a neutron spectrometer in coincidence with the proportional counter. The spectrometer chamber is followed by a second ionization chamber with which it is operated in anticoincidence. Whereas the purpose of the chambers in coincidence is merely to reduce background, the anticoincidence arrangement is a n integral part of the spectrometer. It permits one to discriminate against those proton recoils which are not stopped i n the spectrometer chamber and hence do not produce, in this chamber, ionization commensurate with their total energy. Variation of the gas pressure in the chamber, the use of deuterium as well as hydrogen for the radiator, and the use of absorbers in the path of the recoils give this instrument
390
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
a great deal of flexibility in terms of the energy region accessible to it. The inherent energy resolution of the spectrometer, in terms of the width a t half-maximum for a group of 7-Mev monoenergetic neutrons, is approximately 8%. Using a fission spectrum as the primary neutron source, a large number of elements have been studied over the energy region 3-13 MeV. These experiments were performed under the conditions of -50 % transmission and 10% energy spread. Although the energy resolution was inadequate to resolve narrow resonances, it was completely adequate for the objectives of the experiment. These were: (1) to obtain cross-section data for use in reactor design and shielding calculations and (2) to extend the existing information on cross-section trends as a function of atomic weight. The results of the measurements are summarized in Fig. 9 which is a photograph of a cardboard model constructed to show the trend of total cross section as a function of neutron energy. Although for the light elements, adjacent elements show little resemblance in cross-section characteristics, the heavy elements reveal marked similarities between neighboring elements. Where measurements with monoenergetic neutrons from accelerators have been made, they agree with the above results to within +lo%. 2.7.3.3. Methy#& and Results for Slow Neutrons.* Studies of total cross section6 fciY slow neutrons are carried out in basically the same type of transmission geometry already described. It is the purpose of this section only to remark briefly on the differences in the neutron sources and detectors used, and on certain special problems which arise in the conversion of the measured transmission into the total cross section. A large number of references are available covering the detailed techniques and results obtained in slow-neutron measurements (see refs. 55-58, and previous references quoted therein). 2.7.3.3.1. EXPERIMENTAL CONSIDERATIONS. Sources of slow neutrons for total cross-section measurements are not initially monoenergetic, but require special techniques to study particular neutron-energy intervals. The slow neutrons are produced initially either in a nuclear reactor i
* Also refer to Sections 2.2.2.2 and 2.2.2.3, Vol. 5 A. D. J. Hughes, “Pile Neutron Research,” Addison-Wesley, Reading, Massachusetts, 1953; B. T. Feld, in “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 11, p. 208. Wiley, New York, 1953. Melkonian, W. W. Havens, Jr., and L. J. Raiawater, Phys. Rev. 92,702 (1953). s7F. G. P. Seidl, D. J. Hughes, H. Palevsky, J. S. Levin, W. Y. Kato, and N. G. Sjostrand, Phys. Rev. 96, 476 (1954). SsL. M. Bollinger, D. A, Dahlberg, R. R. Palmer, and G. E. Thomas, Phys. Reu. 100, 126 (1955). s6
W
s
FIG.9. Cardboard
showing variation of average total cross section with energy and atomic weight. Each heavy line represents one barn.
392
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
or by moderating fast neutrons produced by an accelerator. Energy selection is obtained either by reflection of neutrons by a single crystal or by time-of-flight techniques. These various methods allow useful measurements to be carried out over an energy range of to ev or better, a lo8 range in energy.6g In the “crystal monochromator,” an intense polyenergetic beam of neutrons from a reactor is diffracted by a single crystal according to the well-known Bragg condition. I n this way, monoenergetic neutrons are selected with energy determined by the diffraction angle. One important advantage to this instrument is that its monoenergetic neutron beam in any given direction allows the measurement of capture cross sections by activation, which is not possible in time-of-flight systems. The useful energy range covered by crystal monochromators is from about 0.02 ev to 50 ev. Reactors may also be used with time-of-flight techniques to determine the neutron energy, provided mechanical “choppers” are used to serve as a basis for timing. “Slow choppers,” useful in the energy range from about 0.0003 to 0.4 ev, may simply use a rotating cylindrical chopper made up of alternate sheets of cadmium and aluminum. “Fast choppers,” on the other hand, are considerably more complicated to construct since they must turn a t high speed and be heavily constructed to stop neutrons of all energies in the reactor flux. They are useful in the energy range from about 0.005 ev to 20 kev. Thus the choppers provide lower and higher energy ranges than the crystal monochromators, but overlap appropriately for continuity and consistency of data. Pulsed accelerators also provide a very useful basis for time-of-flight measurements. Accelerators employed (and typical references) are : pulse transformer,6o Cockcroft-Walton,6’ Van de Graafflg2 synchrocyclotron, 6o electron linear a c ~ e l e r a t o r ~ ~and ~ ~ ebetatron.B6 4 The neutron-producing reactions usually employed are :SO pulse transformer and Cockcroft-Walton, Ha(d,n); Van de Graaff, Li7(p,n); cyclotron, Be@(d,n) ; and synchrocyclotron, W p or similar reaction. The electron accelerators produce bremsstrahlung, which in turn produces photo-
+
s~ D. J. Hughes, Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1955 4, 3 (1956). W. W. Havens, Jr., Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1956 4, 74 (1956). O * J. H. Manley, L. J. Haworth, and E. A. Luebke, Phys. Rev. 61, 152 (1942); see also, Rev. Sci.Instr. 12, 587 (1941); ibid. 12, 591 (1941). O2 W. M. Good, J. H. Neiler, and J. H. Gibbons, phy8. Rev. 109, 926 (1958). 63 W. W. Havens, Jr. and L. J. Rainwater, Phys. Rev. 83, 1123 (1951). ‘34H. L. Schulta and W. G. Wadey, Rev. Sci. Insfr. 22, 383 (1951); W. G. Wadey, H. L. Schulta, T. F. Godlove, and J. G. Carver, Phys. Rev. 99, 1634A (1955). 6 S E . R. Gaerttner, M. L. Yeater, and R. D. Albert, General Electric Company Report KAPL-1108 (1954).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
393
neutrons from beryllium at low energies or from a heavy element (like uranium) using a giant photoneutron resonance a t high energies. After production, the neutrons are usually moderated and their flight to the detector timed. The usable neutron-energy range available for experiments with pulsed accelerators is roughly 0.001 ev to 20 kev or better. Pulsed accelerators are generally characterized by low background (since the accelerator is “off” while the neutrons are being counted) and short well-defined pulses (since electrical and not mechanical changes are responsible for the bursts). A disadvantage to the pulsed accelerator is that one experiment ties up the machine, a situation in contrast to reactors where many experiments can be performed at once. The highest energies available from pulsed accelerators and fast choppers overlap the lowest energies which can be studied with back-angle Li7(p,n) measurements, described in Section 2.7.3.1.2. In the time-of-flight work the relative instrumental energy spread AE/E normally increases with energy, while in the back-angle Van de Graaff studies AE/E decreases with energy.66 This means that measurements around 5 kev have not been possible in the past with resolution better than about 2.5%. The detectors employed for slow-neutron measurements are determined by the requirements of the particular experimental arrangement. An enriched BFI proportional counter is usually employed for crystal monochromator measurements. For time-of-flight work, it is desirable to use a detector with a fast response time t o minimize the contribution to the time resolution from the detection process itself. In addition, the usual requirements of high efficiency and relative insensitivity to y-rays and fast neutrons must be met insofar as possible.67 Very high counting efficiency may be obtained by the use of a boron-loaded liquid scintillator,58s6s provided background conditions are not severe. Improved y-ray rejection can be obtained by using a multiple array of BFP counters in a single pressure vessel,69 while retaining a moderately high counting efficiency. To reduce the time uncertainty without sacrificing y-ray insensitivity, solid scintillating discs made of mixtures of ZnS and enriched B z 0 3have also been developed.69~70A discussion of the relative advantages and disadvantages of these detectors has been given by B0llinger.~7 66
D. J. Hughes, Progr. i n Nuclear Phys. 4, 330 (1955).
67L.M. Bollinger, Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1966 4, 47 (1956). 68 L. M. Bollinger and G. E. Thomas, Rev. Sci. Instr. 28, 489 (1957). 69 H. Palevsky, N. G. Sjostrand, and D. J. Hughes, Phys. Rev. 91, 451 (1953). 7 0 s . J. Nikitin, N. D. Galanina, K. G . Ignatiew, W. W. Okorokow, and S. I. Suchorutchkin, Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1966 4, 224 (1956).
394
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
2.7.3.3.2. TOTAL CROSS-SECTION DETERMINATION. The total cross section for slow neutrons is determined from the transmission as described in Section 2.7.3.1. However, a few special problems concerned with corrections to the transmission or conversion of the transmission to the cross section are characteristic of slow-neutron measurements. Background measurements depend upon the particular experimental arrangement. For example, in fast-chopper time-of-flight measurements, the background may be measured by inserting a paraffin block in the beam a t the exit stator or by using thick samples for which the transmission is aero a t res0nance.6~Another technique is to determine the counts in a time channel exceeding the time-of-flight of the slowest neutrons that are able to pass through the chopper.71“Second-order effects” can enter to provide difficulties in time-of-flight measurements. These represent the ambiguity as to whether a detector pulse represents a fast neutron or a slow neutron from the preceding cycle. Such effects depend upon the experimental arrangement, the energy range involved, etc. In pulsed accelerator work, it is necessary to determine the sample-in sample-out ratio for equal numbers of neutrons produced in the source rather than equal intervals of time.72These various complications are discussed in det(ai1in the literature references referred to a t the beginning of Section 2.7.3.3. Inscattering corrections described in detail in Section 2.7.3.1.5 for fast neutrons are not nearly as serious a problem in slow-neutron measurements. At this low energy the scattering is isotropic, and the correction may easily be calculated simply from the solid angle subtended by the detector a t the sample. Further, in many slow-neutron measurements this solid angle is negligibly small; e.g., in the Columbia pulsed-cyclotron measurements it is less than 3 X 10-4 of 47r.73 Two very important corrections enter into the determination of the total cross section from slow-neutron transmission measurements. One of these, already described in Section 2.7.3.1.5, causes nonexponential absorption in the sample when the instrumental resolution width is not small compared to the energy region in which rapid cross-section variations occur. I n slow-neutron measurements in heavy elements, where extremely sharp resonances are the rule, it is usually not possible to obtain the true cross section from the experimental transmission by using Eq. (2.7.3.2).66In measurements of this type, it is normally of primary 71V. V. Vladimirski, I. A. Radkewich, and W. W. Sokolowsky, Proc. ist Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1066 4, 22 (1956). 7sL. J. Rainwater and W. W. Havens, Jr., Phys. Rev. 70, 136 (1946). ‘*L.J. Rainwater, W. W. Havens, Jr., J. R. Dunning, and C.S. Wu, Phys. Reu. 75, 733 (1948).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
395
concern to the experimenter to determine the parameters characterizing the resonance, e.g., the resonance energy, the total level width, etc. If the resonance is completely resolved, the total cross section is correctly given by Eq. (2.7.3.2), and the peak cross section (10 and the total level width can be obtained directly from the cross-section curve. If the neutronenergy spread is comparable to but less than the resonance width, the observed transmission curve can be corrected for the effect of finite energy resolution,57and the cross section calculated from the corrected transmission curve. Even if the resonance width is less than the experimental resolution width, information about the resonance parameters can be obtained from the area A above the experimental transmission curve.56 Thus, in the case of a thick sample measurement (mot >> l ) , ~2
= .ImtUor2
(2.7.3.19)
while for a very thin sample measurement (mot<< l),
A
=
?r
-ntaor. 2
(2.7.3.20)
Combining thick and thin sample experiments, Eqs. (2.7.3.19) and (2.7.3.20) again yield the values of uo and I’. The second important correction which must be included to obtain the true cross section is that due to thermal motion of the atoms of the sample nuclei. This causes a “Doppler broadening” and lowering of the peak height of an observed resonance from its true shape. The Doppler broadening is approximately Gaussian in shape with a width given by74 A
=
2(lcTE0m/M)~/~
where m is the mass of the neutron, M the mass of the target nucleus, k the Boltzmann constant, E o the resonance energy, and T the temperature of the target for a gas or somewhat greater than the temperature of the target for a solid.76Corrections for this effect have been calculated using tables from Born,76and graphs of the corrections p ~ b l i s h e d . ~ ~ ~ ~ ~ Details of these methods for determining the true peak cross section and level parameters from observed transmissions are given in a variety of references (see, e.g., refs. 56, 57, 58, and previous references quoted therein). A large volume of slow-neutron total cross-section data has been accumulated and is compiled in ref. 46. Figure 10 shows a representative result obtained by Bollinger et ~ 1 with . a~ fast ~ chopper. For the 74 H. A. Bethe and G. Placzek, Phys. Rev. 61, 450 (1937); H. A. Bethe, Revs. Modern Phys. 9, 69 (1937)--see p. 140. 75 W. E. Lamb, Jr., Phys. Rev. 66, 190 (1939). 78 M. Born, “Optik,” p. 482. Springer, Berlin, 1933.
2.
396
DETERMINATION O F FUNDAMENTAL QUANTITIES
337 OV
3001 0 NEUTRON GROSS S E C T I O N MANGANESE I00
E.
J
333ev
2
'o-1080
2360
3 3
R = 0.50 X
2
22sv 14 336
2360 OV
Id'bm
ln
30 0 .
a
m
z
0 l-
0
w I C10 ln
I
300 I000 U T R O N E N E R G Y (ev)
3
3000
FIQ. 10. A representative slow-neutron total cross-section measurement.68 The solid line represents the theoretical variation of the cross section for the parameters listed in the figure.
same reasons discussed in Section 2.7.3.1, the accuracy of slow-neutron total cross sections can be made high under favorable circumstances, with errors as low as 0.1 % being claimed in some cases.66.77 77
D. J. Hughes, J. A. Harvey, M. D. Goldberg, and M. J. Stafne, Phys. Rev. 90,497
(1953).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
307
2.7.4. Nonelastic Neutron Cross Sections16.78~~9 I n what follows we adopt the following symbols and definitions
+
+ +
total cross section = a, unnS = U. ui 6.. total elastic cross section. total nonelastic cross section (cross section for all processes in which the residual nucleus differs, in structure or excitation, from the target nucleus) =
“i
+
Ua.
total inelastic cross section. total absorption cross section (cross section for all processes in which the absorption of a neutron leads to fission, charged particle emission or de-excitation by ?-emission) = uC uf. capture cross section. fusion cross section. cross section per steradian at angle e. differential elastic-scattering cross section a t angle 8. differential inelastic-scattering cross section a t angle 8. cross section per Mev at energy E . cross section per steradian per Mev at angle e and energy E .
+
elastic transport cross section = rne
+
I0
4s
ue(0)(l
- cos 0) do.
met.
It is quite clear that, for monoenergetic incident neutrons of energy below the threshold for (n,2n) reactions, one method for determining une is to measure ui(e), integrate over 4?r solid angle, and to this add 6,. The latter quantity is in general quite negligible for fast neutrons in all but the lightest and heaviest nuclei. This is a consequence of the barrier suppression of charged-particle emission and the very small widths for y-emission as opposed to particle emission, the ratio of cross sections being -lo3. Another method is to measure a,(@, integrate over 4ir solid angle to obtain ue and subtract this from ut. Of the two methods, the latter is the more practical, b u t even it requires very complete angular distributions and monoenergetic incident neutrons. 2.7.4.1. Sphere Transmission Measurements. The most straightforward and most accurate method for measuring une and one which, in principle, imposes no limitations on the energy distribution of either the source or emitted neutrons and, furthermore, requires only very crude angular distribution measurements of the elastically scattered neutrons, is the so-called sphere transmission method. For simplicity, imagine a n isotropic monoenergetic point source of neutrons placed at the center of a spherical shell with a threshold neutron detector placed at a large 7sL. Cranberg, R. B. Day, L. Rosen, R. F. Taschek, and M. Walt, Progr. in Nuclear Energy [I]1, 107 (1956). Joan M. Freeman, Progr. in Nuclear Phys. 6 , 38 (1956).
398
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
distance from the shell. (See Fig. 11.) We designate the shell thickness r2 - r1 by t and use the macroscopic cross section in units of cm-', i.e., u = 1/X where X is the mean free path for a given process or combination of processes and at represents the number of mean free paths in shell thickness t. Assume further that the detector response has a threshold a t E - AE, where E is the energy of the source neutrons and AE the maximum energy loss due to elastic collisions, and is flat thereafter. ZrT TARGET
SCATTERER
/ 114 C M /
/ T p 2 3 2 PHOTOMULTIPLIER
/
/ /TO STILBENE CRYSTAL
PROPORTIONAL COUNTER
6 MIL REFLECTOR COAT
FIG.11. Experimental arrangement for sphere transmission measurements.
It will be shown that the ratio of detector counts with the shell surrounding and then removed from the source is a measure of the nonelastic scattering in the sphere, where a nonelastic collision is defined as one which reduces the neutron energy by more than AE. To see this, let us first assume une = 0 and t sufficiently small and the atomic weight sufficiently large so as to preclude energy losses larger than AE by elastic collision only. It is now quite clear that the detector will respond identically with and without the sphere of material surrounding the source, for the counter can obviously be imagined to have the shape of a spherical shell concentric with the source and, under the above conditions, every source neutron must eventually penetrate the detector with energy above the detector threshold. This argument indicates that, to first approximation, elastic scattering has no effect on the sphere counts (sphere on) transmission where this quantity is defined by T = counts (sphere
2.7.
DETERMINATION OF NUCLEAR REACTIONS
399
Thus we can, to first approximation, write
T = e-~nd. (2.7.4.1) A little consideration will reveal that Eq. (2.7.4.1) is only accurate when a t least one of the following conditions is fulfilled: (a) une>>u,, (b) t << X, or (c) a,(@) is very strongly peaked in the forward direction. Conditions (a), (b), and (c) are designed to avoid a n effective increase in path length for neutrons as a result of their elastic scattering, for, if such a n increase occurs, the probability for nonelastic scattering is correspondingly increased and Eq. (2.7.4.1) is no longer valid. The transmission of unscattered and elastically scattered neutrons through the shell is then not r u n e t , as was assumed in Eq. (2.7.4.1),but rather e-'ne(t+At) where At is the average increase in path length due to elastic collisions. I t turns out that condition (a) is contrary to nature and condition (b) is impractical from the experimental side, for if the shell is thin compared to A, it will also be thin compared to A,,,. Condition (c) however does prevail, approximately, for incident neutrons of energy > 10 MeV. Generally speaking, therefore, the equations for relating nonelastic cross sections t o transmission measurements are considerably more complicated than Eq. (2.7.4.1). These equations have been worked out in great detail and with high-order corrections by Bethe, Beyster, and Carter.*O We can hardly hope to improve upon their treatment and must content ourselves with a brief description of the general approach to this type of problem. represent the transmission of neutrons which suffer no Let To = collision whatsoever, T1 the transmission of neutrons which undergo one elastic collision, T z the transmission of neutrons which undergo two elastic collisions, etc. Let us also assume 1 source neutron. Then the total transmission of neutrons which have not suffered nonelastic collisions is T = T O T I T 2 . . . Alternatively, if 10= 1 - To represents the number of first collisions, Il the number of nonelastic first collisions, Iz the number of nonelastic collisions following one elastic scattering (second nonelastic collision), I 3 the number of third nonelastic collisions, etc., we have ... (2.7.4.2) T=l-11-I2
+ + +
I1 ne - ulo
=
ne (1 - To) u-
and where
1 2
=
(2.7.4.3)
bt
66
EI(1 - Pl) ; une
(2.7.4.4) (2.7.4.5)
80 H. A. Bethe, J. R. Beyster, and R. E. Carter, J. Nuclear Energy 3 (3), 207 (1956); 8 (4), 273 (1956); 4 ( l ) , 3 (1957); 4 (2), 147 (1957).
400
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
and PI is the probability for escape (without further collision) of neutrons which have suffered 1 elastic collision. Similarly, (2.7.4.6)
E2
where
=
El(1
- PI)
(2.7.4.7)
and P 2 is the escape probability for neutrons which have suffered 2 elastic collisions. 14,16,etc. are given by similar expressions. \
\
\
\
FIG.12. Geometry for evaluating multiple-scattering effects in a spherical shell.80 For shells whose thickness is less than several simplifying assumptions are justifiable. First of all we may assume that a negligible fraction of nonelastic interactions occurs after two elastic scatterings. Secondly, in calculating the number of neutrons undergoing first elastic collisions we may assume that all the source neutrons are available for elastic scattering at every point in the shell, i.e., the probability for a n elastic collision occurring between r and r dr is redr (assuming as usual 1 source neutron). Since the second assumption implies that we are considering a larger number of first elastic and hence second nonelastic collisions than actually take place, while the first assumption eliminates high order elastic collisions and hence the nonelastic collisions which follow these, the errors introduced by the two assumptions tend to cancel.
+
2.7.
DETERMINATION OF NUCLEAR REACTIONS
40 1
Referring to Fig. 12 and invoking our first assumption, we see that the probability of nonelastic scattering following one elastic collision is, to first approximation, aney(r,e).We therefore have, (2.7.4.8) For the special case of isotropic scattering and t << rz, it is found that the y(r,e) averaged over solid angle is almost independent of the location of the elastic scattering, and (2.7.4.9) Equation (2.7.4.8) therefore yields,
I,
=
+u,,,a,t~L
(2.7.4.10)
and from Eq. (2.7.4.2), expanded to order t 2
T
=
1 - anet
+ &,J'(u~ - a&).
(2.7.4.11)
(Under the above assumption L = 4.) Had we not assumed isotropic scattering, Eq. (2.7.4.8) would have had to be evaluated numerically. However, it is shown in ref. 80 that even for highly anisotropic scattering it is a good approximation to assume isotropic scattering if one replaces the elastic cross section by the elastic transport cross section and hence the total cross section by the total transport cross section (since the nonelastic cross section is in all practical cases essentially equal to the nonelastic transport cross section). Under most experimental conditions t X and one must consider multiple elastic collisions. We therefore return to a consideration of Eqs. (2.7.4.2)-(2.7.4.7). The central problem resides in the evaluation of the escape probabilities P,. To simplify this problem, we replace the elastic and total cross sections by the corresponding transport cross sections in order that we may assume isotropic scattering. P , is then a function only of the position r a t which an elastic collision takes place, for this determines the average escape probability as well as the fraction of n-scattered neutrons which suffer their nth collision between r and r dr. We can now see that the number of neutrons which have suffered one elastic collision in the shell is
-
+
El
=
%(l utr
To)
(2.7.4.12)
402
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
and the fraction of first elastically scattered neutrons which undergo elastic collisions between r and r dr is,
+
(2.7.4.13)
A neutron scattered at (r,e) will have an escape probability P(r,e) = ex~[--~,y(r,~)I where y(r,8) is the distance traveled in the shell if there is no further collision. For isotropic scattering the average escape probability for a neutron starting at r is P(r) = P(r,e) dQ (2.7.4.14) 4n
s"" 0
the probability that this neutron does not escape without further collision is 1 - P(r) and the probability that it makes a nonelastic collision is (2.7.4.15) Thus, the average probability of second nonelastic collisions is
since, by definition, /":fl(r)
dr
=
1.
(2.7.4.17)
Pn is thus defined by P,
=
l r f n ( r ) p ( r dr )
(2.7.4.18)
and it is this quantity which must be evaluated. The difficulty resides in the calculation of spatial neutron distributions. PI and Pz have been evaluated by numerical methods,s0 but beyond this the complexity becomes prohibitive. However, it was recognized by Bethe et al. that several simplifying factors may be invoked. Firstly, the spatial distribution of successive collisions very rapidly converges to a limiting distribution fn which corresponds to the spatial distribution of neutrons in a shell of fissionable material which is just critical, i.e., the fundamental normal mode of the shell. Secondly, the escape probability P(r) is not a sensitive function of r and hence the average escape probability P, is not very sensitive to fn(r). As a consequence it can be shown that the higher escape probabilities, even including P p , are approximately equal to that for the normal mode P,.
2.7.
DETERMINATION OF NUCLEAR REACTIONS
403
On the basis of the above we can now write 1 - T = II
+ I2 + I > zfor a thick shell.
(2.7.4.19)
Reproducing Eqs. (2.7.4.13)-(2.7.4.15), using transport cross sections, we have 11
=
(1 - To)%
(2.7.4.20)
(z)(2) ct r
IZ = (1 - To)
(I - PI)
(2.7.4.21)
where T O= We must now write an expression for I>2,the number of nonelastic interactions which occur following 2 or more elastic collisions. We assume Pz = P 3 = P, = P,. The number of inelastic scatterings following two or more elastic scatterings must be equal to the product of the number of second elastic scatterings which do not lead to escape, S = E,(I - P2),and a quantity which represents the fraction of S which undergoes nonelastic interactions. We can arrive a t this quantity as follows : following any given elastic collision starting with the second, the probability for nonelastic scattering is proportional to une while the probability for escape is proportional to aetP,. Since the ratio of probability for nonelastic scattering to the probability for escape is the same for all collisions after the second, the fraction of S undergoing nonelastic interaction is given by une/(une uetPm).We can now write e-el?t.
+
(2.7.4.22) where Ez and El are as defined by Eqs. (2.7.4.15) and (2.7.4.17), but with transport cross sections. Finally,
(2.7.4.23)
P, has been evaluated analytically and numerically for various combinations of geometries and cross sections. The results are available in ref. 80. The arrangement shown in Fig. 11 has been widely useds1 to make measurements with 14-Mev d-T neutrons. For low-energy (<200 kev) deuterons the d-T reaction yields a source which is essentially monoenergetic and isotropic. The detector may be a scintillator, proportional 8 1 D. D. Phillips, R. W. Davis, and E. R. Graves, Phys. Rev. 88, 600 (1952);H.T. Gittings, H. H. Barschall, and G. G. Everhart, ibid. 76, 1610 (1949);H.L. Taylor, 0. Lonsjo, and T. W. Bonner, ibid. 100, 174 (1955).
404
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
counter, counter telescope, or threshold detector. The effective threshold for all the detectors must be such as to exclude the counting of nonelastic neutrons but must be low enough to count all elastically scattered neutrons with essentially constant efficiency. The active element is an organic crystal in good optical contact with a photomultiplier tube. Photomultiplier signals are amplified and fed to a well-regulated amplifier which in turn feeds a multichannel analyzer or ordinary scaler with bias provision. In this way one may count only those pulses due to neutrons of energy 2 E - AE. Alternatively, one can use threshold detectorsa2 to count
3 0
I
2
3
4
5
6
NEUTRON ENERGY, (MEV)
FIQ. 13. Method of determining effective energy threshold of a threshold detector from an excitation function.
neutrons above the effective threshold energy. This energy is determined from the variation of activation cross section with neutron energy. For example, the excitation curve displayed in Fig. 13 requires the choice of 2.2 Mev as the effective threshold since a t this energy the areas A and B are approximately equal. By employing detectors with different thresholds, it is possible to obtain an estimate of the inelastic cross sections for scattering to various energies. Equation (2.7.4.23) can be solved for unsprovided that in addition to T, one measures utrand ual. I n practice one usually does not have utr. However usl/ue = R is readily obtainable to the desired precision from even a crude measurement of the spatial distribution of the elastically scattered *a B. L. Cohen, NucZeonics 8 (2), 29 (1951); B. T. Feld, in “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 11, p. 298. Wiley, New York, 1953.
2.7.
DETERMINATION OF NUCLEAR REACTIONS
405
neutrons, i.e.,
R =
27r J 2 =
ue(e) ( 1
hz*
27r
- cos e) sin e de
a.(e)
(2.7.4.24)
sin B de
Also at is readily determined by a standard good-geometry transmission experiment. Now g e t = (ut - una)R and utr = une uct.We then have only one unknown in our equation, namely uns. However, since the P values as well as T Odepend upon utr which in turn depends upon une,Eq. (2.7.4.23) is usually solved most easily by successive approximation, i.e., by trying values of an, until the measured transmission is achieved. 2.7.4.2. Neutron-Multiplication and Transmission Meas~rements.~3 By measuring both the transmission and the total number of neutrons from source and shell, one can obtain a number of additional cross sections which have important consequences for the design of nuclear energy devices as well as for basic physics. Again considering 1 source neutron we define multiplication as M =T T’, where T‘ represents the number of neutrons arising from nonelastic interactions. Since the number of nonelastic collisions is 1 - T , and T’ = M - T,the number of neutrons generated per nonelastic collision is N = ( M - T)/(1 - T), where
+
+
N =
an,n’
+ 2un,~n+ + + + + + xun,zn
vaf
(2.7.4.25)
Qne
Qns
= un,n’
Qn,Zn
~n,3n
*
*
an,zn
+ + uf uc
(2.7.4.26)
and v = the average number of neutrons generated per fission. For Eq. (2.7.4.25) to be correct one must avoid self-multiplication by keeping the shells quite thin relative to a mean free path for neutrons arising from nonelastic interactions. A first-order correction can be made by using shells of different thickness, and this is especially important in the determination of v for fissionable isotopes since self-multiplication may then be a large effect. By definition,
+
u ~ , ~ gn,zn ’
+
+
+
*
*
+ un,zn= une - uC - uf
(2.7.4.27)
where uc = u , , ~ un,* un.= etc., for all processes removing neutrons. (As previously indicated, this quantity is very small for all but the lightest nuclei.) Also - T 1-T M - 1 N - 1= M __- - - = ~. (2.7.4.28) 1-T 1-T 1-T 83E. R. Graves and R. W. Davis, Phys. Rev. 97, 1205 (1955); N. N . Flerov and V. M. Talyzin, Sovief J . Atomic Energy Trans. 6, 1593 (1958).
406
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
Combining Eqs. (2.7.4.25) and (2.7.4.27), we get -Une + - + U(nv, $ n- ~ - ( Y ) One
“’1
(2.7.4.29)
Qne
One
where a = -0.2
(2.7.4.30)
Of
and combining Eqs. (2.7.4.28), (2.7.4.29) and (2.7.4.30), we get N-l=bn,zn Qne
Qne
M - 1 1-T
(2.7.4.31)
(In the above it has been assumed that un,m = 0 for 2 2 3.) If, by independent experiments one measures any three of the quantities u,, uf, Q,,,~,, or v, one can determine the fourth quantity, as well as une, by multiplication and transmission measurements to the same approximation that one measures une. I n the absence of sufficient experimental data to evaluate all the cross sections in Eq. (2.7.4.31) one can still put limits on these cross sections by postulating maximum and minimum values for the unknown cross sections. For example, the maximum value of un,Znis una in which case = 0. If now, one knows ul and can assume uc negligible, one can place a lower limit on v. Similarly by assuming u,,~,, = 0, one can place an upper limit on v. Fortunately, the quantity which enters most calculations is N , and this is independent of how the partial cross sections are divided. The above analysis assumes that a detector is available for measuring all the neutrons emitted by the sphere. This problem is not simple for it implies either an energy-insensitive detector or slowing down all the neutrons until they are captured in a nuclide whose decay activity can then be precisely related to the initial neutron flux. In practice one usually utilizes a long counter84which is composed of a BF, counter placed along the axis of a paraffin cylinder in such a geometry that the efficiency of the counter is as nearly as possible constant over a large region of neutron energy. For absolute measurements the counter is calibrated with standard sources. Although such counters can readily be made “flat” to + l o % they do exhibit resonance behavior for neutron energies which correspond to levels in the elements which make up the assembly. Everything which has been said thus far assumes a monoenergetic, spherically symmetric source inside a sphere with the detector outside. 84
A, 0. Hanson and J. L. McKibben, Phys. Rev. 72, 673 (1947).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
407
It has been shown by Bethe that the equations developed are equally applicable to an isotropic source and isotropic detector regardless of which is inside and which is outside the spherical shell. It has also been shown that an external anisotropic source can be replaced, for computational purposes, by an internal source which has the same shape and absorption characteristics as the detector it replaces, whereas the internal detector must be replaced, for computational purposes, with a n external one whose shape and angular sensitivity is the same as the source it replaces. Precise corrections for finite distance between source and detector, finite size, absorption and angular asymmetry of the counter, and loss of energy in elastic collisions have been evaluated in ref. 80. Bethe, Beyster, and Carter have carried out a large number of nonelastic-scattering measurements for fission-spectrum neutrons using spherical shells as targets and internal fission-threshold detectors of UZ3*,Th233,and Np237.*6 These results are given in ref. 80. In addition to internal threshold detectors, one can also use biased proportional counters containing hydrogen as the counting gas, or plastic scintillators. Such detectors can be made sufficiently small and sufficiently nondirectional so that they yield precise results when used inside spherical shells even when the external neutron source is of low energy and markedly anisotropic.86 A summary of neutron cross-section data is given in BNL 32V6 and the Brookhaven Conference on Fast Neutron interaction^.^^ 2.7.4.3. Activation, Capture, and Fission Cross Sections. We define “activation” ‘cross section as the cross section for the formation of a residual nucleus which is unstable against beta decay, and we define “capture” cross section as the cross section for the formation of a compound nucleus which decays by gamma emission. Since activation cross sections lead to radioactive end products, they can be measured by detecting the appropriate @- or y-activity. Beta rays are measured by means of a 4~ proportional counter, a thin window ( 2 ~ proportional ) counter or scintillator. Gamma rays are conveniently detected in NaI crystals. 86 The design of the fission counter was essentially the same as that described by B. Rossi and H. Staub, in “Ionization Chambers and Counters.” McGraw-Hill, New York, 1949. * 6 J. R. Beyster, R. L. Henkel, and R. A. Nobles, Phys. Rev. 97, 563 (1955); J. R. Beyster, R. L. Henkel, R. A. Nobles, and J. M. Kister, ibid. 08, 1216 (1955); J. R. Beyster, M. Walt, and E. W. Salmi, ibid. 104, 1319 (1956); M. H. MacGregor, W. P. Ball, and R. Booth, ibid. 108, 726 (1957); R. C. Allen, Nuclear Sci. and Eng. 2 , 787 (1957). 87 Proc. Mev Neutron Cross Sections Conference. Brookhaven National Laboratory R.eport BNL-224 (1953).
408
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
Methods for measuring activation cross sections and some of the more important results of such measurements are given in ref. 88. 2.7.4.3.1. CAPTURE CROSSSECTIONS. Neutron capture cross sections, for fast neutrons, comprise an important class of reactions which do not readily lend themselves to detection by conventional means.80The reason is that the gamma emission is prompt and the cross sections very small, a combination which greatly complicates the problem of y-ray detection, since this detection must be carried out in the field of neutrons and y-rays prevailing during the irradiation. Recent advances in large liquid scintillator techniquesgo have made possible a new approach to this problem, an approach which has already proved most f r u i t f ~ l . ~The ' method involves piping the primary beam of neutrons into the center of a large liquid scintillator where it is intercepted by the target. The liquid scintillator can be made large enough to absorb essentially all the energy of the y radiation resulting from the capture reaction. If now one pulses the primary beam and utilizes low resolution time-of-flight methods, one can readily distinguish the prompt capture y-rays resulting from capture reactions in the target from elastically- or inelastically-scattered neutrons. To understand this let us consider the time-history and magnitude of signals from scattered neutrons. A prompt pulse results from the loss of kinetic energy in the scintillator by nuclear collisions. However, the energy so deposited cannot exceed the incident neutron energy. A delayed pulse results from capture of a neutron which has been thermalized in the scintillator. I n a solution which contains cadmium, the energy thus liberated is -9 MeV. A delayed coincidence produced by pulses of proper magnitude therefore signals an elastic- or inelastic-scattering event while a single prompt pulse signals capture. A typical apparatus for measuring neutron-capture cross sections is schematized in Fig. 14. The neutron beam is pulsed so that neutrons arrive in bursts of -0.1 psec duration with a repetition rate of -104/sec. This is accomplished by electrostatically deflecting the beam from the target except during the 0.1 psec interval when the voltage is removed 88B. L. &hen, NucZeOnics 8 (2), 29 (1951); E.B. Paul and R. L. Clarke, Can. J . Phys. 31, 267 (1953);H. G. Blosser and T. H. Handley, Phys. Rev. 100, 1340 (1955); H.L.Reynolds and A. Zucker, ibid. 101,166 (1956);B. L. Cohen, ibid. 102,453(1956); B. L. Cohen and Eugene Newman, ibid. 99, 718 (1955);S.N. Ghoshal, ibid. 80, 939 (1950);E. Bleuler, A. K. Stebbins, and D. J. Tendam, ibid. 90, 460 (1953);R. E. Bell and H. M. Skarsgard, Can. J . Phys. 34,745 (1956). 89 R. L. Macklin, N. H. Laaar, and W. S.Lyon, Phys. Rev. 107, 504 (1957). 90 F. Reines, C. L. Cowan, Jr., F. B. Harrison, and D . S.Carter, Rev. Sci. Instr. 26, 1061 (1954). 01 B. C. Diven, J. Terrell, and A. Hemmendinger, Phys. Rev. 109, 144 (1958).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
409
and the beam is allowed to enter the target. The collimator defines a narrow neutron beam which passes through the axial tube of the liquid scintillator. This contains 100 liters of liquid within a porcelain-enameled interior. The detectors are six 5-in. photomultipliers with outputs in parallel. The liquid scintillator solution is triethylbenzene and terphenyl, with POPOPs2 to shift the wavelength into a region of high photomultiplier sensitivity. The size of the scattering sample represents a compromise between intensity and transparency to neutrons and y-rays.
PARAFFIN & BORON
SIX 5 " PHOTO-
MULTIPLIERS
RAFFW k BORON
2-1/23, DIAMETER APERTWEJ
LCAPTWE SAMPLE
FIG.14. Experimental arrangement for measuring capture cross sections.g1
A liquid scintillator of the type described can also be used to measure the probability for neutron emission. Diven el aLg3used an arrangement similar to that of Fig. 14 to measure neutron multiplicity from the fissionable isotopes. In this application the scintillator solution is salted with cadmium and the capture sample is incorporated into a fission counter. The prompt fission pulses in coincidence with the beam open a gate for the delayed pulses resulting from the capture in cadmium of neutrons thermalized in the solution. The cadmium concentration may be adjusted to give mean life times for the neutrons in the solution of some tens of microseconds, and under these conditions the resulting pulses are sufficiently spread out in time so that they can be individually counted. By combining the capability for counting y-rays with that for counting neutrons, Diven et aL91have carried out an elegant series of measurements as a function of neutron energy. of the ratio of capture-to-fission for UZs6 92F.N. Hayes, B. S. Rogers, and D. G. Ott, J . Am. Chem. Soc. 77, 1850 (1955). 93 B. C. Diven, H. C. Martin, R. F. Taschek, and J. Terrell, Phys. Rev. 101, 1012 (1956).
410
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
In this experiment the gate is provided by the prompt gammas associated with either capture or fission events while the delayed pulses arise from the thermalization and capture of fission neutrons and the fission counter serves to distinguish between fission and inelastic scattering. The ratio of events involving prompt pulses (R,) to those involving delayed pulses ( R d ) determines the relative probability of the two reaction modes, i.e., R, - Ni Qc (2.7.4.32) Rd - Ni = 1 +-& where N J represents the number of inelastic scatterings which are identified by anticoincidence between the prompt fission and the prompt y-ray pulse when the latter is followed by a delayed pulse due to neutron capture. 2.7.4.3.2. FISSION CROSS SECTIONS.~~ Among the most important cross sections from the standpoint of nuclear energy generation are those for the fission of the heavy nuclides, U, Th, and Pu. The most accurate fission cross sections have been obtained by a direct comparison, at a given neutron energy, of the probability for fission with the probability for n-p scattering since this latter cross section is among the best known of nuclear cross sections. The method for making a comparison measurement is illustrated by Fig. 15. The double counter is constructed so as to expose both fission foil and hydrogenous radiator t o essentially the same neutron flux. One then counts all fission fragments that proceed into 2 r solid angle while simultaneously counting all proton recoils which are above a given energy, which energy determines the fraction of all recoil protons counted. Even for nuclides as heavy as U and for neutrons of energy as low as 10 MeV, some correction must be made for the fact that the center-ofmass motion will produce a counting asymmetry in the forward and backward hemisphere. This correction is calculable on the basis of an isotropic angular distribution, or empirically determined by comparing the counting rates from two different foils with that from the hydrogenous radiator and then comparing the counting rates from the two foils by placing them back-to-back in a double ionization chamber. A knowledge of the ratio of masses provides, of course, an additional check on the initial comparison of fission and n-p scattering cross sections. For detailed consideration on design of detectors for the above type of measurements, see Chapters 1.2, 1.3, Vol. 5 A, and Rossi and Staub, ref. 85. Having established the cross section, as a function of energy, for a 94 W. D. Allen and R. L. Henkel, Progr. in Nuclear Energy [I] 2, 1 (1958); J. A. Hmvey and J. E,Sanders, ibid. [I]1, 1 (1956).
2.7.
DETERMINATION
OF NUCLEAR REACTIONS
FISSION lON1ZATl ON CHAMBER
I
411
C a I CRYSTAL I
,POLYETHYLENE
I
FOIL
PHOTOMULTIPLIER
FISSION FOIL
PROPORTIONAL COUNTFRq
I
FIG.15. Ionization chamber and neutron telescope for measuring 6ssion cross sections. (R. K. Smith, private communication.)
given isotope by comparison with the n-p cross section, it then becomes possible to determine the cross section of other fissionable isotopes by direct comparison. Analogous procedures may be followed to obtain the breakup cross sections for the light nuclei Lie and Bl0. 2.7.5. Differential Interaction Cross Sections The total microscopic cross section ut for the interaction of one nucleus with another may be defined as the probability for a nuclear collision when a single incident particle impinges upon a target containing 1 nucleus per cm2 along the direction of the incident particle. Analogously we may define the partial cross section u p as the number of reactions of a given category which occur under the conditions specified above-thus ut =
zup.
The differential cross section refers to the angular and/or energy dependence of the reaction products from a reaction which is characterized by a unique partial cross section. Assume a parallel beam of N Oparticles which is intercepted by a target containing nt nuclei/cm2 in a plane perpendicular to the direction of the incident beam. n and t refer, respectively, to the atomic density and thickness of the target. I n Section 2.7.4 we have seen that the number of
412
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
particles which remain in the beam after traversal of the target is
N For t
=
N oe-ant.
(2.7.5.1)
<< l/an, the number of particles removed from the beam is simply, (2.7.5.2) N O- N = NNnt.
A physical interpretation of this relationship is that for incident particles very much smaller than the target nuclei, ant is the fraction of each cm2 of target area which is opaque to the incident particles. If we assume a TARGET BEAM
~
f
FIG.16. Schematic of target-detector geometry in scattering and reaction experiments.
one-to-one correspondence between the number of particles removed from the beam and the number emitted by the target, the total number of particles emitted by the target is given by Eq. (2.7.5.2), where No - N now represents the reaction yield, which quantity we designate by Y . The symbols da/dQ, da/dE, and d2a/dQdE as well as a(0), a@), and a(0,E) are used in the literature to specify, respectively, the solid angle dependence (as a function of emission angle), the energy dependence, and the dependence on both solid angle and energy (as a function of emission angle and energy) of the cross section for a given reaction. We adopt the latter symbolism for the sake of brevity. In Fig. 16 the beam is shown as a line source incident on a thin target. A detector D records the number of reaction particles which leave the target a t angle 0 k A0 and pass through the solid angle subtended by the detector a t the source. Differentiation of Eq. (2.7.5.2) with respect to solid angle yields (2.7.5.3)
2.7.
DETERMINATION OF NUCLEAR REACTIONS
413
where u(0) is the cross section per steradian for those reactions whose products are emitted into the solid angle between two coaxial cones of half-angle 0 and 0 d0 respectively. If we now integrate over the solid angle subtended by the detector a t the source we get
+
y(e) = Nonta(e)a.
(2.7.5.4)
The total partial cross section for this category of reaction would then be
IO4*~(e) dn 2r jo2"a(s)sin e =
do.
(2.7.5.5)
Similarly, differentiation of Eq. (2.7.5.2) with respect to energy and with respect to both solid angle and energy defines the other two differential cross sections. If, as often happens, a given reaction yields two or more reaction products which are not identifiable one from the other, the above defined differential cross sections refer to the probability for emission of such reaction products rather than to the interaction probability. In discussing the evaluation of the parameters in the cross-section formula we shall, for the sake of succinctness, confine most of our discussion to the problem of measuring differential cross sections with respect to solid angle, since other variations are readily realized by the expedient of using energy-sensitive detectors. The central problem of measuring differential cross sections is, therefore, to measure the element of yield dY which is encompassed by the solid angle dfl, subtended by the detector at the point of intersection of beam and target. I n Fig. 16 this solid angle is = d A / R 2 and Eq. (2.7.5.3) becomes dY(0) = Nonta(0) d A / R 2 .
(2.7.5.6)
If a(0) is constant over the angular region subtended by the detector, the reaction yield recorded by the detector is simply
Y(B) = N ~ T z ~ u ( B ) A / R *
(2.7.5.7)
and the measurement of the angular dependence of the differential cross section resolves itself into a measurement of the ratio Y(O)/Noas a function of the angle between the incident beam and the line joining target and detector. However, to determine absolute differential cross sections we must also evaluate nt, A , and R2. 2.7.5.1. Differential Cross Sections for the Emission of Charged Particles from Charged-Particle Induced Reactions. 2.7.5.1.1. ENERGYSENSITIVE DETECTORS. * The usual source of charged particles for differential cross-section measurements is an electrostatic or electromagnetic * See also Vol. 4 A, Part 2.
414 2. DETERMINATION OF FUNDAMENTAL QUANTITIES
2.7.
SIDE VIEW
(b)
FIG. 17. Primary and secondary beam collimation geometry for measuring differential cross sections for charged-particle emission from charged-particle induced reactions.
DETERMINATION OF NUCLEAR
I
I
s M
*
Q
8
3
415
416
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
particle accelerator. Such machines are capable of producing microamperes of external monoenergetic beams of light nuclei of intermediate energies. For purposes of the present discussion, we define the intermediate energy region as extending from 1-50 Mev, and it is with this region that we will concern ourselves, In fact, our main emphasis will be the energy region of classical nuclear physics, i.e., below 25 MeV. As already seen, the problem of measuring differential cross sections involves the evaluation of the various parameters in Eq. (2.7.5.4). We consider the general case of a gaseous target since a solid target is but a special, and much simpler case. MAONET INSULATOR
I CAPACITOR
F1J
/
FARADAY
AMPLl FlER
1-1
CUP
I
AL. WINDOW INSULATORS
\ itsov eoo QAUSS FIELD
/
-
REQULATED
Fra. 18.Faraday cup and current integration system.
A charged-particle beam may be diaphragmed so that it takes the form of a circular cylinder of arbitrarily small radius. I n Fig. 17 a collimated beam enters the reaction volume through defining slits A and B. The nondefining apertures between the defining apertures fulfill the criteria necessary to make the collimator effectively wall-less. The first aperture is of inlaid gold sheet which serves to minimize y-ray background by excluding the divergent beam as far in front of the camera as intensity considerations will permit. The final antiscattering slit limits the spray of particles from the second defining slit to a cone of arbitrarily small half-angle. After penetrating the target, the beam is captured in a Faraday cup which serves as the collector for the current integration system. Since absolute cross-section measurements cannot be more accurate than the precision with which one measures the charge which has passed through the target, much time and effort has been devoted to the development of current integration systems. The problems encountered include the prevention of the escape of electrons formed inside the Faraday cup, shielding the cup from electrons formed external to it, and, having overcome these difficulties, precise measurement of the small
2.7.
DETERMINATION
OF NUCLEAR REACTIONS
417
charges collected by the cup. To surmount the problems of electron leakage, one has recourse to electrostatic and magnetic barriers arranged so as to form crossed electric and magnetic fields. An effective precaution is to operate the Faraday cup very close to ground potential. Figure 18 is a schematic of the Faraday cup and integration systeme6 currently used with the Los Alamos multiplate camera. The Faraday cup is insulated by glass from the camera proper. The magnetic barrier is several hundred gauss while the electrostatic barrier is -100 volts positive. The cup feeds a low-leakage high-precision condenser. During the charging cycle the potential on the positive plate is maintained close to ground by the application of a negative potential to the opposite plate. The voltage balance between the two condenser plates is maintained by a circuit built around an electrometer tube which senses any unbalance, and activates the application of a compensating voltage. The charge collected by the cup is then determined from the product of the condenser capacitance and the balancing voltage, the latter being measured by a highprecision potentiometer. Figure 19 exhibits a schematic of the electronic circuitry designed by R. D. Hiebertg6for the above described method of current integration. This integrator has been used t o measure charge in the range 0.5 to 100 pc to a precision of better than 0.5%. From the standpoint of second-order geometry corrections, it is convenient to use rectangular slits to define the product of target thickness t and solid angle Q [Eq. (2.7.5.4)] subtended by the detector. The detector views that portion of the scattering volume defined by apertures D and C (Fig. 17a), the latter aperture also serving to define the solid angle subtended by the detector a t the scattering volume. For the special case of e = go", it is seen that to a first approximation, t = 2bR0/l and
=
2ah/Ro2.
(2.7.5.8)
Furthermore t varies as l/sin 0 giving, at any angle 8, (2.7.5.9) Equation (2.7.5.9) treats the scattering geometry in an idealized manner whereby the scattered beam is assumed t o originate in a line, and the detector solid angle is of infinitesimal extent, and the same distance from all points on the line source. Actually the geometry can usually be so arranged that this picture is an excellent approximation to the real situation and only requires refinement at small and large angles where PP 96
T. M. Putnam, J. E. Brolley, Jr., and L. Rosen, Phys. Rev. 104, 1303 (1956). R. D. Hiebert, Los Alamos Scientific Laboratory, private communication.
i’’~~~‘?
CONTROL SECTION
FIG.19. Electronic circuit for current integration system.96
2.7.
DETERMINATION OF NUCLEAR REACTIONS
419
the first and second derivatives of the cross section versus scattering angle are large. An exact integration for the case of a line source and a n aperture of height h which is very small compared to Ro has been performed b y C. L. Critchfieldg' and this gives
Finite beam size. For reasons of intensity, the beam radius cannot always be <
Taking the first 2 terms of the expansion of the logarithmic term, we have (2.7.5.12) and this quantity replaces 1/Ro in Eq. (2.7.5.10). Finite angular resolution. Since the angular resolution for any given geometry is finite, we must take account of the fact that g(e) is not constant with angle.98 To first approximation one can again consider the reaction products (Fig. 17a) to originate from the intersection of a line source with the gas target. It is required to find the yield from a n element of length dl along the interaction volume into the element of area d u dv on the detector aperture and then integrate over both. The infinitesimal 97 C. L. Critchfield, private communication; see J. C. Allred, L. Rosen, F. K. Tallrnadge, and J. H. Williams, Rev.Sci. Instr. 22, 191 (1951). Q 8 I. E. Dayton and G. Schrank, Phys. Rev. 101, 1358 (1956); E. M. Lyman, A. 0. Hanson, and M. B. Scott, ibid. 84, 626 (1951); H. R. Worthington, J. N. McGruer, and D. E. Findley, &id. 90, 899 (1953).
420
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
yield is
dY =
*
cos
R2
C$
a(@)dl
d u dv
(2.7.5.13)
where R is the distance from dl to d u d v and 9 is the angle between R and the normal to the detector window. By expanding to second order in u(O), dropping linear terms because of symmetry, and integrating, one can arrive a t the desired correction. D. C. Dodderg9has carried out this calculation with the result that Eq. (2.7.5.4) becomes
Y
=
(
NontDa(0) 1
+
__
do)
- -L
a2 + b2 [I) (2.7.5.14) 612
where d ( O ) = da(O)/dO and u”(O) = d2a(0)/d02. To the same approximation as the whole derivation, the derivatives may be obtained by using a(e) uncorrected for finite angular resolution. Combining Eqs. (2.7.5.10)(2.7.5.14), we have the corrected cross-section formula
Y
=
Nono(0) csc 0 4abh RAJ ~
{[
1
+ b2 - a2 212 ~
+
a2 cot2 O ] 3R:v
[ 1 + (3 - k ? ? !(%)I] ~) L do)
(2.7.5.15)
where € 2 is~ given ~ by Eq. (2.7.5.12). It now remains only to evaluate n, the number of scattering centers per unit volume, and this is determined from the pressure and temperature of the gas in the usual way. For example, if the temperature is T°C and the pressure is P cm of Hg, then
n
=
2.688 X l O l e X
P X 0.9955 X 76.0
~
273.2 XK 273.2
+
where K is the number of atoms per gas molecule. The fourth term in the expression is a correction for the density of Hg a t room temperature. Evaluation of particle loss due to multiple scattering in absorbers. Experimental requirements in nuclear scattering and reaction studies dictate that the particles to be detected traverse some amount of scattering material on their way to the detector. In addition, it often happens that it is desirable, because of limited stopping power of the detector, or in order to discriminate between various particles on the basis of range or specific ionization, to interpose an absorber between source and detector. I n any case, the presence of an absorbing medium introduces complica99
D. C. Dodder, private communication.
2.7.
DETERMINATION
OF NUCLEAR REACTIONS
42 1
tions in the evaluation of the absolute cross section as a consequence of the resulting deflections of the particles to be detected. For charged particles the dominant interaction is multiple small-angle scattering, and it is this effect which we shall discuss. Let us direct our attention to a scattering geometry involving a circular target and detector with a common normal axis as illustrated in Fig. 20. The detector which is to determine the flux of charged particles in the effluent beam will see instead superpositions of segments of a two-dimensional Gaussian distribution of particles originating where the beam traverses the scatterer. Dickinson and Dodder loo have developed an elegant graphical method
-
1
d
1
II I
--__----_
0
I--
h
d
W
5-
DETECTOR RADIUS
-L+
FOIL RADIUS
F,
d
FIG.20. Geometry for evaluating particle loss due to multiple scattering in a circular absorber whose normal axis is coincident with that of the detector.*O*
for evaluating the particle loss due to such multiple scattering. The method is briefly as follows: one plots a series of concentric circles which are representative of the two-dimensional Gaussian distributions resulting from the scattering of the individual particles by the stopping medium. The contribution of particles from any point on the scatterer is represented by the number of circles which fall on the detector surface when the center of the two-dimensional Gaussian is superimposed on the scatterer and detector areas so that the center is a t the scatterer position under consideration. Assuming Gaussian distributions, the normalized probability density of multiple scattering through an angle e isloo
P,(O) dB = 2e <e2’aV 100
” (radians)2 4(M/m)ln(297P/Z1/2)
W. C. Dickinson and D. C. Dodder, Rev. Sci. Instr. 24, 428 (1953).
(2.7.5.16) (2.7.5.17)
422
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
where Ei and El are, respectively, the initial and final energies of the scattered particle, M / m is the ratio of its mass to that of the electron mass, Z is the atomic number of the reaction nucleus, and B is defined by the transcendental equation,
+
e E / B = 6710aZ4/S~2/A/12(1 3 . 3 3 ~ ~ )
(2.7.5.18)
where A and a are, respectively, the atomic weight and areal density of the scattering medium, z is the atomic number of scattered particles a n d 7 = Zz/137/l. For small 6, the distribution in terms of particle displacement p will be (2.7.5.19) The graph representing the two-dimensional Gaussian is then drawn so that the radial density of circles is described by
-dN _ - constant x p exp[-p2/AV]. dP
(2.7.5.20)
< p 2 > A v should be chosen so that the size of the graph is convenient for accurate counting of segments when each circle is divided into some convenient number of segments such as 20. The ratio of [ Av]1/2to d [ <e2> Av]1/2 is then the scale factor determining the sizes of areas involved in the summations. It is worth remembering that the maximum percentage loss of particles occurs when F 1 = F z . Multiple scattering of reaction products in gas target.l0' Although scattering into the detector tends to compensate for scattering which removes particles from the solid angle subtended by the detector, complete cancellation occurs only when the first and second derivatives of the variation of cross section with angle are zero. When this is not the case, a calculation similar to that leading to Eq. (2.7.5.14) shows that the particle flux a t the detector is increased, over that given by Eq. (2.7.5.15) by the factor*02 (2.7.5.21)
where Y is computed as in Eq. (2.7.5.15) and E is the rms value of the multiple scattering distribution as given by Williams.1osHowever, this multiple scattering correction is best evaluated empirically. The procedure is to measure the cross section as a function of pressure and to G. Breit, H. M. Thaxton, end L. Eisenbud, Phys. Rev. 66, 1018 (1939). C. T. Chase and R. T. Cox, Phys. Rev. 68, 243 (1940). 105 E. J. Williams, Pmc. Roy. Soc. A169, 531 (1939); Phys. Rev. 68, 292 (1940). 101
101
2.7.
DETERMINATION OF NUCLEAR REACTIONS
423
extrapolate to zero pressure. I n general, for geometries such as are illustrated in Fig. 20, this correction will not exceed 0.5% provided the energy of the reaction products is not decreased by more than 5% during traversal of the distance between reaction volume and detector. E$ect of local heating of gas by the incident beam along its path. This correction may be evaluated experimentally in the same way as the above correction, i.e., by measuring cross sections as a function of beam intensity and extrapolating to zero beam intensity. Local heating of the beam can significantly alter the gas density along the reaction volume
I--------.+ Fro. 21. Geometry for calculating corrections due to slit-edge penetration.
and thus lead to low values of measured cross sections. This effect becomes negligible, however, for protons of energy in excess of 5 Mev when the beam intensities are less than a few tenths of a microampere. Slit-edge penetration. l o 4 Still another common source of error is due to beam penetration of slit edges; however, this effect is readily and accurately corrected for. Assume we are counting particles of energy E with a detector which is biased a t some energy E - AE. The question then arises as to the increase in solid angle subtended by the detector, since the particle can lose a n amount of energy AE by slit penetration and still be counted. The effective widening of the slits is computed as follows. Consider two coaxial slits having a stopping power for the particles being detected of S Mev/cm. Let their half-widths be a and their separation I (Fig. 21), and let the particles originate in a line which lies in a 1°4 J. C. Allred, Los Alamos Scientific Laboratory Report LA-981 (1949); E. D. Courant, Rev. Sci. Znstr. 22, 1003 (1951).
424
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
plane containing the centers of the slits and the length of the slits. The slit penetration possible is AE/S. Assuming E << S T , it is apparent that particles which originate along the reaction volume defined by 2a can only penetrate slit 2. Particles outside this region can penetrate both slits, but the total penetration may not exceed A E / S . Consider P , the path of a particle which misses one edge of slit 1 by a distance C. We now compute the relative probability for traversing slit 2 with no loss of energy and with loss of energy AE. For a loss of energy AE we have Smx= AE/S. From the figure we have:
c+p1
=
pY and for p << 6, y
=
Pl
(2.7.5.22) (2.7.5.23) (2.7.5.24)
and
’=
Similarly
8‘ = P+D’=
and and for AE
CAE -AE +S1‘ (2a - C ) AE -AE Sl 2a AE -AE+S~
+
(2.7.5.25) (2.7.5.26) (2.7.5.27)
<< Sl which is usually the case 2a AE
P + P ’ = T .
(2.7.5.28)
The ratio of the number of particles which penetrate slit 2 to those which pass freely is thus AEISZ. Since this process can occur in a reciprocal manner, i.e., penetration of the edge of slit 1 and free passage through slit 2, the total correction to be applied is 2AE/Sl, i.e., the apparent cross section calculated on the basis of a slit width of 2a must be divided by (1 2AE/Sl). The above derivation is immediately valid for a detector whose vertical dimension is smaller than the slit height and whose horizontal dimension is larger than the slit width. However, identical arguments can be readily carried through for any slit-detector geometry. For example, for circular slits the correction is 2rr AE/ar2Sl. Evaluation of errors. As a rule, differential cross sections are more meaningful in the center-of-mass (c.m.) coordinate system than in the laborqtory system. The relation between the two is
+
u(8)sin $ d$ = u(9)sin qh dqh
2.7.
DETERMINATION OF NUCLEAR REACTIONS
425
where 8 and # refer respectively to the laboratory and c.m. angles of emission. In order to discuss the uncertainties in the c.m. cross section we relate the measured yield to the c.m. cross section by, cos(8 - r$)
1
(2.7.5.29)
where G/sin 0 is the geometrical factor, including corrections, determined by the slit geometry as previously discussed. This geometry depends upon the precision with which one can measure the pressure, temperature, and purity of the target gas, the geometry of the slit system and the integrated current which determines N o . Of these the latter measurement is usually subject to the largest systematic error. The accuracy with which one can measure No depends upon charge-state uncertainty, Faraday cup geometry, and the precision with which one can determine the condenser calibration and leakage. Some of the most common causes of error are: (1) Loss of part of the beam due to scattering in isolation windows and target, (2) Low- and/or highenergy components of the beam for which the corresponding reaction products are not counted, (3) Loss or collection of electrons or ions by the cup and electrical leads as a result of ionization in the region of the cup and photoelectron processes. The angle-dependent uncertainty is also systematic and depends upon the accuracy with which one can position the slit-system axis with respect to the primary beam axis. The uncertainty in cross section produced by a given angular uncertainty depends, of course, on the variation of cross section with angle. For the case of Coulomb scattering, the differential cross section is given by cm2/steradian
(2.7.5.30)
where ml and m2 refer, respectively, to the mass of the incident and scattered nucleus, E is the incident laboratory energy in MeV, and e2 is expressed in Mev-cm. Then, the change in cross section with angle is given by du(4) = -2 (cot#/2)u($) dr$. The angle independent uncertainty, the angle dependent uncertainty and error in Y due to statistical and detection efficiency uncertainties are all independent of each other. The final uncertainty is therefore the r.m.s. sum of the individual systematic and statistical uncertainties. In addition, most cross sections are energy dependent. Again, for the Coulomb scattering the cross-section
2.
426
DETERMINATION O F FUNDAMENTAL QUANTITIES
uncertainty is related to the energy uncertainty by 2dEIE = dolo, and this is a fair estimate for nuclear scattering as well. Although the main features of any experiment designed to measure differential cross sections is as above described, there exist many variants. The chief variant is the type of detector used. This is usually dictated by considerations of reaction energetics, anticipated cross sections, and neutron and y-ray background. The detectors most often used are
FIG.22. Multiplate camera for measuring differential cross sections for the emission of charged particles from charged-particle induced reactions.
scintillation counters, proportional counters, ionization chambers, and nuclear emulsions.* When using electronic detectors it is not feasible to record data simultaneously at all angles, and therefore provision must be made for rotating the counter about an axis through the center of the target and perpendicular to the reaction plane. Electronic detection does, however, permit coincidence counting of reaction products whether they be charged or neutral. Figure 22 pictures one of the experimental arrangements that is used with the Los Alamos variable-energy cyclotron to measure the differential cross sections for light nuclei interactions. lo6 The cyclotron accel-
* See Section 2.7.7 for more recent supplementary remarks. 101
J. C. Allred, L. Rosen, F. K. Tallmadge, and J. H. Williams, Rev. Sci. Znstr. 22,
191 (1951).
427
2 . 7 . DETERMINATION OF NUCLEAR REACTIONS
erates deuterons to energies of 6-14 MeV. The deuteron beam is conducted from the accelerator vault by an ion optical system, having suitable focusing and steering properties, into the muItiplate camera through a beam-defining system which limits the angular divergence to <
l
l
1
1
I
I
I
I
I
I
,
,
)
,
\
,
,
/
\
E d = 14 MEV
0 O0
30
60
90
CENTER-OF-MASS ANGLE,
I20
I50
4I
e
FIQ.23. Typical results obtained with the multiplate camera; D(d,n) and D(d,p) differential cross sections. (Brolley et al., ref. 106.)
are simultaneously detected by 66 nuclear emulsions situated on radii at 2.5" intervals. Each detector views a well-defined portion of the reaction volume through a precision-machined and accurately aligned set of slits. The trajectories of the various charged particles produced in a given reaction are recorded in the emulsion, and the particles are identified through their range and specific ionization. The camera is both versatile and accurate and has consequently been used to measure a large number
428
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
of reaction and elastic-scattering cross sections.lo6 Figure 23 displays an example of the kind of data that have been obtained. The advantage of simultaneous recording of data a t all angles has been utilized in a number of other nuclear plate cameras,107some of which are designed to record all the data on only a few plates,10swhile others are quite similar to the one described. Figure 24 pictures a typical reaction chamber which uses electronic c0unting.1~~ The particle detector is a scintillation crystal and photomultiplier connected by a Lucite light pipe. A rectangular slit defines the solid angle subtended by the detector a t the target. An absorber immediately in front of the detector may be used to attenuate the energy of the reaction products. The entire counting system is mounted on an arm centered on the center of the target inside the evacuated chamber and can be oriented a t any angle with respect to the primary beam. The pulses from the scintillation counter are displayed on a multichannel pulse-height analyzer. The Faraday cup is supported by Lucite rings in a tube connected to the rear of the chamber along the beam axis, and is electrically connected to precision polystyrene condensers, the voltage across which is measured by a vibrating reed electrometer. A rather large number of similar chambers are now in operation."O They differ mainly in the type of electronic detector and targets (gas or solid) used and in the degree of electronic automation involved.* Although it often happens, especially in two-body reactions, that the particles can be identified by their range or energy loss in the detector,
* See also Vol. 2. J. H. Williams and S. W. Rasmussen, Phys. Rev. 98, 56 (1955);T. M.Putnam, ibid. 87, 932 (1952);L. Rosen, F. K. Tallmadge, and J. H. Williams, ibid. 76, 1283 (1949);L. Rosen and J. C. Allred, ibid. 82, 777 (1951);J. C. Allred, D. D. Phillips, and L. Rosen, ibid. 82, 782 (1951);J. C. Allred, D. K. Froman, A. M. Hudson, and L. Rosen, ibid. 82, 786 (1951);J. C. Allred, ibid. 84, 695 (1951);J. C. Allred, A. H. Armstrong, A. M. Hudson, R. M. Potter, E. S. Robinson, L. Rosen, and E. J. Stovall, Jr., ibid. 88, 425 (1952);L. Rosen and J. C. Allred, ibid. 88, 431 (1952);J. C.Allred, A. H. Armstrong, R. 0. Bondelid, and L. Rosen, ibid. 88, 433 (1952);T. M.Putnam, J. E. Brolley, Jr., and L. Rosen, ibid. 104, 1303 (1956);J. E. Brolley, Jr., T. M. Putnam, and L. Rosen, ibid. 107, 820 (1957). 10' H. D. Holmgren, M. L. Bullock, and W. E. Kunz, phys. Rev. 104, 1446 (1956); S. W. Rasmussen, ibid. 103, 186 (1956). E l 0 8 H. B. Burrows, C. F. Powell, and J. Rotblat, Proc. Roy. Soe. A209,461 (1951). 1 0 9 F . S. Eby, Phys. Rev. 96,1355 (1954);H. Palevsky, R. K. Swank, and R. Grenchik, Rev. Sci. Znstr. 18, 298 (1947). 110 P. H. Stelson and W. M. Preston, Phys. Rev. 06,974 (1954);G. E.Fischer, ibid. 96, 704 (1954);R. M.Eisberg and G. Igo, ibid. 93,1039 (1954);P. von Herrmann and G. F. Pieper, ibid. 106, 1556 (1957);J. L.Yntema and M. G. White, ibid. 96, 1226 (1954);R. E. Ellis and L. Schecter, ibid. 101, 636 (1956). 108
2.7.
BEAM COLLIMATOR
DETERMINATION OF NUCLEAR REACTIONS
71
429
TARGET FOIL
TARGET SUPPORT ADJUSTMENTS
BOTTOM HEMISPHERE
FIG.24. Top and right-end cross-sectional views of reaction chamber using electronic counting. (Eby, ref. 109.)
this is obviously not a universal phenomenon. Some of the chambers presently in use have, therefore, been designed to identify the mass as well as the energy of the reaction products. A rather ingenious and relatively simple arrangement using nuclear plate detection with magnetic deflection was developed a t the University of Minnesota for use with their Van de Graaff. l l 1 A wholly electronic method for distinguishing between the masses of reaction products and one which simultaneously discriminates against all types of background has been developed and is now being used with 111 H. D. Holmgren, J. M. Blair, B. E. Simmons, T. F. Stratton, and R. V. Stuart, Phys. Rev. 96, 1544 (1954).
430
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
great success.112Briefly, the arrangement usually consists of a transmission ion chamber and a NaI scintillator in tandem and coincidence. The passage of a particle is recorded when it traverses both the ion chamber and the NaI scintillator. The ion chamber is operated at very high pressure to reduce Landau straggling while still leaving a large fraction of the energy in the scintillator. The output of the ion chamber varies as d E / d x E Z2/v2 while the output of the crystal counter is essentially proportional to the particle energy E = &Mu2. The output pulses are electronically multiplied to yield a pulse height which is proportional to the product of the mass and the square of the charge of the scattered particle. The product spectrum so obtained can be sent through a pulseheight analyzer and into one side of a coincidence circuit while the output from the scintillator is fed into the other side of the coincidence circuit, thus making it possible to obtain the energy spectrum corresponding to a particle group. In order to measure cross sections which are pertinent to energy generation from thermonuclear reactions, it is essential to work at very low input energies. This presents a great many unique problems having to do with the evaluation of incident energy, integrated current, and multiple scattering. At very low energies it is extremely difficult to isolate the reaction chamber from the accelerator without doing violence to the energy definition. Also multiple scattering is obviously a problem, as is measurement of charge, due to the fact that the velocities involved are not large compared to the velocities of the electrons in their atomic orbits, a phenomenon which is conducive to charge-exchange processes. A reaction chamber was therefore developed by J. L. Tuck and coll a b o r a t o r ~for ~ ~the ~ specific purpose of exploring the d-D and d-T reactions at very low deuteron energies. A number of now classic measurements were made with this apparatus. For a description of the apparatus and results obtained with it, the reader is referred to the original paper.113 For a summary of charged-particle cross-section measurements in the energy region under discussion, the reader is referred to the recent Los Alamos compilation. 114 H. E. Wegner, R. M. Eisberg, and G. Igo, P h p . Rev. 99,825 (1955); J . G. Likely and F. P. Brady, ibid. 104, 118 (1956); K. Boyer, H. E. Gove, J. A. Harvey, M. Deutsch, and M. S. Livingston, Rev. Sci. Znstr. 22, 310 (1951); R. H. Stokes, J. A. Northrop, and K. Boyer, Rev. Sci. Znstr. 29,61 (1958); W. Briscoe, ibid. 29,401 (1958); R. H. Stokes, ibid. 31, 768 (1960). llJ W. R. Arnold, J. A. Phillips, G. A. Sawyer, E. J. Stovall, Jr., and J. L. Tuck, Phya. Rev. 93, 483 (1954). 1 1 4 N. Jarmie and J. D. Seagrave, eds., “Charged Particle Cross Sections.” Los Alamos Scientific Laboratory Report LA-2014 (1956); D. B. Smith, ed., and N . Jarmie and J. D. Seagrave, assoc. eds., “Charged Particle Cross Sections-Neon to Chromium.” LOBAlamos Scientific Laboratory Report LA-2424 (1960).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
43 1
2.7.5.1.2. MAGNETIC REACTION-PRODUCT ANALYZERS. I n addition to the various energy-sensitive detectors just described for measuring differential cross sections, magnetic analyzers are often used for this purpose. These analyzers record the magnetic rigidity (Bp) of a particle, or its momentum divided by its charge. Most of the considerations of the previous section apply to the measurement of differential cross sections with a magnetic analyzer. The purpose of this section is to remark briefly on the types of magnetic reaction-particle analyzers usually employed (detailed considerations are given in Section 2.2.1.1)and to discuss crosssection considerations peculiar to such instruments. Experimental considerations. The term “reaction-particle magnetic analyzer” used above includes two basic types: (a) the magnetic spectrometer and (b) the magnetic spectrograph. These differ basically only in the detection system employed, but a number of practical differences arise in their use. In a magnetic spectrometer, particles of a definite magnetic rigidity are brought to a focus a t a set of fixed image-slits placed in front of a suitable detector by adjusting the magnetic field appropriately. Since only a certain momentum increment will be accepted by the slits a t a given field, it is necessary to make repeated runs at different magneticfield settings in order to study a spectrum of particle groups. For this reason, a large acceptance solid angle is desirable to minimize the length of each run. Although some uniform-field single-focusing instruments are employed as spectrometers (see, e.g., ref. 115),the desire for large solid angle has led to the development at the California Institute of Technology of shaped-field double-focusing spectrometers for heavy p a r t i ~ l e s l ~ ~ ~ ~ ~ ’ based on a design by Siegbahn and Svartholm.”* Figure 25 shows a typical insta1lation1lBof a spectrometer of this design. Although the large solid angle increases the intensity, it also reduces the angular resolution, necessitating angular resolution corrections to the cross section of the type described in the previous section, as well as reducing the accuracy of &-value measurements in the lighter nuclei. The mass of the observed particle is usually identified by the use of an energy-sensitive detector, typically a scintillation counter, preceded by thin foils a t the spectrometer focus. Since the magnetic rigidity of the 116 R. S. Bender, E. M. Reilley, A. J. Allen, R. Ely, J. S. Arthur, and H. J. Hausman, Rev. Sci. Znslr. 28, 542 (1952). 116 C. W. Snyder, S. Rubin, W. A. Fowler, and C. C. Lauritsen, Rev. Sci. Instr. 21, 852 (1950). 117 Pi. Whaling and C. W. Li, Phys. Rev. 81, 150 (1951); D. L. Judd, Rev. Sci. Instr. 21, 213 (1950). 118 N. Svartholm and K . Siegbahn, Arkiu Mat. Astron. Fysik 88A, 28 (1947). 119 V. K. Rasmussen, D. W. Miller, and M . B. Sampson, Phys. Rev. 100, 181 (1955).
432
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
particle is known, its range in the foils coupled with its energy recorded by the detector allow unambiguous particle identification. In a magnetic spectrometer the solid-angle determination can be calculated purely from the geometry as described in the previous section, provided the defining aperture widths and positions are accurately known. Because of the shaped field in double-focusing spectrometers, however, the defining slits are frequently not rectangular, which adds to the complexity
BEAM
FIG.25. The Indiana-University heavy-particle spectr~rneter,~~~ a typical installation of a double-focusing spectrometer of the California Institute of Technology design.llB
of the calculation. In this event, the spectrometer solid angle may also be obtained experimentally by a comparison measurement with a geometry whose solid angle is accurately known. For example, the counting rate with a small aperture of known geometry placed in front of the spectrometer can be compared with the counting rate with the aperture removed,120or the counting rate of a separate detector of known solid angle can be compared with the spectrometer counting rate using a-par120
W. A. Wensel and W. Whaling, Phye. Rev. 87, 499 (1952).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
433
ticles from a radioactive source.121The solid angle may also be determined by suitable comparison with Rutherford scattering or any other known cross section if the stopping cross section of the scattering target is known.122,123A typical solid angle for a double-focusing spectrometer is about 6 X low3steradian.lZ2 ,Photomu1 tiplier track plate
FIG. 26. Sketch of the installation of the Massachusetts Institute of Technology broad-range spectrograph.lz6
In a magnetic spectrograph, photographic plate detection is employed to allow simultaneous recording of a considerable momentum range a t a given magnetic field. This type of instrument is normally used primarily for &-value measurements where very high resolution is desired. Although 1 2 1 S. H. Levine, R. S. Bender, and J. N. McGruer, Phys. Rev. 97, 1249 (1955); M. T. McEllistrem, H. J. Martin, D. W. Miller, and M. B. Sampson, ibid. 111, 1636
(1958). 122 C. Y. Chao, A. V. Tollestrup, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 79, 108 (1950). 123 A. B. Brown, C. W. Snyder, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 82, 159 (1951).
434
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
double-focusing analyzers are occasionally used as s p e ~ t r o g r a p h s the ,~~~ single-focusing “broad range’’ s p e c t r ~ g r a p h ’ ~ ~has - ~ ~ ’been more extensively employed. Such an instrument can record simultaneously particles varying in energy by a factor of from 2.5lZ5to 6.’z6 Because of this simultaneous recording, the “effective solid angle” (as far as data gathering capacity is concerned) can be equivalent to a spectrometer with a solid angle of about 3 X lo+ steradian,’Z6 while still retaining good angular resolution. Aside from the necessity of reading photographic plates, probably the only major disadvantage of this type of instrument is that it cannot be used for coincidence experiments, etc. Figure 26 shows a sketch of the Massachusetts Institute of Technology broad-range spectrograph installation (from Browne and B u e ~ h n e r ’ ~ ~ ) . Particle identification in a spectrograph is obtained from simultaneous knowledge of the magnetic rigidity of the particle and its range and track character in the photographic emulsion. I n a broad-range spectrograph, the solid angle varies with position on the photographic plate, a n effect not ordinarily of concern in spectrometers. The accurate determination of cross sections then requires that this solid-angle variation also be known accurately. Figure 27 shows the variation of the relative solid angle of the Massachusetts Institute of Technology spectrograph. l Z 6 This measurement was made by scattering protons from a thin gold target, and shifting the proton group from place to place on the plate by varying the magnetic field. Cross-section determination. Because magnetic reaction-particle analyzers are most often employed to study specific groups of particles emitted in nuclear reactions, some consideration must be given to the methods used for determining the total yield of reaction particles in a particular group. I n a magnetic spectrograph, this can be done simply by counting the tracks in the appropriate group on the photographic plate. In a magnetic spectrometer, this yield measurement may be obtained by the method outlined below, following the procedure given in basic papers by the group a t the California Institute of Technology.116e12a In a double-focusing spectrometer, it is difficult to measure the field by proton-resonance methods because of the shaped field. Therefore, a torsion-type128 or balance-typeltQ fluxmeter is frequently employed. With such a device, the current I necessary to balance the fluxmeter is 1*4 D. R.Bach, W. J. Childs, R. W. Hockney, P. V. C. Hough, and W. C. Parkinson, Rev. Sci. Instr. 27, 516 (1956); R. W . Peterson, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 96, 1250 (1954). l*s C. P. Browne and W. W. Buechner, Rev. Sci. Instr. 27, 899 (1956). 118 T. S. Green and R. Middleton, Proc. Phys. Soc. (London) A09, 16 (1956). 11’B. Elbek, K. 0. Nielsen, and M. C. Olesen, Phye. Rev. 108, 406 (1957). 118 C. C. Lauritsen and T. Lauritsen, Rev. Sci. Instr. 18, 916 (1948).
2.7.
435
DETERMINATION OF NUCLEAR REACTIONS
inversely proportional to the magnetic field, and therefore the momentum of the detected particles. Let y(1) be the yield per incident charged particle (with charge ze) per unit solid angle in the center-of-mass system per unit fluxmeter current, N ( I ) be the number of counts observed in the spectrometer at fluxmeter current I when a total integrated charge Q strikes
1.4
-
1.3 V (r
4
1.2 -
U .-
._ P
1.1
-
L
O -
a" 1.0 -
-
0.9
I
I
I
20
I 40
I
I
60
I
0
Distance olonq Plate(cm)
FIG.27. Relative solid angle as a function of position on the photographic plate for the Massachusetts Institute of Technology broad-range spectrograph.lz5
the target, and a, be the solid angle in the c.m. system corresponding to the spectrometer acceptance solid angle a in the laboratory system. Then,
N(I)
=
- -
y(I)Q, 6 1 ( Q / z e )
(2.7.5.31)
where 61 represents the effective interval on the fluxmeter current scale which corresponds to the momentum interval 6P, accepted by the spectrometer collector (image) slit width. The resolution R, of the spectrometer, appropriate to this collector slit width, is R , = P/6Pc. Since the fluxmeter current I = K / P , d I / I = -dP/P, so that 1611 = I/&.
(2.7.5.32)
Combining Eqs. (2.7.5.31) and (2.7.5.32), one obtains
~ ( =0( W a c ) (ze/Q)[N(I)/II.
(2.7.5.33)
If a thin target is employed, resulting in a discrete momentum profile for the partide group at a particular laboratory angle, one obtains the total yield Y* (at that angle of observation) per unit solid angle in the
436
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
c.m. system per incident charged particle (with charge ze) by integrating over the total area under the peak:
Y*
=
1
y(l) d l
= (R,/Q,)(ze/&)
/
dl.
(2.7.5.34)
The differential cross section in the c.m. system, using the notation of Section 2.7.5.1.1 where Y* = Y(O)/NoQ,is then a(4) = Y*/nt =
(R,/ntQ,)(ze/&)
/
dl
(2.7.5.35)
where nt is the nuclear density times the target thickness parallel to the beam. The solid angle Qc is readily obtained from the solid angle Q by the familiar relation122.123 (2.7.5.36) D, = (1 2a cos 4 a 2 ) 3 ’ 2 11 a COB 41 where
+
+
+
and the subscripts 0, 1, 2, and 3 refer to the target, incident, observed, and residual particles, respectively. 4 is the angle of emission in the center-of-mass system, and El the bombarding energy in the laboratory system. If the incident particles lose energy El in ev in passing through the thin target, and if the “stopping cross section” (or atomic stopping power) (dE/dx)(l/n) in 10-l6 ev cme is €1, then nl = (&/el) X 10l6, and Eq. (2.7.5.35) can be written:
where u ( 4 ) is in millibarns per steradian, and q is the charge striking the target expressed in microcoulombs. In Eq. (2.7.5.37), 2?re in microcoulombs has been replaced by 10-l2, which is accurate to 0.7%. The cross section can actually be determined more accurately in most cases by a thick-target rnea~urernent,’~~ provided the outgoing particle groups involved are separated sufficiently in energy and the cross section does not vary too rapidly with bombarding energy. Suppose the energy increment accepted by the spectrometer collector slit is 6Ez. Then, assuming nonrelativistic particles, dE2/E2 = 2dP/P = 2/R,, so that 6E2 = 2E2/Rc. 1aoL.
1949.
(2.7.5.38)
I. Schiff, “Quantum Mechanics,” 1st ed., p. 99. McGraw-Hill, New York,
2.7.
DETERMINATION OF NUCLEAR REACTIONS
437
If a target is used which is sufficiently thick that the energy spread of the outgoing particles is greater than the total energy spread from all other causes including in particular 6E2,the yield curve will show a flat top of maximum count Nmx. To determine the cross section, it is useful to imagine a target whose thickness along the beam direction t, is just sufficient for the spread in energy of the outgoing particles to be equal to 6E2. If Y,* represents the number of disintegrations in such a target per incident particle (of charge ze) per unit solid angle, then Y,*
=
(2.7.5.39)
y(1) 6 1
where 6 1 is defined in Eq. (2.7.5.32). Combining Eqs. (2.7.5.31) and (2.7.5.39),we then obtain
N,,,
=
(Y,*QJ( Q l z e )
or (2.7.5.40) where q is the integrated beam charge expressed in microcoulombs. If the angle between the incident beam and the normal to the target is el, and between the normal to the target and the spectrometer direction is 02, the cross section can be obtained using the procedure of ref. 123. Considering the observed particle energy E 2 as a function of the bombarding energy El through the usual kinematical equations, a Taylor expansion of E2 in terms of El for a target whose thickness just gives a spread in observed particles of 6E2 yields
6E2=
$I
cos 91 + c2 1 - - ceff. cos 9 2 &I
€1
(2.7.5.41)
€1
Here € 1 is the stopping cross section in 10-'6 ev-cm2 previously defined for the incident beam particles, e2 the same quantity for the observed reaction particles, Ceff the quantity in absolute value signs, and &I the thickness of the target in ev parallel t o the beam direction. The number of nuclei/cm2 viewed by the beam in a target whose thickness is just t, is obtained by combining Eqs. (2.7.5.38) and (2.7.5.39): (2.7.5.42) where E P is in ev. Equation (2.7.5.42) represents the density of nuclei times the effective target thickness in a thick-target spectrometer measurement using a pure element for a solid target (if a compound is used,
438
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
a straightforward modification to Eq. (2.7.5.42) is required130). Finally, the differential cross section in the c.m. system is obtained, from its definition, for this case by
u(4)
=
Y,*/nt,
=
1
1 z N ~ ~ ~ mb R ~ E ~ (2.7.5.43) ~ ~ 4s qQ,E2 steradian
[
where Qc and eeffare given by Eqs. (2.7.5.36) and (2.7.5.41), and Ez is in ev. Equation (2.7.5.43) allows one to determine the cross section as a function of bombarding energy quite readily. Rather than requiring the shape of the appropriate group to be examined in detail a t each bombarding energy, as would be required by Eq. (2.7.5.37) for a thin target, one has only to observe the maximum count Nmsxof the yield from a thick target at each bombarding energy. However, the technique is only applicable t o solid target material which contains no nuclei heavier than those for which the scattering is to be 0 b ~ e r v e d . lThis ~~ procedure has been used with an electrostatic generator by setting the magnetic spectrometer a t a “following” energy for each value of beam energy, such that the effective target thickness represented a layer of the target below surface contaminations.lZa (A typical result of such a measurement will be shown in Fig. 43 of Section 2.7.6.2.) It has also been used with a cyclotron of fixed energy to obtain the yield for a reaction as a function of depth in the target at which the observed particles originate; this can readily be reinterpreted in terms of the cross section as a function of the beam energy.’lDFigure 28 shows a diagram of the angles and the energy relations involved in a thick-target measurement. l 2 3 A correction to Eq. (2.7.5.37) or (2.7.5.43) is necessary if the observed particles may exist in more than one charge state, e.g., He++, He+, and neutral He. A particle group observed a t a particular magnetic field in a spectrometer represents only ions in a particular charge state; a correct yield determination for that particle group should also include those particles in other charge states. To determine the correction, one may in some cases determine the numbers of ions emerging from the target in other charge states directly in the spectrometer.11sIn the most common case of a-particles, the correction may be calculated123from the classic data of Rutherford, of Henderson, and of Briggs. 132 At the energies usually employed in nuclear reaction studies, the neutral component of the exit a-particles is usually negligible, so that the corrected cross section in Eq. (2.7.5.37) or (2.7.5.43) is obtained by multiplying by the G . W. Tautfest and S. Rubin, Phys. Rev. 103, 196 (1956). F. B. Hagedorn, F. S. Moaer, T. S. Webb, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 106, 219 (1957). 13lE. Rutherford, Phil. Mag. 41, 277 (1924); G. H. Henderson, Proc. Roy. Soc. AlOQ, 157 (1925); G. H. Briggs, i&d. A114, 341 (1927). lao
131
2.7.
DETERMINATION
OF NUCLEAR REACTIONS
439
factor (2.7.5.44) The accuracy of the cross section obtained by the use of Eq. (2.7.5.37) or (2.7.5.43) for thin- or thick-target measurements, respectively, is limited primarily by the uncertainty in the st,opping cross section of the target material, which is usually only known to about 5%. I n addition, the solid angles usually are only determined to about 2 % and uncertainties in target composition and uniformity may introduce additional LAMINA IN WHICH PARTICLES PASSING THROUGH SPECTROMETER ORIGINATE
TO CURRENT INTEGRATOR
TARGET
Fro. 28. Energy relations for incident and observed particles in thick-target crosssection measurements with a magnetic s p e c t r ~ r n e t e r . ~ ~ ~
errors of a few per cent. I n regions of rapid cross-section variation, the energy resolution may be insufficient to reproduce the cross section correctly. I n this event, corrections for the finite energy resolution can be made in favorable cases using a procedure outlined in detail by Webb et ~ 1 . The ~ 3 ~over-all probable error in the absolute cross section determined in spectrometer measurements is typically in the vicinity of 6 % (see, e.g., ref. 133). Detailed discussions of the problems of cross-section and &-value measurements with double-focusing magnetic spectrometers are given in refs. 116 and 123. Magnetic spectrographs, because of their high resolution, are generally used for accurate energy measurements rather than to determine absolute cross sections, although relative cross sections are often determined for angular-distribution measurements. Absolute cross sections can be 1 3 3 T. S. Webb, F. B. Hagedorn, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 99, 138 (1955).
440
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
obtained provided the total number of tracks in a given group on the photographic plate are counted, and the solid angle of the instrument is measured carefully at all positions along the plate. Most of the cross sections quoted in the literature which were obtained with spectrographs are actually the result of normalizing to a known cross section (see, e.g., ref. 134). In a few cases an accurate measurement of the absolute cross section has been made with a spectrograph, again with a probable error of about 5
2.7.5.2. Differential Cross Sections for the Emission of Neutral Reaction Products from Charged-Particle Induced Reactions. Some of the considerations required in the determination of cross sections for (charged, neutral) type reactions have already been discussed in Section 2.7.5.1. Since the incident particles are charged in both cases, the problems of current integration, beam scattering, etc., are the same. The important difference arises in the determination of the yield of neutral particles produced in the reactions under consideration here. This yield (and therefore the cross section) is usually more difficult to measure with precision because an additional process is required, namely the conversion of the neutral particle into a charged particle for detection purposes. This conversion process is represented by a nuclear reaction or scattering for neutrons, or the processes of the photoelectric effect, Compton scattering, or pair production for y-rays. Although the details of these various conversions are for the most part accurately known, the actual yield measured depends sensitively upon the exact geometry of the experiment. Furthermore, in the case of y-ray conversion the various processes can usually all occur at once, which obviously complicates the interpretation. I n the present section, no attempt will be made to present these various problems in mathematical detail. Instead, the considerations required will be summarized, and the reader will be referred to typical references which discuss the details of the measurements. It is most logical to discuss the cases of neutron emission and y-emission separately. I n many cases the detection schemes are identical with those discussed in Section 2.7.5.1.3 for (neutron, neutral) reactions, and frequent cross references will be made to minimize duplication. 2.7.5.2.1. NEUTRON EMISSION. The basic problem in the determination of the cross section for (charged, neutron) reactions is the absolute measurement of a fast neutron yield. There are a number of methods for the absolute measurement of fast neutron flux. A complete discussion of these methods is given in Section 2.8. Some of the more popular current 134 C. P. Browne, Phys. Rev. 104, 1598 (1956); D. W. Berger and 0. J. Loper, ihid. 104, 1603 (1956). 156 B. Elbek and C. K. Bockelman, Phys. Rev. 106, 657 (1957).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
44 1
techniques for measurement of the neutron yields from (charged, neutron) reactions are based upon recoil-proton detectors, flat-response counters, and time-of-flight detectors. A few typical examples of these methods are considered below. An example of a recoil-proton detector for this type of cross-section measurement is the nuclear emulsion, which has been used in a variety of (charged, neutral) reactions (see, e.g., refs. 7, 136). A discussion of yield measurements with this type of detection will be given for neutroninduced reactions in Section 2.7.5.3.2, and will not be considered further here. The counter telescope, like the photographic emulsion, operates on the proton-recoil mechanism, but in this case the radiator is external to the energy-sensing mechanism and electronic recording is used. The radiator is usually of CH2 and is followed by two proportional counters and a scintillator of NaI(T1). The three detectors are arranged in tandem and operated in coincidence to discriminate against background effects. As usually operated, the energy loss in the proportional counters is very small, so that the pulse-height spectrum observed in the scintillator gives directly the spectrum of neutrons incident on the radiator. Backgrounds may be obtained by exposing a blank side of the radiator support to the telescope, preferably coated with a layer of carbon containing the same number of atoms as the carbon in the CH2. Typical proton-recoil telescopes for neutron detection are described by Johnson and Trai113’ and by Bame and Perry. 138 In the neutron energy range from 2 to 20 MeV, the telescope described by Johnson and Trailla’ exhibits good resolution (5% a t 13.7 MeV) and low background (about 6%). Although its efficiency is low (3.6 X a t 13.7 Mev), it may be readily calculated to an accuracy limited primarily by the accuracy of the measured n-p cross section. Proton-recoil telescopes are therefore very useful for absolute cross-section measurements. Bame and describe absolute crosssection measurements with a quoted standard error of 5 5 %. A number of measurements of absolute cross sections have been made by means of calibrated neutron sources in conjunction with a ‘‘long already described in Section 2.7.4.2. A typical experiment of this type is described by Jarvis, et aZ.l3OThe efficiency of the long counter may be determined by placing a calibrated Ra-Be neutron source (emitting a total of Q neutrons per minute) a t the target position and observing 186 W. M. Gibson, Proe. Phys. SOC. (London) A62, 586 (1949); A. G. Rubin, F. Ajzenberg-Selove, and H. Mark, Phys. Rev. 104, 727 (1956). 137 C. H. Johnson and C. C. Trail, Rev. Sci. Insir. 27, 468 (1956). 138 S. J. Bame, Jr. and J. E. Perry, Jr., Phys. Rev. 107, 1616 (1957). 139 G. A. Jarvis, A. Hemmendinger, H. V. Argo, and R. F. Taschek, Phys. Rev. 79, 929 (1950).
2.
442
DETERMINATION OF FUNDAMENTAL QUANTITIES
the counting rate per minute (C). When the number of neutron counts per microcoulomb (C') of incident beam is observed, the differential cross section can be obtained from
Q C )
u(e) =
(G)(c)
(A)
(2.7.5.45)
where nt is the number of nuclei/cm2 in the target and b is the number of incident beam charged particles per microcoulomb of integrated beam. I n this m e a ~ u r e r n e n t , the ' ~ ~ quoted accuracy of the absolute cross section obtained for the HS(p,n)He3reaction was j~lo%, half of the error being due to the calibration of the Ra-Be source and half to the tritium composition of the target. Another important method for determining the energy spectrum and flux from a (charged, neutron) reaction makes use of time-of-flight measurements on the neutrons produced in the target by a pulsed (charged) incident beam. Measurements of absolute cross sections can be made by this technique to accuracies of about l O % . l S This method will not be considered further here, since it is discussed in detail for the case of neutron-induced reactions in Section 2.7.5.3.2. 2.7.5.2.2. GAMMA EMISSION. * Before the development of detectors sensitive to the y-ray energy, a t least in detail, absolute y-ray yields were usually measured by the use of quartz-fiber electroscopes or cylindrical Geiger-Muller counters. As previously suggested, these measurements require detailed evaluation of the specific geometry involved, theoretically or empirically or both, in order to be interpretable in terms of absolute yields. Considering only the GM counter, for example, the basic problem is to determine the efficiency of the counter, i.e., the number of counts recorded per incident quantum, as a function of the energy of the incident radiation. This in turn requires detailed consideration of the geometry of the counter, the absorption of the primary radiation in the counter wall, and the production of ionizing secondaries by the photoelectric, Compton, and pair-production processes. Corrections must then be applied for the effects of the buildup of degraded secondary y-radiation, the ionization and radiative losses of the secondary electrons, the scattering of secondary electrons in the counter wall, etc. For cylindrical counters employing specific converters, empirical measurements of the counter efficiency are in agreement with theoretical calculations. A detailed discussion of yield measurements with GM counters and electroscopes is given by Fowler, Lauritsen, and Lauritsen.140
* See Section 2.7.7 for more recent supplementary remarks. 140
W. A. Fowler, C. C. Lauritsen, and T. Lauritsen, Revs. Modern Phys. 20, 236
(1948).
2.7.
DETERMINATION
OF NUCLEAR REACTIONS
443
Typical measurements of (charged, y) cross sections based on these considerations and employing a single G M counter, or two or more counters in coincidence separated by suitable absorbers, are given by ~ the yield Y * ( y ) of y-rays Seagravel4l and by Schardt et ~ 1 . lI n~ general per incident charged particle (charge ze) per unit solid angle is,122 (2.7.5.46) where N , is the number of secondaries a t a given angle of observation registered by the counter when a total charge Q strikes the target, e is the efficiency of secondary-electron production in the converter surrounding the counter,140a is the absorption factor arising from material between target and counter, and fly is the solid angle of the detector subtended a t the target. Equation (2.7.5.46) is the gamma-ray analogy to Eq. (2.7.5.44) of Section 2.7.5.1.2 for charged-particle magneticspectrometer measurements. Chao et al. lZ2 have employed Eq. (2.7.5.34) and Eq. (2.7.5.46) above to compare the yields of a-particles and y-rays from the F19((p,a)016*(y)reaction. Expecting a one-to-one correspondence, this allows an experimental check on the computed GM counter efficiency. The results indicate th at the error in computed counter efficiency obtained by the method of Fowler et ~ 1 . 1 4is~ a t most 10% for y-rays up to 7-Mev energy. Figure 29 illustrates the yield of gammas and alphas obtained in this experiment.140 I n early measurements some idea of the relative contribution to the observed y-ray yield from hard and soft components was obtained by interposing suitable lead absorbers. Gamma-ray energy measurement has since been made precise, however, with the development of a variety of detectors sensitive to the energy of the y-ray (see, e.g., refs. 143-148). With some exceptions, these instruments have been used primarily for energy measurements and only secondarily for yield and cross-section determinations. I n the original Walker and McDaniel pair spectrometer,143electron pairs produced in a thin radiator were analyzed magnetically by a uniform field and counted by suitable Geiger counters in coincidence. I n this arrangement it was possible to obtain the relative intensities of y-rays b y obtaining the areas under the y-ray peaks and correcting for counter resolution and the variation of the pair-production J. D. Seagrave, Phys. Rev. 86, 197 (1952). A. Schardt, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 86, 527 (1952). 143 R.L.Walker and B. D. McDaniel, Phys. Rev. 74, 315 (1948). 144 W. F.Hornyak, T. Lauritsen, and V. K. Rasmussen, Phys. Rev. 76, 731 (1949). 145 R. Hofstadter and J. A. McIntyre, Phys. Rev. 80, 631 (1950). 1 4 e R. D.Bent, T. W. Bonner, and R. F. Sippel, Phys. Rev. 98, 1237 (1955). 1 4 7 D. E. Alburger and B. J. Toppel, Phys. Rev. 100, 1357 (1955). 148 R. D.Bent and T. H. Kruse, Phys. Rev. 108, 802 (1957). 141
142
I
7-
0
100
t-
U
z
“r (3
+ Ip rp
LL
0 0
4
a -I W U
860
I
I
8
FIG.29. Experimental yield of 7-rays and a-particles from the F l g ( p , a y ) reaction, as determined using Eqs. (2.7.5.3.4) and (2.7.5.4.6).122
2.7.
DETERMINATION
OF NUCLEAR REACTIONS
445
cross section with energy. Hornyak et al.144used a magnetic-lens spectrometer to observe internal-conversion electrons produced directly by the emitting target nucleus (an alternative process to the y-emission) or external-conversion photoelectrons produced by the y-rays in a converter foil placed near the target. A complete analysis of the use of this spectrometer in the determination of absolute y-ray yields is given by Thomas and Lauritsen. 149 In particular, the converter efficiencies are expressed in terms of (a) the total atomic conversion cross section for production of a secondary electron of appropriate energy, (b) the probability that the secondary electron is ejected into the solid angle accepted by the spectrometer, and (c) the yield for a particular radiation from the source. These quantities are discussed for the processes of Compton conversion, photoelectric conversion, internal conversion, and internal pair formation. It is concluded that both relative and absolute y-ray intensities can be obtained in favorable cases to an accuracy of 5 to 10% with this technique. The NaI(T1) scintillation counter146has been widely used for y-ray energy measurements since about 1950 because of its simplicity and convenience. Again, however, the determination of absolute yields is complicated by the various processes which can occur in the crystal. A typical experiment to determine this yield is described by Woodbury et al.lbOThe pulse-height spectra for known y-rays of various energies are obtained, and the ratio of the area under the photopeak to the total area under the pulse-height curve is determined. By multiplying this ratio by the calculated probability that a y-ray will interact in the crystal, a semiempirical result for the “photopeak efficiency” as a function of y-ray energy can be obtained for the particular crystal employed. The absolute y-ray yield can then be deduced from the data of any experiment by dividing the measured area under the photopeak by the photopeak efficiency. Corrections must be made for the separation of the photopeak from the Compton distribution, where necessary, and for the absorption of y-rays in the target chamber, Other workers (e.g., ref. 151) employ a calibrated y-ray source of known energy and intensity to determine the photopeak efficiency at one energy, then use theoretical or semitheoretical methodslS2 to determine the photopeak efficiencies at other energies. Further remarks on the determination of absolute y-ray yields using NaI counters for different types of experiments are discussed in Section 2.7.5.3.3. R. G. Thomas and T. Lauritsen, Phya. Rev. 88, 969 (1952). H. H. Woodbury, A. V. Tollestrup, and R. B. Day, Phys. Rev. 95, 1311 (1954). 161 N. P. Heydenburg and G. M. Temmer, Phys. Rev. 100, 150 (1955). 1 6 a D. Maeder, R. Miiller, and V. Wintersteiger, H elv. P h p . A d a 27, 3 (1954). M. J. Berger and J. A. Doggett, P h p . Rev. 99, 663 (1955). 140
446
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
The pair spectrometer of Bent et ~ 1 . 1 4 6employs a lens-type geometry to detect the positron-electron pair (usually internal pairs) in coincidence in two adjacent crystals at the spectrometer focus. It is possible to calculate the peak number of internal pairs transmitted by the spectrometer, per quantum, by taking into account the angular correlation of the positrons and electrons expected163 for the corresponding y-rays of various energies and multipolarities, and the acceptance solid angle of the spectrometer. However, this does not give the number of pairs actually counted because half of the pairs strike the same counter and are not detected in coincidence, and other losses are incurred in an attempt to minimize background, etc.lS4As a consequence, the overall efficiency of this spectrometer is usually determined by comparison with known y-ray yields. A typical application of the three-crystal pair spectrometer to nuclear reaction studies is described by Bent and K r u ~ e . This ~ ~ * spectrometer records the total energy of the pair produced by the incident y-ray in a scintillator, but only when in coincidence with the annihilation radiation in two side crystals. A pulse-height analysis of the pulses in the center crystal then reveals only a single peak (the two-escape pair peak) for each y-ray. Furthermore, the resolution width of this peak is better than that obtained with a single NaI crystal.l47 The efficiency is determined by148(a) the solid angle subtended by the center crystal, (b) the fraction of y-rays incident on the center crystal which produce pairs, and (c) the probability that both annihilation quanta will be detected. A typical efficiency for such a spectrometer installation when detecting 6-Mev y-rays is 2 pair counts per lo6 gammas emitted by the source.148The effectiveness of this instrument for data collection is therefore greatly enhanced by the use of a 100-channel pulse-height analyzer. Figure 30 shows the experimental arrangement employed by Bent and Kruse. An important type of experiment involving the detection of y-rays is the study of Coulomb excitation of target nuclei by passing charged particles.166A complete review of this subject has been given by Heydenburg and Temmer. 168 Considerations concerning the cross-section determination are summarized below. Detection of the de-excitation y-rays from the Coulombexcited nucleus is usually accomplished by means of scintillation counters, although proportional counters and internalls3
M. E. Rose, Phys. Rev. 76, 678 (1949).
164R. D. Bent, T. W. Bonner, J. H. McCrary, W. A. Ranken, and R. F. Sippel, Phys. Rev. 99, 710 (1955). T. Huus and C. ZupanEiE, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 28, 19 (1963); C.L. MoClelland and C. Goodman, Phys. Rev. 01, 760 (1953). lK8 N. P. Heydenburg and G. M. Temmer, Ann. Rev. Nuclear Sci. 6, 77 (1956.)
2.7.
DETERMINATION O F NUCLEAR REACTIONS
447
conversion electron spectrometers have been used. The absolute Coulombexcitation cross section as usual requires the determination of the absolute y-ray yield. This may be done using thin targets provided the target thickness is accurately known; this is most directly determined by observing Rutherford scattering of the charged particles employed in the experiment. Generally, thick-target measurements are more satisfactory for accurate absolute yield measurements. The Coulomb excitation process enjoys the distinction that accurate quantum-mechanical
I
DUMONT 6364 No1
I
511 Kev
DUMONT
DUMONT INCHES
FIQ.30. Experimental arrangement for the determination of y-ray yields and fluxes using a three-crystal pair spectrometer.14*
calculations are possible, so that the predicted yield may be integrated as a function of energy over the range of the slowing-down particle in a thick target. Since the principal contribution to the yield comes near the maximum energy of the incident particle, the thick-target yield is rather insensitive to the details of the stopping power of the target as a function of energy.lhEThus the yield Y of the E2 y-rays (characteristic of the Coulomb-excitation process) from a thick target is given to a good approximation by (2.7.5.47) Y = na(Eo) where n is the nuclear density, and o(Eo)the Coulomb-excitation cross section evaluated at the incident particle energy Eo. The quantity y(Eo) is an integral which may readily be eva1uated.ls6 Measurement of the
448
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
yield may be made161by observing the photopeak in a NaI detector, and dividing by the photopeak efficiency determined by one of the methods described above. Corrections for a small bremsstrahlung continuum which may occur are made by using targets of nearby Z which exhibit negligible yield of Coulomb-excited y-ra~s.1~7 Corrections must also be made for attenuation of the y-rays in the target and target backing.16’ To suppress characteristic K X-rays from the target, the NaI crystal may be preceded by a suitable a b ~ 0 r b e r . When l ~ ~ the corrected absolute yield has been determined, the Coulomb-excitation cross section may be calculated from Eq. (2.7.5.47). The result obtained may be related directly to the E2 transition probability for the state excited, and therefore to its lifetime. Comparisons with the lifetimes obtained by other methods are in good agreement,161-168 thus giving a satisfactory check on this method for the determination of the absolute Coulomb-excitation cross section. 2.7.5.3. Differential Cross Sections for Neutron-Induced Interactions.16,78-79 Much of the discussion bearing upon the measurement of cross sections for charged-particle-induced reactions is also germane to neutron-induced reactions. The chief differences are: (1) in the latter experiments one must generally use a well-defined target8volume, such as the interaction of a thin solid target with the primary beam, as the means of defining the reaction volume seen by the detector, (2) the detectors used must be sensitive to neutrons, either directly or indirectly, and must usually provide a measure of the neutron energy as well, and (3) scattering material must be kept to a minimum in the vicinity of source and detector. The problem of measuring differential fast neutron interaction cross sections is logically divided according to the kind of particles emitted, i.e., whether they be charged or neutral. Interactions leading to the emission of neutral particles are further subdivided according to whether the particles are neutrons or y-rays. Common to the measurement of all three emission probabilities is the evaluation of: (a) the neutron flux a t the scatterer, (b) the number of scattering nuclei in the flux, (c) corrections for multiple scattering of the incident neutron beam with consequent increase of the average path length of a neutron in the scatterer (see Section 2.7.4). In addition to the above, one is faced with the problem of determining the number of emitted particles or quanta per unit solid angle, and, very often, per unit energy interval as well, and this usually requires shielding F. K. McGowan and P. H. Stelson, Phys. Rew. 106, 522 (1957). R. Sherr, C. W. Li, and R. F. Christy, Phys. Rev. 96, 1258 (1954); G. M. Temmer and N. P. Heydenburg, ibid. 104, 967 (1956). 167
168
2.7.
DETERMINATION OF NUCLEAR REACTIONS
449
the detector from the primary neutron flux and from room- and airscattered neutrons. The primary neutron flux a t the scatterer may be deduced from a knowledge of the differential cross sections which are now available for the most useful neutron-producing reactions which lead to intense sources of neutrons of energy up to 30 Mev,114J69or by an absolute flux measurement a t the scatterer. 160 A number of the neutron-producing reactions, among them T(d,n)He4, D(d,n)Hea,and T(p,n)Hea, have the very useful property of a one-to-one correspondence between the emitted neutron and a n energetic charged particle. When this situation obtains, counting of the charged particles emitted into a given solid angle determines the number of neutrons emitted into a corresponding solid angle. By counting the radiations emitted from the scatterer in coincidence with the charged particles associated with the primary neutrons incident on the scatterer, it is possible, at one and the same time, to insure that the reactions observed are associated with the scatterer being investigated and that the neutrons producing the reaction are monoenergetic. Furthermore, it is not necessary to have precise knowledge of the area of the beam at the scatterer, corresponding to the solid angle of acceptance of the charged-particle detector, provided that the scattering material is uniformly distributed over an area larger than the cross-sectional area of the cone of neutrons defined by the associated-particle detector. 2.7.5.3.1. DIFFERENTIAL CROSSSECTIONS FOR REACTIONS LEADING TO THE EMISSION OF CHARGED PARTICLES. A measurement of the differential cross section for the emission of charged particles may be accomplished in any one of a large number of ways. In principle, a thin radiator in the geometry of Fig. 16 may be used to measure the differential cross sections for neutron-induced reactions leading to the emission of charged particles as well as contrariwise. The chief difference arises from the type of detector used and its shielding requirements. One experimental arrangement, 169 J. L. Fowler and J. E. Brolley, Jr., Revs. Modern Phys. 28, 103 (1956); E. Amaldi, Reactions with emission of neutrons and neutron sources. “Handbuch der PhysikEncyclopedia of Physics” (S. Flugge, ed.), Vol. 38/2, p. 87 (1959); A. 0. Hanson, R. F. Taschek, and J. H. Williams, Revs. Modern Phys. 21, 635 (1949); J. E. Brolley, Jr., T. M. Putnam, and Louis Rosen, Phys. Rev. 107, 820 (1957); S. J. Bame, Jr. and J. E. Perry, Jr., Phys. Rev. 107, 1616 (1957); B. C . Diven, Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1966 4, 251 (1956). loo H. H. Barschall, L. Rosen, R. F. Taschek, and J. H. Williams, Revs. Modern Phys. 24, 1 (1952); S. J. Bame, Jr., E. Haddad, J. E. Perry, Jr., and R. K. Smith, Rev. Sci. Inslr. 28, 997 (1957). I61G. Brown, G. C . Morrison, H. Muirhead, and W. T. Morton, Phil. Mag. [8] 2,
785 (1957).
450
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
which is extremely simple and elegant and makes optimum use of the neutron source, is illustrated by Fig. 31 (see also ref. 162). AS illustrated the source is isotropic and monoenergetic, the reaction being T(d,n)He4 at low-deuteron energy. For this reaction the a-particles are sufficiently energetic to be detected and counted in such a way as to determine the , COLLIMATED BEAM OF DEUTERONS
NEUTRONS
TRITIUM
TARGET
FIQ. 31. Experimental arrangement for measuring the energy and angular distributions of protons from neutron-induced reactions.18'
total source strength and hence the neutron flux per unit solid angle. The flux of neutrons a t each point of the scatterer is simply the total yield divided by the product of 4s and the square of the distance to the source. The photographic-plate detectors introduce a minimum of scattering material while recording the range and orientation of the 18aD. L. N a n , Proc. Phys. SOC.(London) A70, 195 (1957); D.L. Allan, Intern. Conf. on Neutron Interactions with the Nuclei, Columbia University, New York, 1957; A. H. Armstrong and J. E. Brolley, Jr., Phys. Rev. 99,330 (1955); V. V. Verbinski, T. Hurlimann, W. E. Stephens, and E. J. Winhold, ibid. 108, 779 (1957).
2.7.
DETERMINA4TION OF NUCLEAR REACTIONS
45 1
charged particles emitted by the scatterer. It is thus seen that a complete evaluation of the angular and energy dependence for the emission of charged particles is possible from a single run. Numerous variations of this method will occur to the reader. The chief difficulty encountered in this experiment is the rather high background count for elements with low cross sections for charged-particle emission. Background must be evaluated by replacing the scatterer with one which is known to have a negligible cross section for emitting charged particles. Another method, which permits somewhat better angular and energy resolution but requires much higher intensities, utilizes a neutron col-
FIG.32. Counter telescope for measuring the differential cross sections for chargedparticle emission from neutron-induced reactions. (Ribe and Seagrave, ref. 164.)
1imato1-I~~ which defines a narrow beam of neutrons for irradiating the scatterer while, a t the same time, providing shielding for the detectors. If simultaneous data are to be recorded a t all angles, then the detectors must again be nuclear emulsions; otherwise they can be electronic. The third general method164for measuring differential cross sections for charged-particle emission is illustrated by Fig. 32. Here one uses a “counter telescope” arrangement consisting of a number of proportional counters and one crystal counter arranged in tandem and operated in coincidence. (Similar arrangements for neutron detection were described in Section 2.7.5.2.1.) The particle loses a negligible amount of energy in the proportional counters. Thus the scintillator responds to the total energy of the emitted particle while the time coincidences with the proportional counters serve to discriminate against background counts. J. C. Allred, A. H. Armstrong, and L. Rosen, Phys. Rev. 91,90 (1953). F.L.Ribe and J. D. Seagrave, Phys. Rev. 94,934 (1954);J. D.Seagrave, ibid. 97, 757 (1955). 183 164
452
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
The scatterer must be thin, relative to the range of the lowest energy particles to be detected, and, if the angular dependence of the differential cross section is to be obtained, the entire assembly must rotate about an axis passing through the center of the scatterer and perpendicular to the plane containing the incident neutron velocity vector and the telescope axis. From the discussion under Section 2.7.5, it will be apparent that the above arrangement not only permits the measurement of the differential cross section with respect to energy and angle but the identification of the particle mass as well. Apart from the general methods above described, there exist a number of specialized techniques which are worth mentioning. One of these is applicable when the element under investigation can be incorporated into the counting gas of a proportional counter or ionization chamber.Is6 I n this case one measures the total number of disintegrations corresponding to a given neutron flux and to a given number of target nuclei in the counter. I n the case of a two-body reaction, the distribution in energy of either reaction product is the same as the distribution in angle in the c.m. coordinate systern.lBBIf, therefore, one and only one of the reaction products is charged, it is possible to use the above method for obtaining angular distributions as well as partial cross sections. A more direct and more detailed approach to the problem for measuring differential cross sections for the emission of charged particles from neutron-induced reactions is to be found in the techniques which involve the introduction of the material to be studied in a photographic emulor cloud chamber.IB8Here one may actually observe the range and spatial orientation of the reaction products so that these instruments are particularly adapted to the otherwise very difficult problem of studying the systematics of reactions involving multiple disintegration products. 2.7.5.3.2. DIFFERENTIAL INTERACTION CROSSSECTIONS FOR REACTIONS LEADING TO NEUTRON EMISSION. The measurement of differential cross sections for nonelastic processes leading to neutron emission is considerably more difficult than for the emission of charged particles from the standpoint of both intensity and background, The intensity problem arises from the circumstance that the former measurements require one to go through three nuclear cross sections corresponding, in 166 C. H. Johnson and H. H. Barschall, Phys. Rev. 80, 818 (1950); R. B. Walton, J. D. Clement, and F . Boreli, ibid. 107, 1065 (1957). I66H. H.Barschall and J. L. Powell, Phys. Rev. 98, 713 (1954). 167 G. M.Frye, Jr. and J. H. Gammel, Phys. Rev. 103,328 (1956);A. H.Armstrong and G. M. Frye, Jr., ibid. 103, 335 (1956);G.M.Frye, Jr., L. Rosen, and L. Stewart, ibid. 99, 1375 (1955). 188 J. P.Conner, Phys. Rev. 89, 712 (1953);Alan B. Lillie, ibid. 87, 716 (1952).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
453
turn, to neutron production, neutron scattering, and neutron detection. It is, therefore, necessary to use scatterers which are not thin for the incident or scattered neutrons, and this entails detailed corrections for multiple-scattering effects. These corrections are usually sufficiently involved, as a result of the energy and angular dependence of elastic and inelastic cross sections, that they cannot be treated analytically. One must therefore resort to Monte Carlo techniques, or to numerical integration methods whereby one divides the scatterer into a number of sections, each section being assigned a macroscopic cross section on the basis of its volume, and determines the fate of neutrons which pass through or are born in each section. The background difficulties derive from the long mean free path of neutrons in all materials. This makes it difficult to separate the signals due to neutrons scattered in the sample from similar signals due to the primary flux and to the scattering of the primary flux by materials other than the sample under study. Where the emitted neutrons are polyenergetic, one must usually measure the energy spectrum of the emitted neut,rons on a n absolute scale, i.e., the cross section per unit solid angle and per Mev as a function of emission angle and energy, which cross section we denote by a(0,E). The two types of detection schemes most used are recoil-proton detectors and time-of-flight. The recoil-proton detectors make use of the n-p cross section which is well known, large, without resonances, and spherically symmetrical in the c.m. system below 14 MeV. The proton recoils can be detected in a proportional counter, ion chamber, organic scintillator, or photographic plate. All of these devices can be used as both radiator and detector since they all contain hydrogen. However, when so used, the pulse-height distribution from electronic detectors must be differentiated in order to obtain the neutron spectrum since the direction of the individual proton recoils is unknown. Organic crystals have, in fact, been used in this way to study the neutron energy distributions from inelastic scattering."j9 More often, however, the crystal is used as a detector in conjunction with a n external hydrogenous radiator. Figure 33 displays the experimental arrangement which has been used to measure a(e,E) for the interaction of 14-Mev neutrons with various element~.17~ The collimator is constructed of iron backed by paraffin. (For fast neutrons, iron has a relatively large macroscopic cross section for nonelastic scattering. Furthermore, the neutrons arising from nonelastic collisions are much degraded in energy.) The source of neutrons M. J. Poole, Phil. Mag. [7] 43, 1060 (1952); [7] 44, 1398 (1953). L. Rosen and L. Stewart, Phys. Rev. 107,824 (1957); S. H. Ahnand J. H. Roberts, ibid. 108, 110 (1957); D. E. Wood and J. B. Singletary, ibid. 114, 1595 (1959). 168
170
454
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
was the T(d,n) reaction for 200-kev deuterons. The cylindrical scatterer is posit.ioned so that its axis of symmetry coincides with the collimator axis, thus making optimum use of the incident neutron flux. The detectors are nuclear emulsions arranged around the periphery of a circle whose center coincides with the center of the scatterer. The neutron collimator permits the irradiation of the scatterer by a narrow beam of monoenergetic neu txons while shielding the detectors from the primary
1
I
I
I
I
LPARAFFIN
F~Q. 33. Experimental arrangement for measuring energy and angular distributions of neutrons from neutron-induced reactions. flux. Background is evaluated by making an irradiation with the scatterer removed. The cross section with respect to energy and solid angle for the emission of neutrons is determined as follows. Imagine a target under investigation (Fig. 33) containing n, nuclei exposed to a time-integrated flux, Fo, neutrons/cm2, of monoenergetic neutrons. Then the yield of secondary neutrons per steradian and per Mev a t angle e and energy E is, (2.7.5.48) where
P , the average primary neutron flux a t the scstterer, is
P
=
Fo
jotecUnzdx t
(2.7.5.49)
The quantity o in Eq. (2.7.5.49) is equated to the tot.al cross section when evaluating the elastic scattering cross section and to the transport cross section when evaluating nonelastic scattering cross sections. t is
2.7.
455
DETERMINATION OF NUCLEAR REACTIONS
the thickness of the scatterer along the direction of the primary neutron beam and n is the density of nuclei in the scatterer. The target now constitutes a new source, and the hydrogen in the emulsion constitutes a new target. The neutron flux per Mev seen by any given detector at angle 0 is simply dF,(O,E) - d2Y(B,E) 1 -(2.7.5.50) dE dEdQ ' 7 where r2 is the average square of the distance from scatterer to detector. The problem is then to measure F , versus E, for once this is accomplished we can immediately determine u(0,E) from Eq. (2.7.5.48). The energy dependence of F , is evaluated by measuring all proton recoils in a volume V of emulsion, which have the properties that they are completely contained in the emulsion and their velocity vectors fall within a rectangular pyramid of half-angles a and p with respect to the line joining the scatterer and detector. dE, The number of proton recoils of energy between E, and E , corresponding to the flux of secondary neutrons whose energies lie between E and E dE is given by
+
+
dNpdE, dE*
=
dF,(B,E) ~ dE
( E )Q ' ~ HV P( E )dE
n ,
(2.7.5.51)
where nH is the density of hydrogen nuclei in the emulsion, u n Vis p the average differential n-p scattering cross section at energy E and over the Q' is the solid angle of acceptance angular range 0" to arc cos(cos a cos of proton recoils, and P ( E ) is the probability that a recoil proton projected into Q' will remain within the confines of V . The exact expression for the solid angle Q' subtended by a rectangular pyramid of half-angles a and is
a),
'
a )]- (2.7.5.52) + arc sin ( d t a n 2 a tan + tan2 p 'OS
In order to reduce emulsion attenuation, the plane of the emulsion is sometimes inclined a t a small angle 6 with respect to the direction of the scattered neutron incident upon it. The space angle between the incident neutron and scattered proton is then given by cos @ = cos a cos 0 cos 6
+ sin /3 sin S
456
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
and it is this angle 4 which must be used in the evaluation of the incident neutron energy. Since the proton-recoil and neutron energies are related by E, = E cos2 4, dE, = cos2 4 dE. Combining Eqs. (2.7.5.48), (2.7.5.50) and (2.7.5.51), we get dNP (cosz 4>rz a(0,E) = (2.7.5.53) d E p an,p (E)Q’nH V P( E)Pn, The application of the time-of-flight method171 to the measurement of cross sections of the type under discussion is illustrated by Fig. 34. The scatterer is a hollow cylinder (to reduce multiple scattering) placed on the axis of the charged-particle beam. The cylinder axis is vertical and coincides with the axis of rotation of detector and shield. Different scattering angles are realized by rotating detector and shield around the axis of the scatterer. The desired resolution and the available yield, which are affected in reciprocal ways, determine the flight path. The detector can be a plastic scintillator cemented directly to the face of the photomultiplier, or a plastic phosphor optically coupled to the photomultiplier by means of a Lucite light pipe. As used in the geometry of Fig. 34, the scintillator has a more or less flat response above 0.5 MeV. I n any case, cross sections are always measured by direct comparison to the well-known n-p differential cross sections, This is accomplished by replacing the scatterer with an hydrogenous one (usually polyethylene) containing a known number of H nuclei and measuring the counting rate as a function of scattering angle and hence neutron energy. The neutrons scattered from the carbon in the polyethylene are easily resolved from those scattered by the hydrogen. If the n-p scattering angle is adjusted so that the energy of the neutrons scattered from the polyethylene is equal to the energy of the inelastic neutrons from the scatterer, a comparison of the detector response to neutrons from n-p scattering with the detector response corresponding to the unknown yield gives a direct comparison of the known to the unknown cross section. The cross section for reactions which result in the emission of neutrons of energy E a t angle 0 is then simply, (2.7.5.54) where CH and CsCrepresent, respectively, the counting rates due to H and scatterer of unknown cross section, n H and NBc are corresponding quantities for the number of nuclei in the two scatterers and u(E)]His the differential cross section for the scattering from hydrogen of the primary neutrons of energy Eo through the angle 4 where 4 is given by 17lL. Cranberg and J. S. Levin, Phys. Rev. 103, 343 (1956).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
457
= EOcosz 9. Background corrections are made on the basis of counting rates with scatterer removed. To reduce background the detector is surrounded by 1 in. of lead which provides shielding from those gammas which do not originate in the scatterer. Enclosing the lead is 6 in. of paraffin loaded with LiC03 to moderate and capture neutrons.
E
POLYETHYLENE
MU METAL SHIELD RCA 581s PHOTOM
FIG. 34. Time-of-flight geometry for the investigation of the angular dependence of the cross section for elastic scattering of neutrons.17'
As already indicated, one of the basic problems in measuring differential cross sections of the type here discussed is the evaluation of multiplescattering effects, since for intensity reasons the dimensions of the scatterer must be a significant fraction of a mean free path for both the primary and secondary neutrons. The methods for making these corrections are basically similar to those described in Section 2.7.4. It is, however, usually necessary to make simplifying assumptions even when using fast electronic computers to evaluate the integrals. Some of the most useful of these, whose validity has been established, by Monte Carlo methods, for scattering dimensions not much longer than one mean free path are: (1) The angular distribution of neutrons leaving the sample is constant after the first collision.
458
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
+
(2) The fraction of n th collision neutrons suffering n 1 collisions is constant. (3) The number of neutrons suffering more than 3 collisions is negligible. Insofar as cross sections for monoenergetic neutrons are concerned, one must be cognizant of the departure from homogeniety of all neutron sources as the c.m. energy of the neutron-producing reaction is increased. Thus the Li'(p,n) reaction generates two neutron groups above a n 28 2,
FIG.35. Cross section as a function of energy and atomic weight for 14-Mev incident neutrons.
incident proton energy of 2.4 Mev as mentioned in Section 2.7.3.1.2. The p-T, d-D, and d-T reactions will produce a continuum of neutron energies above 8.3, 4.4, and 3.6 MeV, respectively. For many purposes the differential cross section with respect to energy, integrated over 4n solid angle is all that is required. A method which has been found extremely useful for carrying out this type of measurement makes use of the geometry of Fig. 11 with a detector which is capable of measuring the neutron energy as well as the neutron flux. The method of measuring the neutron spectrum is identical to th a t used for the evaluation of Eq. (2.7.5.51).The results of a series of such measu r e m e n t ~ using ~ ' ~ nuclear plate detection is illustrated in Fig. 35. 178 E. R. Graves and L. Rosen, Phys. Rev. 89, 343 (1953).
2.7.
DETERMINATION
OF NUCLEAR REACTIONS
459
The proton-recoil counter telescope, which is also very useful in measuring both the neutron energy and absolute flux in experiments of this type, has already been described in Section 2.7.5.2. 2.7.5.3.3. DIFFERENTIAL CROSSSECTIONS FOR PROCESSES INVOLVING THE EMISSION OF T-RAYS.The problem of measuring differential cross sections for processes involving the emission of y-rays from neutroninduced reactions is analogous and the methods quite similar to the problem, discussed in Section 2.7.5.3.2, of measuring cross sections for neutron emission. The chief differences occur in the type of detectors used and in the procedure for discriminating against background. SCATTERING RING GAS TARGET
SHIELDING CONE
Cn4 SCINTILLATION COUNTER
FIG. 36. Typical experimental arrangement, employing the ring geometry, for measuring the energy and angular dependence of the cross section for y-ray emission from neutron-induced reactions. (Day, ref. 173.)
The most precise experiments for determining y-ray energies have utilized the photopeaks from a NaI s ~ i n t i l l a t o r ~and ~ 7 ~this is, in fact, the most generally useful detector for y-radiation. A typical arrangement for measuring the energy and angular distribution of y-rays produced by the inelastic scattering of monoenergetic neutrons is illustrated in Fig. 36. The resemblance to the arrangement for the neutron case (Fig. 40) is immediately apparent. Although the detector can be shielded from the direct neutron flux, by the methods described in Section 2.7.5.3.2, these provide no relief from neutrons scattered by the sample into the detector. Of the techniques for discriminating against such background, the most powerful, suggested by L. Cranberg and developed by R. B. Day and D. A. Lind,174 utilizes a pulsed beam and time-of-flight to effect the separation of the y-rays to be observed not only from the neutrons scattered by the sample but also from the y-rays produced by room-scattered neutrons. A variation of this method avoids pulsing the beam by requiring coincidences between detector signals and signals from the associated particles generated in the neutron-producing reaction.176 The detector response to a scattered neutron will, of course, be delayed (by the neutron 1 7 3 R. B. Day, Phys. Rev. 102, 767 (1956) ; J. J. Van Loef and D. A. Lind, ibid. 101, 103 (1956). 174 R. B. Day, Proc. of t h e Conf. on Neutron Physics by Time-of-Flight, Gatlinburg, Tennessee, ORNL-2309, 90 (1956). 175 P. Shapiro and R. W. Higgs, Phys. Rev. 108, 760 (1957).
460
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
flight time from scatterer to detector) with respect to the anticipated y-ray signal. For measuring cross sections as a function of y-ray energy, Battat and Graves176 have employed a three-crystal pair spectrometer. Since such an instrument yields a single peak, in the pulse-height distribution, for each y-ray energy, the interpretation of the data is quite straightforward as compared, for example, to that from a single-crystal NaI spectrometer. However, the low detection efficiency constitutes a serious limitation on this method. Application of the three-crystal pair spectrometer to (charged, neutral) reactions has already been discussed in Section 2.7.5.2. One of the main problems in measuring differential cross sections for y-ray emission involves the determination of the detector efficiency as a function of y-ray energy. This has been discussed in the case of NaI counters in Section 2.7.5.2. Important corrections arise from self-absorption, multiple scattering, and angular resolution, all due to the finite size of scatterer and detector. DayITahas shown, by means of y-ray transmission experiments, that the first and most important of these corrections can be calculated on the assumption that the total Compton scattering cross section is effective in removing y-rays from the photopeak, and this assumption greatly simplifies the calculation of the correction due to self-absorption in the scatterer. Methods for evaluating the angular resolution corrections have been developed by a number of investigators. Under favorable circumstances it is possible to measure the counter efficiency and scatterer self-absorption a t one and the same time.178 This is accomplished by using a scatterer in the form of pellets which can be coated with a known amount of a radioactive substance. After first measuring the yield of y-rays produced as a result of a given irradiation, the scatterer is coated with a radio-nuclide, the y-rays from which have approximately the same energy spectrum as the ones originating from the neutron reaction under investigation. For energetic neutrons and scatterers of reasonable thickness, each neutron is capable of making more than one inelastic collision so that multiple-scattering effects must be evaluated if accurate cross-section data are to be obtained.l18 A summary of available data on the cross sections for the emission of y-rays from neutron-induced reactions, including angular distributions, is to be found in BNL-32546and BNL-400.”@ 1733177
M. E. Battat and E. R. Graves, Phys. Rev. 97, 1266 (1955). A. M. Feingold and Sherman Frankel, Phys. Rev. 97, 1025 (1955). 178 R. M. Kiehn and C. Goodman, Phys. Rev. 95, 989 (1954). 179 D. J. Hughes and R. S. Carter, Neutron cross sections and angular distributions. Brookhaven National Laboratory Report BNG400 (1956). 176
177
2.7.
DETERMINATION OF NUCLEAR REACTIONS
46 1
2.7.5.4. Cross Sections for y-Induced Reactions. Photodisintegration experiments are in general difficult to perform and in many cases difficult to interpret exactly. The first problem arises because the cross sections are in general small, so that intense y-sources are required. However, the most intense sources are not monoenergetic, for which case the interpretation of the results in terms of cross sections requires unfolding the energy dependence of the source. It is this process which leads to the difficulty in precise interpretation of the data. CONSIDERATIONS. There are two basic types 2.7.5.4.1. EXPERIMENTAL of sources for studies of y-induced reactions: monoenergetic sources and bremsstrahlung sources. The first class includes radioactive sources, which produce y-rays either as the result of natural or artificial radioactivity, and nuclear-reaction sources, which produce monoenergetic y-rays directly in a nuclear (ply) reaction. Included in the second class are electrostatic-generator sources, which produce either thin- or thicktarget bremsstrahlung with an accurately variable but low maximum energy, and betatron and synchrotron sources, which usually produce thick-target bremsstrahlung with a variable and high maximum energy. The advantage of a monoenergetic source is that the cross section under study may be determined quite unambiguously a t a known energy. However, these sources are generally characterized by very low intensity, which introduces the usual difficulties present in low yield measurements. Further, they often actually consist of several monoenergetic y-ray lines, which removes some of the advantage of their use. Radioactive y-ray sources are limited in maximum energy, but if a variety of them are used it is possible to obtain measurements a t a number of low energies. A typical investigation of this type was the determination of the photodisintegration cross section of beryllium in several experiments.180~181 The radioactive sources employed were all artificially produced with the following y-ray energies: ( S b 9 1.70 MeV, (MrP) 1.81 MeV, PI-'^^) 2.185 MeV, (La'4O) 2.50 MeV, and (NaZ4)2.76 MeV. The monoenergetic y-rays from nuclear-reaction sources are generally considerably higher in energy than those from radioactive sources. Some of the more important nuclear-reaction y-ray sources are the F19(p,ay) reaction (6.1-, 6.3-, and 7.1-Mev lines), the Li7(p,y) reaction (17.63- and 14.8-Mev lines), and the H3(p,y) reaction (20.4-Mev line, but very weak). The electrostatic generator producing the proton beam is usually adjusted 180 B. Russell, D. Sachs, A. Wattenberg, and R. Fields, Phys. Rev. 73, 545 (1948); A. Wattenberg, Photoneutron sources. Preliminary Report No. 6, Nuclear Science Series, Division of Mathematics and Physical Sciences, National Research Council, Washington, D.C. Unpublished. 1.91 B. Hamermesh and C. Kimball, Phys. Reo. 90, 1063 (1953).
462
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
to an energy above the appropriate capture-gamma resonance, and a thick target employed to guarantee maximum y-yield from the resonance. Bremsstrahlung sources have the advantage that large intensities are available, but unfortunately a continuous spectrum of X-ray energies up to the maximum electron energy is produced. This spectrum is in general not precisely known, which contributes seriously to the uncertainties of the cross-section determination. I n electrostatic-generator sources, an accurately homogeneous electron beam of known energy strikes a thick or thin target such as gold. By raising the electron energy in small and precise steps, the upper limit of the bremsstrahlung spectrum rises accordingly, allowing photodisintegration experiments to be performed under conditions of precision energy control. m This technique is quite limited in energy, however, being restricted so far to measurements in the range of a few MeV. Experiments with betatrons and electron synchrotrons can be performed with good intensity to much higher energies. However, the energy control is somewhat less precise, although the stability of betatrons has been improved to the point where the energy will remain constant to + 5 kev under favorable conditions.lXs Betatrons and synchrotrons have been used in the accumulation of a large amount of experimental photodisintegration data as a function of energy in the region of the “giant resonances’’ from 12 to 23 MeV. Detection of photodisintegration processes has usually been carried out either by observing the activities produced in the radioactive product nuclei, or by detecting the direct yield of the ejected particles. In the first case, one obtains direct information about a particular photodisintegration process in a particular isotope, provided it is not possible to form the same radioactive product nucleus by a different photodisintegration process involving a different isotope in the target. In experiments designed to observe the shape of the giant resonances, the residualactivity detection method yields only a lower limit to the width of the total resonance photon-absorption cross section, since processes leading to stable isotopes are ignored.**4The residual-activity method imposes certain requirements on the measurements due to the half-life of the product nucleus and the presence of unwanted activities if chemical separation is not employed. If the half-life is not too short, activation curves can be obtained by irradiating thin disks of the target material for some standard dose, and then removing them to a standard counter u*J. C. Noyes, J. E. Van Hoomiasen, W. C. Miller, and B. Waldman, Phys. Rev. 96, 396 (1954). 188 L. Katz, R. N. H. Haslam, R. J. Horsley, A. G. W. Cameron, and R. Montalbetti, Phys. Rev. 96,464 (1954). 184 Karl Strauch, Ann. Rev. Nuclear Sci.2, 105 (1953).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
463
arrangement. If the radioactive product nuclei with disintegration constant X are produced at a constant rate B during the irradiation lasting for a time t l , and the counting begun a t a time 2 2 after the bombardment ceases, then the number of counts observed in the counting time t s is given by the familiar formula:
N,
=
(B/X)(l - e-XLl)e-XLz(l-
e-hL3).
(2.7.5.55)
From the experimental counting data, Eq. (2.7.5.55) allows one to determine B and, therefore, the saturated activity corresponding to the irradiation rate employed. This information, coupled with the number of nuclei in the sample and the photon flux corresponding to that irradiation rate, suffice to determine the cross section in principle, except for complications introduced by the photon-energy spectrum to be discussed later. I n these measurements corrections must be included for absorption in the sample, as well as for the presence of extraneous activities besides the one of interest. If the half-life is short, some method of rapid and automatic sample transfer to the counter is required (see, e.g., ref. 185). A variety of methods are employed for the detection of particles directly ejected in photonuclear processes. Most of these methods do not distinguish between possible processes producing the detected particle; for example, between neutrons from (-y,n), (y,np), or (y12n). As a consequence, detection of photoneutrons will ordinarily make the observed resonance appear wider than the true shape of the (7,n) cross-section curve, unless the thresholds for the unwanted reactions are sufficiently high.l84,186However, the practical advantages of direct particle detection generally outweigh any ambiguities in the source reactions of the particles.187 For the detection of photoneutrons, it is desirable that the detector have a constant efficiency for neutrons from thermal energies to about 10 MeV. For this reason, enriched BFs counters imbedded in a paraffin housing surrounding the neutron-producing target are usually employed (see, e.g., refs. 182, 188, 189). Arrays of several such counters may be used to increase yields near threshold and reduce exposure time.187~190A difficulty with this method is the pileup of secondary elec186L. Kats and A. G. W. Cameron, Phys. Rev. 84, 1115 (1951). 1saP. F. Yergin and B. P. Fabricand, Phys. Rev. 104, 1334 (1956). 187 R. Nathans and J. Halpern, Phys. Rev. 93, 437 (1954). 188 J. Halpern, A. K. Mann, and R. Nathans, Rev. Sci. Instr. 23, 678 (1952); L. E. Lazareva, B. I. Gavrilov, B. N . Valuev, G. N. Zatsepina, and V. S. Stavinsky, Conf. 1966. Part 1, p. 306. Acad. Sci. U.S.S.R. Peaceful Uses of Atomic Energy, MOSCOW, AECTR 2435, Supt. of Documents, U.S. Government Printing Office, Washington, D.C., 1955. 189 R. Montalbetti, L. Katz, and J. Goldemberg, Phys. Rev. 91, 659 (1953). 190 W. H. Hartley, W. E. St.ephens, and E. J. Winhold, Phys. Rev. 104, 178 (1956).
464
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
trons in the counter which occurs with betatrons during the X-ray pulse. This problem can usually be eliminated by gating the counter to count thermal neutrons only between pulses of the m a ~ h i n e . ~To ~ ~determine r~~l the true yield of photoneutrons in this method, the efficiency of the counting arrangement can be determined, for example, by using a calibrated Ra-Be source placed a t the sample p ~ ~ i t i obyn com, ~ ~ ~ ~ ~ paring with the residual activity method [for a sample in a region where the ( y , n p ) and (y,2n) reactions are energetically forbidden189],or in the case of nuclear-reaction sources by comparing with the neutron yields of known nuclear reactions.lgOFigure 37 shows a representative betatron experimental arrangement used a t the University of Pennsylvania for direct detection of photoneutrons with BFa counters (from Nathans and H a l ~ e r n ' ~ ~ ) . A second method for detection of photoneutrons is by means of induced activities due to neutron capture or (n,p) reactions in suitable substances placed in a paraffin block surrounding the target. The induced activity is then measured by Geiger counters located in intimate contact with the substance. Typical examples are rhodium (44second activity, slow neutrons), aluminum (9.6-minute activity, threshold about 3 Mev), and silicon (2.4-minute activity, threshold about 5 Mev). Such detectors with different neutron-activation thresholds give some information about the energies of the emitted neutrons. l g 3Another interesting photoneutroninduced activity method makes use of a NaI(T1) crystal placed in the vicinity of the target sample. Fast photoneutrons produced in the tiarget will then interact in the crystal, producing IlZ8with a half-life for P-emission of 25 minutes. Since this activity is actually produced within the crystal, the detection efficiency for the p r a y s is nearly In addition, the target geometry does not require thermalisation of the photoneutrons. One of the most important aspects of the direct detection of photonucleons is the possibility of obtaining the angular distributions of the emitted particles. A detection method useful for such studies is the nuclear emulsion, which is often used as a detector for yinduced reactions producing charged particles. A complete discussion of photodisintegration experiments with nuclear emulsions is given in a review article by Titterton. lg6 One method is to observe photodisintegrations within the J. Halpern, R. Nathans, and A. K. Mann, Phys. Rev. 88, 679 (1952). D. R. Connors, Ph.D. Dissertation, University of Notre Dame, Indiana, 1956. Unpublished 19sF. Ferrero, A. 0. Hanaon, R. Malvano, and C. Tribuno, Nuovo cimento [lo]4, 418 (1956). 194 K. G. McNeill, Phil. Mag. [7] 46, 321 (1955). 186 E. W. Titterton, Progr. in Nuclear Phys. 4, 1 (1955). 191 18)
Side View.
Betatron
B x.
Target
+
A- A , Ionization Chambers. Lucite Holder for ( r ) meter.
rtSl0t.
d'
12" 6" 0
End View.
FIG. 37. Experimental arrangement for the detection of photoneutrons produced by bremsstrahlung from a betatron.'*'
466
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
volume of the emulsion itself, which is “loaded” with the appropriate target substance. This is particularly useful for experiments with lowintensity sources, such as nuclear-reaction ( p , y ) sources. It has also been used a great deal for multiparticle disintegration measurements, such as C12(y,3a),for example. A second method, particularly useful for angular distributions, is to place the emulsions, shielded from the direct y-ray beam, a t a variety of angles around the target sample. This is particularly desirable in betatron and synchrotron studies, where highly collimated X-ray beams can be attained. The emulsion technique has the advantage over counter studies that the emulsions are relatively insensitive to background electrons, but the disadvantage that the counting techniques are more tedious. A complete discussion of the nuclear-emulsion technique applied to the detection of emitted neutrons is given in Section 2.7.5.3.2. A counter method for observing the angular distributions of photoneutrons190makes use of the Hornyak detector,s0 mentioned previously in Section 2.7.3.1.4. Grains of ZnS are molded in Lucite, which serves to produce recoil protons from the incident neutrons and transmit the emitted light. The higher specific ionization of the protons in the ZnS produces much larger proton pulses than electron pulses, unless severe pileup from the X-ray beam is present. Such a detector, suitably shielded from the main beam and arranged to rotate around the target sample, has been used successfully for photoneutron angular distribution measurements using 70-Mev X-rays from a s y n c h r ~ t r o n . ~ g ~ Another important experimental consideration concerning the determination of cross sections for y-induced reactions is the measurement of the y-ray flux. The method employed depends upon the type of measurement. In photodisintegration measurements with radioactive sources, the y-ray source strength may be determined by comparison with a known source, preferably emitting y-rays of about the same energy. For example, in the measurements of Hamermesh and Kimball,lsl the comparison was a reactor-produced NaZ4 source whose strength was determined from the neutron capture cross section and the known reactor flux employed. In nuclear-reaction source experiments, the y-flux may be determined by using a NaI(T1) scintillation counter whose counting efficiency has been determined. I n the Li7(p,y) work of Hartley et aZ.,19o for example, the efficiency of the NaI(T1) detector was obtained by calculation, by semiempirical comparisons, and by direct comparison with published absolute thick-target yields from lithium. Experiments with an electrostatic-generator bremsstrahlung source by Noyes et al. used aa an X-ray monitor a suitably shielded Geiger counter located behind and in line with the source. The measurements with betatron 196
W. R. Dixon, Cun, J . Phya. 38, 785 (1955).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
467
bremsstrahlung sources normally determine the X-ray flux by a suitable “r-meter’’ placed in a cavity in a Lucite block. The “roentgen” recorded in this way does not correspond exactly to the usual definition of the roentgen, since the ionization is produced in the walls surrounding the cavity which do not have the same absorption coefficients as air.lS7 From the dosage recorded in the r-meter and the assumed shape of the bremstrahlung spectrum, the number of photons per cm2 per unit energy interval striking the target sample during that dose can be determined.’!’? 2.7.5.4.2. DETERMINATION OF CROSSSECTIONS FOR GAMMA-INDUCED REACTIONS. * The cross section for photodisintegration processes can be determined in a straightforward manner if monoenergetic sources are employed. For example, in a measurement of the photoneutron production cross section using a Li7(p,y) source, a BF3neutron counter imbedded in paraffin, and a NaI(T1) y-ray monitor, as described in the previous section, the cross section can be calculated fromlgo u =
(YeJnk,).
(2.7.5.56)
In this equation, Y is the observed number of neutron counts per y-ray count, corrected for background and y-ray attenuation in the sample, and n is the nuclear density. The y and neutron counter efficiencies, er and t,, are determined by methods described in the previous section. Because in such a measurement a cylindrical sample surrounding the source is usually used to increase the yield, the effective sample thickness f i s given by 190 1 t = (1 - e-z) dQ (2.7.5.57)
--/ 4rCL
4*
where p is the linear photon absorption coefficient for lithium y-rays in the sample. The cross section obtained in this way is actually a suitable average for the 17.6- and 14.8-Mev y-rays from lithium. For bremsstrahlung sources, the accurate determination of the cross section is very difficult, because it requires unfolding the bremsstrahlung spectrum from the experimental yield curve. The accuracy obtainable in this way is governed by184(a) the accuracy of the yield curve obtained as a function of peak bremsstrahlung energy, (b) the validity of the assumed bremsstrahlung spectrum used in the unfolding calculations, and (c) the validity of the assumed energy response of the incident flux monitor. The yield curve can be made quite accurate if enough points are taken with sufficient precision, assuming good machine stability.
* See Section 2.7.7 for more recent supplementary remarks. 19’
H. E. Johns, L. Kats, R. A. Douglas, and R. N. H. Haslam, Phys. Rev. 80, 1062
(1950).
2.
468
DETERMINATION O F FUNDAMENTAL QUANTITIES
However, the detailed character of the bremsstrahlung spectrum is not well known, so that this factor gives a major contribution to the uncertainty of any cross-section determination. The theoretical curves of Schifflgs are usually employed, as described in some detail by Katz and Cameron.199 The flux monitor problem is discussed by various authors (see review article by StrauchlS4for references). The Notre Dame group1s2,192~20°has made a number of measurements of photodisintegration processes using thin- and thick-target bremsstrahlung produced by an electron beam from an electrostatic generator. Figure 38 shows their target and counting arrangement used for the 7 BF, COUNTER LEAD PARAFFIN
TARGET ARRANGEMENT ( NOT TO SCALE 1
FIG. 38. Schematic diagram of target and counting arrangement for photodisintegration experiments near neutron threshold using bremsstrahlung produced by an electrostatic-generator electron beam.’$*
photodisintegration of beryllium. l g 2 One interesting result concerns the determination of thin- and thick-target “isochromats.” An isochromat is a curve representing the number of photons having energies between hv and hv dhv as a function of the electron beam energy, i.e., the energy of the upper limit of the bremsstrahlung spectrum. By observing line absorption into particular levels in indium, the thin-target isochromat was foundzooto be represented by a step function, with each new step appearing when a new line absorption process becomes energetically possible, The thick-target isochromat, on the other hand, was found to rise linearly with electron energy, but with sudden changes of slope a t the energies corresponding to new line absorption processes.zoo,201 A somewhat similar behavior has been observed by the group at the University
+
L. I. Schiff, Phys. Rev. 70, 87 (1946); 83, 252 (1951). L. Katz and A. G. W. Cameron, Can. J . Phys. 2Q, 518 (1951). 200 W. C. Miller and B. Waldman, Phys. Rev. 76, 425 (1949). 201 J. L. Burkhardt, E. J. Winhold, and T. H. Dupree, Phys. Rev. 100, 199 (1955) 19B
2.7.
DETERMINATION OF NUCLEAR REACTIONS
469
of Saskatchewan for betatron excitation functions in the light nuclei, la3 where the fine structure in the excitation function is taken to represent absorption into well-defined levels of the target nucleus. I n order to interpret the thick-target excitation functions obtained with electrostatic-generator techniques near photodisintegration thresholds, it is assumed that individual thick-target isochromats are each straight lines of positive slope, at least in the vicinity of threshold. Then the bremsstrahlung distribution can be represented by1g2
P(E,Eo) = (Eo - E ) m
(2.7.5.58)
where P(E,Eo) is the number of photons of energy E per cm2 per unit energy interval per microcoulomb of beam electrons of energy Eo, and m is the slope of the isochromat. If one assumes that the slope of succeeding isochromats is the same, i.e., that m is independent of photon energy E , then the yield Y ( E o )per microcoulomb of beam electrons of energy Eo is given by:lg2
Y(Eo) = Qm /E:: ( E o - E ) u ( E )dE
(2.7.5.59)
where Q is the number of nuclei exposed to the X-ray beam, a ( E ) is the disintegration cross section, and E,, is the threshold energy. The cross section is then obtained by taking the second derivative of the experimental yield curve :Ig2 1 d2Y u(E0) = --* (2.7.5.60) Q m dEo2 The assumption that m is independent of the photon energy E proves to be valid only over a limited energy regi0n.19~8~~~ The betatron and synehroton bremsstrahlung measurements have been interpreted in terms of the photodisintegration cross sections using two principal methods developed by the Saskatchewan group, the “total spectrum method”lg7 and the “photon difference method.”19gIn both of these methods, small changes in the experimental yield curve can introduce large variations in the calculated cross section. Therefore, in both cases “smoothing” processes are involved in evaluation of the data. The total spectrum method requires progressive smoothing of the crosssection curve during the computation, which introduces a certain amount of arbitrariness into the result.199 I n the photon difference method, the yield curves are also smoothed, but according to the criteria that the first and second derivatives should also be smooth and of an appropriate shape.Ig9Because the photon difference method is simpler to calculate and less arbitrary, it is more widely used. The photodisintegration yield Y (Eo) per roentgen of irradiation at
470
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
any betatron operating energy E o is given by:186
Y ( E o )= Q
IOEoa’(E)P’(E,Eo)dE
(2.7.5.61)
where P‘(E,Eo) is the number of photons at energy E per cm2 per unit energy interval per roentgen of irradiation, a’@) is the cross section for a photon of energy E to eject the particle detected, and Q is the number of nuclei exposed to the X-ray beam. Equation (2.7.5.61) actually represents a set of integral equations, one for each yield point obtained a t each betatron operating energy Eo. I n the photon difference method,1Bethe integral of Eq. (2.7.5.61) is replaced by a summation over a series of energy intervals of width E MeV: (2.7.5.62) where E represents the average energy of each interval, a(E) the cross section evaluated at this average energy in each interval, and P(E,Eo) the number of photons per cm2 per E Mev per roentgen of radiation. The difference between the observed yields for betatron energies E O and Eo - E is then
~(Eo - )Y(E0 - E ) = AY(E0) =
a(E) AP(8,Eo) (2.7.5.63) E
where
AP(E,Eo)
P(E,Eo) - P ( E , Eo - E )
(2.7.5.64)
and E takes on the values e/2, 3r/2, 5r/2, . . . , E o - 4 2 . Equation (2.7.5.63) can then be rewritten to obtain the cross section
B(EO- ~ / 2 )= AA(E0) - z a ( E ) A+(E,Eo)
(2.7.5.65)
E
where
(2.7.5.66)
and
(2.7.5.67)
From the observed difference AY in the yields a t successive energies, the cross sections a can be calculated as a function of energy using Eq. (2.7.5.65). Tabular details of the method, as well as a tabulation of the quantities Q AP(E0 - 4 2 , E0) and A+(E,Eo), are given by Xatz and Cameron.lg9Although the details of the cross section obtained by the total spectrum or photon difference methods are uncertain due to the form assumed for the bremsstrahlung spectrum, the peak positions of the “giant resonances,” and especially the cross sections integrated over energy, are relatively insensitive to the bremsstrahlung spectrum shape.lB7
2.7.
~
DETERMINATION OF NUCLEAR REACTIONS
MAX.
BETATRON
ENERQY
47 1
(Mov)
(a)
hv'
(Mrv)
(b)
FIQ.39 (a and b). Representative example of a photoneutron yield curve obtained with bremsstrahlung from a betatron, and the corresponding cross section obtained by a photon difference method.189
472
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
Figure 39 shows a typical photoneutron yield curve and the corresponding cross section extracted by the photon difference method (from Montalbetti et ~ 1 . 1 ~ 9 ) .
2.7.6. Differential Elastic-Scattering Cross Sections 2.7.6.1. Differential Cross Sections for Elastic Scattering of Neut r o n ~ . ~All ’ ~ of the methods described in Section 2.7.5.3 are equally germane to the measurement of differential neutron elastic-scattering cross sections. However, by thus limiting the objective of the measurements, certain simplifications can be realized since one can now utilize R I N G SCATTERER
T-=-
SOURCE
DETECTOR
I
. . ‘.
/’
\
‘.
/‘ I ’ \
Fro. 40. “Ring” geometry for measuring differential cross sections for the elastic scattering of fast neutrons.
the primary source to calibrate the detector response and thus avoid absolute flux measurements. This is especially convenient if the detector has a flat response in the energy region of the elastically scattered neutrons, or if the scattering nucleus is heavy enough not to alter significantly the energy of the scattered neutron, for then the detector may be biased sufficiently high to discriminate against nonelastic neutrons without the necessity for complicated corrections to take account of the energy dependence of detector efficiency. The general approach is to determine the counting rate of the detector when it is in its normal position and again when it is in the position usually occupied by the scatterer.zO2 Figure 40 shows a typical arrange2 0 1 R. N. Little, Jr., B. P. Leonard, Jr., J. T. Prud’Homme, and L. D. Vincent, Phys. Rev. 98,634 (1955);S. E. Darden, R. B. Perkins, and R. B. Walton, ibid. 100, 1315 (1955);H. S. Hans and S. C. Snowdon, ibid. 108, 1028 (1957);J. 0. Elliot, ibid 101, 684 (1956).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
473
ment for measuring elastic-scattering cross sections without recourse to absolute flux determinations. The geometrical arrangement utilizes the scatterer in the form of a ring on whose axis lies the source and detector. This geometry maximizes the amount of scattering sample for a given attenuation and simplifies the problem of shielding the detector from the primary neutron source. Shielding of the detector from the primary neutron flux is accomplished by the introduction of a shadow cone, on the sample axis, between source and detector. The scattering angle e is varied by changing 11 and 1 2 . Ideally, the shadow bar is opaque for source neutrons. However a correction is readily applied for non-opacity and room scattering by determining the counting rate with shadow bar in place and scatterer out. If we denote by P,, the average Aux of primary neutrons in the scatterer, the flux FD scattered into the detector can be represented by (2.7.6.1) where n, is the number of scattering nuclei and u(0) is again the differential elastic-scattering cross section a t angle 0. The counting rate in the detector appropriately corrected for background, as above indicated, will be CI = CFDwhere e represents the detector efficiency. If we now replace the scatterer by the detector, and operate the neutron source a t the same intensity, the counting rate will be C2 = eF. and the ratio of counting rates is simply (2.7.6.2) where the last term represents the average attenuation of the incident neutron flux by the scatterer. The above result is independent of the absolute flux a t the detector and a t the scatterer. For reasonable geometrical configurations, Eq. (2.7.6.2) will give u(0) to an accuracy of approximately 10%. For higher precision it is necessary t o make the usual corrections for finite size of the scatterer, seIf-absorption and multiple scattering in the scatterer, energy dependence of the detector, and finite angular resolution. Disadvantages of the method are that the geometry changes with scattering angle, and one is usually obliged to take account of source and counter asymmetries. A geometrical arrangement which circumvents the above-mentioned disadvantages is illustrated in Fig. 41. Here one uses a small scattering sample around which the detector is rotated in order to vary the scattering angle. The detector is usually an organic scintillator or a proportional
474
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
counter containing hydrogen or propane. Shielding in the form of wedges is placed between source and detector.2°a Where room-scattered background is high, it often becomes necessary to enclose the detector in a shield containing a channel through which enter the scattered neutrons.204 In this geometry (Fig. 42), detector and shield are rotated around an axis passing through the center of the sample and perpendicular to the plane of the scatterer. CYLINDRICAL SCATTERING SAMPLE
NEUTRON SOURCE
-PROTON
amm
SCINTILLATION DETECTOR
FIG.41. “Wedge” geometry for measuring differential elastic-scattering cross sections for fast neutrons. (Walt and Beyster, ref. 203.)
A time-honored method for measuring neutron elastic-scattering cross sections is based on the detection of the recoil nucleus, while still another method utilizes the cloud chamber in the same way as for reaction studies.206The former method has found extensive application in elastic scattering from light nuclei. If the target material is in the form of a gas, and if it is known that only elastic scattering takes place, one can obtain the complete angular distribution of the scattered neutrons by measuring the pulse-height distribution in an ionization chamber. This comes about. as follows: the energy of the recoil nucleus is proportional to (1 - cos +no.m,) 80s H. B. Willard, J. K. Bair, and J. D. Kington, Phys.Rev. 98,669 (1955); M. Walt and J. R. Beyster, ibid. 98, 677 (1955); M. Walt and H. H. Barschall, ibid. 95, 1062
(1964). SorR.
C. Allen, R. B. Walton, R. B. Perkins, R. A. Olson, and R. F. Taschek,
Phys. Rev. 104, 731 (1956). 206 J. R. Smith, Phys. Rev. 96,
730 (1954).
2.7.
DETERMLNATION OP NUCLEAR REACTIONS
475
while the number of recoils in the energy interval AE is proportional to c( $J)A(COS$J~.~.). It is therefore apparent that the elastic-scattering
angular distribution has the same shape as the energy distribution of the recoil nuclei.206Although this method is simple in principle, one must use great care to insure that the pulse height is independent of the position a t which the ions are formed, that recombination is either negligible during the time of ion collection or proportional to the number of ions produced, and that wall effects are small, i.e., the counter dimensions are large compared to the range of recoils. Also, one must exercise particular
SAMPLE-
NEUTRONSOURCE
EXPERIMENTAL ARRANGEMENT
FIQ. 42. Experimental arrangement for measuring neutron elastic-scattering differential cross sections in a high background of neutrons and y - r a y ~ . ~ '
care to minimize and correct for the effects of neutrons scattered in the detector itself. I n deference to this consideration the amount of nonactive material in the detector should be kept as small as possible. A summary of neutron elastic-scattering cross sections with references to the original literature is given in ref. 179. 2.7.6.2. Differential Cross Sections for Elastic Scattering of Charged Particles. Most of the considerations required for the accurate determination of charged-particle elsstic-scattering cross sections, such as current integration, solid-angle determination, corrections for beam size, angular resolution, multiple scattering, etc., have already been discussed in detail in Section 2.7.5. In addition, Sections 2.7.5.1.1 and 2.7.5.1.2 include a discussion of most of the detectors used for such measurements. Therefore, only a brief review of the principal experimental arrangements *oeE. Baldinger, P. Huber, and H. Staub, Helv. Phys. Acta 11, 245 (1938); H. H. Barschall and M. H. Kanner, Phys. Rev. 68, 590 (1940); J. L. Fowler and C. H. Johnson, ibid. 98, 728 (1955).
476
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
employed and a few representative results of these methods will be given in this section. At low energies, the electrostatic generator is usually employed for elastic-scattering measurements. The beam from such a machine is normally passed through a magnetic analyzer to separate various beam components (e.g., protons from diatomic hydrogen ions), and then through a precision electrostatic anaIyzer.42 An electrostatic-generator beam is very useful for precision experiments because it can be made essentially parallel and its spread in energy can be made quite small (typically 0.1 % under routine operating conditions). Its energy is readily variable in arbitrary steps, so that it is extremely useful for the study of elastic-scattering experiments as a function of incident-particle energy. However, the electrostatic generator is limited a t present to proton energies of about 8 MeV* which, because of the Coulomb barrier, prevents its use for many types of experiments in heavier nuclei. The targets employed in electrostatic-generator measurements of absolute elastic-scattering cross sections are ordinarily gas targets (if avajlable), because of the precision with which the number of scattering nuclei may be determined. I n addition, target contamination then can be minimized, and no question of target uniformity exists. To prevent partial loss of the precision energy control available in an electrostaticgenerator beam, the scattering gas may be opened directly to the electrostatic analyzer through a suitable differential pumping column.207This presents several advantages over a foil separating the accelerator vacuum from the scattering (a) continuous flushing of the gas prevents buildup of contaminant gases and vapors; (b) energy degradation and straggling of the incident beam do not occur in an entrance foil (such a foil would have a variable thickness due to carbon buildup); and (c) small-angle scattering effects in an entrance foil are not present. The effects of the gas in the differential pumping column are negligible, since the pressure in the column is much less than in the scattering chamber itself .207 Detailed considerations of the calculation of the cross section from the geometry of gaseous targets have already been given in Section 2.7.5. Of course, many substances are not available in suitable gaseous form, so that solid targets must be employed. For elastic-scattering measurements, the thick-target technique employing a magnetic spectrometer described in Section 2.7.5.1.2 allows measurements to be taken under conditions such that target contamination can be accurately determined and allowed
* See Section 2.7.7 for more recent supplementary remarks. *o7H. L. Jackaon, A. I. Galonsky, F. J. Eppling, R. W. Hill, E. Goldberg, and J. R. Cameron, Phys. Reu. 89, 365 (1953).
2.7.
DETERMINATION
OF NUCLEAR REACTIONS
477
The detectors employed for elastic-scattering experiments with electrostatic generators are usually scintillation counters, proportional counters, or magnetic spectrometers or spectrographs.* A discussion of the application of scintillation counters and magnetic spectrometers and spectrographs to measurements of this type has already been given in Sections 2.7.5.1.1 and 2.7.5.1.2. Proportional counters will give larger pulse heights for a-particles than for protons of the same energy, so they are particularly useful in a-scattering experiments where singly charged reaction products may also be produced. However, the improved energy discrimination of scintillation counters and the fact that they may be mounted directly in the vacuum without the necessity for a thin gas-tight intermediate foil has led to increased use of scintillation counters for elastic-scattering measurements. Figure 43 shows a representative elastic cross section obtained with electrostatic-generator techniques, in this case the elastic scattering of protons by a thick F19 target using the magnetic-spectrometer method described in Section 2.7.5.1.2 (from Webb et ~ 1 . 9 . From the upper energy limit of electrostatic generators (about 8 MeV* a t present) to the upper energy limit included in the present discussion (50 Mev), cyclotrons or linear accelerators are normally employed for the measurement of charged-particle elastic-scattering cross sections. The external beam from a cyclotron normally diverges to some extent, so that one must employ auxiliary equipment with some care in order to focus the beam on the target in a small spot of usable intensity without too much angular divergence. Furthermore, a certain fraction of the beam is lost in the focusing system, which tends to cause background difficulties. I n these respects the cyclotron is at somewhat of a disadvantage to electrostatic generators and linear accelerators, whose beams are seldom far from parallel. However, the cyclotron provides the only method for making measurements in the energy region between electrostatic generators and linear accelerators. For a given number of sections, the linear accelerator (of nuclear particles) is basically a fixed-energy machine, whereas recent developments2°8 in cyclotron techniques permit variation of the beam energy over a wide range (although not as yet with the ease of a n electrostatic generator). I n order to produce a beam with the appropriate characteristics on a remote target (usually from 15 to 30 feet from the cyclotron snout) in an experimental area outside the main cyclotron shielding, a typical cyclotron experimental arrangement will generally include various steering magnets, a strong-focusing pair
* See Section
2.7.7 for more recent supplementary remarks.
R. L. Thornton, K. Boyer, and J. M:Peterson, Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1966 4, 87 (1956). 208
478 2. DETERMINATION OF FUNDAMENTAL QUANTITIES
0
1
’
600
I
700
&
I
9150
060
1100 I200 1300 PROTON ENERGY (KEV)
1 4 0
1500
I
1600
I
1700
1
lm
FIG.43. Typical elastic cross section obtained aa a function of incident proton energy using electrostaticgenerator techniques.13a
2.7.
DETERMINATION OF NUCLEAR REACTIONS
479
of magnetic quadrupole lenses, and an analyzing magnet. The energy spread in the beam spot on the target depends upon many factors, but under routine operating conditions will be around one-half per cent and under very favorable conditions may be reduced to less than O.2%.ll6A typical scattering chamber has already been described in Section 2.7.5.1.1 and shown in Fig. 24. Both gas targets and solid targets are employed in cyclotron elasticscattering measurements. In particular, solid targets have been employed in a rather extensive series of measurements performed to observe the breakdown of Rutherford scattering for a-particles on elements throughout the periodic table (see, e.g., refs. 209-211). In many measurements of this type, the target thickness is sufficient to be determined to good accuracy by weighing of the target foils. I n such a case one actually determines the areal density in mg/cm2, which may be converted into nt = number of nuclei/cm2 if the density is known. This allows the direct determination of the absolute cross section from Eq. (2.7.5.44) when the scattered-particle yield, the solid angle, and the number of incident particles are measured. Where such measurements have been made, the magnitude of the elastic-scattering cross section at forward angles checks with that predicted from Rutherford scattering.212I n many experiments the scattering at forward angles is checked to have the correct Rutherford energy or angle dependence, and then the magnitude assumed to be Rutherford for cross-section normalization. Another technique213 employed in angular-distribution measurements compares the elastic scattering from the target under investigation with that from a gold target inserted before and after each measurement. The ratio of the number of counts from the target under investigation (per monitor count in an auxiliary monitor counter) to the number of counts from the gold target (per monitor count) may then be related simply to the Rutherford cross section.213 The principal advantage to this scheme is its moderate insensitivity to slight systematic errors in the angle determination. The detectors employed in cyclotron elastic-scattering measurements are scintillation counters (discussed above and in Section 2.7.5.1.l), nuclear emulsions (discussed in Section 2.7.5.1.I), magnetic spectrometers or spectrographs (discussed in Section 2.7.5.1.2), proportional counters (e.g., 209
G . W. Farwell and H. E. Wegner, Phys. Rev. 96, 1212 (1954).
N.S. Wall, J. R. Rees, and K. W. Ford, Phys. Rev. 97, 726 (1955). 211 H.E. Wegner, R. M. Eisberg, and G. Igo, ibid. 99, 825 (1955);E. Bleuler and D. J. Tendam, ibid. 99, 1605 (1955); D.D.Kerlee, J. S. Blair, and G. W. Farwell, 210
ibid. 107, 1343 (1957). 212 113
H. E. Wegner, private communication, 1957. W. F. Waldorf and N. S. Wall, Phys. Rev. 107, 1602 (1957).
480
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
ref, 214) or ionization chambers (e.g., ref. 215), and activation foils.216 Gas counters are not often used currently for angular-distribution measurements, because of the greater convenience of scintillation counters; however, they are very useful in proportional counter-scintillation counter telescopes for mass determination (see Sections 2.7.5.2.1 and 2.7.5.3.1). The activation method has been used with internal cyclotron beams.216Representative of elastic-scattering measurements made with
looor 1
TARGET: Th SCATTERING ANGLE :SO"
CORRECTED loot 2
2
5 W
In In v)
0
a v
10-
W
2
G-I W
a
1110. . . .ISI . . . .20I , , . 25 .I....I....I....I.,.J 30 31 40 45 ALPHA-PARTICLE ENERGY(MEV)
FIG.44. Typical elastic cross section obtained as a function of incident a-energy using cyclotron techniques.200 The energy variation was obtained by inserting foils in the cyclotron beam.
cyclotron external beams using scintillation-counter detection are the curves shown in Figs. 44 and 45. Figure 44 shows the failure of Rutherford scattering for a-particles as a function of energy at a fixed angle obtained by Farwell and Wegner,Z09 and Fig. 45 a similar failure for both a-particles and deuterons as a function of angle at fixed energy obtained by Rees and S a m p ~ o n . ~ ~ ' A number of proton elastic-scattering angular distribution measureD. 0. Caldwell and J. G. Richardson, Phys. Rev. 98, 28 (1955). D. A. Bromley and N. S. Wall, Phys. Rev. 102, 1560 (1956). *laB. L. Cohen and R. V. Neidigh, Phys. Rev. 93, 282 (1954). a l r J. R. Rees and M. B. Sampson, Phys. Reu. 108, 1289 (1957). 214
216
2.7.
48 1
DETERMINATION OF NUCLEAR REACTIONS
ments have been carried out using linear accelerators at various fixed energies, for example, a t 9.8 Mev (Minnesota first section218),19 Mev (Physico-Technical Institute of the Academy of Sciences, USSR219), 31.5 Mev (Berkeley220),and 40 Mev (Minnesota second sectionzz1).In a representative experimental arrangement,221 the beam from the Minnesota linear accelerator second section passes through a deflecting magnet, a strong-focusing quadrupole magnet, and then some thirty feet into the
Gold
1.2
0
I
I
I
30
60
90
I
I 20
I
150
IE I
&m.
FIG.45. Representative angular distribution of elastically scattered deuterons and a-particles using cyclotron techniques.*17
target chamber in a shielded experimental area. The beam-energy spread on the target is less than 0.5%, the beam diameter about one-half inch, and the angular divergence less than 1.5 X radian. Most of these experiments have been performed with scintillation-counter detectors. Target and detector considerations have for the most part already been discussed. The accuracy of charged-particle elastic-scattering measurements varies with the type of experiment. The most accurate measurements have in general been determinations of the proton-proton scattering cross section with electrostatic generators. It might be informative to conclude with a brief summary of errors in the cross section obtained in a *1* 219 220
*21
R. H. Lovberg, Phys. Rev. 103, 1393 (1956); N. M. Hintr, ibid. 106, 1201 (1957). R. A. Vanetsian and E. D. Fedchenko, Soviet J. of Atomic Energy 2, 141 (1957). B. B. Kinsey and T. Stone, Phys. Rev. 103, 975 (1956). M. K. Brussel and J. H. Williams, Phys. Rev. 106, 286 (1957).
482
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
representative experiment of this type by Worthington et aLZz2carried out with a gas target and proportional-counter detection. The systematic uncertainties for a particular choice of defining slits were as follows: beam energy 0.1%, proton current 0.1%, integrator capacitance 0.04%, oil density (in measurement of gas pressure) 0.02%, gas contamination 0 to 0.3%, slit-edge scattering 0.06 % maximum, “G-factor” of analyzer slit system f0.04% and multiple scattering 0.01 %. The nonsystematic uncertainties were as follows: statistics k 0.22%, height of oil column k0.02%, gas temperature I f : 0.02%, and counting losses I f : 0.02%. The total error in the cross section resulting when these uncertainties were combined appropriately (except at forward angles) was k0.3%. Acknowledgment The authors are pleased to acknowledge their indebtedness to Leona Stewart for supervising the preparation of the figures, for checking many of the formulae and derivations, and for correcting, proofreading, and assembling the completed manuscript.
2.7.7. Supplementary Remarks The preceding sections (2.7.3 to 2.7.6 inclusive) were completed and submitted in April, 1958. The present Appendix was submitted in August, 1960. Although the general methods for the determination of nuclear cross sections have not been changing rapidly, it seems appropriate to mention briefly some of the more significant improvements in techniques which have developed since the previous sections were written. The energy ranges available for monoenergetic neutrons from various reactions have been considerably extended, from those indicated in Section 2.7.3.1.2, by the advent of the tandem electrostatic accelerator. It is a simple matter to calculate the new upper limits for neutron energies corresponding to the new accelerator energy limits. However, a t these higher energies extra caution is required to insure that low-energy neutron groups or breakup neutrons are biased out appropriately in the neutron detector. A major advance in charged-particle detection methods has been the solid-state detector.” Development of these detectors is proceeding so rapidly that definitive statements at this time about energy resolution, maximum energy capability, and general usefulness will undoubtedly be rapidly outdated. It seems appropriate to indicate briefly their characteristics, however, since they will certainly become one of the more important types of energy-sensitive detectors. The reader is referred to ((
222
H. R. Worthington, J. N. McGruer, and D. E. Findley, Ph98. Rev.90,899 (1953).
2.7.
DETERMINATION OF NUCLEAR REACTIONS
483
papers by Friedland et a1.22s and by McKenzie and BromleyZz4 for details and other references. Solid-state detectors are basically semiconductor junctions which function over a small region as solid-state ionization chambers. Examples of these junctions are Au-Ge junctions cooled to liquid nitrogen temp e r a t ~ r eand ~ ~silicon ~ p - n junctions operated at room temperature.223 A number of important advantages of solid-state counters over scintillation counters for the detection of heavy charged particles can be cited. If the particle range is not too great, solid-state counters produce voltage pulses proportional to the particle energy with an energy resolution which is considerably better than for scintillation counters (0.6% has already been attained). They are small and simple, requiring no high-voltage supplies, associated photomultipliers, gas flow systems, or windows. The only voltage requirement is a small bias, usually less than a hundred volts. They produce pulses with a rise time in the millimicrosecond range or less, making them particularly adaptable to coincidence measurements. There are, of course, certain limitations to the solid-state counters, at least at present. They are most useful for particles with short range because the sensitive “depletion region” of the junction is as yet rather small. Their upper limit for linear response is -35 Mev for a- and He3 particles and -10 Mev for protons. These limits will presumably be increased as junctions with thicker depletion regions are constructed. The resolution capabilities appear t o be limited primarily by noise, so that to attain best results, low-noise, capacity-compensated amplifiers are required. Counters of large area are difficult to construct, so that only about 1 cm by 1 cm counters retaining all of their desirable properties have been used up to 1960. Aside from counting rate considerations, very small counters are often a disadvantage for the determination of absolute cross sections because of difficulties with finite source sizes, multiplescattering when absorbers or d E / d x counters precede the solid-state counter, and other similar considerations. However, because of the rapid improvements being made in the field, the above limitations may soon be considerably reduced. An interesting application of solid-state counters is the construction of multichannel detection systems using a large number of small solidstate eounters placed side by side, as for example, in the focal surface of a magnetic spectrometer. When such systems are combined with small transistorized amplifiers, data accumulation can be greatly facilitated. An interesting new development in particle detection by scintillation 228
914
S. S. Friedland, J. W. Mayer, and J. S. Wiggins, Nucleonics 18, 54 (1960). J. M. McKenzie and D. A. Bromley, Phys. Rev. Letters 2, 303 (1959).
484
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
counters is the possibility of discriminating between electrons and heavily ionizing particles by the difference in decay time of the fluorescence. For example, in the few Mev range, the fluorescent decay time for electrons in Cs(T1) is about 0.70 psec as compared to 0.42 psec for a r - p a r t i ~ l e s . ~ ~ ~ ~ ~ ~ Such decay time differences are also teported for anthraceneZz7 and other organic scintillators.a2*This difference is sufficient to allow discrimination between alphas and electrons in C S I , ok~ ~between ~ fast neutrons (actually proton recoils) and y-rays in anthracene and some other organic scintillators.2ag A large fraction of the (charged particle, y) type of studies continue to concentrate on energy measurements and relative cross sections, without serious attempts at absolute cross-section measurements. One interesting method for determining absolute cross sections, not described in the previous sections, is by measurement of the /3-activity of the residual nucleus of the reaction, a procedure also frequently used in neutron capture cross-section measurements. The very small cross sections for the reactions 0le(p,y) and NeZ0(p,y),both of astrophysical interest, have recently been determinedZ3Oby counting the 66-sec F17 and 23-sec NaZ1positron activities produced in appropriate 0lEand Ne20 targets by a measured integrated proton flux. Unfortunately, the method is somewhat limited by the availability of appropriate activities produced by such capture reactions, and, as usual, accurate absolute cross sections require accurate knowledge of target thicknesses and counter efficiencies. Nevertheless, it is a very useful method for the determination of very small capture cross sections. A number of advances have been made in measurements of cross sections for y-induced reactions. In particular, much more emphasis has been placed on measurements using monoenergetic y-rays. Nuclear y-rays from the T(p,y) reaction have been used to give monoenergetic y-rays with variable energy from 20 to 25 Mev by varying the proton energy. Measurements of the total y-absorption in various elements have been made in this wayza1by attenuation experiments, as well as of the C12(y,n) 226R. S. Storey, W. Jack, and A. Ward, Proc. Phys. SOC.(London) A72, 1 (1958). 226 J. C.Robertson and A. Ward, PTOC. Phys. SOC.(London) A79, 523 (1959). 227 G.T.Wright, Proc. Phys. SOC.(London) B49, 358 (1956). 2~ F. D.Brooks, Progr. in Nuclear Phys. 6,284 (1956);also in “Liquid Scintillation Counting” (C. G . Bell and F. N. Hayes, eds.), p. 268,Pergamon, London, 1959. **o F. D. Brooks, Atomic Energy Research Establishment, Harwell, England, Unpublished Report AERE NP/GEN 8 (1959). 230R. E. Hester, R. E. Pixley, and W. A. S. Lamb, Phys. Rev. 111, 1604 (1958); N. Tanner, Gid. 114, 1060 (1959). *aM.M.Wolff and W. E. Stephens, Phys. Rev. 112, 890 (1958);E.E. Carroll, Jr. and W.E. Stephens, ibid. 118, 1256 (1960).
2.8.
DETERMINATION OF FLUX AND DENSITIES
485
cross section by use of the positron activity of C11.23z+z33 An example of the latter is the interesting experiment by R. B. Day233in which an annular ring of plastic phosphor is placed around the tritium y-source. After a bombardment for an appropriate interval of time, the plastic phosphor is placed directly on a photomultiplier tube and the C" positron activity in the phosphor itself counted. Perhaps the most important advance in the production of monoenergetic y-rays is that which utilizes the annihilation in flight of posit r o n ~ In . ~this ~ ~method, positrons produced a t the target of an electron linear accelerator are momentum-analyzed in a magnetic field and then allowed to annihilate in flight in a thin lithium target. By use of a suitable geometry, the transmitted positrons may be deflected from the photon beam and the photons converged to a focus a t the target of interest. At present, it appears that experiments performed with T(p,y) y-rays, although more limited in accessible energy range, are capable of higher resolution than positron-annihilation y-rays, but that the available intensities from the annihilation method will ultimately be considerably greater. Also, the T(p,y) source is subject to grave difficulties above the threshold for neutron production. Advances have also been made in the bremsstrahlung measurements of y-induced reactions. In particular, the method of Penfold and L e i ~ s ~ ~ ~ for the extraction of cross sections from these measurements is now widely used. Although not fundamentally different from some of the m e t h o d ~ l ~ described, ~ * ~ g ~ it has certain analytical and computational advantages.
2.8. Determination of Flux and Densities 2.8.1. Determination of Flux of Charged Particles* 2.8.1.1. Measurement of Parallel Beams of Charged Particles. We assume we are dealing with a beam of charged particles emanating from an accelerator. We wish to determine the number of particles nB that D . Cohen and W. E. Stephens, Phys. Rev. Letters 2, 263 (1959). R. B. Day, private communication, 1960. 234 J. Miller, C. Schuhl, G. Tamas, and C. Tzara, Cornpt. rend. acad. sci. 249, 2543 (1959); S. C. Fultz, C. P. Jupiter, C. R. Hatcher, and F. D. Seward; C. P. Jupiter, N. E. Hansen, H. W. Koch, and S. C. Fultz, Bull. Am. Phys. SOC.121 6, 36 (1960). 236 A. S. Penfold and J. E. Leiss, Phys. Rev. 114, 1332 (1959). 232L. 233
* Section 2.8.1 is by 0. Chamberlain.
486
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
have passed along the beam line and impinged on a target in a certain exposure. Obviously the beam must be well localized in space and its composition must be well known if it is to be of greatest usefulness. This implies collimator holes and focusing magnets designed to cut out scattered particles or particles whose energy may have been degraded by passage through material. We discuss first the primary monitors, capable in themselves of an absolute beam determination. I n a later section we discuss some useful secondary monitors-those which must be calibrated by comparison with some primary monitor. 2.8.1.2. Primary Monitors. 2.8.1.2.1. FARADAY CUP. The most common instrument for use as a primary standard for measuring beams of charged particles is the Faraday cup. The beam to be measured falls on an electrode sufficiently massive to stop the beam as well as almost all of the secondary particles resulting when the beam particles strike the material of the Faraday cup. The Faraday-cup electrode is electrically insulated from its surroundings, and the measurement of beam hitting the cup consists in measuring the electric charge carried to this electrode by the beam. If Q is the electric charge carried to the Faraday cup, and ze is the electric charge carried by each beam particle, then the total number of beam particles, is clearly nB
= &/ze.
I n this expression e is the magnitude of charge of one electron and z the number of electronic charges carried by each particle of the beam. For example, for a beam of alpha particles we would use z = 2. The basic electrical circuit of the Faraday cup is quite commonly the slide-back circuit shown in Fig. 1. At the end of a run the potentiometer is adjusted to bring the Faraday-cup potential back to its original value. Thus the electrometer amplifier and voltmeter serve only as a null instrument-to show there was no net change in potential of the Faraday-cup electrode. The magnitude of total beam charge must then be equal but opposite to the total charge supplied through C,, namely
Q = -C,AV where C, is the capacity of the condenser C. and AV is the difference in potential as shown by the two readings of the calibrated potentiometer, before and after the interval during which the beam was turned on. Many precautions must be taken if beam charges are to be correctly measured. These are discussed in the following paragraphs. (1) The drift of the instrument (with beam off) must be measured, preferably immediately before and immediately after the beam readings are made, and the drift corrected for, if necessary. Since the drift rate
2.8.
DETERMINATION
487
OF FLUX AND DENSITIES
may be different (before and after the run) the correction is usually based on the average. (2) To be sure the rate of drift used corresponds to conditions as they are during a beam measurement, the potentiometer should be continually adjusted during a run so as to maintain nearly zero reading of the voltmeter at all times. This is sometimes done electronically. The circuit
FARADAY-CUP ELECTRODE
r
.
INCIDENT BEAM
- - - - ---1 I
~
ELECTROMETER AMPLIFIER CALIBRATED POTENTIOMETER
VOLTMETER AT OUTPUT
FIG.1. Circuit diagram of Faraday cup with slide-back potentiometer.
shown in Fig. 2 gives continuous electronic slide back action through the condenser C . The circuit is used just as that of Fig. 1. Since the electrometer is used only as a null indicator when the final manual slideback is made, the condenser C need not be accurately known. The electrometer amplifier shown in Fig. 2 should be a high-gain highinput-impedance amplifier such as that shown in the article of Brown and Tautfestl or (for measurement of very weak beams) a vibrating reed electrometer.* (3) The condensers C, and C must be free from leakage and dielectric “soakage,” which in practice usually means a condenser with polystyrene dielectric,3 and C, should be calibrated on the occasion of its use. For dc capacitance comparison the circuit of Fig. 3 is very good. When the K. L. Brown and G. W. Tautfest, Rev. Sci. Instr. 27, 696 (1956). H. Palevsky, R. K. Swank, and R. Grenchik, Rev. Sci. Znstr. 18, 298 (1947) give an excellent description of the instrument. Good commercially made varieties are available. SAvailable from J. E. Fast Company, Chicago, Illinois, or as type L A “glass mike” of Condenser Products Corporation, Chicago. 1
2
488
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
FARADAY-CUP ELECTRODE
INCIDENT BEAM
- - - - -- - - - - - - - - - - - - - - - -- - - - - - - - - - I I I
I
I
r-----------I
I
I
I
I I
I
I
I
I I I
I
I I
I f
'
I I
'c
I
I
I I
I I
I I I
I
I
I I
I
ELECTROMETER AMPLIFIER (NEGATIVE GAIN)
I I I I
I
- 1 1
L-------_-
circuit is adjusted so that the output voltmeter I'is unaffected by opening or closing switch Sw2, then C z / C , = Rr/Rz. R1 and R z should be wellstandardized variable resistance boxes. Figure 4 shows more detail of the condensers with their containers, and indicates the stray capacities between each terminal and the container. Note that the magnitudes of the stray capacities do not affect the balance point: two of the stray capacities are strictly in the low-impedance part of the circuit and the other two suffer no potential change at the balance point. Figure 5 shows
7
ELECTROMETER
FIQ.3. Bridge circuit for dc comparison of capacitances.
2.8.
DETERMINATION OF FLUX AND DENSITIES
489
FIG.4. Detail of dc bridge, showing stray capacities in addition to the main capacities CI and C2.The balance point is independent of the stray capacities.
FIG. 5. Preferred electrometer circuit showing stray capacitances in connection with the standardized condenser CS.
490
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
the same detail of the condenser C, in the circuit of Fig. 2. Again one stray capacitance is in the low-impedance potentiometer circuit, and the other stray capacitance returns exactly to its original charge when the potentiometer is adjusted for zero reading (or the original reading) of voltmeter V . Comparison of Fig. 4 with Fig. 5 shows that in each case the effects of stray capacities are similarly eliminated, indicating the suitability of the bridge circuit for intercomparing condensers. Since absolutely calibrated condensers are usually available calibrated at finite frequency (often 1000 cps) one polystyrene condenser must be compared with a n absolute condenser a t this frequency. By choosing a polystyrene condenser very nearly equal in capacitance to the absolute condenser one can arrange to get an accurate comparison from a standard ac bridge such as the General Radio type 650-A impedance bridge. Effects of stray capacities can be minimized if a standard capacity a t least as large as 0.01 pf is chosen. (4) It is of course essential that, to the desired accuracy, all of the beam current and only the beam current reach the Faraday cup. The beam itself should be well-defined and free from any spray of scattered particles such as those which might emerge from a collimator. Ideally, the beam should be focused on a collimator and then the beam emerging from the collimator should again be magnetically analyzed and focused on the entrance to the Faraday cup, although it is frequently not convenient to use this procedure. More usually the beam emerging from the collimator falls directly on the Faraday cup. Carefully made exposures with pairs of X-ray films can be used in many cases to place limits on the fraction of the beam lying outside a certain geometrical area.4 Gas around the Faraday-cup electrode would give rise to ionization and hence to gain or loss of charge. The Faraday cup should be in vacuum, and experiments should be performed to make sure the charge collected is independent of pressure for the quality of vacuum used. A diffusion pump is usually required. Backscattering of beam particles can cause significant loss of beam charge, especially for low-energy electron beams. * This may be minimized by constructing the surface of the Faraday cup (where the beam strikes) of material of low atomic number and by allowing the beam to strike the Faraday cup a t the bottom of a deep hole. Backscattering may
* See also Vol. 4 B, Part 10. ‘The method is described in G . P. Thomson and W. Cochrane, “Theory and Practice of Electron Diffraction,” p. 244. Macmillan, London, 1939. The X-ray films determine accurate exposure ratios for different regions of the beam. Spatial integration then tells what fraction of the film exposure occurred outside any prescribed area.
2.8.
DETERMINATION OF FLUX AND DENSITIES
49 1
further be reduced if the region of the deep hole has a magnetic field, transverse to the beam direction, sufficient to deflect the incident beam appreciably. The magnetic field serves to trap charged particles, particularly if they are of lower energy than the beam. Secondary electron emission from the Faraday cup can cause a loss of negative charge, and secondary emission from the surroundings of the Faraday cup (especially the entrance foil) may cause an unwanted gain in negative charge by the Faraday cup. Most of this effect is eliminated if sufficient magnetic field is maintained in the region of the deep hole and in the region between the Faraday cup and any thin film where the beam may enter the vacuum region of the Faraday cup. A test for residual secondary emission may be made by biasing the Faraday cup (and all its electrometer circuit) with respect to the surrounding conductors, or by biasing the surroundings with respect to the Faraday cup. I n either case the bias voltage supply must be very stable, for any variations in biassupply voltage induce unwanted charges on the Faraday cup. Bias voltages typically used go up to a few hundred volts, and sometimes to more than 1000 volts. By comparing Faraday-cup charge with some secondary monitor a t various bias voltages (both positive and negative), it is possible to measure how much charge transfer is due to low-energy charged particles. With proper magnetic trapping it should in every case be possible to reduce this as low as desired, certainly always well below 1%. It is more difficult to test for high-energy charged particles backscattered off the cup or knock-on electrons from the entrance windows. A real test for backscattering may involve varying the depth of the hole into which the particles fall in the Faraday cup and varying the position of the entrance window. While proton beams with energies less than 20 Mev can be stopped in graphite without generating secondary particles, either electron beams or high-energy proton beams generate many secondary particles. The Faraday cup must be massive enough to stop a sufficient fraction of the secondary particles. For electron beams the reader is referred t o the articles of Tautfest and Panofskys and of Brown and Tautfest.'* Rest practice is to make a mock-up of the Faraday cup prior to actual construction, and measure with nuclear emulsions the numbers of charged particles entering and leaving the skin of the Faraday cup at various points over its surface. By making the conservative assumption that all entering particles are of one sign and all leaving particles are the opposite sign, it is possible to set a firm upper limit for the loss of charge due to secondary charged particles. This procedure has been described by * See also Vol. 4 B,Chapter 10. 6 G.W.Tautfest and W. K. H. Panofsky, Phys. Rev. 106, 1356 (1957).
-TO
ELECTROMETER
/ /-.ooe
>
SOLID BRASS
I J P
TO PUMP
I
I
II
FOILS
-
t
I
BEAM
/CONNECTED
-FILLING
LTEFLON
Be -cu
3mg &‘A1
TO IOOOV
TUBE
-
0
1
2
3
4
INCHES
FIG.6. Faraday cup of Peterson, as arranged for the calibration of an ionization chamber for 340-Mev protons. Permanent magnets, not shown, have been used outside the vacuum wall to provide a transverse magnetic field of 100 gauss across the face of the Faraday cup.
2.8.
DETERMINATION OF FLUX AND DENSITIES
493
rELECTRONlCS PLATE VACUUM
POLYSTYRENE BIAS RING
MOLDED RUBBER GASKET
-POLYSTl‘RENE
-12
INCHES+
FIQ.7. Faraday cup of Brown and Tautfest for electrons up to 300 MeV. Permanent magnets, not shown, were inserted in the cavity of the collector electrode to give a transverse field of 300 gauss.
INTEGRATOR BIAS VOLTAGE IN KV
FIG.8. Curve of response versus dc collector-electrode bias, as obtained with the Faraday cup of Brown and Tautfest. On the basis of this cwve the gain or loss of charge due to low-energy electrons could be guaranteed less than 0.2%.
494
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
Aamodt, Peterson, and Phillipsa in constructing a Faraday cup for 340Mev protons, and a similar procedure is described by Brown and Tautfest for a Faraday cup for electrons. Figure 6 shows the Faraday cup of Peterson, as set up for the calibration of a n ionization chamber for 340-Mev protons. Figure 7 shows the Faraday cup of Brown and Tautfest as used by Tautfest and Panofsky for electrons, up to 300 MeV. Figure 8 shows a curve of Faraday-cup response versus dc bias as given by Brown and Tautfest. 2.8.1.2.2. ABSOLUTE BEAMMEASUREMENT BY COUNTING. With the development of very fast scaling circuits7 it has become practical to measure charged-particle beam intensities by counting the individual beam
-1 U
00-
- - - - - - --
7-
LEAD SCATTERING
COUNTER TELESCOPE NUMBER 2
COUNTER CHAMBER TELESCOPE
NUMBER
I
FIG.9. Schematic diagram of experimental arrangement for absolute beam monitoring by counting, as used to calibrate an ionization chamber.
particles. Since in most experiments the primary beam is much more intense than any beam which could be counted directly] it will usually be necessary to calibrate a secondary beam monitor, such as an ionization chamber, using several intermediate steps. The method described here presupposes the availability of a t least two fast scaling circuits (resolving time less than 10-7 second) and a duty cycle of the accelerator which produces the beam of a t least 1%. Using these parameters one would estimate about 1 % counting loss at counting rates of about 1000/sec. Figure 9 shows a suitable experimental arrangement. One scaler would record counts from a telescope of two counters placed directly in the beam, to be used at beam rates near lo3 per second. The other scaler would be used to monitor a second telescope placed behind a lead foil (for multiple scattering) so as to count only about 1% of the beam, the second telescope to be used for beam rates of 1O3-1Ob/sec, approximately. Crudely speaking, the ratio of counting rates of the two telescopes would be determined a t lo3 particles per second, and the ratio of counting rate 6
R. L. Aamodt, V. Peterson, and R. Phillips, Phys. Rev. 88, 739 (1952).
Model 520A, high-speed decade scaler of the Hewlett-Packard Co.; V. Fitch, Rev. Sci. Instr. 20, 942 (1949); J. Marshall and J. Fischer, Proc. Nall. Electronics Conf. 9, 491-500 (1953). 7
2.8.
DETERMINATION OF FLUX AND DENSITIES
495
in the second telescope to the ionization-chamber current would be determined a t lo6 particles per second (a beam level well suited to the ionization chamber). Actually, in each comparison several different beam levels would be used, and a n extrapolation of the ratio to zero beam intensity would be performed. Somewhat better accuracy might be attainable if more intermediate steps were used. Calibrations to 1% or 2% should be obtainable with this method in counting times of the order of one day. 2.8.1.2.3. BEAMMONITORING BASED ON INDUCTION ELECTRODES. When a beam of charged particles passes near an electrode it induces an apparent charge on the electrode. If the electrode geometry is carefully chosen,
FIG. 10. Arrangement of induction electrodes for measuring total accelerated charge in a synchrotron.
the apparent charge is easily related to the beam density. Although this method of beam measurement is not a t present applicable to external beams unless they are extremely intense, the method has been used for some time in connection with large synchrotrons such as the Cosmotron and Bevatron to measure the circulating beam charge within the machine. The measurement is greatly facilitated by the bunching of a synchrotron beam during acceleration. Present practice allows beam charge measurements to 5 or 10% on an absolute basis, with indication that this accuracy can be improved. Beam charges of a t least lo7 electron charges are needed in most cases, this minimum being determined by rf pickup from the rf system of the synchrotron. A suitable arrangement of electrodes is shown in Fig. 10. Each of the three electrodes shown is a hollow rectangular conducting box without ends, so the beam can pass through all three electrodes. The two end electrodes are grounded guard electrodes, necessary to give a well-defined effectivelength to the center electrode. The effective length will be called L, and is the distance from center to center of the two gaps between the electrodes. These electrodes would most frequently be installed within a straight section of the synchrotron.
496
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
If the circumference of the synchrotron is called K , the average charge induced by the beam on the (center) electrode is qL/K, where p is the beam charge. The average potential of the electrode must be measured with respect to the potential when there is no beam in the vicinity of the electrode, that is, when the beam bunch is on the other side of the synchrotron. A block diagram of a suitable circuit arrangement is shown in Fig. 11. A pulse generator is also included. By calibrating with the pulse generator one may make measurements that are independent of many of the circuit parameters. The pulse generator is assumed to have a very low output impedance and the amplifier to have a high input impedance.
PULSE GENERATOR
C II
‘
LINEAR
AMPLl F I ER
-
FAST
A
DC
- INTEGRATOR -OUTPUT
L
FIG.11. Block diagram of induction-electrode circuit.
The amplifier might well be a triode amplifier of approximately unit gain, followed by distributed amplifiers. It is assumed that the rise time of the amplifier system is appreciably shorter than the time between pulses during which the induction electrode is not affected by beam charge, so that the potential between pulses may be well established at the amplifier output. Furthermore the amplifier must be linear to considerable precision.* We assume that there is no overshoot a t the amplifier output. (The whole pulse structure should be of one sign.) Any overshoot, if present, would be compensated by a passive linear network inserted in the amplifier input or output circuit. The fast integrator circuit must serve the function of determining the “area” under the induction-elec trode pulses. A number of techniques are currently in use, At the Bevatron, only the fundamental frequency component is measured, and one obtains the absolute charge by applying a correction factor based on the Fourier analysis of the pulse shape as photographed on an oscilloscope screen. Th at method has been described by Heard.0
* See also Vol. 2, Chapter 6.2.
* Harry G . Heard, University of
California Radiation Laboratory Reports UCRL3609 and UCRL-8092. Unpublished.
2.8.
DETERMINATION OF FLUX AND DENSITIES
497
A different method will be described here.g Within the limitations described above, the fast integrator circuit may be that of Fig. 12. I n this circuit t,he diode serves to establish the minimum or quiescent potential (between beam bunches) as ground. The output is then obviously proportional t o the time average of the pulse, averaged over a complete rf cycle. The constants in this circuit must be chosen to minimize the imperfections of the diode (e.g., type OA85). Reasonable values are R 1 = R z = R S = 200,000 ohms, C1 = Cz = C3, RICl = lOO/f, where f is the radiofrequency of the machine. However, for the higher frequency
FIG.12. Circuit of fast integrator.
synchrotrons the capacities given here may be too small, becoming comparable to stray capacities, or giving insufficient sensitivity. The slow integration-integration over many acceleration pulses of the synchrotron-may be accomplished by switching the condenser Cs out of the fast integrator circuit and discharging it into a dc electrometer such a s that of Fig. 2 in which the input is always very nearly a t ground potential. The switching operation is to be performed once each acceleration cycle, a t the time in the cycle when the circulating beam charge is to be measured. The switching circuit must be well shielded, and resistors may be required in series with switch contacts to avoid undesired charge changes as the contacts open. A vibrating-reed electrometer may be called for. The pulse generator to effect the calibration is shown here only in a form suitable for representing negative beams, although circuits can be devised for positive beams based on the same principles. The circuit is shown in its essential components only in Fig. 13. The vacuum tube must be operated so that it is guaranteed to be cut of between pulses. Care must also be used to avoid overloading the tube. The time average of the output voltage is t,hen - I R , when measured with respect to the voltage between pulses, where I is the current indicated by the (well-calibrated) ammeter in the plate circuit, and R is the (well-calibrated) resistance 9 The fast integrator described here is similar to that used at the Cosmotron. C. E. Swartz (private communication, 1957).
498
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
indicated in the plate circuit. When coupled to the induction electrode through the known capacitance C (see Fig. 11), it imitates a beam whose charge is just
Q
= -(K/L)IRC.
This is the equation through which the calibration is effected. Although the calibration of the system described here should be independent of the frequency of pulses from the pulse generator, errors will of course be minimized if the pulse generator is used very close to the radio frequency of the accelerator. To the extent that the fast integrator works properly, the response should be independent of the width of the beam bunch. The
R
VARIABLE BIAS NEGATIVE
FIG.13. Basic pulse-generator circuit for calibrating induction electrodes.
bunching of the beam represented by the pulse generator may be varied by changing the bias and rf amplitude, but care must be used to make sure the tube is always cut off between pulses. It may in fact be advisable to add a bit of circuitry that will allow continuous monitoring of the minimum grid voltage on a separate meter. More complicated arrangements of induction electrodes can be used to measure the radial or vertical beam position. One such apparatus, as used in the Cosmotron, has been described by Swarta,'* but will not be described here. 2.8.1.2.4. OTHERABSOLUTEBEAMMONITORS. Other primary methods of measuring fluxes of charged particles include calorimetric methods, and the counting of tracks in nuclear emulsion. A full discussion of these subjects is omitted because they are believed to be of less general usefulness. lo C. E. Swartz, Rev. Sci. Instr. 24, 851 (1953).
2.8.
DETERMINATION
OF FLUX AND DENSITIES
499
2.8.1.3. Secondary Monitors. 2.8.1.3.1. IONIZATION CHAMBERAS RELATIVE BEAMMONITOR. * I n most work with charged-particle beams it is important to have some secondary standard for monitoring beam intensities, chosen to interfere as little as possible with the beam and to be useful over wide ranges of beam intensity. By far the most common instrument of this kind is the parallel-plate ionization chamber. The ionization chamber can be constructed with sufficiently thin foils so that the beam is little affected by its presence, especially for proton beams above 30 Mev or electron beams above 2 MeV. Since each beam particle that traverses the ion chamber produces many ion pairs in the gas of the chamber, electrometer drift currents rarely cause trouble. A number of precautions should be taken to achieve reliable operation. To decrease the tendency of ions, once formed, to recombine, one fills the ion chamber with a gas, such as argon, in which the negative charges may be collected (quickly) as electrons rather than (slowly) as negative ions. Variations of barometric pressure may be prevented from affecting the internal pressure by using a pressure well above (or well below) atmospheric pressure. To prevent drift due to leakage currents along insulators, one should always arrange that the insulators responsible for the electrical insulation of the electrode connected to the electrometer input (sensitive electrode) should have negligible voltage drop across them-always less than one volt. The chamber should have a well-defined volume of gas from which ions are collected, determined by two highvoltage electrodes, one on each side of the sensitive electrode. This last measure is necessary since contact potentials of the electrodes will always give rise to some ion collection, even in a region between two electrodes a t the same electrical potential. (The author has experienced a considerable degree of nonlinear operation of an ion chamber which had a grounded foil on one side of the sensitive electrode-there was a partial collection of ions from this region, with very different fractions of ion recombination a t different beam levels.) Figure 6 shows a very satisfactory form of ion chamber used by Chamberlain, SegrB, and Wiegand for monitoring proton beams in the energy range 100-300 MeV. Whenever the ion chamber is used, a check should be made that the electric field in the ion chamber is sufficient to collect all the ion pairs formed by the beam. This is done by comparing the ion chamber with some other monitor while using various voltages on the high-voltage electrode of the ion chamber (various collection voltages). Voltages of 300-1000 are quite typical for the ion chamber of Fig. 6, filled with slightly over one atmosphere of argon. If the collection voltage is SUE* See also Vol. 4 A, Section 2.1.5.
500
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
cient, the ion chamber will give a response independent of collection voltage. In judging whether the collection voltage is adequate (which in principle means extrapolating to infinite voltage) the following ruleof-thumb is useful for parallel-plate ion chambers that are reasonably near the point of complete charge collection: when the collection voltage is doubled, the recombination is cut to one-half or less (at any given beam intensity). Typically the rule might be applied as follows: If raising the collection voltage from 400 volts to 800 volts causes 1% change in ion chamber response (charge collected by the dc electrometer) then there is no more than 1%of charge loss by recombination at the higher collection voltage. Jesse and Sadauskis” have found that impurities of the order of 0.1 % can quite appreciably affect the response of an ion chamber. From this we infer that unless special precautions are taken it will not be possible to compare beams in different laboratories by the use of ion chambers without transporting the chambers and gas-filling apparatus from one laboratory to another. I n practice, less trouble seems t o be experienced than might be expected from the work of Jesse and Sadauskis, in getting reproducible results with argon-filled ionization chambers. Probably the argon used (believed 99.5% argon and the remainder mostly nitrogen) is already “saturated” with impurities in the sense of Jesse and Sadauskis. Methane, ethane, and ethylene are other promising gases for ionization chambers. Although they have been used less often than argon, they may be more reproducible because they are less sensitive to small amounts of impurity. Nitrogen may also be less sensitive to impurities than is argon. Oxygen is to be avoided, even as a small impurity, for electrons attach to oxygen. Thus, there is much more ion recombination when oxygen is present. Effects due to radioactivity induced by the beam in the materials of the ion chamber are not usually noticeable unless the ion chamber is accidentally exposed to a very high beam level shortly before a very weak beam is to be measured. Even in these cases there will be little error if ion chamber drift (with beam off) is measured immediately before and immediately after a measurement. The use of the electrometer with an ionization chamber is just the same as with a Faraday cup-except possibly that the drift corrections are usually smaller, fractionally, than with the Faraday cup. Ionization chambers have proved most useful for average beam intensities between lo3 and lo9 particles/sec-cm2, for a 300-Mev proton beam with a duty cycle of somewhat less than 1%. At lower intensities elec‘1
W. P. Jesse and J. Sadauskis, Phys. Rev. 88, 417 (1952); ibid. 100, 1755 (1955).
2.8.
DETERMINATION
OF FLUX A N D DENSITIES
50 1
trometer drift becomes troublesome and a t higher intensities it is difficult to avoid recombination of ions within the gas. A number of other methods for secondary beam monitoring will be mentioned briefly. RADIOACTIVITY. Reactions such as C12(p,pn)C" 2.8.1.3.2. INDUCED have frequently been used'2 to give a measurement of charged-particle beams through measurement of the radioactivity induced. If, as in this example, the product measured is ,&radioactive, it is difficult to make a n absolute measure of the yield. Thus it is important to be able to calibrate such a monitor (against a primary monitor) in the same laboratory where i t is to be used. For proton beams of energy higher than 1 Bev the production of the alpha-active Tb149by bombardment of gold promises to be a useful m0nit0r.l~The 4-hour lifetime of the activity is convenient for many purposes, and i t is a n activity easily counted accurately. The reaction has a high threshold, about 600 MeV. 2.8.1.3.3. SECONDARY-EMISSION MONITOR. Tautfest and Fechter14have described a practical secondary-emission monitor consisting of 20 thin aluminum foils placed in the beam, in vacuum. It has the disadvantages t,hat it yields a very small current (comparable to that from a Faraday cup) and that it must be operated in a good vacuum (pressure less than 10-6 mm of mercury). However, it has the important advantage th a t it gives a linear response to very high peak beam intensities (as tested with high-energy electron beams, 16 ma/cm2, and probably much higher values). Surface effects may alter the sensitivity, so frequent calibration may be necessary. MONITORS. Any substance that scintillates 2.8.1.3.4. SCINTILLATION when the beam passes through it can be placed in the beam, and the yield of light measured with a phototube or photomultiplier connected to a dc electrometer integrator. KitchenI6 has described a secondary monitor based on the weak scintillation of air being detected by a photomultiplier tube, the monitor being proved useful in the range 10-lo10-7 amp for 30-Mev protons. Such a monitor is capable of measuring larger beam currents than an ionization chamber, but its calibration may need t o be remeasured frequently as a protection against changes in photomultiplier response. 12 See, for example, R. L. Aamodt, V. Peterson, and R. Phillips, Phys. Rev. 88, 739 (1952). 1s Private communication from B. J. Moyer and S. Parker (1958). Moyer and Parker have used this monitor, based on work of R. B. Duffield, G. Friedlander, and L. Winsberg. See Lester Winsberg, Bull. A m . Phys. SOC.3 (No. 8), 406, F5 (1958). l4 G. W. Tautfest and H. R. Fechter, Rev. Sei. Znstr. 26, 229 (1955). l 6 Summer W. Kitchen, Rev. Sci. Znstr. 26, 234 (1955).
502
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES
2.8.1.3.5. CERENKOV MONITORS. Where relativistic particles compose the beam to be measured, eerenkov light may be used, in much the same way as scintillation light. The Cerenkov radiator may be a solid or liquid substance, or (for extremely relativistic beams) a gas. The comments about scintillation monitors apply also to Cerenkov monitors. 2.8.1.3.6. RELATIVEMONITORS BASEDON COUNTER TELESCOPES. A thin target may frequently be placed in the beam to be monitored, and a counter telescope arranged to count the charged particles ejected from the target in some convenient direction. The precautions applicable to any counter telescope are also applicable here. By judicious choice of target, angle of the telescope, size of telescope, etc., one can have a monitor capable of handling an arbitrary beam intensity. Unfortunately these monitors are sensitive to many parameters, so their calibration tends to be different each time they are placed in the beam. If left undisturbed, these monitors can be very useful for short periods, however. 2.8.1.3.7. DETERMINATION OF FLUXDENSITIESOF PARTICLES SCATTERED FROM A TARGET. To determine a differential scattering cross section we must measure the number of (charged) particles R scattered into a detector of solid angle w. These quantities will be used in the usual expression for the differential cross section
where n B is the number of beam particles traversing a thin target with target atoms per unit area, measured normal to the beam direction. 2.8.1.3.7.1. Determination of the Scattered Flux R. I . Counters.* Very similar considerations can be applied to a great variety of different types of counter. Pulse ionization chambers, proportional counters, and scintillation counters (either with gaseous, liquid, or solid scintillator) are intended to give as signals electrical pulses whose sizes are proportional (ideally) to the energies deposited in the active counter material by charged particles. Geiger counters are intended to give a uniform response to any ionization within their gas. Cerenkov counters*6 may be used where it is desired to count only very fast particles or where only particles within a certain velocity interval are to be counted.lBJ7Essentially the same considerations will apply to the use of all of these counters.
NT
*See also Vol. 9 A , Chapter 2.1. For general information on Cerenkov Counters aee John Marshall, Cerenkov counters. Ann. Rev. Nuclear Sci. 4, 141 (1954), and Sections 1.5 and 2.2.1.3.1 of Vol. 5 A. ' 1 0. Chamberlain and C. Wiegand, The velocity-selecting Cerenkov counter. Proc. CERN Symposium on High-Energy Accelerators and Pion Phys., Vol. 2, p. 82. CERN, Geneva, 1956. 16
2.8.
503
DETERMINATION OF FLUX AND DENSITIES
TABLE I. Approximate Characteristics of Various Types of Counter
Assumed active material
Type of counter Geiger counter Proportional counter Pulse ionization chamber Scintillation counter Scintillation counter Cerenkov counter
Uncertainty in determining the time of an event (set)
By volume 90% argon, 10% alcohol, 10 cm Hg total pressure By volume 96% argon, 4 % COS,1 a t m Argon, 1 atm
Dead time (see)
Type of particle for which counter is most useful
10-6
10-4
Any
10-8
2 X 10-6
Any
10'0
2
Stilbene or good plastic scintillator NaI (T1 activated)
10-9
10-8
Any
10-8
10-6
Any
Glass
2
4 x 10-9
Relativistic particles
x
10-9
x
10-6
a, heavy ion
Table I gives some very approximate information about the characteristics of a few counter varieties. The data given are intended only for orientation, and are made up on the assumption that the active regions of the counters have linear dimensions of the order of a few centimeters. Many special arrangements have been used to count only one kind of particle in the presence of a background of other particles. We will not attempt to include here these more special devices, but will restrict our attention to the case in which, apart from background effects, only one kind of particle emerges from the target in the direction of the detector. A typical counter arrangement is shown schematically in Fig. 14, PHOTOMULTIPLIER
LIGHT PIPE SCI NTlL LATOR OF AREA A TARGET
SCI NTILLATOR
LIGHT PIPEPHOTOMULTIPLIER-
FIG.14. Geometry of scintillation-counter telescope. By placing the photomultiplier tubes on opposite sides as shown, one may eliminate effects due t o pulses from particles that hit the photomultipliers rather than the scintillators.
504
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
which shows as a n example a telescope of two scintillation counters. We leave to another section the problem of determining the effective solid angle subtended a t the target by the counter arrangement and center our attention now on the measurement of the number of counts of both counters in coincidence. A typical coincidence system is assumed, such as that described by Garwin. l 8 This system measures the number of particles passing through both counters, subject to these restrict,ions. (1) Some accidental coincidences will be recorded, due to particles which traverse only one or other of these counters, two such events sometimes occurring within the coincidence circuit resolving time. For a given integrated beam passed through the target, the number of accidental coincidences is not constant with respect to beam intensity, but is proportional (in first approximation) to the beam intensity. One corrects for accidental coincidences by extrapolating the number of counts per unit integrated beam to zero beam intensity. (Plot number of counts per unit integrated beam versus beam intensity, and make a straight-line extrapolation to zero beam intensity.) The theory of accidental coincidences is included in an article by Feather. l9 Accidental coincidences become a very prominent problem in experiments involving pulsed accelerators. (2) Some true coincidence events will be lost because two scattered particles, each traversing both counters, pass through the counters at substantially the same time. This effect is also corrected for, a t the same time as (1) above, by performing the extrapolation to zero beam intensity. This correction is also accentuated when the beam is derived from a pulsed accelerator. (3) I n some experiments there may be a small background, a s from cosmic rays, which is not derived from the beam, hence the rate of these events must be measured with the beam off, and this background rate subtracted from all observed rates measured with the beam on. (4) Some coincidences are due to particles that do not come from the target, although they are derived from the beam. I n most experiments it is sufficient to measure the number of these counts per unit integrated beam, by counting with the target removed, and subtract this number from the corresponding number measured with the target in place. However', it is always possible that the presence or absence of the target may affect this background, hence the source of the background should be carefully investigated to establish whether a simple subtraction process is adequate. l8 '9
R. L. Garwin, Rev. Sci. Znstr. 21. 569 (1950). N. Feather, Proc. Cambridge Phil. SOC.46, 648 (1949).
2.8.
DETERMINATION
OF FLUX AND DENSITIES
505
(5) Some of the pulses from the counters will be too small to actuate the coincidence circuit in the usual way. These small pulses may arise from particles that pass only through the edge of one counter and hence do not lose much energy in the scintillator, or from many other causes. The understanding of these events and the correction for the counts missed usually depend to a large extent on the so-called “bias curves.” The “bias” originally referred to discriminator bias a t the output of a linear amplifier. At the present it, is customary to express the bias in terms of the input a t the counter necessary to actuate the whole circuit in the normal way. The bias is proportional to the reciprocal gain of the
a W
-0
BIAS OF ONE COUNTER
FIG. 15. Bias curve of scintillation counter. The straight line used t o extrapolate t o zero bias is shown dashed.
amplifier. A fairly typical bias curve is shown in Fig. 15. I n the region of the plateau the circuit is counting nearly all the coincidences. When the bias is adjusted to higher values (lower gain), a n appreciable fraction of counts are lost. When the bias is set very low, additional counts usually are seen, due largely to photomultiplier noise and overloading of parts of the circuits. Using the plateau region only, a straight line may be matched to the bias curve. Where this straight line intersects the vertical axis one may read the number of counts per unit beam “extrapolated to zero bias.” One method of correcting for small pulses missed b y the coincidence circuit is to use the extrapolat,ed value of counts per unit beam. This is not, however, the only method. I n practice, one must choose a method commensurate with the accuracy desired and the information available about the det,ailed nature of the bias curve. II. Photographic emulsions. Nuclear photographic emulsions are frequently used t o determine the flux of scattered particles. The emulsions are exposed in an accurately known geometrical arrangement at known
506
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
distance from the target while a known number of beam particles traverses the target. The emulsions are usually held in an accurately machined holder designed to minimize errors due to beam misalignment.20 The emulsions are oriented so the particles to be counted enter the emulsion surface at a small grazing angle because the tracks, after development, are more easily seen and counted if they run nearly parallel to the emulsion surface. The numbers of scattered particles are determined by counting the tracks entering a known area of emulsion. For many purposes the method is convenient because many pieces of emulsion, placed to measure the scattering at various angles, may be exposed simultaneously. However, up to the present time it is necessary t o scan the emulsions by handoperated microscope, so the scanning process becomes very tedious whenever many scattering events are to be recorded.* Apart from the statistical errors inherent in any counting process, the most troublesome source of error is in the determination of the scanning efficiency. Experience has shown that different individuals have different average scanning efficiencies, and that, furthermore, a given individual may have a different scanning efficiency at one time than another. The accurate determination of particle fluxes requires constant vigilance and imagination in repeatedly determining the average scanning efficiency. While emulsions rarely suffer from errors due to accidental coincidences, virtually the same corrections must be made for backgrounds as have already been discussed for counters. 2.8.1.3.7.2. Determination of Solid Angle. The solid angle is usually defined by a detector of effective area A located a t a distance 1 from the target. In many cases the greatest dimension of the area A is much smaller than the distance 1 and the solid angle may then be given by w =
A/P.
Two extreme cases will be discussed, the first is that in which the full sine of the detector determines the area A . The second is the case in which a slit or aperture determines the area A . I. Full detector area as sensitive area. For definiteness this case will be discussed using the example of high-energy charged particles detected by a telescope of two scintillation counters in the geometry of Fig. 14. Assuming that the second scintillator is sufficiently large to intercept all
* Refer to Vol. 5 A, Chapter 1.7. See, for example, J. C. Allred, L. Rosen, F. K. Tallmadege, and J. H. Williams, Rev. Sci. In&. 22, 191 (1961) as well a8 G . W. Tautfest and W. K. H. Panofpsky, Phya. Rev. 106, 1356 (1957). 10
2.8.
DETERMINATION
OF FLUX AND DENSITIES
507
particles that have traversed the first scintillator (even allowing for multiple scattering in the first scintillator), the solid angle is determined by the first scintillator. Strictly speaking, the solid angle w is a function of the sensitivity of the counter. One will use a slightly different figure for I (in the expression for solid angle), depending on how much of the scintillator must be traversed to cause a useful count. Knowledge of this is, in turn, dependent upon the bias curve, which has already been discussed. In general one may say that the solid angle determination is most accurate if the thickness t of the scintillator is very small compared to the distance 1 from target to scintillator.
FIQ. 16. Geometry of counters with aperture used to determine the solid angle, showing the trajectory of a particle reflected from the inside of the aperture. This arrangement is seldom satisfactory for a precise determination of solid angle.
I I . Solid angle determined by slit or aperture. It is sometimes more convenient, especially when counting low-energy particles, to define the solid angle by a slit or aperture in a shielding wall. The aperture is then followed by a larger counter system, as in Fig. 16. A persistent difficulty with this arrangement is that it is difficult to calculate accurately the corrections for particles that traverse the edges of the aperture or reflect from the inside faces of the aperture. This arrangement should be satisfactory only if the shield thickness t can be made so small compared with target-to-slit distance 1 that there is a negligible difference between A/Z2 and A / V 2 (see Fig. 16), and furthermore only if the dimensions (width or diameter) of the aperture can be made large compared to the range of the particles in the shield material. It is rarely practical to satisfy these criteria. Many other arrangements will be used in practice. I n all of them it is well to consider in detail the effects of scattering from dense materials and the proper analysis of the bias curves.
508
2.
DETERMINATION O F FUNDAMENTAL Q U A N T I T I E S
2.8.2. Determination of Differential X-ray Photon Flux and Total Beam Energy*
2.8.2.1. Introduction. The choice of a particular technique for X-ray photon measurements will depend on the nature of the source, the desired information, the range of photon energies and intensities, the required accuracy, and the investment of time and effort contemplated. This section consists of a description of the available photon sources, followed by descriptions of particular measurement techniques. t High-energy measurements are stressed, although some information which is useful at low energies is also given. For the present purposes the dividing line between high and low energies is arbitrarily drawn a t 3 MeV. The discussion is limited t o photon beams of small divergence. If the source emits photons of only one energy, flux determination is a simple counting problem, which is not discussed here explicitly. For a source which emits photons with several energies, the differential photon flux, N ( k ) , is defined as the number of photons/sec with energy k, per unit energy interval. The quantities which can be derived from N ( k ) , and the units in which they will be expressed, are: Photon flux
=
s
N ( k ) d k , photons/sec
Differential energy flux Beam power
=
Beam intensity
1
=
k N ( k ) , watts/Mev
k N ( k ) d k , watts
‘I
=-
A
k N ( k ) d k , watts/cm*
A is the area of a beam cross section. Another quantity which is often useful is the total beam energy ( T B E ) defined as the product of beam power and exposure time. The unit of total beam energy used here is joules, even though the use of Mev would be more convenient for converting t o “equivalent quanta,” a term occasionally used by high-energy physicists. The number of “equivalent quanta’’ is defined as the ratio of total beam energy to maximum photon energy. The determination of N ( k ) for a given X-ray beam can be separated into a determination of the relative dependence of N ( k ) on k , plus an absolute normalization. The dependence on k may be known from theory, but in most cases it must be determined with a photon spectrometer. If
t The techniques described are illustrative examples of methods which have proven useful; there has been no attempt to be comprehensive. ~
* Section 2.8.2 is by J.
S. Pruitt and H. W. Koch.
2.8.
DETERMINATION
OF FLUX AND DENSITIES
509
the absolute efficiency of the spectrometer is known, the measurements can also be used for normalization. In the case of thin-target high-energy bremsstrahlung beams, for which the shape of the differential flux distribution is assumed known from theory, absolute normalization, in principle, reduces to determination of the photon flux, which is a counting problem. However, the great preponderance of photons with energies less than 1 Mev makes a simple count extremely insensitive t o the flux of high-energy photons. The alternative customarily used when the bremsstrahlung spectral shape is known is t o measure the total beam energy, a quantity which is more sensitive to high-energy photons and can be evaluated by several techniques which do not require the use of a spectrometer. 2.8.2.2. Sources of Photon Beams. The most common photon beams are bremsstrahlung beams, discussed below. Their major characteristic is their broad energy spectra, which limit their value in physical research. For instance, use of a bremsstrahlung beam to determine a reaction cross section as a function of photon energy requires the untangling of the effects of photons of different energies, which greatly complicates the analysis. There are, however, several techniques which show promise of allowing direct experimentation with monoenergetic photons whose energy can be varied at will. One of these, positron annihilation in flight, is also discussed in this section. The last part of this section describes two examples of sources of monoenergetic photons of fixed energy. 2.8.2.2.1. BREMSSTRAHLUNG. Bremsstrahlung are produced when electrons bombard matter and are decelerated in the atomic fields. For electrons of kinetic energy T , photons will be generated with all energies between k = 0 and k = T . Theoretical calculations of the elementary process have been made by several authors, and their theories have been compared with experimental measurements of the differential bremsstrahlung cross section.' For T > 10 MeV, theory and experiment agree within the experimental errors. Theory should be accurate to within about 2% a t these energies, but the experiments are only good to about l o % , so the shape of a differential photon flux distribution can, in principle, be calculated relatively precisely and easily. For T < 10 MeV, however, the theoretical approximations lose their validity and th e differences between the approximate theory and experiment may be larger than 10%. The available sources of high-energy bremsstrahlung include betatrons, electron synchrotrons, and linear accelerators. They are usually sources of thin-target spectra, where the target is too thin t o significantly degrade the electron energies but thick enough for multiple scattering to occur. The experimental spectra in such a case can be approximated by those 1
H. W. Koch and J. W. Mots, Revs. Modern Phys. 31,920 (1959).
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
m
5
20
I
a
a
In ' v)
za m
O
.2
0
.4
.0
.6
1.0
k k,,
FIG.1. Product of theoretical integrated-over-angles thin-target bremsstrahlung cross section and photon energy.' The formula used is listed as 3BS(e) in reference 1.
i
2
5
10
20
50
100
200
500
k, Mev
FIG.2. Theoretical integrated-over-angles thin-target bremsstrahlung cross section.* The formula used is listed as 3BS(e) in reference 1.
2.8.
DETERMINATION
10 2.0
km=(Mev) Photons/sec X
19 1.3
511
OF FLUX AND DENSITIES
36 0.9
68 0.6
130 0.3
260 0.2
480 0.1
260 0.2
480 0.1
I
TABLE11. 15 Mev Photons/sec/Mev/watt 1
kmsr(Mev) Photons/sec X 10+
19 1.6
36 1.2
68 0.7
130 0.4
lists the number of photons/sec with energies k 2 0.9km,,, and Table I1 lists the photons/sec with k between 14.5 and 15.5 Mev, both for a bremsstrahlung beam power of 1 watt. The shape of thick-target differential flux distributions, where electron energy loss is no longer negligible, depend upon the target thickness and are much more difficult to evaluate. The available calculations are reviewed in reference 1. Specific examples based on simple assumptions can be found in references 4 and 5. 2.8.2.2.2. POSITRON ANNIHILATION IN FLIGHT. The annihilation of positrons with atomic electrons in a medium produces two photons. If it occurs when the positron is at rest, these photons are isotropic with opposite momenta and with energies of 0.511 MeV. If the annihilation occurs in flight, a less likely event in a thick medium, one of the photons has a larger energy, which depends only on the emission angle and positron energy. In recent experiments,s thick tantalum targets were L. I. Sohiff, Phys. Rev. 83, 252 (1951). 8 A . S. Penfold and J. E. Leiss, Phys. Rev. 114, 1332 (1959). E. Hisdal, Phys. Rev. 106, 1821 (1957); E. Hisdal, Arch. Math. Nuturvidenskub 54(3), 1 (1957). N. E. Hansen and S. C. Fultz, UCRL 6099 (1960). C. Tzara, Compt. rend. 246, 56 (1957);C.P. Jupiter, N. E. Hansen, R. E. Shafer, and-S. C. Fultz, Phys. Rev. 121, 866 (1961);F. D.Seward, C. R. Hatcher, and S. C. Fultz, ibid. 121,605 (1961);F. D.Seward, S. C. Fultz, C. P. Jupiter, and R. E. Shafer, UCRL 6177 (1960). a
512
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
bombarded with high-energy electrons from a linear accelerator. Positrons produced in the resultant shower were magnetically analyzed, and those of a given energy were allowed to annihilate in a thin beryllium foil. The photons observed at a given angle were monoenergetic within the limitations discussed below. This technique for producing monoenergetic photons of variable energy has a very low efficiency because it involves two processes, positron production and annihilation in flight. Each has an efficiency of the order of 10W when the width of the observed photon spectrum is just a few per cent of the peak energy, and only one photon is observed for lo1*electrons incident on the tantalum. Specifically, it has been shown’ that for a tantalum thickness, X , between 0.05 and 0.3 radiation lengths, with a n electron kinetic energy, T , between 10 and 100 MeV, the number of positrons observed in a solid angle, a, <0.03 steradians with kinetic energies between T+ and T+ dT+ is:
+
positrons = F(a)T(ln T - 0.58)X(1 - 1.15X)Q dT+ (2.8.2.1) electron where a = T+/T, and F ( a ) is a function whose calculated values are shown in Table 111.
TABLE 111. F ( a ) for Equation (2.8.2.1) ~
T+ a=-
T
F ( a ) X lo6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
4.43
6.98
7.78
7.46
6.34
4.96
3.24
1.54
0.62
The annihilation photons observed at a laboratory angle 0 have a n energy : me2 T+ k = where p2 = (2.8.2.2) 1 - p cos e T+ 2mc2
+
The cross section per Mev for annihilation in flight can be written:
-
2b4 sin4 p
(1
+ p2 - 28 cos
e)2
]
*
(2.8.2.3)
The angle e and the differential cross section are plotted as functions of photon energy in Fig, 3 for T+ = 10.2 Mev positrons. 7
L. Katz and K. H. Idkan, Nuclear Instr. & Methods, 11, 7 (1961).
2.8.
OF FLUX A N D DENSITIES
DETERMINATION
513
The requirements for producing a narrow photon spectrum are a small positron energy spread (dT+ << Tf), a thin beryllium target to minimize muhiple scattering and positron energy degradation, and a detector which intercepts only a small range of 8. In practice, the electron beam power is too small to allow all of these conditions t o be satisfied, and a
AS A FUNCTION OF PHOTON ENEROY ANNIHILATION PHOTON ENERQY VERSUS LA9ORATORY ANOLE . 6
I I I
I
I
I
I
I
?-9+I12
2 3 4 5 6 7 8 9 1 0 ANNIHILATION PHOTON ENERG'
0
MeV)
FIG.3. Relative differential photon flux and emission angle for photons generated by annihilation-in-flight of positrons with kinetic energies of 10.2 Mev (20 mcl).
spectrum of photon energies is observed. The annihilation differential photon flux distribution drawn in Fig. 4 was calculated for T+ = 10.2 Mev positrons, using a 0.5-Mev thick beryllium target and a detector which intercepts 0.0163 steradians in the forward direction. It does not include the effects of multiple scattering. Bremsstrahlung will also be produced by the positrons when they strike the beryllium target, and the relative number will be appreciable even though beryllium has a low atomic number. The theoretical bremsstrahlung differential photon flux for the preceding case is also shown in Fig. 4. Although the annihilation peak can be easily resolved, the low-energy bremsstrahlung tail complicates the analysis of experimental data.
514
2.
DETERMINATION O F FUNDAMENTAL QUANTITIES I
I
I
I
EREMSSTRAHLUNG
z E 2
t v)
0
a
ANN IHI LATION
IW
z
D u
z\
>
z \
ln
z
P ’ 0 I
n
0
I I
2
4
6 k , Mev
I0
12
FIQ.4. Composite differential photon flux distribution generated by positrons with kinetic energies of 10.2 Mev in a 0.5-Mev thick beryllium target. The detector intercepts a solid angle of 0.0163 steradians at the target, considerably larger than the experimental solid angles reported in reference 6.
2.8.2.2.3. MONOENERGETIC PHOTONS OF FIXED ENERGY. Monoenergetic photons are usually obtained from processes associated with transitions between nuclear energy levels, like radioactive decay, neutron and proton capture, and resonance fluorescence. The most common radioactive and resonance fluorescence sources are listed in reference 8. The intensities of these sources depend upon many factors, but the two examples discussed below illustrate the tremendous range of intensities encountered. One of the most common radioactive sources is Co60,which emits two gamma rays in cascade with energies of 1.17 and 1.33 MeV. A 1-curie source emits 14.8 mw of gamma ray power. For a large source of 1000
* J. B. Marion, in “1960 Nuclear Data Tables” (K. Way, ed.). Part 3. National Academy of Sciences, Washington, D.C., 1960.
2.8.
DETERMINATION
OF FLUX AND DENSITIES
515
curies, the intensity a t a distance of 1 meter is about 120 pw/cmZ if there is no self-absorption. Carbon-12 bombarded by bremsstrahlung is a source of resonance fluorescence photons with energies of 15.12 M ~ VIn. ~a 20-Mev bremsstrahlung beam the flux of 15.12-Mev photons at 90°, from a 2-gm/cm2 polystyrene target, is 0.3 photons/sec per steradian per microwatt of bremsstrahlung power. For a typical situation using a 2 0 0 - p ~20-Mev ~ betatron beam, the intensity of fluorescence photons 1 meter from the pw/cm2. This is obviously a very low carbon is only about 1.5 X intensity source, but promises to have experimental uses with a highpower linear accelerator as the source of bremsstrahlung. 2.8.2.3. Photon Spectrometers. The primary problem in designing a photon spectrometer is that, like neutrons, photons are uncharged and cannot be separated in energy by magnetic deflection. Differential flux distributions must be determined from studies of the behavior of secondary electrons generated by the photon beam. There is seldom a one-to-one correspondence between photon energy and secondary electron energy, so that a high-efficiency spectrometer, which counts every photon and detects all of the secondary electrons, is characterized by poor energy resolution (of the order of 10%). High-resolution spectrometers, on the other hand, which detect only those secondary electrons which satisfy rigid criteria, are in general characterized by low detection efficiency (of the order of Photon spectrometer measurements are correspondingly either difficult to make or hard t o interpret, or both. There are two main types of photon spectrometers in use today. Total absorption spectrometers are designed to absorb most of the incident photon energy, and to generate a signal which can be used t o count the photons and to sort out their energies. They are in general high-efficiency, Iow-energy-resolution instruments, although the recent development of anticoincidence and coincidence techniques has improved the energy resolution without too great a sacrifice of detection efficiency. Magnetic spectrometers, on the other hand, are capable of energy resolutions of the order of 1 or 2% but are invariably low-efficiency instruments. 2.8.2.3.1. TOTAL ABSORPTION SPECTROMETERS. Total absorption spectrometers usually contain either sodium iodide crystals which generate scintillation light, or transparent media which generate cerenkov light. In both cases, the light is viewed by photomultipliers which generate voltage pulses. The problem of determining a differential photon flux distribution is reduced to the problem of analyzing an electronic pulse height distribution.* If N ( k ) is the incident differential photon flux, and
* See also Volume 11, Chapters 9.6 and 11.1. 9
E. Hayward and E. G . Fuller, Phys. Reu. 106, 991 (1957).
516
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
P(e) is the corresponding pulse height distribution, the determination of N ( k ) from P(e) requires solution of the integral equation: P(c) de
=
dc
lo"""
N ( k ) e ( k ) K ( k , t )dk
(2.8.2.4)
where e(k) is the spectrometer detection efficiency, the probability that a photon of energy k will produce a voltage pulse. K ( k , t ) is a crystal response function, the pulse height distribution generated by monoenergetic photons of energy k, normalized so that :
JK(k,e)de = 1.
(2.8.2.5)
Several techniques for solving Eq. (2.8.2.4)for N ( k ) are discussed below in connection with specific applications. 2.8.2.3.1.1. Single-Crystal Scintillation Spectrometers. The probability that a photon will interact while passing through a large sodium iodide crystal is relatively large (e.g., greater than 0.94 for a 9-in. crystal), and each interaction produces a voltage pulse. If the shape of the differential photon flux distribution is known approximately, the number of photons transmitted without int,eraction can be calculated reliably, so that a large-crystal spectrometer can be used to determine the photon flux (photons/sec) both efficiently and accurately. The probability that a photon will deposit all of its energy in a large crystal is much smaller, since the initial interaction generates a penetrating shower of secondary electrons and photons. Scintillation light is generated isotropically along the electron paths and the amount generated depends on the photon energy, k , and the fraction of the energy absorbed by the crystal. If all of the energy is absorbed in the crystal, the voltage pulse height is proportional to k , but, in general, some of the energy leaks from the crystal, and the pulse height is deficient. Single-crystal spectrometer response functions for monoenergetic X-rays usually look like those shown in Fig. 5,'O and are characterized by a peak which corresponds to total energy absorption, and a low-energy tail. For the low-energy curve in Fig. 5, the width of the main peak is determined by statistical fluctuations in the amount of light seen by the photomultipliers and the subsidiary peak signifies the escape of a photon which has undergone a single Compton scattering. At higher energies pair production is significant and the details become more complicated, but the subsidiary peak signifies the escape of a positron annihilation photon.1' lo
11
H. W. Koch and J. M. Wyckoff, IRE Trans. on Nuclear Sci. NS-6, No. 3 (1958). H. W, Kooh and J. M. Wyckoff, J . Research Natl. BUT.Standards 66, 319 (1956).
2.8. DETERM~NATION OF
FLUX AND DENSITIES
517
Response functions depend on photon energy and spectrometer geometry. They have been calculated by Monte Carlo methods with crystals of various sizes for photon energies up to 5 Mev,12 and have been determined experimentally u p to 20 Mev with monoenergetic photons from ) and with measurements radioactive sources and from ( p , ~reactions10,13'14 with electron beams.11 At higher energies, response functions have been
--7
PULSE HEIGHT B , M e V
FIG.5. Typical response functions for single-crystal scintillation spectrometers, normalized to unity at the peaks.10 The 1.12- and 4.43-Mev distributions were measured with a 5-in. diameter, 4-in. long sodium iodide crystal, using monoenergetic gamma rays from Zn66 and RaD(a,Be) respectively. The 19-Mev distribution was synthesized for a 5-in. diameter, 9-in. long crystal from measurements with monoenergetic electron beams.
deduced from measurements with bremsstrahlung beams with known photon flux distributions, by solving Eq. (2.8.2.4) for K(k,e).l0 2.8.2.3.1.1.1. Applications. The cross section for elastic scattering of photons by atomic nuclei has been measured a t both hight6 and lowL6 energies using a technique which requires only relative measurements of photon flux, and which does not require knowledge of spectrometer response functions. The high-energy measurements were made with the 12 M. J. Berger and J. Doggett, J . Research Null. Bur. Standards 66, 355 (1956); W. F. Miller and W. J. Snow, Rev. Sci. Znstr. 31, 39 (1960); C . D. Zerby and H. S. Moran, ORNL-CF-60-5-72 (1960). 1s J. Kockum and N. Starfelt, Nuclear Znst. & Methods 4, 171 (1959). 1 4 Proc. Total Absorption Gamma-Ray Spectrometry Symposium, OTI-TID-7594 (1960). 16 E. G. Fuller and Evans Hayward, Phys. Rev. 101, 692 (1956). 16 K. Reibel and A. K. Mann, Phys. Rev. 118, 701 (1960).
518
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
apparatus of Fig. 6, using a betatron bremsstrahlung beam of peak energy km,,, and a spectrometer voltage discrimination level set so that only pulses produced by photons with energies within 10% of k,, were recorded. The scattering cross section per steradian is: (2.8.2.6) where p is the number of target atoms/cm2 along the photon path, and D is the detector solid angle. N0(0.95km,,) and No(0.95kmJ are respectively photon fluxes in the direct X-ray beam with the scatterer removed, and at the angle e with the scatterer in position. BETATRON
rn .TERED X-RAYS
IONIZATION CHAMBER MONITOR
FIG. 6. Schematic experimental arrangement for measurements of nuclear elastic scattering cross sections with a single-crystal scintillation spectrometer. Dimensions and shielding details are given in reference 9.
The peak cross sections measured at 6 = 120" are of the order of a few millibarns for most scattering materials, and this small cross section, plus the need to eliminate pulse pile-up, resulted in small counting rates and long experiment times. The statistical accuracy of the measured cross sections is consequently about lo%, and the energy assignments are accurate t o within 5%. Photon total attenuation coefficients have been measured for several materials'' using a technique which required only relative measurement of differential photon fluxes. The experimental arrangement is shown in Fig. 7. For an absorber of length 2 (gm/cm2), the differential photon flux transmitted by the absorber is :
N ( k ) = N ( k , x ) = Ni(k)e+k)z 1'J.
M. Wyckoff and H. W. Koch, Phys. Rev. 117, 1261 (1960).
(2.8.2.7)
-
I
!
I
~
+:
SOURCE
ABSORBER ~.
REED IELECTRO. EFEII] VIB.
FIG.7. Experimental arrangement for single-crystal spectrometer measurements of the photon total attenuation coefficient of carbon as a function of photon energy." The photon source was a 180-Mev electron synchrotron operated at 90 Mev. The experiment was performed in good geometry to insure that scattered photons were removed from the beam.
520
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
where N i ( k ) is the differential flux of incident photons and p ( k ) is the mass attenuation coefficient (cmz/gm). p ( k ) can be calculated from Eq. (2.8.2.7) if the transmitted spectrum is known for at least two different values of 2. The pulse height distributions generated by 90-Mev bremsstrahlung transmitted by seven different lengths of graphite absorber are shown in Fig. 8. These distributions were analyzed to obtain p(k) with an accuracy of about 1% at 60 MeV. Absolute measurements of the cross section for production of thintarget bremsstrahlung have been made with single-crystal spectrometers at both highls and energies using a technique which required absolute measurements of the differential photon flux. The high-energy measurements used the experimental arrangement shown in Fig. 9, with bremsstrahlung produced in the target by an extracted betatron electron beam. The measurements were made with several targets and electron energies and at several angles. The data shown in Fig. 10 were obtained with 4.54-Mev electrons and a gold target.'s The discrepancy between theory and experiment is more obvious after integrating over the photon emission angle, as shown in Fig. 11.l The experimental data were estimated to be more accurate than 20% in all cases and more accurate than 10% in most cases. In the bremsstrahlung cross section and attenuation coefficient experiments discussed above, Eq. (2.8.2.4) was solved for N ( k ) with the help of matrices.l8 First, the pulse height and photon energy scales were divided into intervals Aej and Aki respectively. Then assuming that P ( E )and N ( k ) are constant across any interval, Eq. (2.8.2.4) can be rewritten:
P(€J =
1
MjiN(ki)
(2.8.2.8)
i
where
M 1% .. =
02 /AkL
K(k,4(1 - e-L(k))dk A€j
(2.8.2.9)
The detection efficiency has been written (1 - e-L(k)), where L(k) is the crystal thickness for photons of energy k, in mean-free-paths. The differential photon flux, N ( k ) , can then be obtained by inverting the M matrix10J8J0and forming the sum: (2.8.2.10) N. Starfelt and H. W. Koch, Phys. REV.102, 1598 (1956). J. W. Motz, Phys. Rev. 100, 1560 (1955). 20 W. R. Burrus, Paper 17 in Proc. Total Absorption Gamma-Ray Spectrometry Symposium OTI-TID-7594 (1960); W. R. Burrus, I R E Trans. on Nuclear Sci. NS-7, NOS.2-3 (1960). 19
1 l
10
0
40
~
I
~
l
'
120 160 200 CHANNEL NUMBER
80
~
,
[
240
FIG. 8. Experimental pulse height distributions obtained with a single-crystal scintillation spectrometer, with 90-Mev bremsstrahlung transmitted by seven different lengths of carbon." The longest absorber comprised 698.9 gm/cm2 of carbon. 52 1
522
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
BETATRON
LUCITE COUIMATORS
FOCUSINQ MAGNET
BENDING MA TO PUMP-
TARGET
TARGET CHAMBER.
8.
BENDINQ MAGNET
I
ALUMINUM WINDOW FARADAY CAGE LEAD COLLIMATOR
NoL(Tt) SCINTILLATION SPECTROMETER LEAD SHIELD
100 cm
FIQ. 9. Experimental arrangement for single-crystal scintillation spectrometer measurements of the cross section for production of thin-target bremsstrahlung.'*
' 1 1 AulZ.79) Eo-moct m4.54 Mev
0
EXPERIMENT
--SAUTER -SCHIFF
GLUCKSTERN
a
HULL I ITH
t l
10" I o 0
z
B L 3 PHOTON ENERGY, M e V
FIQ.10. Experimental single-crystal scintillation spectrometer measurements of the differential cross section for production of bremsstrahlung by 4.54-Mev electrons in a thin gold target.18 The curves are theoretical predictions. 623
524
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
This technique for determining N ( k ) from P ( E )has proven quite satisfactory below 10 MeV, where the response functions are strongly peaked and the largest elements of the matrix M are along the diagonal. The largest elements of M are displaced from the diagonal for single-crystal spectrometers at higher energies, and application of M-’ to P(e) often predicts values of N ( k ) which oscillate violently with changing k in a
I
1
4.54 M c v SAUTERFANO
-
CORRECTED SAUTER-FANO
-
-
I
/
I
l
l
I
l
rnuiuis ciscnui,
l
1
mev
FIQ.11. Experimental single-crystal scintillation spectrometer measurements of the product of integrated-over-angles bremsstrahlung cross section and photon energy, using 4.54-Mev electrons incident on a thin gold target.’ The solid curve is the theoretical prediction of Schiff, and the dashed curve was fitted by eye to the experimental points. The arrows show the peak energy cross sections predicted by Sauter and Fano.
totally unphysical manner. These oscillations are apparently due to an amplification of the inherent statistical errors in the experimental data.Z0 In such cases, N ( k ) can be determined from Eq. (2.8.2.8) by trial and error. This can be done by assuming values of N ( k ) and calculating P ( E ) from Eq. (2.8.2.8). The differences between calculated and true values of P(e) can then be removed by changing the assumed N ( k ) . 2.8.2.3.1.2. Anticoincidence Scintillation Spectrometers. The low-energy tail of a spectrometer response function can be practically eliminated by surrounding the spectrometer crystal with a liquid, plastic, or crystalline
2.8.
DETERMINATION OF FLUX A N D DENSITIES
525
0
l J A L L L 2 S c a l e in Inches
FIG. 12. Schematic diagram of crystals, photomultipliers, and shielding in a n anticoincidence scintillation spectrometer.22 The center crystal produces the pulse, and the shaded crystal plus t h e crystal on t h e right provide an anticoincidence shield. Only events in which no energy leaks out of the center crystal are recorded.
_I
w
z z a
I 0
\ v)
I-
z 3
0
u
PULSE HEIGHT E , M e v
FIG.13. Pulse height distributions obtained from an anticoincidence scintillation spectrometer and produced by gamma rays from neutron capture in hydrogen.22 Curve a is without anticoincidence, curve b with anticoincidence, and curve c is bnckground. Curve b minus curve c is a response function for 2.22-Mev gamma rays.
526
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
scintillator connected in anticoincidence.21p22 Any event in which part of the photon energy escapes from the crystal will produce an anticoincidence signal and will be discarded. An instrument which has been used at energies below 10 Mev22 is shown in Fig. 12. The main crystal, 2.4 in. in diameter and 6 in. long, is surrounded by an annular crystal 2.0
-
-
1.5
.-
N
-I > a8
I _15
-
-
1.0 -
-
U
0
5 Resolution
10
15
(%I
FIQ. 14. Experimental energy resolution versus photon energy for the anticoincidence scintillation spectrometer of reference 22. The resolution varies inversely as the square root of photon energy.
which together with the crystal at the rear acts as the anticoincidence shield. The great improvement in the response function shape can be inferred from the curves of Fig, 13, which show the pulse height distributions produced by 2.22-Mev gamma rays from neutron capture in hydro21 R. E. Connally, Rev. Sci. Instr. 24, 458 (1953); R. D. Albert, Rev. Sci. Instr. 24, 1096 (1953); P. R. Bell, Science 120, 625 (1954); K. I. Roulston and S. I. H. Naqvi, Rev. Sci. Znstr. 27,830 (1956); R. C. Davis, P. R. Bell, G. G. Kelley, and N. H. Lazar, IRE Trans. on Nuclear Sci. NS-S,82 (1956); C. C. Trail and S. Raboy, Rev. Sci. Instr. SO, 425 (1959); also see papers by R. W. Perkins, J. M. Nielsen, and R. N. Diebel, and by W. H. Ellett in Proc. Total Absorption Gamma-Ray Spectrometry Symposium, OTI-TID-7594 (1960). (2 C. 0. Bostrom and J. E. Draper, Rev. Sci. Insfr. 32, 1024 (1961).
2.8.
527
DETERMINATION OF FLUX AND DENSITIES
gen without anticoincidence (a) and with anticoincidence ( b ) . Curve c is background. Figure 14 is a summary of the energy resolutions obtained and Fig. 15 is a plot of the crystal efficiency. The efficiency drops rapidly with increasing energy, and is only 7% at 10 Mev. This decrease is caused by the increasing range of secondary electrons and photons, which make total absorption events increasingly rare. Figure 15 indicates that the value of the anticoincidence spectrometer is limited at energies above 10 MeV. 1.0
0.5 >.
0 Z
w - 0.2
0 LL LL
w 0.1
0.05 0.1
0.2
0.5
1.0
2.0
5.0
10.0
k, M e V FIG. 15. Experimental detection efficiency of the anticoincidence scintillation spectrometer of reference 22 as a function of photon energy.
2.8.2.3.1.2.1. Applications. There have been attempts to measure both relative and absolute gamma ray fluxes with single-crystal spectrometers.23 If photons of more than one energy were present, these determinations were greatly complicated by the mixing of the peak of one component of the pulse height distribution with the tails of other components. The pulse height distribution obtained with the spectrometer of Fig. 12 for chlorine is shown in Fig. 16, and indicates that the anticoincidence spectrometer is a valuable tool for sorting out complex spectra. A matrix solution of Eq. (2.8.2.4) is extremely impractical when the pulse height, spectrum is complex, as in Fig. 16. In such a case, it is more reliable t o successively subtract response functions fitted to the highest energy peak, the next highest, and so 0 n . ~ For ~ 3 ~the anticoincidence spectrometer, the response function tail (curve b minus curve c in Fig. 13) is so small that this peeling process will not reveal structure in the photon 28
N. H. Lazar, I R E Trans. on Nuclear Sci. NS-6, No. 3 (1958).
528
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
2000
I
I
I
I
I
PULSE HEIGHT
1
I
6,
I
I
1
Mev
FIG. 16. Anticoincidence scintillation spectrometer pulse height distribution produced by gamma rays from neutron capture in chlorine.2'
spectrum which was not apparent in the pulse height distribution, but this is not necessarily the case for a single-crystal spectrometer. 2 . 8 . 2 . 3 . 1 . 3 . Coincidence Scintillation Spectrometers. Figure 17 is a cutaway diagram of the two crystals used in a high-resolution coincidence spectrometer recently developed for use with high-energy X-rays.24 Incident photons pass through the axial hole in the front crystal and are absorbed by the 9-in. rear crystal. Only voltage pulses produced by pair production events which occur close to the entrance surface are recorded. Most of the volume of the big crystal is consequently available for absorbing the showers produced by these events, and the energy leakage is much smaller than with a single-crystal spectrometer which accepts pulses produced by events occurring at all depths. 2* B. Ziegler, H. W. Koch, and J. M. Wyekoff, Bull. Am. Phys. SOC. 6,11 (1961), and paper to be published.
2.8.
DETERMINATION
OF FLUX AND DENSITIES
529
The selection of pair production events is made by requiring coincidence of the rear-crystal pulses with pulses from the front crystal produced by 0.511-Mev gamma rays. These photons originate from annihilation of positrons at rest in the rear crystal (see Section 2.8.2.2.2). The total attenuation coefficient of sodium iodide for 0.511-Mev gamma rays is large enough (0.09 cm2/gm) to ensure that both the annihilation photons and the positrons were generated within a few centimeters of the entrance to the rear crystal.
FIG. 17. Cutawa,y diagram of the crystal arrangement in the coincidence scintillation spectrometer. Voltage pulses produced by absorption of photons in the 9-in. rear crystal are recorded only if a 0.511-Mev photon from positron annihilation simultaneously produces a pulse in t h e crystal with the axial hole.
Response functions for this spectrometer have been measured with 6.14- and 17.64-Mev gamma rays from F l g ( p , y )and Li7(p,y) respectively, and these are shown in Fig. 18. Comparison with the curves of Fig. 5 shows that the coincidence spectrometer has much better energy resolution than a single-crystal spectrometer. The detection efficiency of this instrument is only a few per cent for 10-Mev photons, but increases with increasing energy. 2.8.2.3.1.3.1. Applications. The coincidence spectrometer has been used to measure photon total attenuation coefficients in an experiment similar to the one described in Section 2.8.2.3.1.1.1 with the same experimental arrangement (Fig. 7). The data have not yet been analyzed, but
2.
530
DETERMINATION OF FUNDAMENTAL QUANTITIES
the accuracy of attenuation coefficient measurements will still be about 1 %. The anticipated improvement over the single-crystal measurements concerns the dips in the curves in Fig. 8, which are produced by nuclear absorption processes. The coincidence spectrometer has distinguished detail in the nuclear absorption cross section which the single-crystal instrument could not resolve.
-
1.0-
-
I
W
s
I
?L
-
+- .8I
2
-
W
I
W
.6
-
-
-
v)
J
-
3
n
w
-
-2.8 %
-2.0%
.4-
L a
-
2
.2-
k
-I
-
0
.
0
I
I
I
I
1
5
l
l
l
l
10
l
l
I
l
l
15
FIG.18. Coincidence scintillation spectrometer response functions measured with 6.14- and 17.64-Mev gamma rays from FIg(p,y) and LiT(p,y),respectively, normalized to unity at the peaks. The peaks correspond to capture by the crystal of all except 0.51 Mev of the incident photon energy.
When a high-resolution spectrometer is used to analyze a continuous photon flux distribution with little or no structure, there is a simpler method of solving Eq. (2.8.2.4) for N ( k ) than the matrix solution described in Section 2.8.2.3.1.1.1. The product N ( k ) e ( k ) in the integrand of Eq. (2.8.2.4) can be written as a Taylor series in the neighborhood of a particular photon energy, ko: P(k) = N(k)e(k) = P(ko)
1 dT(k0) Wko) +(k - ko)* dk (k - ko) + 2 7
+... Therefore, Eq. (2.8.2.4) can be rewritten :
(2.8.2.11)
2.8. For a given
E,
DETERMINATION OF FLUX AND DENSITIES
531
if ko is chosen so that :
ko(a)
(2.8.2.13)
=
the second term on the right of Eq. (2.8.2.12) vanishes, and the equation may be rearranged to yield: (2.8.2.14) where
+
higher order terms. (2.8.2.15) 6 = d2P’dk2(ko) J K ( k l r ) ( k - k o ) 2dk 2P(kO) J K ( k , 4 dL Therefore, the number of counts/sec in channel e of a pulse height distribution is approximately proportional to the differential photon flux at LO, the center-of-gravity energy of the response functions for E. If the differential photon flux distribution changes slowly with photon energy, as in the attenuation coefficient distributions of Fig. 11, 6 can be neglected in Eq. (2.8.2.14). In the region of the nuclear absorption dips, however, this technique could be used only if 6 was included in the calculations. The determination of N ( k ) would have to be made by successive approximations. 2.8.2.3.1.4. cerenkov Spectrometers. Large blocks of dense material like lead glass emit Cerenkov light when irradiated with X-rays. 26 The advantages of a Cerenkov spectrometer are that blocks large enough to absorb almost all of the photon energy can be manufactured with little difficulty, and faster response times are possible than with sodium iodide, since the Cerenkov light is emitted by the secondary electrons themselves rather than by the crystal atoms. The great disadvantage is that Cerenkov light is generated only by electrons moving with velocities greater than the velocity of light in the block. A Cerenkov spectrometer cannot detect electrons with energies below about 200 kev, and, since there are usually a large number in this category, the spectrometer resolution is relatively poor. Kantz and Hofstadter have made measurements which indicate that a large spectrometer will have an energy resolution of about 17% for 180-Mev electrons.26 2.8.2.3.2. MAGNETIC SPECTROMETERS. Magnetic spectrometers in general consist of a thin foil converter, where the secondary electrons and positrons are generated, a magnetic field to separate the particles in energy, and detectors t o count the number in a given energy bin. There are two main types, Compton and pair spectrometers. The former meas26 26
See Vol. 5, A, Section 1.5.4 by Lindenbaum and Yuan. A. Kante and R. Hofstadter, Nucleonics 13(3), 36 (1954).
532
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
ure the number of electrons generated in the foil in Compton scattering interactions, and the latter are coincidence instruments which measure the number of electron-positron pairs generated. There are two basic reasons for the low detection efficiency of highresolution magnetic spectrometers. In the first place, the converter foil must be very thin, so that multiple scattering of the particles is not significant, with the result that only a small fraction of the incident photons generate secondary particles. Second, most of the secondaries generated by photons of a given energy must be discarded in order to obtain a one-to-one correspondence between photon energy and electron positron) energy. (or electron
+
y BFAM I
FIG.19. Schematic diagram of a single-channel Compton spectrometer.27 F is a Be converter foil, P is the magnetic field outline, and B is a baffle which defines the spectrometer acceptance angle, 8.
2.8.
DETERMINATION OF FLUX AND DENSITIES
533
2.8.2.3.2.1. Compton Spectrometers. Compton scattering is a simple two-body process which is well understood theoretically, and the spatial and energy distributions of the electrons can be accurately predicted over a large energy range. This permits evaluation of the absolute detection efficiency and of its variation with photon energy, a prerequisite for accurate determination of the shape of a broad differential photon flux distribution. For example, it should be possible to predict the absolute efficiency of the simple single-channel spectrometer shown in Fig. 1g2' to within 10% for all photon energies between 1 and 30 MeV.
B e TARGET THICKNESS, m m
FIG. 20. Theoretical energy resolution (dashed curves) and detection efficiency (solid curves) of a Compton spectrometer with an acceptance angle of 1.4"and an exit slit width of 1 % of the electron energy.Z8
The electron counter cannot distinguish between Compton and pair electrons, and a correction must be made for the contribution of the latter. This is customarily done by repeating the measurements with the magnetic field reversed so that positrons are counted, and subtracting the positron count from the electron count. Detailed investigations of the characteristics of Compton spectrometers have been made below 30 Mev.28 The calculations of detection efficiency and energy resolution for a particular spectrometer are shown as a function of foil thickness and photon energy in Fig. 20. This indicates that 2% resolutions are possible, and also illustrates the rapid decrease of detection efficiency with decreasing photon energy. Z'J. W. Mots, W. Miller, H. 0. Wyckoff, H. F. Gibson, and F. S. Kirn, Rev. Sci. Znstr. 34, 929 (1953). z8 E. Keil and E. Zeitler, Nuclear Znslr. & Method8 10, 301 (1961).
534
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
A revealing comparison between a typical Compton spectrometerze and an anticoincidence scintillation spectrometer has been made by the designers of the instrument of Fig. 12.z2 They state that if the two spectrometers are placed as close to the photon source as possible without materially changing their resolutions, the counting rate in a pulse height peak will be larger for the crystal by five or six orders of magnitude. I n addition, the Compton spectrometer usually collects data at only one
FIQ.21. Experimental Compton spectrometer measurements of the relative bremsstrahlung differential energy flux distribution generated by 30-Mev electrons incident on a platinum target.81 The solid curve is predicted by Schiff*for thin-target bremsstrahlung at 0’ and the dashed curve has been corrected for electron energy loss in the target.
energy at a time and the crystal spectrometer measures the differential flux distribution over a wide energy range in one exposure. 2.8.2.3.2.1.1. Applications. Compton spectrometers have been used mainly to identify and measure the energies of nuclear gamma rays,29 but some measurements on bremsstrahlung spectra have also been reported.sOJ1Some relative bremsstrahlung measurements are shown in Fig. 21 as a function of relative photon energy for 30-Mev electrons incident on a platinum 2.8.2.3.2.2. Pair Spectrometers. Pair spectrometers are usually symmetrical instruments with identical electron and positron detectors which count only events where the photon energy (minus 2 m e ) is equally L. V. Groshev, A. M. Demidov, D. L. Lutsenko, and V. I. Pelekhov, “Atlas of Gamma-Ray Spectra from Thermal Neutron Capture.” Pergamon Press, New York, 1959. so J. W. Motz, W. Miller, and H. 0. Wyckoff, Phys. Rev. 86,968 (1953). H. Kulenkampff, M. Scheer, E. Schriifer, and J. Seyerlein, Naturwissenschaften 21, 1 (1958).
2.8.
DETERMINATION OF FLUX AND DENSITIES
535
divided between electron and positron. Coincident detection minimizes background and improves the energy resolution. The two main disadvantages of pair spectrometers are that it is difficult to calculate the absolute efficiency and its variation with photon energy, and that extremely long experiment times are required to obtain high-resolution data. This latter is best illustrated by the following simple arguments. Consider a spectrometer bombarded by N ( k ) dk photons of energy k per second. If the spectrometer only counts electrons and positrons with kinetic energies between T and T dT, the singles counting rates for electrons and positrons will be :
+
+
re = e(T)p dT JN(k)[u,(k,T) a,(k,T)I dk rP = e ( T ) p dT JN(k)a,(k,T) dk
(2.8.2.16)
where p is the foil thickness in atoms/cm2, and e ( T ) is the efficiency for detecting a particle of energy T ; a,(k,T) and o,(k,T) are respectively the differential cross sections per Mev for a photon of energy k to generate particles with energy T by pair production and by Compton scattering in the foil. The counting rate for true coincidences will be: rt where k’ mately :
=
2(T
=
e2(T)pd T 2 N(k’)a,(k’,T)
(2.8.2.17)
+ mc2). The random coincidence rate will be approxir r = 2rrerp
(2.8.2.18)
where 7 is the detector resolving time. The time required for n true coincidences is: time =
n Drt
__
(2.8.2.19)
where D is the duty cycle of the source (D = 1 for a steady source). With the help of Eqs. (2.8.2.16), (2.8.2.17), and (2.8.2.18), this time can be expressed as a function of the counting errors, random coihcidence errors, and energy resolution without knowing any of the spectrometer parameters except the resolving time, r , and the collection efficiency, e : time
=
87
De26,6,26,2f @‘I.
(2.8.2.20)
6, is the ratio of random coincidences t o true coincidences, 6, is the statistical counting error ( l / x h ) , and 6, is the spectrometer energy resolution (dk‘/k‘).f(k’) is the dimensionless factor:
536
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
f(k’) has been calculated for the case where N ( k ) varies as I l k for
which approximates a bremsstrahlung spectrum (Fig. 2). The approximation was made that u,(k,T) is independent of T , for T < k - 2mc2. The values of f(k’) listed in Table IV were obtained for a platinum foil and should be somewhat larger for lower atomic numbers. Table V lists the predicted experiment times as a function of energy resolution for 25-Mev photons in a beam with a peak energy kmax= 50 MeV. TABLE IV. f ( k ’ ) for Equation (2.8.2.20), with Bremsstrahlung Beams k’(Mev) k,,..(Mev) 25 50 100 200
25
50
100
200
0.8 2.4 3.7 4.6
0 0.3 2.4 3.5
0 0 0.1 2.4
0 0 0 0.02
TABLE V. Pair Spectrometer Experiment Time Versus Resolution &(%)
t(hours)
8 4 2 1 0.5
0.7 3 12 46 185
The other parameters used were 1% random coincidences, 1% counting error, 0.01 psec resolving time, e = 0.8, and D = 0.018, the latter correspond& t o 100-psec bursts of X-rays from an accelerator with a repetition rate of 180 bursts/sec. The numbers in this table indicat,e that high-resolution pair spectrometer measurements are impractical under normal conditions with a broad photon spectrum. The most obvious ways to decrease the experiment time are to reduce T , increase e, or increase D. 2.8.2.3.2.2.1. Applications. As with Compton spectrometers, pair spectrometers have been used principally with gamma rays below 20 Mev.32 A few pair spectrometer measurements with bremsstrahlung beams have s* R. L. Walker, acd B. D. McDaniel, Phys. Rev. 74, 315 (1948); see also the article by D. E. Alburger, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed ), pp. 767-794. Interscience, New York, 1955.
2.8.
DETERMINATION OF FLUX AND DENSITIES
537
been reported in the 1iteratu1-e.~~ In addition, a pair spectrometer with an energy resolution of +% at 20 Mev has recently been used t o determine The data from this latter the nuclear absorption cross section of 016.34 experiment are plotted in Fig. 22. They were obtained by studying the total attenuation coefficients of water, and did not require absolute knowledge of the spectrometer efficiency (Sections 2.8.2.3.1.1.1 and 2.8.2.3.1.3.1).This work was made possible by stretching the X-ray pulse until it was 3000 psec long, but it still required approximately 3 months of work at a rate of about 144 hours per week to obtain the data shown in Fig. 22. The long experiment time was justified in order to demonstrate and measure the structure in the tot(a1 photonuclear absorption cross section. 2.8.2.4. Total Beam Energy Measurements. The techniques for measuring total beam energy ( T B E ) differ for high- and low-energy X-rays partly because of differences in the available beam intensities. In units of pw/cm2 at a distance of 1 meter from the source, typical low-energy intensities range from about 0.5 a t 50 kev up to several hundred at 3 MeV. At high energies, corresponding numbers for betatrons and electron synchrotrons are 500 at 20 Mev up to 10,000 at a few hundred Mev. In both ranges, however, recently developed accelerators, the direct accelerators at low energies and linear accelerator at high energies, promise to increase these numbers by several orders of magnitude except near 50 kev. There were early attempts to use calorimetry to determine the total beam energy with low-energy X-rays, but the low beam intensities made accurate measurements impossible. The alternative which was adopted was to measure what is technically called expos~re-dose,~~ in roentgens. The roentgen is defined in terms of ionization in air, which can be accurately measured with low-intensity beams. The exposure-dose is proportional to the energy/cm2 in an X-ray beam, and the proportionality factor varies slowly with photon energy. One roentgen is equivalent to 3360 ergs/cm2 and one roentgen/min to 5.6pw/cm2 both to within 10% between 0.06 and 2 MeV. One advantage of the roentgen at low energies is that both the total beam energy and the energy locally absorbed in an irradiated medium (the absorbed-dose) can be calculated from the exposure-d~se.~~ This duality 8* J. W. DeWire and L. A. Beach, Phys. Rev. 83, 476 (1951); G. Diambrini, A. (3. Figuere, A. Serra, and B. Rkpoli, Nuovo cimento [lo] 16, 500 (1960). SIN. A. Burgov, G. V. Danilyan, B. S. Dolbilkin, L. E. Laeareva, and F. A. Nikolayev, to be published. *6 Report of the ICRU. Natl. Bur. Standards (U.S.) Handbook No. 78, 2 (1959). 86 Protection against Betatron-Synchrotron Radiations up to 100 MeV. Natl. Bur. Standards (U.S.) Handbook No. 66, 26 (1954).
50
40 L
O
e.E
..
0
30
I-
u W m u) m
g
20
0
2
0 I-
a U
::
10
m
a LT a W A
0
0
3
2
I 19
I 20
I
1
1
I
I
I
21
22
23
24
25
26
k, M e v FIG.22. Experimental pair spectrometer messurements of the cross section for photonuclear absorption in
0'' obtained from studies of the attenuation of a 200-Mev bremsstrahlung beam by 100 gm/cm2 of water.84
2.8.
DETERMINATION
OF FLUX AND DENSITIES
539
arises because at low energies X-rays essentially deposit their energy where they interact with the medium, At high energies, the roentgen is not a very useful unit of either exposure-dose or of absorbed-dose, because the ranges of secondary particles and primary photons are comparable. The exposure-dose in roentgens cannot be measured directly, and the absorbed-dose at a point depends on the incident energy at many points in the medium. These reasons, plus the higher intensities, have led to a return to exposure measurements in absolute energy units a t high energies. However, at least one of the techniques which has been used for this purpose at high energies, the calorimeter, has also been used for some low-energy measurements, as described below. 2.8.2.4.1. LOW-ENERGY FREE-AIRIONIZATION CHAMBERS. The roentgen is defined below 3 Mev as “the exposure-dose of X- or gammaradiation such that the associated corpuscular emission per 0.001293 gm of air produces, in air, ions carrying 1 electrostatic unit of quantity of electricity of either sign.JJ36For photons of energy k, the beam energy content in ergs/cm2 can be calculated from the exposure-dose in roentgens with the e q u a t i ~ n ~ ~ , ~ ~ : ergs/cm2 roentgen =
[g]
1 0.001293pa(k)
(2.8.2.22)
where W is the average energy in ev required to produce one ion pair in air, yn(k) is the energy absorption coefficient for air in cm2/gm, the sum of total attenuation coefficients describing photoelectric, Compton, and pair events, if each is multiplied by the fraction of the photon energy given to secondary electrons. Calculated values of the ratio of erg/cma t o roentgens are plotted in Fig. 23, using W = 33.6 ev.asThe experimental points at 0.66 and 1.25 Mev were measured with calorimeter^.^^ Exposure-dose in roentgens can be measured directly with a free-air ionization chamber, or more indirectly with a thimble chamber which has been calibrated with a free-air chamber. Figure 24 is a schematic diagram of a typical free-air chamber for photons with energies less than 500 k e ~ . ~ ~ It collects ionization equal t o the ionization produced by electrons gens’ G. J. Hine and G. L. Brownell, “Radiation Dosimetry,” p. 13. Academic Press, New York, 1956. ** Z. Bay, W. B. Mann, H. H. Seliger, and H. 0. Wyckoff, Radiation Research 7,567 (1957). 88 P. N. Goodwin, Radiation Research 10,6 (1959); I. T. Myers, W. H. LeBlanc, and D. M. Fleming, unpublished data (1961). 40 H. 0. Wyckoff and F. H. Attix, Design of Free-Air Ionization Chambers. Nail. Bur. Standards (U.S.) Handbook No. 64 (1957).
540
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
erat,ed in the volume V , which is defined by the aperture area and the Iength of the collecting electrode. The large dimensions of this chamber are necessary because the electrons generated in the beam must lose all of their energy in air before striking the plates, and because of the requirements of electronic equilibrium (the number of electrons and the electron energy entering the
0
0.01 0 0 2
0.05
0.1
0.2
0.5
I
2
5
10
k, Mev FIG. 23. Theoretical ergs/cm' per roentgen for photon energies between 0.01 and 10 MeV. The curve wm calculated for W = 33.6 ev required to produce an ion pair in air.asThe points at 0.66 and 1.25 Mev were measured calorimetrically with Cs1a7 and Coeo sources, respeotively.39
collecting region must equal the number ,and energy leaving). These requirements set an upper limit on the energy of photons which can be measured with a free-air chamber of practical size, but the same general type of chamber has been successfully used at energies as high as 2.6 Mev by filling it with air at high pressure to reduce the electron ranges.41 At still higher photon energies, electron ranges become comparable with 4 1 H. 0. Wyckoff, J . Research Natl. Bur. Standards 64C, 87 (1960); K. K. Aglintsev and G . P. Ostermukhova, Proc. Mendeleeu All-Union Sci. Research Znst. Metrology No. 66 (1961).
2.8.
DETERMINATION
OF FLUX AND DENSITIES
54 1
photon mean-free-paths, and it is difficult to correct for the lack of electronic equilibrium. 2.8.2.4.2. ABSOLUTEHIGH-ENERGY MEASUREMENTS. There are five techniques which have been used for measuring total beam energy in absolute units with errors of only a few per cent. Most of the measurements have been used to calibrate standard ionization chambers in lightly filtered bremsstrahlung beams with peak photon energies, kmax, between 6 and 1080 MeV.
FIQ.24. Schematic cross section of a free-air ionization chamber for measuring exposure-dose in roentgens at energies below 500 k e ~ . ~This O instrument is used t o determine the amount of ionization produced in the shaded volume V.
2.8.2.4.2.1. Calorimeters. X-ray calorimeters are designed t o absorb most of the energy in an incident photon beam and to detect the resultant temperature rise. The incident energy is determined by comparing this rise with the rise produced by dissipating an accurately measured quantity of electrical energy in the calorimeter. This technique has been used with low-energy bremsstrahlung beams with k,,, between 400 kev and and 1.4 Mev,42.43with CsI3’ and Co60 radioactive with high-energy bremsstrahlung beams with k,,, between 18 and 42 J. S. Laughlin, J. W. Beattie, W. J. Henderson, and R. A. Harvey, Am. J . Roentgenol. 70, 294 (1953). 4 s G. S. Dolphin and G. S. Innes, Phys. in Me& Bid. 1 , 161 (1956); J. MeElhinney, B. Zendle, and S. R. Domen, Radiation Research 6, 40 (1957).
542
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
300 Mev.42.4"41The advantage of this technique is that total beam energy can be measured without knowledge of either the photon spectral distribution or the atomic parameters of the calorimeter. It has the disadvantages of low sensitivity and the need for obtaining corrections for the amount of energy escaping from the calorimeter, which is sometimes as large as 10%.
?X-RAYS
FIG.25. Schematic diagram of a typical X-ray calorimeter, used for measurement of total beam energy.
Figure 25 is a generalized diagram of a typical X-ray calorimeter. It is designed to be adiabatic, with the absorbing elements, the lead cylinders, thermally isolated from their surroundings. The cylinder supports are made of a thermally insulating material (the use of threads is common) and the box is evacuated t o minimize exchange of heat by conduction. The surfaces of cylinder and box are normally plated with a metal of low thermal emissivity (such &g gold) to reduce the exchange of thermal radiation. Even with these precautions, it is necessary t o maintain the 44 P. D. Edwards and D. W. Kemt, Rev. Sci. Znstr. 24, 490 (1953). 4 6 s . P. Kruglov, Zhur. Tekh. Fiz. 28, 2310 (1958);Soviet Phys.-Tekh. Phys. 5, 2120 (1958). 48 J. S. Pruitt and S. R. Domen, J . Reeearch Natt. Bur. Standards 66A, 371 (1962). 47 J. S. Pruitt and W. Pohlit, 2. Nalurforsch. lLb, 617 (1960).
2.8.
DETERMINATION O F FLUX AND DENSITIES
543
calorimeter box at a constant temperature. Good thermal isolation and good temperature control are well worthwhile since they minimize the corrections for heat transfer and permit longer exposures at lower intensities. Extreme precautions were taken with the NBS ~ a l o r i m e t e r , ~ ~ for instance, with the result that 90-min exposures to an X-ray beam with an intensity of 2 pw/cm2, which used a 13.8-cm2beam, produced a temperature rise of 2 X 10-40C,and yielded determinations of total beam energy with a rms deviation less than 4%. A calorimeter of the type shown in Fig. 25 detects changes in the reIative temperature of the two cylinders when one of them is irradiated with X-rays. This is done with thermistors, resistors with large temperature coefficient, which are embedded in the cylinders. The thermistors are connected in a Wheatstone bridge so that the output voltage is proportional t o an increase in temperature of the bombarded cylinder. Calibration is performed by dissipating power in separate resistive elements embedded in the cylinders. 2.8.2.4.2.2. The Scintillation Spectrometer Method. Large-sodium-iodidecrystal spectrometers have been used to measure total beam energy with the same experimental arrangement used to determine total attenuation coefficients (Fig. 10).48849 As pointed out in Section 2.8.2.3.1.1, a large-crystal spectrometer counts the total number of incident photons efficiently and accurately, if a correction is made for photons which traverse it without interaction. On the other hand, a counter of this type would be swamped with counts from low-energy photons if it was used in a lightly filtered bremsstrahlung beam, as mentioned in the introduction. This experiment avoids jamming by filtering out low-energy photons with a long absorber. The total energy in a beam irradiating the crystaI with the absorber removed can be calculated from the number of spectrometer counts with the absorber in position, n, using the equation:
TBE =
nJN(k)kd k JN(k)e-p(k)zdk
(2.8.2.23)
where N ( k ) is the differential flux distribution of photons incident on the absorber, and x and p ( k ) are the length and total attenuation coefficient of the absorber, respectively. This technique has the advantage of high sensitivity and has been used at bremsstrahlung energies as low as 6 MeV. It does require knowledge of the incident photon spectra and the total attenuation coefficients. However, Eq. (2.8.2.23) is relatively insensitive t o errors in the shape of N ( k ) , 45
49
E. 0.Fuller and E. Haywsrd, J . Research Natl. Bur. Standards 66A,401 (1961). J. E.Leks, J. S. Pruitt, and R. A. Schrack, unpublished data (1958).
544
2.
DETERMINATION OF FUNDAMENTAL QUANTITIES
and the validity of the assumed attenuation coefficients can be tested by varying the absorber length. 2.8.2.4.2.3. The Balanced-Converter Method. Measurements of ionization behind thin foils of different atomic number in a high-energy bremsstrahlung beam can be used to determine total beam energy. Using foils of atomic number Z1 and Z z and identical thickness, measured in electrons per cm2, it has been shown50 that the total beam energy is given by:
where p is the ionization measured behind the foils in a thin-walled parallel-plate chamber with an air gap thickness t, p e is the foil thickness in electrons/cm2, and I,,, is the minimum specific ionization produced in air by an electron. R ( 2 ) is a slowly varying function of atomicnumber which has the units of a cross section. It is essentially the pair production cross section for hydrogen averaged over the photon flux distribution, but contains factors which describe the deviation of the pair-plus-triplet cross section from a Z 2 dependence on atomic number, and the deviation of the secondary electron-positron pairs from minimum ionization. The values of R ( 2 ) originally calculated for carbon, aluminum, copper, and lead for 330-Mev bremsstrahlung are listed in Table VI. TABLE VI. R(2)for Equation (2.8.2.24), at 330 Mev
z R ( Z ) , barns
0.0254
13 0.0255
29 0.0250
82 0.0220
The great advantage of this technique is its experimental simplicity, but this advantage is at least partially offset by the requirements of detailed knowledge of the differential photon flux distribution, the pair production cross sections, and the specific ionization of electrons. 2.8.2.4.2.4. The Shower Method and the Quantameter. Measurements of ionization at different depths in a large absorbing medium irradiated by a photon beam can also be used to determine the total beam energy. If q(x) is the ionization measured in a thin-walled parallel-plate ionization chamber with a gap thickness t at depth x in the medium, the incident energy can be expressed ass1:
TBE
W, @
q(x) dx. (2.8.2.25) dI3t W , is the energy required t o produce unit ionization in the gas of the =
6o
W. Blocker, R. W. Kenney, and W. K. H. Panofsky, Phys. Rev. 79, 419 (1950).
61
A. P.Komar and S. P.Kruglov, Zhur. Tekh. Fiz. SO, 1369 (1960);Soviet Phys.-
Tech. Phys. 6, 1299 (1961).
2.8.
DETERMINATION
OF FLUX AND DENSITIES
545
chamber, d, and dg are the densities of the medium and the gas, and S is the ratio of the mass electron stopping power of the medium (Mevcm2/gm) to that of the gas, averaged over the energy spectrum of electrons entering the chamber. S can be predicted theoretically if the electron spectrum is known. It is a slowly varying function of energy. There have been several measurements of bremsstrahlung beam energy with the shower technique in the energy range between 16 and 1080 HIGH VOLTAGE r l O N COLLECTOR
/-
f S T A I N L E S S STEEL 6 rnm INSULATION
2 mm X-RAYS
10 m m
%-dx+I mm
2 mm
QUANTAMETER
FIG.26. Schematic cross section of the quantameter, a copper ionization chamber with twelve ionization gaps arranged to evaluate in a single measurement the total ionization produced in B high-energy penetrating shower.54 Mev.60-53The most r e ~ e n have t ~ ~ been ~ ~made ~ with a quantameter, the copper ionization chamber shown in cross section in Fig. 26.54 The quantameter determines the integral in Eq. (2.8.2.25) in a self-contained copper medium with a single measurement, automatically summing the ionization at several depths to evaluate the integral by Simpson’s rule for parabolic interpolation. The quantameter is not large enough to absorb all of the incident energy but compensates for the energy leaking from the sides of the chamber with peripheral air cavities, and compen62
53
D. C. Oakley and R. L. Walker, Phys. Rev. 97, 1283 (1955). J. W. DeWire, private communication (1959). R. R. Wilson, Nuclear Znst. 1, 101 (1957).
2.
546
DETERMINATION OF FUNDAMENTAL QUANTITIES
sates for transmitted energy with an oversized ionization gap at the rear. These corrections are usually of the order of a few per cent. The quantameter constant, the ratio of incident enetgy t o measured ionization, has been theoretically evaluated for the normal gas filling of argon to a pressure of 800 mm of mercury at 2 0 T , and for air filling to a pressure of 760 mm of mercury at 20°C. The calculated values of this ,~~ 4.80 and constant for argon are 4.79 X 1018 k 3% M e v / ~o u l o rn b and 4.72 X 10'8 k 1.8% Mev/coulomb,6' and for air, 7.74 and 7.81 X lo1* k 2% Mev/coulomb.61 2.8.2.4.2.5. Pair Spectrometer Method, Pair spectrometers (Section 2.8.2.3.2.2) have been used t o determihe total beam energy from measurements of the differential photon flux distributions (Eq. 2.8.2.17).62*63 There is no need for a high-resolution spectrometer in this application, so the arguments about long experiment times given in Section 2.8.2.3.2.2 are not very pertinent. The accuracy is usually limited by uncertain knowledge of the absolute detection efficiency and energy resolution of the spectrometer. 2.8.2.4.3. RELATIVEHIGH-ENERGY MEASUREMENTS. There are two techniques which have been used to determine total beam energy in relative units, to investigate the variation of ionization chamber calibrations with the peak energy of a bremsstrahlung beam. 2.8.2.4.3.1. Pair Spectrometer. Pair spectrometers have been used to measure the ratio of the total beam energy at two different bremsstrahlung energies by comparing the numbers of pairs produced by If n ( f ) is the number of pairs counted, photons of a fixed energy, f.66,66 the ratio of total beam energies is simply:
T B E , - n1(k)C1(5) -where C ( f ) = TBEa n&G)Cs(f)
3
kdk.
(2.8.2.26)
N ( k ) is the differential photon flux distribution of the incident photon beam. C ( f ) can be calculated as a function of the peak energy of the bremsstrahlung beam for a given 5. This method yields accurate measurements of the energy ratio without knowledge of either the spectrometer efficiency or energy resolution, although it does depend upon knowledge of the shape of the differential photon flux distribution. 2.8.2.4.3.2. Copper Activation. A second method for relative measurements of total beam energy requires counting positrons emitted by Cuez produced by the reaction Cu6a(y,n)Cuazin a copper foil in the photon s 6 F .J. Loeffler, T. R. Palfrey, and G . W. Tautfest, Nuclear Instr. & Methods 6, 50 (1959).
w.P. swanson, private communication (1961).
2.8.
DETERMINATION OF FLUX AND DENSITIES
547
beam.66 The cross section for this reaction, acu(k>,has a peak at 18 Mev and is zero below about 10 Mev and above about 38 If Ac is the measured positron activity produced by a bremsstrahlung exposure, the ratio of total beam energies in beams with two different peak photon energies is:
and N ( k ) is again the differential photon flux distribution. This method requires knowledge of the shape of the reaction cross section, acU(k), but, at high bremsstrahlung energies, N ( k ) varies slowly over the region where acu is large, and 6 is insensitive to errors in the assumed values of ucu. However, 6 is sensitive to the assumed shape of the differential photon flux distribution. 2.8.2.4.4. HIGH-ENERGY IONIZATION CHAMBERCALIBRATIONS. The four standard ionization chambers shown in Fig. 27 have been calibrated with measurements of total beam energy which cover the energy range from 6 to 1080 MeV. The three copper chambers were designed at Cornell University,68 the University of Illinois,44 and the Leningrad Institute of Physics and Technology (LPTI).46 The chamber designed at the United States National Bureau of Standards (NBS)4shas walls of an aluminum alloy, 2024 Dural (nominally 93.4% Al, 4.5% Cu, 1.5% Mg, and 0.6% Mn). The chambers are all air-filled and open t o the atmosphere. The calibrations of these chambers in bremsstrahlung beams with different peak energies, k,,,, are listed in Tables VII-X, which also show the method used and the place where the data originated. In addition to the institutes where the chambers were designed, the data comes from the California Institute of Technology (Cal. Tech.),62s62the General Electric Company (GE),63the Max Planck Institute for Biophysics (MP1),47 and Purdue University.66 Some of the Purdue and Illinois measurements were relative, and these are marked with asterisks. A few of the calibrations were obtained by intercomparison of the chambers. Those obtained by quantameter measurements are all based on the original calculation of the quantameter constant, 4.79 X 10l8 Mev/coulomb (7.51 X lobjoules/coulomb). The calibrations are all listed in units of joules/coulomb at 2OoC and 760 mm of mercury, although most were reported in other units. The Illinois measurements with the Illinois chamber were originally reported in e r g - ~ m / e s uand ~ ~ have been converted to joules/coulomb for a chamber with the design air gap thickness of 0.104 in. The listed k,, in this case 57 68
A. I. Berman and K. L. Brown, Phye. Rev. 96, 83 (1954). D. R. Corson,J. W. De Wire, B. D. McDaniel, and R. R. Wilson, NP-4972 (1953).
HIGH VOLTAGE
3 8 m m A1
HIGH VOLTAGE ,
\
HIGH VOLTPGE
64mm
I
O I‘ N
COLLECTOR
ILLINOIS
CORNELL
P LPTl
COPPER
ALUMINUM A L L O Y
STAINLESS S T E E L
NBS INSULATION
FIG.27. Schematic cross sections of four standard ionization chambers which have been experimentally calibrated to determine total beam energy in high-energy bremsstrahlung beams. The name below each chamber is the institute where it waa d&gn4.4w6.aw
2.8.
DETERMINATION
OF FLUX AND DENSITIES
549
TABLEVII. Calibration of the Cornell Ionization Chamber in a Small-Diameter Bremsstrahlung Beam, a t 20°C and 760 mm of Hg kmar(Mev) Institute 60 90 90 120 125 130 144 150 170 175 197 200 200 220 225 250 250 250 270 275 300 308 315 315 318 320 325 497 500 500 600 780 789 1000 1080
NBS NBS Purdue Purdue Purdue NBS GE Purdue Purdue Purdue Cornell GE Purdue Purdue Purdue Purdue Cornell Purdue Purdue Purdue Purdue Illinois Cornell Cornell Purdue Purdue Purdue Cal. Tech. Cal. Tech. Cal. Tech. Cornell Cornell Cal. Tech. Cornell Cal. Tech.
Methoda
Calibration (joule/coulomb)
Reference
Comparison Comparison Cu activation* Spectrometer * * Cu activation* Comparison Spectrometer Cu activation* Spectrometer * * Cu activation* Spectrometer Spectrometer Cu activation* Spectrometer * * Cu activation* Balanced converter Spectrometer Cu activation* Spectrometer * * Cu activation * Cu activation* Comparison Spectrometer Quantameter Balanced converter Spectrometer* * Cu activation* Quantameter Spectrometer Shower Quanta meter Quantameter Quant ameter Quontameter Quantameter
6.01 X l o 6 f 2.7% 5.82 2.7 5.67 5.62 5.54 5.54 2.7 5.79 3 5.58 5.71 5.67 5.96 5 6.00 3 5.84 5.81 5.94 6.00 4.3 6.17 5 6.16 6.26 6.31 6.58 6.45 2.5 6.43 5 6.64 3 6.67 3.8 6.62 6.64 7.27 3 8.17 2.7 7.08 4.9 7.65 3 8.17 3 8.32 3 8.75 3 8.99 3
55 55 55 55 55 55 53 56 55 55 53 53 55 55 55 55 53 55 55 55 55 53 53 53 55 55 55 53 52 52 53 53 53 53 53
KEY: * Normalized to curve at kmax = 250 MeV; kmaX= 270 MeV.
** Normalized to curve at
2.
550
TABLE) VIII.
DETERMINATION OF FUNDAMENTAL QUANTITIES
Calibration of the Illinois Ionization Chamber in a Small-Diameter Bremsstrahlung Beam, at 20°C and 760 mm of Hg
kmax(Mev)
Institute
Method
Calibration (joule/coulomb)
Reference
25 30 35 40 45 50 60 70 90 110 130 146 150 170 195 244 293
NBS NBS NBS NBS NBS NBS NBS NBS NBS NBS NBS Illinois NBS NBS Illinois Illinois Illinois
Comparison Comparison Comparison Comparison Comparison Comparison Comparison Comparison Comparison Comparison Comparison Calorimeter Comparison Comparison Calorimeter Calorimeter Calorimeter
39.8 X lo6 f 2.5% 40.0 2.5 39.8 2.5 39.9 2.5 40.7 2.5 41.4 2.5 43.1 2.5 44.6 2.5 48.6 2.5 52.8 2.5 56.7 2.5 60.9 3 60.5 2.5 63.8 2.5 64.9 3 72.3 3 83.3 3
46 46 46 46 46 46 46 46 46 46 46 44 46 46 44 44 44
T A B LIX. ~ Calibration of the LPTI Ionization Chamber in a 40-mm Diameter Bremsstrahlung Beam Filtered by 4.0 gm/cma of Aluminum, at 20°C and 760 mm of Hg kmax(Mev)
Institute
Method
Calibration (joule/coulomb)
Reference
16 22 25 29 40 53 53 68 72 a5 85 95
LPTI LPTI LPTI LPTI LPTI LPTI LPTI LPTI LPTI LPTI LPTI LPTI
Quantameter Quantameter Quantameter Quantameter Quantameter Calorimeter Qu ant amet er Qua ntameter Calorimeter Calorimeter Quantameter Quantameter
8.52 X lo6 f 3% 3 8.31 8.15 3 8.07 3 3 7.90 7.75 1.7 7.80 3 7.88 3 7.87 1.5 1.o 7.95 7.99 3 8.09 3
51 51
51 51 51 51 51 51 51 51 51 51
2.8.
551
DETERMINhTION O F FLUX AND DENSITIES
TABLE X. Calibration of the NBS Ionization Chamber in a 42-mm Diameter Bremsstrahlung Beam Filtered by 4.5 gm/cm* of Aluminum, at 20°C and 760 mm of Hg k,,(Mev) 6 8 10 13 16 19 20 20 25 26 30 31 35 35 36 40 41 45 45 50 50 60 60 70 70
90 90 110 110 130 130 146 150 150 150 170 170 170 200 230 260 a
CaIibration (joule/coulomb)
Institute
Methoda
NBS NBS NBS NBS NBS NBS NBS NBS NBS NBS NBS NBS MPI NBS NBS NBS NBS NBS N BS NBS NBS NBS NBS NBS NBS NBS NBS NBS NBS NBS NBS Illinois NBS NBS Illinois Illinois NBS NBS Illinois Illinois Illinois
Scintillator Scintillator Scintillator Scintillator Scintillator Scintillator Scintillator Calorimeter Scintillator Calorimeter Scintillat or Calorimeter Calorimeter Scint illator Calorimeter Scint illator Calorimeter Calorimeter Scintillator Scintillator Calorimeter Calorimeter Scintillator Scint illator Calorimeter Calorimeter Scintillator Scintillat or Calorimeter Calorimeter Scint illator Comparison Calorimeter Scintillator Pair spectrometer * Pair spectrometer* Calorimeter Scint illator Pair spectrometer * Pair spectrometer* Pair spectrometer*
KEY: * Normalized to curve a t k,,
=
170 MeV.
4.10 X lo6 4.17 3.99 4.07 4.09 4.17 4.04 4.16 4.14 4.11 4.10 4.12 4.04 4.10 4.11 4.07 4.09 4.02 3.96 3.86 3.99 3.94 3.82 3.73 3.86 3.80 3.71 3.60 3.84 3.81 3.64 3.84 3.82 3.69 3.75 3.81 3.87 3.71 3.87 3.98 4.09
+ 2% 2 2 2 2 2
3 2.1 3 2.5
3 1.6 1.5 3 1.3 3 1.7 1.3 3 3 1.5 1.5 3 3 2.4 2.3 3 3 2.4 1.3 3 3.6 1.3 3 1.6 3
Reference 48 48 48 48 48 48 49 46 49 46 49 46 47 49 46 49 46 46 49 49
46 46 49 49 46 46 49 49 46 46 49 46 46 49 56 56 46 49 56 56 56
2.
552
DETERMINATION O F FUNDAMENTAL QUANTITIES
are 2.5% lower than the values originally reported, because of an error in the original betatron energy c a l i b r a t i ~ n . ~ ~ If the calibrations of each chamber are plotted as a function of energy, it is possible to fit each with a smooth curve passing through most of the points to within the quoted error. The only large deviation is the 500-Mev pair spectrometer calibration of the Cornell chamber. The calibration
v; W
0
OUANTAMETER
l-
a [r
m a
/
0
W
m
a I 0
105
I II
1
11 kMAx,
Mev
FIG.28. Experimental calibration curves of four standard ionization chambers in air at 20°C and 760 mm of mercury as a function of the maximum photon energy of a bremsstrahluug beam (Tables VII, VIII, IX, X). The experimental calibration curve of the quantameter, filled with argon at 20°C and 800 mm of mercury, is also shown (Table XI).
curves are shown in Fig. 28, along with a curve labeled quantameter. The quantameter curve comes from Table XI, which was derived from Tables VII and IX by reversing the procedure and considering the quantameter calibration unknown. The slight decrease in quantameter calibration with increasing energy is well within the quoted errors, so that there is no real experimental evidence for variation of the calibration between 53 and 317 MeV. The calibrations of the four chambers of Fig. 27 must depend on the 6s
L. J. Koester, private communication (1958).
2.8.
DETERMINATION OF FLUX AND DENSITIES
553
shape of the spectrum of incident photons, since the calibrations vary with ksx. They will consequently also vary with beam filtJration,a variation which has been studied in detail for the NBS chamber.46It has been shown
TABLEXI. Calibration of the Quantameter in a Small-Diameter Bremsstrahlung Beam, for Argon Filling of 800 mm of Hg at 20°C.
kwx(Mev)
Institute
Method
53 72 85 308 315
LPTI LPTI LPTI Illinois Cornell
Calorimeter Calorimeter Calorimeter Calorimeter Spectrometer
Calibration (joule/coulomb) 7 . 6 3 X lo6 k 7.64 7.64 7.50 7.43
1.8% 1.6 1.1 2.5 5
Reference 51 51 51 53 53
for this chamber that the change of calibration with filtration is negligible at energies above 100 Mev for l o w 4 absorbers. The change becomes noticeable at lower energies, but is still only 2 % for a filtration change from 4.5 to 10 gm/crn2 of aluminum at 6 MeV.
This Page Intentionally Left Blank
3. SOURCES OF NUCLEAR PARTICLES AND RADIATIONS 3.1. Radioactive Sources* At some time in nearly all investigations of experimental nuclear physics, a need arises for a radioactive source. The source may be required for instrument tests, for the standardization of detector efficiency, or for the calibration of the energy scale of a spectrometer. It is desirable to combine, if possible, the functions of disintegration rate standard and energy standard in a single source; however, many of the nuclides suited for one purpose are unsuitable for another. The following discussion treats the preparation of sources of neufrons, a-particles, /3-particles, and y-rays, with some comments on the calibration of these sources for use in typical laboratory situations. 3.1.1. Radioactive Neutron Sources
The discovery of the neutron and the subsequent study of its properties and interactions are intimately connected with the development of radioactive neutron sources. Although the earliest work in neutron physics used neutrons from the bombardment of light elements with a-particles from polonium or radon, it was soon recognized that the photodisintegration of deuterium and beryllium could also be used to prepare neutron sources. Accelerators and nuclear chain reactors have now largely supplanted radioactive neutron sources in detailed studies of neutron reactions and in production of neutron-induced radioactivities. Nevertheless, radioactive neutron sources possess several advantages which make them very important in a nuclear laboratory. They are compact, and therefore portable; many have a constant output over many years; they require no maintenance, and are relatively inexpensive. Such features make them suited to the study of the slowing down and diffusion of neutrons in various media, and the determination of neutron scattering and absorption cross sections. They are also widely used as primary and secondary standards of neutron flux which, because of their small size, may be conveniently exchanged between laboratories. For more extensive data on the fundamental processes involved, the methods of determining absolute neutron yields, and results of inter-
* Chapter 3.1 is by G. D. 0 Kelley. 555
556
3.
SOURCES O F NUCLEAR PARTICLES AND RADIATIONS
national intercalibrations, the reader is referred to review articles by Anderson,’ Wattenberg,Z Feld,3 curt is^,^ and Hansoa6 3.1.1 .l.a-Particle Neutron Sources. Useful thick-target neutron yields have been observed when a number of light elements are irradiated with a-particles from the disintegration of heavy radionuclides. Yields obtained from thick targets, and yields obtained when the a-emitters were intimately mixed with different target materials, are summarized in Table I.6-EThe data are sufficiently accurate to show that the yield is highest for beryllium, with boron, fluorine, and lithium or sodium following in order. 3.1.1.1.1. THEB ~ ( a , n )REACTION.The high (a,n)yield of beryllium has been responsible for its extensive use as a target material. Naturally occurring a-particle sources such as Pozx0,RaZz6,AcZz7,and Em222 have long been popular as components of Be(cr,n)Cl2 neutron sources. More recently, Pu239,Am241,and Cm242 have been used successfully when alloyed with beryllium. The maximum neutron yields for these transuranium sources were measured by Runnalls and Boucher,’ whose data fit the following empirical expression :
nmax= 0.152E:66 neutrons/lOs alphas.
(3.1.1)
Table I shows also that Eq. (3.1.1) fits other cases well enough to be used for extrapolating to other a-particle energies for which no experimental yield data are available. The principal nuclear reaction in sources using a beryllium target is Be9
+ He4 = C12 + n + 5.71 MeV.
(3.1.2)
For PoZ1Oa-particles (5.30 MeV) on a thin beryllium target, the energy of the emergent neutrons would be expected to range from about 11 Mev (outgoing neutron and incoming a-particle in the same direction) to 6.7 Mev (outgoing neutron and incoming a-particle in opposite direcH. L. Anderson, Neutrons from alpha emitters. National Research Council, Nuclear Science Series, Preliminary Report No. 3, Washington, D.C. (1948). a A. Wattenberg, Phys. Rev. 71, 497 (1946). 8 B. T. Feld, in “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 11, Part VII. Wiley, New York, 1953. 4 L. F. Curtiss, “Introduction to Neutron Physics,” Chapter 111. Van Nostrand, Princeton, New Jersey, 1959. 6A. 0. Hanson, Radioactive neutron sources. I n “Fast Neutron Physics” (J. B. Marion and J. L. Fowler, eds.), Vol. I, Chapter IA. Interscience, New York, 1959. 6 R. J. Breen and M. R. Hertz, Phys. Rev. 98, 599 (1955). 7 0. J. C. Runnalls and R. R. Boucher, Can. J . Phys. 54, 949 (1956). 8 J. H. Roberts, Neutron yields of several light elements bombarded with polonium alpha particles. U.S. Atomic Energy Commission Report MDDC-731 (1944).
3.1.
557
RADIOACTIVE SOURCES
tions). In a practical source, where the Po*loand beryllium are intimately mixed, the spectrum is more complicated for a number of reasons. One complication is that the final nucleus can be left in its ground state, or in the excited states at 4.43 and 7.65 MeV. Less than 7 % of the Po-a-Re reactions lead to the 7.65-Mev state of C1*, but about 60% excite the 4.43-Mev state.g The 3.22-Mev cascade y-ray has an intensity of only TABLE I. Summary of Neutron Yields from a-Particle Neutron Sources Neutron yield per 106 a-particles
a-Source
Target
Measured thick targetsa
Measured actual sources
2.6 80 24
1 .o 60 16 2.8 1.1 0.80 0.53 46lC 6.gd 677 60 f 2 74 2 112 3
1.5 1.4 0.74
Ace27
Puzs9 Am2441 Cm242
Be Be Be Be
**
Reference 6 6 6 6 6 6 6 1 1 1
7 7 7
Calculatedb 66
485-552" 709 58 75 110
Roberts.* Runnalls and Boucher.7 c Measured for a source with no Poalopresent. d For a source in equilibrium with its decay products. Depends on PoZ1Ocontent of source. 0
b
about 3% that of the 4.43-Mev y-ray;IO y-ray transitions between the 7.65-Mev level of spin zero and even parity and the O+ ground state are, of course, strictly forbidden. Further, because the beryllium usually is thick with respect to the range of the incident Po2lo a-particles, a spread of neutron energies in the forward direction is expected to extend from the maximum of 11 Mev down to 5.3 Mev (for zero-energy a-particles) . An additional complication of the neutron-energy spectrum arises from the spread in a-particle energy introduced by the thick beryllium target. The variation of reaction cross section with incident a-particle 9K. G. Steffen, 0. Heinrichs, and H. Neuert, 2. Physik 146, 156 (1956); K. C. Steffen and H. Neuert, i b i d . 147, 125 (1957). 10L. E. Beheian, H. H. HaIban, T. Hussain, and L. C . Sanders, Phys. Rev. 90, 1129 (1953).
558
3.
SOURCES O F NUCLEAR PARTICLES AND RADIATIONS
energy is influenced by both the potential barrier of the Be9 nucleus (3.7 MeV) and the resonances caused by the level structure of the Clacompound nucleus. Low-energy neutrons have been observed , which may be due in part to the multibody breakup processes He4
+ Be9 = He4 + Be8 + n - 1.67 Mev = 3He4 + n - 1.57 MeV.
(3.1.3) (3.1.4)
Experimental determinations of the neutron energy spectrum from Po-a-Be sources have been made by measuring proton recoils in emulsionsl1I12 and counter teles~opes,'~~'4 and by measuring the energy released in the Li6(n,a)Ha reaction, using LiEI(Eu)as a scintillator.16The results of these experiments suggest two prominent peaks at about 3.5 and 5 MeV, with small "humpsJJ a t 8 and 10 MeV. Several workers also report considerabIe intensity below 2 MeV. A typical spectrum is shown in Fig. 1. Using thin-target excitation functions, Hessls calculated the spectrum expected for a Po-a-Be source. I n one case he assumed that all of the neutron yield not going to the levels at 0 and 4.43 Mev went via the 7.65-Mev state, and in another case he assumed that 80% of the excess yield went into the multibody breakup reaction. Although the experimental neutron-energy spectra are not in very good agreement with each other, none agrees very well with either spectrum calculated by Hess. Because of its long half-life (1622 years), Razzshas been used to prepare standard neutron sources. However, the Ra-a-Be neutron-energy spectrum is not as well studied as that of Po-a-Be, chiefly because of experimental difficulties associated with the intense y-radiation from radium. The energy spectrum from Ra-a-Be is very complicated, owing to the variety of a-particle energies emitted in the decay of radium and its products. After a few weeks, all of the a-emitting decay products have attained their steady-state concentrations except PoZ1O, whose growth is determined by 19.4-year Pb2'0. The neutron energy from Ra-a-Be peaks at about 4 MeV, and extends to a maximum of about 12 MeV. There seem to be more intermediate- and low-energy neutrons than in Po-a-Be sources, which may be due either to an enhancement of B. G. Whitmore and W. B. Baker, Phys. Rev. 78,799 (1950). L. L. Medveczky, Acta Phys. Acad. Sn'. Hungar. 6,261 (1956). 18 R. J. Breen, M. R. Hertz, and D. U. Wright, Jr., The spectrum of poloniuml1
l*
beryllium neutron sources. U.S. Atomic Energy Commission Report IVILM-1054 (1955). 14 R. C. Cochran and K. M. Henry, Rev. Sn'. Instr. 26,757 (1955). 16 R. B. Murray, Nuclear In&. 2,237 (1958). l6 W. N. Hess, Neutrons from (a,n)8ources. U.S. Atomic Energy Commission report UCRL-3839, revised (1957).
NEUTRON ENERGY (Mev)
2
3
I
I
Y
20
30
t 40
50
60
70
80
PULSE HEIGHT
FIG.1. Neutron spectrum of a l-curie Po-a-Be source. The measurement was made by using a LiBI(Eu) scintillator to measure the energy released in the Li6(n,a)Hsreaction (Murray16).
560
3.
SOURCES O F NUCLEAR PARTICLES AND RADIATIONS
the multibody reaction by the 7.68-Mev a-particles of or to (y,n) reactions from the large number of high-energy y-rays. The spectrum of Ac2Z7-a-Be is similar to that for radium sources, but has only about oneseventh the y-ray i n t e n ~ i t y . ' ~ Plutonium alloyed with beryllium makes a very convenient long-lived neutron source with a low y-ray intensity. The neutron-energy spectrum1* shows a considerable low-energy component, a prominent peak at about 4 MeV, followed by a much lower peak at about 7 MeV, and an upper limit of -11 MeV. IN OTHER ELEMENTS. As will be seen 3.1.1.1.2. THE(a,n) REACTION from Table I, several light elements beside beryllium can be used as targets in (a,n)neutron sources. The (a,n)reactions on boron, fluorine, and lithium appear most promising. The reactions in baron are:
+ He4 = + n + 0.5 Mev BI0 + He4 = N13 + n + 1.07 Mev
B" and
N14
(3.1.5) (3.1.6)
where most of the yield from natural boron comes from the reaction with B11 (81.2% abundance). The energy spectrum is simpler than that of Be (a,n)sources. Measurements of the Po-a-B spectrum1J4 indicate a single broad peak at -2.7 MeV, and a practical end point at -5 MeV. The spectrum of a Ra-a-B source is similar. The F(a,n) reaction is of interest because it produces a reasonable yield for Po a-particles, and, aIthough the spectrum from a pure fluorine source has not been measured, such a pure source is expected to exhibit a simple spectrum which peaks a t about 1.5 Mev.16 The Li7(a,n) reaction is useful as a source of lowenergy neutrons which is relatively free from y-rays. Because the threshold of the Li7(a,n) reaction is only 4.38 Mev and that for Lis(a,n) is 6.64 MeV, only the Li7 contributes when natural lithium is used with PoZl0 a-particles (5.30 Mev). The mean energy is probably about 0.4 Mev.16 OF a-PARTICLE NEUTRON SOURCES. A technique 3.1.1.1.3. FABRICATION for the preparation of Ra-a-Be sources developed for the Manhattan District has been described by Anderson.' Sources were made by evaporating to dryness a suspension of Be metal powder in a solution of RaBrz. The resulting dry, intimate mixture of RaBrz and Be was then pressed hydraulically into a pellet having a density of 1.75 grnjcmz by the conventional techniques of powder metallurgy. To prevent loss of radioactive gas, which would not only cause the neutron yield to vary, but which would also create a serious hazard, the pellets currently are sealed in an inner container, which is in turn sealed in another container. Such pressed sources are very convenient because of their small size, and their neutron 17
W. R. Dixon, A. Bielesch, and K. W. Geiger, Can. J . Phys. 86,699 (1957). Phye. Rev. 88,740 (1955).
18L. Stewart,
3.1.
RADIOACTIVE
561
SOURCES
emission rates are very constant with time because of the physical stability of the mixture. Transuranium elements are very useful constituents of a-particle neutron sources, as they can be readily alloyed with beryllium. These alloyed sources are very homogeneous and can be made quite reproducibly. Several reviews describe the fabrication methods applicable to P U ~ ~ ~ - ~ - B sources,18*and Runnalls and Boucher’ discuss extension of the alloy techniques to Am241and Cm242as well. TABLE 11. Y-Ray Yields from
(a,n) Sources
in Milliroentgens per Hour at
One Meter19
Source RazzB-a-Be PbZlo-a-Be PoZ’O-a-Be A~2~7-a-Be Pu zs9-~-Be Thzz*-,-Be
Tiiz
7-Yield per curie (mrhm)
Y-Yield per 106 neutrons /sec (mhm)
1622 yr 1 9 . 4 yr 138.4 day 21.6 yr 24 ,360 yr 1 . 9 1 yr
825 22 0.1 146 3.7 900
55-82 8.8 0.04 7.3-8.6 1.8 45-53 -
A 0.5-gram Ra-a--Be source typically may be 2 cm in diameter and 2 cm long; a 1-gram source would be about 25% larger. On a weight basis, polonium sources require less active material and so have much smaller dimensions. In practice a doubly sealed capsule of Po-a-Be may have over-all dimensions of 1 cm or more. High yield Pu-a Be sources are very bulky, because 16 grams of Pu are required per curie of a-radiation. 3.1.1.1.4. ?-RADIATION FROM (a,n) NEUTRON SOURCES. A recent survey of the y-radiation from a number of Po neutron sources has been made by Breen and Hertz,s who used a NaI(T1) scintillation spectrometer. The y-ray yields from all polonium sources are low, and are about one y-ray per neutron. Table 1119shows y-ray yields for a variety of sources, from which it is seen that most sources constitute a serious y-ray hazard, and that special care must be exercised if the average radiation level is to be kept below a tolerance dose. 188 M. J. F. Notley and J. Sheldon, Manufacture of Pu-Be Neutron Sources. British Atomic Energy Research Establishment Report AERE-R-3670 (1961) ; R. E. Tate and A. S. Coffinberry, “Plutonium-Beryllium Neutron Sources, Their Fabrication and Neutron Yield,” i n Proc. 2nd Intern. Conf. on the Peaceful Uses of Atomic Energy, P/700, 14, 427 (1958). Is Neutron Sources and Their Characteristics, a bulletin distributed by the Commercial Products Division, Atomic Energy of Canada Ltd., P. 0. Box 93, Ottawa, Canada; Private communication from M. R. Fleming, Atomic Energy of Canada Ltd., October 27, 1959.
562
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
TABLE 111. Spontaneous Fission Neutron Sources Nuclide U232 U'as pu*aa Pusas
PU240 Pu94Z Pu944 Cmer2 Cmar4 (3262
Cf264 a
b
a-Decay Tina
Fission Ti/za
Fissions per lo1 alphas
74 yr 4.51 X 109 yr 2 . 8 5 yr 86.4 yr 6580 yr 3 . 7 9 X 1Obyr -7 .G X 107 yr 162.5 day 1 7 . 9 yr 2 . 2 yr
8 X l O I 3 yr 8 . 0 X l O l S yr 3 . 5 X 1 0 9 yr 3 . 8 X 10IOyr 1.22 X 10I1yr 7 . 1 X 10IOyr 2 . 5 X 1Oloyr 7 . 2 X 106yr 1 . 4 X 10Tyr 66 yr 56 day
9 . 2 X lo-' 5 . 6 X lo-' 8 . 1 X lo-' 2.3 X 5.4 X 5.3 3 . 0 X lo8 G.2 X 1.3 3 . 3 X lo4 Large
-
Neutrons per fissionb
1.89 2.04 2.09 2.32
i0.20 f 0.13 f 0.11 k 0.16
2.33 f 0.11 2.61 f 0 . 1 3 3.51 0 . 1 6
Strominger, Hollander, and Seaborg.ll Crane, Higgins, and Bowman.22
3.1.1.2. Fission Sources. In certain studies pertinent to nuclear reactor physics, it is desirable to employ sources with a neutron-energy spectrum similar to that of the fission neutrons. Such "mock fission" sources have been made using Po2lo a-particles on a mixture of light-element target materials.2O Now that heavy isotopes of the transuranium elements can be produced by multiple neutron capture in high-flux nuclear reactors, it has become possible to use heavy nuclides which decay by spontaneous fission as sources of fission neutrons. The use of spontaneous fission neutron sources should increase as reactors of higher flux become available for production of larger quantities of these heavy nuclides. A summary of pertinent data on several nuclides which may prove useful as spontaneous fission sources is given in Table III.21~22 Partial half-lives for a- and spontaneous-fission decay were taken from the nuclear data tabulation of Strominger, Hollander, and Seaborg.21The ratio of a-tofission half-life was multiplied by lo6 so that various nuclides could be compared with each other, and with the yields of Table I. It is important t o note that a high yield of neutrons per a-particle implies that the neutron spectrum will be relatively free from (a,n) reactions. 3.1.1.3. Photoneutron Sources. Photons of energy greater than the neutron binding energy can induce neutron emission from nuclei. The y-energies from nuclides of useful half-life are less than 3 MeV. Since the 1 0 E. Tochilin and R. V. Aves, Neutron spectra from mock fission sources. U.S. Report USNRDL-TR-201 (1958). *I D. Strominger, J. M. Hollander, and G. T. Seaborg, Revs. Modern Phys. SO (2), Part 11, 584-904 (1958). 59 W. W. T. Crane, G. H. Higgins, and H. R. Bowman, Phye. Rev. 101,1804 (1956).
3.1.
563
RADIOACTIVE SOURCES
(y,n) thresholds for deuterium (2.226 0.003 MeV) and beryllium (1.666 k 0.002 MeV) are the only ones known below about 6 MeV, radioactive photoneutron sources always use either deuterium or beryllium. Recent reviews of this subject have been made by Wattenberg,2s23 Hanson16and Curtiss.4 The energy of neutrons resulting from the bombardment of deuterium or beryllium by y-rays is, for a given laboratory angle e between the y-ray and neutron, given byg3
En =
[y]
]+
[E, - Q - E'2 2(A - 1)931 931A3
6 cos 0
(3.1.7a) (3.1.7b)
where En is the neutron energy, E, the y-ray energy, Q the neutron binding energy, all in MeV; A is the mass number of the target. To obtain the greatest intensity from practical sources, the y-ray source is usually surrounded by heavy water or beryllium, so the neutrons have an inherent spread AE, corresponding to an isotropic distribution in 0, of AEn = 26. ' (3.1.8) For example, a t En = 100 kev, AEn/Encli 4% for beryllium, and ~ 2 5 % for deuterium. I n most sources the considerable quantities of beryllium or heavy water introduce the most serious energy spread. Neutron scattering by the target materials may cause, in addition to a significant energy spread, a downward shift in the average energy of the emerging neutrons. The y-rays may lose energy by Compton scattering and then produce lowenergy neutrons, introducing a further uncertainty in the neutron spectrum. The characteristics of several radioactive photoneutron sources are summarized in Table IV.24-31 23 A. Wattenberg, Photo-neutron sources. National Research Council, Nuclear Science Series, Preliminary Report No. 6. Washington, D.C. (1949). 2 4 B. Russell, D. Sachs, A. Wattenberg, and R. Fields, Phys. Rev. 73, 545 (1948). 26 D. J. Hughes and C. Eggler, Phys. Rev. 7 2 , 902 (1947). 2 6 A. 0. Hanson, Phys. Rev. 76, 1795 (1949). 37 E. SegrB, unpublished work reported in reference 5, page 33. z8R. D. O'Neal, Phys. Rev. 70, 1 (1946). 29 L. G. Elliott, unpublished work reported in reference 5, page 33. 30 R. L. Walker, Los Alamos Scientific Laboratory Report MDDC-414 (1946); 0. R. Frisch and R. L. Walker, Los Alamos Scientific Laboratory Report LA-400 (1945). 3lL. F. Curtiss and A. Carson, Phys. Rev. 76, 1412 (1949).
3.
564
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
TABLE IV. Properties of Some Photoneutron Sources EW (Mev)
Source
y-Rays Per decay
(Mev)
Reference
En,*
Na24+Be Na24 f D
15.0hr 15.0hr
2.754 2.754
1.00 1.00
0.83 0.22
2 2
Mn66 + B e
2.58hr
1.81 2.13 2.65 2.98 2.65 2.98
0.28 0.20
0.15 0.30
25 25
Mn66+D
2.58hr
Gal3 + B e
14.3 hr
+D
14.3hr
Ga72
0.005 0.005
Standard yield, ReferX lo-' ence 29 14
24 24
2.9
24
0.31
24
5.9 3.7
24 23
6.9 4.6
24 23
(1-08)
0.22
2
1.87 2.21 2.51 2.51
0.08 0.33 0.26 0.26
(0.18) (0.48) (0.75) 0.13
2
0.99 0.01 0.01
0.16 0.22 (0.26)
26 28 -
+ Be Yaa + D
104day
1.83 2.73 2.73
In116m + B e
54.0min
2.09
0.25
0.30
25
Sb124 + B e
60.9 day
1.69 2.09
0'51} 0.07
0.024
26
La140+ B e La14'+D
40.2day 40.2day
2.52 2.52
0.01 0.01
0.62 0.13
2 2
0.23 0.68
24 24
(Th228)*+ B e
1.91 yr
(Th228)b + D
1.91 yr
1.7-2.4 <0.01 2.62 0.35 2.62 0.35
0.83 0.20
26 26
3.5 9.5
29 29
1622 yr 1622yr
Complex 2.42
-
-
-
0.02
0.12
28
3.5 0.1
31 30
YES
(Raaz6)b
+ Be
(Ra2z6)b+ D a
104day
~~
10
27
0.3
27
0.82
24
19
24
~
Energies in parentheses are calculated values.
* In equilibrium with its decay products.
3.1.
RADIOACTIVE SOURCES
565
Because neutron yields may be measured for a variety of source arrangements, it is convenient to compare sources in terms of the yield for one gram of beryllium metal or one gram of heavy water at a distance of one cm from a one-curie y-source. Neutron yields from practical sources can be calculated in terms of this standard yield by
Y
=
4lrptY,
(3.1.9)
where t is the effective thickness of target surrounding the source, p is the density, and Y , is the standard yield. Table IV includes standard yields measured for several y-ray sources. The Sblz4-y-Be source is one of the most convenient and important photoneutron sources. It has the highest yield of any source except NaZ4-y-Dz0;its half-life of 60 days is long enough to be useful, and the Sb neutron-capture cross section is sufficient to permit reactivating the Sb y-source by irradiation in a nuclear reactor for a few days. Before strong, artificially-produced y-sources were available, Ra-y-Be sources were used as sources of intermediate-energy neutrons. Although t,he neutron spectrum is complicated, the long half-life and reasonable yield make Ra-y-Be sources very attractive as permanent standards, especially since such a source does not require mixing the radium and beryllium, and so the neutron intensity should not vary with time because of physical changes in the source. The sources of the National Bureau of Standards contain a pressed RaBrz pellet at the center of a 4-cm diameter sphere of beryllium metal. An added advantage of these sources is that their neutron yields are not affected by the PoZl0content, and should exhibit only the 1622-year half-life of Razzs. 3.1.1.4. Calibration of Radioactive Neutron Sources. It is frequently desirable to know the absolute yield of a neutron source. If a standard source is available, an accurate comparison between the standard and unknown can be made, if they are both counted under identical conditions by a detecting system whose efficiency is independent of energy. Should it become necessary to determine the absolute number of neutrons emitted from a source without the help of a standard, either a water bath method or a critical reactor method is usually appropriate. One of the associated narticle methods may be used if it is possible to assemble a neutron source of special design. These methods will be described briefly. In the water bath methods, a source is located at the center of a thermalizing medium (frequently water or an aqueous solution) which is large enough to capture nearly all the neutrons, and the capture rate throughout the medium is determined by absolute counting of samples placed within the medium. When a boric acid solution is used, most of the neutrons are stopped in boron, and absolute measurements depend
566
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
on counting the total number of disintegrations occurring in a sample of BF3gas of known volume,30 or on quantitatively measuring the helium evolved in the B10(n,a)Li7reaction.32Other variations of this technique involve determining the absolute disintegration rates of radioactive nuclides such as MnS6,Inl16, or AulgS.Corrections must be made for loss of neutrons to hydrogen and oxygen. The loss of neutrons by the (n,a) reaction in oxygen has been reported to be 3.4 f 2.1% for Po-a-Be sources,33and 2.5 f 0.3% for Ra-a-Be.34 A more detailed discussion of water bath methods may be found in an account of neutron standardizations at the Los Alamos Scientific Lab0ratory.3~ When a neutron source is inserted into an operating reactor, it will have two effects: the material from which it is made will absorb thermal neutrons, and the source will emit fast neutrons. The power will rise if there is a net emission of neutrons, and, conversely, will decrease if there is a net absorption. This is the basis of the critical reactor method. The effect on the multiplication factor of introducing a source into the reactor is not very sensitive to the energy of the neutrons from the source, although a correction must be made for the resonance escape probability, i.e., the probability that a fast neutron will escape resonance capture in UzSsand become a thermal neutron. A further correction is due to slow neutron absorption by the source and its container. The technique depends upon absolute beta counting, which may be performed to better than 2%; however, the over-all error is probably 15%.38 The associated particle method measures the yield by counting the particles associated with the reaction which produces the neutron. An early proposal for such a technique is due to Glueckauf and Paneth,*’ who suggested that an accurate, absolute yield measurement could be performed on a beryllium photoneutron source by measuring the helium from the Bee(r,n)2He4 reaction. Martin reportedas that it should be possible to calibrate photoneutron sources to better than 1% by this method. I n calibrating the Oxford photoneutron source,88the number F. G. P. Seidl and S. P. Harris, Rev. Sci. Instr. 18, 897 (1947). asL. G. Elliott, E. P. Hincks, and A. N. May, Can. J . Research 26A, 386 (1948). a4 A. DeTroyer and G. C. Tavernier, Bull. acad. roy. Belg. [5] 40, 150 (1954). ssR. W. Dodson, A. C. Graves, L. Helmhola, D. L. Hufford, R. M. Potter, and J. G. Povelites, i n “Miscellaneous Physical and Chemical Techniques of the Los Alamos Project” (A. C. Graves and D. K. Froman, ed.), Chapter 1. McGraw-Hill, New York, 1952; A. C. Graves, R. L. Walker, R. F. Taschek, A. 0. Hanson, J. H. Wdliads, and H. M. Agnew, ibid., Chapter 2. D. J. Littler, Proc. Phys. SOC.(London) A64, 638 (1951). E.Glueckauf and F. A. Paneth, Proc. Roy. SOC.A166, 229 (1938). a* D. J. Littler, ed., Report on a Conference on Neutron Source Preparation and Calibration, July, 1954. British Atomic Energy Report AERE NP/R1577 (1956).
3.1.
RADIOACTIVE SOURCES
567
of protons emitted by the D2(y,n)H1reaction was measured in a deuterium-filled ionization chamber. This technique may be applied to a-particle sources if the product nuclei can be determined absolutely. For example, Martin has proposed3* that a Po-a-F source could be calibrated by measuring the yield of either NaZ2or stable NeZ2which are formed in the reaction
5 NeZ2.
Flg(a,n)Na22
(3.1.10)
Because of a small uncertainty in the ratio of electron capture to positron emission in the decay of Na22, the determination of NeZ2is preferred. 3.1.2. Radioactive a-Particle Sources
The energy or momentum scale of a heavy particle spectrometer is frequently calibrated using a-particles of accurately known energy. Alpha-particle sources with a standardized disintegration rates are often needed for calibrating the efficiency of a-counters. Because of these important applications, the need for standard a-sources has shown a rapid increase in recent years. 3.1.2.1. Source Preparation. Because of their high mass and charge, a-particles exhibit a short range in matter; therefore, sources of a-particles should be as thin as possible. Backscattering from a thick source mount is small and can easily be determined, so an a-source is usually mounted on a metal plate for ease of preparation and handling. The most obvious method of source preparation is evaporation of a dilute solution on a metal plate. A known volume of stock solution is transferred to the plate and carefully evaporated t o dryness under an infrared heat lamp. To produce a thin source by this technique, it is essential that the solution contain no chemical compounds which will contribute more mass to the final deposit than the radioactive material itself; thus the choice of the chemical form for the radioactive element and the chemical purity of all the reagents used are most important. If the volume of solution to be evaporated is very large, it can be confined to the desired region by a border of Zapon lacquer. Later, the Zapon and any small amounts of volatile impurities can be removed by ignition in an induction heater or open flame, providing that the radionuclide is present in a nonvolatile form. Reviews by Dodson et ~ 1and. by~ Hufford ~ and contain useful information on extensions of this technique to solutions of low surface tension, sulfuric acid solutions, and the use of spreading agents. One very successful spreading technique (Dodson 39 D. L. Hufford and B. F. Scott, in “The Transuranium Elements” (G. T. Seaborg, J. J. Katz, and W. M. Manning, eds.), National Nuclear Energy Ser. Div. IV, Vol. 14b, Paper 16.1. McGraw-Hill, New York, 1949.
568
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
et ~ 1 . 3 ~calls ) for mixing an alcoholic solution of the nitrate of the substance to be deposited with a dilute solution of Zapon lacquer in Zapon thinner, alcohol, or acetone. The mixture is spread or painted on the foil backing, allowed to dry, and then ignited to destroy the organic material and convert the nitrate to oxide. After each ignition, the deposit is rubbed with tissue to insure that subsequent coats will adhere. Uniform deposits with smooth vitreous surfaces can be prepared by application of many thin coats, the only requirements being that the element deposited have a nonvolatile compound, soluble in alcohol, and that the foil have a melting point high enough to permit ignition. Some of the thinnest and most uniform deposits are prepared b y vacuum sublimation. This method can be applied whenever it is possible to obtain a compound of the radioelement whose vapor pressure is a t least 0.1 mm a t a temperature below that for rapid decomposition. Typical procedures for some heavy elements may be found in references 35 and 39. Most procedures for subliming a deposit onto a source plate are time-consuming and have a low yield. A device for subliming from a crucible to a thin film with nearly 100% yield has been described b y Pate and Yaffe.40 Their design offers the possibility th a t a-sources can be made by sublimation which are not only thin and uniform, but also contain known aliquots of a stock solution. The most convenient technique, apart from simple evaporation of a solution on a backing, is electrodeposition. Excellent uniform films of a number of heavy elements ranging from trace amounts to a few mg/cm* have been electrolyzed onto a number of metals and alloys. The procedure used to obtain good deposits depends on the element in question. Some of the procedures used for polonium, thorium, uranium, neptunium, and plutonium have been reported in the published records of the Manhattan Project,36~a0*41 and a recent article by Ko42gives electrodeposition procedures for all the actinide elements through curium. 3.1.2.2. Methods of Assay. The disintegration rate of a n a-particle source is generally determined using either a counter of large solid angle for weak sources or a low geometry counter for assay of intense sources. I n a typical counter for sources of low intensity, the flat sample foil is placed inside a proportional counter or parallel-plate ionization chamber; thus, the sample exhibits a 2~ geometry and, except for small corrections, half of the emitted a-particles will be counted. The necessary B. D. Pate and L. Yaffe, Can. J . Chem. 94, 265 (1956). C. C. Casto, in “Analytical Chemistry of the Manhattan Project” (C. J. Rodden, N. H. Furman, E. H. Huffman, L. L. Quill, T. D. Price, and J. I. Watters, eds.), National Nuclear Energy Ser. Div. VIII, Vol. 1, Chapter 23. McGraw-Hill, New York, 1950. 4 1 R.KO, Nucleonics 16 (I), 72 (1957). 40
41
3.1.
RADIOACTIVE
SOURCES
569
corrections arise from self-absorption in the source and scattering from the backing. With reasonable care it is usually possible to keep the selfabsorption correction very small or even negligible; however, whenever a foil is counted in a 2s chamber, more particles are found entering the counting volume than expected from the geometry and self-absorption. This excess of counts is due to back-scattering, and will depend upon the particular backing used, the state of the backing surface, and the a-particle energy. A typical counting yield for polished platinum backing in a 2a counter is about 51.3%43 although it is not uncommon to measure counting yields in excess of 52%.44 The determination of backscattering and self-absorption corrections for absolute alpha counting is discussed by Curtis el al.44 A logical extension of 2a counting is the use of a 4~ (100% geometrical efficiency) counter. Pate and Yaffe,46 using a 4n proportional counter, obtained good counting-rate plateaus for a Po21o source mounted on a thin plastic film. In this situation, it is only necessary to apply a small absorption correction, since all the particles are counted, regardless of direction. Lyon and Reynolds43have applied a 4s a-y coincidence counting to the calibration of a-sources. The coincidence method has the advantage that it is independent of self-absorption effects and counter efficiencies, but has the disadvantage of requiring detailed knowledge of the decay scheme of the a-emitter. A 4a geometry can also be attained by incorporating the a-radioactive material in a liquid scintiIIator.4 6 A variety of medium-geometry a-detectors with end-window proportional counters or scintillation counters are used in various applications. These are generally calibrated against a standard a-source of accurately known disintegration rate. Low-geometry counters are used for assay of high-level a-sources. These consist of an evacuat,ed tube containing several antiscattering baffles, with a plug on one end for accurately positioning the source. The detector end contains a carefully machined collimator through which the a-particles pass into the detector, which may be a proportional counter, ionization chamber, or scintillation counter. If the critical dimensions are large, then it is possible to construct a chamber whose geometry can be calculated to better than 0.1% from accurate measurements of the 43
44
W. S. Lyon and S. A. Reynolds, Nucleonics 14 (12), 44 (1956). M.L. Curtis, J. W. Heyd, R. G. Olt, and J. F. Eichelberger, Nucleonics 13 (5),
38 (1955). 45
B. D.Pate and L. Yaffe, Can. J . Chem. 33, 610 (1955).
W. H. Langham, in “Liquid Scintillation Counting’’ (C. G. Bell, Jr., and F. N. Haynes, eds.). Pergamon, New York, 195s; see also International Atomic Energy Agency, Proc. Symposium on Melrologu of Nuclides, Vienna, Austria, Oct. 14-16, 1969, International Publications, New York, 1960. 46
570
3.
SOURCES OF N U C L E A R PARTICLES AND RADIATIONS
components. For thin, uniform sources, the counting rate a t the detector depends only upon the geometry factor, because significant numbers of backscattered a-particles emerge only at large angles to the axis and thus are stopped by the baffles. At present, low-geometry a-counting is the most accurate technique for determination of a-disintegration rates: with care, an accuracy of about 0.1% can be obtained. For example, assays of the same sample by workers a t the AERE, Harwell, England, and UCRL, Berkeley, California using low-geometry counters of different design, agreed to 0.1 %.47 Information on the design of low-geometry counters and some practical considerations about their use will be found in articles by Curtis et aZ.44 and Robinson.48 Because a-particles have a short range in matter and readily transfer all their energy to the source container, the heat output of an intense a-source is related to its disintegration rate. Corrections are necessary if y-rays accompany the a-decays, for y-ray absorption will contribute to the measured heat. Although the calorimetric method is, in principle, very accurate, measurement of the activity depends on the average energy release of the nuclide, which may not be known very well. The technique is very well-suited t o assay of high-level a-sources whose average energy release is known, and t o precise comparisons between samples of the same nuclide. A comprehensive bibliography on calorimetric applications to nuclear measurements has been compiled by Myers4gin his 1949 review article. 3.1.2.3. a-Particle Standards. The National Bureau of Standards a t present supplies two types of standards for determining a-particle counting yield. One such standard is a uranium oxide source with an emission rate of about 10 a-particles per second into the forward hemisphere, as counted in a 2r geometry. The other type of standard consists of Po21° electrodeposited onto palladium foil, and may be obtained up to strengths of a thousand a-particles per second into the forward hemisphere. Both standards are intended as reference sources for which the accuracy is & 5 % , although by special arrangement the Bureau will prepare Eources to f 2 % . It is possible that other standards such as AmZ4l and Pu23s will be made available in the future. The a-energy standard used most frequently is Potlo. Wapstra, Nijgh, and Van Lieshout60have evaluated the published measurements of the K. M. Glover and G. R. Hall, Nature 173,991 (1954). H. P. Robinson, i n “Measurements and Standards of Radioactivity, Proceedings of an Informal Conference.” Natl. Acad. 8ci.-Natl. Research Council Publ. No. 678 (1958). 49 0. E. Myers, Nucleonics 6 (ll),37 (1949). a* A. H. Wapstra, G . J. Nijgh, and R. Van Lieshout, “Nuclear Spectroscopy Tables,” p. 128. North Holland Publ., Amsterdam, 1958. 47
46
3.1.
57 1
RADIOACTIVE SOURCES
TULE V. Primary a-Energy Standards Nuclide
Energy, (Mev)
Reference
Po210 BiZ12(ThC)
5.3042 f 0.0016 cui6.0498 f 0.0018 (~06.0897f 0.0018 7.6895 f 0.0023 8.7864 L- 0.0028 5.8012 f O.O0Zo
50 50 50 50 50 51
Po214(RaC') Pozlz(ThC') Cm244
a Value for Cmz44 was recalculated from reference 51, by using 5.3042 Mev for the Po21O energy.
TABLE VI. Useful Secondary a-Energy Standards Nuclide POZ'O
E,, Mev Abundance, % ' 5.304
100
Half-life 138 day ~~
Production" Nat.
~
~h23y10) 4.682 4.615
76 24
8.0 X lO4yr
Nat., sep. isotope
UP34
4.768 4.717
72 28
2.48 X 106 yr
Nat., sep. isotope
PUP38
5.495 5.452
72 28
86.4 yr
Daughter Cm242,sep. isotope
Pu239
5.147 5.134 5.096
72.5 16.8 10.7
2.436 X lo4yr U238(n,y)
Pu"2
4.898 4.858
76 24
3.79 X 106 y r
Cm242
6.110 6.066
73.7 26.3
162.5 day
Cmz44
5.801 5.759
76.7 23.3
17.9 yr
Mult. n capt.
E263
6.633 6.592
90 7.7
20 day
Mult. n capt.
~
~
_ _ _ _ _ ~ ~
~-
Mult. n capt. AmZ4l(n,y)
-
Methods of production: Nat. = a member of a natural decay series. Sep. isotope available as an electromagnetically enriched isotope. (n,y) = radiative neutron capture, usually followed by one or more beta decay steps. Mult. n capt. = multiple neutron capture, starting with Pu*a9. a
=
572
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
Po2l0a-energy and suggest 5.3042 L- 0.0016 Mev as the best value. Five nuclides whose energies are known with sufficient accuracy for use as primary standards are included in Table Vf~lin decreasing order of reliability. All energies were determined by magnetic deflection. Where primary standard accuracy is not required as, for example, in the calibration of pulse ionization chambers, the nuclides listed in Table VI will be found useful. These secondary standards were selected, mainly from the review by Strominger, Hollander, and SeaborgZ1by application of the following criteria: (a) (b) (c) (d)
Half-life greater than 1 week. Accurately known energy. Availability. Simplicity of decay scheme. (1) Few alpha groups > 1% ’ abundant. (2) Absence of intense internal conversion electrons which would tend to broaden a peak when using a pulse ionization chamber.
3.1.3. p- and y-Ray Sources Nuclides which decay by emission of nuclear p- or y-rays frequently must be counted in experimental situations which are far from ideal. Standard sources are then needed to determine either the counting yield or the energy scale of the apparatus, or both. 3.1.3.1. Source Preparation. The method used to prepare a source for beta counting depends on the purpose of the measurement. If the active material is in a carrier-free solution, an aliquot may be evaporated to dryness on a thin plastic film, and the source may then be counted using a 4xp counter or a counter with calibrated geometry. Pate and Yaffe40 have described a device for vacuum evaporation of aliquots of certain radioactive compounds onto thin films with almost a 100% yield. In some instances the source can be made very thin by using an electrodeposition procedure, but backscattering from the source mount becomes important. Frequently, the radioactivity is present in solution with large amounts of carrier. Sources are then prepared by precipitating the carrier in an appropriate chemical form, after which the precipitate is transferred onto either a filter paper, a watchglass, or a metal planchet. The chemical yield is determined by weighing and the counting geometry is calibrated for this type of source mount, taking into account the self-scattering and 6 1 F. A. White, F. M. %urke, J. C. Sheffield, R. P. Schuman, and J. R. Huizenga, Phys. Rev. 109, 437 (1958).
3.1.
RADIOACTIVE SOURCES
573
self-absorption of the thick source, and the scattering from the thick backing. An extensive literature has become available on the preparation of beta sources; some valuable suggestions may be found in references 52, 53, and 54. Although any of the methods of preparing beta sources can be used to prepare y-sources as well, liquid or other unusually thick sources can be used to advantage with y-counters because y-rays are absorbed and scattered to a lesser degree than ,&particles. A convenient mount for y-sources is made from a disc of blotting paper or a chemical “filter accelerator” taped to a card. An aliquot of the solution to be counted is simply transferred to the absorbent paper and allowed to saturate it. After the liquid is evaporated under an infrared heat lamp, the source may be covered by cellophane tape. Solutions contained in small, biological test tubes may be assayed very conveniently in a well-type NaI y-ray scintillation counter, or in a y-sensitive ionization chamber. 3.1.3.2. fi-Assay Methods. If an absolute determination of a 0-activity is to be made using a n end-window counter, the accuracy obtained will depend greatly on the details of the counting arrangement. The “counting yield” of an essentially weightless source on a thin backing can be standardized to 5% or better, provided that an absolute standard of the activity is available, or if the geometry, absorption, and scattering effects for the activity have been determined. An additional correction for y-ray background is necessary when counting 0-y sources. By determining an absorption curve for the sample, it is possible to obtain the y-counting rate with the p-rate excluded, and to correct the y-rate beyond the &range to zero absorber thickness. This method generally leads to a high value for the y-background, because all of the y-counting rate has been used, even though some coincident y-rays are detected simultaneously with the &particles, and hence would not increase the beta counting rate. Another objection is that y-scattering and production of secondary electrons by the @-absorberlead to a further uncertainty in the y-background. These corrections usually are small, except for nuclides such as CoB0,for which the 0-energy is low and the y-energy and intensity are high. Calibration standards must be used when assays of thick &sources are desired to better than 10%. A carrier-free source of the activity should be prepared and its absolute disintegration rate determined. Aliquots of this source are then processed in the same way as the unknown, taking B. P. Bayhurst and R. J. Prestwood, Nucleonics 17 (3), 82 (1959). G. Friedlander and J. W. Kennedy, “Nuclear and Radiochemistry” (revised ed.), pp. 278-282. Wiley, New York, 1955. 6 4 Oak Ridge National Laboratory, “Master Analytical Manual,” Sections 2, 3, and 5. U.S.Atomic Energy Commission Report TID-7015 (November, 1957). 62
63
574
3.
SOURCES OF NUCLEAR PARTICLES A N D RADIATIONS
care that the amounts of carrier, sample mounting procedures, and other experimental details reproduce the treatment of the unknown. An over-all counting yield may be obtained by comparing the counting rate of the calibration standard with the known disintegration rate of the activity added. Routine @-countingusing a 27r geometry is very convenient, especially if weak sources requiring high detection efficiency are to be measured. An additional advantage is that the anisotropic scattering effects are of less importance in this high geometry case. The internal sample (or “windowless”) counters are often necessary if low-energy 8-rays are to be determined. The general procedure for calibration of the counting yield is similar to that described above for end-window counters. The most widely used technique of primary standardization of 8emitters is the 4 ~ counter, 8 operating either in the Geiger or in the proportional regions. Since with the 4n/3 counter it is possible to record the total rate of emission of @-particles in all directions, coincident y-rays such as annihilation radiation are always detected with a 8-particle, and result in a single count. Likewise, the measured disintegration rate is not affected by scattering, because any discharges which are caused by repeated scattering of the primary particle, and which follow the main event, will occur within the resolving time of the counter. The same is true of secondary radiation produced within the counter. Besides the usual 47r counter using twin, facing 27r counters, various attempts have been made to achieve a 4?r geometry free of source absorption by introducing the source either as a gas in a gas-filled counter or as a liquid in a liquid scintillation counter. In applying either of these techniques, a counting efficiency correction is required because some of the 8-particles strike the walls of a gas counter without producing any primary ionization; an analogous “wall-loss” occurs in a scintillation counter with dissolved source, unless the scintillator dimensions are very large with respect to the 8-range. A t present, the most accurate and reliable 47rP counters operate in the proportional region. A carefully designed proportional counter, used with low-noise amplifiers having good overload characteristics, can yield counting rate-voltage plateaus 500 to 800 volts wide, with a slope of less than 0.1%/100 volts. Geiger counters, on the other hand, require less electronic instrumentation, but because they exhibit relatively narrower and steeper plateaus, 47rp Geiger counters are not generally considered to be as precise as the proportional type. A carefully designed 47rP proportional counter has a counting efficiency in excess of 99.5%, so the principal corrections arise from absorption in the source and in the mounting film. Indeed, self-absorption in the source
3.1.
RADIOACTIVE SOURCES
575
is and shall probably remain the factor which limits the accuracy of 47r counting. Seliger and SchwebelS6reported that even samples which gave the best 477 counting results were shown by microscopic examination to consist of lumps of varying size. They did not, however, make a systematic attempt to measure absorption losses. The source problem has also been treated by Pate and Yaffe140who measured the self-absorption of Nina,&radiation (67 kev) in nickel dimethylglyoxime of 0-250 pg/cm2 superficial density, and showed how quantitative corrections could be applied in other 4?r counting problems. Three techniques have been used to determine the source-mount absorption correction : (1) A so-called “sandwich” procedure used by Hawkings et a1.,66in which the counting rate of a source on a known thickness of backing is measured, followed by a determination made with an identical film covering the sample; (2) Seliger and co-workersS7proposed the calculation of the absorption correction using measurements of 27r and 4?r single film and sandwiched film counting rates, together with mathematical relationships developed using simplified assumptions of scattering and absorption effects; (3) A determination of the counting rate as a function of the actual mount thickness was first made by SmithlS8and this “absorption curve” approach has been the subject of an extensive study by Pate and Yaffe.69Any of these methods is satisfactory for counting /3-rays of energy greater than a few hundred kev, but a t the lower energies the absorption curve technique appears to be the most accurate. Pate and Yaffe estimate that the error in the disintegration rate arising from their source mount absorption correction does not exceed 0.2%, even a t 67 kev. Films to be used as source mounts should be rendered electrically conducting by a metallic coating a t least 2 pg/cm2 surface density applied by vacuum evaporation. This treatment will serve to prevent distortions in the electric field of the counter, which may arise either from source charging or from penetration of the electric field of one counter through the source aperture into the other, thus creating a region of low collecting field near the source. The general technique of 4?r@counting has been systematically studied H. H. Seliger and A. Schwebel, Nucleonics 12 (7),54 (1954). R. C. Hawkings, W. F. Merritt, and J. H. Craven, Proceedings of Symposium on Maintenance of Standards, National Physical Laboratory, 1951. H. M. S. O., London, 1952. 67 H. H. Seliger and L. Cavallo, J . Research Natl. Bur. Standards 47, 41 (1951); W. B. Mann and H. H. Seliger, J . Research Natl. Bur. Standards 60, 197 (1953). 6s D. B. Smith, 4 Pi Geiger Counters and Counting Technique. British Atomic Energy Research Establishment Report AERE-I/R-1210 (1953). B. D. Pate and L. Yaffe, Can. J . Chem. 33, 929 (1955). 66
SE
576
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
by Pate and Yaffe.40.46J9060Other helpful information may be found in papers by Seliger and c o - w o r k e r ~ ~ ~ who ~~7 describe ~ ~ 1 measurements performed at the National Bureau of Standards and the results of international comparisons of samples. It is possible a t present to obtain a reproducibility of about 0.5%, and an accuracy nearly as good in favorable cases. In cases where a p-particle and a y-ray are emitted in sequence, 8-7 coincidence counting can be a very convenient and accurate method of absolute disintegration rate measurement. Consider a radioactive source which decays by emission of a single beta transition followed by a single unconverted y-ray, placed between a 8-counter and a y-counter. The counting rate of the p-counter Np is given by
Np = Noeg
(3.1.11)
where NOis the disintegration rate, and q is the counting yield of the 8-detector. Similarly, the y-counting rate N , is given by N y
=
NO!,
(3.1.12)
where e,. is the counting yield of the y-detector. The coincidence rate N , is No = Nocge, (3.1.13) which by use of Eqs. (3.1.11) and (3.1.12) reduces to
No
=
NgN,/N,.
(3.1.14)
Neither the source self-absorption nor the counting yields of the detectors need be known for this simple case, an attractive experimental simplification. The counting rates in the 8- and y-channels must be corrected for dead-time losses, if any, and for background rates. The background rate of the @-channelmust include the contribution arising from the y-sensitivity of the 8-detector, and is measured by use of a @-absorberas described above for conventional 8-counting. The coincidence counting rate must first be corrected for random coincidences in the customary way, i.e.,
N,
=
2rNgN,
(3.1.15)
where 27 is the resolving time of the coincidence circuit, and Ng and N , B. D. Pate and L. Yaffe, Can. J . Chem. 33, 1656 (1955). W. B. Mann and H. H. Seliger, Preparation, maintenance and application of standards of radioactivity. Natl. Bur. Standards (U.S.) Circ. NO.694 (1958). 60
61
3.1.
RADIOACTIVE SOURCES
577
are the uncorrected counting rates. Further, a coincidence background must be removed, which may be due solely to environmental radiation, or, in more complicated cases, to y-y coincidences from y-ray cascades in the source. A correction must also be applied for angular correlations, if they exist. Although this discussion used a simple example, the method may be adapted to other situations, such as complex decay schemes, 6-electron coincidence counting, X-y coincidence counting, and y-y coincidence counting. The P-y coincidence method was shown by P u tma n 6 z 6 3to be valid for nuclides with several @-groupsprovided th a t the sensitivity of either the P- or y-detector was the same for all branches of the decay scheme. The 4743 counter is the logical approach to this condition. Because 4 ~ 0 - y coincidence counting is nearly independent of adsorption in the source and its mount, the technique has been used to calibrate the counting . ~ ~ counting ~~~ nuclides with yield of thick sources in a 4 ~ cp o ~ n t e rWhen complex decay schemes by the 4rp-y coincidence technique, the corrections frequently turn out to be relatively insensitive to the details of the decay s ~ h e m e . So ~ ~ with * ~ ~ quantitative information about a decay scheme, it may be possible to standardize sources by 474-7 coincidence counting to a few tenths of a per cent.B4 3.1.3.3. y-Assay Methods. Gamma-ray counting by spectrometry a t a defined solid angle has been discussed in Section 2.2.3.3. Integral counting arrangements using y-counters a t arbitrary geometries or NaI wellcrystal counters without differential pulse-height analysis must be calibrated using sources of known disintegration rates. High-pressure ionization chamber^^^,^^ make excellent devices for the secondary standardization of y-emitters. Gamma-gamma coincidence counting has been applied a t the National Bureau of Standards to the calibration of Gos0 sources, and results which are repoducible to 0.1%have been obtained.Bs J. L. Putman, Brit. J . Radiol. 23, 46 (1950). L. Putman, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), Chapter 26. Interscience, New York, 1955. 6 4 P. J. Campion, in “Measurements and Standards of Radioactivity, Proceedings of an Informal Conference.” Natl. Acad. Sci.-Natl. Research Council Rept. No. 24,24 (1958);Intern. J . Appl. Radiation and Isotopes 4 , 232 (1959). 66 R. Gunnink, L. J. Colby, Jr., and J. W. Cobble, Anal. Chem. 31, 796 (1959). C. C. Smith and H. H. Seliger, Rev. Sci. Znstr. 24, 474 (1953). 67 J. L. Putman, in “Measurements and Standards of Radioactivity, Proceedings of a n Informal Conference.” Natl. Acad. Sci.-Natl. Research Council Rept. No. 24, 69 (1958). 68 R.W.Hayward, in “Measurements and Standards of Radioactivity, Proceedings of a n Informal Conference.” Natl. Acad. Sci.-Natl. Research Council Rept. No. 24,76 (1958). 82
O3J.
578
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
3.1.3.4. 8- and y-Ray Standards, The calibration of the counting yield for a particular counter is generally accomplished using a disintegration rate standard. The National Bureau of Standards (NBS) produces such radioactivity standards for distribution to interested laboratories. A list of the NBS sources, and details of their standardization and use, were published by Mann and Seligersl in June, 1958; the latest information on available radioactivity standards can be obtained by requesting one of the periodic NBS bulletins on the subject. Most of the NBS standards TABLE VII. Standard Electron Lines
Source Th(B Th(B ‘I’h(B Th(B Th(B Th(B Th(B Th(R
+ C + C”)-A line + C + C”)-B line + C + C”)-F line + C + C”)-I line + C + C”)-Ia line + C + C”)-J line + C + C”)-Lline
+ C + C”)-X line
Electron energy, (kev) 24.510 36.151 148.08 222.22 222.90 234.61 422.84 2526.3
f 0.003 5 0.003 f 0.02
193.59 481.70 624.21 975.9 1164.5 1324.3
f 0.05
kO.02 rtO.02 50.02 rtO.06 f0.4
f0.15 f 0.10 f0.3 f0.3 f0.2
Magnetic rigidity of electron line, (gauss-cm) Reference 534.21 652.40 1388.45 1753.91 1757.07 1811.11 2607.18 9986.7
f0.03 ?c 0 . 0 4 f 0.10 f 0.14 f 0.14 f 0.15 f 0.35 f 1.5
69 69 69 69 69 69 70 71
1618.10 2838.9 3381.28 4657.9 5322.5 5879.4
f 0.20 k0.4
50 72 73 74 75 75
f 0.50
f 1.0 2 1.0 & 0.7
are solutions of long-lived a-, p-, and y-emitters; in addition, the Bureau can also supply standard a-sources mounted on metal disks and pointsource y-standards mounted between layers of polyester tape. Standardized samples of short-lived nuclides are now available commercially, and distribution of these materials has been discontinued by the NBS. The commercial standards are prepared in the same way as the NBS standards, and their accuracy is better than 3%. The KBS continues to maintain primary standards of both long- and short-lived nuclides and to compare them with samples from other national standardizing laboratories. Because these standards are at hand, the NBS also calibrates radioactive samples from laboratories requesting this service, provided that the samples conform to certain physical, chemical, and activity level specifications. Sources emitting internal conversion electron or y-ray lines of accii-
3.1.
RADIOACTIVE SOURCES
579
rately known energy are needed for the calibration of electron or y-ray spectrometers. The data on standard electron lines presented in Table VI169-76 may be used to calibrate various types of electron spectrometers. TABLEVIII. Standard 7-ray Energies" Source Am241 I131
Cd"39 ThB(Pb212) ThB(Pb212) Hg203 1"' I101
Au198 Th C" ( T l z O S ) Positron annihilation Biz07 Cs'37
Mn64 Biz07 c o 60 Na2z coao
Na24 Biz07 ThC"(TI2OS) Na24
7-ray energy, (kev) 59.568 80.164 87.7 115.03 238.62 279.12 284.31 364.47 411.77 510.84 510.976 569.7 661.65 837.9 1063.9 1172.8 1276 1332.6 1369 1771 2614.3 2754
f 0.017 f 0.009 f0.2 f 0.01 f 0.02 f 0.05* f0.05 f 0.05 zk 0.04 f 0.07 f 0.007 f0.1 fO.10 f 0.3" f0.3 f 0.5 f4 f 0.4 f1 f5 f 0.4 f1
Unless otherwise noted, the energy values were taken from data referred to in the Table of Isotopes by Strominger, Hollander, and Seaborg.21 6 From Wapstra, Nijh, and Van Lieshout.60 c From Katoh, Nozawa, and Yoshisawa.76 (1
For calibration of instruments which are linear with energy, such as scintillation spectrometers, the electron energies have been given. Magnetic spectrometers require a calibration of a momentum scale, and for such applications the magnetic rigidities of the lines in gauss-cm are included K. Siegbahn and K. Edvarson, Nuclear Phys. 1, 137 (1956). G. Lindstrom, Phys. Rev. 83, 465 (1951). 71 G. Lindstrom, Phys. Rev. 87, 678 (1952). 72 G. Backstrom, Arkiv Fysik 10, 393 (1956). 73 G. Lindstrom, K. Siegbahn, and A. H. Wapstra, Proc. Phys. SOC.(London) B66, 54 (1953). 74 D. Alburger, Phys. Rev. 92, 1257 (1953). 76 G. Lindstrorn, A. Hedgmn, and D. E. Alburger, Phys. Rea. 89, 1303 (1953). 69
70
580
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
as well. A list of standard 7-ray energies is given in Table VIII,'" and will be found especially useful for calibrating scintillation spectrometers. No energy standards below 60 kev were included, because X-rays are the preferred standards in this energy region, and such energies are well known.
3.2. Artificial Sources* The first use of an artificial source in nuclear physics was in 1932 when Cockcroft and Walton successfully disintegrated lithium by protons of about 400 kev.' Further expansion in energy of direct-voltage accelerators and the development of the cyclotron provided sources for the study of the gross structure of the nucleus. For the study of nuclear level schemes and the structure of liquids and solids, the nuclear reactor has proved to be a prolific source of neutrons of energies ranging from a fraction of a n electron volt up to about 1 MeV. With the development of the synchronous type of accelerators, first a t energies of several hundred MeV, came the production of mesons and then, with the modern multi-Bev machines, the production of antiprotons and hyperons. 3.2.1. Low-Energy Sources
Charged particles and neutrons of energies up to a few Mev are usually obtained from dc (direct potential drop) accelerators which comprise about half of all existing accelerators.2 Although accelerators can serve usefully as thermal-energy neutron sources, neutrons of these energies are more frequently derived from nuclear reactors. The chief advantage of dc accelerators lies in their precision, the voltage spread being of the order of 0.1 % ' or less. They fall into two main classes: (1) those where high voltages are obtained by a cascade process such as the cascade rectifiers or voltage-multiplying circuits and (2) those where charge is actually transported to a terminal as in the belt-charged electrostatic generators. Many high-voltage devices were investigated around 1930 and voltages M. Nozawa, and Y. Yoshizawa, J . Phys. SOC.Japan 13, 1419 (1958). D. Cockcroft and E. T. S. Walton, Proc. Roy. SOC.(London) A137, 229 (1932). 2 G. A. Behman, Nuclear Instr. 3, 181 (1958);6, 129 (1959). 76T.Katoh,
* J.
* Sections 3.2.1.1 and 3.2.1.2 are by M. H. Blewett.
3.2.
ARTIFICIAL
581
SOURCES
of 2 or 3 million volts were reached but, in most cases, it was not possible to connect a discharge tube across the terminals without breakdown a t much lower voltages. Tesla coils and other types of resonant transformers have been found useful for producing X-rays for hospitals but have been supplanted by other machines in nuclear physics. Surge generators, wherein a stack of capacitors is charged in parallel and discharged in series, are not particularly useful since the pulses are so short and the voltage too irregular. For many years, cascade transformers were used for a considerable amount of research a t California Institute of Technology by Lauritsen and his co-workers3but, more recently, these have been replaced by electrostatic generators. 3.2.1.1. Cascade Rectifiers. Direct voltages up to 100-200 kv are rather easily obtained from a simple transformer-rectifier system such as
Fro. 1. Standard transformer-rectifier circuit. The capacitor serves as a filter to maintain constant voltage output.
that illustrated in Fig. 1. Higher voltages can be reached by voltage doubling, usually in several stages as in the Greinacher, or CockcroftWalton, type of circuit shown in Fig. 2. With the rectifiers acting as switches, the two capacitors in each stage are charged on alternate half cycles from the ac source and double the voltage appears across the two capacitors when they are connected in series. In modern installations, a separate stack of filter capacitors, charged through a series resistance, is usually included to reduce the voltage ripple. This circuit, also known as a cascade rectifier, together with an appropriate ion source and discharge tube, is usually termed a Cockcroft-Walton accelerator. Such accelerators can provide dc currents up to about 10 ma and, in air, have been operated up to about 1.5 MeV, each stage contributing about 200 kv. By using a pressure tank, somewhat higher voltages can be achieved but this is not very prevalent. The main advantage of this type of accelerator, compared H. R. Crane, C. C. Lauritsen, and A, Sdtan, Phys. Rev. 46, 507 (1934).
582
3.
SOURCES O F NUCLEAR PARTICLES AND RADIATIONS
with the belt-charged generator, is in the lack of moving parts and in the higher possible currents. Ripple and regulation in voltage multipliers have been analyzed by several authors, e.g., Lorrain et aL4 Ripple is inversely proportional to frequency of excitation and to capacity of the capacitors in the multiplier stack. It is directly proportional approximately to the square of the number of stages of multiplication. Hence, it is desirable to use capacitors of 6V
T
E
V sin w t
1
T Cf
FIQ.2. Voltage-multiplying circuit with R-C filter.
large value, high frequency of excitation, and as few stages as possible. Typical parameters for a 750-kv multiplier are: number of stages. . . . . . . . . . . . . 5, capacity of stack unit.. . . . . . . . 15,000 pF, frequency of excitation. . . . . . . . 600 cycles. Ripple at the terminal of this system is about 400 v, peak to peak, per milliampere of current drawn. This ripple can be reduced to negligible values by the resistance-capacity filter shown in Fig. 2. Typical values for the filter units would be R, = 5 megohms, Rf’= 20 megohms, and C, = 20,000 pF. Voltage measurement and information for voltage regu4 P. Lorrain, R. Beique, P. Gilmore, P. E. Girard, A. Breton, and P. PichB, Can. J . Phys. 58, 299 (1957).
3.2.
583
ARTIFICIAL SOURCES
lation systems is usually accomplished through a wire-wound resistor of high quality connected from the high-voltage terminal to ground. The most serious problems encountered in the use of these systems are associated with the design of the evacuated accelerating tube. Although many successful designs of equipment for generation of high voltage were available in 1932, it remained for Cockcroft and Walton to demonstrate that maintenance of high voltage across an evacuated accelerating tube Voltage dividing resistor
/
Axis
\ t/
Corona rings FIG.3. Section of accelerating tube.
required very careful voltage division and shielding of the insulating walls of the tube from discharges in the vacuum. Modern accelerating tubes follow the design of Cockcroft and Walton and generally have the form shown in Fig. 3. Voltage multipliers can be operated with either polarity and can be used for the acceleration of electrons or positive ions. However, they are most frequently used in nuclear physics laboratories as positive-ion sources. The ion source, which is housed in the terminal of the filter stack, can be of several forms,Kthe most prevalent being a radio-frequency source,6 a cold-cathode P.I.G. ~ o u r c e or , ~ a duo-plasmatron source.8 With any of 6 M. S. Livingston and J. P. Blewett, “Particle Accelerators,” Chapter 4. McGrawHill, New York, 1962. 6 R. N. Hall, Rev. Sci. Znstr. 19, 905 (1948). 7 J. Backus, “National Nuclear Energy Series,” Div. I, Vol. 5. McGraw-Hill, New York, 1949. * C. D. Moak, H. E. Banta, J. N. Thurston, J. W. Johnson, and R. F. King, Rev. Sci. Znstr. SO, 694 (1959).
584
3.
SOURCES O F NUCLEAR PARTICLES AND RADIATIONS
these sources, several milliamperes of protons, deuterons, helium, or other ions can be extracted by a suitable ion-optical s y s t e m g ~and ~ ~then carried through a n accelerating tube along the axis of the filter stack to the ground, or in a horizontal direction to a terminal connected to the ground terminal of the machine. Cockcroft-Walton accelerators of conventional types are commercially available from the Philips Company of Eindhoven, Holland, or from Haefely of Basel, Switzerland. Variants of the cascade-rectifier system have been evolved in the United States by Radiation Dynamics of Westbury, N.Y., and by the High Voltage Engineering Corp. of Burlington, Mass. The Radiation Dynamics “Dynamitron” operates a t a high frequency of about 300 kc and has been redesigned to use much smaller capacitors and a coupling system suited to the higher frequency. Units for various voltages up to 3 Mv are listed in the catalogues. The HVEC “Insulating-Core Transformer” consists of a stack of 30-kv three-phase transformer rectifiers with magnetic cores aligned and insulated from each other. Present ratings are for output voltages to 600 kv. I n 1962, both of these systems were in preliminary operation and showed promise of success. 3.2.1.2. The Electrostatic (Van de Graaff) Generator. Electrostatic or “Van de Graaff”” generators are very widely used for dc acceleration of charged particles and, with suitable targets, as neutron sources to energies in the range between 1 and 5 million volts with a precision of somewhat better than 0.1 %. Currents are available in this range up to about 500 pa for positive-ions and up t o about 1ma for electrons. I n a new I [ tandem” variation, positive-ion energies over 20 Mev are possible but at drastically reduced currents (20 pa or less). Electrostatic accelerators have also found wide industrial application and are available commercially from two firms. The Van de Graaff, belt-charged, machine can be purchased from the High Voltage Engineering Corporation of Burlington, Mass. A modern version of the Wimshurst machine, charged by a rotating, insulating cylinder, is manufactured by S.A.M.E.S. of Grenoble, France. The general principles of the belt-charged generator were known in the latter part of the last century but its practical value for nuclear physics was first demonstrated by R. J. Van de Graaff. l1 Since that time, many
* The name of Van de Graaff, often used as an adjective for these machines, is now used as a trademark by the High Voltage Engineering Corporation. R. W. Allison, el al., Rev. Sci. Instr. 32, 1331 (1961). 10 A. Yokosawa, “Proceedings, International Conference on High Energy Accelerators, Brookhaven, 1961,” p. 385. U.S.Government Printing Office, Washington 25, D.C., 1961. 11 R. J. Van de Graaff, Phys. Rev. 38, 1919A (1931).
3.2.
585
ARTIFICIAL SOURCES
Generating voltmeter
/
c
- Corona point for regulation
Charging belt
J
/ /Housing
Motorf
FIG.4. Schematic drawing of Van de Graaff generator.
developments and modifications have been made which have been summarized by Herb12 and by Van de Graaff, Trump, and Buechner.13 The method of operation is illustrated by Fig. 4.Charge from a 20- or 30-kv supply is sprayed from a row of corona points onto a moving belt which carries it to the field-free region inside a spherical metal terminal. Here, the charge is removed by another row of corona points and flows to the outer surface of the terminal. If this amount of charge is Q, and the la
R. G. Herb, “Handbuch der Physik,” Vol. XLIV, pp. 64-104. Springer, Berlin,
1959. l3
R. J. Van de Graaff, J. G. Trump, W. W. Buechner, Repts. Progr. in Phys. 11, 1-18
(1948).
586
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
capacitance of the terminal to ground is C, the terminal is raised to a potential V = Q / C . As the charge-carrying process continues, the potential of the terminal rises until either the process is stopped or electrical breakdown occurs between the terminal and its surroundings. By means of an accelerating column, the terminal is connected to ground through a series of corona rings either by resistors or corona points. Inside the terminal is located an ion source (or filament if electrons are desired) and, after a small amount of preacceleration to pull the ions to the mouth of the accelerating column, they are accelerated down a beam pipe inside the column by the electric field therein. Early Van de Graaff generators were operated in air a t atmospheric pressure. This required a great deal of space around the terminal and the highest voltages attained were about 1 Mv. Today, the apparatus is enclosed in a pressure tank and operated at pressures of 10 or more atmospheres. Since electrical-breakdown voltage is proportional to pressure, this modification has made it possible to reach potential differences of 5 million volts, and more, in equipment of reasonable size. A modern 5-Mev accelerator is enclosed in a pressure tank about 20 ft long and 8 ft in diameter and the structure may have either a horizontal or vertical axis. I n either case, the terminal is supported from the ground end on insulating rods. As the insulating gas, ordinary air is unsatisfactory since it will support combustion in case of spark breakdown to a combustible component. Instead, a combination of nitrogen and carbon dioxide is frequently used. The superior insulating properties of the electronegative gases of the freon and sulfur-hexafluoride families are offset by the corrosive products they liberate in an electrical breakdown. In systems where breakdown is not expected, sulfur hexafluoride is now sometimes used. The voltage of the terminal is measured to a precision of 1% or better by a generating voltmeter mounted in the wall of the housing opposite the terminal. In this device a rotating vane alternately is exposed to and shielded from the electric field of the terminal. Charge induced on the vane provides an externally available signal whose amplitude is a measure of terminal potential. This signal can now be fed back to the charging supply to control the amount of charge carried to the terminal. For more precise control the energy of the accelerated beam is measured by deflecting it in a magnetic field to fall on an insulated slit system which gives a precise indication of beam deflection and hence of particle energy. Information from this system is fed back through amplifiers to corona points in the wall of the housing opposite the terminal. Charge from these points sprayed toward the terminal can be used to cancel part of its charge and so to drop its potential. With this arrangement terminal voltage can be controlled to better than one part in a thousand.
3.2.
ARTIFICIAL SOURCES
587
The current available from a Van de Graaff machine is determined by the amount of charge which can be carried by the belt. Originally made of paper, belts now in general use are of rubberized fabric, between 20 and 60 cm wide, and with speeds of the order of 1200 ft/min. The charge that can be sprayed on the belt is limited by gas breakdown and so increases with gas pressure. The charging rate for a 60-cm belt at 10 atm is usually about 500pa. The over-all charging rate can be doubled by using the downgoing half of the belt to carry charge from the terminal to ground. Belts also serve another function: to drive a generator in the terminal to provide power for the ion source and the extracting and focusing lens systems. For acceleration of electrons, the terminal contains a conventional electron gun. For positive ions, the more complicated ion-sources enumerated in the preceding s e c t i ~ n ~ must * ~ ~ be ~ . *mounted in the terminal. Control and monitoring of these components can be accomplished by light beams or by microwave links. However, it is more usual, and almost as effective, to control by pulling insulating strings and to monitor by watching meters in the terminal through a telescope. Sources of difficulty in this type of accelerator have been electrical breakdown across the insulating gas or along the support members, or discharge inside the evacuated accelerating tube, the latter being the major offender. This tube, which has the same design as that described in the preceding section, is mounted beside the charging belt between the terminal and the ground plane below the charging supply. Potential along the tube is divided either by a resistor string or by corona points between the corona rings that shield the accelerating tube. Another array of corona rings surrounds the belt and serves to distribute potential between the terminal and ground. It is important that all of these parts be carefully made, kept clean, and precisely placed. Small imperfections can cause loss of available voltage of as much as 50%, or even complete failure. A serious limitation in positive-ion machines has been electron loading from electrons emitted in the accelerating tube. These travel back to the terminal end to produce large quantities of X-rays which then ionize the insulating gas and cause leakage currents to the outside. By the addition of lead shielding at the ion source, TurnerL4has been able to effect considerable reduction in these effects. The positive ions that are usually accelerated are protons and deuterons; hydrogen-3 can also be accelerated but is more frequently used as a target; helium ions and alpha particles require a more elaborate ion source and preacceleration system. With a thick beryllium target, through the Beg(d,n)BLo reaction, the Van de Graaff generator provides a source 14
C. M. Turner, Phys. Rev. 96, 599 (1954).
588
3.
SOURCES OF NUCLEAR PARTICLES AND RADIhTIONS
of neutrons of either thermal or higher energies. For 3-Mev bombarding energy and a current of 200 pa, about 10’2 neutrons/sq cm are obtained. To slow these neutrons down to thermal energies (about 0.02 ev) the target is enclosed in moderating material which may be paraffin, water, or some other hydrogenous material. A great many other reactions have also been studied which produce monoenergetic neutrons over a wide energy range; many details of the various targets and reactions may be found in a review by Fowler and Brolley.16 Energies of accelerated particles in Van de Graaff accelerators can be extended into the 10-30 Mev region (and, perhaps, even higher) through the use of the “tandem” principle proposed by Bennett16 and Alvarez” and improved and reduced to practice by the High Voltage Engineering Corp.18~~~ Figures 5 and 6 illustrate the principles of two-stage and threestage tandem accelerators. In the two-stage tandem, positive ions from a conventional source are passed through an “electron-adding canal” containing hydrogen gas where a fraction of the positive ions are converted to negative ions by the addition of two electrons. The resultant negative ions are separated from the residual beam by an analyzing magnet and accelerated to a belt-charged, positive high-voltage terminal. In the positive terminal, the beam again passes through a gas-filled channel where the gas strips most of the negative ions of the added electrons. The resultant, high-energy, positive-ion beam is again accelerated as it passes from the terminal to ground. The accelerated beam thus has twice the energy of a beam originating a t an ion source in the terminal. However, because of the inefficiency of conversion to negative ions, intensities are only of the order of a few microamperes. Three-stage tandem accelerators are of two types, and Fig. 6 shows the “neutral-beam injection’’ version. The beam from the usual type of positive-ion source is first neutralized by gas and then drifts to a negatively charged terminal. In the negative terminal, electrons are added to make negative ions which are accelerated from the negative terminal to ground. At this high energy, they then enter a second pressure tank for a procedure like that described for the two-stage tandem machine, resulting in a positive-ion beam of three times the energy produced by the terminal voltage. I n principle, this positive-ion beam could be returned to the first machine and accelerated to its negative terminal, thus giving a fourth stage of acceleration, but termination of acceleration within a high-voltage termiJ. L. Fowler and J. E. Brolley, Jr., Revs. Modern Phys. 28, 103 (1956). W. H. Bennett and P. F. Darby, Phys. Rev. 49, 97, 422, 881 (1936). 17 L. W. Alvares, Rev. Sci. Instr. 22, 705 (1951). 1s R. J. Van de Graaff, Nuclear Instr. & Mefh. 8, 195 (1960). 19 P. H. Rose, Nuclear Znstr. & Meth. 11, 49 (1961). 16
18
Positive high-voltage terminal
ou
FIG.5. Two-stage tandem Van de Graaff generator.
Positive high-voltage terminal
Negative high-voltage terminal
I Neutraliz
source
FIG.6. Three-stage tandem Van de Graaff generator.
590
3.
SOURCES O F NUCLEAR PARTICLES A N D RADIATIONS
nal is not very suitable for experimentation. In the alternate, “negativeion injection” form of three-stage tandem, a negative-ion source is housed inside the terminal of a vertically mounted Van de Graaff machine and accelerated from the negative terminal to ground. The negative beam is then bent 90” to enter a two-stage tandem machine for two more accelerations. Two-stage accelerators have already reached over 15 Mev and three-stage machines are expected to provide 25-30 Mev in the near future.
Insulating rotor Negative high-voltage terminal
I
L charging supply
-
FIQ.7. SAMES electrostatic generator for negative polarity.
The principle of the SAMES electrostatic generator is illustrated in Fig. 7. In this generator, the belt of the Van de Graaff is replaced by a ceramic cylinder revolving around its axis in an atmosphere of hydrogen. SAMES generator or accelerator units are available a t voltages up to 2 Mv with currents up to 5 ma and are to be found in many European laboratories for nuclear physics.
3.2.1.3. Nuclear Reactors.* This is a descriptive section which should acquaint the experimentalist a t the reactor with the main properties of the machine which supplies radiation for his use. A more adequate discussion of design methods would draw heavily on the fieIds of metaIlurgy, chemical engineering, and mechanical engineering, with emphasis on
-
* Section 3.2.1.3 is by H.
Kouts.
3.2.
ARTIFICIAL SOURCES
59 1
material properties and heat transfer. It would require a treatment of the theory of neutron transport which is beyond the need of those who regard the reactors as a tool.’ 3.2.1.3.1. ‘GENERAL FEATURES. The steady operation of the chain reaction in a nuclear reactor is a consequence of a balance between competitive yields on the average 2.42 neutrons. When processes. The fission of U236 on the average this many neutrons lead to one further fission in the course of their absorption or loss in any other way from the reactor, the critical condition exists. Only the average behavior will be discussed here. The fluctuations in the fission rate do not have any great influence on the operating research reactor. The nuclear processes which compete with fission in destroying the neutrons in the reactor are: radiative capture in the fissionable nuclei, radiative capture in other nuclei, (n,2n), (n,a), (n,p) processes to a n almost negligible degree. The one other process which must be taken into account in establishing the steady chain reaction is the leakage of neutrons from the reactor. This leakage is the diffusion of the neutron gas outward from the high-intensity source region where it is produced. Leakage could only be avoided in an infinite reactor with a spacially constant source strength and therefore no variation in the neutron density. The operating level of the nuclear reactor is controlled by adjusting the competitive process to a balance, once the desired point is reached. Usually control is accomplished by adding or subtracting material which has a high-capture cross section. This material is contained in so-called poison control elements. More rarely, the amount of fissionable material in the core can be varied by means of insertion or removal of fuel control elements. Sometimes the fuel and the poison form successive sections of a single control element. These elements are said to have fuel followers. As poison is inserted, fuel is removed, and also conversely. Some reactors achieve neutron balance control through variation of the leakage probability. These are usually fast reactors, in which the mean energy a t which neutrons are absorbed is several kev or more. I n this region, poison elements are less useful, because the radiative capture cross sections are not especially large. 1 A partial bibliography of this subject is as follows: A. M. Weinberg and E. P. Wigner, “The Physical Theory of Neutron Chain Reactors.” Univ. of Chicago Press, Chicago, Illinois, 1958; S. Glasstone and M. Edlund, “The Elements of Nuclear Reactor Theory.” Van Nostrand, New York, 1952; H. Etherington, Ed., “Nuclear Engineering Handbook,” 1st Ed., McGraw-Hill, New York, 1958.
592
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
Most research reactors are of the thermal variety, with the adjective referring to the energy distribution reached by the neutrons before they are absorbed. In a thermal reactor, the neutrons are slowed down (moderated) by nuclear scattering until they are nearly in thermal equilibrium with the kinetic motion of the nuclei in the reactor core. In this region, the large fission cross section of U2s6 (the common research reactor fuel) is relatively favorable for the fission process. The real reason for the dominance of thermal systems among research reactors is not this advantage, which is minor. It is that most of the research of present interest which can use the high intensities of neutrons from these machines is done with neutrons of relatively low energy. Many kinds of experiments use thermal reactors as neutron sources. Some examples are: neutron diffraction studies of solid state structure, low-energy neutron cross-section measurements using crystal spectrometers or time-of-flight energy analysis, nuclear level scheme studies based on radioactive decay or capture gamma ray spectroscopy, liquid state structure studies based on measurements of lowenergy inelastic scattering, and a host of applications to other fields which use radioactive tracer techniques. The neutron energy degradation takes place primarily through elastic scattering by light nuclei, but in some reactors inelastic scattering by heavier nuclei also plays an important part. The probability of radiative capture must be kept small during moderation. The useful moderators are therefore those which are efficient in reducing the neutron energy, but which have low-capture cross sections. The dominant materials for this purpose in research reactors are light water, heavy water, beryllium, and graphite. The change in nuclear binding energy during fission is given mostly to kinetic energy of the fission fragments. The remainder appears as kinetic energy of the neutrons, and energy of beta particles and cascade gamma rays from the radioactive decay of the fission products. Some is carried away by neutrinos. The breakdown is as follows:2 fission fragment KE neutron KE /!?-particleKE 7-rays neutrinos
168 MeV, 5, 7J
13J
11.
Additional energy, a few Mev per fission, depending on the design of the reactor, is liberated by neutron radiative capture.
* H. Etherington, ed., “Nuclear Engineering Handbook,” 1st Ed., p. 2-2. McGrawHill, New York, 1958.
3.2.
ARTIFICIAL SOURCES
593
All of this energy except that carried by neutrinos is eventually transformed to heat in the reactor. This heat must be removed if the temperatures of reactor components are to be kept reasonable. The heat removal methods are those commonly used when large amounts of energy must be dissipated. That is, a fluid is circulated past the fuel-bearing elements, and the heat transferred is carried away to a heat exchanger whose secondary circuit contains air, or which uses ground water or water circulated through a cooling tower. Other, but similar, methods are occasionally used. The excess neutrons and the gamma rays from fission and fission products complicate the design problem because they must be shielded against. The source strengths are very large. A rule of thumb is that about 10 curies of gamma rays are produced by 1 watt of steady reactor operation. These gamma rays have energies from about 12 Mev downwards, with the peak at a few MeV. The reactor itself absorbs some of this radiation, but enough escapes to lead to severe personnel hazard and interference with experiments unless thick shielding is provided. The shield thickness required for most reactors varies from about 3 to 5 meters of ordinary concrete or from 1.5 to 2.5 meters of special high-density concrete. Pool reactors use thick water layers for the same purpose. The shielding thickness required is not very sensitive to the power level or the type of reactor. In addition, the fission products themselves would be severe health hazards if they were freed. Much of the engineering design of a reactor is aimed at making such a release very improbable. Fission produces neutrons over the entire fueled region of the reactorthat is, throughout the reactor core. The only practical limitation on the source strength is the ability to remove the heat generated by the fission process. Standard methods now in use permit heat removal of up to about 1.5 kw/cm3, which provides a maximum steady source strength of about 1014neutrons/cm3-sec. Improvements in design should permit this yield to be increased by a sizable factor in the near future. Altogether, a l-Mw research reactor produces 7.2 X 10l6neutrons/sec, of which about 3 X 101s/sec are needed to produce further fissions. Typically, about 2 X 10I6/sec are lost in radiative capture in the fuel, moderator, coolant, and structural material. The rest are available for research uses. In fact, though, it happens that the intensities of available neutron beams are by no means comparable to those which would be available from point sources of the same strength. The intensity is reduced considerably by the diffuseness of the reactor core, and by the need for collimation. Furthermore, the energy distribution of the neutrons is continuous. Any experiments requiring monoenergetic neutrons suffer further losses by the energy selection process.
3.
594
SOURCES OF N U C L E A R PARTICLES A N D RADIATIONS
The experimental facilities are of two kinds: those which remove neutron beams to equipment outside the reactor shield, and those which expose material to radiation in the reactor core. Recent advances in design have led to considerable improvement in the efficiencies of both kinds of facilities. Beams have been increased in output and have had the intensity of unwanted components of the radiation reduced. Irradiation facilities are now placed in regions where the neutron spectrum is best for the experiments to be done. 3.2.1.3.2. FEATURES OF THE PROCESSES INVOLVING NEUTRONS. The only fissionable material commonly used in research reactors is U236.U23s must be synthesized in reactors by neutron capture in Th232.It is, therefore, more expensive. Furthermore, it is difficult to handle, being an unusual health hazard because of its high alpha decay rate and also because of large gamma activity accompanying decay products of U232 which is also produced through the (n,2n) process on the U233as it is made. Pu239is also relatively difficult to use because of the large alpha decay rate, and it must be synthesized by irradiation of UZ3* in a reactor. Only a few research reactors are not of the thermal type. We shall, therefore, concentrate here on only this class. The nuclear properties of U236of primary interest for reactor use are the following :s of
=
582 barns (fission cross section),
oc = 100 barns (capture cross section), a = 0.092 (capture to fission ratio), v = 2.43 (neutrons emitted per fission), = 2.074 (neutrons emitted per neutron absorbed).
These numbers are those applicable t o 2200 m/s (room temperature) neutrons; the averages over the distribution of neutron absorptions in a reactor are slightly different. The neutrons are mostly evaporation products from the highly excited fission products, emitted immediately after the fission.4 About 0.65% of the neutrons appear a considerable period of time after fission.s These delayed neutrons are emitted from fission products which are a t first cooled by evaporation until further neutron emission is no longer energetically possible. After a beta decay, emission of another neutron 3
R. Sher and J. Felberbaum, “Least Squares Analysis of the 2200 m/s Parameters
of U*ss, lJ236, and Pu*3@. BNL-722, June, 1962. 4 1. Halpern, Ann. Rev. Nuclear Sn’. 9, 245-342 (1959). 6
G . R. Keepin and T. F. Wimett, Nucleonics 16, No. 10, pp. 86-90 (1958).
3.2.
595
ARTIFICIAL SOURCES
10 9-
076-
5-
3 0)
.d
4-
h
-
.$ Y
d +
2-
1
0
1
3
2
I
4
5
I
6
Neutron energy (mev)
FIG. 1. Energy distribution of prompt fission neutrons.
is sometimes possible and so takes place. The delayed neutrons are then emitted with a half-life characteristic of the beta decay preceding them. The energy distribution of the prompt neutrons is continuous, as shown in Fig. 1. The average energy is approximately 2.5 MeV; the most probable energy is about 0.6 MeV. The distribution can be accounted for by the evaporation model. The delayed neutrons have lower energies, in the range of a few hundred kev. Six separate groups have been identified,6with characteristics shown in Table I. The existence of the delayed neutrons makes control of a nuclear reactor a much simpler matter. The neutron balance can be regulated more coarsely, because there is a partially delayed response to any deviation from the steady state. The kinetic behavior of a reactor is best discussed using the multiplicaTABLE I . Delayed Neutron from U*as Fission Half-life (sec-1) 54.51 f 0 . 9 4 21.84 rt 0 . 5 4 6.00 f 0 . 1 7
2.23 f 0.06 0.496 f 0.029 0.179 k 0.017
Relative yield 0.038 0.213 0.188 0.407 0.128 0.026
f 0.003 & 0.005
0.016 *f* 0.007 0.008 f 0.003
3.
596
SOURCES O F NUCLEAR PARTICLES AND RADIATIONS
100
5 0
J
; 10
1 10-3
10-2
Ak Reactivity (F)
FIQ.2. Asymptotic reactor period for positive reactivity.
tion factor k , which is the factor by which Y would have to be divided to cause steady operation. k is also very nearly the relative return of neutrons in a single fission cycle. The reactivity is defined as k - l / k = p." If the reactivity is positive, the fission rate in the reactor increases. If the reactivity is negative, the fission rate decreases. In either case the change in fission rate assumes an asymptotic exponential form, with a 6
S. Glasstone and M. Edlund, ibid., p. 298.
3.2.
597
ARTIFICIAL SOURCES
stable period T given by the ((inhourformula” P = 1/T
+ 2 Pi/(l + XiT)
(3.2.1.3.1)
i
where 1 is the average time between generations of prompt neutrons.
I varies in value in thermal reactors from about 50 psec for some light water moderated systems to about 1 msec for most graphite or heavy
Note: Curve has asymptote at 80.6 secondsseconds ~
5 x 10-5
lo-‘
10-3 Ak -Reactivity (-
r)
FIG.3. Asymptotic reactor period for negative reactivity.
water moderated systems. For normal reactivity changes associated with reactor control, the first term in Eq. (3.2.1.3.1)is negligible, and the relation between reactivity and stable period is seen to invoIve only the delayed neutron properties. Figure 2 is a plot of the asymptotic positive period vs. reactivity. Figure 3 is a similar plot for negative reactivities. Inspection will show that controllable increases of reactor power accompany values of p less than the total delayed neutron fraction 8. When p = 8, the reactor is
598
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
critical on prompt neutrons alone, and above this point the braking effect of the delayed neutrons is lost. The previous statements on the relation between the reactivity and an asymptotic period must be modified to take into account feedback effects. The temperatures of reactor components have noticeable effects on the reactivity of the system, these effects being associated with several causes. Among the most important are the following: (a) As the moderator temperature is changed, the average energy of neutrons in near equilibrium with the moderator thermal motion also changes. Relative absorption cross sections of components are altered, and the neutron balance is upset. The effect is usually that of reactivity reduction with increasing temperature ; the absorption cross section of UZs6diminishes faster with energy increase in this region than do the capture cross sections of most other substances used.’ These generally have capture cross sections which are nearly 1/v. Thus the fraction of neutron absorptions causing fission decreases with rising moderator temperature. (b) The rate a t which neutrons leak from the reactor core is proportional to their speed. The increase of average thermal neutron speed with moderator temperature leads to an increased leakage fraction and thus a reduced fission fraction. The effect is a reduction of reactivity with increasing moderator temperature. (c) Reactors with low uranium enrichment have appreciable capture of neutrons in the UZ3* resonances. An increase in fuel temperatulje leads t o increased resonance capture and thus reduced fission probability, through doppler broadening of the resonances. (d) Most coolant materials capture neutrons. I n reactors with solid fuel, the greater expansion coefficients of coolants cause expulsion of neutron absorbers from the core region when the temperature is increased. The relative probability of fission increases as the temperature is raised; so does the reactivity. A number of other effects act on the reactivity as the temperature changes. There are influences of structural expansion, of changing spacial neutron distribution, etc. Each of these has an associated temperature coefficient of reactivity] which is the derivative of reactivity with temperature. It is considered essential for safety that the net coefficient be negative. Any unwanted increase in reactivity which causes power and temperature rises will then be opposed by a natural restoring action. There are also time changes in reactivity which are associated with burnup of the uranium. As the amount of uranium decreases, the reactivity falls for two reasons. The first is a decrease in the number of nuclei available for fission. The second is the production of new nuclei which BNL-326, Suppl. No. 1 to 2nd Ed., p. 118 (January 1, 1960).
3.2.
ARTIFICIAL SOURCES
599
compete for neutron capture. These are the fission products, a number of which have large cross sections for neutron capture. The fission product which affects reactivity the most is Xe136.It is primarily a daughter product of decay although there is evidence of a small direct yield also. and the precursor Te136are products of 6 % of fissions. Te136has a half-life of only 2 min. For this reason, the II36 can be considered to be formed directly, because it has a half-life of 6.7 hr. The daughter xenon has a half-life of 9.2 hr. The xenon thermal cross section is 2.7 X los barns, associated with a resonance strategically located a t 0.084 ev. The combination of high yield and cross section causes appreciable disturbance to the neutron balance and the reactivity a t reactor powers above a few hundred kev. At about 1 Mev a typical research reactor operating at steady state for periods greater than the half-lives concerned will have somewhat over 0.01 in reactivity tied up in the Xe136 capture. At higher powers this effect increases, to a theoretical maximum between 0.04 and 0.05 in most cases. The precise maximum differs from one reactor to another, depending on the neutron spectrum, the neutron flux shape, and other details. It is a true maximum, though, because a t the higher powers the burnup by the higher neutron intensity reduces the xenon content. The limiting case occurs when everywhere the xenon production rate is equal to its rate of destruction by neutron capture, with decay being negligible. cross section prevents this fission product from being The smaller destroyed significantly by neutron capture. Therefore, even when the Xe136 density is capture-limited, there is a large production rate. If the reactor is shut down, the xenon continues to be formed a t the same rate, while its rate of destruction falls to the level set by radioactive decay. The xenon density therefore rises, and the reactivity effect increases. A xenon transient occurs; if the reactor is left shut down, the xenon reactivity effect can rise to a very high value, 30% or more not being uncommon in very high powered systems. Since few reactors have provision for compensating such extreme changes through control rod motion, the reactor must then stay shut down until the iodine and the xenon have both decayed enough. A large xenon transient will last for from 1 to 2 days, Figure 4 shows the appearance of such a reactivity transient. Xenon transients become important at reactor powers above about 1 Mw. The second most important fission product is SmL42.This nuclide is stable. Therefore its reactivity effect increases steadily with reactor life until the burnup and production rates are equal. At this point about 0.012 of reactivity is taken by samarium capture. After reactor shutdown, there is an increase in the reactivity held by the samarium, because of the loss of burnup.
2
ln
2
0
600
0 0
-!
8
? N
? rl
9 rl
2
0
0
h In
rl 0
X 0
0
9
!!i
3.2.
ARTIFICIAL SOURCES
601
The other fission products have effects which are usually lumped into an average capture cross section of about 100 barns/fission. The burnup of uranium and the increased poisoning of the fuel combine to limit the duration of use of reactor fuel elements. Once the system cannot be held critical a t the steady power desired, refueling is needed. Fuel burnup in research reactors which use highly enriched uranium is commonly taken t o from 20% to 40% of the initial uranium charge. These reactors use alloy fuel which suffers damage from irradiation and from fission product gas evolution beyond this point. Reactors with natural uranium metal fuel are limited by radiation damage to only a few per cent burnup. Good design makes the reactivity limit and the metallurgical limit coincide. 3.2.1.3.3. ENGINEERING. 3.2.1.3.3.1. Fuel Elements. The fuel elements in most research reactors contain the uranium in an aluminum alloy. A sandwich construction is usedls the alloy being a thin layer faced with uranium-free aluminum. All edges are also protected by a picture framing which is integral with the surface cladding. The most common elements are made of plates. The clad, alloy, and picture framing are rolled as a unit, the whole being held together by edge welding and pressure bonding. Occasional elements of other shapes, such as cylinders, are coextruded sandwiches. The cladding serves to keep the fission products in the metal matrix. It is, therefore, thick compared with the path length of fission products (about 0.02 mm in aluminum), and even more important, is thick enough so that corrosion and erosion will not bare the alloy core to the coolant stream. The clad thickness is also chosen to be compatible with manufacturing tolerances, so that it is not too thin in any one place. A common choice is an alloy core about 0.5 mm thick in a cladding of the same thickness. Alloys of from 14% to 30% uranium by weight are common. At the higher value and beyond, it becomes necessary to soften the alloy with silicon, t o make the rolling properties compatible with those of the clad. Uranium metal fuel rods must be canned, because the metal is chemically active. The canning material is usually aluminum tubing, though British reactors use a magnesium alloy (Magnox). Some reactors are fueled with sintered UOz pellets, which are clad in a way similar to the metal. Some research reactors are homogeneous or semihomogeneous, with the fuel dissolved in or mixed with the moderator. These will be discussed separately. 8 J. E. Cunningham and E. J. Boyle, PTOC. 1st Intern. Cord. on the Peaceful Uses of Atomic Energy, Geneva 1965, P/953 9, pp. 203, 234 (1956).
602
3.
SOURCES OF N U C L E A R PARTICLES A N D RADIATIONS
3.2.1.3.3.2. Coolant. The common heat transfer agents in research reactors are light water, heavy water, and air. Water is the most efficient agent, having a high heat transfer coefficient and a high heat capacity. It is therefore preferred for high-performance reactors. Air is used only to cool some of the large graphite moderated reactors. Separated Li’ would be an excellent coolant, although the reactor it is used in would be expensive. The corrosion problem would be difficult to overcome. The limit on heat removal is set by the surface temperature of the fuel elements. This must be kept below the melting point. The limit is called the burnout point;Bin water-cooled systems it is expressed in terms of heat generation rate rather than temperature, which is more difficult to estimate. The burnout heat flux depends on the water temperature, the pressure, the channel flow rate, the channel dimensions, and perhaps other variables as well. The physics of the burnout process is poorly understood; thus it is standard practice to design the heat transfer aspects of the reactor conservatively. A substantial burnout margin is allowed for. High performance systems are often pressurized to raise the boiling point, since burnout will not occur until some point beyond the onset of surface boiling. The reason for avoiding fuel element melting is primarily safety. It is desirable to prevent the presence of fission products in the coolant. Furthermore, there appear to be no grounds to rule out possible exothermic metal-water reactions which could propagate. These could lead to pressures which might rupture the reactor vessel, causing severe radiation hazard. Though there has been no evidence of such a sequence in the cases of accidental burnout which have taken place, the policy has been t o remain conservative. 3.2.1.3.3.3.Moderalor. The neutron moderating material is usually the same as the coolant, though there are notable exceptions. The material must contain an abundance of light nuclei, to slow down the neutrons. It must have a lowcapture cross section in both the thermal and the epithermal regions. It must not be subject to severe radiation damage, which is caused principally by fast neutrons. It must, of course, exist as a manageable solid or liquid. The suitable nuclides are hydrogen, deuterium, carbon, beryllium, and oxygen. The materials which have been found acceptable are light water, heavy water, beryllium, beryllium oxide, and graphite. All have been used to some extent, though beryllium and its oxide are found only in a few high-performance systems. C. F. Bondla, in “Nuclear Engineering Ilandbook” (H. Etherington, ed.), 1st Ed. Chapter 9-3. McGraw-Hill, New York, 1958.
3.2.
ARTIFICIAL
SOURCES
603
3.2.1.3.3.4. Material Strength. The maintaining of structural integrity of a reactor is of paramount importance. Any breaking, bending, or melting of any part of the reactor core can cause quick reactivity changes which could lead to accidents liberating fission products. Therefore, careful attention is given to pertinent design details. The problem is complicated by the high ievels of radioactivity induced in all materials in the reactor proper. These cause repair of faulty components to be always difficult and sometimes impossible. Components must be overdesigned with respect to corrosion and stress. Allowance must be made for progressive radiation damage over the history of the system. l o Fabrication of reactor vessels and of associated pipes must conform to approved standards which have been set by the appropriate technical societies. 3.2.1.3.3.5. Control Rods and Drives.11 The materials used for control rods have large thermal neutron cross sections. In low-flux systems (<<1014/cm2-sec)canned boron carbide rods are often used. These must be well sealed where they are in contact with water, because water leakage leads to gas generation which deforms the control element. It could then stick in its channel. In some low-flux reactors, a low-boron steel is used. This material suffers severe radiation damage through the Bl0(n,cr)Li7reaction, and its use is recommended only in very low power reactors. Hafnium metal is occasionally used, because it has a high cross section, a high resonance integral, and excellent corrosion resistance to water. Its expense limits its use to certain very high flux reactors. Cadmium-silver alloys are suitable, and have been used. Rare earth cermet mixtures have become promising (e.g., oxides of europium, gadolinium, dysprosium, and yttrium). These are made as wafers clad in stainless steel. The positions of control rods are regulated by control rod drive systems. These usually consist of electric motors which transfer motion through mechanical linkages such as rack and pinion drives. Usually the rods fall into the core region under gravity if a “scram” signal is given or if there is an electrical power failure. Sometimes the speed of insertion is enhanced by air pressure or springs. Shock absorbers are used to avoid physical damage to the reactor. 10 L. Grainger, “The Behavior of Reactor Components under Irradiation,” Review Series No. 6. I.A.E.A., Vienna, 1960. 11 W. K. Anderson and J. S. Theilacker, eds., “Neutron Absorber Materials for Reactor Control.” Naval Reactors, Division of Reactor Development, U.S.A.E.C. Superintendent of Documents, Washington 25, D.C., (1962).
604
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
3.2.1.3.3.6. Instrumentation. Instrumentation can be divided into three classes : neutron sensing, process, and radiation sensing. The neutron level instruments are used to monitor the power level of the reactor, and to govern its control. The sensing instruments are usually compensated boron-sensitized ion chambers, whose currents are measured Boroncoated ion chamber
,
Micromicro ammeter
~
I/ Boron coated ion chamber
-
-
Recorder
~
switching
Micromicro ammeter
-
Recorder A
+ l switching
Boron coated ion chamber
-
Fission
-
counter
DC Amplifier log N converter
--
-
Period circuit
Count
I
Recorder
n
< Scaler
FIG.5. Typical neutron instrumentation.
by micromicroammeters. At low power levels during startup, pulse counters and count rate meters are used. These are normally fission counters. Additional instruments (period instrumentation) measure the logarithmic derivative of the power. All neutron instrument outputs are displayed, usually on chart recorders, and most or all are wired into the scram bus circuit so that power levels which are too high or periods which are too small cause the control rods to be inserted some fractional distance (setback) or completely (scram). The time constants of the neutron cir-
3.2.
ARTIFICIAL SOURCES
605
cuitry must not exceed a few microseconds, because rapid response to power changes is essential. Figure 5 shows a block diagram of typical neutron instrumentation. Most research reactors have servo systems which drive a control rod in response to an error signal which is the deviation of the current in an ion chamber from a preset level. This device maintains constant power. Some, but not many, research reactors have fully automatic startup systems. The process instruments measure quantities associated with heat removal. These are values of pertinent temperatures, pressures, and flow rates. Conventional detection methods are used. The more vital quantities, such as flow rate, reactor coolant outlet temperature, and reactor pressure, are also included in the scram bus. The radiation detection instruments measure activities in the experimental areas, the coolant loop, etc. 3.2.1.3.3.7. Fuel Handling. The fission products in spent fuel elements remain very radioactive for long periods. They must be handled carefully to avoid radiation overexposure to personnel. A typical fuel element from a 1-Mw swimming pool reactor has a source intensity of about lo5 curies when it is used up. Heavily shielded apparatus is therefore needed during transfer of such elements, even after allowance for some decay of the activity. These transfer casks are used to move the fuel to deep waterfilled canals, where storage under adequate water shielding takes place. The water serves the secondary purpose of removing the fission product heat, which in some cases would be enough t o melt the fuel elements. If the reactor flux level is much above additional cooling is needed inside the transfer cask to prevent fuel melting there. Reactor canals are always likely sites of critical mass accidents, because a large number of fuel elements usually gather there between shipment to reprocessing plants. 3.2.1.3.4. HAZARDS. The hazardous aspect of a nuclear reactor is a result of its stored fission products. The radiation burst of an uncontrolled power excursion is not likely to cause excessive irradiation of personnel, even those nearby, because the shielding is adequate to protect them. But if the fuel elements melt and the reactor vessel ruptures, the released fission products can do severe damage to people both by irradiation and ingestion. Therefore, extreme precautions are taken in both design and operation to ensure that fission product release does not take place. As discussed earlier, conservative mechanical design criteria are applied to make the probability of fuel melting or of a nuclear excursion as small as possible. All instrumentation is designed to shut down the chair reaction if evidence of abnormal behavior appears. The instruments and con-
606
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
trol elements are designed to be fail safe: on loss of electrical power, a scram occurs. It is also common practice to design the system in such a way that accidents of unforeseeable nature cannot cause severe consequences to people. Precautions are taken which would cope with any size of credible accident. This is interpreted to mean any size of accident which cannot be ruled out completely using well-established physical arguments. Such precautions usually extend to the choice of site and the design of the building housing the reactor. Site selection depends on such questions as meteorology, hydrology, soil conditions, earthquake and storm histories, flood probability, and the density of nearby population. Buildings are designed to higher standards than is common if earthquakes are probable. Furthermore, it is common practice to house the reactor in a leaktight building which will retain the fission products which the reactor might free in the maximum accident. These containment buildings are made of steel or reinforced concrete which will withstand credible pressure buildups. All penetrations for pipes and wiring use leak-tight seals. Personnel or equipment access is through airlocks. The construction and operation of all nuclear reactors in the United States must be reviewed and approved by the Atomic Energy Commission for hazard aspects. Similar practices are followed in most other countries. 3.2.1.3.5. EXPERIMENTAL FACILITIES. 3.2.1.3.5.1. Beams. Neutron beams are extracted through holes in the reactor shield. The holes may extend into the reflector or the core of the reactor. Where a reactor vessel is used, or in a pool reactor, this penetration is made by means of a beam tube or thimble which prevents fluid leakage through the shield hole (see Fig. 6). If there is a possibility of escape of activated air out through the beam hole, a gasketed aluminum cover pIate is used. The low cross sections of aluminum cause minimum interference with the exiting neutron beam. Beam tube diameters are chosen according to experimental needs and engineering feasibility. The crystals in diffraction spectrometers are usually kept small for good instrument resolution. The rotors of time-offlight instruments are limited in size by strength considerations. Therefore, there is seldom any need for beam apertures greater than about 10 cm in diameter. This size of aperture is usually about the largest desired in a beam hole in or near the reactor core. Larger sizes perturb the neutron distribution too much in a small reactor. Also, any penetrations through a reactor vessel cause regions of potential weakness which must be specially strengthened, and the required extra vessel thickness in the neighborhood rises rapidly with thimble diameter. Estimates of the uncollimated neutron flux from a beam hole can be
3.2.
ARTIFICIAL SOURCES
607
made if the neutron flux at the point of extraction is known. I n most cases, knowledge of the interior flux unperturbed by the beam tube is adequate. The angular distribution of the neutrons entering the beam tube can be assumed to be isotropic in such a n estimate, and the exit beam strength can be found simply from solid angle considerations. As a n example, a beam tube 10 cm in diameter, penetrating 200 cm of shielding and 100 cm into a reactor region where the thermal neutron flux is 10*3/cm2sec will emit an uncollimated beam of approximately 7 X lo8 thermal neutrons/sec/cm2 of beam hole aperture.
Vessel wall
Ref lector
FIQ.6. Beam tube thimble through a vessel.
Some beam tubes pass completely through the reactor core or reflector and on out through the other side of the shield, as in Fig. 7. These through holes are accessible from either side. They allow access to the collimator without disturbing the experimental facilities a t the pile face, or they can be used to supply separate experiments a t the two ends. Either way, a scattering block is needed in the hole inside the reactor, to serve as a source of neutrons. Graphite is commonly used for this purpose. The neutron beam has a content of fast neutrons and gamma rays which are a health hazard, and which also create unpleasant background levels for the experiment and neighboring ones. Crystal spectrometers use deflected beams, and the direct beam past and through the diffraction crystal is heavily shielded t o stop the unwanted radiation. l 2 Borated I* G. E. Bacon, “Neutron Diffraction,” 2nd Ed., pp. 8-10. Oxford Univ. Press. London and New York, 1962.
608
3.
SOURCES O F NUCLEAR PARTICLES AND RADIATIONS
paraffin or heavy bakelite is used to stop neutrons. Lead, iron, or bismuth are used to attenuate the gamma rays. When the apparatus is away from the pile face, a portable beam stopper made of similar material is placed in the line of the beam, to prevent personnel hazard. Unfortunately, shielding thicknesses must still be found by trial and error. Not only does the beam contamination vary with the reactor, but it also depends strongly on the design of the collimator, which introduces a sizable attenuation.
$
block
Beam tube
\\/ FIG.7. A through beam hole.
Some research reactors have shutters built into the shield, to attenuate the beam when it is necessary to work on the apparatus. I n other cases, the collimator can be flooded with water, for the same purpose. 3.2.1.3.5.2. Irradiation Facilities. The most common irradiation facilities are those used to activate material by thermal neutron capture. The samples to be activated are placed in a region where the neutron flux is high, either in the core or the reflector, the transfer usually being done by means of a mechanism which can be used even when the reactor is operating. One commonly used facility of this kind is the pneumatic tube. This is a device which drives the sample into its irradiation position and back by air pressure from a storage tank. Air valves switch the pressure from one end of the transfer tube to the other. A system of this kind can be designed
3.2.
ARTIFICIAL SOURCES
609
to have short sample transit time, a feature useful in work with shortlived activities. A typical diagram is shown in Fig. 8. Actually, almost any kind of drive principle can be used in sample loading. Commonly used are endless belts and flexible cable drives. I n a great many reactors, the sample is inserted on a string or a rod. Usually the sample must be cooled if it is to remain in the neutron flux for very long. It is heated both by gamma ray absorption and by neutron capture.
Shield ~~
Reactor core and reflector
~
Sample conveyer
Compressed air supply
,
Fig. 8. Schematic drawing of pneumatic tube irradiation facility. Air pressure t o the top section drives the “rabbit” into the reactor. Pressure to the bottom drives it out. Samples are loaded into t h e rabbit, which is then inserted in the out-of-pile section of the tube.
Reactors with tightly closed vessels (e.g., those moderated and cooled by heavy water) must have tubes or thimbles allowing access to the highflux region without breaking the fluid seal. Otherwise, samples which are irradiated must be inserted and removed during shutdown, when the vessel can be opened. 3.2.1.3.5.3. Thermal Column. A thermal column is a large penetration through the shield, filled with a material which moderates neutrons well without absorbing them excessively. Usually graphite is used as the thermal column material; heavy water offers a slight advantage at much greater cost. The thermal column is meant to receive leakage neutrons from the reactor at its inner face, supplying well-moderated neutrons at the outer face.
610
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
Common sizes are about 2-meter cubes. The best design has this structure as close as possible to the reactor core, in an arrangement shown in Fig. 9. Most of the neutron losses in the thermal column are the result of leakage from the sides into the surrounding shield. The attenuation caused by leakage and by the smaller effect of absorption in reactor grade (low impurity) graphite is by about a factor of ten in 30 cm when the thermal column width is 2 meters.
1 I Shield
I
FIQ. 9. Thermal column location. The stepped structure avoids radiation leakage through cracks.
Since the graphite attenuates the core gamma rays very poorly, it is usual to place from 8 t o 16 cm of lead or bismuth between the thermal column and the reactor face. This gamma shielding must be cooled. The usefulness of a thermal column in basic physics research is not great. Most applications are in fields of biology or applied physics, where sometimes there is a real need for an isotropic flux of well-thermalized neutrons with little fast neutron contamination. 3.2.1.3.5.4. Fast Neutron Irradiation Facilities. Some irradiation facilities are used in the study of radiation damage to solids. Since most such damage is done by the higher energy neutrons (upwards of a few kev), the
3.2.
ARTIFICIAL SOURCES
61 1
facilities are located where the fast-to-thermal neutron ratio is high, or they have features which accentuate this ratio. A typical facility of this kind is a thimble through the reactor vessel, allowing the sample to be placed in the core region. The fast neutron intensity can be increased by augmenting the fuel loading nearby, or the facility can contain its own fast neutron booster as internally contained uranium fuel, as in Fig. 9. 3.2.1.3.6. SPECIFIC REACTOR DESIGNS. 3.2.1.3.6.1.Water The water boiler reactor core is a vessel containing a solution of a uranium salt in light water. The salt is the nitrate, the fluoride, or the sulfate. The solution must be kept slightly acid to prevent precipitation of uranium oxide or uranium hydroxide; the vessel and all other parts in contact with the liquid must therefore be made of a corrosion-resistant metal such as stainless steel. Even so, the possibility of corrosion is always present. Once a leak has occurred, it is difficult to repair. The vessel is surrounded by a reflector of graphite blocks; this is surrounded, in turn, by shielding. Beam tubes and irradiation tubes can be included as penetrations through the shield and reflector up to the vessel. An irradiation thimble penetrates the vessel into or through the core. Cooling is provided by the circulation of water through coils of tubing immersed in the fuel solution. The power level is then limited by the convective transfer of heat to the cooling coils. The maximum reactor power is usually about 10 kw, although careful design has permitted a power of nearly an order of magnitude greater. Control is managed partly by long-term control of the U236concentration, and partly by the insertion of control rods in thimbles through the core. The liberation of fission products directly into the water causes dissociation and the formation of hydrogen and oxygen gas. A recombiner (such as a hot platinum wire) is maintained in a gas space above the liquid to prevent the accumulation of a n explosive mixture. The limitation on power causes the usefulness of the water boiler to be very limited. Only a few exist. Figure 10 is a sketch of the main features of the system. 3.2.1.3.6.2.Graphite Moderated Reactors.I4 Most existing graphite moderated research reactors are patterned after the Oak Ridge X-10 reactor. They were designed to make use of the ability to generate a chain reaction with natural uranium rods in a graphite moderator structure. The critical size of such a system is, however, quite large, and the size necessary to 13 J. R. Huffman and A. M. Weinberg, U.S. Research Reactors. Progr. in Nuclear Energy, Reactors 1, 96-109 (1956). 1 4 I. Kaplan and J. Chernick, Proc. 1st Intern. Conf. on the Peaceful Uses of Atomic Energy, Geneva 1966, P/606 6, p. 295 (1956).
612
3.
SOURCES OF NUCLEAR PARTICLES A N D RADIATIONS
permit overcoming reactivity coefficients and the principal fission product poisons (Xe1as,Sm14a)is even greater. A typical reactor consists of a cube of graphite about 7-8 meters on edge, with the fuel-bearing region a cylindrical volume of about 5-6 meters in diameter. The natural uranium used is made of cylindrical metal slugs with an optimum diameter of about 2.8 cm. These are protected by metal jackets or cans to prevent contamination of the coolant stream by uranium or
FIQ.10. Schematic view of a water-boiler reactor.
fission products. The fuel is placed along lines in channels in the graphite, the optimum spacing between fuel channels in a square array being about 20 cm. The coolant is air, forced a t high speed past the fuel in the channels. This air is not recirculated. It is drawn from the outside and returned to the outside. During its passage through the reactor, the neutron flux produces activation by way of the reactions A40(12,y)A41and N'4(n,p)C14. The activated air is freed to the atmosphere a t the top of a high stack. This precaution assures dilution to safe concentrations before the radioactive mixture returns to ground level.
3.2.
ARTIFICIAL SOURCES
613
Control of the reactor is through poison rods inserted in channels in the graphite, passing through the uranium-loaded region. Ordinary commercial graphite is not suitable for such a nuclear reactor, because it has too high a content of impurities (particularly boron) to permit criticality. The graphite used is a high-density material (specific gravity between 1.6 and 1.7), with an impurity content amounting to well under 1 millibarn absorption cross section per carbon atom at 2200 m/s. Graphite with these properties is said to be reactor grade. The physical placement of experimental facilities in a graphite reactor of this kind is a relatively simple thing. Experimental beam holes are simply passages through the shield wall and the graphite. There is no pressure vessel to contend with. If through holes are used, a graphite scattering block is placed in the beam tube channel, This must be thick compared with the scattering mean free path of the neutrons, 10 cm or more being adequate. Irradiation facilities may similarly be located in the graphite structure. The size of this kind of reactor causes it to have a relatively low neutron flux density, since the neutrons must be dispersed over the large volume. Typically, the peak neutron flux (at the reactor center) is about 2 X 10" neutrons/cm2-sec, per megawatt. This feature reduces the usefulness of the reactor for research requiring high neutron intensity. On the other hand, the large size also allows many experiments to be done a t once, with minimum mutual interference. The Brookhaven Graphite Reactor, which is the most thoroughly used device of this type, has sixty beam tubes. Floor space limitations a t the outside of the shielding reduce the effective number to about forty. I n addition, though, there are numerous irradiation facilities, two thermal columns, and several other experimental facilities. The available thermal neutron flux of this reactor has been increased to about 2 X 1013neutrons/cm2-sec at 20 Mw, by conversion to enriched uranium fuel. This has removed the neutron capture by the U238and also the amount of U235required to compete for thermal neutron absorption. By this tactic, the thermal neutron density has been increased by a factor of about four. A large graphite reactor is expensive to build, and the inefficiency of air cooling makes it expensive to operate. These points, along with the relatively small thermal flux which is obtained, combine to make this kind of reactor somewhat obsolete. The ones which exist will continue to be used heavily, particularly for routine radioisotope production, solid state physics survey studies, and other research which does not require the high neutron intensities which today's methods provide. But comparable neutron intensity can be had from small light water or heavy water moderated reactors, a t much lower cost. The feature of plentiful
6 14
3.
SOURCES O F NUCLEAR PARTICLES AND RADIATIONS
research facilities is one which can only be exploited by the very large laboratories which have an abundance of research groups. 3.2.1.3.6.3. Light Water Moderated Pool Reactors.I6 The swimming pool reactor in its various forms is the most common research reactor. The original was built a t Oak Ridge, for use in research on the shield problem. The design is well suited to this purpose. The small reactor core is located in a large pool of water which serves as the reactor moderator, coolant, reflector, and shield. Radiation attenuation measurements can be made in the water directly, or they can be made in shield material lowered into the water near the core face. The design is also readily adapted to studies on the effect of radiation on material placed in the water. For more basic research, modification of the original Oak Ridge design has had to be made. While irradiation facilities are easily added to the design, well-designed beam tubes are not. The reactor core is normally suspended by a rigid framework from a bridge structure which crosses the pool (see Fig. 11). The core is about 8 meters below the pool surface, this being the amount of water necessary to attenuate the gamma radiation to a tolerable level. The fuel elements are box constructions containing plates of aluminum clad uranium-aluminum alloy, as shown in Fig. 12. End fittings allow elements to be located accurately in a grid plate which has mating holes. Though the properties of the fuel elements vary from one reactor to another, the following are typical. The uranium-aluminum alloy core of a sample fuel plate is 0.5 mm thick, and the clad has the same thickness. The water gap between plates is 2.6 mm thick. The plates are 7.5 cm wide by 61 cm long; they are attached a t the edges by brazing to a thick side plate of aluminum which lends structural rigidity to the in 93.5% enriched uranium (though box. A plate contains 10 gm of U2s6 some pool reactors use 20% enriched uranium, there is no really good reason for this choice, and there are metallurgical problems which argue against it). With eighteen plates in the structure, a box then contains 180 gm of U236.Commonly, the plates are all given a slight curvature in the same direction, so that thermal expansion effects will all act in the same direction and the water gap thickness will be nearly preserved. This curvature causes an element to deviate from a strictly rectangular shape, and the core loading must be nested. The control rods are usually aluminum boxes lined with cadmium and filled with boron carbide. These must be carefully sealed against water in-leakage, because subsequent gas generation will cause the aluminum box to deform. The rods usually are inserted into special fuel elements 16 W. M. Breazeale, R. G . Cochran, and K. 0. Donelian, Proc. 1st Intern. Cmf. m the Peaceful Uses of Atomic Energy, Geneva 1966, P/489 2, p. 420 (1956).
3.2.
ARTIFICIAL SOURCES
615
which lack the central nine plates. The rod drives are located on the bridge above the reactor, and these transfer motion to the rods by racks and pinions. An electromagnet holds a rod onto its rack; this electromagnet is part of the scram bus, and cutting the current allows the rod to be dropped into the core by gravity. Whenever the rod and rack are Rod drives
A
FIG.11. Suspension of a pool reactor.
connected, the rod position is transmitted to the reactor console by a selsyn generator attached to the rack. Normally four rods are used; three are safety rods, worth several per cent in reactivity apiece, and one is a fine control rod which is worth less than 1% in reactivity and which is used to shim to criticality. The pool reactor built in this way has a critical mass of about 2500 gm of U236.When aluminum canned graphite blocks are placed around the core, the critical mass is reduced by a few hundred grams. A single fuel
616
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
Lattice plate fits this diameter
W 01
Fuel zone
'-I
/
/
10.060"
m
,0.11 7
"
cladding material is aluminum
FIQ.12. Bulk shielding facility fuel element. Fully enriched uranium is alloyed with aluminum to form a 0.020 in.-thick fuel slab. This slab is clad with 0.020 in. aluminum on both sides in a "picture frame" type fuel plate. There are 18 plates per element; 140 gm of total.
element added to the edge of the critical loading increases the reactivity by about 0.01, When the water in the pool is allowed to circulate by natural convection through the hollow end fittings and up through the water gaps between fuel plates, the reactor can safely be operated at a few hundred kilowatts. Above about 100 kw, however, the activity in the air above
3.2.
617
ARTIFICIAL SOURCES
the pool becomes unpleasantly high. This activity is mainly produced by fast neutrons through the 0l6(n,p)N16reaction in the water, though other activity also can be present, depending on the impurities in the water. The activated gas is brought quickly to the surface by the convection currents and released there. Measures can be taken to retain it longer in the water (for instance, the placement of baffles above the core), and then the power can be increased to the limit set by convective heat transfer. By using forced circulation and treating the water of the pool as a heat sink, steady power levels of about 1 Mw can be achieved. The heat is eventually rejected to the air by convection at the pool surface. Thermal column
Rail
Reactor core
I‘ Pool
i 11 py
11 1 1
jl
Shield
Graphite
reflector
I
L--3 ‘Rail Pool position
Stall position
FIG.13. Pool reactor with stall.
At 1-Mw operation, a pool reactor has a peak thermal neutron flux in the center of the core of about l o L 3neutrons/cm2-sec. The pool is modified as shown in Fig. 13 when beam tubes are installed. A narrow section is added at one end. This “stall” position is lined with graphite on three sides to a thickness of perhaps 50 cm. When the reactor is placed inside this graphite structure, it is reflected partially by the graphite. The beam tubes are penetrations through the shield and the graphite, lined with aluminum pipe and sealed at the reactor core face to prevent water leakage. These beam tubes therefore transmit, neutrons from the reactor face. The graphite is a more efficient reflector than the water surrounding the core in the pool position. Therefore the reactor has a lower critical mass in the stall position. Naturally, there are safety procedures to follow
618
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
in transferring the core from the pool to the stall by bridge motion. These consist in unloading the core partially to a safe size before the transfer, and then restoring t o a critical loading by accepted methods. A thermal column can be included by extending the graphite through the shield on one side. The prototype of the pool reactor with a stall position is the Naval Research Laboratory Reactor. 3.2.1.3.6.4. Tank-Type Light Water R e a c t ~ r . ~A~simpler J ~ ~ ' ~light water moderated reactor dispenses with the pool and uses only a tank to contain the core. There is no heat reservoir naturally available; therefore the heat is removed by forced circulation of coolant water through the fuel elements, then through an external loop to a heat exchanger, and afterwards back to the core tank. The secondary heat exchange fluid is usually water which is either taken direct from a supply such as ground water and returned, or circulated through a cooling tower or a spray pond. As much as about 50 Mw can be generated and cooled in this way in a reactor core of the size used in a swimming pool reactor. Most tank-type reactors generate considerably less power than this, being in the range from 1 to 5 Mw. The higher powers require pressurization to suppress boiling on the fuel surface. The control system is essentially the same as that of the pool reactor, except that the rod drives are mounted on a cover plate over the high tank. In some cases the core size is reduced by the use of partial graphite reflector. This can be aluminum-clad graphite, as in the pool reactor, or it can also consist of graphite outside the core tank. The graphite within the tank is then carefully fitted to reduce the amount of water between the core and the tank wall, because this layer considerably lowers the effectiveness of the graphite. Quite rarely, beryllium is used in place of graphite. The entire reflector of this material is then held in the core tank. 3.2.1.3.6.5. Heavy Water Enriched Uranium R e a c t ~ r A . ~ reactor ~~~~ somewhat similar to the tank-type light water reactor uses heavy water la A. M. Weinberg, T. E. Cole, and M. M. Mann, Proc. 1st Intern. Conf. on the Peaceful Uses of Atomic Energy, Geneva 1966, P/490 2, pp. 402, 430 (1956). l7 J. B. Godel, Description of Facilities and Mechanical Components Medical Research Reactor (MRR). BNL-600 (February, 1960). l* W. H. Zinn, G. A. Anderson, T. Brill, 0. W. Childs, J. A. DeShong, J. J. Dickson, E. E. Hamer, J. M. Harrer, L. C. Livesey, B. I. Spinrad, S. Untermyer, J. T. Weills, and J. M. West, Proc. 1st Intern. Conf. on the Peaceful Uses of Atomic Energy, Geneva 1966, P/861 2, p. 456 (1956). 19 T. J. Thompson, Proc. 2nd Intern. Conf. on the Peaceful Uses of Atomic Energy, Geneva 1968, P/417 10, p. 375 (1958). Po D. G. Hurst and A. G. Ward, Progr. in Nuclear Energy, Reactors 1, 1-48 (1956).
3.2.
ARTIFICIAL SOURCES
619
instead for cooling and neutron moderation. The fuel elements are of plate construction, somewhat similar to those used in pool reactors. They are not nested close together, though, and some irradiation facilities can be placed in the space between them to use neutrons from high flux intensity regions. The peak flux intensity is only a little greater than that in a light water moderated and cooled tank reactor, but the experimental facilities use the neutron output more efficiently. On the other hand, the heavy water is expensive, and the system must be built tight to reduce loss of this liquid and to prevent its rapid degradation by exchange with atmospheric moisture. The tightness requirement increases the cost of most components of the vessel and of the primary cooling circuit. It complicates the problem of refueling the reactor, and so makes this procedure more expensive. The first reactor of this kind was the CP-5, at Argonne National Laboratory. I n open tank-type reactors (no graphite) the beam tubes are piped into the tank, up to the core face or near it. 3.2.1.3.6.6. Heavy Water, Natural Uraniumz0Several versions of research reactors use natural uranium rods and heavy water moderator. The most common type is patterned after the NRX reactor, a t Chalk River. This is a callandria design, with the coolant and the moderator being separate. The fuel elements are inside concentric cylinders of aluminum, and the heat is removed by light water passing between the cylinders. The coolant is fed in through an entrance manifold, and passes out into an exit manifold. The optimum fuel rod spacing is about 20 cm between elements. This leaves space in between elements for beam tubes and irradiation tubes, so the experimental facilities can be placed where the neutron flux is high. A typical reactor of this kind is a right circular cylinder about 2 meters in diameter. Typical peak thermal neutron flux intensities are about 4 x 1OI2 neutrons/cm2-sec at 1 Mw. 3.2.1.3.6.7. Zirconium Hydride, Enriched Uranium.21The TRIGA reactor has fuel elements made of a mixture of zirconium hydride and uranium. The heat is removed by water circulating outside the zirconium cladding of these elements. Since a large fraction of the neutron moderation is produced by collisions with the hydrogen in the fuel elements, this reactor has a prompt negative temperature coefficient, a feature which lends a considerable improvement in safety compared with nonhomogeneous enriched uranium reactors. The peak power at steady state is about 100 kw, and the peak thermal 2 1 S. L. Kouta, T. Taylor, A. McReynolds, F. Dyson, R. S. Stone, H. P. Sleeper, Jr., and R. B. Duffield, Proc. 2nd Intern. Conf. on the Peaceful Uses of Atomic Energy, Geneva 1968, P/1017 10,p. 282 (1958).
620
3.
SOURCES OF NUCLEAR PARTICLES A N D RADIATIONS
flux a t the core center is about 10l2 at this power. Later designs of the reactor permit somewhat higher power and flux levels. The prompt negative temperature coefficient allows the reactor to be operated in a pulsed mode. That is, the injection of a step increase in reactivity is followed by a high transient which is ended by the temper-
FIU.14. HFBR core.
ature reactivity feedback. Operated this way, the reactor can be used in research on phenomena with short relaxation times (e.g., production of short-lived isomers). 3.2.1.3.6.8. High Flux Beam Reactor. Two high-flux reactors are now being built. The first, being constructed a t Brookhaven National Laboratory, is primarily designed to deliver high neutron flux to beam tubes. This reactor will have a concentrated core of enriched uranium-aluminum alloy plate elements somewhat similar to the ones used in pool re-
3.2.
ARTIFICIAL SOURCES
621
actors. However, they will contain a more concentrated alloy (30% P6 by weight) than do pool reactor fuel plates. The alloy is to be 0.5 mm thick; the clad 0.375 mm thick. The core will contain twenty-eight elements in the geometry shown in Fig. 14 with a total core volume of 90 liters and a U236 content of 7.5 kg. The reactor power will be 40 Mw; the fuel will last 40 days, at 20% P6 burnup. The moderator, coolant, and reflector will be heavy water. Since the design provides too little heavy water in the core to accomplish good
FIQ.15. HFBR flux distribution.
neutron moderation, most of the neutron slowing down will take place in the reflector. As a result, the thermal neutron flux density will peak in the reflector, where the beam tubes are placed. Figure 15 shows the neutron flux a t several energies as a function of position in the reactor. The thermal flux at the peak has been measured to be 7.5 X l O I 4 a t 40 Mw. The beam tubes will generally be oriented so that they do not face the core region (see Fig. 14). This stratagem will not change the supply of thermal neutrons very much, but it will reduce the accompanying fast neutron and gamma ray intensities by a large factor. Furthermore, the
622
3.
SOURCES O F NUCLEAR PARTICLES A N D RADIATIONS
beam tubes will be placed at different positions relative to the core face, so that some will selectively supply more neutrons in the low epithermal range. These beams will be especially useful for epithermal neutron experiments . Within the core and near the core-reflector interface, the high-energy neutron flux will be very great. These regions are set aside for irradiation tubes in which solid state damage studies will be done on small samples such as single crystals. A few isotope production facilities will be placed well out in the reflector, where the neutron flux is well thermalized and high. The HFBR is scheduled for completion in the latter half of 1963. 3.2.1.3.6.9. High Flux Isotope Reactor. The High Flux Isotope Reactor is being built at Oak Ridge National Laboratory. Its primary purpose will be to irradiate CfZK2, forming transuranic elements of higher nuclear mass and atomic number by successive neutron capture. These resultant nuclei will then be irradiated further in other reactors, or they will be used as accelerator targets, to produce high atomic number nuclides. The reactor will have a unique core design, containing only two fuel elements. These will each be concentric cylinders of uranium-aluminum fueled, aluminum-clad plates. The plates are to be curved as evolutes, this being the shape which maintains constant spacing in a water channel. The two fuel elements are themselves to be concentric, so that a safety cylinder of absorber material can be inserted between them. The moderator and coolant are to be light water. The reactor power is to be 100 Mw. The cylindrical space within the inner fuel element is called a flux trap. This is a region where there is no uranium to absorb neutrons, and so the thermal neutron flux is sharply increased. The transuranic material to be irradiated will be placed in this flux trap, where the thermal flux will be about 3 X 10l6 neutrons/cm2-sec at the rated power with the sample in place. A beryllium metal reflector will be placed outside the core. A few beam tubes will be built into this reflector, with the same geometric considerations used as were discussed above in connection with the High Flux Beam Reactor. The HFIR is also scheduled for completion in the second half of 1963.
3.2.
ARTIFICIAL SOURCES
623
3.2.2. Medium- and High-Energy Sources* Artificial sources for medium and high energy are provided by the particle accelerators which give well-defined beams of radiation and bombarding particles whose energy and direction are much more accurately known and with higher intensity than the high-energy radiation available from cosmic rays. Acceleration of charged particles is always accomplished by means of a n electric field. Mechanical forces or gravitational fields are too weak; magnetic fields only deflect the particles. The methods of acceleration are merely variations of the way the electric field is applied. Application of the required amount of direct voltage between two terminals is limited to a few Mev and accelerators of this type have been described in the previous section (3.2.1.1 and 3.2.1.2). A semidirect method of applying an electric field is used in the betatron where the field is that associated with a time-varying magnetic field in a central core. All the other methods are various forms of using radio-frequency electric fields whereby the charged particles are accelerated many times by a relatively small voltage and are shielded from the applied field when it is reversed. This repeated application and shielding scheme was used, in 1928, in what was probably the first particle accelerator: a two-stage linear accelerator built by Wideroe' to accelerate positive ions to about 70 kev. It is also used in modern high-energy linear accelerators. A magnetic field can be used to bend the charged particles into a circular or spiralling path in order to pass and repass an rf source (or sources) a great many times in order to obtain a final energy that is a large multiple of that given by the rf electric field. I n the cyclotron, or magnetic resonance accelerator, the electric field has a fixed frequency and relativistic effects set an upper limit of around 25 MeV. With the evolution of the phase-stability principle and a greater knowledge of the particle orbits, it is possible in the synchronous accelerators to keep the particles in resonance with the radio-frequency source to as high an energy as one wishes. The limit is the cost of the accelerator. In synchrotrons, the magnetic field increases with time and not only keeps the particles moving on an approximately circular path but, by the field's spatial distribution, keeps the particles focused within a narrow toroid. In the alternating-gradient type of synchrotron, wherein the radial gradient of the magnetic field alternately changes sign along the orbit, the radial and vertical excursions of the beam (due to scattering by residual gas and imperfections in the injection optics and magnetic field) 1
R. Wideroe, Arch. Elektrotech. 21, 387 (1928).
* Section 3.2.2 is by M. H.
Blewett.
624
3.
SOURCES O F NUCLEAR PARTICLES AND RADIATIONS
are only a few centimeters whereas the orbit can be several miles in circumference. Recent developments in accelerator design, such as the AVF (azimuthally varying field) cyclotrons and FFAG (fixed-field alternating-gradient) accelerators, have extended the principles of azimuthal variations in the magnetic field to provide more highly intense beams. These various methods of applying the accelerating electric field and ways of using magnetic fields are shown in Table I, together with the maximum energy which has been achieved to date. Fairly complete descriptions of all types of accelerators, of theoretical studies of orbit dynamics, of special components such as ion sources, magnets, shielding, etc., and quite comprehensive bibliographies are available in recent books by Livingston and Blewett2 and by Livingood. During the past three decades, the maximum particle energy available from particle accelerators has risen by a factor of ten every 6 or 7 years. A recent catalogue4 lists as complete, or very nearly so, eleven proton synchrotrons (a twelfth, a t 7 Bev, has since been put into operation in the U.S.S.R.), seven electron synchrotrons, and two linear accelerators, all 1 Bev or more in energy. Design studies have already been made6,“and are continuing in several laboratories for accelerators which will reach hundreds of Bev, or perhaps 1000 Bev. The limit, particularly in the alternating-gradient type of design, seems to be purely economic. Formerly it was believed that an increase in energy would be accompanied by a loss of intensity of several orders of magnitude. However, in a 1000-Bev AG accelerator, merely because of its larger circumference, it should be possible to have intensities higher by a t least a factor of ten than the presently existing 30-Bev machines. FFAG accelerators can provide intensities another factor of ten, or one hundred, higher but above a few tens of Bev their cost becomes prohibitive. I n the energy range of a few Bev, it is possible for linear accelerators to be able to give currents up to a milliampere, or so, but again they are expensive. One limitation arises in using more highly relativistic particles to bom2 M. S. Livingston and J. P. Blewett, “Particle Accelerators.” McGraw-Hill, New York, 1962. 3 J. J. Livingood, “Principles of Cyclic Particle Accelerators.” Van Nostrand, Princeton, New Jersey, 1961. 4 M. Q. Barton, “Catalogue of High-Energy Accelerators,” Brookhaven National Laboratory Report BNL-683, 1961. 6 M. Sands, “Proceedings, International Conference on High Energy Accelerators, Brookhaven, 1961,” p. 145. U.S. Government Printing Office, Washington, D.C. “Design Study for a 300-1000 Bev Accelerator,” Brookhaven National Laboratory Report BNL-5701; “Experimental Requirements for a 300-1000 Bev Accelerator,” Brookhaven National Laboratory Report BNL-5700, 1961. @
TABLEI Accelerator type
Particle orbit
Magnetic field
Proton linear acceler- Straight line None ator None Electron linear accel- Straight line erator Circle of constant Increasing in time; decreasing Betatron slightly with radius radius Circle of increasing Constant in time; decreasing Cyclotron slightly with radius radius Circle of increasing Constant in time; decreasing Synchrocyclotron slightly with radius radius AVF cyclotron Circle of increasing Constant in time; large changes in azimuth radius Electron synchrotron Circle of constant Increasing in time; decreasing slightly in radius radius Circle of constant Increasing in time; decreasing Proton synchrotron slightly in radius radius AG synchrotron Circle of constant Increasing in time; alternating-gradient azimuthally radius Fixed-field alternating- Circle of increasing Constant in time; large changes in azimuth gradient synchrotron radius
Method of acceleration
Rf of constant frequency (about 200 Mc) Rf of constant frequency (1000-3000 Me) Electric field due t o changing flux in a magnetic core Rf of constant frequency (about 20 Mc) Rf of decreasing frequency (20-10 Mc) Rf of constant frequency
Top energy achieved (1962) 70 Mev 1 Bev
300 Mev -30 Mev
700 Mev
80 Mev
Rf of constant frequency
1 . 2 Bev
Rf of increasing frequency
10 Bev
Rf of increasing frequency Rf of increasing frequency
33 Ekv (protons); 6 Bev(e1ectrons) (Models only)
626
3.
SOURCES O F NUCLEAR PARTICLES AND RADIATIONS
bard stationary targets. The energy available in the center-of-mass system, for the production of new particles and interesting reactions, increases approximately as the square root of the energy of the bombarding particle since, with increasing energy, a larger fraction is associated with the motion of the center of mass. For example, with a 1000-Bev primary beam, the available center-of-mass energy is only 45 Bev. This has led to increased interest in providing beams of equal and opposite momenta which can intersect in some way and, thus, provide their total kinetic energy in the center-of-mass system. 3.2.2.1. The Betatron. That the induced transverse electric field produced by a changing magnetic flux could be used to accelerate charged particles, which could then be considered to behave somewhat like a secondary circuit in a transformer, was contemplated by many workers in the early days of trying to produce artificial sources for nuclear studies. However, the influence of the shape of the magnetic field, particularly its possible focusing effects and resultant limitations, was not appreciated until the analytical orbit studies of Kerst and Serber.’ On the basis of these studies, and with very careful magnet design, Kerst was successful in constructing the first betatron in 1941.8 In principle, the betatron can be used for the acceleration of either electrons or positive ions, but its application in practice has been exclusively as an electron accelerator (Kerst named it for its acceleration of beta particles). The highest energy achieved is 300 Mev at the University of Illinoisg and betatrons to 100 Mev are commercially availabIe,’O but the majority in use are in the range around 25 MeV. The useful output of these machines is usually in the form of X-rays whose intensity may run as high as 14,000 r/min at a distance of 1 meter from the target, as in the 300-Mev betatron. It is also possible to extract the accelerated electron beam and currents of the order of amp are available. Higher intensities are possible through the application of the FFAG principle (see Section 3.2.2.7). A fairly complete discussion of the theory and practical details of betatron construction and operation can be found in a recent article by Kerst.” In all types of magnetic accelerators where the charged particles are constrained to move in a circular path by the magnetic field, the electro7
D. W. Kerst and R. Serber, Phys. Rev. 60, 53 (1941).
D.W.Kerst, Phys. Rev. 60, 47 (1941). 9 D.W. Kerst, G. D. Adams, H. W. Koch, and C. S.Robinson, Phys. Rev. 78, 297 8
(1950). 10 W. F. Westendorp and E. E. Charlton, J . A p p l . Phys. 16, 581 (1945). 11 D. W. Kerst, “Handbuch der Physik” (S.Flugge, ed.), Vol. 44, pp. 193-217. Springer, Berlin, 1959.
3.2.
ARTIFICIAL SOURCES
627
magnetic force, a t every instant, must balance the centrifugal force; i.e.,
_ -- evB mu
r
(3.2.2.1)
where e is the charge, m the mass, and v the velocity of the particle, r is the radius of the path, and B is the magnetic flux density. In a betatron, the particles’ orbit has a fixed value of the radius, To, and the magnetic field increases with time to keep this relation satisfied. A t the same time,
FIG. 1. Cross-sectional view of bet.atron magnet with central flux core and poles which are shaped to produce a field proportional to T - ~ .
the electric field El which appears when the circular orbit is linked by the changing magnetic flux, is given by 2woE = rro2B
(3.2.2.2)
where B is the average flux density within the orbit. But, d d E =dt (mu/e) = at (Bore)
(3.2.2.3)
where Bo is the value of B at the orbit T o . Thus, B = 2Bo,
(3.2.2.4)
i.e., for stable acceleration at a fixed radius, the rate of change of the average flux density within the orbit must be twice the rate of change of the flux density at the orbit. This condition can be satisfied in a magnet whose configuration in cross section is like that shown in Fig. 1. The magnet gap is adjusted until the condition shown in Eq. (3.2.2.4) is satisfied.
628
3.
SOURCES OF NUCLEAR PARTICLES A N D RADIATIONS
During acceleration, the electrons make several hundred thousand revolutions in their orbit. I n order that small field perturbations, or other disturbances, do not cause deflections that result in eventual loss of electrons, it is essential that forces be included to restore the particles to their orbit and to limit excursions from the orbit to the inner dimensions of the vacuum chamber. Numerous early attempts t o build magnetic induction accelerators failed because this problem was not solved. Restoring forces are introduced by shaping the magnet pole, and thus the magnetic flux density, to the form:
B
=
B,(r/ro)-n
(3.2-2.5)
where Bo and roare the flux density and radius at the electron orbit and n is a number between 0 and 1. With a field shape which satisfies this condition, the radial and vertical equations of motion (for motions of small amplitude) become : P (1 - n ) w 2 p = 0 (3.2.2.6) Z nu22 = 0
+ +
where w/2n is the frequency of revolution of the electrons in their orbit; p and z are the radial and vertical displacements from the central orbit. It is evident that if n is less than zero the motion becomes unstable in the vertical direction; if n is greater than unity instability occurs in the radial motion. Within the prescribed limits, the field will provide restoring forces in both directions under whose influence the electrons will oscillate about their “equilibrium orbit” with a frequency slightly less than the frequency of revolution. These oscillations are known as the “betatron” (or “free”) oscillations and occur in all other types of magnetic accelerators. Their expected amplitude sets a lower limit to vacuum-chamber dimensions. Equations (3.2.2.6) do not include the slow changes which occur during the acceleration process in the velocity or mass of the particles (and, perhaps, in the field-shape index, n ) . When a more precise solution of the equations is made, it is found that the vertical oscillation amplitude is proportional to n-1/4B-1’2and the radial amplitude to (I - n)-1/4B-1/2. If no perturbations are introduced during the acceleration period, the amplitude of these oscillations will be determined by the injection conditions, since the particles are usually injected with both angular and radial spread, or a t a radial position somewhat different from the “equilibriumorbit” position. However, this “adiabatic damping” will produce a beam of much smaller cross-sectional area a t the final energy, a n advantage for targeting or experimentation.
3.2.
ARTIFICIAL SOURCES
629
Destructive resonances are possible between the betatron oscillations and the revolution frequency.12 For example, if the field index n is 0.75, the radial oscillation takes exactly two revolutions. Small irregularities in the field pattern can provide coupling between the betatron oscillation and the frequency of circulation and the beam can be lost very rapidly. In most betatrons, n is greater than 0.5 which gives stronger vertical restoring forces than radial restoring forces. Thus, the vacuum chamber can have smaller vertical than radial dimensions and the magnet gap can be reduced. The vacuum chamber is a glass or ceramic torus, flattened in the vertical plane and with a radial extent of the order of 10% of the orbit radius. The magnet tends to be rather massive, ranging in weight from 3.5 tons for a 20-Mev machine to 340 tons for the 300-Mev betatron. It must be laminated because of the rapidly varying field and is usually built of standard transformer laminations. The injector for a betatron is a simple electron gun located outside the orbit. Electrons of energy in the range between 50 and 100 kev are sprayed in a direction generally tangent to the orbit and, by a process that is rather incompletely understood, a small fraction of these is captured into stable orbits and accelerated. The acceleration period is usually of the order of 10 msec. This corresponds to the time required to drive the magnet to full field. I n most installations, the magnet is resonated with a capacitor bank and driven a t the resultant resonant frequency. Injection occurs shortly after the field crosses zero and acceleration continues to the peak of the sine wave of the magnetic flux density. By means of dc biasing of the field or the flux pattern, it is possible to increase the acceleration period from approximately a quarter wave to almost a half sine w a ~ e . ’ ~ , ~ ~ A t the end of acceleration, auxiliary coils expand or contract the electron orbit until the electrons collide with a target which may be simply a tungsten wire. If the electron beam is to be extracted, a “peeler” is introduced. This is a U-shaped magnetic shield which is mounted a t the outer extremity of the vacuum chamber; the orbit is expanded until the beam enters the open side of this shield. Inside the peeler, the field is sufficiently low that the electrons travel in almost a straight line and are able to escape from the field of the main magnet and to emerge as a fairly sharp beam. The upper limit to the energy achievable in a betatron is set by radiation loss. Relativistic electrons, constrained t o a circular orbit, radiate electro12 J. J. Livingood, “Principles of Cyclic Accelerators,” Chapter 5. Van Nostrand, Princeton, New Jersey, 1961. 13 D. W. Kerst, Phys. Rev. 68, 233 (1945). 11 W. F. Westendorp, J . A p p l . Phys. 16, 657 (1945).
630
3.
SOURCES OF NUCLEAR PARTICLES A N D RADIATIONS
magnetic energy at a rate that increases very rapidly with electron energy. 1 6 , 1 6 The energy, U , lost per revolution is given by
U
=
8.8W4/r (evlrevolution)
(3.2.2.7)
where W is electron energy in hundreds of Mev and r is the orbit radius in meters. A t energies above 100 MeV, this loss disturbs the relation between average flux and orbit flux density and causes the orbit to contract. Compensation can be achieved up to about 300 Mev but above this the betatron becomes impractical. In the 100-Mev range, the radiation takes the form of an almost continuous spectrum, including the visible, which emerges tangentially in the direction of motion of the electrons. Thus, in the higher energy machines] the beam can be seen as a bright spot at the point where the line of sight is tangent and the intensity of the light can be used as a monitor of beam intensity. The chief advantages of the betatron lie in its simplicity of construction and low cost. However, at higher energies, the electron synchrotron (see Section 3.2.2.5.1) is more to be preferred, first, because its annular magnet requires less iron, and second, radiation losses can be compensated more easily, through additional power in the rf accelerating cavity. 3.2.2.2. The Cyclotron. The cyclotron is a positive-ion accelerator which, in its classical constant-frequency form, is most useful in the energy range between 5 and about 20 Mev for protons and deuterons. With multiply ionized carbon, nitrogen, and other heavy ions, energies up to 100 Mev can be reached. Circulating beam currents of the order of 1 ma can be obtained which can then be used on an internal target, or a fraction, of the order of one tenth, can be extracted for use on external targets. I n this section only the classical cyclotron will be discussed. Extension to higher energies by frequency modulation will be described in Section 3.2.2.4 (the synchrocyclotron), and modern improvements in cyclotron design, using azimuthally varying magnetic fields, will be covered in Section 3.2.2.7. Historically, the cyclotron was the first artificial source to produce particles for nuclear-physics studies above 1 Mev and the first to apply the magnetic resonance principle whereby a magnetic field bends the particles to cause them to use repetitively a single, relatively lowvoltage, source of p ~ t e n t i a lThe . ~ ~general ~ ~ ~ principle of the cyclotron is illustrated in Fig. 2 where a uniform magnetic field is maintained in the direction normal to the plane of the diagram. In this field, the angular velocity, V/T, of a J. P. Blewett, Phys. Rev. 69, 87 (1946). J. Schwinger, Phys. Rev. 76, 1912 (1949). E. 0. Lawrence and M. S. Livingston, Phys. Rev. 40, 19 (1932). I* E. 0. Lawrence, M. S. Livingston, and M. G . White, Phys. Rev. 42, 150 (1932). 16 16
3.2.
ARTIFICIAL SOURCES
631
charged particle is given by (cf. Eq. 3.2.2.1)
v/r
=
eB/m
(3.2.2.8)
where e is the charge and m is the mass of the partic., and B is the magnetic flux density. Because the angular velocity is independent of radius, it is possible for the particles to remain in phase with an rf accelerating field applied between the two D-shaped electrodes shown in the figure. Particles which
Deflector + = L / - -
-L
FIG.2. Plan view of the cyclotron showing the shape of the dee electrodes. The magnetic field is perpendicular t o the plane of the diagram, The dashed path is a representative particle orbit.
arrive a t the gap between the dees a t an accelerating phase of the rf electric field will continue to arrive a t accelerating phases as they travel on an orbit that consists of half circles of ever-increasing radius. At full energy, the particles emerge, as shown, into a deflecting field that can convert part of the circulating beam into an external beam. For particles which are nonrelativistic, the kinetic energy can be expressed as
(3.2.2.9) or T (electron volts) e
er2B2 2m
= __
where B is in webers/sq m (= l o 4 gauss), e in coulombs, m in kilograms,
632
3. SOURCES
O F NUCLEAR PARTICLES AND RADIATIONS
and r in meters. The frequency of the radio-frequency oscillator matches the frequency of revolution of the particle and is given by f = - . eB
27rm
(3.2.2.10)
From these relations, it can be seen that the charge-to-mass ratio of the particle plays an important role; one cannot, for a given fixed frequency and a given magnetic field, accelerate both protons and deuterons. However, if a cyclotron has been designed to accelerate deuterons, it can also accelerate doubly ionized helium and protons in the form of singly ionized molecular ions, since these have almost exactly the same value of elm. From the difference in charge, the kinetic energies will be different; for example, in a cyclotron designed to give 20-Mev deuterons the maximum energy for helium ions would be 40 Mev and for the hydrogen ions 10 Mev per proton. Because of this flexibility, and because of their usefulness in neutron studies, the great majority of existing cyclotrons are designed to accelerate deuterons. A small modification of the magnetic-field pattern is required for successful operation of a cyclotron. In a perfectly uniform field, there would be no restoring forces toward the median plane and the beam would be lost to the upper and lower poles of the magnet before full energy could be reached. The necessary focusing is achieved, as in the betatron (Section 3.2.2.1), by making the field decrease slightly with radius. In order to do as little violence as possible to the cyclotron resonance relation, the rate of decrease is kept as low as possible and the vertical focusing is, therefore, very weak. If the magnetic field is expressed in the form given in Eq. (3.2.2.5), typical n-values over most of the accelerating region are near 0.02. At the maximum radius, fringing effects increase the n-value to almost unity and this produces a small amount of damping in the vertical oscillations. There can be some focusing or defocusing effects produced by the electric fields at the accelerating gaps which must also be taken into consideration. A fairly complete analysis of the effects of the magnetic and electric fields has been made by Cohen. l9 A cyclotron magnet has the general configuration shown in Fig. 3 with fields between the poles of the order of 15-20 kgauss. The vacuum chamber including the accelerating-electrode structure is mounted in the gap between the cylindrical pole pieces. Some designers prefer to taper the poles to give a higher value of uniform field, but this usually means a reduction in useful area. The pole faces are shaped very carefully and must be precisely machined in order to give the correct, very slight, decrease in 19
B. L. Cohen, Rev. Sci. Instr. 24, 589 (1953).
3.2.
ARTIFICIAL SOURCES
633
field with radius. It is usually necessary t o add magnetic shims to produce the exact field shape and to correct for inhomogeneities. Excitation coils, whose cross section represents a balance between initial cost and power consumption, are liquid-cooled and wound around both pole pieces. The charged particles originate in an ion source located near the center of the accelerator and this source can be of several special designs, depending on the type of particle.20,21*22 Essentially, the ion source consists of a vertical pipe in which a n arc discharge is maintained and the ions are extracted through a n aperture in the side and are then immediately captured by the accelerating field.
FIG.3. Structural shape of a cyclotron magnet; exciting coils are wound around the cylindrical poles. Carefully shaped shims are attached to the pole tips to give the proper dependence of field with radius and to correct for azimuthal inhomogeneities.
The radio-frequency power supply must deliver some tens of kilowatts at frequencies that are usually between 10 and 20 Mc/sec. Of this power, an appreciable fraction is needed for the beam (1 ma a t 20 Mev corresponds t o a power of 20 kw). The remainder is dissipated in copper losses in the resonant accelerating structure. Voltages maintained across the accelerating gap are of the order of 100 kv and, in the larger machines, can run as high as 500 kv. For best operation, it is desirable that this voltage be as high as possible; this permits acceleration to take place in as few revolutions as possible and assists in maintaining the phase relation between the particles and the accelerating field. Two effects set an upper limit in energy for a given cyclotron design. First, the necessary decrease with radius of the magnetic field eventually M. S.Livingston, Revs. Modern Phys. 18, 293 (1946). R. S. Livingston and R. J. Jones, Reu. Sci. Instr. 26, 552 (1954). 2* R. J. Jones and A. Zucker, Rev. Sci. Znstr. 26, 562 (1954). Zo
21
634
3.
SOURCES O F NUCLEAR PARTICLES A N D RADIATIONS
causes the particles to fall out of resonance with the accelerating field. Second, the relativistic increase in the mass of the particle as its energy increases also causes the particle to fall behind the correct phase of the accelerating field. One can raise the limits due to these effects, somewhat, by increasing the voltage across the dees, and reducing the number of revolutions for which the phase must be correct, but this will require larger magnet-gap size and thus increase the magnet power. The two limiting effects in combination make it difficult to design a constant-frequency cyclotron for energies higher than about 30 Mev for protons and 40 Mev for deuterons. Practical considerations, such as costs of magnet and rf system, probably set the limits somewhat below this. Although the beam obtainable from a cyclotron is generally more intense than that from an electrostatic generator, its quality is not nearly so good. Extracted beams have dimensions of centimeters in comparison with the millimeter-diameter beams obtainable from the electrostatic accelerators. Energy spreads of several per cent are found in cyclotron beams whereas beams from electrostatic generators are essentially monochromatic. If magnetic analyzers are used to obtain well-defined and monochromatic beams from cyclotrons, the resultant beam will be orders of magnitude lower in intensity than that from an electrostatic generator. Hence, the electrostatic generator has been preferred for precise measurements of nuclear energy levels in its energy range up to about 5 MeV. The new tandem Van de Graafl accelerators will provide much less intensity but will begin to compete in energy with the classical cyclotron. Proton linear accelerators are other competitors but, with their short duty cycle, provide lower average intensity. The advantages of the cyclotron lie in its lower cost, simplicity of construction, and reliability of operation. 3.2.2.3. The linear Accelerator. Although the linear accelerator was the first accelerator to use several low-voltage accelerating steps rather than a single potential dropll the lack of high-frequency rf sources of sufficient power prevented the exploitation of this type of accelerator for many years. Then, too, the early success of the cyclotron had provided a copious source of particles of many MeV. However, since World War 11, the development of high-power high-frequency sources and microwave techniques has led to renewed interest and to the construction of linear accelerators for protons and heavy ions in the tens of Mev range and for electrons to the billion-volt range. The chief advantage of the linear accelerator is, of course, its ability to provide emergent beams that are well collimated in the forward direction and, at high energies, with small energy spreads. Early linear accelerators consisted of a series of hollow electrodes, successive ones being charged by an ac source with opposite polarity to give electric fields in the direction of the beam in the gaps between the elec-
3.2.
ARTIFICIAL
SOURCES
635
trodes. The gaps were spaced so that the particles, with increasing velocity, would arrive at the gap when the electric field was accelerating. However, a modern linear accelerator can be considered to be a large waveguide (usually a cylindrical pipe) which carries an electromagnetic wave in its interior and the axial electric-field component of the wave provides the accelerating force. The structure of the waveguide differs depending upon whether or not the particles are highly relativistic. Since electrons a t relatively low energies have very nearly the speed of light (at 2 Mev the ) can travel in step with the wave in a relatively simvelocity is 0 . 9 8 ~they ple pipe. Protons and heavy ions travel much more slowly and, therefore, require a series of “drift tubes” to shield them from the rf wave during the part of the cycle not suitable for acceleration. Since the parameters and problems of the two types are quite different, they will be treated separately. Theoretical and construction details of both types have been covered fairly comprehensively in a review article by L. Smith.23 3.2.2.3.1. THE PROTON LINEARACCELERATOR. The prototype for modern proton linear accelerators was the 32-Mev machine designed and built at the University of California under the direction of Alvarez and P a n o f ~ k yResearch .~~ machines have also been built a t the University of Minnesotaz6and the Rutherford Laboratory, Great Britain. Other proton linear accelerators are in use as injectors for high-energy proton synchrotrons (see Section 3.2.2.5.2). A proton linear accelerator is capable of delivering currents of the order of 1 ma, or more, in a sharply focused beam at energies which, in principle, are unlimited; however, in practice no linear accelerator for protons has been built to date (1962) for energies higher than 70 MeV. Although the well-focused beam emerging from a linear accelerator is more easily available and has a higher intrinsic intensity than the extracted beam from a cyclotron, the linear accelerator has not been popular because of its relatively high cost. This cost is, in turn, associated with the high fields (of the order of 2 x 106 volts/meter) that must be maintained in the accelerator cavity to attain Mev energies in reasonable distances. The maintenance of such high fields results in power losses of the order of 60 kw/Mev of acceleration : in a 50-Mev machine the power required is about 3 Mw. For this reason, the linear accelerator is operated in short pulses with a duty cycle that rarely surpasses 1 %. Successful acceleration in the linear accelerator (together with the synchrocyclotron-Section 3.2.2.4, and the synchrotron-Section 3.2.2.5) 23 Lloyd Smith, “Handbuch der Physik” (S.Flugge, ed.), Vol. 44, pp. 341-388. Springer, Berlin, 1959. 24 L. W. Alvaree, et al., Rev. Sci. Znstr. 26, 111 (1955). 26 E. A. Day, et al., Rev. Sci. Znstr. 29, 457 (1958).
3.
636
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
depends upon the principle of phase stability discovered independently by McMillanZ6and Vek~ler.~’ According to this principle, radio-frequency electric fields can be applied in accelerators in such a fashion that particles undergoing acceleration will continue to experience this field a t or near a correct accelerating phase. This accelerating process is self-correcting over a certain phase region and can continue indefinitely without the particle falling out of phase with the field as do the particles in the classical cyclotron. This synchronizing mechanism can be readily understood in the case of the linear accelerator where stable acceleration is possible for the E sin w t --r
FIQ. 4. Representation of a series of accelerating gaps, separated by a distance ds. Across each gap an electric field E sin wt is applied.
rising part of the accelerating-field wave, that is, between 0” and 90”. If the particle arrives a t an accelerating region too soon, it will receive less energy than it should and so will take longer to arrive a t the next accelerating region. If it reaches the accelerating gap after the correct time, it will gain too much energy, will reach the next gap too soon, again closer to the desired phase. Phase stability can be expressed in more general terms as follows. Let an accelerating system consist of a sequence of accelerating gaps, as shown in Fig. 4, each separated from the next by a distance ds. If a particle, having a prescribed “equilibrium” energy, passes a gap when the phase of the accelerating field is +,, the “equilibrium” phase, it will gain exactly the correct amount of energy. Thus, its velocity will change by the correct amount so that it will arrive a t the next gap when the field is a t the same phase (provided the gaps are spaced correctly) and so it will continue on a prescribed acceleration cycle. We shall now investigate the behavior of a particle that arrives at an incorrect phase @. If W is the energy of this particle and W o is the energy of the equilibrium particle, the energy change as the particle crosses the gap will be given by
d(W - W,) 25E. 27
= =
eV(sin @ - sin 4,) eE ds(sin - sin 40)
+
(3.2.2.11)
M. McMllan, Phys. Rev. 68, 143 (1945).
V. I. Veksler, Compt. Rend. Acad. Sci. U.S.S.R. 43, 444 (1944);44,393 (1944);
J. Phys. (U.S.S.R.) 8, 153 (1945).
3.2.
637
ARTIFICIAL SOURCES
where V is the peak voltage across the gap and E is the average accelerating field. We now assume that the equilibrium particle travels from one gap to the next in exactly one full period of the accelerating signal whose frequency is w/27r. Then, if v is the velocity of the equilibrium particle, w
ds/v
=
27r.
+
If the nonequilibrium particle has a velocity v 6v, it will traverse the 6v). If it passed the preceding distance between gaps in a time w ds/(v gap when the rf phase was 4 it will arrive a t the next gap a t a phase (4 d 4 ) given by d 4 = w ds 1 -
+
+
(m- 1)-
(3.2.2.12)
Equations (3.2.2.11) and (3.2.2.12), in combination, describe the behavior of the phase of arrival of the nonequilibrium particle. One can now apply these equations t o the proton linear accelerator. It is assumed that the particles can be considered to be nonrelativistic and that the kinetic energy can be expressed as & mu2.Also, the variations in phase and velocity from those of the equilibrium particle are assumed to be relatively small. Then the equations become: d
- (v
ds
6v)
=
(eE/m)cos
40
64
(3.2.2.13)
and d ds
- (64)= -w6v/v2
(3.2.2.14)
where 64 = 4 - 40.Over a range where v changes relatively slowly, these equations can be combined to give
d2 &5
(64)
+
eEw cos mv3
(bo
64
=
0.
(3.2.2.15)
If cos 4o is positive, this represents a stable oscillation in phase. Since sin 4o must also be positive to obtain acceleration, this means that, for the linear accelerator, the equilibrium phase must lie between 0" and 90". If this condition is satisfied, particles that wander from the equilibrium phase will undergo stable oscillations about this phase and will continue indefinitely to be accelerated. The amount of phase error or, in other words, the range of "phase acceptance," depends upon the value chosen for the equilibrium phase and is larger the closer the equilibrium phase is to 0". A larger range of phase acceptance means, of course, a larger fraction of the injected beam
638
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
can be successfully accelerated. However, the voltage across the accelerating gap will be lower for a choice of equilibrium phase near 90'. A compromise value of about 60" is typical for a linear accelerator; this results in a phase-acceptance range from about 30" to 120' and acceleration of about one quarter of the injected beam. With the stable phase oscillations are associated oscillations in velocity or energy as indicated by Eq. (3.2.2.12). These oscillations are usually relatively small. A more precise analysis indicates that the phase oscillation is damped during acceleration with an amplitude approximately proportional to the -ith power of the particle energy. Typical parameters for a proton linear accelerator might be as follows: w = = v = 40 =
E
1.25 X lo9 (200 Mc/sec), 2 X lo8 v/meter, 1.4 X 10' (1 MeV) or 7 X lo7 (25 Mev), 60".
Then, from Eq. (3.2.2.15), the wavelength of the phase oscillation is approximately 1 meter at 1 Mev and approximately 10 meters at 25 MeV. From Eqs. (3.2.2.14) and (3.2.2.15), the fractional energy error a t 1 Mev is 15%/radian of phase oscillation and a t 25 Mev is about 7%/radian of phase oscillation. But a phase oscillation of 1 radian amplitude a t 1 Mev will have damped by a factor of (25)-3/u when the protons have been ' energy oscillation at 1 Mev will accelerated to 25 MeV. Therefore, a 15% correspond to an energy oscillation of only 2% a t 25 MeV. This is about the energy spread observed in proton linear accelerators, and at 50 Mev energy spreads of less than 1% are obtained. The accelerating field is usually the axial component of the transverse magnetic (known as TMolo) mode in a cylindrical waveguide. The frequency of operation is determined from the requirement that the radial variation of the field across the aperture be small. From an exact solution of Maxwell's equations for the pertinent field pattern in the waveguide, it can be shown that, for a field variation of 10% across the aperturelzs (3.2.2.16) where rmaxis the half-aperture. If the injection energy is about c/30 and the aperture is 1 cm, or more, the wavelength must be greater than 1.5 meters and the frequency must be no greater than 200 Mc/sec. This is the frequency chosen for most proton linear accelerators. As previously stated, the protons are shielded from the field during the 28 M. S. Livingston and J. P. Blewett, "Particle Accelerators," p. 315. McGraw-Hill, New York, 1962.
3.2.
ARTIFICIAL
639
SOURCES
nonaccelerating part of the cycle by a series of drift tubes, as shown in Fig. 5 . The lengths of the drift tubes are such that the equilibrium protons require exactly one full rf period to traverse them. Hence, the proton that arrives at the gap between drift tubes at an accelerating phase will continue to experience accelerating fields at succeeding gaps. As indicated in the figure, the drift-tube length increases as the proton velocity increases.
Drift tube st:em
Drift tube
FIG.5. Schematic diagram of interior of a proton linear accelerator showing d;ifttube structure. The protons are traveling from left to rightdand the drift tubes-are made longer as the velocity of the protons increases.
FIG.6. Pattern of the electric field between drift tubes which will produce a radial defocusing effect.
Supports for the drift tubes take the form of radial rods attached to the wall of the cavity, but these rods make negligible deformations in the basic field pattern. Radial focusing is essential in a proton linear accelerator for reasons illustrated in Fig. 6. As an off-axis proton enters an accelerating gap it experiences a radial focusing force and as it leaves the gap it feels a radially defocusing force. But, in order to satisfy the requirements of phase stability, the field must be rising as the proton passes the gap.
640
3.
SOURCES OF NUCLEAR PARTICLES A N D RADIATIONS
Hence, the defocusing force is the stronger. This problem was solved in the early machines by the introduction of grids to change the pattern as shown in Fig. 7. If the grids are made to intercept as little as possible of the proton beam, their focusing properties are poor; even with the best compromise, losses are high. Modern linear accelerators use alternatinggradient focusing (see Section 3.2.2.6) obtained by the inclusion of quadrupole magnets inside the drift-tube structure. With this method, there are no appreciable losses during acceleration. Protons are injected into the linear accelerator a t energies in the range of 500 kev to 1 MeV. In almost all cases, the injector is a Cockcroft-Walton Drift tube
FIG.7. Elect,ric-field pattern after the insertion of a grid; this pattern will be focusing.
generator (Section 3.2.1.1). Of the injected beam, about one quarter is accepted in the phase-stable region, the rest is rejected a t the low-energy section of the accelerator. The acceptance can be approximately doubled by the inclusion of a “buncher.” The buncher, inserted between the injector and the linear accelerator, is an rf cavity wherein a small modulation in energy is given to the injected beam. Also included is a drift space where the protons which have been accelerated in the buncher catch up with those that have been decelerated so that the injected beam is then bunched in phase. When the buncher’s field is correctly phased with respect to the field in the linear accelerator, these injected bunches fall in the region of phase acceptance and the final output intensity is increased. The dimensions of a linear-accelerator cavity for 200-Mc excitation are about 1 meter diameter and 50 cm/Mev length. In early machines, the rf structure was a lightweight copper cavity supported in a steel vacuum tank. Newer designs combine the rf cavity and the vacuum tank in a structure of copper-clad steel. The steel wall necessary to maintain rigidity under vacuum is lined with copper to decrease rf losses. Much electrical conditioning of the cavity is necessary before the structure will support the high fields which reach values of the order of lo7 volts/meter between drift tubes. Figure 8 shows a photograph of the 50-Mev proton linear accelerator that serves as an injector for the Brookhaven alternating-gradient synchrotron.
3.2.
ARTIFICIAL SOURCES
641
FIG.8. The 50-Mev proton linear accelerator which serves as an injector for the Brookhaven 33-Rev AGS. Protons are injected a t 750 kev from a Cockcroft-Walton accelerator situated off the bottom of the photo. Rf power is fed into the linac near the center of the 110-ft long tank through the large triode amplifiers shown near the top, center. Protruding from t h e right of the tank are connectors to the “ball tuners” which are used for final tuning.
642
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
Linear accelerators, similar in basic design to the proton machines, have been used to accelerate heavy ions to energies as high as 10 Mev/nucleon. Machines of this type are in operation a t the Lawrence Radiation Laboratory and at Yale University.29 Since heavy ions travel even more slowly than protons the frequency is lower, of the order of 70 Mc, with a corresponding increase in the tank-cavity diameter to about 10 ft. In recent years, there has been considerable interest in extending the energy of proton linear accelerators into the Bev range, both for research purposes and as injectors for super-high-energy synchrotrons. Although, for a few Bev, a proton synchrotron is much less expensive, the linear accelerator would give higher intensity by several orders of magnitude. Proposed designs have included an extension of the drift-tube structure, described above, to about 200 Mev and then a change to a microwave structure (as in the electron linear accelerators) to reach higher energy. However, even at several Bev protons are still not very relativistic, they would still need transverse focusing, and the complexity of the beam dynamics poses many problems. Some of the difficulties associated with such an extension in energy have been recently summarized by L. Smith.30 3.2.2.3.2. THEELECTRON LINEARACCELERATOR. The structure of an electron linear accelerator differs markedly from that of a proton linear accelerator, as mentioned in the introduction to this section, primarily because electrons travel a t speeds so close to that of light. This results, for one thing, in a quite different choice of frequency. From Eq. (3.2.2.16) it can be seen that the problem of field variation across the aperture is no longer a determining one. As frequency is increased, dimensions of the accelerating cavity decrease and losses per unit surface area are increased. Since the dimensions are inversely proportional to frequency and losses rise only as the square root of frequency (skin effect), net losses decrease in proportion to the square root of the frequency. Hence, it is desirable to increase frequency to the point where the dimensions of the system begin to limit the aperture available to the accelerated beam. The frequency chosen for electron linear accelerators has, in most cases, been about 3000 Mc (10 cm wavelength). Since the electrons travel with the accelerating wave, the drift-tube structure is no longer necessary. The transverse-magnetic-field pattern is still essential, since it has an axial electric-field component, and the desirable condition is approached by a cylindrical waveguide supporting this mode in a traveling-wave system. Empty waveguides have phase velocities greater than the velocity of light but this phase velocity can E. L. Hubbard, et al., Rev. Sci. Znstr. 39, 621 (1961). Lloyd Smith, “Proceedings, International Conference on High Energy Accelerators, Brookhaven, 1961,” p. 73. U.S.Government Printing Office, Washington, D.C. 20
$0
3.2.
643
ARTIFICIAL SOURCES
be decreased to the desired value by loading the system with a series of irises (hollow disks), illustrated in Fig. 9. Appropriate choices of the dimensions of the waveguide, the dimensions and spacing of the irises, will result in a phase velocity equal to that of light. It should be mentioned that it would be extremely inefficient to add this sort of loading to reduce the velocity to that suitable for protons or heavy ions. Electron and proton linear accelerators differ in at least two other important aspects. With the velocity of the electrons so close to c and so effectively constant, phase stability ceases to be effective. However, in order that the electrons remain a t an accelerating phase of the rf wave, the phase velocity must be exactly correct which means that the dimensions of the guide must be maintained very precisely. Machining tolerances
FIQ.9. Iris-loaded waveguide used in electron linear accelerators.
are of the order of 0.0002 in. and operating temperatures must be precisely controlled. Again, a t extremely relativistic velocities, the radially defocusing forces mentioned in the previous subsection virtually disappear,”and focusing is no longer required. Most modern electron linear accelerators are designed according to the principles evolved by a group a t Stanford University originally under the direction of W. W. Hansen. This group was responsible for the development of 3000-Mc klystrons with outputs of more than 20 Mw (in pulsed operation) and, using such power tubes, has developed the design of the basic accelerating structure. Presently operating at Stanford is the l-Bev electron linear accelerator, known as “Mark 111,” based on these s t ~ d i e s .Average ~ ~ . ~ ~currents of the order of 1 pa in a beam about in. in diameter are produced. The use of traveling waves rather than the standing, resonant-cavity mode used in proton machines, is not essential to linear acceleration of electrons but depends to a great extent on the type of power source, in particular, on the type of load which is optimum for the klystrons presently used. The traveling-wave mode provides the driver with a load that appears to be almost purely resistive rather than highly reactive as does a resonator of high Q. In the traveling-wave system, it is necessary to choose a group velocity that is low compared with the velocity of light, but not
+
31 32
M. Chodorow, et al., Rev. Sci. Inslr. 26, 134 (1955). R. F. Post and N. S. Shiren, Rev. Sei. Znstr. 26, 205 (1955).
644
3.
SOURCES O F NUCLEAR PARTICLES AND RADIATIONS
zero as in the case of the resonant system. Low group velocities are necessary for the maintenance of high power density and, hence, high field in the guide for the input power levels available. In the Stanford Mark I11 machine, the group velocity is about 0.01 c. With this group velocity, the accelerating field is damped to l / e of its input value in a distance of about 3 meters. At this point the wave is reflected and a new input is introduced. The complete system, about 90 meters long, is powered by thirty 20-Mw klystrons distributed along the accelerator a t separations of about 3 meters. Operation of the electron linear accelerator is pulsed and the Stanford 1-Bev machine accelerates during l-psec pulses, 60 times/sec. Injection into an electron linear accelerator is from a simple electron gun, giving electrons of energies of tens of kilovolts. These could be accelerated in a waveguide where the phase velocity is that of light, since the electrons gain velocity so quickly, but many particles would be lost because they would not be a t exactly the correct phase. In the Stanford Mark 111accelerator, following the injector gun, there is a special bunching section where the beam is bunched around a particular phase. This section is also an iris-loaded waveguide, of special dimensions and taper, where the particles are not only bunched but also accelerated by an increasing accelerating field. A large number of electron linear accelerators, based on the Stanford design, have been built in the energy range up to 100 Mev and many of these are in use for medical and radiographic applications. At higher energy, together with the Stanford Mark 111 1-Bev machine, there is a somewhat improved version, aimed a t 2 Bev, partially in operation a t the Orsay Laboratory, in France. The most recent development is a 2-mile long electron linear accelerator which has been under design for several years by the Stanford group3a,34,3K and for which construction was approved in 1961. This machine, in its first stage, should have an energy around 20 Bev. Eventually, by the addition of more power sources, it is hoped that this accelerator will reach energies of the order of 40 Bev. There is one further noteworthy advantage in an electron linear accelerator. In connection with the betatron (Section 3.2.2.1), it has already been mentioned that magnetic accelerators, in which electrons are accelerated in circular paths, are limited in peak energy by radiation losses. This 8s
R. B. Neal and W. K. H. Panofsky, “CERN Symposium Proceedihgs,” Vol. I,
p. 530. CERN, Geneva, 1956. 34 R. B. Neal, “Proceedings, International Conference on High Energy Accelerators and Instrumentation,” p. 349. CERN, Geneva, 1959. 8 6 K. L. Brown, et al., “Proceedings, Conference on High Energy Accelerators, Brookhaven, 1961,” p. 79. U.S.Government Printing Office, Washington, D.C.
3.2.
ARTIFICIAL SOURCES
645
limitation does not apply to linear machines. Consequently, the only electron accelerator under construction, in 1962, for energies above 10 Bev is an electron linear accelerator. 3.2.2.4. The Synchrocyclotron. It had been recognized during the period before 1945 t ha t the basic limitation to cyclotron energy, set by the relativistic increase in mass (see Section 3.2.2.2, and Eq. 3.2.2.10), could be removed by variation of frequency of the accelerating signal during acceleration. A t the point where the relativistic effects began to decrease the frequency of revolution of the circulating protons, the frequency of the accelerating system could be reduced and the protons could continue to be accelerated. But it had not appeared possible to do this with sufficient precision to keep the resonance conditions; until the principle of it was not realized th a t the phase stability was enunciated in 1945,26r27 protons would remain, automatically, a t or near the correct accelerating phase. It then became evident that phase-stable acceleration would not only carry the particles through the effects of relativity but could also compensate the decrease in magnetic field with radius necessary for axial focusing of the beam, With the modification to frequency variation over an accelerating period, it is, of course, no longer possible to maintain continuous acceleration. The output of the “synchrocyclotron” (sometimes called the “frequency-modulated” cyclotron) must be pulsed and its average current output will be decreased by a factor of the order of one thousand below that of the conventional cyclotron. Development of the synchrocyclotron was initiated a t the University of California where model tests verified the principle in 1946.38The 184-in. cyclotron, originally planned to be operated as a constant-frequency cyclotron with enormous dee-voltage, was then modified for synchronous acceleration and operated successfully within the year.37 This success stimulated the construction of other similar machines; together with several machines of lower energy, about a dozen synchrocyclotrons have , ~ above ~ 300 Mev been completed with energies from 100 to 700 M ~ vthose finding major application in studies of the production and properties of mesons. Above 700 MeV, the magnet of a synchrocyclotron becomes so large and the increased tuning range of the rf system presents such serious design problems that it seems improbable th a t this type of accelerator will be extended into the Bev energy range. For these energies, the proton synchrotron (Section 3.2.2.5.2) is more suitable. For the synchrocyclotron, one must rewrite the equations of Section J. R. Richardson, et al., Phys. Rev. 69, 669 (1946). W. M. Brobeck, et al., Phys. Rev. 71, 449 (1947). 38 M. S. Livingston and J. P. Blewett, “Particle Accelerators,” p. 354. McGraw-Hill, New York, 1962. 3.3
3’
646
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
3.2.2.2 in relativistic form. The orbit radius and particle energy are then related as follows: (3.2.2.17) r(meters) = [ T ( T E O ) ] ’ ’ ~ 300B
+
where the kinetic energy T and the rest energy E oof the particle are given in Mev and B , the magnetic flux density, is in webers/sq meter (units of 10,000 gauss). The revolution frequency, for the same units, is given by f(Mc/sec)
=
B 14,320 ___.
Eo
+T
(3.2.2.18)
The behavior of the particles with respect to phase stability is somewhat different from that previously described for the linear accelerator. In any circular accelerator, a particle which receives too much energy will travel on a path with a larger radius than the equilibrium particle, will take a longer time to make one revolution, and so will arrive at the accelerating gap later in phase; the particle which receives too little energy will travel on a path of smaller radius and will arrive earlier than the equilibrium particle. Thus, the “equilibrium” phase must be on the falling side of the wave or between 90‘ and 180”. Derivation of the equation describing the phase motion is similar to that presented in Section 3.2.2.3.1. The azimuthal equation of motion for a particle traveling on a circular orbit in a constant magnetic field is combined with a relation connecting phase and position of the particle.39 The resulting equation can be written in the form:
where
W is the total energy of the nonequilibrium particle, is the phase of arrival of the equilibrium particle, 64 is the error in phase of arrival (assumed to be small) of the nonequilibrium particle, V is the voltage across the accelerating gap, B is the magnetic flux density, e is the charge of the particle, c is the velocity of light. 6 0
Equation (3.2.2.19) represents a stable oscillation if cos rpo is negative but sin domust be positive to give acceleration, hence, domust lie between 89
D. Bohm and L. Foldy, Phys. Rev. 73, 649 (1947).
3.2. ARTIFICIAL SOURCES
647
90” and 180”. The instantaneous frequency of phase oscillation follows from the phase equation: phase-oscillation frequency
Be@ eV cos $0 21r nW3
= __
(
)
112
-
Typical values for insertion in this expression would be (for protons):
B = 20 kilogauss = 2 webers/sq meter, W / e = 1.038 X lo9 ev (100 Mev kinetic), V = 20 kv, 40 = 150”. For these values, the frequency of phase oscillation would be about 50 kc. Since the frequency of revolution of the particles (Eq. 3.2.2.18) is about 28 Mc, the phase oscillation requires many revolutions of the particles for one full period. The requirements for the accelerating system in a synchrocyclotron differ in two main respects from those in a constant-frequency cyclotron. First, the dee-to-dee voltages in the synchrocyclotron can be relatively low (a few tens of kilovolts) since phase stability preserves the accelerating phase relation. This makes it possible to apply rf to a single deethe other dee is grounded and becomes a so-called “dummy dee.” Second, the dee system must remain resonant over a wide frequency range; during acceleration to 700 MeV, for example, the frequency of excitation must drop to less than 60% of its initial value. This requirement is met by mechanical tuning of the excited dee. Most synchrocyclotrons perform this tuning by inclusion of a rotating capacitor connected, in some cases, a t the end of the dee-stem and in other cases directly across the accelerating gap. The latter solution presents some difficult problems associated with eddy currents induced in the moving metal parts by the high magnetic field, Recently, the large synchrocyclotrons have adopted a vibratingreed type of variable capacitor which takes the form of a large, magnetically excited, tuning-fork structure mounted around the dee structure so that vibrations of the blades of the tuning fork vary the dee capacity to ground and so change the resonant frequency of the dee ~ y s t e m .I ~n ~ all. ~ ~ cases the mechanical tuner repeats its cycle a t a rate between 30 and 100 cycles/sec. This cycling rate governs the rate at which accelerated beam pulses are yielded by the machine. Magnets for synchrocyclotrons are basically scaled-up versions of conventional cyclotron magnets. I n the largest synchrocyclotrons, operating 4oR. L. Thornton, “CERN Symposium Proceedings,” Vol. I, p. 413. CERN, Geneva, 1956. 4 1 F. Krienan, Nuclear Instr. and Methods 6, 280 (1959).
648
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
a t energies up to 700 MeV, magnet weights are as high as 7000 tons. Pole faces must be accurately parallel and the field must be very carefully measured and corrected for any azimuthal nonuniformities. Betatron oscillations (cf. Section 3.2.2.1) in synchrocyclotrons are similar in character to those in constant-frequency cyclotrons. But, because of the slower rate of acceleration, synchrocyclotrons are somewhat more sensitive to resonances between the various possible modes. Particularly important is the resonance that occurs when n = 0.2 (see Eq. 3.2.2.5) where the frequency of radial oscillation is exactly twice the frequency of vertical oscillation. The n = 0.2 point occurs a t the edge of the magnet where the effects of fringing begin to make themselves felt and, at this point, there is important if not total beam loss in all synchrocyclotrons. Particles which succeed in passing this resonance are usually lost at a very slightly greater radius where n reaches 0.25 and the frequency of vertical oscillation becomes exactly half of the revolution frequency. When internal targets are used, the effects of the various resonances can be avoided simply by locating the target at a smaller radius where the n-value is still less than 0.2. The pulsed internal beam intercepted by such a target will be of the order of 1 pa. The extraction methods used in constant-frequency cyclotrons are not effective in the synchrocyclotron because the increase in radius per revolution, already very small, is smeared out completely by the radial oscillations associated with the phase oscillations. However, extracted beams, of the order of about 10% of the circulating beam, can be obtained by a method proposed by Tuck and Teng42and for which extensive analysis of the beam dynamics was carried out by L e C ~ u t e u r This . ~ ~ is the “peelerregenerator” method in which the beam expands to pass, in succession, through a region where the magnetic field is weakened by a magnetic shield and then through a region where the field is increased. Successive passages through this combination induce a large betatron oscillation which eventually throws the beam into a magnetically shielded channel through which it can be extracted. This method of extraction was first n ~ ~ has ~ ~ ~been ~~~ successfully applied in the Liverpool s y n c h r o c y ~ l o t r o and adopted by most of the other large synchrocyclotrons. 3.2.2.4.1. THEMICROTRON. In Veksler’s first paper on phase stabilitylZ7 he proposed a variation of the cyclotron for electrons which is now called the “microtron.” This machine, sometimes also known as the “electron J. L. Tuck and L. C. Teng, Phys. Rev. 81, 305 (1951). (Abs.) K. J. LeCouteur, Proc. Phys. Soc. (London) B64, 1073 (1951). 4 4 A. V. Crewe and K. J. LeCouteur, Rev. Sci. Znstr. 26, 725 (1955). 46 K. J. LeCouteur, Proc. Roy. Soc. (London) A232, 236 (1955). 4 6 A. V. Crewe and J. W. G. Gregory, Proc. Roy. SOC.(London) A232, 242 (1955). 4z
4s
3.2.
ARTIFICIAL SOURCES
649
cyclotron,” has a constant magnetic field and a constant accelerating radio-frequency, like the cyclotron, but the particles behave as if there were a stepwise variation in frequency so that, in behavior, it is somewhat like a synchrocyclotron. The general principle of operation can be seen if the fundamental cyclotron equations are rewritten (cf. Eq. 3.2.2.10) as follows: 1-
-
f
-r=-
w
2r Bec2
(3.2.2.20)
where f is the revolution frequency of the particles, T is the period of revolution, W is the total energy of the particles, B is the magnetic flux density, e is the charge of the particle, c is the velocity of light. Thus, the time required to complete one revolution is, for a constant magnetic field, linearly proportional to the total energy of the particle, and the increase in period from one revolution to the next is directly proportional to the energy gain, i.e.,
If the energy gain per turn is adjusted to give a n increase in period th a t is an integral multiple of the rf frequency, i.e., if the time of revolution in each orbit is one or more periods longer than in the previous orbit, the particles will return to the accelerating region a t the same phase for each turn. Of course, the time taken for the first turn must also be a n integral number of cycles of the radio-frequency and it can be shown that the minimum number is 2 cycles. Each succeeding orbit can take one or more cycles longer than the previous one and all orbits have a common tangent a t the accelerating region as shown in Fig. 10. Veksler showed2’ th a t the motion was stable in phase for very small changes in the frequency or the magnetic field. A more complete analysis was carried out later by Henderson el al.47 Several microtrons, with average currents of about f pa, have been built for energies of a few MeV, including one to serve as a n injector for the 1.2-Bev electron synchrotron a t Lund, Sweden. The largest one built, to date, is a 29-Mev microtron completed a t University College, London, in 1958.48This machine has a magnet with a pole diameter of 80 in. and 4’ 48
C. Henderson, et al., Proc. Phys. SOC.(London) B66, 41 (1953). D. K. Aitken, el al., Proc. Phys. SOC.(London) 77, 769 (1961).
650
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
uses a 2-Mw magnetron, operating a t 3000 Mc/sec, as a source of rf power. It produces an average current of about 0.1 pa and an extracted beam of about one quarter of this intensity. The chief advantage of the microtron lies in the relative simplicity with which one can extract the beam because the orbits are widely spaced. A t the point opposite the rf cavity, the orbits will be spaced a distance apart of nX/r, where X is the rf wave-length and n is the number of rf cycles between turns. For n = 1, and X = 10 cm, as in the 29-Mev machine, the
FIQ. 10. Representation of the electron orbits in a microtron; the magnetic field is perpendicular to the plane of the diagram.
distance is about 3 cm. A fairly simple pipe can serve as a magnetic shield to allow the electrons to emerge but care must be taken that it does not disturb the main magnetic field enough to change the orbit periods. For an accelerator using internal targets or for the production of X-rays, the microtron has no particular advantage over the betatron. Although the frequency is constant and the rf system can, therefore, be relatively simple, large amounts of power are required because the steps in energy must be a fairly large fraction of the total energy. In a high-energy microtron, the first revolution of the beam does not clear the rf cavity and special slots have to be incorporated in the cavity. A high degree of azimuthal uniformity is required in the large magnet and the radial de-
3.2.
ARTIFICIAL SOURCES
651
crease in magnetic field required for axial focusing must not be so large that it disturbs the orbit period. Thus, extension to higher energies introduces many complexities of design. The possibility of designing a microtron for protons in the Bev energy range has been suggested by Roberts.49 However, this would involve something like a 20-Mev linear accelerator as an accelerating cavity and a very complicated magnet structure to provide the required focusing. Although it might be possible to obtain high currents in such an accelerator, it would probably be very expensive. 3.2.2.5. The Synchrotron. I n the synchrotron, particles undergoing acceleration are constrained by a magnetic field to travel on a n orbit which is approximately circular and whose radius remains fixed. From Eq. (3.2.2.17)it can be seen that, to maintain a constant radius, the magnetic field must be increased during the time taken for acceleration from a low value corresponding to the injection energy to a final value corresponding to maximum energy. Since the maximum value for magnetic fields of the necessary good quality is of the order of 10-15 kilogauss, this value plus the maximum energy desired fixes the radius of the accelerator. The annular magnet has a field which decreases with radius like (r/ro)-n to provide radial and vertical restoring forces as in the betatron (Section 3.2.2.1). Accelerating fields are provided by one or more rf stations distributed around the magnet ring and the frequency must increase to keep in step with the increasing velocity of the particles. Invention of the synchrotron was announced approximately simultaneously by VekslerZ7and McMillan.26 The name “synchrotron” was suggested by McMillan and is followed in the western world whereas Veksler proposed the name “synchrophasotron” by which this type of accelerator is known in the Soviet Union. This section will discuss only “constant-gradient” synchrotrons, i.e., those where the radial variation in magnetic field is kept constant in azimuth. Alternating-gradient (change in radial gradient with azimuth) synchrotrons will be covered in the next section. The majority of constant-gradient synchrotrons in existenceK0 accelerate electrons to energies which range from a few tens of Mev up to somewhat over 1 Bev. For energies above 50-100 MeV, they are preferable to betatrons because they require less steel and because radiation losses can be compensated more easily. Since the synchrocyclotron can provide protons of energies of several hundred MeV, proton synchrotrons have been built for energies above 1 Bev and are the only proton accelerators to penetrate this energy region. At present (1962) the highest energy constant-gradient proton synchrotron is the 10-Bev synchrophasotron a t Dubna, U.S.S.R., but a 12.5-Bev machine, a t the Argonne National Labo6o
A. Roberts, Annuls of Physics 4,115 (1958). G. A. Behmrtn, Nuclear fnstr. S, 181 (1958);6, 129 (1959).
652
3.
SOURCES O F NUCLEAR PARTICLES A N D RADIATIONS
ratory, U.S.A., is expected to be completed in 1963. Electron synchrotrons and proton synchrotrons differ appreciably in design, chiefly because electrons above a few Mev have a velocity almost equal to that of light, and the two types are treated separately, below. Phase stability in both types of synchrotron is governed by the same considerations as those previously outlined for the synchrocyclotron. Particles which receive more energy than the “equilibrium” particle travel on orbits of somewhat larger radius and take a longer time to return to the accelerating station. However, in a synchrotron, three additional factors must be included. First, the magnetic field varies more rapidly with radius than in the synchrocyclotron. In the synchrocyclotron, the field was assumed to be very nearly constant so far as the phase motion is concerned whereas, in the synchrotron, the radial field index, n [ = - ( d B / d r )( r / B ) ] ,may have values between 0.25 and 0.75 and these effects must be included. Second, the magnetic field varies with time; thus, a changing flux links the particle orbit and provides a field of the betatron type. In the synchrotron this is usually a decelerating field which, alt,hough not large compared with the rf accelerating field, must be included in the derivation of the phase equation. Finally, in the electron synchrotron, radiation losses a t high energy become very important. These losses have already been mentioned in Section 3.2.2.1, in connection with the betatron, and Eq. (3.2.2.7)shows that the energy lost per revolution can become appreciable above 100 MeV. Such losses must be compensated by suitable increases in the accelerating field. If radiation effects are neglected, the phase equation for the synchrotron is : d dt
where
W is the total energy of the particle, n is the field index = - ( d B / d r ) ( r o / B ) , B is the magnetic flux density, moc2 is the rest energy of the particle, h is the “harmonic number” of the accelerating system, i.e., the ratio of the frequency of the accelerating field to the frequency of revolution of the particle, V is the peak voltage on the accelerating gap, 4ois the phase of arrival of the “equilibrium” particle, 64 is the phase error of a nonequilibrium particle, r a is the radius of the “equilibrium orbit.”
3.2.
ARTIFICIAL SOURCES
653
In constant-gradient synchrotrons, n is always less than unity (see Eqs. 3.2.2.6). Thus, since W is always greater than moc2,Eq. (3.2.2.21) will As in the synrepresent a stable oscillation for negative values of cos chrocyclotron, the stable phase, must lie between 90" and 180". For relativistic energy ranges, the phase oscillations are damped like (W)-ll4. In the nonrelativistic range, the phase is very weakly damped or undamped depending on whether n is greater or less than 4. Detailed treatments of the orbit dynamics in electron synchrotrons have been made by Bohm and Foldy5' and in the proton synchrotron by Twiss and Frank. Comprehensive review articles, including constructional details of many existing machines, have been written on the electron synchrotron by Wilson63and on the proton synchrotron by B l e ~ e t t 5 ~ and by Green and Courant.66 SYNCHROTRON. In physical appearance, the 3.2.2.5.1. THE ELECTRON electron synchrotron is similar to the betatron. The heavy central core necessary for betatron acceleration is absent but the ring magnet, with its , similar poles shaped to give a field falling off with radius like ( T ~ / T ) - ~ is to the ring magnet used in the betatron; a cross-sectional view is shown in Fig. 11. Since, in most cases, operation is pulsed a t frequencies between 20 and GO cycles/sec, the magnet structure is laminated and is usually made of transformer steel, with lamination thickness of about 0.014 in. In many electron synchrotrons, the magnet is resonated with a shunt capacitor bank and is driven directly from the ac power line or by a poweroscillator circuit. Continuous excitation of the magnets for the larger synchrotrons calls for power dissipation of several hundred kilowatts and, to avoid excessive heating of the magnet structure, these machines are often operated a t lower pulse rates and use grid-controlled rectifiers to discharge the capacitor bank into the magnet coils. The magnet gap is usually shaped to give a field index, n, of the order of 0.7. This choice results in stronger focusing forces in the vertical (axial) direction than in the horizontal (radial) direction and so results in a shorter and wider magnet gap. Such a choice proves to be more economical in total steel, copper, and power than would result with a smaller n-value. In the synchrotron, radial aperture is also required for the radial motions which are associated with the phase oscillations; these are smaller, as a D. Bohm and L. Foldy, Phys. Rev. 70, 249 (1946). R. Q . Twiss and N. H. Frank, Rev. Sci. Inslr. 20, 1 (1949). 63 R. R. Wilson, "Handbuch der Physik" (S. Fliigge, ed.), Vol. 44, p. 170. Springer, 61
61
Berlin, 1959. 6 4 J. P. Blewett, Repts. on PTOgr. in Phys. 19, 37 (1956). 66 G. K. Green and E. D. Courant, "Handbuch der Physik" (S. Fliigge, ed.), Vol. 44, p. 218. Springer, Berlin, 1959.
654
3.
SOURCES OF NUCLEAR PARTICLES A N D RADIATIONS
rule, than the “betatron” oscillations arising from errors in injection position and angle (cf. Section 3.2.2.1). The total aperture, i.e., the extent of suitable field pattern available inside the vacuum chamber, is usually about 5-10% of the orbit radius in the radial direction and about half this amount in the vertical direction. To avoid undesirable effects of eddy currents in the vacuum chamber, this component is usually constructed of glass or ceramic sections. The interior insulating surface is coated with a thin conducting layer of high
Magnet
i
i
Orbit
centerline I
i
i FIG. 11. Cross-sectional view of the magnet structure in an electron synchrotron. The complete magnet has the form of a ring with its center at the left of the diagram.
resistance to prevent the collection of charge that might produce high local electric fields. The injector is an electron gun, similar to that used in the betatron, that provides particles a t energies of 50-100 kev, i.e., at velocities about half the velocity of light. In some electron synchrotrons, attempts have been made to initiate rf acceleration directly after injection. However, this procedure has the disadvantage that the radio-frequency must be matched to the electrons’ velocity and increased rapidly as this velocity increases. A t a few MeV, the velocity of the electrons is so close to that of light that the frequency can be held constant; the small change in velocity as the energy is increased further will result in negligible variation in the radius of the equilibrium orbit. But the initial rapid frequency change results in such technical difficulties associated with wide-band circuits that it has seemed desirable to avoid this problem and to use betatron acceleration during the first few Mev of acceleration. This is rather easily accomplished by the inclusion of a few flux bars shunting the magnet gap inside
3.2.
ARTIFICIAL SOURCES
655
the orbit radius, as shown in Fig. 11. These bars are designed to saturate when the beam energy reaches the desired value a t which the constantfrequency rf can be turned on. There is no appreciable loss of beam during the transition from one type of acceleration to the other. A few machines actually inject at sufficiently high energy that constant frequency can be used; for example, the 1.2-Bev synchrotron of the Frascati Laboratory, Italy, has a 3-Mev Van de Graaff injector. I n the early electron synchrotrons, the rf accelerating unit was a C-shaped pipe supported in the vacuum chamber and similar in its operation to the cyclotron’s dee system. Such a structure required valuable space in the magnetic aperture and has since been superseded by a quarterwave resonator constructed as shown in Fig. 12. A section of appropriate length is cut from the vacuum chamber and is silvered inside, outside, and
b-
e=@ coaxid input FIG.12. Typical quarter-wave cavity used for acceleration in an electron synchrotron.
around its ends. A ring of the coating is removed, as shown, near the end of the structure. If the length has been chosen correctly, with the dielectric constant of the chamber’s material having been taken into account, this system will act as a quarter-wave resonator. It can be excited externally by conventional rf techniques and the desired accelerating voltage will appear across the gap in the silver coating. A few lines are scribed azimuthally in the coating to reduce eddy currents to negligible levels; these do not seriously impair the rf properties of the system. The resonator can be designed for operation a t the frequency of revolution of the electrons or a t a multiple, h, of that frequency. High values of h have the advantage of reducing the size of the resonator and of reducing the radial oscillations associated with the oscillations in phase. If h is greater than unity, the phase acceptance process during acceleration will produce h bunches of electrons around the azimuth of the machine. I n the electron synchrotron, an important innovation has been the introduction of field-free “straight sections,” first proposed by a group at
656
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
the University of Mi~higan.66,~’ It was demonstrated that the inclusion of four such straight sections would require a slightly larger magnetic aperture but would otherwise not have deleterious effects on the various motions of the electrons. From the shape of its circumference, this machine was called the “race-track” synchrotron. These straight sections have proved to be very useful as locations for injection mechanisms, rf accelerating units, and targets. From an internal target the output of an electron synchrotron, like that of the betatron, is a continuous spectrum of X-rays with quantum energies up to the energy of the circulating beam. Targets can be located a t radial positions either inside or outside the particles’ orbit. Usually, a t a time just before the maximum magnetic field has been reached, the rf is turned off and, under the influence of the still rising magnetic field, the beam spirals radially inward to strike the target. Under such conditions the X-ray bursts are of the order of 10 psec. Or, to obtain maximum energy, the magnetic field may be allowed to rise to a value where saturation effects cause instability in the betatron oscillations and the beam can oscillate to strike a target at an outer radial position. For bursts of longer duration, up to 2500 psec, the rf voltage is slowly reduced; this causes large synchrotron oscillations, the electrons gradually fall out of step with the rf field and have large enough radial excursions to strike the target. Very short bursts, of the order of 0.1 psec, can be obtained by inducing rapid betatron oscillations with pulsed coils which produce local disturbances in the magnetic field. The beam can be extracted from an electron synchrotron by methods similar to those used in the betatron but such practice is not common. 3.2.2.5.2. THEPROTON SYNCHROTRON. Although the principles governing the operation of the proton synchrotron are identical with those applicable to the electron synchrotron, the existing machines differ radically in appearance and in ease of construction. Because protons do not reach velocities approaching that of light until they have energies of several Bev, it is not possible in the proton synchrotron to avoid the problems associated with varying the frequency of the accelerating system over a wide range. This has created serious difficulties in design. Since all existing proton synchrotrons have been built to reach energies of 1 Bev or more, these machines are very large in size, their magnets require very large amounts of power, vacuum chambers are large in extent and difficult to design, injectors must have high energies, and thousands of tons of shielding are required. b6 67
H. R. Crane, Phys. Rev. 69, 542 (1946); 70, 800 (1946). (Abs.) D. M. Dennison and T. H. Berlin, Phys. Rev. 69, 542 (1946).
3.2.
ARTIFICIAL
SOURCES
657
In 1962, seven constant-gradient synchrotrons are in existence and two more are almost completed. Their locations and energies are as follows: Delft (Holland). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Birmingham (Great Britain). . . . . . . . . . . . . . . . . . . . . . . . . . Brookhaven (U.S.A.) (“Cosmotron”) . . . . . . . . . . . . . . . . . . Princeton (U.S.A.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saclay (France) (“Saturne”) . . . . . . . . . . . . . . . . . . . . . . . . . . Lawrence Radiation Lab. (U.S.A.) (“Bevatron”). . . . . . . . Dubna (U.S.S.R.) (“Synchrophasotron”) . . . . . . . . . . . . . . .
1 Bev 1 Bev 3 Bev 3 Bev 3 Bev 6.2 Bev 10 Bev
Scheduled for completion in 1963: Rutherford Lab. (Great Britain) (“Nimrod”) . . . . . . . . . . . 7 Bev 12.5 Bev Argonne (U.S.A.) (“Z.G.S.”) . . . . . . . . . . . . . . . . . . . . . . . . . Design parameters for these accelerators and pertinent bibliography have been collected by B a r t ~ nOne . ~ of the primary distinguishing characteristics is the difference in magnet design. The machines a t Birmingham, Brookhaven, Princeton, Rutherford Laboratory, and Saclay have C-shaped magnets; the magnetic return path is inside the particles’ orbit and the outer edge of the vacuum chamber is exposed. Figure 13 is a cross section through the magnet of the Cosmotron and is typical of the design in this group of machines. The other synchrotrons have H-shaped or “picture-frame” magnets with magnetic returns bothinside and outside the orbit as shown in Fig. 14 which is a cross section through the magnet of the Bevatron. Such a design is more efficient magnetically and somewhat higher fields can be reached before the effects of saturation destroy the field pattern. But, in theexperimental use of such accelerators, this magnet design is inconvenient since the beam is relatively inaccessible and it is more difficult to observe or extract primary and secondary beams. The magnets in the synchrotrons a t Delft and the Argonne Laboratory differ from the others in that the field does not decrease with radius but is uniform; focusing is achieved by shaping the ends of the magnet sectors. Although this method of focusing would appear to be a radical departure, its effects are almost identical with those achieved by the radially decreasing field and the betatron oscillations have characteristics almost indistinguishable from those observed with the more conventional methods. 58 In all of these machines, except the one a t Birmingham, four or more straight sections are included whose lengths are of the order of one third of the radius of curvature. These are used for injection, for rf accelerating stations, for beam-observation mechanisms and for targeting. 68 J. J. Livingood, “Principles of Cyclic Particle Accelerators,” Chapter 4. Van Nostrand, Princeton, New Jersey, 1961.
658
3.
SOURCES O F NUCLEAR PARTICLES AND RADIATIONS
LOCATION OF
PLATE
LOCATION OF V A ~ U U MCHAMBER
FIG.13, Cross section of the magnet of the Brookhaven Cosmotron. Twelve plates, +-in. thick, are welded into blocks as shown at the left. The whole magnet structure is composed of four separate quadrants of a ring whose center is situated to the left of the diagram, 30 ft from the central orbit position.
The radius of curvature of a proton in a field of 15 kilogauss, a typical peak value in a proton synchrotron, is about 3 rneters/Bev (cf. Eq. 3.2.2.17). Thus, the magnets for the highest energies require many thousands of tons of steel. It is important, therefore, to keep the aperture as small as possible and the magnet design must be the result of very careful
Coil Vacuum chamber
Magnet yoke
FIQ.14. Cross-sectional diagram of the magnet of the Berkeley Bevatron. The poles are 5 ft wide and the gap height is 12 in.; the radius of the central orbit is 50 ft.
3.2.
ARTIFICIAL SOURCES
659
and detailed analysis of the orbit dynamics including estimates of the size of the oscillations which will be produced by imperfect injection conditions, by scattering from residual gas in the vacuum chamber, from errors in construction and erection of the magnet structure, and from errors in the frequency of the accelerating system. For example, the aperture inside the vacuum chamber of the C o s m ~ t r o nis~25 ~ in. wide by 6 in. high and the magnet gap is 36 in. by 9.35 in. ; the magnet’s cross section is about 8 ft by 8 ft. The poles are shaped to give an n-value of about 0.6 and correcting windings are included on the poles to correct the field’s shape a t the higher values when saturation effects set in. Stored energies in such large magnets reach values of tens of megajoules a t full field and rapid pulsing would result in very high power requirements. For this reason, most proton synchrotrons are operated with an acceleration period of the order of 1 sec with a repetition rate of 10-20 pulses/min. To store such large amounts of energy in capacitors is expensive and to provide large bursts of power from a normal distributing or generating system would result in a considerable dip in voltage. The usual power supply consists of a motor-flywheel-generator combination ; energy stored in the flywheel is supplied to the magnet during the acceleration period and is returned to the flywheel via the generator, acting as a motor, when the current in the magnet is restored to zero. From the observations made on beam behavior in the earlier machines, the proton synchrotron at Princeton has been designed with a smaller aperture and is pulsed a t 20 cycles/sec, to obtain higher average intensity.60 The large amount of aperture required in a constant-gradient proton synchrotron creates serious difficulties in the design of the vacuum chamber. The walls must have sufficient strength to withstand evacuation without distortion yet must be sufficiently thin that the magnet gap is not extended unduly; the materials must be nonferromagnetic and must not produce eddy currents large enough to disturb the magnetic field but the accumulation of charge must be prevented; the outgassing rates should be very low. The vacuum chambers in the various machines represent compromises in these conflicting requirements. For example, in the Cosmotron, the vacuum chamber consists of solid front and back walls of stainless steel; the upper and lower surfaces are composed of narrow bars of stainless steel insulated from each other and from the front and back walls, the whole surface being then covered with a sheet of special rubber (“Myvaseal”) of low vapor pressure. The chamber is evacuated by twelve 20-in. oil diffusion pumps distributed around the circumference of the Cosmotron Staff (M. H. Blewett, ed.), Rev. Sci. Instr. 24, 723-870 (1953). M. G. White and F. C. Shoemaker, “Proceedings, International Conference on High-Energy Accelerators and Instrumentation,” p. 362. CERN, Geneva, 1959. F,~
60
660
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
machine. I n the Bevatron, the vacuum chamber is integral with the magnet structure, as shown in Fig. 14, and is evacuated by diffusion pumps located in the straight sections. Variations in the magnetic properties of the steel in the magnets in proton synchrotrons produce large variations in the remanent field, the field left in the magnet gap when the current in the exciting windings has been reduced to zero, Thus, unless extensive correcting windings are included, the magnetic field at injection should be several times the value of the remanent field, i.e., 100 gauss or more. This requires that the injector have a n energy of several MeV. The injectors for the various 3-Bev machines are horizontal Van de Graaff generators which yield protons of between 3 and 4 MeV. The larger synchrotrons use proton linear accelerators with energies of 10 Mev and higher as injectors. The injected beam is deflected into the synchrotron’s vacuum chamber, usually by a n electrostatic inflector, at a time when the magnetic field corresponds to a n orbit which is larger than the central orbit for the machine. Before acceleration is applied, the magnetic field rises and the orbit corresponding to the injection energy shrinks toward the central value. During this time, injection can continue over several revolutions until the betatron oscillations, resulting from somewhat incorrect injection conditions, become so large that they fill the aperture. The frequency of the accelerating system must cover a range given, approximately, by the ratio of the velocity of light to the velocity a t injection. I n the Cosmotron, for example, this ratio is about 11:1. The change must take place rather rapidly during the initial part of the acceleration cycle; later, the frequency changes more slowly as the protons’ velocity becomes close to that of light. Generation of a low-level signal with the correct frequency cycle is not difficult and can be achieved by saturation of a ferromagnetic core in the resonant circuit of a n oscillator or by various mechanical tuning or heterodyning procedures. It is more difficult, however, to provide the rf level of 1000, or more, volts required a t the accelerating gap. In the Cosmotron, the power amplifier is designed for broad-band operation and the accelerating unit, a ferrite-loaded cavity, is operated untuned. I n the Bevatron, and in most newer machines, the high-level stages of the power amplifier and the accelerating unit are continuously tuned to the operating frequency. This introduces a certain amount of mechanical and electrical complexity but results in a saving in power and reduction in power ratings of components. In the earlier proton synchrotrons, the acceleration system operates at the frequency of revolution of the protons but in the newer machines a higher harmonic is used. This results in a reduction in the radial motions associated with tke phase oscillations and in a reduction in the amount of radial aperture required.
3.2.
ARTIFICIAL SOURCES
661
The location of the beam and its behavior can be continuously observed by “pickup electrodes”; these are shaped plates that are located adjacent to the beam’s orbit and on which charges are induced as the beam passes by. These electrodes can be used to indicate beam intensity, bunch structure, and radial and axial position of the beam. Information derived from pickup-electrode signals can be used for continuous correction of the radiofrequency program or, from observations of the beam’s behavior, as an indication of defects in the magnetic-field pattern. In principle, acceleration of deuterons and heavier ions is possible in proton synchrotrons but this involves major modifications in the acceleration system. To date, only the synchrotron a t Birmingham has been converted for use with deuterons. Internal targets, for the production of secondary beams, can take a variety of forms and produce bursts of particles of various durations. If the accelerating system is turned off while the magnetic field is still rising, the particles’ orbit will shrink to a smaller radius and the beam can strike a target permanently located in an internal position that does not interfere with injection. Or, targets can be moved into a position inside the inner or outer walls of the vacuum chamber late in the acceleration cycle when the betatron and phase oscillations have been damped to small values and the beam has small dimensions. The particles’ orbit can then be expanded or contracted by a small variation in the accelerating frequency. Small wire targets or foils can be quickly thrust directly into the beam or flipped through it to give short bursts, of low intensity. Or, a wire target can be left in position and a small portion of the beam can intercept it over many thousands of revolutions (continually accelerated) to give bursts that are up to 50 msec long. Synchrotrons with C-shaped magnets have the advantage that the secondary beams can emerge at any point around the azimuth whereas the machines with H-shaped magnets can locate targets only in, or very close to, the straight sections. For observations of collisions at small angles and for studies of shortlived particles, it is preferable to extract the primary high-energy proton beam. A technique used at the Cosmotron, and evolved by Piccioni et al.,sl first reduces the protons’ energy by passage through a target and the beam is displaced sharply toward an inner radius; distortion of the magnetic field by the pole-face windings increases this displacement to several inches. The beam then enters an auxiliary magnet placed in a field-free region and is bent sharply outwards to emerge from the synchrotron. By this method, about 50% of the circulating beam can be extracted. In 1962, a t the Cosmotron three external beams are available and very little use is made of internal targets. At the Bevatron, beam extraction is more diffie.1
0.Piccioni, et al., Rev. Sci.
Znstr. 26, 232 (1955).
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3.
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cult because of the magnet structure but a design similar to that described above is being incorporated and tests, early in 1962, showed good promise of success. Because there are no field-free straight sections in the Birmingham machine, this extraction technique cannot be used. However, an extracted beam of about 15% of the circulating one has been produced by using a target to give an outward oscillation to the beam which then enters the deflecting magnetic field of a coil carrying a high current. The coil is inserted pneumatically during the accelerating cycle and the beam emerges tangentially to the accelerator through an exit port.s2 Billion-volt protons produce a large variety and copious quantities of secondary particles so that heavy shielding is required for protection of personnel and for the elimination of spurious particles from experiments.s3ss4Iron-loaded concrete or steel are preferred for this purpose and most proton synchrotrons are enclosed in walls of these materials up to 10 ft thick. Computation of beam trajectories through the fringing fields of the synchrotron’s magnet then permits the design of channels through these walls for particles of various momenta. Similar shielding is required for extracted beams and their transport systems of analyzing and focusing magnets. One of the most serious problems connected with the operation of large accelerators is associated with the rapid movement of the large masses of shielding structure as the experimental program changes and develops. The maximum intensity that can be accelerated is limited by spacecharge effects arising from the electromagnetic forces present in large aggregates of charge. These forces consist of two terms: the electrostatic repulsive force between like charges and an attractive magnetic force due to the motion of the column of charge which acts like a current. These forces must be added to the focusing force provided by the shape of the magnetic field and finally become large enough to cause instability of the betatron oscillations. There will be partial neutralization of some of the space-charge effects by the residual gas in the vacuum chamber but it is difficult to estimate the degree of neutralization. The maximum number of particles which can be accepted is directly proportional to the crosssectional area of the beam and to its injection energy and inversely proportional to the radius of the orbit and to the fraction of the circumference into which the beam is bunched. In the Cosmotron, for example, the 82
83
G. A. Doran, et al., Nuclear Instr. and Methods 7 , 225 (1960). S. J. Lindenbaum, Ann. Rev. Nuclear Sci. 11, 213 (1961).
64 M. S.Livingston and J. P. Blewett, “Particle Accelerators,” Chapter 14. McGrawHill, New York, 1962. 66 J. J. Livingood, “Principles of Cyclic Particle Accelerators,” p. 71. Van Nostrand, Princeton, New Jersey, 1961.
3.2.
ARTIFICIAL SOURCES
663
maximum intensity is probably about 1013particles/pulse ; in the Bevatron, with its new 20-Mev injector, somewhat higher values should be possible. However, this limit has not been reached and intensities of only a few times 10“ have been observed; the reason is not yet completely understood. Barton and Nielsenls6in studies a t the Cosmotron, have found that the unaccelerated injected beam exhibits azimuthal instabilities and rapidly forms clusters of varying density. When the accelerating field is applied, these density modulations persist and appear to result in loss of beam. The limitation of intensity by space-charge effects has also been studied a t the Saclay accelerator, S a t ~ r n e . ~ ~ I n principle, there is no limit to the energy possible with the proton synchrotron but, for the constant-gradient type, the magnet’s cross-sectional dimensions increase approximately in proportion to the final energy. For energies above 10-15 Bev the magnet sizes and weights and the necessary power requirements become so large that it seems improbable that accelerators of this type will be built for higher energies than the 12.5-Bev machine being constructed a t Argonne. The successful operation of the alternating-gradient synchrotrons, which require much smaller aperture and smaller magnet dimensions, will be discussed in the next section. Their relative ease of construction and operation leads to the expectation that higher-energy proton synchrotrons will follow this design. The constant-gradient synchrotron, because of the ability to inject over many revolutions, can provide somewhat higher intensity per pulse but the smaller magnets of the alternating-gradient type can be pulsed more rapidly. To obtain orders of magnitude higher in intensity, the fixed-field alternating-gradient synchrotrons appear more promising. 3.2.2.6. The Alternating-Gradient Synchrotron. Alternating-gradient focusing is a method for providing strong restoring forces to particles which stray from a desired orbit, forces much stronger than those described in the preceding sections. The method was presented in 1952 by Courant, Livingston, and Snyder,B*and B l e ~ e t t ; ~itghad also been suggested earlier by Christofilos but had not been published.70With strong restoring forces, the particles need less aperture for their oscillations, the cross section of the magnet can be reduced from that in the constants* M. &. Barton and C. E. Nielsen, “Proceedings, International Conference on High Energy Accelerators, Brookhaven, 1961,” p. 163. U.S. Government Printing Office, Washington, D.C. 87 H. Bruck, et al., “Proceedings, International Conference on High Energy Accelerators, Brookhaven, 1961,” p. 175. U.S. Government Printing Office, Washington, D.C. 68 E. D. Courant, M. S. Livingston, and H. S. Snyder, Phys. Rev. 88,1190 (1952). 89 J. P. Blewett, Phys. Rev. 88, 1197 (1952). ‘O E. D. Courant, et al., Phys. Rev. 91, 202 (1953).
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gradient accelerators, and it becomes economically feasible to build machines of much higher energy. In fact, as energy is increased even to the range of hundreds of Bev, the magnet’s aperture remains almost constant16so that the cost of an AG (alternating-gradient) accelerator varies approximately linearly with energy. The general principle of alternating-gradient focusing follows from the fact that an alternation of focusing and defocusing lenses (electromagnetic or optical) of equal focal length is a focusing combination independent of which lens comes first. This becomes evident when it is realized that the greatest excursion of a ray (or a particle if ion lenses are concerned) must take place after a defocusing lens and, hence, in a focusing lens. Since restoring forces in ordinary lenses vary linearly with the distance from the lens axis, this means that the strongest forces on the erring ray are focusing forces and, as a result, the combination is a focusing one. It has been realized for many years that strong focusing forces can be maintained in one plane in an electric or magnetic field which has a strong transverse gradient. But it follows, from Maxwell’s equations and the equations of motion of particles in electromagnetic fields, that an equally strong defocusing force must be experienced in the other plane through the lens axis. In alternating-gradient focusing systems, the focusing plane alternates from the horizontal to the vertical plane by periodic reversals of the gradient in the field. Thus, focusing and defocusing alternates in both planes and net focusing results in both planes. Although the focusing forces in both planes are not nearly so strong as the focusing or defocusing forces in the individual lens elements, they can be made much stronger than the forces previously introduced in cyclic accelerators where the magnets were shaped to give a field that decreases with radius but not more rapidly than inversely with radius, i.e., an n-value between 0 and 1 (cf. Eq. (3.2.2.5)]. It would be possible to provide strong alternations in gradient by the insertion of quadrupoles71 between constant-gradient (or zero-gradient) bending magnet sections. But in synchrotrons it has been customary to combine the functions of bending and focusing in a magnet with the crosssectional form shown in Fig. 15. The poles have a shape that is approximately hyperbolic, modified where necessary to take account of the proximity of current-bearing coils. The radial gradient is of theorderof 4%/em. If the lens lengths are too great, or if the alternating gradients differ too much in value, the system becomes unstable and is no longer a focusing system. Small imperfections in the system can produce serious distortion in the orbits of the particles, as first pointed out by Adams, Hine, and 7 1 M. S. Livingston, “High Energy Accelerators,” Chapter 7. Wiley (Interscience), New York, 1954.
3.2.
ARTIFICIAL SOURCES
665
FIG. 15. Typical cross section for a magnet of an alternating-gradient synchrotron. Magnets with poles shown by the solid lines are alternated with magnets whose poles are shaped as shown by the dashed lines. In the Brookhaven AGS, the external dimensions of the steel laminations are 33 in. wide by 39 in. high; t h e pole width is 12.5 in. and the gap height a t the central orbit position is 3.5 in.
L a w s ~ nThe . ~ ~conditions for stability can be derived from a study of the solutions of the equations of motion with a periodic restoring force. This is a form of Hill's equation and its solutions have been discussed by Courant and Snyder,73L U d e r ~and , ~ ~Green and Courant.76 The orbit has a form, in either plane, described by 2 =
A/iW2cos(J ds/P)
(3.2.2.22)
where
x represents displacement from the equilibrium orbit, A is a constant, 0 is a periodic function repeating over each pair of focusing and defocusing lenses, s represents distance along the orbit.
P has a n approximately sinusoidal behavior and contributes a superimposed wiggle on the slower sinusoidal motion given by the cosine term. The basic oscillation corresponds to the betatron oscillations discussed in previous sections. Because of the stronger restoring forces, the betatron oscillations have a shorter wavelength and smaller amplitude. Several betatron oscillations will take place during one revolution of the particles J . B. Adams, M. G. N. Hine, and J. D. Lawson, Nature 171, 926 (1953). E. D. Courant and H. S. Snyder, Annuls of Physics 3, 1 (1958). '4 G. Luders, Nuovo Cimento [lo] 2, Suppl., 392 (1955). 76 G. K. Green and E. D. Courant, "Handbuch der Physik" (S. Flugge, ed.), Vd. 44, p. 300. Springer, Berlin, 1959. 2 '
73
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3.
SOURCES O F NUCLEAR PARTICLES A N D RADIATIONS
in an AG synchrotron whereas, in the CG synchrotron, the wavelength of the betatron oscillations is between one and two revolutions. It is important that the number of betatron oscillations per revolution, denoted usually by the symbol v (& in European laboratories), be nonintegral. For integral values, or values too close to integral, small imperfections in the magnet structure and placement would cause the oscillations to build up resonantly and the beam would be completely lost. Very detailed computations of the orbits over many revolutions are necessary to determine the values of the magnetic-field gradient, the value of v (known as the ‘(working point” or “tune”), and the azimuthal arrangement of magnet structure. Figure 16 shows a typical stability, or “necktie,” diagram for an AG magnet system of sixty focusing and sixty defocusing sectors with no straight sections. The lines represent regions of instability, or “stopbands.” If the effects of straight sections were included, the pattern would be shifted slightly toward smaller values of n. The strong restoring forces in AG accelerators have an additional effect : the variation of radius with particle momentum is strongly decreased. This so-called momentum compaction is described by the approximate relation between variations in orbit radius, 6r, and momentum error, 6 p : (3.2.2.23)
In CG synchrotrons, the term v 2 is replaced in this equation by (1 - n). Thus, for v-values of the order of 5, the new focusing method has a momentum compaction a factor of one hundred larger than that in the weaker focusing case. This high momentum compaction has several consequences, one of the most serious being connected with the phase stability in AG synchrotrons. The equation describing the phase oscillations can be written, d
1
d
w& (64)
di: L l / Y 2 - (m,c2/W)2
]
=
hec2V cos 40 2m02
64
(3.2.2.24)
where
W is the total energy of the particle, moc2is the rest energy of the particle, h is the harmonic order of the accelerating signal (ratio of its frequency to the frequency of revolution of the particle), V is the maximum energy gain possible per revolution, ro is the equilibrium radius, @o is the equilibrium phase, 64 is the phase error.
3.2.
667
ARTIFICIAL SOURCES
The value of Y is usually 5-10 and initially W is close to moc2;thus, this equation indicates that phase stability exists if cos &, is positive. But when W becomes greater than vmoc2,the stable phase shifts discontinuously to a value such that cos 4ois negative. This phenomenon is known as phase transition and the particle energy a t which it occurs is known as
0.05
0.10
0.15 n: N2
0.20
0.25
0.30
0.35
\ FIG.16. Stability diagram for an alternating-gradient synchrotron with a magnet of N sectors having a field index n’ and N sectors with a field index -n”[n = ( R / B , ) (dB/dR)J.The lines show the integral resonances for a system with N = 60 and the values of Yare shown along t h e edge of the region of stability. The small black diamond shows the region for stable operation of the Brookhaven 33-Bev accelerator whose Y values are both about 8.75.
the transition energy; a t this energy, there is no phase stability. There was much concern, at first, about the possibility of passing this energy but analysis showed that it should be possible to do so by making a phase jump in the accelerating system, using fairly conventional electronic techniques. At Brookhaven, a 10-Mev electron “analog” alternating-gradient synchrotron, with electrostatic bending and focusing lenses, was built to investigate transition phenomena. It was found that the particles could be
668
3.
SOURCES O F NUCLEAR PARTICLES AND RADIATIONS
accelerated past this region without serious beam In the Soviet Union, a method was proposed77 of avoiding the transition energy through the addition of compensating magnets where the sign of the magnetic field, but not its gradient, is reversed. However, the success of the highenergy AG proton synchrotrons in passing the transition point has shown that such compensation is unnecessary. 3.2.2.6.1. AG PROTON SYNCHROTRON. I n 1962, three AG proton synchrotrons are in operation. The highest proton energy yet achieved is 33 Bev in the Brookhaven AGS which was completed in 1960.78At the CERN Laboratory in Geneva, Switzerland, a 28-Bev accelerator has been in operation since 1959.79980 In 1961, it was reported that a 7-Bev machine had been brought into operation in the U.S.S.R.; this was built as a model for a 70-Bev accelerator that is under construction.8LThese accelerators are reasonably similar in general characteristics; their parameters and some operating characteristics are given in the references quoted above ; details of the Brookhaven machine are presented as being typical. Figure 17 shows a schematic drawing of the Brookhaven AGS. The magnet consists of 240 units, connected in series and housed in a tunnel almost exactly one-half mile in circumference. The magnet sections are about 7 ft long, about 3 ft by 3 ft in cross section, and the radial gradient in the gap alternates every second unit. The magnetic-field pattern has the correct value over an aperture about 3 in. high and 7 in. wide from the injection value of 120 gauss to the maximum of 13,500 gauss (on the central orbit). Interspersed among the magnet sectors are twenty-four quadrupole magnets and thirty-six sextupole magnets to make small corrections in the field pattern a t injection and to offset the effects of saturation in the main magnet at top fields. The main magnet sectors are arranged in groups of ten, each group being separated by field-free straight sections 10 ft long. The return legs of the C-shaped magnets are located inside the orbit for ten units, then outside for the next ten units, and so forth; this alternation is shown in Fig. 18. Such an arrangement leaves long clear runs alternately inside and outside the orbit for the emergence of secondT 8 G. K. Green, “CERN Symposium Proceedings,” Vol. I, p. 103. CERN, Geneva, 1956. A. A. Kolomenskij, Soviet Phys.-Tech. Phys. 1, No. 4, 721 (1957). 7* AGS Staff, “Proceedings, International Conference on High Energy Accelerators, Brookhaven, 1961,” p. 39. U.S. Government Printing Office, Washington, D.C. J. B. Adams, Nature 186, 568 (1960). *O M. G. N. Hine and P. Germain, “Proceedings, International Conference on High Energy Accelerators, Brookhaven, 1961,” p. 25. U.S. Government Printing Office, Washington, D.C. V. V. Vladimirskij, “Proceedings, International Conference on High Energy Accelerators and Instrumentation,” p. 371. CERN, Geneva, 1959.
3.2.
669
ARTIFICIAL SOURCES
BUBBLE CHAMBER HOUSE
LINEAR
X DENOTES POSITIONS OF
240 MAGNETS SERVICE BUILDING [ADMINISTRATION,POWER, CONTROLS, LABORATORIES. MACHINE SHOP, ETC.]
UNDERGROUND MAGNET TUNNEL
FEET
0 I
100 I
200 I
400 I
I
600 I
I
800 I
I
I000 I
I
FIG. 17. Plan view of the Brookhaven alternating-gradient synchrotron (AGS). Protons are accelerated to 750 kev by a Cockcroft-Walton accelerator and then to 50 Mev in the linear accelerator before entering the synchrotron. The tunnel housing the 240 magnets and twelve accelerating stations is about one-half mile in circumference and is covered with about 12 f t of earth shielding. At the top of the diagram, a small spur tunnel breaks the ring for emergence of beams to be detected by a n 80-in. liquid hydrogen bubble chamber.
ary beams. Figure 19 shows a closer view of the magnet units with the vacuum chamber visible inside the gap. The precision of alignment and the uniformity in field of magnets in an AG accelerator must be very high since the focusing or defocusing strengths of the individual lens elements are much higher than their net strength. Displacement of a lens element, or azimuthal inhomogeneity in
670
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
FIG.18. View, from above, of some of the magnets in the Brookhaven 33-Bev AGS and showing one of the twelve rf accelerating stations located in a 10-ft field-free straight section. The radius of curvature of the particles’ orbits is 280 ft but the inclusion of straight sections increases the over-all radius of the accelerator to 421.5 ft.
Q,
Y
FIG. 19. Another view of the magnets in the Brookhaven AGS showing the vacuum chamber located inside the C-shaped jaws. Two magnets providing radial focusing are followed by two magnets providing axial focusing, in succession around the ring. At the extreme right of the photo is shown one of the thirty-six sextupole magnets which can make small corrections in the second derivative of the magnetic field. Next t o the s e x t u p l e is one of the forty-eight vacuum pumping stations and it,s control box. Between the second and third magnets is shown the location of one of the thirty-six pairs of pickup electrodes whose signals can indicate the radial and vertical position of the beam a t any time during the acceleration period.
672
3.
SOURCES O F NUCLEAR PARTICLES A N D RADIATIONS
field strength, may result in an orbit displacement some twenty or more times the amplitude of the lens error. The magnets in the Brookhaven AGS were aligned to better than 0.010 in. after very careful magnetic measurements had established their magnetic characteristics and the location of the magnetic median plane and the orbit center line. The magnetic field strengths in the sectors are identical within about 2 parts in 10,000,the result of careful control of mechanical construction and several years of model studies. The stability of the underlying subsoil was investigated in detail and its mechanical properties were taken into account in the design of the magnets’ supporting structure. The vacuum chamber is a fairly simple pipe, approximately elliptical in cross section, and fabricated of Inconel, an alloy of high resistivity (about 120 pohm-cm). The wall thickness is 0.078 in., a minimum capable of supporting atmospheric pressure. Vacuum requirements are not severe : operating pressures are of the order of mm Hg. Eddy currents in the vacuum chamber not only produce small changes in the shape of the magnetic field a t injection, they also reduce the value of the field a t the orbit and small variations in the resistivity and thickness of the walls can produce serious azimuthal inhomogeneities. To diminish these effects, the rate of rise of the magnetic field is reduced by a factor of about four at the beginning of the acceleration period. The power supply for the magnet is an alternator, motor-generator, flywheel combination similar to that described in the previous section. The rate of rise of the field is reduced initially by including, in series with the magnet, a 70-ton inductor which saturates at a time when the main magnetic field is about four times the injection field. Also included in the magnet circuit is an extra magnet sector which, by field-sensing devices in its gap, provides accurate timing signals for the injection pulses and the rf starting oscillator. The injection energy of the AGS is 50 MeV, corresponding to a n injection magnetic field in the synchrotron of 120 gauss. This was felt to be the lowest value a t which the field might be reasonably free of the variations associated with the remanent field left in the magnet gap a t the end of each pulse. The injector is a 50-Mev proton linear accelerator, 110 ft long, and including 124 drift tubes (see Fig. 8). Alternating-gradient focusing is provided in the linear accelerator by the insertion, inside each drift tube, of a pulsed quadrupole and the polarities are reversed every second quadrupole. Protons are supplied to the linac by a 750-kev Cockcroft-Walton accelerator of conventional design (cf, Section 3.2.1.1). A prebuncher cavity is inserted between the preinjector and the linear accelerator and increases the output intensity of the linac by a factor of about two over that obtained in its absence.
3.2.
ARTIFICIAL SOURCES
673
From the linear accelerator, the beam is carried through a series of steering magnets and focusing lenses (quadrupole triplets) for a distance of about 140 ft to the inflection point, almost tangential to the synchrotron ring. An electrostatic field, of about 160 kv across 6 in., is applied directly across the vacuum chamber to bring the beam into its equilibrium orbit. After the beam has filled the complete circumference of the synchrotron, the inflector field is turned off very rapidly (in about 0.1 psec) SO that it does not disturb the beam on succeeding revolutions. Because of the high momentum compaction, mentioned above, multiple-turn injection has not been attempted. It may be possible to inject two or three turns by distorting the equilibrium orbit over a small region and with somewhat different injection techniques. The radio-frequency acceleration system a t the AGS operates a t the twelfth harmonic of the revolution frequency of the protons. At 50 MeV, protons have a velocity about one third the velocity of light; thus, at injection the frequency of the acceleration system is about one third its final value. The frequency range is 1.5-4.5 Mc/sec. Tolerances on frequency control in AG synchrotrons, particularly around the transition energy, are much more severe than in the CG machines and it is no longer possible to program the frequency cycle. The primary accelerating signal is derived by pickup electrodes from the beam itself. This signal is amplified and fed to twelve accelerating stations distributed around the circumference in some of the 10-ft long straight sections. The proper initial frequency is provided by a starting oscillator but the beam-control is applied after only a few revolutions of the beam. The accelerating units are ferrite-loaded cavities kept in tune by servo-controlled saturating currents. The final important control element is derived from pickup electrodes that sense the radial position of the beam. Information from this source corrects the phase of the accelerating signal and keeps the beam a t any desired radius. This control can also be used to move the beam, near the end of the acceleration period, to any desired radial position to suit targeting arrangements. The transition energy, for the AGS, is about 7 Bev; no loss of beam is observed even for some misadjustment in the phase-shifting equipment. The acceleration period to reach the top energy of 33 Bev is about 1 sec and the entire process can be repeated every 3 sec. However, a wide variety of cycling rates is available, depending on the maximum energy desired. Lower energies are obtained by stopping the acceleration process and the rising current in the magnet a t the desired time during the magnet cycle. Thus, pulsing a t 30 Bev can be repeated every 2.4 sec, a t 20 Bev every 1.6 sec, and at 10 Bev every 0.8 sec. With one-turn injection, the intensity depends on the intensity available from the injector. At present,
674
3.
SOURCES OF NUCLEAR PARTICLES AND RADIATIONS
the output from the linear accelerator is of the order of 10 ma; about 75% of this can be captured in the synchrotron and accelerated to full energy to give an intensity of 3 to 5 X 10“ protons/pulse. It is expected that the space-charge effects mentioned in the previous section will become evident when the intensity is of the order of 10l2protons/pulse. Whether it is easier to reach this limit by increasing the output from the linac or by trying to inject two or more turns is not yet apparent. Secondary beams are derived from targets moved rapidly into the region of the beam a t the end of acceleration. At these high energies, the secondary beams are strongly peaked in the forward direction so that they are brought out a t angles as small as possible with respect to the primary beam. Typical momentum spectra of the various secondary particles observed at the AGS have been reported by Cool.82Bursts from the target can be of various durations depending on the detection method. For very short bursts, useful for taking pictures in bubble chambers, a “rapid beam deflector’’ has been inserted in one of the 10-ft straight sections. This is a two-turn coil whose rapidly pulsed magnetic field distorts the orbit of the circulating protons by amounts up to 2& cm, and can provide bursts of about 50 psec duration. By locating this deflector a t a suitable position, the maximum distortion of the orbit can occur at several possible azimuthal target locations. It is also possible by this means to give only a small portion of the beam to one target and then allow the remainder of the beam to hit another target a t another azimuthal location so that two or more targeting areas can be used and several experiments can be carried out simultaneously. Figure 20 shows a typical arrangement of experimental equipment at the AGS. At the left, one target (F-10) provides a beam for use with a 20-in. liquid-hydrogen bubble chamber. The beamtransport system includes two electromagnetic velocity spectrometers to give high-purity separated beams of antiprotons, positive and negative kaons, and pions. In the central region, another target (G-10) can provide beams for use with counters or spark chambers. During the acceleration, the beam is very tightly bunched into pulses of 15-20 nsec duration and, if a target is directly inserted, this bunch structure remains and makes detection, particularly by counters, very difficult. Some of the bunch structure can be removed by programming an exponential decay in rf amplitude a t the end of the acceleration period and allowing the beam to drift inward in radius to hit the target. Bursts of 10-50 msec duration can then be obtained. For longer bursts, however, it is necessary to keep the magnetic field constant a t a value corresponding t o the desired energy, to turn the rf off, and to allow the beam to circulate 82 R. L. Cool, “Proceedings, International Conference on High Energy Accelerators Brookhaven, 1961,” p. 15. U.S. Government Printing Office, Washington, D.C.
FIG.20. One of the arrangements of experimental apparatus at the Brookhaven AGS (Summer, 1961). At the extreme left a target a t F-10 provides secondary beams which traverse the transport system (D are deflecting magnets, Q are quadrupoles) including two electromagnetic separators (BS) to provide separated antiprotons and K-mesons t o t h e 20-in. liquid-hydrogen bubble chamber (BC). The target located at G-10, near the center of the picture, provides many secondary beams for use with large Cerenkov counters (C) and spark chambers. Liquid-hydrogen targets (H), 10 ft long, are shown in two of the secondary beams. The shaded areas are shielding blocks of heavy concrete and long concrete beams form a roof over the accelerator ring in this area.
676
3.
SOURCES O F NUCLEAR PARTICLES A N D RADIATIONS
until the bunch structure disappears; then the beam can pass through a thin target many times and produce secondary beams over periods of 100 msec or longer. For many experiments, an extracted proton beam is desirable. This can be obtained with a loss-target scheme similar to the one described in Section 3.2.2.5.2 and in use at the Cosmotron. Another method, which can possibly bring out almost the entire beam in one revolution, makes use of two or three rapidly pulsed magnets, located in appropriate straight sections. At the AGS, it is planned that the first magnet will distort the orbit a sufficient amount so that the beam can jump inward past a thin current sheet which forms the return winding of a second magnet; this magnet, in its turn, gives a further distortion to drive the beam to the outer edge of the vacuum chamber where a third magnet then deflects it outside the synchrotron. In 1962, the apparatus is under construction but extraction has not yet been attempted. As shown in Fig. 20, very massive shielding of heavy concrete or steel is required both around the accelerator and around experimental equipment. Large numbers of magnets and quadrupoles are needed for transporting the secondary beams to the detectors. This requires a great deal of space and much special handling equipment for efficient operation. At present, at the AGS, experimental areas of over 100,000 sq ft are available but there is a continual need for expansion. 3.2.2.6.2. AG ELECTRON SYNCHROTRON. The highest energy yet achieved for electrons was reached in August, 1962, when the 6-Bev Cambridge alternating-gradient electron accelerators3attained its design energy. Two other machines of very similar design and for about the same energy are under construction : the Deutsches Elektronen Synchrotron (DESY), a t Hamburg, Germany, is scheduled for completion in 1963;s4 the progress of the other one a t Yerevan, Armenia, U.S.S.R., is not known at this time. The first alternating-gradient synchrotron to be operated was the 1.3-Bev electron machine a t Cornell University which was completed in 1955.86Other AG electron synchrotrons for about l-Bev energy are located at the University of Lund, Sweden, and a t the University of Tokyo, Japan;4 a 500-Mev machine is a t Bonn, Germany. The electron AG synchrotron, unlike its proton counterpart, has a practical upper limit in energy which is set by the radiation losses; as previously noted (Section 3.2.2.1), radiation effects cause a loss of electron 88 M. S. Livingston, “Proceedings, International Conference on High Energy Accelerators and Instrumentation,” p. 335. CERN, Geneva, 1959. 84 W. Jentschke, “Proceedings, International Conference on High Energy Accelerators and Instrumentation,” p. 339. CERN, Geneva, 1959. 8 6 A. Silverman, et a!., Bull. Am. Phys. SOC., Ser. 11’1, 59 (1956).
3.2.
ARTIFICIAL SOURCES
677
energy that is given by 88.5 W4/r kev/revolution where W is the electron energy in Bev and r is the radius in meters. For an orbit radius of 30 meters and a peak electron energy of 6 Bev, this energy loss is 3.8 Mev/revolution. Even in this example, the radius of the orbit has been made sufficiently large that the maximum value of the magnetic field, corresponding to the peak energy, is only 7000 gauss. The radiation losses could be reduced somewhat by a further increase in radius (which would increase the cost of the magnet structure) and a corresponding decrease in the magnetic field but it is apparent that the practical limit in energy is near when the losses are above a few Mev per revolution. In general principles, the electron AG synchrotrons are somewhat simpler than the proton machines. Because the electrons can be injected at an energy where their velocity is so close to that of light, the accelerating system can be of constant frequency. For typical parameters, the transition energy is a few Mev and, if injection occurs a t an energy above this, the problems associated with transition can be avoided. The magnets are of similar design to those in the proton machines but, because they are smaller, it is possible to pulse them a t rates up to 60 cycles/sec. Since the Cambridge 6-Bev Electron Accelerator (C.E.A.) is typical of the higher energy machines, it will be described in further detail. The magnet of the C.E.A. consists of forty-eight units alternately focusing and defocusing and separated by field-free straight sections. Its cross section is similar to that shown in Fig. 15 b u t the pole width is 6.5 in. and the magnet gap is 2 in. high at the orbit. The gradient is about 3.3%/cm. The magnets are constructed of 0.014-in. laminations of transformer steel, suitable for pulsing a t 60 cycles/sec, and are bonded into blocks about 11 in. long. Twelve such blocks make up one unit and special end pieces are laminated in a direction normal to the ends. Exciting coils are of water-cooled stranded cable. The magnet is resonated with capacitors and is excited from the ac power line through a transformer and an isolating circuit which delivers power to compensate for the magnet power losses. The vacuum chamber for an electron synchrotron of such high energy presents special problems associated with the radiation mentioned above. This radiation, which takes the form of soft X-rays, is very destructive, particularly to the insulating materials that would be convenient to use for vacuum-chamber construction in rapidly varying magnetic fields. In the C.E.A. the chamber, roughly elliptical in cross section, has been fabricated of stainless steel deeply slotted in the radial direction to minimize the effects of eddy currents which might distort the magnetic field. The
678
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slots are sealed with an external coating of epoxy-loaded glass cloth but the slot geometry is such that the glass cloth is shielded from the X-rays by the metal wall of the chamber. The injector for the Cambridge machine is a 25-Mev linear accelerator of standard design. The injected pulse has an intensity of about 200 ma and lasts for 0.7 psec, just long enough to fill the circumference of the synchrotron. Inflection into the synchrotron orbit is accomplished by a pulsed magnetic field in one of the straight sections. Radio-frequency acceleration is at a high (360th) harmonic of the revolution frequency of the electrons. This corresponds to a frequency of about 500 Mc/sec, a frequency convenient for the design of resonant cavities and waveguide power lines. The whole system includes sixteen high-& resonant cavities and is capable of supplying a total energy gain of 6 Mev/revolution. The frequency is also a subharmonic of the frequency of the Iinac injector so that the injected pulses can be synchronized with the rf system, to obtain higher capture efficiency. It is expected that the intensity of the Cambridge accelerator will be about 1011electrons/pulse or a time-average of about 1 pa a t the 60-cycle operation. Targets can be mounted in several of the straight sections and secondary beams of gamma rays and various charged particles can emerge into a large experimental area. Methods for distorting the orbit or exciting oscillations of the beam so that it can strike the targets would be like those already described. It is probable that, eventually, a fairly large fraction of the circulating beam can be extracted; it could be deflected by pulsedmagnetic-field methods like those mentioned above. 3.2.2.7. Fixed-Field Alternating-Gradient Accelerators. It has already been mentioned that alternating-gradient focusing results in a high degree of momentum compaction [cf. Eq. (3.2.2.23)]; i.e., particles with wide divergence in momentum will travel on orbits which are spaced quite closely together. It follows, then, that it is possible to contain all momenta from some initial injection value to the final accelerated value in a radial space that is considerably smaller than the full radius and make an accelerator with an annular magnet somewhat like the synchrotron but with a field that does not vary with time. The problems associated with pulsing large magnets can be avoided, acceleration can take place at a faster rate, and higher average intensities can be obtained than in either the CG or AG synchrotrons. Alternating-gradient focusing is provided in ways that are of two general forms: the radial-sector type and the spiral-sector (spiral-ridge) type.86 The simplest form, the radial-sector FFAG (fixed-field alternatinggradient) accelerator, consists of a series of wedge-shaped magnet sectors 86
K. R. Symon, et al., Phys. Rev. 103, 1837 (1956).
3.2.
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679
arranged in a circle as shown in Fig. 21. In all sectors the magnetic field increases rapidly with radius and alternating-gradient focusing results from having the field reverse in direction in alternate sectors. Thus, the particle orbits have a scalloped form, as shown in the diagram. It is usual for the sectors giving the inward bend to be made longer, in the azimuthal direction, than those giving the outward bend. However, Ohkawa8’ showed that it was possible to have a closed orbit even if the sectors are
FIG.21. Plan view of the shape of magnet sectors in a radial-sector FFAG accelerator. The particles are injected at a small radius and, as they are accelerated, travel on orbits with ever-increasing radius. The dotted line shows the “scalloped” orbit at an intermediate energy.
made the same size. From Fig. 22, it can be seen that the particles are bent inward a t a radius larger (higher field value) than when they are bent outward so that with identically sized sectors there will be a net effect of inward bending. In this type of accelerator, the direction of circulation of the particles is immaterial and one can even have two beams circulating in opposite directions, simultaneously. The general principles of the radial-sector FFAG were suggested independently by several personsSsbut the most intensive work on theoretical T. Ohkawa, Rev. Sci. Znstr. 29, 104 (1958). M. S. Livingston and J. P. Blewett, “Particle Accelerators,” p. 627. McGraw-Hill, New York, 1962. a7
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and construction problems has been carried out, in the United States, a t the MURA (Midwestern Universities Research Association) Laboratory in Madison, Wisconsin. Members of this group constructed a small model of the radial-sector type for accelerating electrons to about 400 kev.89 This was first brought into operation at the University of Michigan and was later moved to Madison for study of the behavior of the orbits and t o complement theoretical work being carried out on a computer. More recently, a 40-Mev electron model with equally sized sectors, suitable for
FIG.22. A radial-sector FFAG accelerator with equally sized sectors for “two-way” operation. Typical orbits for particles traveling in either direction are shown.
two-way operation, has been completed a t Madison and successfully operated.90Both of these models have given results that are in close agreement with the extensive theoretical computations carried out by the MURA group and provided corroboration of the fundamental principles. In a spiral-sector FFAG accelerator, there are no reversals in magnetic field, but the field is changed by spiral sections or ridges, shaped as shown in Fig. 23. Again, the magnetic field increases rapidly with radius and the ridges provide regions of stronger field that spiral outward in radius. When F. T. Cole, et al., Rev. Sci. Instr. 28, 403 (1957). MURA Staff, “Proceedings, International Conference on High Energy Accelerators, Brookhaven, 1961,” p. 344. U.S. Government Printing Office, Washington, D.C. 88
go
3.2.
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the particles cross one edge of the spiral, they are radially focused and vertically defocused; as they cross the other edge, they are radially defocused and vertically focused. The degree of focusing is stronger than in the radial-sector type and the “scalloping” of the orbits is considerably less. This type of design was first suggested by Kerst and has been the subject of a great deal of theoretical study by the MURA group;*6a small ~‘ model was constructed to accelerate electrons to about 120 k e ~ . This model, like the radial-sector one, also operated successfully and in accordance with predictions. Further studies at MURA, both computational and
ridge
FIG.23. Plan view of magnet poles of a spiral-sector FFAG accelerator. The shaded areas represent ridges where the magnetic field (perpendicular to the plane of the diagram) is much stronger than in the unshaded region.
with models, have led to a proposed design for a 10-Bev proton synchrotron of the spiral-sector FFAG type.92 The theory of particle orbits in FFAG accelerators is extremely complicated because of the complex nature of the magnetic fields. For example, the expression for the magnetic field in the median plane of a spiral-sector machine takes the form (3.2.2.25)
where rois a reference radius and Bothe value of the magnetic flux a t th a t 91
D. W. Kerst, et al., Rev. Sci. Znstr. 31, 1076 (1960).
s* MURA Staff, “Proceedings, International Conference on High Energy Acceler-
ators, Brookhaven, 1961,” p. 57. U.S. Government Printing Office, Washington, D.C.
682
3.
SOURCES OF NUCLEAR PARTICLES A N D RADIATIONS
radius, e is the azimuthal coordinate, f is a periodic function of period 27r, k is the relative gradient of the average field, N is the number of spirals per revolution, and l / w describes the angle of the spiral. If y is the angle a field spiral makes with a radial line, then tan y = l/(wiV). Detailed treatments of the behavior of the orbits necessarily involve the use of large digital computers. However, the general nature of the motion of the particles can be expressed in terms similar to those for other AG accelerators. For small motions, the number of betatron oscillations in the radial and vertical direction, in the spiral-sector type, can be expressed approximately as follows:
+ 1)ll2 vY (vertical) = [ - k + w 2 N 2 vr (radial) = ( k
E1112*
(3.2.2.26)
F is a term known as the “flutter” which is related to the coefficients of a Fourier expansion of the function f of Eq. (3.2.2.25). The equations of motion of the betatron oscillations contain nonlinear terms which are much larger than those for the other AG accelerators and give rise to many other types of instability. Resonances occur, not only for integral values of the number of oscillations per revolution but also for many linear combinations of such values. These and other more complex instabilities lead to stringent requirements on magnet tolerances and placement. The magnet in an FFAG accelerator will be considerably larger than in the conventional CG or AG machine. In the radial-sector type, the circumference will be larger because of the inclusion of the reversed-field magnets. The ratio of the orbit radius to the magnetic radius of curvature, called the circumference factor, cannot be much smaller than a factor of six. Although there are no negative-field sections in the spiral-sector type of FFAG accelerator, practical designs result in a circumference factor of about two. The 10-Bev design has a radial aperture of 2.5 meters and a total magnet weight of steel of 33,000 tons (compared with 3500 tons of steel in the 33-Bev AGS). I n FFAG synchrotrons, acceleration is provided by radio-frequency systems similar to those used in ordinary synchrotrons. The frequency rises rapidly as the energy increases and then becomes approximately constant as the particle velocity approaches that of light. As in the other AG synchrotrons, there is a transition energy where the stable phase shifts from the rising side of the rf wave to the falling side; however, beyond this transition energy the frequency of revolution of the particles decreases very slightly rather than continuing to increase.
3.2.
ARTIFICIAL SOURCES
683
The intensity in these accelerators is limited by the same sort of spacecharge effects discussed in previous sections. However, the rate of pulsing is now limited, not by the rate at which large magnets can be pulsed, but by the rate of the rf system. It is estimated that the proposed 10-Bev protons/sec. Because the magnetic field is machine could yield 1014-1016 constant in time, it is also possible to leave an accelerated group of particles circulating in the orbit corresponding to its energy and to accelerate further pulses to this orbit, thus “stacking” a highly intense beam which could be extracted quickly or might be able to be brought out as an almost continuous beam. In the 40-Mev electron model, at MURA, beams have been stacked to give circulating currents of several ampere^.^" 3.2.2.7.1. AVF (AZIMUTHALLY VARYING FIELD)CYCLOTRONS. In Section 3.2.2.2, it was shown that two factors lead to a decrease in angular velocity with energy in a conventional cyclotron. The first is the relativistic increase in mass and the second is the radial decrease in magnetic field that is essential for stability of the betatron oscillations. As early as 1938, Thomasg3had shown that axial focusing could be achieved in a radially increasing magnetic field through the introduction of azimuthal variations in the field. However, his suggestions were not put to test until much later.94The development of the synchrocyclotron (Section 3.2.2.5) circumvented the difficulty by periodic variation of rf frequency but a t the sacrifice of the continuous output of the conventional cyclotron. The studies of FFAG magnet design, by the MURA group,86showed that even stronger focusing was possible than had been contemplated by Thomas (his design is a variant of the radial-sector type) and have led to the development of many varieties of AVF (also called “sector-focused)’) cyclotrons. By application of the FFAG principles, it is possible to design cyclotrons with strong axial focusing in a magnetic field whose average value increases with radius in proportion to the particles’ energy, thus cancelling the effects of the relativistic increase in mass ;continuous acceleration a t constant frequency is feasible to energies of several hundred MeV. One of the forms of pole geometry for the magnet of an AVF cyclotron is shown in Fig. 24. Typical values for the gap height in the “hill” region might be 7.5 in. while, in the “valley” regions, the gap would be increased to 28 in. This would result in azimuthal variations in field of about three to one. Other forms including fourfold, or even higher fold, sectored structures have been designed. Either the hill or the valley regions, or both, must be shaped as a function of radius and special trimming coils must be added to maintain the condition of isochronism and to give the proper 93L.
94
H. Thomas, Phys. Rev. 64,580 (1938).
E.L. Kelly, et al., Rev. Sci. Znstr. 27, 493 (1956).
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average azimuthal field along the particles’ orbit. Several designs incorporate much more pronounced spiral shapes than that shown in Fig. 24. For a given machine, the pole shape and the current-bearing windings included on the pole faces are the result of extensive calculations and, in many cases, of scaled models.
__
~
__._
Median plane
Section AA
FIQ.24. Schematic diagram of the magnet for an AVF (azimuthally varying field), or sector-focused, cyclotron. The top is a plan view where the shaded areas represent ridges in the magnet’s poles to provide regions of stronger field. Below is a cross-sectional view showing the hill and valley structure of the gap.
One particular advantage of the AVF cyclotrons is that they can be designed for variable energy and to accelerate various kinds of charged particles. This is because the fields can be suitably altered by means of changing the currents in the main windings and in windings on the pole faces. In the conventional cyclotrons, the magnet poles are very carefully shimmed for the given maximum energy and change of energy would require changing the shims. Operation for variable energy and different particles also requires that the frequency of the acceleration system can be changed. In some of the
3.2.
ARTIFICIAL SOURCES
685
designs, the system can be tuned over a range of about three to one; in others, the rf system operates at a frequency that is a multiple of the particles’ revolution frequency. The shape of the magnet poles in AVF cyclotrons presents obvious mechanical problems in mounting the accelerating system. In some cases, a dee-structure like that of conventional cyclotrons has been fitted between the poles; in others, the rf system has been made to fit into the spiral shape of the valley region. A considerable number of AVF cyclotrons are now being designed and constructed. Many of the parameters and design details of these machines together with discussions of theoretical and constructional problems were presented a t a conference held in 1959 at Sea Island, Georgia.96Several of these cyclotrons are now completed and in successful operation. At Oak Ridge, a 76-in., spiral-ridge, variable-energy, variable-particle cyclotron has produced protons up to 80 MeV. At the University of California, Berkeley, an 88-in. machine can give protons to 50 MeV, deuterons to 60 MeV, and also accelerate heavy ions. A t the University of California, Los Angeles, a fixed-energy, spiral-ridge cyclotron for 50-Mev protons has been in operation since 1960. The operational characteristics of these and several other machines were described at a recent conference, held a t U.C.L.A.96 One item of particular interest came from studies with a 500-kev electron cyclotron which is in operation a t Oak Ridge and has been built as an analog model for a contemplated very high energy (800 Mev or more) proton cyclotron. I t had been believed that an integral resonance in the radial betatron oscillations would set a limit in energy near 900 Mev in proton machines but, at Oak Ridge, a fraction of the beam was successfully accelerated past this type of resonance in the electron model. It has appeared to be difficult to extract the beam from an AVF cyclotron through the falling magnetic field a t the edge, chiefly because the successive orbits are spaced very closely together. However, Blosser and his associates at Michigan State Universityg7have made a detailed study of a method which uses the instability of the beam under the influence of properly located disturbances in magnetic field. The build-up of a resonant condition in the beam’s oscillations would give the necessary separation between turns for insertion of a deflector. The computations indicate that $6 “Sector-Focused Cyclotrons, Proceedings of an Informal Conference,” Nuclear Science Series, Report No. 26. Publication 656, National Academy of SciencesNational Research Council, Washington, D.C., 1959. 96 D. J. Clark, Phys. Today 16, No. 8,32 (August, 1962). (Proceedings of this conference are scheduled to be published in Nuclear Instr. and Methods.) 97 H. G. Blosser, et al., Internal Reports, Dept. of Physics, Michigan State University, East Lansing, Michigan.
686
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SOURCES O F NUCLEAR PARTICLES AND RADIATIONS
a well-focused extracted beam of good intensity might be possible from the 50-Mev cyclotron being designed at Michigan State University. At the University of Colorado a beam has been extracted by an ingenious method. In this spiral-ridge machine, designed to give 30-Mev protons, a beam of negative hydrogen ions was accelerated, at the maximum radius it was stripped of electrons by an aluminum-oxide foil, and then brought outside the magnetic field. Negative-ion acceleration has also been successful to energies of about 50 Mev in the cyclotron at U.C.L.A.98Recently, Kelly has reported successful extraction of beam from the Berkeley cyclotron by means of an electrostatic d e f l e c t ~ r . ~ ~ 3.2.2.8. Colliding Beam. In all the previous sections, the production of new particles and nuclear reactions has been assumed to be the result of an accelerated beam striking a stationary target. However, as a beam of particles becomes more relativistic, the fraction of energy which is available for particle production becomes relatively smaller, the remainder of the energy going into kinetic energy of the resultant particles and the target nucleon. One can derive an expression for the available energy to produce a particle or particles, of rest mass ml, obtained from bombardment of a proton at rest by a proton of total energy W , and rest energy moc2,by a consideration of the situation in center-of-mass coordinates. The greatest amount of energy will be available for particle production if all particles have zero velocity in the c.m. system; thus, in the laboratory system all of the particles have the same velocity. Then, from the equations for conservation of energy and momentum, it can be derived that
From this expression, one obtains the result that the total energy, mlc2, available for particle production is only 5.86 Bev if protons with kinetic energy 30 Bev strike a fixed target. However, if the incident particle and the target particle had equal and opposite momenta, all of the kinetic energy of both particles would be available for particle production. For this reason, much attention has been given recently to the possibility of providing collisions between countercirculating beams of electrons or protons. But the density of such intersecting beams must be high enough that a sufficient number of collisions can be observed. For cross sections of 4 X ern2 (as for protons) and for countercirculating beams with a density of 1 amp/sq cm, the num88 g9
J. R. Richardson, et al., Bull. Am. Phye. SOC.,Ser. I1 7, 431 (1962) (Abs. AAIO). E. L. Kelly, Bull. Am. Phys. SOC.,Ser. I1 7, 431 (1962) (Abs. AA11).
3.2.
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ARTIFICIAL SOURCES
ber of collisions will be about 8000/sec in a region whose cross-sectional dimensions are 1 sq cm and whose length is 1 meter. Since observations in a reasonable solid angle around this region would detect only a small fraction of these interactions, it would be desirable to raise the density of each of the colliding beams by a t least a factor of ten for feasible experiments. This now appears to be possible. The first proposal for providing intersecting beams came from the group at MURAloOwho suggested the use of two high-intensity FFAG
EXTERNAL EXTERNAL? PROTON BEAM
A
‘\L
TARGET BUILDING
FIG.25. A possible arrangement for two magnetic storage rings for use with the Brookhaven AGS. The solid rings represent a tangential arrangement to provide one interaction region of the colliding beams; the dotted ring could give two possible interaction regions.
accelerators situated tangent to each other. Later, the two-way radialsector ac~elerator,~’ mentioned in Section 3.2.2.7, appeared to be somewhat less expensive; in fact, the radial-sector electron model at MURA has successfully accelerated beams in both directions simultaneously to O after the original MURA proposal, energies above 25 M ~ V . ~Shortly O’Neill‘O’ and others suggested another, more economical alternative. One could store many pulses of beam, extracted from an existing accelerator, in external magnetic storage rings where the pulses would be kept circulating until sufficient density had been accumulated. A possible arrangement of such storage rings for use with the Brookhaven AGS is shown in Fig. 25, where the rings could be either tangent a t one point, or could intersect at two or more locations. There is, at present, much interest in storage rings but considerable controversy exists over their usefulness, particularly compared with accelloo
Io1
D. W. Kerst, et al., Phys. Rev. 102, 590 (1956). G. K. O’Neill, Phys. Rev. 102, 1418 (1956).
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SOURCES O F NUCLEAR PARTICLES A N D RADIATIONS
erators of high primary energy.Io2Storage rings are certainly less costly: two storage rings for use with the AGS would probably cost about twice as much as the accelerator itself, i.e., about 60 million dollars, whereas an accelerator with the same available particle-production energy could cost well over half a billion dollars. However, there are certain limitations to colliding-beam systems. It would not be possible to place a detector, such as a bubble chamber, in the collision region since this would destroy the circulating beam. The collision processes would have to be deduced from ELECTRON BEAM FROM LINAC
RF
RF
-MAGNET
I
2
FIG.26. Diagram of the storage-ring apparatus for the experiment of Princeton and Stanford Universities t,o obtain two colliding beams of electrons provided by the Stanford electron linear accelerator.
observations of secondary particles a t some distance. An extremely high vacuum (better than mm Hg) is necessary in the storage rings to keep down background from collisions between the circulating beams and nuclei of residual gas. Storage of sufficiently dense beams from the present high-energy synchrotrons would take many pulses and would require a n accumulation time of at least several minutes. This would preclude the storage of beams of unstable particles for studies such as those involving collisions between K-mesons and protons; it might be possible, however, to store antiprotons. Finally, the secondary beams from such collidingbeam systems are of low energy and not useful, therefore, for the production a t a distance of tertiary beams. More information on the usefulness of storage rings will be available in the near future as a result of some experiments with colliding beams of lo* Session VI, “Proceedings, International Conference on High Energy Accelerators, Brookhaven, 1961,” pp. 247-290. U.S. Government Printing Office, Washington, D.C.
3.2.
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689
electrons. A pair of storage rings has been built as a joint project of Princeton and Stanford Universities t o be used with the Stanford linear accelerator and a plan view of the arrangement is shown in Fig. 26. Io2 The rings can accommodate electrons up to 500-Mev energy and electronelectron scattering experiments can be conducted that would correspond to bombarding a stationary target with electrons of about 1000 Bev. I n 1962, final tests of this equipment are in progress. At the Frascati Laboratory, in Italy, a single storage ring, called AdA (for “anello di accumulazione”), has been built for use with the l-Bev electron synchrotron. I n this ring, positrons and electrons of 250 Mev can circulate in opposite directions and are obtained from targets bombarded by gamma rays from the synchrotron.l02 The group at Frascati is also constructing a large ring (“Adone”), capable of storing particles of energies over 1 Bev, which will probably be used with the linear accelerator a t Orsay, France. At the CERN Laboratory, in Geneva, a 2-Mev electron storage ring is being constructed to study problems connected with stacking highly intense beams. The group undertaking this project is also studying the feasibility of constructing a pair of storage rings for use with the 25-Bev proton synchrotron.Io2
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4. BEAM TRANSPORT SYSTEMS
4.1. Introduction" For experiments in high-energy physics, large accelerators are used to produce intense beams of charged particles of high energies with which various types of experiments can be performed. The beams may be of two types: (1) external beams which have been extracted from the accelerator, such as for instance the 3.0-Bev external proton beams from the Brookhaven Cosmotron; (2) secondary particle beams, which are produced when the accelerated particles hit a target, either inside the machine or outside the accelerator (for a n external beam). The secondary beams may consist of pions, K mesons, or antiprotons. Both types of beams are generally required to travel a large distance, since the experiments are performed at a. relatively large distance from the accelerator, partly because of the necessity of a thick shielding wall (-3-5 meters thick) between the machine and the experimental area, and partly because by locating the experiments at a large distance from the machine and spreading out different external beams by means of bending magnets, it is possible to have available more space in which the experiments can be carried out, and also a larger number of experiments can be simultaneously accommodated. In order to have a beam travel over a large distance without a considerable decrease of intensity due to its divergence, it is necessary that the particles of the beam be focused along the path length for both directions perpendicular to the main direction in which the beam is traveling. These two types of focusing are generally referred to as horizontal and vertical focusing. The horizontal focusing refers to the lateral direction (y direction) perpendicular to the main beam direction (taken along the 2 axis). If a beam is diverging from a target (secondary beam), the focusing may consist, for instance, of making the beam parallel to the x axis along both the y and z directions so that it can be refocused to a small image at the location of the experimental area, where the experiment is to be performed. The simultaneous focusing of particles along the y and z directions at the same image location is generally called double focusing.
* Chapters 4.1 and 4.2 are by R. M. Sternheimer. 69 1
4.2. Beam Bending and Focusing Systems For a long time, focusing was achieved by means of bending magnets with a uniform magnetic field. By uniform field is meant a‘field that does not vary with position inside the magnet. The vertical focusing in these magnets is obtained by making the beam enter and leave the magnet a t an angle to the normal to the entrance face and to the exit face, respectively. The horizontal focusing arises from the difference in the path lengths of different trajectories through the magnet. I n order to achieve double focusing with a bending magnet for reasonable values of the object distance S and image distance T (for instance, S 5 3R, T 5 3R, where R is the radius of curvature of the particles in the magnetic field), it is necessary that the angle of deflection cp in the magnet be a t least of the order of 30”. This, in turn, requires a very long magnet (length - R / 2 ) for high-energy particles, for which R is large for a typically attainable magnetic field, of the order of 15,000 gauss. [For a field of 15,000 gauss, R (in meters) = 2.22p3,,/,, where p B e v l o is the particle momentum in Bev/c.] * In 1952, it was shown by Courant, Livingston, and Snyder‘ that a beam of charged particles can be focused very effectively by means of a pair of quadrupole magnets. Each quadrupole magnet consists of four poles with hyperbolic cross section. The cross section of such a magnet is shown in Fig. 1. With the polarity of the four poles as shown in this figure, the magnetic field H is zero along the central axis (y = x = 0), whereas for points off the axis, the following relation holds: (4.2.1) where k is a constant independent of y and z ; here y and z are displacements perpendicular to the x axis, which is along the direction of the particle beam; Huand H , are the corresponding components of H. Courant et at.’ have shown that a set of two quadrupole magnets with values of k: k, and kb, such that kb = -ka, can produce very effective focusing of the incident beam along both the vertical and horizontal directions. The decisive advantage of quadrupole magnets is that for the same amount of focusing, they can be made considerably shorter than the corresponding bending magnet which would give double focusing for the same object and image distances. This type of focusing, by means of
* For convenient computation see also “Electron Physics Tables.” Natl. Bur. Standards ( U . S . ) Circ. No. 671 (1956). 1 E. D. Courant, M. S. Livingston, and H. S. Snyder, Phys. Rev. 88, 1190 (1952). 692
4.2.
BEAM BENDING AND FOCUSING SYSTEMS
693
quadrupoles, has been called strong focusing or alterna ting-gradient focusing, since the two magnets have opposite signs for the gradient of the magnetic field [Eq. (4.2.1)]. The proposal of Courant et al.' and of Christofilos2 for alternating-gradient focusing was actually advanced primarily for high-energy accelerators, in order to reduce the amplitude of the horizontal and vertical oscillations about the equilibrium orbit, which in turn enables one to decrease drastically the cross section of the
FIG.1, Cross-sectionalview of a quadrupole strong focusing magnet. With the poles as shown in the figure, the magnet is vertically focusing and horizontally defocusing for positively charged particles traveling out of the paper.
magnet required to bend the particles. I n a n alternating-gradient synchrotron, there is a large number of magnet sections, which are successively focusing and defocusing in each direction a t right angles to the direction of the particle beam ( g and 2 ) . A given sec tion is either vertically focusing (VF) and horizontally defocusing, or horizontally focusing (HF) and vertically defocusing, as will be discussed below [see Eqs. (4.2.5) and (4.2.10)]. Thus the sequence of focusing magnets around the circumference of the machine will be of the type: VF, HF, VF, HF, etc. Between the focusing sections, there are bending magnets in which there is little N. Christofilos, unpublished report (1950).
694
4.
BEAM TRANSPORT SYSTEMS
or no focusing of the beam. Several accelerators in various stages of construction are based on the strong focusing principle, notably the 3-Bev Pennsylvania-Princeton high-intensity proton synchrotron, the 6-Bev Harvard-MIT electron accelerator, the 30-Bev proton accelerator a t Brookhaven, and a similar 28-Bev machine which has just been completed at the CERN Laboratory a t Geneva. In this chapter, we will give the basic equations which determine the focusing action of a strong focusing system consisting of two quadrupole magnets with opposite field gradients. The focusing action of a wedgeshaped magnet with nonuniform field will be discussed at the end of this chapter. In Section 4.2.1, the equations of the trajectories of particles in a single quadrupole magnet are determined. Section 4.2.2 contains a discussion of the lens equations304for a single quadrupole magnet and for a system consisting of two quadrupoles. The lens equations are used to obtain the conditions of simultaneous vertical and horizontal focusing of the beam (double focusing). We will also discuss the double focusing by a system of three quadrupoles, and will compare the focusing properties of this type of system with those of the two-magnet system. The matrix method6 of calculation for strong focusing magnet systems is described in Section 4.2.3. This method is mathematically equivalent to the method of the lens equations of Section 4.2.2 and can be used assan alternative to the lens equations depending upon the particular focusing problem considered. A particular example of double focusing by a twomagnet system is treated in Section 4.2.2 by means of the lens equations, and, for comparison, in Section 4.2.3 the solution for the same example is given by means of the matrix method. Finally, in Section 4.2.4, we give the basic equations for the vertical and horizontal focusing by a wedge-shaped magnet with nonuniform field. 4.2.1. The Trajectories of Particles in a Strong Focusing (Quadrupole) Magnet
In order to derive the equations of motion which result from the field (4.2,1), we note that the force in the z direction, F,, is given by
F,
=
ev,H,/c = (ev2k/c)z
(4.2.2)
where v2 is the component of the velocity v along the x direction (v, = v), and e is the charge of the particle. We have F, = m ~ ( d 2 ~ / d t 2 ) (4.2.3) R. M. Sternheimer, Rev. Sci. Znsfr. 23, 629 (1952). R. M. Sternheimer, Rev. Sci. Instr. 24, 573 (1953). 6 E. D. Courant and H. S. Snyder, Brookhaven AGS Internal Report EDC-HSS 1 (1953); Ann. Phys. ( N . Y . ) 3, 1 (1958). a 4
4.2.
BEAM BENDING AND FOCUSING SYSTEMS
695
where m is the rest mass of the particle and y = (1 - V ~ / C ~ ) - ' ~Upon ~ . writing dt = dx/v, in Eq. (4.2.3), one obtains from Eq. (4.2.2): myv,
Since myv, = myv (4.2.4) becomes
=
d2z - ek - - -z dx2 c
(4.2.4)
p , where p is the momentum of the particle, Eq.
d2z/dx2 = Kz
(4.2.5)
where K is defined as: (4.2.6)
where rois the radius of curvature of the particles in an arbitrary constant field Ho. If K is negative (e.g., with dH,/dz < 0, for particles with positive charge), the force F , is restoring the particles toward the xy plane ( z = 0 ) , and the magnet action is focusing. The solution of Eq. (4.2.5) is then 2 =
COS[(Z/K)
+ 61
(4.2.7)
where K is defined as IKI-l'2 and has the dimensions of a length ( K is to be taken as positive); Z and 6 are constants which are determined by the initial conditions (the values of z and dz/dx at the entrance of the magnet). If, on the other hand, K is positive, the magnet action is vertically defocusing, and z is given by Z =
cosh[(z/rc)
+ 51
(4.2.8)
where Z and f are again constants determined by the initial conditions. In a similar manner, the y component of the force, F,,
F,
=
-w,H,/c
= -(w&/c)Y
(4.2.9)
leads to the following equation of motion: d2y/dx2 = - Ky.
(4.2.10)
It may be noted that Eq. (4.2.10) is identical with Eq. (4.2.5) for the z motion, except for the change of sign. Thus if the magnet is vertically
focusing, it will be horizontally defocusing, and vice versa. The solution of Eq. (4.2.10) for y is again given by a cosine or hyperbolic cosine function, similar to (4.2.7) or (4.2.8). I t is evident from this discussion that a single quadrupole magnet cannot produce double focusing of a beam of charged particles, ie., simultaneous focusing in the vertical and horizontal directions. I n order to achieve double focusing, it is necessary to use a system of (at least) two
696
4. BEAM
TRANSPORT SYSTEMS
magnets, one of which is vertically focusing and the other horizontally focusing. l--B
4.2.2. The lens Equations for a Single Quadrupole Magnet and for a Two-Magnet System. Conditions of Double Focusing by a TwoMagnet System
Before discussing the properties of a two-magnet system, we will consider the focusing action of a single quadrupole magnet. As was shown by Courant et aLll the focusing action of the magnet is equivalent to th a t of a thick lens. The efiective length of the magnet will be called 1. I n addition to the length of the magnet iron, 1 should include a correction for the fringing fields a t the entrance and exit of the magnet, as will be discussed below. The following treatment of the lens equations for a single quadrupole magnet and for a two-magnet sys tem follows closely that given by Sternheimer.3s4 4.2.2.1. Focusing Action of the Magnetic lens. Let us first consider the direction (y or z ) for which the magnet is focusing. This direction will be labeled by the subscript 1. The distance from the object (source of the beam) to the effective field boundary at the entrance of the magnet will be called S1. The image distance as measured from the effective exit boundary of the magnetic field will be called 7'1. It can be shown4 [see Eq. (4.2.126)below] th at the object and image focal planes F o , and ~ FI,~ are a t a distance A : K Cot (4.2.11) from the effective boundaries of the magnetic field, where a = Z / K . The convention with respect to the sign of X is that positive X means that the object focal plane is to the left of the magnet entrance, and the image focal plane is to the right of the magnet exit. Here and in the following, it is assumed that the particle beam travels from left to right (see Fig. 2). In general, a < ~ / 2 so , that X > 0. The focal length f l of the lens is given by4 [see Eq. (4.2.11411: f l
=
K
CSC
(Y
=
(K2
+
where the sign of the square root is to be taken as positive for a negative for T < a < 2 ~ . The Newtonian form of the lens equation is given by: ~ o . i ~ r= . 1fi2
(4.2.12)
< T , and (4.2.13)
where xo,lis the distance from the object to the object focal plane F o , ~ , and x I , 1 is the distance from the image to the image focal plane F1.1. The 6
F. C. Shoemaker, R. J. Britten and B. C. Carlson, Phys. Rev. 86, 582 (1952).
4.2.
BEAM BENDING AND FOCUSING
SYSTEMS
697
convention with respect to the signs of xo,1 and xx.1 is that x0.1 is positive if the effective object is to the left (in front) qf Fo.1, and is positive if the resulting image is to the right of F I , I . I n view of Eq. (4.2.11), we have (4.2.14) (4.2.15) Upon substituting Eqs. (4.2.12), (4.2.14), and (4.2.15) in Eq. (4.2.13), one obtains the following lens equation:
The principal planes are a t a distance f l from the focal planes. The object principal plane Po,l is a t a distance f l to the right of the object focal plane Fo,l, while the image principal plane P I . 1 is a t a distance f l to the left of the image focal plane F I , l (for a < T , f l > 0). EXAMPLE : Figure 2 shows the location of the focal planes and principal planes for the first magnet traversed by the beam in the two-magnet system discussed below (see Table I). The value of K = K-lI2 is taken as 1.396 m. Thus K = 0.513 m-2. Taking as a n example 3.0-Bev protons ( p = 3.825 Bev/c, e / c p = 0.784 X lo-'), one finds from Eq. (4.2.6) th a t dH,/dz = 654 gauss/cm. For a magnet of semi-aperture A/2 = 15.24 cm, this gives a field of 9,970 gauss a t the pole tips. The corresponding value of Hero for 3.0-Bev protons is: c p / e = 1.276 X lo7 gauss-cm. It will be assumed that the magnet has an effective length I = 1.19 m. One thus finds: X = 1.222 m, fl = 1.856 m. I n Fig. 2, the origin of the 5 axis is a t the object (source) which is at a distance of S1 = 4.57 m from the magnet entrance. The magnet entrance and exit are therefore located a t x = 4.57 m, and x = 5.76 m, respectively. The coordinates of the focal planes and principal planes are as follows: x F , O = 3.35, T F J = 6.98, x p , = ~ 5.21, X P J = 5.13 m. It is seen from Fig. 2 th a t the principal planes are very close to each other. Moreover, the average of .cp,o and X P . I [ ~ ( ~ P ,X~P J ) = 5.17 m] corresponds to the center of the magnet. These results have general validity, and can be easily proved as follows. If f p . 0 and Z P , denote ~ the coordinates of the principal planes Po,r and with respect, to the magnet entrance, we have
+
z p , o = K(CSC a zp,I = K ( a
- cot a) - CSC a).
+ cot
(4.2.17) (4.2.18)
698
4. I
BEAM TRANSPORT SYSTEMS
I
I
I
I
1
0
-fl-:-
I
2
4
3
7
8
9
+
b
DISTANCE x
1
-1-
D
A
0
1
21-
-1-
sl-
1
6
5
(IN M E T E R S )
FIG. 2. Schematic view of magnet showing focal planes and principal planes for direction 1 for which magnet action is focusing. The magnet shown in this figure has a length I = 1.19 m, and I[ was taken aa 1.396 m. The origin of the z coordinates cor' = 4.57 m from the entrance of the magresponds to an object placed a t a distance 51 net. The image I1 is located a t x = 8.01 m, corresponding to an image distance TI= 2.25 m.
From Eqs. (4.2.17) and (4.2.18), one obtains +(zp,o
+ z p , ~=)
iCYK
I n order to prove that the difference we note that
61
41.
= z p , ~- 5p.1
(4.2.19) is very small,
In the above example, a = 0.852, and Eq. (4.2.20) gives: a1 = 0.077 m, in good agreement with the value obtained above (5.21 - 5.13 = 0.08 m). Figure 2 shows the image 11produced by the magnet. Ilis located a t 5 = 8.01 m (image distance T I = 2.25 m). It should be noted that the Newtonian form of the lens equation used above (Eq. 4.2.16) is, of course, equivalent to the Gaussian form: (1/4
+ (1/d
=
l/fi
(4.2.21)
4.2.
BEAM BENDING AND FOCUSING SYSTEMS
699
where ul is the distance from the object to the object principal plane Po.1, and T~ is the distance from the image to FIvl.In the present example, we have: c1 = 5.206, T~ = 2.886, f , = 1.856 m, and Eq. (4.2.21) can be easily verified. 4.2.2.2. Defocusing Action of the Magnetic lens. Now we consider the direction along which the magnet action is defocusing. This direction (y or z ) will be labeled 2. The object and image distances will be denoted by S2 and T z , respectively. The convention with respect to the signs of Sz and Tz is the same as for S1 and TI. Thus if the effective object is to the left of the entrance face and the image is to the left of the exit face, S2 is positive and Tz is negative. I t may be noted that in the lens equations (4.2.16) and (4.2.24), we have in general S1 = Sz,provided that the object is anastigmatic, i.e., the effective source in the vertical direction coincides with the effective source in the horizontal direction. This condition is automatically satisfied if the particles to be focused are secondaries produced in a target which then acts as the source for both the vertical and horizontal directions. On the other hand, the beam of particles may emerge from a region of magnetic field having considerable focusing action, for which the effective vertical and horizontal objects presented to the focusing system are separated by a large distance. An example of this situation is encountered in the focusing properties of the external proton beam of the Brookhaven Cosmotron.’ The proton beam which emerges from the external magnet shims is approximately parallel in the horizontal plane (object a t infinity), whereas in the vertical direction, the particles appear to come from a point (vertical crossover7) inside the external shims. For the defocusing case, the object and image focal planes are a t a distance p : (4.2.22) p = K coth a from the field boundaries4 [see Eq. (4.2.127)]. I n contrast to the situation ~ t o the right for direction 1 (for a < 7r/2), the object focal plane F o , lies ~ to the left of the entrance boundary, and the image focal plane F I , lies of the exit boundary (see Fig. 3). The focal length f 2 is given by4 [see Eq. (4.2.115)]: fi =
- (p2
-
,2)1/2
=
- K csch
a.
(4.2.23)
In the same manner as in Eqs. (4.2.13)-(4.2.16)one obtains the following lens equation from Eqs. (4.2.22) and (4.2.23): (4.2.24) ( 8 2 -k p)(Tz -k p ) = p2 - K ~ . R. L. Cool, G. Friedlander, 0. Piccioni, S. L. Ridgway, and R. M. Sternheimer, Brookhaven National Laboratory Report BNL 498 (1-24) (1958).
4.
700
BEAM TRANSPORT SYSTEMS
The object principal plane Po,z is located at a distance / fi/to the left of the object focal plane FO,*.Similarly, the image principal plane is a t a distance lfzl t o the right of Fr,z.
EXAMPLE : Figure 3 shows the location of the focal planes and principal planes for the direction 2 for the magnet considered above in connection with Fig. 2. I
I
I
I
I
I
I
I
I
4
I 5
1
2
I 3
I
I
6
I 7
8
I
0
1
I 9
DISTANCE x (IN METERS)
FIG. 3. Schematic view of magnet showing focal planes and principal planes for direction 2 for which magnet action is defocusing. The magnet shown in this figure is the same as for Fig. 1 ( I = 1.19 m, K = 1.396 m), and the origin of the z coordinates is defined in the same manner as in Fig. 1 (object a t distance SZ= 4.57 m from magnet entrance). The image Zz is located at z = 4.065 m, corresponding to an image distance Ta= -1.695m.
The origin of the x axis is again taken a t the location of the object so th a t Sz = 4.57 m. With I = 1.19 m, K = 1.396 m, one obtains p = 2.018 m, f 2 = -1.457 m. The resulting coordinates of the focal planes and principal planes are as follows: X F , O = 6.59; X F J = 3.74; xp.0 = 5.13; xp#r= 5.20 m. Similarly as in Fig. 2, we again have x p j o = x ~ ,and ~ ,the average: +(xp.g z p , f ) = 5.17 m, corresponds to the center of the magnet. These results can be proved in the same manner as in Eqs. (4.2.17)(4.2.20). By expanding the hyperbolic functions (coth a and csch cy), one obtains
+
62 E
XP,O
- XPJ
%K
(-$+&)
(4.2.25)
4.2.
BEAM BENDING A N D FOCUSING SYSTEMS
701
which is approximately the negative of ti1 for the focusing case (direction 1). Thus 8 2 is also expected to be very small. The result given by Eq. (4.2.25) is: ti2 = -0.067 m, in good agreement with the value obtained above (5.13-5.20 = -0.07 m). The image I2produced by the magnet is located a t x = 4.065 m, corresponding t o a n image distance T2 = -1.695 m measured from the exit of the magnet. The fact that Tz is negative implies, of course, that the image is virtual, i.e., the outcoming rays effectively diverge from the image point. I n similarity to Eq. (4.2.21), we can also obtain the image distance from the following Gaussian form of the lens equation [as a n alternative to Eq. (4.2.24)]: (4.2.26) (l/UZ) (1/72) = l/fZ
+
where uz and r 2 are the distances from the object to the object principal plane and from the image t o the image principal plane PI,^, respectively. As discussed above, we have S1 = Sz in Eqs. (4.2.16) and (4.2.24), unless the effective source is astigmatic. However, the image distances T I and T 2 are, of course, quite different. Indeed T1 is generally positive because of the focusing action in direction 1, whereas T2 is negative (defocusing). The condition TI = T2 would correspond to double focusing, and this cannot be achieved with a single quadrupole lens, as was discussed in Section 4.2.1. 4.2.2.3. Focusing Properties of a Two-Magnet System. We shall now obtain the focusing properties of a system of two quadrupole magnets with field gradients of opposite sign. The two magnets will be denoted by a and b. The beam enters the system through magnet a and leaves the system through magnet b. The effective lengths of the two magnet sections will be called 1, and l b . The separation between the inner field boundaries (exit of a and entrance of b) will be denoted by d. The values of K for the two sections will be called K, and Kb. According t o an elementary theorem of geometrical opticsls a system of two lenses a and b, with focal lengths fa and f b , respectively, acts as a single thick lens with focal length f = - f a f b / q , where q is the distance between the internal focal planes of the system, i.e., between the image focal plane of a, FI,, and the object focal plane of b, Fob. The sign of q is taken as positive if Fob is to the right of Fro (assuming that the particle beam travels from left to right). The object focal plane Fo of the equivalent lens is a t a distance f u z / q from the object focal plane FOoof magnet a, 8 See, for example, G. S. Monk, “Light, Principles and Experiments,” p. 21. McGraw-Hill, New York, 1937.
702
4.
BEAM TRANSPORT SYBTEMS
and is in front (to the left) of Foaif q is positive. The image focal plane FI of the combined system is a t a distance f b 2 / qfrom the image focal plane F I b of magnet b ; if q > 0, then FI is to the right of Fib. EXAMPLE : Figure 4 shows the focal planes for the two-magnet system which is discussed below, for the direction 1 for which the first magnet (a) is focusing. The characteristics of magnet a have already been discussed in
0
I
2
3
4
5
6
i
7
8
9
DISTANCE x ( I N METERS)
FIG. 4. Schematic view of two-magnet system showing focal planes and principal planes for direction 1 for which magnet a is focusing. The two-magnet system shown in this figure is discussed in the text (la = h = 1.19 m, d = 0.280 m, I C ~= 1.396 m, Kb = 1.478 m). The origin of the x coordinates corresponds to an object placed at a distance S1. = 4.57 m from the entrance of magnet a.
connection with Fig. 2. Magnet b has the same length as a ( l b = 1.19 m), and the value of Kb is 1.478 m. The effective separation d is 0.280 m. For magnet b, one finds: p b = 2.219 m, and f l b = - 1.655 m. The coordinates of the focal planes of a and b (with respect to the object at a distance S1, = 4.57 m from the entrance of a) are as follows: ( x F , O ) ~ = 3.35; ( ~ p . 1= ) ~ 6.98; ( X F , O ) b = 8.26; ( x F , I ) ~ = 5.01 m. The distance q1 is equal to: ( X F , O ) b - ( x F , I ) ~= 8.26 - 6.98 = 1.28 m. The focal planes and principal planes of the equivalent single lens are also shown in Fig. 4. The x coordinates of these planes are as follows: X F . O = 0.652, X F , I = 7.154, x p . 0 = 3.057, X P J = 4.749 m. We note that in this case, the principal
4.2.
703
BEAM BENDING AND FOCUSINQ SYSTEMS
planes are separated by a considerable distance, unlike the situation for a single quadrupole magnet (Figs, 2 and 3). The focal length of the equivalent system is: f l = 2.405 m. It may be noted that in Fig. 4, we have omitted for simplicity the subscript 1 from the designation of the focal planes and principal planes; e.g., Foa instead of ( F o , ~ ) ~ . In view of the preceding results for a single lens (magnet) and the theorem for combining the action of two lenses, we obtain the following equation4which determines the image distance for the direction 1 (y or z ) , defined as the direction along which the first magnet (a) is focusing: (s1a
f
Ula>(Tlb
+
=
Ulb)
(4.2.27)
J1
+
where J 1is the square of the focal length f l of the system; Sla U 1 , and Tlb U l b are the distances X O . ~and of the object and image from the object focal plane and image focal plane of the system, respectively. Thus Eq. (4.2.27) is equivalent to Eq. (4.2.13) for the Newtonian form of the lens equation. S 1 , is the object distance measured from the entrance face of a, and T , b is the distance from the exit of b to the image. The constants J1,UI., and u 1 b are given by
+
J1
=flz =
u~a Ulb
(Ka2
E=
-ha -
=
Pb
-
-k
(Ka2
(h2-
- Kbz)/ql2
Xa2)(Mb2
+
XaZ)/ql
Kb2)/ql
(4.2.28) (4.2.29) (4.2.30)
where q1is the distance between the internal focal planes, and is given by: =
41
d
- X u -I- p b .
(4.2.31)
In Eqs. (4.2.28)-(4.2.31), A, 3 K . cot a! and C(b = Kb coth p, where = l a / ~ aand p = Ib/Kb. UIa is the distance from the object focal plane F o , of ~ the system to the effective entrance of a, and U l b is the distance from the exit of b to the image focal plane RS1. A similar set of equations describes the focusing action in the direction for which magnet a is defocusing. The lens equation is given by? a
(825
Uza)(Tzb
+
Uzb)
= Jz
(4.2.32)
where Sza and Ts are the object and image distances measured from the entrance of a and the exit of b, respectively. The constants J z , U2.,and U z b are obtained from the following expressions : J z = fzz = (Pa2 U 2 a = Pa
-
(Pa2
Ka2)(Kb2
- Ka2)/q2
+ hb2)/qZ
-Xb
+
-
(Kb2
d
Pa
- Xb
where pa = K . coth a, and
hb
= Kb cot p.
u2b
qz
= =
+
Xb2)/qz2
(4.2.33) (4.2.34) (4.2.35) (4.2.36)
704
4.
BEAM TRANSPORT SYSTEMS
In order to obtain double focusing, we must have: Tlb
=
(4.2.37)
T2b.
For simplicity, i t will be assumed in the following that the object is anastigmatic, so that S1, = S z a = S,. We shall now show that for a given particle momentum, given values of K~ and Kb, and for a given geometry of the magnets (la, l b , d ) , there exists a t most one set of values of Saand Tb (object and image distances) for which the system is double focusing. In order t o vary the focusing conditions for a given momentum, i t is then necessary to vary the currents in the two magnets, so as to change K~ and Kb. For simplicity, we consider the case where the two sections are equal in length and have field gradients of the same magnitude (though, of course, with opposite signs). Thus 1, = l b , and Ka = Kb. We then have: U1, = u 2 b ; Uza = Ulb; and J 1 = J 2 . Equations (4.2.27) and (4.2.32) for the focusing become:
(8, (Sa
+
ula>(Tb
Ulb)(Tb
+
Ulb)
=
u~a)=
(4.2.38) (4.2.391
J1 J1.
From Eq. (4.2.38), one finds Tb = - U l b
-I- J l / ( S a -I- UU).
(4.2.40)
Upon inserting this result into Eq. (4.2.39), one obtains after some algebraic reductions :
Sa2 which gives4
8a
+
+ U,) + + Ulb) + $[(U,a -
Sa(U1a
= -+(ula
UlaUlb
- J1
ulb)2
=
0
(4.2.41)
~ J I ] ' ' ~ . (4.2.42)
If this expression for S, is positive, it corresponds to a case of double focusing. We note that in this arrangement, we have Sa = T b , i.e., equal object and image distances, so that the focusing is symmetrical with respect to the center of the magnet system (point midway between the two magnets). This result is also obvious on general grounds, since we have assumed that ua = Kb, and Ia = la. The same procedure as in Eqs. (4.2.38)-(4.2.42) can also be used when the two magnets have unequal lengths and/or different field gradients. In this case, there is again only at most one set of values of S, and Tb for which double focusing occurs. Thus one obtains S a < T b by making the quadrupole magnet a stronger than magnet b, i.e., a! > p. Similarly double focusing with Sa > Tb is obtained by makihg LY < 0. If either of the magnets, e.g., b, is considerably weaker than the other
4.2.
BEAM BENDING
AND FOCUSING
705
SYSTEMS
I ~
I
/
sKh ‘ 8 , 3 W 8 )
I
~
sK(7/2,r/2)l
1 0.5
1.0
1.5
2.0
2.5
3.0
3.5
dK
FIG.5. Object distance S K and image distance T K for double focusing by various two-magnet systems, as a function of the separation dK. The numbers a f i e d to each curve are the values of (Y = l . / ~and @ = I ~ / Kfor the two sections (a and b). SK,T K , and dK represent the values of S, T,and d, respectively, in units K-lIp = K .
(a), then there may not be any possibility of double focusing if magnet b is unable to compensate for the strong focusing and defocusing produced by magnet a in the directions 1 and 2. Figure 5 shows the results of calculations for various double focusing systems, as previously obtained in reference 4.The curves give the object or image distance S K or TK as a function of the separation d~ between the two magnets. Here S K , T K ,and dK are the distances S,, Tt,,and d in units of K-”2 = K . It was assumed that the two sections have field gradients of the same magnitude (K,, = Kb = K ) but that they may have
706
4. BEAM
TRANSPORT SYSTEMS
different lengths. The different curves correspond to different values of L/tC = LY and & / K = p. The notation used is ~ K ( L Y p) , and TR(LY, p). For the cases with a = p (1, = &,), SK and T K are equal. Figure 5 shows that SK and T K generally decrease rapidly as d K is increased. This illustrates the well-known fact that for a two-lens system consisting of a convergent and a divergent lens, the focusing power is increased by increasing the separation of the lenses. Of course, eventually one may lose in effective aperture if the separation is made too large [see below, Eqs. (4.2.46) and (4.2.47)]. For the system with a = 7r/4, p = 37r/16, the lens b is rather weak compared to a. As a result, for d K < 0.5, there is no longer any double focusing position. Actually, for d K = 0.5, the image distance T K becomes infinite, and it remains very large as d~ is increased up to 0.7. Even for d K = 1.0, we have T K = 5.9 as compared to SIC = 2.3, which shows that the relatively small difference a - /3 = x/16 leads to widely different values of SK and T K ,even when the separation dK is appreciable. I t should be noted that the equations for the focusing in both the vertical and horizontal directions are quite complicated, so that it has not been possible to obtain expressions which give the required values of LY and p for double focusing in terms of the object and image distances S and T . I n general, a and p must be obtained by a procedure of successive approximations, i.e., by assuming a set of values a! and 8, and calculating T for both the vertical and horizontal directions. These values of T will be denoted by T , and Ti,, respectively. In general, one finds T , # Th, and both T , and TI, are different from the desired value of T . Then a and 0 must be suitably changed until a set (a$) is obtained for which T , and Th are both equal to the required T . If one has a strong focusing system a t one’s disposal, with a given geometry (la, la, and d fixed), the general focusing properties can be obtained by calculating the object and image distances S and T for double focusing for various assumed values of K,, and Kb. Then, on a graph of K. versus p , where p = K ~ / K . , one can plot a set of curves of constant S (but varying T ) and another set of curves each of which has constant T (but varying S). The intersection of the curve S = SOwith the curve T = To gives the required values of K. and p for double focusing a t S = Soand T = T o . An example of such a plot is shown in Fig. 6. These curves were calculated for the large strong focusing system in use a t the Brookhaven C o s m o t r ~ nThis . ~ system has an aperture A = 30.5 cm. Each quadrupole has a length & = ib = 1.012 m, and the separation of the iron of the two 0 R. M. Sternheimer, Brookhaven Cosmotron Internal Reports RMS-53 and 55 (1955).
4.2.
BEAM BENDING AND FOCUSING SYSTEMS
707
FIG.6. Plot of K, vs. p for double focusing a t various object distances S and image distances T for a two-magnet system with I, = b = 3.90 ft. = 1.189 rn, d = 0.92 f t = 0.280 m. The abscissa p is defined as: p = Kb/Ke. The intersection of curve of constant S with a curve of constant !/' gives the required values of K, and p for double focusing. The solid curves are curves of constant T,while the dashed curves pertain to constant S. The values of S, T,and K~ in this figure are given in feet (conversion factor: 1 f t = 0.305m).
magnets is d = 0.457 m. As was noted above, the effective length 1 of each magnet should include a correction for the fringing field a t the entrance and exit of the magnet. According to the work of Dayton et a1.,l0 the correction A1 a t each end of the magnet is: A1 = 0.29A1where A is the aperture of the quadrupole, i.e., the distance between opposite pole tips (see Fig. 1). Thus the effective length of the quadrupoles is: 1. = lb
=
1.189 m
and the effective separation is: d = 0.280 m.ll This system has been used extensively to double-focus the external proton beam of the Cosmotron, with an effective object distance S i% 4.6 m, and for various values of the image distance T. As an example, for S = 15 ft = 4.57 m, T = 30 f t = 9.14 m, Fig. 6 gives K~ = 4.58 ft = 1.396 m, p = 1.060, so that Kb = 1.480 m. This is the double focusing system which has been considered I. E. Dayton, F. C. Shoemaker, and R. F. Moeley, Rev. Sci. Instr. 26,485 (1954). However, according to recent measurements of F. Eisler (Cosmotron Internal Report FE-4, January, 1960), A1 is only 0.17A. If thia value of the fringing field correction is used, the effective length of thc magnets becomes I, = .!b = 1.116 m, and the effective separation d = 0.354 m. ID 11
708
4.
BEAM TRANSPORT SYSTEMS
above in connection with Fig. 4. In the discussion concerning Fig. 2, it was shown that for 3.0-Bev protons, the required field H a in magnet a is 9,970 gauss. Similarly, the field in section b is given by: H b = (Kb/Ka)Ho =
9,970 X (1.396/1.480)’
=
8,870 gauss.
(4.2.43)
The required field strengths H , and Ha are proportional to the momentum p . Thus for 2.0-Bev 7 mesons ( p = 2.135 Bev/c), one finds:
Ha
=
9,970 X (2.135/3.825)
=
5,565 gauss,
H b = 4,951 gauss.
The curves of Fig. 6 are applicable to any system with I, = zb and with the same value of d / l a = 0.280/1.189 = 0.235. It is necessary only to scale all distances appropriately. TABLEI. Values of the Pammeters for the Vertical and Horizontal Focusing by a Two-Magnet System’ Parameter A, P6
91 u 1 a
Ulb Jl1’2
= f,
Tib Pa Ab
92
uz. U2b 521’2
Tzb
= fz
Value 1.222 2.219 1.277 -3.920 0.075 2.405 8.778 2.018 1.423 0.875 -0.408 -6.245 3.420 9.053
1, = l b = 1.189 m, d = 0.280 m, K. = 1.396 m, K ) = 1.478 m. The object distance Sa is taken &B 4.57 m, and the system gives double focusing at an image &tance Tb S 8.9 m. All lengths are given in meters. 0
As an example of the focusing equations [Eqs. (4.2.27)-(4.2.36)], we have given in Table I a list of the values of the various focusing parameters for the two-magnet system described above (Za = zb = 1.189 m, d = 0.280 m, ua = 1.396 m, Kb = 1.480 m). It should be noted that the same type of strong focusing action which is obt,ained with quadrupole magnets can also be achieved b y means of electrostatic lenses. Since the force on a particle in a n electric field E is: F, = eE, as compared to the magnetic force e(v X H)/c [Eqs. (4.2.2)
4.2.
BEAM BENDING AND FOCUSING SYSTEMS
709
and (4.2.9)], the equivalent electric field must be of the form: E , = key, E. = kez, where k, is a constant given by. k, = ( u / c ) ( d H , / d z ) = ( u / c ) k . Since the electrostatic unit (e.s.u.) of potential, the statvolt, equals 300 volts, the field gradient k , in volts per cm per cm which is equivalent to a magnetic field gradient dH,/dz, is given by ke.volts/crn/orn=
~ O O ( V / C( d) H y / d Z ) g o u s s / c r n .
In several applications, magnetic or electrostatic strong focusing systems have been used to focus external beams from cyclotrons and Van de Graaff generators, with a resultant large increase of intensity at the image position (by a factor of the order of 10 to 30).12 In designing a magnet system for double focusing, one is generally interested in the lateral magnification x1and xz in the directions 1 and 2. The lateral magnification x, is defined as d y I / d y o or dzl/dzo, where dyo and dzo are small horizontal and vertical displacements at the object, and dyr, dz1 are the resulting displacements of the trajectory a t the image. The absolute magnitude lxal is equal to b,/a,, where a, is the size (height or width) of the object (source) and b, is the size of the image along the direction 2. xI (i = 1, 2) is given by x 1
= -"ft/Xa
=
--J:fz/(S%a-k Um)
(4.2.44)
where fi is the focal length for the direction i, and x, is the distance between the object and the object focal plane Fo., Qf the system. We note , follows that for equal lengths and equal field gradients (1, = la,K, = K ~ ) it directly from Eqs. (4.2.38) and (4.2.39) that xlxz = 1. For such a system, x1 and x 2 are considerably differentin general. As defined above, x1is the magnification for the direction along which magnet a is focusing. As an example, for a = 6 = 7r/4, dx = 0.5, for which SK = T K = 3.6 (see Fig. 5 ) , we have: XI = -2.4, xz = -0.42. Similarly, for a = p = ~ / 4 , d K = 1.5 (SK= T K = 2.2), one finds: X I = -3.7, xz = -0.27. (Here the minus sign merely means that the image is inverted.) I n the strong focusing systems which have been used a t the Cosmotron in the past few years for secondary particle beams from a target placed in the proton beam, magnet a is in general vertically focusing ( = direction 1). This has the advantage that the horizontal magnification (xz) is quite small, l* See, for example, B. Cork and E. Zajec, University of California Radiation Laboratory Report UCRL-2182 (1953); F. B. Shull, C. E. MacFarland, and M . M. Bretscher, Rev. Sci. Znstr. 26, 364 (1954); C. J. Strumski, D. I. Cooper, L). H. Frisch, and R. L. Zimmermann, d a d . 26, 514 (1954); E.L. Hubbard and E. L. Kelly, zbzd. 26, 737 (1954); D.A. Bromley and J. A. Bruner, University of Rochester Report NYO3823 (1954). See also A. Septier, Advances an Electronics and Electron Phys. 14, 85 (1961).
710
4.
BEAM TRANSPORT SYSTEMS
thus giving a small horizontal image width which does not interfere appreciably with the energy resolution obtained from the analyzing magnet traversed by the beam. Thus, for a secondary beam, one will have in general a bending magnet behind the quadrupole focusing system in order to momentum-analyze the particles. If one assumes, for simplicity, that the bending magnet (with uniform field) has negligible focusing action, then the location and the width of the image past the bending magnet will be primarily determined by the quadrupole system. In particular, the image width b h is given by X h a h , where a h is the width of the object (target) “seen” by the quadrupole system. If the bending magnet has a momentum dispersion D , then the range of momenta accepted inside the image width is: A p = X h a h / D . Here D is defined by: D 5 d y / d p , where y is the lateral position of the image as a function of p. It is seen that A p is proportional to X h . Thus the momentum resolution will be improved if X h is decreased, and for this reason, it is advantageous to make magnet a vertically focusing and b horizontally focusing, instead of the reverse. Another consideration in the design of a strong focusing system is the effective aperture B at the entrance of magnet a ; B is defined as follows. We consider the extreme trajectories which just graze the magnet iron in either section a or b. The values of y and z a t the entrance of a, to be denoted by ye and ze, are then compared to the maximum allowable values of y and z inside the magnet system, ym and zm (which are of order A/2). Then B is defined by: B 3 Yeza/(YmZm). For magnet systems in which the first section, a, is vertically focusing, one finds y,/ym 0.4, ze/Zm 0.9 for typical cases, SO that B is of the order of 0.3-0.4. The effective area a t the entrance of magnet a is - € A 2 . The intensity a t the image is proportional to the solid angle subtended a t the source which is -eA2/S2, where S is the object distance. If the detector at the image collects all of the focused particles, the intensity gain G due to the use of the magnet system is:
-
-
G LZBA~L~/(LS~C%) (4.2.45) where L is the total distance between the source and image, and Q. is the area of the detector. For the case that the first section (a) is vertically focusing, one can easily show that ZJZm and ye/ym,to be denoted by E# and eY, respectively, are given by the following equations: (4.2.46)
4.2.
711
BEAM BENDING AND FOCUSING SYSTEMS
We have: c = c,C., so that E can be calculated from Eqs. (4.2.46) and (4.2.47), if the values of K. and Ki, for double focusing have been determined. If the first section (a) is horizontally focusing, E . will be given by Eq. (4.2.47) [instead of Eq. (4.2.46)], whereas eY will be given by Eq. (4.2.46). However, the value of E = cue. is obviously not affected by this interchange of the equations for e, and 6,.
-5 -5
I
I
22.5
1
20.0 17.5
N
15.0 h
+ 12.5 z w
5
10.0
U -I
a
2
7.5
0
5.0 2.5
0
0
2
4
6
8
DISTANCE x
10
12
14
16
18
(IN M E T E R S )
FIG.7. Trajectories with maximum amplitude yn = zm = 15.2 cm for the twomagnet system discussed in the text, which gives double focusing for S = 4.57 m, T 8.9 m (see Table I). It is assumed that magnet a is vertically focusing. We note that the angles of acceptance a t the object, 8, and 8. for the y and z directions are given by: 8, = 2cYym/Sand 6, = 2e,zm/S. The solid angle a t the object is: A n = OY8.. For the two-magnet system considered above (Table I), we have: E . = 0.956, E , = 0.409, so that B = 0.391. In order to obtain the intensity gain G from Eq. (4.2.45),we assume that the detector is a 10-cm diameter counter, so that a = 78.5 cm2. With A = 30.5 cm, S = 4.57 m, and L = 16.06 m, one finds G = 57.2. Figure 7 shows the y and z trajectories with maximum amplitude ym = z, = 15.24 cm for the two-magnet system discussed above. I t has been assumed that magnet a is vertically focusing. The values of y
712
4.
BEAM TRANSPORT SYSTEMS
and z a t the entrance of magnet a are: ye = 6.25 cm, ze = 14.58 cm, giving cu = ye/ym = 0.410, e, = ze/zm = 0.957, as was obtained above. The focal planes of the equivalent thick lens are also shown in Fig. 7. The locations of the focal planes (with respect t o the object a t S, = 4.57 m) are as follows: Z O . ~= 0.652; xi," = 7.154; x 0 , h = 4.164; X i , h = 13.48 m. The magnet system gives approximate double focusing at 2 6Z 16.1 m, corresponding to an image distance T !Z 8.8 m from the exit boundary of magnet b. 4.2.2.4. Quadrupole Magnets with Rectangular Aperture. Recently, Panofsky and Piccioni have independently proposed the use of quadrupole magnets with rectangular aperture in order to increase the solid angle of acceptance AQ a t the source for the same magnet gap area as for a system of conventional quadrupoles (with hyperbolic pole faces). The reason for this improvement is as follows. Figure 7 shows that the aperture required in the y direction for magnet a is appreciably smaller than the aperture required in the z direction. Thus by using for magnet a a quadrupole with rectangular cross section, and w, < ha (w,= width, ha = height), and correspondingly for magnet b, wb > hb, one can increase the solid angle subtended a t the source, and hence the intensity a t the image, for the same magnet area as for a conventional quadrupole. Hand and Panofsky13 have designed and built a rectangular quadrupole for use in conjunction with a conventional (15.2 cm diameter) quadrupole which precedes the rectangular magnet and produces the vertical focusing of the beam. I n the rectangular quadrupole, the desired magnetic field is obtained by means of current sheets on the four sides of the magnet, with the current flowing along the same direction on the two lateral sides, and along the opposite direction on the magnet faces which are above and below the beam. It can be shown that this configuration gives the required constant field gradient inside the aperture (dH,/dz = dH,/dy = k ) . The magnet constructed by Hand and Panofsky13 has a width w = 58.4 cm, height h = 10.2 cm, and length I = 102 cm. The maximum attainable field gradient is -430 gauss/cm. The problem of the increase of intensity obtainable with rectangular quadrupoles has been investigated by Sternheimer.14 It was found th a t the increase of intensity (as compared t o conventional quadrupoles havwhere ing the same field gradient) is given by a factor tl = ymax,a/yeXit.,, ymax,b is the maximum horizontal displacement in magnet b, and yexit,, is the value of y at the exit of a, for a given trajectory. q is given by the following expressions : la 14
L. N. Hand and W. K. H. Panofsky, Rev. Sci. Instr. 30, 927 (1959). R. M. Sternheimer, Brookhaven AGS Internal Reports RMS-6 and 7 (1959).
4.2. (1) for Sza
BEAM BENDING AND FOCUSING SYSTEMS
713
> K. (4.2.48)
where E is defined by (la/Ka)
6
(2) for SZa < K ~ : 7 =
E
E
[I
+
(la/Ka)
+ COth-'(S2a/Ka)
coth 5 Ka
+
( d 2 +2Kb2)
coth2 E
Ku
(4.2.48a)
]
'I2
+ tanh-'(Sza/Ka).
(4.2.49)
(4.2.49a)
In order to realize the intensity gain b y a factor 7, it is necessary to make Y b = q Y,, where Y, and Y b are the available horizontal apertures of a and b, respectively. (Ya,b equals the width w for a rectangular quadrupole.) The z aperture of magnet b can be made correspondingly smaller than that of a. We have: zb = Z o / { , where l is given by the following expression for Tlb > Kb: (d2 + Ka2) tanhZ81'"
(4.2.50)
Kb2
with
8E
(lb/Kb)
+ COth-'(Tib/Kb).
(4.2.50a)
A similar expression applies for Tlb < Kb. I n reference 14, the values of 7, eg, and e, were calculated for various two-magnet systems as a function of the length I ( = 1, = h,) for three values of the ratio d / l : namely, 0 , 0.4, and 0.8. As a n example, for 1 = K, one finds '1 = 1.32 for d = 0 , 7 = 1.63 for d = 0.41, and = 1.97 for d = 0.81.
A disadvantage of the rectangular quadrupoles is that they require considerably more power than conventional quadrupoles for the same aperture area and the same field gradient. 4.2.2.5. High-Current Focusing Lens. I n another recent development, the use of high-current focusing lenses has been considered. The basic idea underlying this device is that a uniform cylindrically symmetrical current flowing parallel to the direction of an incident beam of particles would produce a restoring force towards the axis of the cylinder proportional t o the distance T from the axis. I n practice, such a uniform current can be approximated by an array of thin parallel wires. A given particle will then experience the required field ( a T ) , except when it comes close to one of the conductors, in which case there may be appreciable deviations. If the particle hits a conductor, it will be scattered and thus will not contribute to the intensity a t the image.
714
4.
BEAM TRANSPORT SYSTEMS
A focusing lens of this type has been built and tested by Luckey16 at M.I.T. He used a regular hexagonal array of nineteen conductors. The spacing between adjacent conductors was 3.8 cm. The lens had a length of 100 cm, and was operated at currents up to 1000 amp per conductor which gives a gradient of -46 gauss/cm. The present limitation on the lens arises from the force on the conductors. However, it is expected that with additional mechanical constraints on the conductors, considerably larger gradients can be obtained. As a test of the focusing properties of the lens, 100-Mev electrons produced by the bremsstrahlung beam of the M.I.T. Synchrotron were focused by the lens with an object distance (from the source to the front of the lens) of 200 cm. At a distance of 200 cm behind the exit of the lens, the intensity was observed with a scintillation counter. As the lens current was varied, a maximum was obtained a t the value (-500 amp) calculated for lOO-Mev/c particles. For a 5 cm diameter source, the size of the image was -7.5 cm high by 10 cm wide. The image size did not diminish appreciably when a smaller (1.3 cm diameter) source was used. It is likely that the deviations from the required field near the conductors give rise to an image size of the order of the spacing between the conductors. The main advantage of the wire lens as compared to standard quadrupoles is that one can obtain a higher solid angle of acceptance A n at the object for the same stored energy. On the other hand, the finite size of the image and imperfect focusing constitute a serious drawback to the use of the high-current lens. 4.2.2.6. Momentum Dispersion at the Image. In connection with the design of a strong focusing system, it is useful to determine the momentum dispersion at the image, i.e., the values of dT,/dp and dTh/dp, where T , and Th are the image distances for vertical and horizontal focusing, respectively. A large value of these derivatives indicates that particles whose momentum p deviates slightly from the desired momentum po will be focused a t an appreciabIe distance from the focus for p = PO,and can therefore be largely eliminated by means of appropriate collimators. We will first obtain d T / d p for a single quadrupole magnet which acts as a convergent lens. From the definition of K [Eq. (4.2.6)], one finds dK
K
=
(4.2.51)
zp'
Similarly, dX
&=
x 4-
1 csc2 a 2P
D. Luckey, Rev. Sei. Znstr. 31, 202 (1960).
(4.2.52)
4.2.
BEAM BENDING AND FOCUSING SYSTEMS
715
From Eq. (4.2.16), one obtains
T
=
(SX
+
- A)
K')>/(&
(4.2.53)
whence
dT dX ~ )+ _ - ( P + Kdp
( S - X)'dp
2~ ~
dK -.
(S - X)dp
(4.2.54)
Upon inserting Eqs. (4.2.51) and (4.2.52) into Eq. (4.2.54), one finds (4.2.55) which is the desired result. I n connection with the calculation of d T / d p for a two-magnet system (see below), it is useful to express d T / d p in terms of the image distance T , instead of S. By means of Eq. (4.2.16), one obtains (4.2.55a) which is equivalent to Eq. (4.2.55). In order to derive d T / d p for a single divergent lens, we note that Eq. (4.2.24) gives: (4.2.56) T = - ( K2 SP)/(S PI.
+
+
By proceeding in the same manner as in Eqs. (4.2.51)-(4.2.55a), one obtains the following expressions for d T / d p :
+ d2[p(S' + + 2 ~ ' s+ (8' - K')Z csch' a]. (4.2.57) dT 1 T ( T 2 - K')Z csch2 a]. (4.2.57a) dp - 2p(K2 - p 2 ) [ p ( T 2+ + ~ K ' + dT
1
K')
d p = - 2P(S
K')
For a two-magnet system, the momentum dispersion at the image dTb/dp is given by
where ( a T a / a p ) bis the value of d T b / d p for magnet b alone, for a fixed object distance Sb from the entrance face of b. Here sb refers to the distance of the object seen by magnet b, i.e., the image formed by magnet a. Thus we have: dSb = -dT,, where d T , is the change in the position of the image formed by a due to a small change of momentum d p . This leads to the relation (4.2.59)
716
4.
BEAM TRANSPORT SYSTEMS
which has been used in Eq. (4.2.58); here aT,/ap is the momentum dispersion of magnet a alone. I n Eq. (4.2.58), aTb/aSb is to be evaluated for a constant momentum p. We first consider the case where magnet a is focusing, and magnet b defocusing (direction 1). I n order to obtain dTb/aSb, we differentiate Eq. (4.2.24) with the following result dSb(Tb f Pb) so that
+ dTb(sb + Pb)
=
0
(4.2.60) (4.2.60a)
The subscript b denotes that all quantities pertain to magnet b. Upon inserting Eqs. (4.2.55), (4.2.57a), and (4.2.60a) into Eq. (4.2.58), one obtains
where 8, is the object distance from the entrance of a, and Tb is the distance from the exit face of b to the final image. For the direction 2 for which magnet a is defocusing, one obtains by means of Eqs. (4.2.55a), (4.2.57), and (4.2.58) :
It may be noted that Eqs. (4.2.61) and (4.2.62) involve only the field gradients and the distances S, and Tb, but no intermediate quantities such as T, and sb. Thus (dTb/dp)i can be calculated directly after the parameters K, and Kb of the strong focusing system have been determined. For the two-magnet system considered above (Table I), for the direction 1, one finds (dTb/dp)l = (115.6/p)m/(Bev/c), where p is the particle momentum in Bev/c. Similarly, for direction 2, one obtains (dTb/dp)z = (27.8/p)m/(Bev/c).
As was expected, both values of (dTb/dp)i are positive, meaning that the image distances increase with increasing momentum. 4.2.2.7. Width of the Image Due to a Momentum Acceptance Ap. For a point source along the axis of the strong focusing system, the expressions
4.2.
BEAM BENDING AND FOCUSING SYSTEMS
717
for ( d T b / d p ) land ( d T b / d p ) zenable one to determine the width of the image along the y and z directions due to the momentum acceptance Ap. These widths will be denoted by w, and w,,respectively. In the present discussion, we assume that Ap is given. Ap is determined by the aperture of the quadrupoles and the bending magnet which is used to momentumanalyze the particles and which generally precedes the quadrupole pair. A rough estimate of Ap for such an arrangement will be given below (see Section 4.2.4). The average momentum of the particles will be denoted by PO, and the range of accepted momenta is assumed to extend from po - ( A p / 2 ) to p a ( A p / 2 ) . We will assume th at magnet a is vertically focusing, and consider first the horizontal direction, in order t o obtain the width wy.We consider particles of momentum PO ( A p / 2 ) . These particles will be focused a t a distance ATb,
+
+
(4.2.63)
behind the central image l o pertaining to momentum P O . For the particles with p = pa, we denote by f y ~ , othe values of y a t t,he exit of magnet b for the extreme trajectories which just graze the ( A p / 2 ) , the corresponding pole faces of magnet b. For momentum po values of y a t the exit of b can be written as follows:
+
AYE
=
5 YE.O & y ~ , i ( A p / 2 )
(4.2.64)
where YE,^ is of the order of ~ E , ~ / p The o . second term of Eq. (4.2.64) is thus very small, being of the order of Ap/po times the first term ( Y E , o ) . It can be easily shown that the extreme y coordinates for
P
=
Po
+ (AP/2)
a t the central image ZO are given by:
We will henceforth neglect the second term of Eq. (4.2.65), since it is of order ( A P ) ~ . The total width w, at the central image 10 due to the particles with momentum p a ( A p / 2 ) is obtained b y taking the difference between the two y values of Eq. (4.2.65).One finds:
+
(4.2.66)
For particles of momentum po - ( A p / 2 ) , the image lies a t a distance ATb in front of T o , where ATb is given by Eq. (4.2.63). It can be easily
718
4.
BEAM TRANSPORT SYSTEMS
shown that the resulting width w, at I0 is again given by Eq. (4.2.66). For particles whose momentum lies inside the range from P O - ( A p / 2 ) to po ( A p / 2 ) , the width of the beam a t IO will be less than the value given by Eq. (4.2.66). Therefore, Eq. (4.2.66) for w,represents the total width of the beam at lo, when the momentum acceptance is Ap. The width w,in the vertical direction is similarly given by:
+
(4.2.67)
where 5-Z E , O are the values of z at the exit of quadrupole b, for the extreme trajectories for momentum P O . The values of yg.0 and z E , ~in Eqs. (4.2.66) and (4.2.67) are given by: YE,O = luYm
ZE.O
= l z ~ m
(4.2.68)
where ym and zm are the maximum possible values of y and x inside magnets a and b. Thus ym and zm are the semi-apertures in the y and z directions: ym = A , / 2 , zm = A , / 2 , where A , and A , are the corresponding apertures of the quadrupoles, which are of the order of the distance A between opposite pole faces. In Eq. (4.2.68), and l, are quantities pertaining to the exit of b, which are similar t o the ratios c, and e, referring to the entrance of a. The expression for can be simply obtained from Eq. (4.2.47) for e, by replacing S a by Tb, p a by P b , K. by Kb, and a by p:
r, r,
In a similar manner, the expression for 3; is obtained from Eq. (4.2.46) for e, with the following result:
+
lu= Tb/(Tb2
Kb2)'I2.
(4.2.69a)
Upon inserting Eqs. (4.2.68)-(4.2.69a) into (4.2.66) and (4.2.67), one finds : (4.2.70)
For the example considered above (Table I), one obtains A , = A . = 30.5 cm, this gives:
r. = 0.499,
{, = 0.987. With
w, = 46.2(Ap/p0)cm
w, = 100.l(Ap/po)cm.
Thus for Ap/po = 0.02, the widths at the image lo (at Tb C 8.9 m) are: w, = 0.92 cm, w, = 2.00 cm.
4.2.
BEAM BENDING AND FOCUSING SYSTEMS
719
The widths w, and w, obtained above must be added to the image widths produced by the finite dimensions of the source (target) a s a result of the magnification of the system ( x h and xv). I n addition, if a bending magnet for momentum-analysis is located in front of the quadrupoles a and b, there will be an additional term in w,due to the momentum dispersion of the bending magnet. This effect will be discussed below (see Section 4.2.4).
4.2.3. The M a t r i x Method of Calculation for Strong Focusing Magnet Systems
I n connection with the design of appropriate strong focusing systems for various experimental arrangements, it is often convenient to use the matrix method of calculation rather than the method of the lens equations which has been described above. These two methods are, of course, mathematically equivalent, an d it is only a question of convenience of the calculation that determines which method is best suited to a particular problem. The matrix method was first proposed by Courant and Snyder.6 As a n illustration of the matrix method, we shall first consider a single strong focusing (quadrupole) magnet, which acts as a converging lens in the horizontal (y) direction. We consider the focusing action in the y direction. For any particular trajectory through the magnet, we denote by yo and yo' the values of y and dy/dx, respectively, a t the effective entrance boundary of the magnet. The corresponding values of y and d y / d x a t the exit boundary will be called y1 and yl', respectively. Courant and Snyder6 have shown that y1 and yl' are related t o yo and yo' by a 2 X 2 matrix, to be called M , such th at (4.2.72)
where the matrix elements Mi, depend only on the properties of the magnet (effective length I and field gradient parameter K ) . The matrix equation (4.2.72) represents the following linear relations: yi
=
Milyo
y11 = Mziyo
+ Mnyo' + M22yo'.
(4.2.73) (4.2.74)
4.2.3.1. Matrix for Focusing Lens. It can be easily shown that the matrix M for the case of a converging (focusing) lens is given b y =
where a
=
I/K.
(
cos a sin a
- K-l
K
sin a cos LY
(4.2.75)
720
4.
BEAM TRANSPORT SYSTEMS
In order to derive Eq. (4.2.75), we note that the displacement y inside the magnet is given by (4.2.76) y = Y COS[(X/K) 61
+
where Y and 6 are constants, and we assume that x = 0 at the entrance of the magnet. We consider a trajectory for which the slope y' ( = dy/dx) a t the entrance (x = 0) is zero. This gives 6 = 0, Y = yo. The values of y and y' a t the exit of the magnet are given by y1 =
yl' =
Y cos = M l l Y - (Y/rc)sin LY = M z l Y .
(4.2.77) (4.2.78)
Equations (4.2.77) and (4.2.78) yield: iM11 = COB a
Mzl
= -K-~
sin a.
(4.2.79)
In order to determine the other two matrix elements, M12and M z 2 ,we consider a trajectory for which yo = 0 (but yo' # 0). I n this case, 6 = 90" in Eq. (4.2.76), and the values of, YO' yl,and yl' are related as follows to Y , K, and a, yo' = -Y/K (4.2.80) y1 = Mlzyo' = - Y sin a (4.2.81) y11 = Mzzyo' = - (Y/K)COS a. (4.2.82) Upon solving Eqs. (4.2.80)-(4.2.82) for Mlz and MI2 M22
M22,
sin a = cos a.
=K
one obtains (4.2.83) (4.2.84)
It may be noted that the determinant IM( of the matrix M is unity: IMJ=
MllM22
- M12M21= 1
(4.2.85)
as can be easily verified from Eq. (4.2.75). This is a general property of the matrix M for any type of strong focusing system, and it can be used in numerical calculations as a check on the accuracy of the matrix elements M,. I n the vertical ( z ) direction, the focusing (or defocusing) action of a magnet system can also be represented by a matrix, which will be denoted (4.2.86) where 20 and are the values of z and dz/dx a t the entrance of the field region, and zl, 211 are the corresponding values a t the exit of the field region.
4.2.
BEAM BENDING AND FOCUSING SYSTEMS
72 1
4.2.3.2. Matrix for Defocusing lens. In the example considered above of a single quadrupole, the magnet action is defocusing in the vertical direction, and one can easily show th at the matrix N is then given by =
cosh a sinh a
(~-1
sinh a cosh a
K
(4.2.87)
The expressions for the mat,rix elements Nij in Eq. (4.2.87) can be determined in the same manner as the Mi, [Eqs. (4.2.76)-(4.2.84)] by considering two particular trajectories through the magnet, which are of the general form Z = cosh[(z/K) E] (4.2.88)
+
where the constants Z and f depend on the initial conditions, i.e., on the values of zo and z{ a t the entrance of the magnet. From Eq. (4.2.87) it is easily verified that the determinant IN1 = 1, which is a n example of the general property discussed above. The matrices M and N of Eqs. (4.2.75) and (4.2.87) transform initial values at the entrance of the magnet to final values pertaining to the exit boundary of the magnet. I n addition, one may consider a field-free region (drift space) of length zo following the magnet (see Fig. 8). We wish to find the matrix M for the horizontal displacements which transforms initial values at the entrance of the magnet to final values pertaining to the exit of the field-free region past the magnet. 4.2.3.3. M a t r i x for Field-Free Region. For this purpose, we shall use the matrix Mo which describes the action of the field-free region alone. M Ois given by:
Mo
=
(A To)-
(4.2.89)
M o relates the final values y2 and yz' at the exit of the field-free region to the initial values y1 and yl' at the entrance of this region. The matrix elements ( M & of M O can be easily derived from the equations [see Eqs. (4.2.73) and (4.2.74)l: Yz Y2'
+ xoy;
= y1 = y1I.
I n order to obtain the matrix (see Fig. 8) we note th at
A?i which transforms
(4.2.90) (4.2.90~~) yo, yo' into yz, yz'
Hence M is obtained by straightforward matrix multiplication of Mo and
722
4.
BEAM TRANSPORT SYSTEMS
M [as given by Eq. (4.2.75)]. One finds i l Z = MOM
=
(cos
a
- K - ~ x O sin a
-K-l
K
sin a
+
).
sin a 20 cos a cos a
(4.2.92)
= 1. It can be easily verified that the determinant As an illustration of the preceding discussion, Fig. 8 shows the y trajectory of a particle traversing a focusing section (of length I ) followed by a field-free region (of length 2 0 ) . The value of K is such that Z / K = s/2, FOCUS1N G SECTION
-MATRIX -MATRIX
FIELD -FREE REGION
M -MATRIX MOM= -M
Mo-
FIG.8. Schematic diagram showing horizontal (y) trajectory in a focusing section followed by a field-free region. The horizontal displacements at the three boundaries arc denoted by yo, y,, and ~ 2 The . angles qo, 71,and 72, shown in the figure, are given by: ~i= tan-’ yi’ (i = 0, 1, 2), where yo’, yl’, and y ~ ’arc the values of dy/dx at the boundaries.
and the phase angle 6 [Eq. (4.2.76)] is -30°, so that the trajectory inside the magnet is given by: y = Y COS[(?~X/~Z)
- (a/6)].
(4.2.93)
The horizontal displacements yo, yl, y2 at the three boundaries are shown in Fig. 8, together with the angles V O , ql, q2, which are given by: q; = tan-’ g i (i = 0, 1 ,2 ), where yo’, y i , yz’ are the values of the derivative cEy/dx at the boundaries. In the present example, the lens which precedes the field-free region has a defocusing action in the vertical (2) direction [matrix N , Eq. (4.2.87)]. In this case, the combined matrix which transforms the initial values zo and z< into the final values of E and d z l d x , to be denoted
m
4.2.
by
z2
iV
and
=
221,
MoN
723
BEAM BENDING AND FOCUSING SYSTEMS
is given by the matrix product of N and Ma:
=
cosh a
+
K-*
K-IXO sinh a sinh a
K
+
sinh a xo cosh a cosh a
).
(4.2.94)
We note that if the horizontal action is defocusing (instead of focusing as was assumed above), the matrix M for the horizontal motion will be given by Eq. (4.2.87) (involving the hyperbolic functions) instead of Eq. (4.2.75). Conversely, the vertical action will then be focusing, so that the matrix N for the vertical motion will be given by Eq. (4.2.75) instead of (4.2.87). I n other words, the roles of M and N are just interchanged in this case. 4.2.3.4. Matrices for Two-Magnet System. We will now obtain the transformation matrix for the system of two magnets ( a and b) separated by a field-free region, which has been discussed in Section 4.2.2. The matrix M u which relates the values of y and dy/dx a t the exit of magnet b to y and dy/dx a t the entrance of magnet a, is obtained by matrix multiplication, as follows :
Mu
=
Mu,bMdMy,a
(4.2.95)
where My,aand M y , bare the matrices M u pertaining to magnets a and b, respectively; Md is the matrix for a field-free region of length d ; thus Md = M Oof Eq. (4.2.89), with xo = d . An equation similar to (4.2.95) gives the matrix M , for the vertical motion in the two-magnet system. I n the following, we will denote the matrices M , and M , by MI and Mz, where M I pertains to the direction 1 for which magnet a is focusing, and M2 refers to direction 2, for which magnet a is defocusing. I n view of Eqs. (4.2.87) and (4.2.92), one finds = =
Ml,bM&l,a
( coshsinhp K;'
sinh 0) (cos a - (d/K,)sin -1 cosh -K, sin a
Kb
ff
K,
+
sin a d cos a cos a >. (4.2.96)
The first matrix on the right-hand side of Eq. (4.2.96) represents MI,* [Eq. (4.2.87)];the second matrix is Md2Ml,,,, as obtained from Eq. (4.2.92). By straightforward matrix multiplication, one obtains the following expressions for the matrix elements of M1: (4.2.97) cash @[COS CY - (d/K&&l a] - (Kb/K,)Sinh p Sin ff (MI)IZ= cash @ ( K a Sin a d COS a) Kb sinh p cos (4.2.98) (4.2.99) ( M 1 ) n = K:' sinh ~ [ C O Sa - (d/u,)sin a] - K ; ~ cosh p sin a (M1)22 = K;' sinh @(Ka sin a d cos a) cosh /3 cos a. (4.2.100) (M1)11 =
+
+
+
+
724
4.
BEAM TRANSPORT SYSTEMS
The matrix M Zpertaining to the direction 2 is given by
M Z= M ~ ~ b M d M 2 . a =
(-KC'cossin/3 /3
@) (cosh
sin cos @
Kb
a
4- (d/K,)sinh
K , ~
sinh a
a
K~
sinh a 4- d cosh a cosh a (4.2.101)
where the first matrix on the right-hand side is M2,b, as obtained from Eq. (4.2.75),and the second matrix represents M d M 2 , a[see Eq. (4.2.94)]. The resulting matrix elements of M z are as follows:
+
+ +
(MZ)II COS @[cash (d/Ka)sinh a] (Kb/Ka)Sin @ sinh a ( M ~ ) I= z cos @ ( K ~sinh a d cosh a ) Kb sin @ cosh a (Mz)z~ = -K:' sin @[cosha (d/K,)sinh a] K,' cos @ sinh a ( M Z ) Z= Z -K;' sin @ ( K ~sinh a d cosh a) cos @ cosh a.
+
+
+
+ +
(4.2.102) (4.2.103) (4.2.104) (4.2.105)
4.2.3.5. Matrices for Three-Magnet System. In a similar manner, one can obtain the matrices for a three-magnet system (triplet). We denote the quadrupoles by a, b, and c. The distance between magnets a and b will be called d, as before, and the distance between b and c will be denoted by g. The matrix Mi for the transformation from the entrance of a to the exit of c is given by: Mi =
Mi,CM,Mi,bMdMi,,
(i = 1, 2)
(4.2.106)
where M i , a l Md, and M i . b have their previous meanings, M , is the matrix for a field-free region of length g, and Mi,, is the matrix pertaining to magnet c. For direction 1, it will be assumed that magnets a and c are focusing, while magnet b is defocusing. Equation (4.2.106) can then be written as follows : M1 = M I , c M , ( M l , b M d M l , a ) . (4.2.107)
The matrix elements of (Ml,CMdMl,a)can be obtained from Eqs. (4.2.97)(4.2.100);M , is given by Eq. (4.2.89),and M1,,is given by Eq. (4.2.75), in which K must be replaced by K , and a by l C / K C . Similarly, for direction 2, one obtains:
M Z= M Z , C M ~ ( M Z , & ~ M P , ~ )
(4.2.108)
where ( M Z , b & f d M Z , a ) is given by Eqs. (4.2.102)-(4.2.105), and M z , , can be obtained from Eq. (4.2.87). A symmetric triplet is frequently used to obtain double focusing. In this configuration, magnets a and c are identical and have the same values of the field gradient. Moreover, the distances d and g are equal. If the
4.2.
BEAM BENDING A N D FOCUSING SYSTEMS
725
gradients are so chosen that the object and image distances are equal (S = T), then the two magnifications are equal: x1 = x2 = 1, which is desirable in some experimental arrangements. In order to make a comparison of a typical three-magnet system with a two-magnet system having essentially the same field gradient and the same length of magnetic field, we consider the following two configurations which can be used to focus low-energy particles ( p 5 300 Mev/c) from a target: (1) System A is a symmetrical triplet, with magnet lengths 1,
=
1, = 20.3 cm
b#!
=
40.6 cm
separations d = g = 10.2 cm, so th at the total length of the system is: L, = 101.6 cm. It is found that for values of the field gradients such that K~ = K~ = 33.3 cm, Kb = 36.4 cm, one obtains double focusing for object and image distances S = T = 146.3 cm. The magnification x is unity for both the horizontal and vertical directions. Assuming that magnets a and c are vertically focusing and b is horizontally focusing, one finds tg = 0.524, c, = 0.975, so that t = 0.511. The intensity a t the image is determined by the quantity
e/S2 = 0.511/(146.3)2 = 2.39 X lo-'. (2) For comparison, we consider system B consisting of two quadrupoles with lengths I, = &, = 40.6 cm, separation d = 20.4 cm, and K~ = Kb = 34.8 cm. Thus system B has the same gradient as the average gradient of system A , and moreover, the combined length of the magnets is the same (81.2 cm), as well as the total length of the system (La = 101.6 em). From Eq. (4.2.42), one finds that the double focusing condition is satisfied for S = T = 49.3 em, which is considerably smaller than the value S = T = 146.3 cm for system A . The magnifications are: l x h l = 0.239 and lxvl = 4.18. One finds tha t eII = 0.195, c2 = 0.817, so that e = 0.159. Thus t is considerably smaller than for the triplet ( t = 0.511), but because of the reduced object distance,
t/S2 = 0.159/(49.3)'
=
6.54 X lopb
is appreciably larger than for the three-magnet system. Thus the gain in intensity for system B as compared to A is a factor GO= 6.54/2.39 = 2.74. The gain Go will be further increased if the particles have a short lifetime, e.g., for K mesons. The total object-to-image distance L is:
L
=
2(146.3) 4- 101.6
=
394.2 cm
726
4.
B E A M TRANSPORT SYSTEMS
+
for the triplet system, as compared to:L = 2(49.3) 101.6 = 200.2 cm for system B. For K mesons of momentum p~ = 300 Mev/c, the fraction F of particles which do not decay in traversing a distance L is: F = e-v, with v = L r n ~ / ( ~ ~where p ~ ) ,the lifetime TK = 1.2 X sec., mK = K meson mass = 494 Mev/c2. For system A , F = 0.165, whereas for system B, F = 0.400. Thus the total intensity gain at the image, GK for K mesons is: GK = 2.74 X (0.400/0.165) = 6.64.
It can be concluded from these results that for secondary beams of short-lived particles of low momenta, it is more advantageous to use a two-magnet system as compared to a three-magnet system with the same average gradient and length, since one gains both from the larger solid angle Afl subtended a t the target and from the smaller object-toimage distance L , which reduces the decay probability 1 - F. The fact that the horizontal magnification I ~ h lis appreciably less than 1 for the two-magnet system may also be useful in order to obtain good energy resolution using a bending magnet, as was discussed above. On the other hand, for focusing external beams over large distances, where the beam incident on the system is not very divergent, it may be advantageous to use a triplet system, as was done in the external proton beams at the Cosmotron and for the antiproton beams from the Berkeley Bevatron. I n this case, the fact that we can make X h = xv = 1 is convenient, since it enables us to preserve the size and shape of the source, so that the image can then serve as an object for a subsequent similar three-magnet focusing system. 4.2.3.6. Focal length and Position of Focal Planes of a Magnet System. From the matrices M and N for a general magnet system we shall now obtain the focal length jof the magnetic system and the position of the focal planes for both the horizontal and vertical directions. We consider the matrix M of Eq. (4.2.72). The focal length f is defined as follows.' Consider a trajectory which enters the strong focusing system at a unit distance from the central axis, yo = 1, and with zero slope, yo' = 0 (i.e., the effective object is at infinity). Then the focal length f is that distance along the x axis over which the displacement y of the outcoming trajectory decreases by one unit. With the initial conditions yo = 1, y4 = 0, the values at the exit of the system yl and y; are given by : yi = Mi1 y l l = Mzi. (4.2.109) The displacement y after exit from the system is given by: 1/ = y 1 +
y1)2 = Mi1
+ Mzls
(4.2.110)
4.2.
BEAM BENDING A N D FOCUSING SYSTEMS
727
where x is measured from the exit boundary of the system. The focal length f is given by Ax, where Ax is obtained from the following equation : Ay = - 1 = M ~ ~ A x
(4.2.111)
whence
f
=
AX
=
-1/Mzl.
(4.2.112)
We note that this derivation is independent of whether the action of the magnetic system is focusing or defocusing. For a defocusing system, Ax, and hencef, is negative, i.e., the displacement y increases with increasing x [the outgoing ray (trajectory) is diverging]. Equation (4.2.109) gives the horizontal focal length, which will be denoted by f h . The vertical focal length fu is obtained in a similar manner from the matrix N, and is given by fv
=
-1/Nzi.
(4.2.113)
In the derivation of Eq. (4.2.112),we have not made use of any specific property of the strong focusing system. Hence Eqs. (4.2.112)and (4.2.113) (as well as the following equations for the location of the focal planes) are applicable t o a general strong focusing system which may consist, for instance, of several quadrupole magnets separated by field-free regions. For the case of a single magnet considered above, Eqs. (4.2.112) and (4.2.113) reduce t o the usual expressions [see Eqs. (4.2.12) and (4.2.23)]: f h = K c8C a! (4.2.114) f v = --K csch a. (4.2.115) I n order to obtain the distance d I , h of the horizontal image focal plane FIVhfrom the exit boundary of the system, we use again the initial conditions y o = 1, y< = 0 for the horizontal motion. The distance d 1 . h is then determined by the position x where the y displacement of the outcoming ray is zero, Thus yi whence
+ 91'2 d1.h
=
=
Mi1
x
=
+ M 2 1 ~= 0
-Mii/Mzi.
(4.2.116) (4.2.117)
I n Eqs. (4.2.116) and (4.2.117), 2 is measured from the exit boundary of the strong focusing system. A positive d z , h means that the horizontal image focal plane F Z , h lies to the right of the exit boundary. For the distance dr,,, of the vertical image focal plane FI,,, from the exit boundary, one obtains in a similar manner: dr." = -Nii/Nzi.
(4.2.118)
728
4.
BEAM TRANSPORT SYSTEMS
For the example discussed above, where the matrices M and N are given by Eqs. (4.2.75) and (4.2.87), one finds dr,h = dr,, =
K
cot a coth a.
-K
(4.2.119) (4.2.120)
The distance dr,h for the focusing (horizontal) direction corresponds to A, (Eq. 4.2.11), whereas the distance dr,, for the defocusing (vertical) direction corresponds to - p (Eq. 4.2.22). We note that dr.o is negative, i.e., the vertical image focal plane lies to the left of the exit face (see Fig. 3). In order to obtain the location of the horizontal object focal plane FO,h, we consider the case where the exit boundary conditions are: y1 = 1, yl' = 0 (image at infinity). The resulting equations for yo and yo' as obtained from Eq. (4.2.72) are given by M11yo Mziyo
+ MlZYOI = 1 + Mzzyo' = 0.
(4.2.121) (4.2.122)
The object focal plane Fo,h is determined by the distance x (as measured from the entrance of the magnet system) for which y = 0. The corresponding distance do,h is given by do,h
=
x
= yo/yo'.
(4.2.123)
In view of Eq. (4.2.122), we obtain do,h = -Mzz/Mzi
(4.2.124)
where a positive sign of do,h means that the horizontal object focal plane
FO,his in front (to the left) of the entrance boundary. In the same manner, the distance do,%from the entrance of the system to the vertical object focal plane Fo,, is given by do.,
=
-Nzz/Nzi.
(4.2.125)
For the case of a single magnet considered in Section 4.2.2, one finds (4.2.126) (4.2.127)
In this case, we have: do,h = dr.h, and do,, = dr,,, as was already discussed above (see Figs. 2 and 3). Referring to the matrices M and N [Eqs. (4.2.75) and (4.2.87)], we see that this result arises from the fact that Mll = M z 2and N11 = N z z for these matrices. However, for more complicated strong focusing systems, these matrix elements are in general not equal, and as a result, d0.h # dr,h and do,, # dr,v. In particular, for the
4.2.
BEAM BENDING AND FOCUSING SYSTEMS
729
strong focusing system consisting of two quadrupole magnets (with gradients of opposite sign) separated by a field-free region [see Eqs. (4.2.27), (4.2.32)],the distances do,i and d1.i to the focal planes are given by : do,; = - uia (4.2.128) dr,i = - Uib. (4.2.129) In general, do,l and df,l are different since U1, # Ulbl and similarly do.2 # d1,2. Correspondingly, for the matrices M and N for the complete two-magnet system, we have M l l # Mz2 and N11 # N Z Z . We will now obtain the expression for the lateral horizontal magnification X h in terms of the matrix elements Mij. For a ray which starts with a finite slope and zero displacement a t the object, the initial conditions yo and ya' are related as follows to the object distance 8 h (distance from object to entrance boundary of system) : ?/O/yO' =
From the equations for yl and
(4.2.130)
8h.
(Eq. 4.2.72), one finds
~ 1 '
where Th is the distance from the exit boundary of the system to the horizontal image. Now we consider a ray starting from the object with a finite yo and with zero slope (go' = 0). Thus we have (4.2.132) (4.2.133)
y1 = Mnyo Yl'
=
M21yo
a t the exit of the magnet system. The horizontal displacement D h of the trajectory a t the position of the image is given by
From Eq. (4.2.134),we find for the IateraI magnification Xh
=
DdYo
where we have made use of IMI as follows:
=
=
1/(Mzi&
+ M2z)
Xh:
(4.2.135)
1. Equation (4.2.135)can be rewritten
(4.2.136)
730
4.
BEAM TRANSPORT SYSTEMS
where X 0 , h is the distance from the object to the object focal plane FoJ,. The last expression in Eq. (4.2.136) is identical with Eq. (4.2.44) given above. For some applications, it may be useful to have an expression for X h in terms of the image distance Th (rather than Sh). By means of Eqs. (4.2.131) and (4.2.135), one obtains: ~h
= Afii
(4.2.136a)
-k MziTh.
For the vertical direction, the lateral magnification xt, is given by xu =
l/(NZlSr,
+ Nz2).
(4.2.137)
By means of Eq. (4.2.131),we can obtain the expression for the horizontal longitudinal magnification A h , which is defined as aTh/dSh, where the partial derivative is evaluated for constant momentum p of the particle. Thus one finds:
(Mz1Tn
Mil)'
(4.2.138)
where we have used the property that /MI = 1. A similar expression holds for the vertical longitudinal magnification A, = a T , / a s , :
Equations (4.2.138) and (4.2.139) can be used, for example, to determine the extent of the image along the direction of the outcoming beam when the target which constitutes the object has a large longitudinal dimension (e.g., for a liquid hydrogen target whose length may be of the order of a meter). EXAMPLE : As an example of the use of the matrix method, we give the matrices for the strong focusing system considered in Table I. From Eqs. (4.2.97)(4.2.100),one obtains for the matrix Icfl for the direction 1 : -0.0315
M I = M & k f J l l . a= -0.4160
2.526 1.630
where it is assumed that all distances are expressed in meters. From the matrix elements of M 1 ,one obtains by means of Eqs. (4.2.112), (4.2.117), (4.2.125), (4.2.131), and (4.2.135) assuming that the object distance Sla = 4.57 m: f l = 2.404 m, do,l = 3.918 m, dr,l = -0.076 m,
4.2.
BEAM BENDING AND FOCUSING SYSTEMS
73 1
Tlb = 8.76 m, and x1 = -3.68. As expected, these results are in agreement with the corresponding values obtained from the lens equations (Table I), within the small rounding errors of the numerical calculations. We note that do.1 = - Ula, and dI,l = - u l b . Similarly, upon using Eqs. (4.2.102)-(4.2.105)~one obtains for the matrix M n pertaining to direction 2 :
From the matrix elements of M z , one finds: fz = 3.421 m, do,2 = 0.411 m, dr,z = 6.244 m, Tzb= 9.06 m, and xz = -0.822. These results are in good agreement with the corresponding values of Table I.
4.2.4. The Focusing Equations for Deflecting Wedge-Shaped Magnets with Finite Field Index n In the preceding discussion, we have considered only quadrupole focusing magnets. For these magnets, the field H is zero along the central axis (y = z = 0). We shall now discuss the focusing action of deflecting magnets with nonuniform fields, which can be used simultaneously for double focusing and for a determination of the momentum of the particles. For such magnets, it is customary to consider the field index n,first introduced by Kerst and Serber,lB which is defined by: n = -(R/H)(dH/dr), where R is the radius of curvature of the normal trajectory (in the central region of the particle beam), H is the field along this trajectory, and dH/dr is the radial derivative of H. A schematic view of a wedge-shaped magnet is shown in Fig. 9. The central trajectory traverses the path MM’VQ; the path NN’V’UU’ represents a second trajectory, which will be discussed below. The center of curvature of the central trajectory is P ; the distance ( P M ) = ( P V ) = (PQ) is the radius of curvature R of the particles. The entrance and exit angles at the magnet boundaries are denoted by s and t. These angles are to be taken as positive if the path of the particles at the entrance or exit of the magnet is on the same side of the normal to the boundary as the center of curvature of the central trajectory. Thus in Fig. 9, s is positive and t is negative. The focusing depends critically on the entrance and exit angles s and t. For the case of a uniform field (n = 0 ) ,the equations for the focusing action have been derived by Camac” and C r o s ~ . ~These ~ * ’ ~equations were extended by D. W. Kerst and R. Serber, Phys. Rev. 60, 53 (1941). M. Camac, Rev. Sci. Instr. 22, 197 (1951). W. G. Cross, Rev. Sci. Znstr. 22, 717 (1951). l9 References to earlier works on the focusing action of wedge-shaped magnets are given in the review article of K. T. Bainbridge, in “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 1, pp. 566-614. Wiley, New York, 1953. l6
l7
732
4.
BEAM TRANSPORT SYSTEMS
Sternheimerato the case of a wedge-shaped magnet with nonuniform field (n # 0). As will be discussed below, the resulting focusing equations can be written in the form of the lens equations (4.2.16) and (4.2.24), provided that &, Ti,K , A, and p are appropriately defined.4 I n practice, the field is made nonuniform inside the magnet by tilting the upper and lower pole faces with respect to each other, so that the gap height g increases or decreases along the width of the magnet (from A to B in
P
FIG. 9. View of a wedge-shaped magnet showing the central particle trajectory MM'VQ and a second particle trajectory NN'V'UU'.
Fig. 9). Positive n (radial falloff of the field) corresponds to the case of increasing g, as one goes radially outward with respect to the center of curvature P of the central trajectory. Magnetic spectrometers with nonuniform fields to provide double focusing have been extensively used in mass spectroscopy experiments.20-26 4.2.4.1. Vertical Focusing by the Fringing Field of the Magnet. For the case of a single magnet with n = 0, the only vertical forces acting 80 N. Svartholm and K. Siegbahn, Arkiv Mat. Astron. Fysik SSA, No. 21 (1946); N. Svartholm, ibid. SSA, No. 24 (1946). F. B. Shull and D. N. Dennison, Phys. Rev. 71, 681 (1947); 72, 256 (1947). 1) D. L. Judd, Rev. Sci. Znstr. 21, 213 (1950). 9s E. S. Rosenblum, Rev. Sei. Znstr. 21, 586 (1950). a 4 N. Svartholm, Arkiv Fysik 2,115 (1951); E. Arbman and N. Svartholm, ibid. 10, 1
(1955). 26 N. E. Alseevsky, G . P. Prudkovsky, G . I. Kusourov, and S. I. Filimnov, Doklady Akad. Nauk S.S.S.R. 100, 229 (1955). Z6 A. V. Dubroniv and G. V. Balabina, Doklady Akad. Nuuk S.S.S.R. 102, 719 (1955).
4.2.
BEAM BENDING AND FOCUSING SYSTEMS
733
on the particles occur at the entrance and the exit of the magnet. We will first consider the situation at the magnet entrance. Referring to Fig. 9, we introduce a coordinate system xyz with origin a t M , the x axis perpendicular to the entrance face ( A B ) and pointing towards the interior region of the magnet, the y axis along the magnet entrance ( M B ) , and the z axis pointing out of the paper. It will be assumed that at the magnet entrance, the vertical component of the field varies from 0 to its value H , inside the magnet over a distance Ax along the x axis. From Maxwell’s equations, we obtain : (4.2.140)
Equation (4.2.140)shows that there is a field component H , given by:
H,
=
(4.2.141)
(H,/Ax)z
which extends over the region of the fringing field (of width Ax). The resulting vertical force F:’ acting on the particle is given by:
FP) = - e v ~ ’ H , / c = -e(v/c)sin s ( H , / A x ) z
(4.2.142)
where v:) = v sin s is the y component of the velocity v. The force FF) acts during a time At = A x / ( v cos s), so that the resulting change Ap:’ of the z component of the momentum is given by: Apr’
=
F:’ At
=
- ( e H , tan s / c ) z
=
- ( p tan s / R ) z .
(4.2.143)
I n a similar manner, one finds that at the exit of the magnet, the z component of the momentum changes by the following amount: Apr’
=
- ( e H , tan t/c)z
=
- ( p tan t/R)z.
(4.2.144)
Aside from the terms (4.2.143) and (4.2.144) which describe the action of the fringing fields, there exists also a vertical force F , inside the magnet, if the field H is nonuniform (n # 0). The force F, arises from the radial component of the field H , inside the magnet, which is determined by Maxwell’s equations: (4.2.145)
One thus obtains :
H,
(4.2.146)
= -(nH,/R)z.
The force F, is then given by:
F,
=
evH,/c
=
-[(evnH,)/cR]z
=
-(pvn/R2)z.
(4.2.147)
734
4.
BEAM TRANSPORT SYSTEMS
It is seen that the terms A p r ) , Ap:", and F , will act as restoring forces provided that s, 2, and n, respectively, are positive. Thus in Fig. 9, the magnet action is vertically focusing a t the magnet entrance (s > 0) and vertically defocusing a t the exit ( t < 0). 4.2.4.2. Horizontal Focusing by a Wedge-Shaped Magnet. For the horizontal focusing, the situation is somewhat different. There is no horizontal focusing or defocusing action at the entrance or the exit of the magnet. Nevertheless, the sign and magnitude of the angles s and t determine the focusing (or defocusing) action of the magnet in the following manner. Referring to Fig. 9, we see that for a second trajectory NN'V'UU' (originating a t the object a t a finite S), which arrives above the central trajectory M M ' V Q a t the entrance of the magnet, the deflection angle Q' inside the magnet should be larger than the angle (p for the central trajectory in order that the particle shall be focused at the image. If re' is the distance between the second trajectory and the central trajectory at the magnet entrance, the second trajectory will traverse an additional path length: Al, = -re' tan s in the magnetic field. Similarly, at the exit of the magnet, if rt' is the distance between the two trajectories, the second trajectory will traverse an additional path length : Alt
=
-rt' tan t.
In Fig. 9, r,' is given by the distance ( M N ) .The distance IAl,] is given by: ( M M ' ) = ( N N ' ) = r,' tan s. Similarly, a t the exit of the magnet, we have rt' = ( U Q ) and AZt = ( U U ' ) = rt' tan 111. It may be noted that for the first-order theory here presented to be applicable, it is necessary that r'/R be <
4.2. BEAM
BENDING AND FOCUSING SYSTEMS
735
(for s = t = 0) is exactly compensated by the decrease of the field H 1 - (r8'/R),where H and H' for the second trajectory since H ' / H are the values of the field along the central trajectory and the second trajectory, respectively. This result enables one to understand qualitatively why the restoring force F, inside the magnet has a factor (1 - n). Specifically, F, is given by
F,
= -[m(1
- n)H,/cR]r' = -[pv(l - n)/R2]r' (4.2.148)
where r' is the radial displacement from the central trajectory; thus r' = r - R = ( V V ' ) in Fig. 9. We note that the factor in the square bracket of Eq. (4.2.148) for F, differs from that of Eq. (4.2.147) for F, only by the replacement of n by (1 - n). The equations for F , and F, were first derived by Kerst and Serber.ls The restoring forces F, and F, give rise to characteristic vertical and horizontal oscillations in various particle accelerators. These oscillations are usually called betatron oscillations, since they were first discussed by Kerst and SerbeP in connection with the theory of the operation of the betatron. The restoring forces F, and F, are sometimes referred to as the betatron forces. We note that the special case of a quadrupole magnet discussed above (Sections 4.2.1-4.2.3) can be obtained from the case of finite R and n by letting R and n both go to infinity, in such a manner that Rlnl-1'2 remains finite. In the limit In1 -+ w , the quantity Rln[-1/2is equal to K-112. Alternatively, we have the correspondence,
n(p2++ -K12
(4.2.149)
where K is given by Eq. (4.2.6). In Eq. (4.2.149),the angle (p, of course, decreases to zero in such a manner that n(p2remains finite as In1 -+ 0 0 . 4.2.4.3. Vertical Focusing for n > 0. We shall first discuss the case where 0 < n < 1, so that the action of the field inside the magnet is focusing both horizontally and vertically. I n this case, the reduced distances S1 and T1 for the vertical focusing are defined as follows:4
= AS& - (S,/R)tan s] TI = T,/[1 - (T,/R)tan t ] S1
(4.2.150) (4.2.151)
where S, is the object distance and T, is the image distance for vertical focusing; R is the radius of curvature of the particles, s and t are the entrance and exit angles a t the magnet boundaries (see Fig. 9). We define A,, and K , as follows: A, 3 Rn-'12 cot(n112(p) (4.2.152) K , 3 Rn-112. (4.2.153)
736
4.
BEAM TRANSPORT SYSTEMS
where Q is the angle of deflection in the magnetic field. With S1, T1, K,,, and A, as defined by Eqs. (4.2.150)-(4.2.153), it can be shown that the following lens equation similar to Eq. (4.2.16) holds:
(Si - Av>(T1- A,) whence Ti
=
= K,,~
+ Xu2
(six,+ Ku')/(#I - A").
(4.2.154) (4.2.155)
From the value of TI, one can obtain T, by means of Eq. (4.2.151). One finds T, = T1/[l (TI/R)tan t ] . (4.2.156)
+
From Eqs. (4.2.150)-(4.2.156) one can also obtain the following alternative expression3 for T,: T,
=
+ + tan t]-'
R[n1/2tan(n1/2p 5 )
(4.2.157)
where the angle 4 is defined by: tan 6 = -n-'/Z[(R/S,)
- tan s].
(4.2.157a)
4.2.4.4. Horizontal Focusing for n < 1. A similar situation exists for the horizontal focusing. I n this case, the reduced distances S2and T2 are defined as follows:4 (Sh/R)tan s] (4.2.158) Sz = S h / [ l Tz Th/[l (T&/R)tant ] (4.2.159)
+
+
where s h and Th are the horizontal object and image distances, respectively. The lens equation is given by
(Sz- b)(Tz - A h ) where
Ah
and
Kh
=
Kh2
+
Ah2
(4.2.160)
are defined by: Ah 3 Kh
R(1 - n)-'" cot[(l - n)112$0] R(1 - n)-'".
(4.2.161) (4.2.162)
From Eq. (4.2.160), we obtain
+ Kh2)/(SZ -
Ah).
(4.2.163)
Th = Tz/[l - (T2/R)tan t ] .
(4.2.164)
Tz
= (SZAh
Finally, from Eq. (4.2.159),
From Eqs. (4.2.158)-(4.2.164), one can also obtain the following alternative expression for Th: Th = R ( (1
- n)l/Ztan[(l
+ 31 - tan t]-1
- n)1/2p
(4.2.165)
4.2.
BEAM BENDING AND FOCUSING SYSTEMS
737
where q is defined by: tan q
= - ( 1 - n)-1'2[(R/Sh) + tan
s].
(4.2.166)
EXAMPLE : As a n example of the preceding results, we discuss the case of a single wedge-shaped magnet for which n = 0.4, cp = 40°, s = 30°, and t = -24.8'. For an object distance S ( = S, = sh) = 3.OR, this magnet gives double focusing a t a n image distance T = 4.9612. Considering first the focusing action in the vertical direction, one finds from Eqs. (4.2.150)-(4.2.156) : S1 = -4.098R, Xu = 3.345R1
K , = 1.581R TI = 1.506R
and
T,
=
4.96R.
For the horizontal direction, Eqs. (4.2.158)-(4.2.164) yield the following results: 82 = 1.098R, Kh = 1.291R1 Ah = 2.150R1 TP = -3.829R1 and Th = 4.96R. For a magnet with given n and cp and specified values of the object distance S ( = S, = 8,) and the entrance angle s, there is (at most) one angle t which will give double focusing; the image distance T (= T,= T h ) is completely determined by the values of n, cp, S, and s. In order to obtain the values of t and T for double focusing after T, and T 2have been determined from Eqs. (4.2.155) and (4.2.163),we make use of the following relation [cf. Eqs. (4.2.156) and (4.2.164)]:
From Eq. (4.2.167), one obtains tan t T
= =
R(T1 - Tz)/(2TlT2) 2TiT2/(Ti 7'2).
+
(4.2.168) (4.2.169)
Figures 4-6 of reference 3 show plots of T and t (for double focusing) as a function of the entrance angle s for three values of the angle of deflection: cp = 20°, 40°, and 90'. The curves were calculated for S = l R , 2R, 3R, and w , and for the following n values: n = 0 and 0.4 for all three p values, and, in addition, n = -0.4 for lp = 90". Similar calculations of the double focusing for cp = 180' with n = 0.8 and 0.9 have been carried out by Karmohapatro.21 S. B. Karmohepatro, Indian J . Phys. 32, 26 (1958).
738
4.
BEAM TRANSPORT SYSTEMS
4.2.4.5. Horizontal Defocusing for n > 1. For the case n > 1, the equations for the vertical focusing (Eqs. 4.2.150-4.2.1574 are unchanged. However, in the horizontal direction, there is now defocusing, so that Eqs. (4.2.160)-(4.2.166) must be suitably modified. The reduced distances Sz and T z are still given by Eqs. (4.2.158) and (4.2.159). I n place of Eq. (4.2.160), the lens equation is given by Eq. (4.2.24), in which p and K are replaced by p h and Kh defined as follows: /lh Kh
R(n - I)-'/' coth[(n - 1)'Izp] R(n - 1)-"'.
(4.2.170) (4.2.171)
From Eq. (4.2.24) one obtains the following equation which determines the reduced horizontal image distance Tz:
Tz =
-(Kh2
+ SZPh)/(sZ +
Ph).
(4.2.172)
The actual horizontal image distance T h can now be obtained from T z by means of Eq. (4.2.164). 4.2.4.6. Vertical Defocusing for n < 0. For the case n < 0, Eqs. (4.2.158)-(4.2.166) apply for the horizontal focusing, but Eqs. (4.2.152)(4.2.157a) must be modified, since the magnet action is vertically defocusing in this case. S1 and T 1are still given by Eqs. (4.2.150) and (4.2.151). In analogy to Eq. (4.2.24), the lens equation is now given by (S1
where p,, and
K~
+
+ p.1
= pU2-'.K
(4.2.173)
= Rlnl-'/2 coth(ln11/2p)
(4.2.174) (4.2.175)
pY)(T1
are defined by: pv I(,
~%(n(-'/~.
The reduced vertical image distance
Ti =
-(G'
-k
T1
is obtained from
sipv)/(si
f
pv).
(4.2.176)
The actual image distance T , is then obtained from 2'1 by means of Eq. (4.2.156). As an alternative, T , can be obtained from the following equation^,^ which are appropriate for n < 0, provided that 1 - (S,/R) t a n s < (S ,/R)In11123 T , = R[tan t - lnI1/ztanh(ln11/2p t')]-l (4.2.177)
+
where the angle
[' is defined by: tanh 5'
E
Inl-1/2[(R/S,)- tan s].
(4.2.178)
In comparing Eqs. (4.2.150) and (4.2.151) with Eqs. (4.2.158) and (4.2.159), we note that the sign of the terms involving tan s and tan t is
4.2.
BEAM BENDING AND FOCUSING SYSTEMS
739
negative for vertical focusing, whereas it is positive for horizontal focusing. Aside from this result, there are the usual differences concerning the dependence on n:16the factor n1/2for the vertical focusing is replaced by (1 - n)II2 for the horizontal focusing. Similarly, Eqs. (4.2.165) and (4.2.166) for Ta can be obtained from Eqs. (4.2.157) and (4.2.157a) for T, by merely reversing the signs of tan s and tan t , and replacing n by (1 - n). In reference 4, Sternheimer has obtained the expressions for the momentum dispersion D and the magnifications xu and Xh for a wedgeshaped magnet with arbitrary field index n and arbitrary entrance and exit angles s and 1. For the momentum dispersion D, defined as d y l d p , where d y is the lateral displacement of the image due to a small change of momentum p , one finds: (1) F o r n < 1:
(2) For n
> 1:
+ (Tn/R)tan 21
{[l - cosh(n - 1)1'2p][l -
(Th/R)(?Z - l)%inh(n - 1)112p] (4.2.180)
where Th is the horizontal image distance (as measured from the exit face of the magnet). For the case n > 0, the vertical magnification at the image xv is given by
+
xu = sec & { c o ~ ( n l / ~ pt;ll)[l - (T,/R)tan t] where the angle
+
- n1/z(Tu/R)sin(n1/2p 5,))
(4.2.181)
I,,, is defined by: Ern tan-'(n-'"%an s). (4.2.182) < 0, the appropriate expressions for xu can be found in 3
For the case n Table 111 of reference 4. For n < 1, the horizontal magnification Xh =
+
Xh
is given by
+
sec 6,(cos[(l - n)'/zp 6,][l (Th/R)tan t] - (1 - n)1/2(Th/R)sin[(l- n)'12p
+
6m]}
(4.2.183)
where 6, is defined by: ,6
= - tan-l[(l - n)-'4an s].
(4.2.184)
740
4.
BEAM TRANSPORT SYSTEMS
The expressions for Xh for the case n > 1 are given in Table I11 of reference 4. 4.2.4.7. Focusing Properties of Wedge-Shaped Magnets with Uniform Field. The expressions given above for n # 0 cannot be directly used for the case of a uniform field (n = 0). The image distances T,,o and Th,ofor vertical and horizontal focusing by a wedge-shaped magnet with uniform field are given byl7~l8
+
R sin cp cos s cos t s h cos(p - S)COS t T h s o = (&/R)sin(p - s - t ) - cos(cp - t)cos s' For n
D
=
=
0, the expressions for D,xu, and
(R/p)((l- cos p)[l
= SeC S{COS(p- S ) [ l
+
are as follow^^^^^^:
+ (Th/R)tan t ] + (Th/R)sin
xa = (1 - p tan s)[l - (T,/R)tan Xh
Xh
(4.2.186)
p]
(4.2.187)
- (T,/R)tan s (4.2.188) (Th/R)tan t ] - (Th/R)Sin(p - 8)). (4.2.189) t]
We note that Eqs. (4.2.187) and (4.2.189) for D and Xh can be obtained directly from Eqs. (4.2.179) and (4.2.183),respectively, by setting n = 0. In order to derive Eq. (4.2.188) from Eqs. (4.2.181) and (4.2.182),one must expand lm and xu for small n, and go to the limit n -+ 0. The longitudinal magnification A h and Aw for n = 0 can be obtained by straightforward differentiation of Eqs. (4.2.185) and (4.2.186). One finds: A
-
- cos2
cos2 t/[(Sh/R)sin(cp - s - t ) - cos((p - t)cos
a T h 1 ~ s
=
ash
A , , = - a= T v o
as,
s12
(4.2.190)
-l/((S,/R)[tans
+ (1 - (otans)tant] - (1 - (otant)}2. (4.2.191)
In connection with some applications, it may be useful to evaluate the derivatives dTh,o/dpand dT,,o/dp, which give the momentum dependence of the image distances Th.Oand T,,o for a wedge-shaped magnet with uniform field. These derivatives are analogous to the quantities dT/dp (i = 1, 2) which have been evaluated in Eqs. (4.2.51)and (4.2.62) for the case of a single quadrupole magnet and a system of two quadrupoles. In order to derive the expression for dTh,o/dp,it is convenient to start from Eqs. (4.2.165) and (4.2.166), which give for n = 0: R
= tan((o
q =
R
+ q ) - tan t -= -Dh
- tan-'(tan
s
+ R/Sh).
(4.2.192) (4.2.193)
4.2.
BEAM BENDING AND FOCUSING SYSTEMS
741
Here and in the following, we denote the denominator of Th,O by Dn. Thus
Dh = tan(q
+ 7) - tan t.
(4.2.194)
As shown by Eqs. (4.2.192) and (4.2.193), Th.0 is a function of the five variables: R, s h , s, t , and p. As the momentum p is varied, R , t , and q will vary, while sh and s remain unchanged since they pertain t o the incident beam. Therefore we obtain :
From Eq. (4.2.192), we find: (4.2.196)
Equation (4.2.193) gives : (4.2.197)
whence (4.2.198)
It can be easily shown that: aR/ap = R / p (for n = 0). In connection with the second and third terms on the right hand side of Eq. (4.2.195),we note that
(4.2.200)
The derivative a q / a p can be obtained from Eqs. (86) and (91) of reference 4 by setting n = 0. One finds 1 - = - -[sin
a(a
aP
P
(a
+ (1 - cos (a)tan t].
(4.2.201.)
It is easily shown that for a small change of momentum which results in an increment dq, the corresponding increment dt of the exit angle is equal to dq, so that: &/ap = a q / a p .
742
4. BEAM
TRANSPORT SYSTEMS
Upon inserting Eqs. (4.2.198)-(4.2.201) into Eq. (4.2.195), one obtains the following result for dTA,o/dp:
+ S,R cos2q sec2(cp+ q ) + [sec*(v + 7) - sec2 t ]
DA
X [sin cp
+ (1 - cos cp)tan t ] }
(4.2.202)
where 7 and Dh are defined by Eqs. (4.2.193) and (4.2.194), respectively. In order to derive the expression for dT,,o/dp, one proceeds essentially in the same manner as for dTh,o/dp, starting from Eqs. (4.2.157) and (4.2.157a) for T,. However, in the present case, one must first obtain aT,%/dR,aT,/dcp, and aT,/dt for arbitrary n, and then go to the limit n --f 0. One thus obtains
--dp
Ra2 pDv2 sec2 1
] [sin
+
cp
+ (1 - cos cp)tan t ]
where a and D, are defined by: a
= S,/(Sv tan s - R )
D, = ( a - q)-'
+ tan t.
(4.2.204) (4.2.205)
4.2.4.8. Discussion of a Focusing System with Momentum Analysis. A commonly used focusing arrangement consists of a bending magnet B to momentum-analyze the particles, followed by a pair of focusing quadrupoles Q. and Qb. This system is shown schematically in Fig. 10. For this arrangement, one is interested in the momentum acceptance Ap and the resulting width w of the final image (formed by QaQb). In order to obtain an estimate of the momentum acceptance Ap of the system, we will assume that the limiting momenta, po - (Ap/2) and po (Ap/2) are those for which the central trajectory just grazes the pole faces of the quadrupole magnet (Q. or Qb) which is horizontally focusing ( @ , in Fig. 10). According to the definition of ey [see Eqs. (4.2.46) and (4.2.47)], the available horizontal aperture at the entrance of QG is given by e,A,, where A , is the maximum available aperture, i.e., A , is of the order of the distance between opposite pole faces. We now have: EVA, = D Ap Sal&/dpI Ap (4.2.206)
+
+
where D is the momentum dispersion at the image I B formed by magnet B , dq/dp is the derivative which enters into D [see Eq. (4.2.201)], and S,
4.2.
BEAM BENDING AND FOCUSING SYSTEMS
743
is the effective object distance for quadrupole Q., i.e., the distance from the image I g to the entrance face of Q.. Thus we have: S, = dB - T B , where d B is the distance between the exit face of B and the entrance face of Q., and T B is the distance of the image I B from the exit face of B. The first term on the right hand side of Eq. (4.2.206) gives the width of the beam at la due to the momentum dispersion D . The second term of Eq. (4.2.206) gives the additional broadening of the beam in traversing the distance S, between I g and the entrance of Q..
0
FIG.10. Schematic view of focusing system consisting of a bending magnet B and two quadrupole magnets Qa and &a. The figure shows the central trajectories for two momenta: PO and P O - ( A p / 2 ) .
Upon using Eq. (4.2.187)for D,and Eq. (4.2.201) for dcpldp, we obtain from Eq. (4.2.206): AP { ( l- coa q)(R egA, = -
Pc
+ T s tan t ) + T g sin cp + S.[sin cp + (1 - cos cp)tan t ] )
(4.2.207)
where cp is the angle of deflection, R is the radius of curvature in the bending magnet B , and t is the exit angle from B , as shown in Fig. 10. Upon T B = ds, we obtain from Eq. (4.2.207): using the relation: S,
Q
Po
=
e,A,/i[sin
(a
+ + (1 - cos cp)tan t]dB + ~
( -1 cos
cp)].
(4.2.208)
For the case that Q. is vertically focusing and Qb is horizontally focusing, is given by Eq. (4.2.47), whereas if Q, is horizontally focusing, eg is given by Eq. (4.2.46). It should be noted that the momentum acceptance A p given by Eq. (4.2.208) represents only an estimate. In Fig. 10, the upper trajectory
e,
744
4.
B E A M TRANSPORT SYSTEMS
pertains t o the average momentum pa, while the lower trajectory is tangent to the lateral boundary of Qb, and thus represents the central trajectory for momentum pa - (Ap/2). For this momentum, particles emitted by the target a t a small angle to the central trajectory will be transmitted by QoQb, if the angle is such that the particle path lies above the central trajectory in Fig. 10. On the other hand, if the trajectory lies below the central trajectory of Fig. 10, the particles will not be accepted by QoQb. Thus for momentum p a - (Ap/2), roughly half as many particles as for momentum PO will be accepted by the present focusing arrangement. (The actual fraction will, of course, depend on the other aperture stops and collimators which may be present in the system.) It can also be seen that even if the momentum is somewhat below pa - (Ap/2), a small fraction of the particles will be accepted by the system, namely those for which the (upward) angle at the target is sufficiently large so that they will not hit the lower lateral boundary of Qb in Fig. 10. We note that the preceding discussion does not include the effect of the finite dimensions of the target and detector. A complete discussion of the various factors which enter into the width w of the final image will not be given here. However, we will describe the three effects which arise directly from the momentum spread Ap. (1) As a result of the momentum dispersion D of the bending magnet, there will be a lateral displacement at the image I B formed by B , which is given by ;AYB = D(Ap/2) (4.2.209)
+
for the particles of momentum pa (Ap/2). Here D is given by Eq. (4.2.187),if the bending magnet has uniform field, i.e., n = 0. A ~ repreB sents the width of the effective object for the quadrupole pair Q.Qb. Thus as a result of the horizontal magnification XA,Qof QoQb, there will be a width a t the final image which is given by AlY/ =
IXh.QI
AyB
(4.2.210)
where X h , Q can be obtained from Eq. (4.2.135) or (4.2.136a). The matrix elements Mij which enter into these equations are given by Eqs. (4.2.102)(4.2.105) for the arrangement of Fig. 10, in which the first quadrupole, Q., is vertically focusing, so that the horizontal direction is direction 2 in the notation of Eqs. (4.2.95)-(4.2.105). I n the reverse case, in which Q. is horizontally focusing, the Mi, are given by Eqs. (4.2.97)-(4.2.100). (2) As a result of the momentum spread, the position of t h e image formed by QaQb will be spread out, even if we would assume that the position of the image formed by the bending magnet does not depend on p . This effect has been discussed in Section 4.2.2 [see Eqs. (4.2.51)-(4.2.71)].
4.2.
BEAM BENDING AND FOCUSING SYSTEMS
745
The corresponding spread in the distance from the exit of Qb to the final image, ATa, is given b y : (4.2.211)
where i = 1 if magnet Q. is horizontally focusing, i = 2 if Qa is vertically focusing, and (dTb/dp)i is given by Eq. (4.2.61) or (4.2.62). (3) As a result of the momentum spread A p , the image formed b y the bending magnet B will also be spread out longitudinally (Le., along the direction of the outcoming beam). If this spread in the image distance is denoted by A T B , we have: A S , = A T R . We note that A T B is given by ( ~ l T h , ~ / dApp), where d T h , a / d pis obtained from Eq. (4.2.202).The resulting spread in the location of the final image is given by:
A T @ = lAil AS,
=
l A i l ( d T h , ~ / dA~ p)
(4.2.212)
where i = 1 or 2 as discussed after Eq. (4.2.211),and Ai is given b y Eq. (4.2.138), in which the matrix elements M i j can be obtained from Eqs. (4.2.97)-(4.2.100) for i = 1, and from Eqs. (4.2.102)-(4.2.105) for i = 2. The total spread ATb in the location of the final image is given by:
ATb
=
ATa
+ AT@.
(4.2.213)
As discussed in Section 4.2.2 [see Eqs. (4.2.63)-(4.2.71)], the width a t the image due to ATb is given by: AZY/ = (ATb/Tb)ra,(Au/2)
(4.2.214)
where Tb is the image distance from the exit face of Qb, A , is the effective maximum available aperture of the quadrupole pair QaQb, and {, is given by Eq. (4.2.69) for i = 1 and Eq. (4.2.69a) for i = 2. We note th a t { , A , represents the width of the particle beam a t the exit of Qb. For i = 2, we have : (4.2.215)
The total width of the final image due to A p is approximately given by:
ApYf
=
A1Yi -t A z Y / .
(4.2.216)
I n connection with Eq. (4.2.213), we note that both A T a and A T @are positive, and that they must be added to obtain the total ATb. This result arises from the fact th at with an increase in momentum by a n amount A p , A T a is such as t o increase T b , as discussed after Eq. (4.2.62), and moreover, A T @ will also act in the same direction, since a n increase of momentum will result in an increase of the image distance from B , T B , and
746
4.
BEAM TRANSPORT SYSTEMS
hence a decrease of the object distance S, from the entrance of quadrupole Q.. As is shown by Eq. (4.2.138), aTh/aS,, is always negative, so that for focusing (Tb > 0 ) , Tb will be increased as a result of the decrease of S,. Equation (4.2.216) for Apyf does not include the width a t the image due to the width of the source, i.e., the target which is in front of the bending magnet B. If the width of the target is denoted by at, the resulting width of the image, Atyr, is given by I x ~ , B Q ~ u ~ , where Xh,BQ is the horizontal magnification for the entire system BQ.Qb. Atyf must be added to Apyf to obtain the total width w of the final image. 4.2.4.9. Comparison of Strong Focusing Quadrupoles with WedgeShaped Focusing Magnets. The question arises as to whether in a particular experiment it is more advantageous t o use a system of strong focusing quadrupoles combined with an analyzing magnet with uniform field, or as an alternative, one or two bending magnets with nonuniform field and appropriate entrance and exit angles s and t. In general, for experiments with high-energy particles ( p 2 1 Bev/c), it is desirable t o use the first alternative, because the large radius of curvature R of the particles in a reasonably attainable field (-15,000 gauss) would force one to use very long bending magnets ( I 2 R / 2 ) and large entrance and exit angles (e.g., s = 50°, t = -50'). The magnetic saturation effects will be important and may make it difficult to define the effective magnetic field boundaries a t the entrance and exit of the magnet, because of the irregular shape of the fringing field. This is particularly true if the pole faces are tapered, giving a large n value to increase the focusing. In this case, model measurements are necessary to determine the effective field boundaries, and hence the effective entrance and exit angles s and t . By contrast, the equivalent quadrupole magnet system will in general be considerably shorter, and the effective field boundaries are more easily determined, because of the symmetry of the quadrupole field, and its rapid falloff outside the magnet iron. However, in certain cases, it may be desirable to use the same magnet for focusing and analyzing the particles, if one is dealing with relatively low-momentum particles with a short lifetime so that the distance L from the object to the image must be minimized. As an example of this possibility, we mention the experiment of Meyer et ~ 1on .the~scattering ~ of K+ mesons by protons, a t energies up to 90 Mev ( p = 310 Mev/c). In this experiment, the Kf mesons were focused and momentum-analyzed by a single bending magnet (45.7 cm wide, 91.4 cm long), whose pole faces were modified as follows. Each pole face was divided into two sections of equal length (45.7 cm). In the first section traversed by the beam, the pole faces were tapered so as t o give increasing gap height with increasing 28
D. I. Meyer, M. L. Perl, and D. A. Glaser, Phys. Rev. 107, 279 (1957).
4.3. BEAM
SEPARATORS
747
r , so that n was positive, which leads to vertical focusing. I n the second section, the pole faces were tilted so as to make n negative, thus giving horizontal focusing. Obviously, the two sections correspond to magnets a and b of the previous discussion, the distance d being zero in this application. Double focusing of the Kf mesons was obtained for a total object-to-image distance of 3 meters. It should be emphasized that this type of system is practical mainly for relatively low-energy particles ( p 5 0.5 Bev/c). ACKNOWLEDGMENT
I wish t o thank Dr. Luke C. L. Yuan for several helpful suggestions concerning this chapter.
4.3. Beam Separators* The production of secondary beams of particles depends upon the phase space available due to high energy collisions. The result is that high momentum secondary beams from accelerators are produced predominantly in the forward direction. Various methods of selecting the momentum, focusing the secondary particles, and separating the various charged particles have made it possible to produce “clean” beams of particles. The design of secondary beams is very dependent upon the available flux of particles, the types of detectors t o be used, and the lifetime of the desired particles. For bubble chambers and emulsion experiments, spatial separation of the background particles is required ; whereas for counters and spark chamber experiments, temporal separation is required. 4.3.1. Degrader Type Separation Early experiments were done’ using a d E l d x degrader to separate the particles of a given momentum and different masses. The low velocity 1 L. T. Kerth, D. H. Stork, R. P. Haddock, R. W. Birge, J. R. Peterson, J. Sandweiss, and M. Whitehead] Positive heavy mesons produced at the Bevatron. UCRL3031 (June, 1955); 0.Chamberlain, W. W. Chupp, A. G. Ekspong, G. Goldhaber, S. Goldhaber, E. J. Lofgren, E. SegrB, C. Wiegand, E. Amaldi, G. Baroni, C. Castagnoli, C. Franzinetti, and A. Manfredini, Phys. Rev. lo%, 921 (1956);W.H.Barkas, R. W. Birge, W. W. Chupp, A. G. Ekspong, G. Goldhaber, S. Goldhaber, H. H. Heckman, D. H. Perkins, J. Sandweiss, E. SegrB, F. M. Smith, D. H. Stork, L. Van Rossum, E. Amaldi, G. Baroni, C. Castagnoli, C. Franzinetti, and A. Manfredini, ibid. 106, 1037 (1957);and L. E. Agnew, Jr., T. Elioff, W. B. Fowler, R. L. Lander, W. M. Powell, E. SegrB, H. M. Steiner, H. 6. White, C. Wiegand, and T. Ypsilantis, ibid. 118, 1371 (1960).
* Chapters 4.3 and 4.4 are by Bruce Cork.
748
4.
BEAM TRANSPORT SYSTEMS
particles have a greater energy loss in the degrader and are deflected a greater amount in the magnetic field a t the exit of the degrader. This method has many disadvantages, including loss of the desired beam due to interactions and scattering by the degrader. For low energy beams, electromagnetic separators are much better. 4.3.2. Electromagnetic Separators
A coaxial static-electromagnetic velocity spectrometer has been used2 to give a 450 Mev/c K- and p beam. Although the solid angle for accept-
copper conductor
FIG.1. The electrostatic plates p are supported by cylindrical insulators. The side walls N, 8 are the “pole tips” of an H-shaped electromagnet. The vacuum enclosure is made by welding stainless steel plates SS to the pole tips.
ance of transmitted particles was small, several experiments were done with this spectrometer. A parallel-plate velocity spectrometer has been built3 to reject negative pions from a beam of antiprotons. The electromagnetic separator, shown in Figs. 1, 2, and 3, consisted of two parallel plates 20.3 cm wide and 580 cm long separated by a mean distance of 10.2 cm and maintained 9 J. J. Murray, A coaxial static-electromagnetic velocity spectrometer for high energy particles. UCRL-3492 (May, 1957); and N. Horwitz, J. J. Murray, R. Ross, and R. Tripp, 450 Mev/c K - and $ beams in the northwest target area of the Bevatron separated by the coaxial velocity spectrometer. UCRL-8269 (June, 1958). a C. A. Coombes, B. Cork, Pi. Galbraith, and G. R. Lambertson, Phye. Rev. 112, 1303 (1958).
4.3.
BEAM SEPARATORS
749
a t potentials of f 180 kv. The crossed magnetic field was adjusted so that antiprotons would be undeflected by the combined electric and magnetic fields, while pions were deflected and did not go through the slit. The magnetic field of approximately 200 G along the full length of the separator was supplied by water-cooled conductors. The pole tips were also a part of the vacuum vessel and the return path for the magnetic field was in iron; thus the spectrometer was an H magnet.
FIG.2. Bev-1795. End view of parallel plates of t.he crossed electric and magnetic field velocity spectrometer. The sides of the vacuum tank are made of soft iron and are the pole tips of an H magnet. The top and bottom are made of stainless steel.
A particle of velocity PO is transmitted through the slit when the magnetic field is adjusted so that P o = E / B , where E is the average electric field and B is the average magnetic field. Other particles of charge e and momentum p are displaced a t the output of the spectrometer by a distance 1J =
g[
-
;]
and have an angular deflection
The effective length of the electric field is 1.
750
4.
BEAM TRANSPORT SYSTEMS
FIG.3. Bev-1806. The 6.1 in long parallel plate velocity spectrometer. Charged particles enter from the right. High potential is applied through the coaxial feedthrough insulators on the left.
4.3.3. Radiofrequency Separators
For energies greater than a few Bev, the spatial separation of pions from K mesons becomes very difficult. Since the K meson lifetime is approximately one-half that of the pion, and 4 is less for a given momentum, the system cannot be allowed to become arbitrarily long. Also, the angular separation using a dc separator,
is proportional to l / p 3 . Very good optics will be required in the multi-Bev region. A mass sensitive deflector for high energy particles has been proposed by Panofsky.4 Such a device has been built6 and used to select 350 Mev/c 4 6
W. K. H. Panofsky, HEPL 82 (May, 1956). R. P. Phillips, HEPL 171 (June, 1959).
4.3. BEAM
SEPARATORS
751
electrons. A microwave cavity was located at the center of the optical system of a triple focusing spectrometer. This cavity was coherent with the microwave linear accelerator and deflected the electrons that were scattered from a target. The amount of deflection depended upon the time of transit between the target and the deflecting cavity. The desired particles can be selected by changing the relative phase of the microwave cavity and the accelerator. With the radio frequency spectrometer, the angular separation is
eEl
A0 = -
PPC
where E is now some effective electric field. Various systems6 have been proposed so that the net force due to the electric and magnetic fields of the cavity not be zero. It is not practical to make most accelerators coherent with a microwave cavity, so that either longer wavelengths must be used as proposed by Veksler' or the secondary beam has to be made coherent by means of a series of cavities. Rejection of unwanted particles at all phase angles can be obtained by'using circular polarization of the deflecting forces in both cavities. The deflection of particles of one velocity would be zero while other velocities would be deflected by both cavities. Since the power requirements are difficult, and the solid angle of acceptance is small, Montague8 has suggested that plane polarization be used but that an absorber be so located that all particles which pass through the separator undeflected are stopped on this central beam absorber, located behind the second cavity. The desired particles are deflected around the absorber and are brought to focus in a useful beam. A spectrometer of this type is being designed a t CERN to separate 10 Bev pions from K r n e s ~ n . ~
4.3.4.
Separation by Nuclear Interactions
At energies greater than 10 Bev, the spatial separation of particles becomes very difficult. Fermi has shownL0that the relativistic increase 8 W. K. H. Panofsky and W. A. Wenzel, Rev. Sci. Znstr. 27,967 (1956); J. P. Blewett and J. D. Kiesling, BNL/ADD/JPB-JDK (August, 1959); and M. L. Good, A radiofrequency separator for high-energy particles. UCRL-8929 (October, 1959) ; E. D. Courant and L. Marshall, Rev. Sci. Znstr. 81, 193 (1960). 7 V. I. Veksler, PTOC. Intern. Conf. on High Energy Accelerators, CERN, Geneva, 1969 p. 426 (1959). 8 B. W. Montague, CERN PS/INT. AR/P SEP/60-1 (July, 1960). 9 W. Schnell, CERN 61-5 (1961). 10 E. Fermi, Phye. Rev. 66, 1242 (1939); 67, 485 (1940).
752
4.
BEAM TRANSPORT SYSTEMS
in ionization loss above the minimum, which is caused by the relativistic lateral extension of the electric field of the moving particle, is appreciable. For values of /3 = u / c > 0.97, the ionization increases as log /3r to a maximum that is dependent upon the density effect." This increase of ionization has been used12 with a n argon-filled cloud chamber to distinguish pions from protons by droplet counting. The pions have approximately 20 % greater ionization loss than protons at lOLo eV, and good separation in the energy range of 1O1O to 10l2 eV has been achieved. Other separation methods, by means of strong interaction processes, have been used. Pure p-meson and neutrino beams have been obtained by filtering out nucleons, K mesons, and 7r mesons. The long lifetime of the p meson and the large interaction cross section of the nucleons and K mesons makes this possible. At high energy the differences in the behavior of the various types of particles again allows separation schemes to be devised. Several methods that include charge exchange of a beam of high energy particles followed by magnetic separation have been suggested. l3
4.4. Some Examples of Beam Transport Systems 4.4.1. High Momentum Beams Using Counters as Detectors A beam of 10" protons of 6 Bev energy incident on a 15.2 cm long beryllium target produces approximately three antiprotons and lo6 negative pions per millisteradian in the forward direction for a f6 % momentum interval a t 2.8 Bev/c. This beam, using a system of magnetic quadrupole focusing magnets, a parallel-plate velocity spectrometer, time-of-flight scintillators, and a gas cerenkov counter, has been used14 to measure total, differential and charge-exchange antiproton-proton cross sections in the momentum interval from 1.7 to 2.8 Bev/c. The K--p total cross section was also measured in the same beam. For experiments using scintillation counters and spark chambers as detectors, the duty cycle of the beam should be as long as possible, thus reducing the accidental rate in the counters. A beam duration of 300 msec R. Sternheimer, Phys. Rev. 88, 851 (1952);81, 256 (1953);103, 511 (1956). R. G.Kepler, C. A. d'Andlau, W. B. Fretter, and L. F. Hansen, Nuovo cimento [lo] 7, 71 (1958);and L. F. Hansen, and W. B. Fretter, Phys. Rev. 118, 812 (1960). G. Goldhaber, S. Goldhaber, and B. Peters, CERN 61-3 (January, 1961). 14 R. Armanteros, C. A. Coombes, B. Cork, G. R. Lambertson, and W. A. Wenzel, Phys. Rev. 119, 2068 (1960). 11
I*
4.4. SOME
EXAMPLES O F BEAM TRANSPORT SYSTEMS
753
was achieved by steering a portion of the circulating proton beam of the Bevatron into a thin foil a t the outer radius of the orbits. Small angle collisions caused a portion of the proton beam to become phase unstable. The increasing magnetic field of the Bevatron caused the “equilibrium orbits” to move to a smaller radius, and finally the proton struck a n inner radius target. Antiprotons produced in the forward direction in the Bevatron were deflected in the magnetic field of the Bevatron. The secondary beam
-
0
10
20
30
40
50
Scale (feet)
FIG.4. MU-18068. Antiproton beam for p - p scattering experiment 1.0 to 2.0 Bev beam. Elements are described in text. [From Phgs. Rev. 119, 2068 (1960).]
channel (Fig. 4) consisted of strong focusing magnetic quadrupoles Q1to Q,, deflecting magnet M , and steering magnets C , and Cz. The momentum was selected by locating the target at the appropriate place in the magnetic field of the Bevatron and then tuning the magnets of the channel to the corresponding momentum. The momentum width, +6%, was determined by the horizontal image of the target a t the entrance of Q2. The magnetic quadrupoles Q1 and QZ were 20.3 cm diameter triplets. Antiprotons from the target were focused a t the slit in front of M . The height of the target could be made small, 0.28 cm, and thus the height of the image at the slit was small. Separation was then possible with the
754
4.
BEAM TRANSPORT SYSTEMS
electromagnetic velocity spectrometer-a crossed electric and magnetic field, with the electric field vertical. The system of 10.2 cm diameter magnetic quadrupoles focused the antiproton beam along a 30.5 meters time-of-flight channel. Particles produced in the 183 em long methane cerenkov counter were deflected by magnet Cz,while antiprotons continued on to the hydrogen target. A t a point 4.5 meters beyond Q,,the focused beam was an image 5.1 cm in diameter.
2.0 Bev
I
127
Time of flight ( m p e r e )
FIG.5. MU-19940. Antiproton time of flight delay curve. Both scintillators and a threshold gas Cerenkov counter (in anticoincidence) were used to reject pions.
The separator, described in Chapter 4.3,consisted of parallel plates 11.6 meters long separated by 10.2 cm and at potentials of 5 180 kv. At 1.7 Bev/c the separator rejection factor of pions to antiprotons was 3. This was sufficient to reduce the pion rate in the scintillation counters so that the accidental rate in the time-of-flight coincidence circuits was low. At 2.8 Bev/c the difference between /3 and PO is small, so the electromagnetic separator was not useful. Fortunately the pion rate is lower at high momentum so that accidental counts in the scintillators are less of a problem. Identification of antiprotons was made by means of eight time-of-flight scintillation counters and one gas Cerenkov counter. The delay curve is given by Fig. 5. Tuned either to antiprotons or to K - mesons the back-
4.4.
755
SOME EXAMPLES OF BEAM TRANSPORT SYSTEMS
ground was negligible. Approximately 60 antiprotons per minute were detected at the hydrogen target at 1.7 Bev/c and 15 per minute at 2.8 Bev/c.
4.4.2. Separated Beams for Bubble Chambers The parallel-plate velocity spectrometers described above have been used to obtain good spatial separation of charged particles for bubble chamber and emulsion experiments.l6 A 1.17 Bev/c K- beam, Fig. 6 , consisted of two spectrometers 6.1 meters long and a system of focusing lenses and slits, Fig. 7. Particles of negative charge were produced by 6 Bev protons striking a 3 mm high target inside the Bevatron. Care was taken to keep the lens aberrations small so that the height of the image at the first slit was less than 5 mm. "
HYDROGEN BUBBLE \ \
QUADRUPOLE BENDING
FOCUSING
MAGNET
MAGNETS
J
J
!? i
\
FIQ.6.The 1.17 Bev/c K- beam a t the Bevatron. [From Rev. Sci. Instr.31,1054 (1960).]
With the parallel plates separated a distance of 6.35 cm and operated at a potential difference of 380 kv, the attenuation of pions when tuned to K- mesons was approximately loK,a factor of 30 a t the first slit and 3000 a t the second. A considerable amount of the background through the first slit is due to the decay of pions in the first part of the system, producing muons that are transmitted by the slit. Most of these are not transmitted by the second system. The yield was approximately 9 K- mesons per 10" protons on the target, and the background was approximately 10 beam muons, probably due to the decay of pions admitted to the second stage. The pion background a t the bubble chamber was 8%. H. K. Ticho, Proc. Intern. Cmf. on High Energy Accelerators, CERN, Genera,
lb
1969 p. 387 (1959); Rev. Sci. Znslr. 31, 1054 (1960).
Feet Qbject principal plane
.nOF-I--
5
10
Image principal plane
II
HORIZONTAL PLANE"
''
Bending magnet
Puodrupole 3
Puodrupole 2
Puodrupole I
T\
\
El
\ \
Spectromete
$1,.
I I
I
-c VERTICAL PLANE
0
,,
,
,
,, ,
,
5 10 Feet
IM
1-
BEAM
FIG. 7. MUB-226. Focal propertias of the magnetic quadrupoles used in the 1.17 Bev/c K - separated beam. [From Rev. SCi. Znstr. 31, 1054 (1960).]
Ern E-
-4
01
-4
FIG.8. MU-19685. Plan view of 1 to 4 Bev/c K - beam.:Two high pressure Cerenkov counters were used to reject pions and antiprotons. [From Phys. Rev. 123, 320 (1961).]
4.
758
BEAM TRANSPORT SYSTEMS
The beam was also tuned t o antiprotons and gave a yield of 0.5 antiprotons per 101' protons, with a 20% pion background. A similar system with three 6.1 meters long separators has been used'" to obtain a 1.65 Bev/c beam of antiprotons, with 0.8 p per 10" protons and a background of 10% pions and 250% muons. Beam
0.2"A1 hemispherical window 6810A photomultiplier tube
$" Fe magnetic shielding Lucite light pipe
I$' quartz window
Scale (inchas)
FIQ.9. MU-19686. Cross section of one of the differential high pressure Cerenkov counters.
Two separators 3.05 meters long have been used" with similar optics to give a n 800 Mev/c K + meson beam. Another beam, taking 800 Mev/c K - mesons or antiprotons from an internal target of the Bevatron, a t zero degrees production angle, has been separated'* using two 3.05 meters 1 6 J . Button, P. Eberhard, G. R. Kalbflehch, J. E. Lannutti, G. R. Lynch, B. C. Maglic, M. L. Stevenson, and N. H. Xuong, Phys. Rev. 121, 1788 (1961). 17H. Bradner, W. Chinowsky, G. Goldhaber, S. Goldhaber, T. Stubbs, D. H. Stork, and H. Ticho, The K+-p interaction at 455 MeV. UCRL-9745 (June, 1961); to be published in Phys. Rev. 1*P. Bastien, 0. Dahl, J. Murray, M. Watson, R. G. Ammar, and P. Schlein, in "Proceedings of the International Conference on Instrumentation for High Energy Physics," p. 299. Interscience, New York, 1960.
4.4.
SOME EXAMPLES OF B E A M TRANSPORT SYSTEMS
759
long separators. The yield of this beam was 10 K- mesons per 10" protons, and the background was pions and muons. When tuned to antiprotons the yield was 0.5 ?? and one pion or muon per 10" protons.
4.4.3. Special Beams At high momentum, where both spatial separation and time-of-flight separation becomes difficult, it is possiblelg to use high pressure gas
1.0
lo-' N
lo-'
I -b Y
10-3
IO-~
IO-~
'o-60
200
400 600 8 0 0
1000 1200 1400 1600 1800
Methane pressure ( p s i )
FIG. 10. MU-19707. Selection of particles by varying the pressure in the high pressure methane eerenkov counter, 1.95 Bev/c. The counting rate of all particles down the channel is MI.Mode A operation is coincidence-anticoincidence operation and mode B is anticoincidence operation only. Methane pressure in psi (equivalent to 0.07 kg/cm*).
cerenkov counters to select the desired particles. A 1 to 4 Bev/c Kmeson beam is shown in Fig. 8. The two high pressure methane cerenkov counters are shown in Fig. 9. These were constructed so th a t cerenkov light from K mesons was focused onto the photomultiplier located on the axis and cerenkov light from the pions was focused on the photocathodes of the outer ring of photomultipliers. Th e signals from the pions were then l 9 V . Cook, B. Cork, T. Hoang, D. Keefe, L. Kerth, W. Wenzel, and T. Zipf, K--p and K--n cross sections in the momentum range 1-4 Bev/c. UCRL-9386 (January, 1961); Phys. Rev. l2S, 320 (1961).
760
4.
BEAM TRANSPORT SYSTEMS
placed in anticoincidence. With two counters of this type, the rejection of pions was as shown by Fig. 10. Total and differential K--p and K--n cross sections in the momentum range 1 to 4 Bev/c have been measured. This type of temporal separation of charged particles is adequate for many high energy experiments where counters or spark chamber detectors are used.
5. STATISTICAL FLUCTUATIONS IN NUCLEAR PROCESSES* It is well recognized that we can never measure any physical magnitude exactly, that is with zero error. Progressively more elaborate experimental or theoretical efforts result only in narrowing the range of the possible error. Therefore, in reporting the result of any measurements, it is obligatory to specify the probability that the result is in error by some specified amount, because a gamble on reladive correctness is always involved in all physical determinations. The theory of statistics and fluctuations, presented synoptically here, describes the mathematical procedure involved in the reduction of experimental data of the type encountered in nearly every measurement in nuclear physics.
5.1. Frequency Distributions In any series of measurements, the frequency of occurrence of particular values is expected to follow some “probability distribution law” or “frequency distribution.” There are about a half dozen distributions which are needed most often in the statistical appraisal and interpretation of nuclear data. Of these, the Poisson distribution, or distributions derived from it, is the most common. 5.1 .l. The Binomial Distribution The binomial distribution is the fundamental frequency distribution governing random events. The normal and the Poisson frequency distributions can be derived from it. If the probability that an event will occur in a single trial has the constant value p , then the probability that it will not occur in the same single trial is q = 1 - p . In a group of z independent trials the probability P, that the event will occur x times is given by that term in the binomial expansion of ( p 4)“ in which p is raised t o the x power. For example, if z = 3 identical radioactive atoms are observed over a time
+
1 This part is chiefly an epitome, with some additions and rearrangements, of the treatment given in R. D. Evans, “The Atomic Nucleus.” McGraw-Hill, New York, 1955. See especially Chapters 26, 27, and 28, and Appendix G for documentation of statements given here without proof, and for illustrative examples.
* Part 5 is by Robley D. Evans. 761
5. STATISTICAL
762
FLUCTUATIONS
+
interval during which the probability of decay is p = for each atom, then the chance that three, two, one, or zero atoms will decay during the observation is given by the individual terms of the corresponding binomial expansion :
The probability P o of zero successes is often a very useful statistic. For example, the total proability of one or more successes is simply 1 - PO. The general analytical form of the binomial expansion is (P
+ PI’
z=o
=
1 PZ
(5.1.2)
z=z
in which any individual term can be written as
P, =
Z!
z!(z
- z)! p=(l - p ) - .
(5.1.3)
The binomial distribution, Eq. (5.1.3),cont,ains two independent parameters: the constant probability of success p , which can have any vaIue from 0 t o 1, and the t,otal number of independent trials z. It applies rigorously to those phenomena in which the total number of trials z and the total number of successes 5 are both integers, and 0 5 x _< z. In Eq. (5.1.3)the coefficient z!/z!(z - z)! is the number of unordered selections or combinations of z things (trials) taken z (successes) at a time. Figure 2 contains an example of a binomial dist,ribution, for the special case of p = 0.4,z = 40. The values chosen for p and z determine the mean value (Section 5.2.1)as m = p z = 16, and the standard deviation * ) = 3.10. (Section 5.2.4)as u = d
5.1.2. The Multinomial Distribution The binomial distribution is a special case of the multinomial distribution. The multinomial distribution describes random processes in which several independent results, each having constant probabilities p l , p 2 , . . . , p a , are possible. The separate probabilities are then given by terms of the expansion of ( p l p2 .* * where
+ +
Pl+P2+
+
* . . +p,=1
and z is the number of trials. The general term in this expansion of (pl PZ . palais Z! ,p;’pq’ . . . p;‘ (5.1.4) PZ,.,,. . . . .Z. -
+ +
+
Zl!ZZ!
.
*
.
zr.
5.1.
763
FREQUENCY DISTRIBUTIONS
in which xi is the number of occurrences of the event whose probability is pi for one trial, and x1 xz * * x8 = z is the number of trials. The multinomial distribution applies to nuclear disintegrations in which several processes compete; e.g., for Cu64nuclei let pl = probability of 0- emission, p 2 = probability of p+ emission, p 3 = probability of electron capture, and p4 = probability of no transition during a fixed interval. Note that O ! = 1. This follows from the elementary definition of the factorial n! (5.1.5) _n -- (n - I ) ! hence-I!1 = (1 - l ) ! = O!
+ + - +
5.1.3. The Normal Distribution
The normal distribution (sometimes erroneously referred to as the Gaussian error curve) is an analytical approximation t o the binomial distribution when z is very large. In the normal distribution the observed m . The quantity x is a continuous variable with limits of - CQ and dx is differential probability dP, that x will lie between x and x
+
+
(5.1.6)
. .
where e = 2.7183 , , m = true mean value of x, and u = standard deviation of the distribution of x about m. The standard deviation is a fundamental statistic (see Section 5.2.4) whose value depends upon the mean value for most frequency distributions. But in the normal distribution, u is independent of m, and u and m act as two independent and freely adjustable parameters. Note that the normal distribution of x is always symmetrical about the mean value m. The smooth curve in Fig. 2 illustrates the shape of a typical normal distribution. The points of maximum slope, at which d2(dP,/dx)/d2x = 0, fall a t 5 = m k u where the slopes have the values
d(dP,/dx)/dx
=
f l/a2
G.
Tangents t o the distribution curve at these inflection points (m k u ) intersect the z axis at x = m f 2u, and this simple geometrical relationship offers a convenient method of estimating a graphically from an experimentally determined distribution curve. The full width of the normal distribution a t half-maximum height is 2a 2/2i-n2 = 2 . 3 5 4 ~ .The full width is 2a at 0.6066 of the maximum height. Figure 1 gives the integrals of the normal distribution. These play an important role in the statistical theory of errors, which is ordinarily based
764
5.
0
STATISTICAL FLUCTUATIONS
1.0
0.5
1.5
2.0
2.5
w/a
FIG.1. Integral of the normal distribution. The ordinate P, is the fraction of the total area of the symmetric normal distribution which falls farther from the mean value than a distance w ,measured in units of the standard deviation u. Particular numerical values which find frequent use are w/u
0
Pw
1.000
0.5 0.6171
0.6745 0.5000
1 0.3173
2 0.0455
3 0.00272
on normally distributed variables. In Fig. 1 the shaded areas marked &Pwcorrespond to either of the integrals
pw= J-”.-” dP,
=
/mLI1.d P ,
(5.1.7)
and Pw is the probability that a single observation of 2 will differ from the mean value m by more than w. From Eq. (5.1.7), the probability that a single observation of x will lie between x1 = m - w1and x2 = m - w 2 is (5.1.8) P(x1 to 2 2 ) = +lPw, - P w 2 1 .
5.1.4. The Poisson Distribution Poisson’s distribution describes all random processes whose probability occurrence is small and constant. It applies to a large proportion of all observations made in experimental nuclear physics. The general conditions under which the Poisson distribution is valid can be visualized by considering the illustrative case of radioactive disintegration. Then the necessary and sufficient conditions are : of
5.1.
FREQUENCY DISTRIBUTIONS
765
1. The chance for an atom to disintegrate in any particular time interval is the same for all atoms in the group (all atoms identical). 2. The fact that an atom has disintegrated in a given time interval does not affect the chance that other atoms may disintegrate in the same time interval (all atoms independent). 3. The chance for an atom to disintegrate during a given time interval is the same for all time intervals of equal size (mean life long compared with the total period of observation). 4. The total number of atoms and the total number of equal time intervals are large (hence statistical averages significant). Let a be the true average rate of appearance of disintegration particles from such a random process. In a very short time interval, di, the probability of observing one particle is P I ( & ) = a dt; and the probability of observing no particle is Po(&) = 1 - a d t , if the time dt has been chosen so short that the probability of observing two or more particles in dt is negligible. dt), of observing x particles We now write the probability, P,(t dt. This is the sum of two mutually exclusive probabilities: in the time t either z particles in t and none in dt, or (z - 1) particles in t and one in dt. Then (5.1.9) P z ( t d t ) = P,(t) Po(dt) P,-l(t) * P l ( d t ) = P,(t) (1 - a d t ) P,_l(t) a d t . (5.1.10)
+
+
+
+
+
Rewriting Eq. (5.1.10) in differential form, we have
The solutJion2of Eq. (5.1.11) is (5.1.12) as can be verified by differentiation. But at is simply the true mean value for the number of events in time t, m = at. Writing m for at in Eq. (5.1.12) we obtain the usual general form of the Poisson distribution mz
P , = - ecm X!
(5.1.13)
in which P, is the probability of observing x events when the average for a large number of tries is m events. Although m may have any positive value, z i s restricted to positive integer values only. Note that the Poisson distribution has but one parameter, m. It can be shown easily that the
* H. Bateman, PhiZ. Mag. [6] 20, 704 (1910).
766
5.
STATISTICAL FLUCTUATIONS
binomial distribution Eq. (5.1.3) with two parameters, z and p , becomes identical with the Poisson distribution in the limiting case of p << 1, that is when p - + 0; but with a large number of trials z + 00, so that p z = m. For the numerical evaluation of individual terms in a Poisson distribution it is often convenient t o note from Eq. (5.1.13) that P-1
=
X
m P,
m
P2+1= X+lP,
(5.1.14)
and to use Stirling’s approximation to the factorial (5.1.15)
Then the probability of observing the mean value is
P,=
2 ~ mI + % + - [4 --( 1
.)]-I*
(5.1.16)
The Poisson distribution is asymmetric, always favoring values of x which are less than m. For example, substitution of x = m in Eq. (5.1.14) shows that i f the mean value is a n integer the probability of observing one less than the mean value is the same as the probability for the mean value. Figure 2 illustrates the Poisson distribution for m = 16. The Poisson asymmetries which are evident in Fig. 2 are associated with the small mean value, m = 16. For large mean values, say m 2 100, the Poisson distribution becomes much more nearly symmetric about x = m, and is well approximated by a normal distribution which is given the same mean value, m, and a standard deviation of u = 4%. For comparison, Fig. 2 also shows a normal distribution, Eq. (5.1.6) with m = 16 and u = 4, and a binomial distribution, Eq. (5.1.3) with z = 40 and p = 0.4 (hence m = p z = 16). A special result of exceptional importance in physics follows at once from Eq. (5.1.13). The probability Po of no event (x = 0) is simply
P O = e-m.
(5.1.17)
This is the statistical basis for the law of radioactive decay, in which e-kt is the probability that an atom will survive without decay for a time t when on the average it should have decayed m = At times, and also for the law of y r a y transmission, in which e-rl is the probability that a photon will travel a distance 1 without a collision when on the average it should have suffered m = pZ collisions.
5.1.
FREQUENCY DISTRIBUTIONS
767
FIQ.2. Comparison of binomial (Eq.5.1.3), normal (Eq.5.1.6), and Poisson (Eq. 5.1.13) distributions, each for a mean value m = 16, calculated for the parameters shown on the figure. The binomial and Poisson distributions are histograms because they are defined only for integer values of 2, wherem for the normal distribution I is B continuous variable. The area under each distribution is unity. The binomial distribution is higher and narrower than the others because its standard deviation (Section 5.2.4) of c = d m ( 1 - p ) = 3.10 is less than the va.lue u = 4 chosen arbitrarily for this normal distribution and the value o = 4; = 4 which is inherent in the Poisson distribution. 5.1.5. The Interval Distribution
In a series of random events, such as disintegrations in a very large group of identical and long-lived radioactive atoms, or pulses in a GeigerMiiller or scintillation counter which is exposed to a constant source of radiation, the distribution of intervals between successive events is often of experimental importance. 5.1 5 . 1 . Intervals Between Successive Random Events. If the primary process is random, and follows Poisson's distribution, we can derive easily the distribution in size of the time intervals between successive events. Let the true mean rate have the constant value of a events per unit time. Then the probability that there will be no event in the time interval t, during which we should expect at events on the average, is, from Eq. (5.1.17), Po = e-at. The probability that there will be one event in the time dt is PI(&) = adt. Then the combined probability for no
768
5.
STATISTICAL FLUCTUATIONS
+
event in the time interval t and one event between t and t dt is the product of these t w o independent probabilities Po(t) and Pl(dt).Therefore the probability dP, that the duration of a particular interval will be dt is between t and t dP, = ae-at dt (5.1.18)
+
if the random process is one which meets the conditions required for Poisson's distribution, as listed in Section 5.1.4. If the random process does not have a sufficiently small p and large z to validate the Poisson distribution, then the interval distribution will still correspond to
dP,
=
Po(t) . Pi(&)
but with Po(t) as given by the binomial distribution Eq. (5.1.3). We see at once from Eq. (5.1.18) that short intervals have a higher probability than long intervals between randomly distributed events. This accounts for the high incidence of short intervals, or "close doublets," and the infrequency of very long pauses, which characterize the output of scale-of-1 counters in nuclear physics. Integration of Eq. (5.1.18) gives the probability that a particular interval will be longer than T , and hence the fraction of the intervals that are longer than T , which is (5.1.19) where aT is the number of events that would have been expected, on the average, in the interval T . (An experimental example is considered in. Eq. (5.4.3) and Table 11.) Then the fraction of the intervals that are shorter than T , which is equivalent to the probability that any particular interval will be shorter than T , is
P,,
=
1 - P L T = 1 - e-aT.
(5.1.20)
5.1.5.2. Intervals Involving s Random Events. In nuclear physics, many counters employ scaling circuits in their output. If the scaling factor is s (commonly s = 2, 4, 8, 16, 32, . . . or 10, 100, 1000, . .) then we are interested in the statistical distribution of the intervals of time which involve a total of s random events. The distribution of time intervals produced by an s-fold scaling of a Poisson distribution is here called the s-fold interval distribution. An s-fold interval contains s - 1 events in a time t, and one event during the additional time dt, thus a total of s events in a time t dt. Therefore the probability dP, that an s-fold dt is, using Eq. (5.1.12), interval will have a duration between t and t
.
+
+
,-
dPt = P,-l(t) * Pl(dt) = (at)"-' e-Ot . a dt = (at)"-' e-mtd(a2) (5.1.21) (s - 1). (s - l ) ! ~
5.1.
769
FREQUENCY DISTRIBUTIONS
FIG. 3. Examples of s-fold interval distributions, plotted against at, the average number of events in a time t . The area under each curve between time zero and at is the probability that an interval containing s events will be shorter than t . The average interval f is at = s, and t h e most probable interval fo is ato = s - 1. The standard deviation u, is ant =
4;.
where a events per unit time is the true average rate of the random process, at is the expected mean value of the number of events in time t (or t dt), and s is the actual number of events in the time t dt. The probability t h a t an s-fold interval will be shorter than T is the integral of Eq. (5.1.21) from t = 0 to t = T , and can be shown3 to be
+
+
P< -T = P s f P , + I +
*
’
*
+POY
(5.1.22)
+
where P,, Pa+*,. . . , are the Poisson probabilities for s, s 1, . . . events in the time T , as given by Eq. (5.1.13) with x = s, and m = aT. Equation (5.1.22) says simply that, the probability that s events will R. D. Evans, “The Atomic Nucleus,” p. 795. McGraw-Hill, New York, 1955.
770
5.
STATISTICAL FLUCTUATIONS
0.12 0.11 0.IC
0.0s O.OE
0.Oi
d4 d@t)
0.a 0.a
0.W
0.0: 0.0:
0.0 C
2
FIG.4. The s = 16 interval distribution, showing the regularizing action of a scaler, and the asymmetries of the interval distribution, with the normal distribution from Fig. 2 repeated for comparison purposes. Both distributions have the same mean value and the same standard deviation.
occur in less than a time T is the Poisson probability of s, or more, events in the time T. Of course, the probability that an s-fold interval will be longer than T is simply P >T = 1 - P g . For numerical work, convenient tables of the individual Poisson terms for m = 0.001 to m = 100, and their summations, have been published by Molina. Figure 3 shows the s-fold interval distributions for s = 1, 2, and 4. The s = 1 curve is the basic interval distribution of Eq. (5.1.18) for the intervals between successive events in a Poisson distribution. Note that the most probable interval is zero, although the average interval is t = l / a , or at = 1 on the horizontal scale of Fig. 3. The s = 2 curve demonstrates the suppression of very short intervals which is a basic sta4 E. C. Molina, “Poisson’s Exponential Binomial Limit.” Van Nostrand, Princeton, New Jersey, 1947.
5.2.
77 1
STATISTICAL CHARACTERIZATIONS O F DATA
tistical characteristic of scaling circuits, and is ordinarily called the regularizing action of a scaler. For any s, the average interval is I = s/a, whereas the most probable interval is always shorter and equal to to = (s l)/a. As s is increased, the regularizing action of a scaler becomes even more pronounced. Figure 4 shows the s-fold interval distribution for s = 16, together with a symmetric normal distribution as a comparison curve. The asymmetry of the s-fold interval distribution is still apparent at s = 16, but a normal distribution begins to be a useful approximation if it is assigned the same mean value I = s/a and the same standard deviation ut = I/& (Section 5.2.4) as is inherent in the s-fold interval distribution.
-
5.1.6. Summary of Frequency Distributions Table I collects the results of the previous sections and, for ease of reference, anticipates results on mean values and standard deviation$. TABLE I. Summary of Frequency Distributions" Mean values Frequency distribution
Distribution Z!
Binomial
P, =
Normal
dp, =
x!(2
- x)! P"(l 1
~
u
2/zr
Sample True
- py-=
e - ( ~ - n ) ~ / dx Zu~
mz
p I -- - - - I n
Poisson
z
pa
dP41 - P)
z
m
Q
5
m
i
-
X!
Interval s-fold intervnl (1
dP1
=
(at)a-le-at d(at)
~
(8
- I)!
Standard deviation
t
1
a
6 -1 a
s -
z/5
a
a
For ease of reference, mean values (Section 5.2.1) and standard deviations (Section
5.2.4) are anticipated here.
5.2. Statistical Characterization of Data The over-all results of a series of n independent measurements xll
.
22,
. . , x,,of any physical quantity will usually be summarized in the form
of a few statist,ical parameters. Usually these are: 1. The experimenter's estimate of the true value of z (expressed as the mean value Z, Section 5.2.1), and
772
5. STATISTICAL FLUCTUATIONS
2. His estimate of the chance that his mean value is in error by less than some specified amount (expressed as the standard error UI of the mean value, Section 5.2.5). In some cases he will also wish to quantify: 3. The breadth or dispersion found among his individual measurements (expressed as the variance u2, Section 5.2.2, or as the standard deviation u, Section 5.2.4), and 4. The extent to which the individual measurements agree with some reasonable frequency distribution (expressed as a test of goodness of fit, Chapter 5.4). 5.2.1. Mean Value
In any Jinite series of measurements we can never find the exact value of the true mean value m which characterizes the “infinite population” (i.e., an infinite amount) of data. Although the true mean value is constant, our individual measurements should be distributed about this mean value in a manner given by the particular frequency distribution which describes the process being studied. The true mean is that value of x for which the first moment of a frequency distribution is zero. Thus
2
( x - m)P, = 0
(5.2.1)
defines the mean, where P, is the probability of the observation x. Geometrically, m is the “center of gravity’’ of the distribution. A uniformly thick sheet of metal cut in the shape of the histogram of P, vs. x , or the curve of dP,/dx vs. x for a continuous distribution, would balance on a horizontal knife edge placed parallel to the frequency axis, at x = m. For all the frequency distributions discussed in Chapter 5.1 it can be shown quite generally that the best (i.e., least mean-square deviation) estimate of m in a finite series of observations is the simple arithmetic average z of the n independent measurements, xl,x2, . . . , x,; that is:
(5.2.2)
The “sample mean” Z approaches the true mean m as the number of observations n is increased. Table I lists the sample mean and true mean
5.2.
STATISTICAL CHARACTERIZATIONS OF DATA
773
evaluated from Eq. (5.2.1) for each frequency distribution. The “sample mean” 3 is synonymous with the terms “average value” 2, <x>, or as used in physics, and “expected value” or “expectation value” E [ z ] ,as used in statistics. In any distribution, the mean value (i.e., the average value) is to be distinguished from the median value (i.e., the value which is as frequently exceeded as not), and from the modal value (i.e., the most probable value). The mean, median, and mode do not coincide except for a symmetric distribution such as the normal distribution. In the 4-fold interval distribution of Fig. 3, for example, the mean is at = 4,the median (evaluated through Eq. (5.1.22)) is at = 3.7, and the mode is at = 3. In general, the median always lies somewhere between the mean and the mode; thus the mean, median, and mode always occur in alphabetical order in any distribution.
5.2.2. Variance The breadth of the statistical distribution of our individual readings about the true mean value is expressed quantitatively by the variance u2, or by the square root of the variance which is called the standard deviation u, of the distribution. For a particular mean value m, a small variance u2 corresponds to a sharply peaked distribution, whereas a large u2 denotes a broad, flattened distribution. For any frequency distribution, the variance is defined as the average value of the square of the individual deviations (x - m) from the true mean, for an infinitely large number of observations. Thus (5.2.3)
or, in terms of a very large series of n independent measurements (n >> 1) of 2, i=n
(5.2.4) i=l
The variance is thus seen to be simply the second moment of the frequency distribution taken about the true mean, m. Geometrically, if a sheet of metal is cut in the shape of the distribution, the variance is the “moment of inertia” of the sheet taken about an axis at z = m and parallel to the frequency axis.
774
5.
STATISTICAL FLUCTUATIONS
The variance for each of the frequency distributions of Chapter 5.1 can be derived using Eq. (5.2.3) over appropriate limits. This gives z=z
(x - pz)*z!pz(1 - ')"-" 21(z - x)!
u2 (binomial) =
=
pz(1 - p)
(5.2.5s)
(5.2.5~)
1-(1 Lm(t
2=b
u2 (interval) = u2 (s-fold
int.)
=
-
-
k)' :)
2
ae-nLdt = 1 a
(5.2.5d)
dt = -.S a2 ( s - l)!
(5.2.5e)
a"t"-'e-"!
5.2.3. Sample Variance I n a finite series of n observations we can never determine u2 exactly, just as we can never determine m exactly. Our estimate of the true variance u2 of the distribution must therefore be based on our imperfect estimate Z of m. Accordingly, we define a "sample uariance," S2, by analogy with Eq. (5.2.4), but with m replaced by 2. Thus the experimental data give (5.2.6) i=l
With the help of the substitution (xi - 2 ) = [(xi - m ) - (Z - m ) ] , it is easy t o show quite generally that the expected value E[S2]of S2, averaged over a number of repeated samples of n observations, is
E[SZ] = u2
-1 - u2 - = n___ d. n
n
(5.2.7)
Then our best estimate of the true variance u2 of the distribution, in terms of our total finite number n of observations, xl,x2, . . . , xn, is i=n
'c
n n - s2= uz=-7 n-1
(Xi
- f)Z.
(5.2.8)
i= 1
5.2.4. Standard Deviation * The breadth of a distribution of data is best visualized in terms of some stat,istic which has the same dimensions as the mean value. The
5.2.
STATISTICAL CHARACTERIZATIONS OF DATA
775
best suited parameter is the square root of the variance, which is called the standard deviation, u, of the distribution. In the normal distribution Eq. (5.1.6) the standard deviation is just one of two arbitrary parameters, m and u. In other distributions u depends upon p, z, a, s, or m in the manner derived in Eqs. (5.2.5) and summarized in Table I. Because of its symmetry, the normal distribution contains the fraction 0.3413 of its area between x = (m - u) and x = m, and the same fraction between x = m and x = ( m cr). Thus 68% of the area of any normal distribution lies in the domain x = m _+ u, as shown in Fig. 1. None of the other distributions is symmetric. For them, the mean differs from the median, m does not divide the area into halves, the area between m - u and m is different from the area between m and m u, and both fractional areas depend upon the other parameters of the distribution. For example, for the s = 4 interval distribution of Fig. 3, Eq. (5.1.22) will show that the domain of at below the mean value, at = 4, contains not one half but 0.567 of the area, the domain between the mean value and one standard deviation below the mean contains 0.424 of the area, while the domain between the mean value and one standard deviation above the mean value contains only 0.282 of the area. With these obvious asymmetries in mind, it is clearly risky in physics to express experimental results as a sample mean plus or minus a quantity such as the standard deviation or even the standard error unless the degree of asymmetry of the distribution is known to be small. Otherwise it is best to state the sample mean, and to state the standard deviation or the standard error, but to avoid the commonly used f sign. The expected value of the standard deviation (Table I) corresponds to an infinite population of data. In a finite sample of n independent observations, the best estimate of this standard deviation is to be obtained by evaluating the individual deviations (xi - 3) from the sample mean 3 and then computing from Eq. (5.2.8) the estimated standard deviation of the parent distribution. This is
+
+
I
i=n
(5.2.9a) or I
i=n
(5.2.9b)
n-1 in the case of interval distributions. In computing the standard deviation
u
of a series of observations xl,
776
5.
STATISTICAL FLUCTUATIONS
. . . , x,, the arithmetic can often be greatly simplified, and progress can be made on the computations while the experiment is still in progress by referring the individual readings to some arbitrary value xo, usually chosen as a round number near 2. Then it is easy t o show that Eqs. (5.2.2) and (5.2.9) become
22,
i=n
z
=
[A
‘c
20
+ -n
(xi
- x , , ) ~ ]-
(Xi
(5.2.10)
- ro)
[* (z
-
i=l
I
x0)2
(5.2.11)
in which the first square bracket can be evaluated before the final value of 2 has been determined, and the second square bracket often will be only a small correction term. A form which is often arithmetically useful when xi contains only one or two digits, but which is always analytically valid, is
c
i=n
i=n
i=l
(xi -
$2
=
(2 $2) -
n(z)2.
(5.2.12)
i=l
Where only one observation x is made n = 1, and then r n z z = x, but u is indeterminate because the spread of the data has not been sampled. If the process is known to be Poisson distributed, as in many nuclear counting experiments, then the best estimate of u is the expectation value U N & (5.2.13) as given by Eq. ( 5 . 2 . 5 ~ ) .
5.2.5. Standard Error The usual objective of a series of n independent measurements xl,z2, . . . , xn of a physical magnitude is to estimate the true mean value m, and to state the chance that the estimate 2 differs from m by less than some specified amount. The usual “specified amount” is the standard error of the mean. To evaluate the standard error we imagine that repeated samples of size n are taken, and ask how the mean values from each of these samples would be distributed about the grand average of the sample means. It is found that a series of k mean values, Z1,Z2,. . . , z k , each based on n measurements of x, will tend to exhibit a normal distribution about their grand average 2. This is true, if n is sufficiently large, even if the
5.2.
777
STATISTICAL CHARACTERIZATIONS O F DATA
parent population x is not normally distributed but, for example, is an asymmetric Poisson or interval distribution. In general, the distribution of mean values tends to be much more nearly normal than the parent population. It can be shown that in a large series of k hypothetical measurements of the sample mean 2, each based on n independent measurements of x, the grand average Z approaches the true mean m, and the sample mean which is value 3T: is normally distributed about m with a variance given by i=k
(5.2.14)
where uz is called the standard error of the mean Z, and u is the usual standard deviation of the distribution of x about 2. Because the distribution of 2 about m is nearly normal, and therefore symmetric, the chance that 2 lies within (m I az) is about 68%, by Fig. 1. The mean of a single series of n independent measurements xl, x2, . . . , x, is then to be reported as ( 2 f G), where i=n
(5.2.15) and the standard error of the mean is I U
i=n
1
(5.2.16)
n(n - 1)
Experimentalists must be sure to distinguish clearly between standard error (of the mean) and standard deviation (of the distribution). An everyday illustration occurs in nuclear counting, and can serve also as a derivation of the basic Eq. (5.2.14) for any Poisson distribution. Suppose that n independent measurements of a constant nuclear process have given xl, x2, . . . , zn in a set of n 1-minute intervals. Considered as a whole, the total number of events is v = x1 x2 * * * x, = nl. Then if the distribution is Poisson, the expected value of the standard deviation of v is l / V in a series of hypothetical repetitions of the n observations. Then the fractional standard error of v is
+ +
+
(5.2.17)
778
5.
STATISTICAL FLUCTUATIONS
This is a basic result, applicable to all counting experiments on Poisson distributions and hence nearly every counting experiment in nuclear physics. For example, if u = 104 counts, then the fractional standard error of our knowledge of the average counting rate is l/di@ = 0.01 = 1%. Because v comprises all our data, it furnishes our estimate of the mean, and its standard deviation 4;is in this case also the standard error 4; of the mean v, because we have lumped all the data as though we had made just one observation over an n-minute period. But the fractional uncertainty is
4:
where u = is the standard deviation of the distribution of xi about 3. The fractional uncertainty G / v of our over-all result must be the same, whether expressed as a ratio involving v or 2. Therefore the standard error uz of the mean 2 of n observations xl,x2, . . . , x,,is u / d i , where u is the standard deviation of xi about 2, as in Eq. (5.2.14). 5.2.6. Probable Error
The earlier literature of physics made frequent use of a statistic called the probable error. In a symmetric normal distribution, the probable error r defines a domain ( m f r ) which should include exactly one half of the observations of x. For the normal distribution, Fig. 1 shows that P, = 0.500 for w/u = 0.6745. The probable error r of a single observation was therefore taken as r = 0.67451s (5.2.19) and the probable error of a mean value as
rz
=
0.6745~.
(5.2.20)
These relationships are meaningful only for a normal distribution. They fail, and Eq. (5.2.19) is generally invalid, for any of the asymmetric frequency distributions (binomial, Poisson, interval) which are met most often in nuclear physics. The use of probable error should be discouraged, and the concept abandoned, except as a means of interpreting earlier literature where Eqs. (5.2.19) and (5.2.20) may be used to convert probable error to standard deviation and standard error. 5.2.7. Dimensions
From consideration of Eq. (5.2.13), it is evident that both x and (I must be dimensionless quantities, because u has the same dimensions
5.2.
STATISTICAL CHARACTERIZATIONS OF DATA
779
as x and fi.It is generally true that all such quantities in the distribution functions and in other statistical expressions are without dimensions. For example, 3 may be physically the average number of counts per minute, but statistically the time unit chosen is only an arbitrary interval or classification by which the data have been taken. It is to be regarded statistically as dimensionless. This can be visualized by considering time intervals measured off on a chronograph tape, in which case the particular interval used for classification might equally well be one second, or an equivalent length of tape, or even an equivalent mass of tape. The interval itself does not have the dimensions of time, length, or mass but is always statistically dimensionless, as are all the other basic statistical quantities. Although the interval distributions Eqs. (5.1.18) and (5.1.21) contain the rate a in events per unit time, it always occurs in the product at or a dt, which is again dimensionless.
5.2.8. Precision vs. Accuracy In the statistical evaluation of measurements, precision and accuracy have entirely different meanings.
-
High Accuracy
Low Accuracy
High
Precision I I
X-
3
m
X-
-x m
FIG. 5. Technical definitions of precision (small statistical error) and accuracy (small systematic error).
Precision relates only to repeatability. A result of high precision is one which has a small standard error attributable to statistical fluctuations, even if it has a large bias due to some systematic error. In the earlier statistical literature the “precision” h = 1/u 4 was used as a parameter of the normal distribution before the adoption of the present nomenclature of variance and standard deviation. Thus high precision means small UE. Accuracy relates to correctness with respect to some absolute scale. High accuracy implies freedom from bias due to systematic errors in calibrations and standards. Thus, because of the incorrect calibration of a millimeter scale, the mean range obtained for a given group of LY rays
780
5.
STATISTICAL FLUCTUATIONS
could be seriously in error (i.e., low accuracy) even if the individual measurements had a high degree of reproducibility (high precision). Thus high accuracy implies small (2 - m ). The two concepts are illustrated in Fig. 5.
5.3. Composite Distributions Most measurements or calculations in physics involve more than one source of error or of statistical fluctuations. The joint effect of simultaneous but independent sources of statistical fluctuations is considered in this chapter. 5.3.1. Combined Probabilities
Two processes A and B are said to be “independent” if the probability of A occurring is unaffected by the occurrence or nonoccurrence of B, and vice versa. The combined probability of two or more observations which are independent of each other is the product of the individual probabilities. An example occurred in the derivation given earlier for Eq. (5.1.9), in which PZ(t)and Po(dt) are clearly independent probabilities. Two processes A and B are said to be “mutually exclusive” if when one of the events occurs the other cannot occur. The combined probability of two or more observations which are mutually exclusive is the sum of the individual probabilities. The derivation of Eq. (5.1.9) again provides a clear example when the mutually exclusive probabilities P,(t)Po(dt) or P,-l(t)Pl(dt) are added to obtain their combined probability. Consider a nuclear example of combined probabilities. Suppose a certain radioactive sample contains a mixture of an a-ray emitter and an unrelated, independent, @-ray emitter, the average activities being A a-rays and B 8-rays per unit time. What is the probability M ( z ) of just 5 ( x = 0, 1, 2, . . .) 8-rays in the time interval between any two successive a rays? The probability dPt of an interval of duration 1 between any two a-rays is dPt = e-AtA dt, and the probability of x 8-rays in the time t is P, = (Bt)Ze-Bt/z!The combined probability, dPz,r,of these two independent circumstances, an a-ray interval t and x D-rays during t, is the product
(5.3.1) Now, intervals of various durations t are mutually exclusive, hence the probability M ( z ) of 2 8-rays in any interval is the sum, or integral, of
5.3.
78 1
COMPOSITE DISTRIBUTIONS
dP,,t over all possible intervals. This gives (5.3.2)
If, for example, B / A
=
M(0) =
2, then the probabilities sought are:
M(3) M(4) M(5)
0.333 0.222 = 0.148
= =
M(1)
=
M(2)
= $7
$
= = =
=
0.099
A*%= 0.066 & = 0.044
.
Finally, the occurrence of x = 0, 1, 2, . . &rays in any interval are themselves mutually exclusive processes. Hence the combined probability for a n y x in a n y interval between a-rays is
c
Z=O
M(x) =
B/A
1
1
+B/A
(1
+ B / A ) 24-(1 + B / A ) 3
+
...
=
1. (5.3.3)
The sum of all possible x in all possible intervals is unity, which in probability theory is the maximum possible probability, and represents certainty, 5.3.2. Superposition of Several Independent Random Processes The complete generalization of the Poisson distribution is usually required in nuclear problems because several independent types of radiation will actuate most detection instruments. For example, in ionization chamber measurements of a-rays, there will be present a background composed of a- and @-raysfrom radioactive contamination of the walls of the instrument, of cosmic rays, and of y-rays from the earth and surrounding building materials. If each of several such processes is itself random and follows a Poisson distribut,ion, the resulting over-all fluctuations may be predicted.6 Let z, y , z, . . . be the actual number of particles from the several independent random processes, in the interval chosen. Let them respectively produce specific effects (such as ion pairs) of a, 0, y, . . . Per particle. Then the actual effect, u, on the instrument is u=ax+py+yyz+
.
*
*
(5.3.4)
.
The true mean value, mu,of u is
mu = am, 6
+ + ym, + Om,
*
R. D. Evans and H. V. Neher, Phys. Rev. 46, 144 (1934).
.
(5.3.5)
782
5.
STATISTICAL FLUCTUATIONS
From a series of n independent measurements of u, our best estimat,e of mu is the sample mean i=n
(5.3.6)
Ui.
n
In any single observation each process produces x, y, z, . . . particles, instead of the true mean value mz, mu, m,, . . . particles. Then the net deviation due to the joint action of the separate processes will be 4 z - m,) P(y - mu) Y(Z - ma> * * . , if the physical effects of the separate processes are additive in the detection instrument. The probability that z, y, z, . . . particles will be produced by the several independent physical processes is the product of the Poisson probabilities : P, = (m,)ze-m*/z!,P, = (m,,)ye-mU/y!, P , = (mJze-mz/z!,. . . . Hence the variance of the distribution of u about its true mean mu is
+
+
+
x [ C Y ( X - m,)
+ p(y - mu) + Y(Z - m,)+
*
. - 1'.
(5.3.7a)
Upon expansion, Eq. (5.3.7a) reduces to simply uU2= a2mu
+ @'mu+ y2mz+ - -
(5.3.7b)
which gives the expected value of the standard deviation, uu, of u about the sample mean fi. In accord with the principles of Sections 5.2.4 and 5.2.5, our best experimental estimate of u u will be given by I
i=n
n-1
(5.3.8)
and of the standard error, uc, of the sample mean, fi, by (5.3.9) Equation (5.3.7) is of far-reaching importance in the statistical evaluation of all types of nuclear observations. The number of independent events x, y, z, . . .. are dimensionless quantities, but their specific effectiveness a,p, y, . . . may have any dimensions, such as ion pairs or statcoulombs per particle, or scaling-circuit output pulses per input
5.3. COMPOSITE
DISTRIBUTIONS
783
event, etc. If the weighting factors a, p, y, . , . are unequal, then the events with large specific effectiveness can easily dominate the fluctuations, u,, even if they contribute only a small share of the over-all mean value ti. In single-channel counters, the weighting factors are usually equal, a! = p = 4 = . . . Then the fluctuations are determined simply by the sum of the average number of random events from all the independent processes, that is, u, II (Y(Z g z * -)]I2.
-
+ + + -
5.3.3. Propagation of Errors Where a physical magnitude is to be obtained from the summation or the differences of independent observations on two or more physical quantities, the standard error, u 1 , 2 , .. . , of the sum or difference is given by (u1,2,...)Z
+ +.
= u1*
a22
*
*
(5.3.10)
where UI, u2, . . . are the absolute values of the standard errors of the mean values of the several quantities, expressed, of course, in the same units. Thus, (100 f 3) (6 f 4) = (106 f 5) (100 f 3) - (105 f 4) = - ( 5 f 5).
+
An everyday example is the estimation of the true counting rate, a, due to any source of radiation, when the counter has a true background counting rate, b. If the background has been determined by observing Btt, counts in a time tb, then our estimate of the true background rate is
(5.3.11) Note that the statistical uncertainty in our evaluation of the background depends inversely on the square root of the duration of the observation, as would be expected for any Poisson distribution. With the source added, ( A B)t, counts are observed in a time to, and our estimate of the true counting rate (a b) for the source plus background is
+
+
(A
I:
+ B)ta k d ( A + B)ta
-=A
+ (5.3.12)
Subtracting Eq. (5.3.11) from Eq. (5.3.12) gives the observed counting rate of the source as A B B (5.3.13) ta ta tb
+-+--
784
5.
STATISTICAL FLUCTUATIONS
Note that the background uncertainty enters twice, once as B/t, for its statistical fluctuation during the measurement of ( A B ) and once as B/tb for its fluctuation during the measurement of B alone. I n the design of counting experiments, it is clear from Eq. (5.3.13) that the uncertainty in the source strength, a, measured for a fixed time t,, can always be reduced by prolonging the independent background measurement, t b . But if laboratory conditions impose a restriction on the total time available, t, tb, then it can be shown that the fractional statistical uncertainty in the determination of a has its minimum possible value when the fraction of the time spent on the background measurement is
+
+
If a physical magnitude Y is to be obtained by multiplication or division of results of several independent observations on two or more physical magnitudes yl, yz, . . . , the fractional standard error U Y / Y in the resulting value of Y depends upon the fractional standard errors ul/yl,az/y2, . , . in the measurement of yl, y2, . . . and is given by (5.3.14)
Equation (5.3.14) is a good approximation whenever the fractional standard errors are small, that is, when n(ui/yJ2 << 1, as often happens. Then, for example (3.00 f 0.09)(2.00
0.08)
=
6.00 f 6.00
dt$y
+
(q)’ =
6 & 0.3.
5.3.4. Difference of Two Mean Values When some feeble radiation effect is comparable only with the statistical fluctuations in the natural background of the detecting instrument, special care must be taken in interpreting the results of the measurements, and standard tests of the “significance of the difference of means” may need t o be applied. Let mu and mo be the true mean values of two independent populations of normally distributed data, while (27. f UE) and (B f ur) are measured values of samples from the two populations, each measurement being based on a sufficient number of observations so that the uncertainty in the standard errors ug and u: is small. Then our best estimate of the difference, (mu - mv),of the two means is
-
mu - m, - (ti
*
uc)
- (5 f UB) = (ti - 5)
f due2
+ a?’.
(5.3.15)
5.3.
COMPOSITE DISTRIBUTIONS
785
It can be shown that (ti - 8) is normally distributed about the true mean value (mu- mu)with a standard deviation of U(6-Z)
=
l/aii*+ a;*.
(5.3.16)
If, for example, the true mean value is (mu- mu)= 0, then as seen from Fig. 1 there is about a 32% chance that the absolute value of (zi - 8) will be numerically greater than the standard deviation of its own measurement, U+Q. 5.3.5. Significance levels and Confidence Intervals
It is customary but arbitrary in the theory of errors to reject any hypothesis which falls below a “signijicance level” of 5 %. The hypothesis being tested is usually rejected if it predicts that the observation made was so unusual that it should occur in less than 5 % of a large number of trials. Because Fig. 1 shows P , = 0.0455 for w/u = 2, there is only about a 5% chance that the observed value of (zi - 8) would exceed 2u(a-q if in reality (m,- m,) = 0. Accordingly, an observation of a difference of at least twice the standard deviation of the difference between two mean values, i.e., (ti - 8) 2 2u(~-;,, would be said to be “statistically signijicant” at the 5% level and usually would lead to rejection of a tentative hypothesis that the two mean values were identical. If a 1% significance level is desired, we see from Fig. 1 that w/u = 2.6 for P, = 0.01 and therefore choose (C - 8) 2 2.6u+:). In 95 out of 100 experiments, the true difference (mu- m,) will lie within the interval (ti - 8) f 2u+;), hence this interval is called the 95% conjidence interval for (mu- m,). However from one experiment alone it would be entirely incorrect to claim that the probability is 0.95 that the confidence interval actually contains the mean (mu- m,). These simple confidence intervals and significance levels, based on normally distribukd variables, are satisfactory for most nuclear physics data. This is because most measurements in nuclear physics follow the Poisson distribution and, if the mean value m is not too small, the Poisson distribution is nearly symmetric and is approximated quite well by a (recall Fig. 2). The normal distribution having mean m and u = population variance u2 is often known better than, or at least as well as, the sample variance S z , Eq. (5.2.6), because the data are known to follow the Poisson or the interval distribution. This happy situation is met much less commonly in the biological sciences and in engineering, where many of the normal distributions have a variance u2 which cannot be predicted, but must be estimated from the data. Then it often happens that the sample variance S2is more accurately known than the population variance u2, Under these conditions Student’s t distribution provides the pre-
6
786
5.
STATISTICAL FLUCTUATIONS
ferred basis for tests of the significance of the difference of means of small samples. See Section 5.4.3 for other comments on the estimation of u2 in a normal distribution.
5.4. Tests for Goodness of Fit In physics the most generally useful statistical tests for the “goodness of fit” between data and hypothesis are based on the so-called x2 distribution, the major development of which is due to Karl Pearson. 5.4.1. Pearson’s Chi-square Test The chi-square test is a nonparametric test which may be used to quantify the validity of any proposed distribution law. Pearson’s x 2 test determines the probability P that a full repetition of the observations would show greater deviations from the frequency distribution which is assumed to govern the data. Pearson’s x2 test may be most simply stated as follows: First, we define the quantity
c
i=n
x2=
i-1
[(observed value)i - (expected ~ a l u e ) ~ ] ~ (5.4.1) (expected va1ue)i
where the summation is over the total number of independent classifications, n, in which the data have been grouped. In general, the observed values should be subdivided into at leaat Jive claaaiJications,each containing at leaat five evenla. If there are fewer than five classifications, each should contain appreciably more than five events. It is best to combine classifications containing fewer than five events with an adjacent classification. See Table I1 for an example. The “expected values” are computed from any a priori assumed frequency distribution, e.g., binomial, normal, Poisson, interval, etc. Secondly, we determine the number of degrees of freedom, F, which is the number of independent ways in which the observed values may differ from the expected values. In general, one degree of freedom is lost for each parameter of the observed series which is used in calculating the expected series, so F is the number of classifications minus the number of parameters used to obtain the expected values. The x2 distribution (5.4.2)
5.4.
787
T E S T S FOR GOODNESS OF FIT
TABLE 11. Montgomerys’ Distribution of Cosmic-Ray Bursts in Time Classification
Total
(oi - ei)*
ei
tl
10
sec
sec
(obs)
(exptd)
0 50 100 200 500 1000 2000 5000
50 100 200 500 1000 2000 5000
22 17 26 68 47
19.4 17.9 30.5 63.5 50.5
+2.6 -0.9 -4.5 +4.5 -3.5
0.35 0.05 0.66 0.32 0.24
0
31.2
+1.8
0.10
m
Oi
3q
(oi
- ei)
-
--
-
213
213.0
0
ei
Average interval between bursts = I / a = 521.6 sec; degrees of freedom, F = 6 = 4; P = 0.8.
-2
:.
gives the differential probability, dP, that x 2 will lie between x 2 and x2 dx2, as a function of x 2 and F, as is indicated schematically in the inset of Fig. 6 . The shaded area P then gives the probability that x2 should exceed its observed value. For all practical purposes, the usual statistical tables of x2, F , and P are adequately summarized in the curves given in Fig. 6. Therefore, with x2 from Eq. (5.4.1), and F , enter Fig. 6 ,
+
FIG.6. Integrals of the x z distribution, Eq. (5.4.2). For F > 30 use the normal distribution integrals of Fig. 1, with x = m = 4 2 F - 1, u = 1, and ( i ) P w = P. In actual practice F seldom exceeds 30.
4/22,
788
5.
STATISTICAL FLUCTUATIONS
and determine P , the probability that, on repeating the series of measurements, larger deviations from the expected values would be observed. In interpreting the value of P so obtained, we may say that, if P lies between 0.1 and 0.9, the assumed distribution very probably corresponds t o the observed one, whereas if P is less than 0.02 or more than 0.98 the assumed distribution is extremely unlikely and is to be questioned seriously. The x2 test, and especially the determination of F , may be better understood through an example. I n Table I1 we apply the x2 test to the hypothesis that the observed time intervals between successive cosmic-ray bursts are in agreement with the interval distribution Eq. (5.1.19). Experimentally, Montgomery and Montgomerye observed the time of occurrence of 213 bursts in a total of 30.8 hr; thus the average interval was T = 521.6 sec = l/a. The observed intervals, t , were then classified into eight arbitrary ranges of time intervals t l 5 t 5 t z , as shown in the table. These observed frequencies oi are given in column 4 of the table, e.g., 22 intervals had a duration of less than 50 sec, etc. The expected frequencies ei are calculated from Eq. (5.1.19), in the form e . - 213(e-at1 I
-
e-ats
1
(5.4.3)
with a = 1/521.6 sec, and are given in column 5 of the table. Because oi and ei are less than 5 for the two longest intervals, these classifications are combined with the 1000 t o 2000 sec classification before applying the x2 test. There remain six classifications in which the observed distribution can differ from the assumed interval distribution. But these six classifications are not all independent because they contain two implicit restrictions in which the observed and expected distributions are required t o match. These are the two parameters from the observed series which are used to predict ei in Eq. (5.4.3), namely Zoi = Zei = 213 events, and a = Zo@ = 1/521.6 sec. The number of degrees of freedom F , or independent ways in which oi may differ from ei, is then:
F
=
6 (classifications) - 2 (parameters)
=
4 degrees of freedom.
The calculation of x2 = 1.72 is shown in column 6 and 7 of Table 11. For F = 4, Fig. 6 shows that the probability is P = 0.8 that, in a repetition of this entire set of observations, x2 would exceed 1.72. Thus in 80 out of 100 similar experiments, the deviations from the interval distribution would be expected t o be greater than here observed. There is therefore strong support for the conclusion that the observed cosmic-ray C. G . Montgomery and D. D. Montgomery, Phys. Rev. 44, 779L (1933).
5.4.
TESTS FOR GOODNESS O F F I T
789
bursts obey the interval distribution and are therefore randomly distributed in time.
5.4.2. Poisson Index of Dispersion Often, a n experimenter obtains a small sample of data which he presumes is from a Poisson population, but which contains such a small number of independent measurements that it would be pointless to attempt t o match the data with any assumed distribution. Fortunately for nuclear physicists it has been shown that, in a series of n independent measurements xl, x2, . . . , zn on a Poisson population for which Z >> 1, the quotient (n times the sample variance)/(mean value) follows the x2 distribution with F = n - 1 degrees of freedom. We therefore define the Poisson index of dispersion as i=n x 2 = c (Xi
- 2)2
(5.4.4)
i=l
with x 2 as given by Eq. (5.4.2) for F = n - 1. The numerator is nS2 of Eq. (5.2.6); the denominator is our best estimate of u2, by Eqs. (5.2.2) and (5.2.5~).Hence the Poisson index of dispersion is a n estimator of the ratio of the sample variance t o the population variance. This ratio has a x2 distribution, with F = n - 1 degrees of freedom because only one observed parameter, the sample mean, 3, has been utilized in comparing the observed values xi with the expected dispersion. As a n illuminating illustration, compare the two sets of y-ray counting data, taken with two Geiger-Muller counters called A and X , given in Table 111. The values of xi are the total number of counts observed in successive 5-minute intervals. For counter A , the population standard deviation estimated from the = 13, and this is actually a little residuals, by Eq. (5.2.9a), is u ‘v smaller than the estimate based on the mean value by assuming a Poisson distribution] u ‘v & = 15. The value of x2 = 4.37, with F = 6, gives, from Fig. 6, the probability P = 0.6 that, in repetitions of these seven measurements, the sample variance would exceed the dispersion observed here. Note that these measurements extend from 248 t o 209, and therefore have a range of 248 - 209 = 39, or about 2 . 5 ~ .Also these two observations, 248 and 209, out of the seven observations, fall outside the domain Z f u, which is about the proportion expected from Fig. 1, i.e., P, = 0.32. These data then meet th e statistical tests for small samples. Because the primary process is surely Poisson, the counter A is judged to be operating satisfactorily.
2/-q
5.
790
STATISTICAL FLUCTUATIONS
Now consider the data from counter X. Here x 2 = 0.58, F = 5 , and hence in P = 0.99 of the repetitions we would expect greater variability among the independent observations. This dispersion is grossly subnormal and would be expected with good apparatus only once in 100 tries. We conclude that either (1) a very unusual observation has been made or (2) the instrument is faulty and devoted t o periodic spurious counts. The seasoned experimenter will surely choose t o be suspicious of the TABLE 111. Analysis of Y-Ray Counting Data
I
Counter A
1
2 3 4 5 6 7
Total
- 18
209 248 217 235 224 233 223
21 - 10 8 -3 6 -4
1589 2 = 227
F=7-1=6;
0
Counter X
1 2 3 4 5 6
324 441 100 64 9 36 16
-
Total
990 x= = 4 . 3 7
.'.P=O.6
242 241 249 246 236 250
-
-
1464 Z = 244
0
IF=6-1=5;
4 9 25 4 64 36
-2 -3 5 2 -8 6
-_ x2
142 0.58
=
:.P=O.99
counter or its electronic circuits and will proceed with further examination of the instrument. Note also that the standard deviation estimated from the residuals is only u N 4- = 5, which is much smaller than u N 6= 4%= 16, and moreover that none of the readings fall outside Z f 16. A naive experimenter would usually choose the counter X as superior to counter A because of the self-consistency and reproducibility of its readings. These are false clues. Variability comparable with or even greater than that shown by counter A must be exhibited by a reliable instrument operating on a random process. When a small sample is thought to have come from a binomial population, the binomial index of dispersion is analogous t o Eq. (5.4.4) except that in accord with Eq. (5.2.5a) the population variance estimator 2 is replaced by 2(1 - p ) , so that i.= n..
xz =
2
i-i
(Xi
2(1
- 2)'
- p)
(5.4.4a)
5.4.
791
TESTS FOR GOODNESS OF FIT
has the x2 distribution for F = n - 1 if xI has a binomial distribution with mean p z and variance pz(1 - p ) N 2(1 - p ) . 5.4.3. Confidence Interval for the Standard Deviation of a Normal Distribution
In some nuclear experiments we wish t o determine and make practical use of the standard deviation of a distribution. We recognize that our inability to measure exactly the true mean m gives rise to our everyday use of the standard deviation, and the standard error, as an estimate of our uncertainty in the mean value. But if our experimental goal is the measurement of the standard deviation, then we realize that there must also be a standard deviation of the standard deviation of the distribution. The problem arises particularly when the parent distribution is normal, rather than Poisson, for then u is a free-floating parameter which is totally unrelated to m. The x2 distribution leads directly to the estimation of confidence intervals for the standard deviation u of a normal population. It can be shown, as would be expected from the form of Eqs. (5.4.4) and (5.4.4a), that if x is normally distributed the ratio nS2/u2has a x2 distribution with F = n - 1 degrees of freedom, where S2 is the sample variance among n measurements, as given by Eq. (5.2.6), and u2 is the true variance of the normal distribution. Then, with F = n - 1, i=n
(5.4.5) i= 1
It follows that if x~.gsis the P = 0.98 value, and xi.02is the P = 0.02 value of x2 for F = n - 1, then in 96% of the cases tested, nS2/u2will lie within the interval (5.4.6) which, on rearrangement gives the 96% confidence interval for u2 as (5.4.7) Obviously any other confidence intervals can be selected by suitable choice of the x2 values in Eq. (5.4.6). As an illustration, suppose that from an aqueous radioactive source a number of supposedly equal portions have been prepared by pipetting, and are to be distributed among many laboratories as interlaboratory reference standards. It is desired to test the uniformity among n of these
792
5.
STATISTICAL FLUCTUATIONS
portions and from these measurements to quantify the inhomogeneity of the standards. From Eq. (5.4.7) we can say, with 95% confidence, that the true standard deviation u is not greater than that given by
s
lin
~ > XO.96
L
>
O
.
(5.4.8)
Then if precise measurements of a set of n = 5 standards show a sample standard deviation of S = 0.6% we can say that in 95% of such cases the true standard deviation, u, should not be greater than
On the other hand, if n = 30 standards give the same S = 0.6%) then we can put a much lower limit on u and can say with 95 % confidence that u 5 0.6 -\/30/17.7= 0.8%.
5.5. Applications of Poisson Statistics to Som Instruments Used in Nuclear Physics’
Most of the primary processes which are measured in a nuclear physics laboratory follow a Poisson distribution, or its counterpart, the interval distribution. The binomial and the normal distribution are met somewhat less frequently. Some of the more common questions of experimental design and of statistical treatment of data can be put into standard forms ready for routine use on Poisson distributions.
5.5.1. Effects of Resolving l i m e in Single-Channel Counting The resolving time p of a counter channel causes the very short intervals to be missing in the output. All counting systems are really interval counters, not particle counters. Paralyzable (Type I) counters are unable to provide a second output pulse until there is a time interval of at least p between two successive true events. Then if a is the true rate, the observed counting rate A will be or
+--
A = ae-’p ‘v a(1 - ap .) a N A(l Ap) when ap << 1.
+
(5.5.1) (5.5.2)
7 We give here only some of the principal results, and without proof. For a more detailed treatment see R. D. Evans, “The Atomic Nucleus,” Chapters 26, 27, 28, and Appendix G. McGraw-Hill, New York, 1955.
5.5.
APPLICATIONS O F POISSON STATISTICS
793
Paralyaable counter systems have a maximum possible counting rate of A,, = l / e p , which occurs when the true rate is a = l / p . For an infinitely strong source, A -+ 0, as a .+ C O . Nonparalyzable (Type 11) counters are unresponsive for a resolving time p after each recorded event, but are not affected in any way by events which occur during this recovery interval p. The true rate a, and the observed rate A are then related by
A or
a 'v A ( l
=
~ (-l Ap) when up << 1.
+ Ap)
(5.5.3) (5.5.4)
Nonparalyzable counters have a maximum possible counting rate of A,,, = l / p , which is approached asymptotically as a -+ CO. Although scintillation and proportional counters can approximate the behavior of a n ideal nonparalyzable counter, most counting systems exhibit an intermediate behavior somewhere between the two limiting cases of Types I and 11. I n practice the counting losses due t o finite resolving time must be evaluated empirically. If possible, maximum counting rates should be kept low enough so that up 5 0.05. Then the exact limiting expressions Eqs. (5.5.1) and (5.5.3) give correction ratios a / A which differ by less than O.l%, and Eq. (5.5.2) can be used for any counting channel. 5.5.1 .l. Two-Source Method for Measuring Resolving Time. The effective resolving time of a single-channel counter can be measured most satisfactorily by comparing the response of the apparatus to the radiation from two approximately equal sources, taken separately and taken simultaneously. Let b be the true average background when no source is present, and let U A and U B be the true elevation of the counting rate for each of the two approximately equal sources. Then the observed rates for each of the two sources, including background, are A A and An, where and Then when both sources are measured simultaneously, the sum of their b = U A ae b, but the radiation will elevate the true rate to as observed rate will be only A s , where
+
UA
+ +b UB
=
As(1
+
+ +
Asp)
provided all the counting rates are small enough so that Asp << 1. Solving these three equations for the resolving time, p, in terms of the observed counting rates, gives
(5.5.5)
794
5.
STATISTICAL FLUCTUATIONS
or, to an approximation which is usually satisfactory, (5.5.6) The observations should be taken in the sequence A A , A S , AB, so as to avoid the positioning errors which would be unavoidable in the sequence AA, A B , A S . The source positions for A A and A B must be well separated so that when A s is measured neither source can scatter radiation from the other source into the counter. It can be shown that, if a total measurement time of 3T is approximately equally divided between measurements of A A , As, and AB, and if A A 'v A B , the standard error in the determination of p is 1
(5.5.7)
if the background b is known in advance with negligible error. 5.5.2. Effects of Resolving Time in Coincidence and Anticoincidence Circuits
In a two-channel coincidence circuit, let channel 1 receive random pulses (e.g., from 8-rays) at an average true rate all and have a resolving time pl. Let channel 2 receive randomly distributed pulses from a time-coincident radiation (e.g., prompt y-rays associated with the @-rays from the same source) at an average true rate a2, and have a resolving time pz. Let a1,2 be the true coincidence rate. Then the total randomcoincidence rate, for false coincidences, is arandom
- al,Z)('& - a l , 2 ) ( P l + P2) a l p 1 << 1 and azpz < < 1. In practice,
= (a1
(5.5.8)
the singles provided as usual that rates a1 and a2 are usually large compared with the true coincidence rate a1,2. Then Eq. (5.5.8) becomes approximately arandom 'v
ala2(Pl -k
(5.5.9)
P2).
If a1 and a2 are each proportional to source strength, we note from Eq. (5.5.9) that the random coincidences increase with the square of the source strength. This condition imposes an upper limit on the useful source strength in any coincidence experiment, because at least half of the total observed coincidences should be true coincidences. Then the source strength, in disintegrations per unit time, should be less than the reciprocal of the sum of the resolving times ( p l pz).
+
5.5.
APPLICATIONS O F POISSON STATISTICS
795
In anticoincidence circuits, pulses are recorded from channel 1 provided that there is no coincident pulse in channel 2. The number of random anticoincidences is also given by Eq. (5.5.8). 5.5.3. Scaling Circuits In a single-channel scale-of-s (s = 2, 4, 8, 16, . . . , or 10, 100, . . . , etc.) one output pulse is registered for each s randomly distributed input pulses. If a is the true average input rate, the distribution of scale-of-s output intervals is given by the s-fold interval distribution of Section 5.1.5.2. The “regularizing action” of a scaler, mean value, modal value, variance, and standard deviation of the s-fold interval distribution have been treated in the corresponding earlier sections. We discuss here only the treatment of data from the output of the scale-of-s. 5.5.3.1. Constant Time Interval. I n a series of n equal time intervals, tConst,let the number of s-fold output pulses be ul, u2, . . . , un (u; can be a noninteger if interpolation has been resorted to, thus ui = xis, where xi is the number of input pulses), Then the average scale-of-s output is i=n
(5.5.10) The average number of input events during toonat is then 3 = Qs, and the best estimate of the true average rate, a, of the random input events, which is usually the quantity of experimental interest is a=-.
QS
(5.5.11)
tconst
The variance uu2 of the output, ui,about the mean ii, is i=n
(5.5.12) i=i
and the expected value of uU2,if the input process is Poisson, is
(5.5.15 ) The standard error of ii is UlC
ug
=
l/n
(5.5.14)
and its expected value is
(5.5.15)
796
5.
STATISTICAL FLUCTUATIONS
The fractional standard error in a is the same as in its expected value is
ti, u,/a = u d G ,
and
(5.5.16) The value of the Poisson index of dispersion x 2 in terms of the output data is, with F = n - 1 degrees of freedom, i=n
(z,
i=n
-3
- 2)2 -
s
J
x 2 =i =c l
i=l
(Ui - J ) 2 .
(5.5.17)
5.5.3.2. Constant (Preset) Number of Counts. Scalers in which one measures the time, ti, required to accumulate exactly a preset number, s, of input counts follow the s-fold interval distribution. In a series of n independent observations on a constant source whose true average input rate is a, let tl, t 2 , . , . , 1, be the time required to accumulate s input counts. Then the average interval for s input counts is i=n
(5.5.18) and the best estimate of the true average rate, a, of the random input events is S
a N =-
(5.5.19)
t
The variance
at2of
the output intervals
ti
about the mean interval 5 is
i=n
(5.5.20) i=l
and the expected value of from Eq. (5.2.5e) to be
ut2, if
the input process is Poisson, is found
E[ut2] =
S' (0
(5.5.21)
The standard error in 5 is ut
=
and its expected value is
z -
(5.5.22) (5.5.23)
5.5.
APPLICATIONS O F POISSON STATISTICS
The fractional standard error in a is value
u,/a =
797
u,/l, and has the expected (5.5.24)
The Poisson index of dispersion x2, in terms of ti, can be estimated t o sufficient accuracy by taking 3 = s, and xi E sf/&. Then
2 (f
i=n
i=n
(Xi - 3)2 'v s
- 1)'
(5.5.25)
i=l
or, if s >> 1, more simply a=n
- (l)?(t, - f)*.
(5.5.26)
5.5.3.3. Varying Time Interval and Number of Counts. I n order to avoid both interpolation and the use of present mechanisms, it is often most convenient experimentally to allow both the number of counts, 2; = uis, and the time intervals, ti, to vary by modest amounts from one run t o the next. Then in a series of n similar but not necessarily equal time intervals tl, 1 2 , , . . , t,, we observe u l , u2, . . . , u,,similar but not necessarily equal integer output readings from a scale-of-s. Then the number of input events in each of the n observations is xi = uis. The observed average counting rate in each of the n readings is Ai = xi/ti = uis/ti, and the counting rates A l l Az, . . . , A , become the predominant experimental parameters. There are then two estimators available for the true mean counting rate. The estimate i=n
(5.5.27) i=l
is usually t o be preferred when nonsystematic errors due t o positioning of apparatus or sources may be present among the n observations. But if all experimental conditions are exactly repeatable, and therefore the variations among the individual values of A ; are due entirely t o statistical fluctuations in the nuclear events, then the best estimate of the true mean counting rate, a, is
(5.5.28)
798
5.
STATISTICAL FLUCTUATIONS
for Poisson distributions. The variance of individual observations of the counting rate about the mean value, A, is i=n
‘c
aA2 N -
n-1
The standard error in
i=l
( A <- A)?
(5.5.29)
A is UA
=
and its expected value is
E[u;il =
(5.5.30)
z/n
&$
The fractional standard error in a is u,/a value 1
=
(5.5.31)
a,-/A and has the expected (5.5.32)
As in Eqs. (5.5.16) and (5.5.24), the expected fractional standard error is always simply the reciprocal of the square root of the total number of events counted. TABLEIV. Statistical Treatment of Scale-of-400 Counting Data, on a Constant Source of Radiation, with Varying Number of Counts, u i s , and Varying Counting Time, ti ui scaleof-400
i
136 217 297 272 208 198 255
3040 4878 6694 6120 4673 4440 5752
1583
35,597
Total Eq. Eq. Eq. Eq. Eq. Eq.
ti
aec
Ai counts per sec scale-of-1
(Ai - A)z
ti(Ai - A ) z
17.895 17.796 17.747 17.778 17.804 17.838 17.733
0.0092 0.0000 0.0027 0.0004 0.0000 0.0015 0.0044
28 0 18 2 0 7 25
(5.5.27) (I Y 2 = 17.799 coants/sec (5.5.28) a = A’ = 17.787 counts/sec (5.5.29) U A cv 0.055 counts/sec (5.5.30) u z ‘v 0.021 counts/sec (5.5.31) E[UZ]= 0.022 counts/sec (5.5.33) xz = 80/17.799 = 4.5; F = n
0.0182
- 1 = 6; P ‘vO.6
80
5.5.
APPLICATIONS OF POISSON STATISTICS
For the counting situation in which neither ui nor
ti
799
is held constant,
x 2 is to be expressed in terms of the observed counting rates A<. Then, going back to Pearson’s x2, Eq. (5.4.1), rather than to the concept of Poisson index of dispersion, we can write
The statistical treatment of counting data, according to the methods of this section, is illustrated by Table IV. The best estimate of the true counting rate would be 17.79 jz 0.02 counts/sec. The x2 test shows that the statistical fluctuations in this set of seven measurements are reasonable and would be exceeded in about six out of ten such sets of measurements ( P = 0.6).
5.5.4. Counting-Rate Meters In counting-rate meters each pulse from a counter is converted electronically into a charge q which is added to the charge Q coulombs on a condenser C farads. A resistance of R ohms shunts the condenser. At equilibrium, the average charge 0 on the condenser is proportional to the input counting rate, a. The charge Q is observed continuously, either by reading the potential difference, V = Q/C volts, across the condenser with a vacuum-tube voltmeter, or by reading the current, i = V / R = Q/R C amperes, flowing through the shunt resistance. The statistical interpretation of the counting-rate-meter output readings due to Poisson-distributed input pulses requires a special statistical theory because the integrating and averaging circuit RC produces an exponential interdependence of successive observations on all preceding observations. 5.5.4.1. M e a n Counting Rate. The mean number of pulses in the time dt will be a dl, and the expected increment of charge interval from t to t on the condenser in dt is aq dt. At a later time to,this increment will have decayed to aqe-(tO-t)’RC dt, because of leakage of the charge through the resistance, with the time constant RC seconds. Then if Q = 0 at t = 0, the expected value Qmof the charge at any later time t o is
+
&,,,(to)
=
jot’ aqe-(to+’RCdt = aqRC(1 - e--to’RC)
(5.5.34)
and the expected equilibrium charge, after the counting-rate meter has been operating for a time t o >> RC, is Qm
=
aqRC.
(5.5.35)
800
5. STATISTICAL
FLUCTUATIONS
Thus Qm is simply the specific charge y per pulse times the average number of pulses a R C occurring in one time constant RC seconds. 5.5.4.2. Statistics of a Single Reading. The counting-rate meter stores statistical information but has an exponentially decaying memory. After an equilibrium time, t o >> RC, a single instantaneous observation of Q is an estimator of Qmand hence of the input counting rate, a = Q,/qRC. The variance of Q about Qm is to be computed by recognizing that the only statistically independent contributions to Q are the increments of charge ay dt due t o a dt random events in the small independent time intervals dt. If a is Poisson distributed, then the standard deviation of a dt counts is (a d t ) 1/2 counts. At a later time to,the charge increment due t o these (a dt)'l2 counts will have decayed to (a dt)1/2qe--(fo--t)/RC. All such contributions to the standard deviation of Q at t o are statistically independent. Hence their net effect is to be obtained, as in Eq. (5.3.10),as the sum of their squared values. Then the variance C+"Q), of a single reading of Q at to, is a2(Q) =
htoay2e-2(fO-f)'RC
dt = &aq2RC(1 -
e-2to/RC)
(5.5.36)
and at equilibrium, t o >> RC, the variance of Q is
u2(Q) = ;aq2RC.
(5.5.37)
The fractional standard deviation of a single instantaneous observation of Q, about its true mean Qm, is then
(5.5.38) or the same as would be expected in a direct counting observation over a time 2RC. 5.5.4.3. Average of n Independent Readings. Suppose that a t t o >> RC a single reading Q1 is taken, followed by additional readings Q 2 , Q3, . . . , Qn,taken at subsequent times ORC, 28RC, . . . , (n - 1)ORC. For each of these n readings taken at equally spaced time intervals, ORC, the expected value of Q is the same. Hence the best estimate of the true mean is simply i=n
-
1 Qm 'v Q = n
Qi.
(5.5.39)
i=l
If 0 is small, these successive readings of Q are not statistically independent of one another, because of the exponential memory of the RC circuit. The statistically independent information consists only of those
5.5.
APPLICATIONS OF POISSON STATISTICS
801
parts of the various readings which are due t o charge accumulated since the preceding reading. It can be shown th a t the standard error u9(&) of the mean value & depends upon n and 0, and is given by
where u ( Q ) = q d m is the standard deviation of a single reading, Eq. (5.5.37). If e >> 1, so that the n readings become really statistically independent, then Eq. (5.5.40) reduces to
(5.5.41) which is of course the usual relationship for independent readings, Eq. (5.2.14). 5.5.4.4. Average of Continuous Readings. Frequently the output of a counting-rate meter is a continuously recording galvanometer or voltmeter. Continuous observation of Q over a finite time T corresponds t o an infinite number of single readings, with infinitely close spacing, that is, n + 00 , e + 0, but such that no = T/RC. For continuous observations, the standard error UT(&) of the average charge 0 is obtained from Eq. (5.5.40) subject to the conditions e << 1, no = T/RC, and is
where u ( Q ) = q reduces t o
d
m from Eq. (5.5.37).When T >> RC, Eq. (5.5.42) (5.5.43)
which is equivalent t o the usual Poisson relationship, where aT is the total number of pulses in time T , and T is so much longer than RC that the information stored at the beginning of the observation can be neglected. I n the practical use of a continuously recorded output, the mean value & divides the trace into two equal areas, as illustrated in Fig. 7. The fractional standard deviation of a single observation u ( Q ) / & can be estimated directly from the recorded output. If aRC >> 1, as it does in any reasonable design, then the frequency distribution of Q about 0 will approximate the shape of a normal distribution, but with the standard deviation as given by Eq. (5.5.37). Then by Fig. 1, (Q - &) will exceed
5.
802
STATISTICAL FLUCTUATIONS
tFIG.7. Schematic counting-rate-meter output, with base line o f f bottom of page. The dotted lines locate and the domain of *%(Q), for visual estimation of the standard deviation u(&).
24Q) only about 4.6% of the time. Then if we draw dotted lines, as in Fig. 7, which include all but about 2% of the highest observations and 2% of the lowest observations, we shall have made a reasonable direct evaluation of the band width *2u(Q) and hence of the standard deviation u ( Q ) . From this, the standard error m(0) of the mean value can be obtained from Eq. (5.5.42) and it is plotted in Fig. 8.
1
2
4 T/RC
6 810 20 40 Duration of Observotion
60 80100
FIG.8. Ratio of standard error U T ( ~ of ) the mean 8, to the standard deviation o(Q) of Q, for continuous observation8 of duration TIRC, Eq. (5.5.42).
5.6.
USEFUL INEFFICIENT STATISTICS
803
5.6. Useful Inefficient Statistics Sections 5.1 through 5.5 deal with “efficient” statistics, that is, those which extract the maximum amount of information from the data. Many situations arise in which it is desirable to use an “inefficient” statistic, that is, one which removes only part of the maximum amount of information but which does it much more simply and quickly. There are many types of inefficient statistics. Nuclear physicists find the following very simple ones the most generally useful for quick estimates of the validity of data when the number of independent observations n is small. These are nonparametric tests, that is, they are applicable to any distribution. In addition, the prevalence of the Poisson distribution in nuclear physics always affords the quick estimate of the fractional standard deviation a/x = l/di. 5.6.1. Estimate of the Mean Value The median (middle value when n is odd; average of the two middle values when n is even) has an efficiency of 20.64 for all n,as an estimator of the mean 2. That is, the median of n = 10 independent measurements is aa good an estimator of the true population mean m as is the average 3 of six measurements. The median is a more efficient statistic than the mid-range (average of largest and smallest readings) for all n except 3 and 5. i=n
median
-
‘v
2
=1 n
xi.
(5.6.1)
i=l
5.6.2. Estimate of Standard Deviation
For n
2 t o 20 an estimate of
u
can be based simply on the range.
range = (largest reading) - (smallest reading) Z ( r - 2 ) 2cv-.range n-1 dG
(5.6.2)
The efficiency falls from 99% for n = 3 to about 85% at n = 10. When > 12 it is best to divide the data into a number of equal groups each containing about seven or eight observations, and to use as the effective range the average of the ranges of these groups. n
5.6.3. Estimate of Standard Error (5.6.3)
804
5.
STATISTICAL FLUCTUATIONS
5.6.4. Estimate of x2 If x is dimensionless and scale-of-1, then the range and median can be used for a quick estimate of the Poisson index of dispersion, Eqs. (5.4.4) and (5.5.17). Thus X2
=
z1 q $- Z)Z
X
nS2
= -=
z
(n - l ) a 2 = n - 1 (range)Z -. z n median ~
(5.6.4)
Because x2 must be dimensionless, caution must be used in applying Eq. (5.6.4), which refers only t o cases in which the range and median are both dimensionless. Thus Eq. (5.6.4) would apply to the data on xi in Table 111, but not t o Ai in Table IV. When the range and median refer t o time, t, as in the interval distribution for scale-of-s outputs, Eq. (5.5.25), then it is still proper t o use the range of ti as an estimator of the sample variance, S2,of t, as in Eq. (5.6.2). That is, Z(ti -
f)2
=
nS2
=
n-1 n
(n - l)u2'v -(range of t J 2
(5.6.5)
and a quick estimate of the Poisson index of dispersion in the form of Eq. (5,5.26) is
When the measured variable is the counting rate, Ail as in Eq. (5.5.33), the range of Ai can be used as an estimator of the sample variance, S2, of A as in Eq. (5.6.2), thus
z(Ai- A)z = nS2
=
(n -
n-1 n
l)uA2 N -(range
of
(5.6.7)
and a rough estimate of x 2 in the form of Eq. (5.5.33) is x 2
=
1
A
f
Zti(Ai - A ) 2N T Z(A, - A)' A
(range of Ai)'].
(5.6.8) (5.6.9)
In a nuclear counting experiment, regardless of the dimensions of the measured quantity, a simple recipe for determining quickly whether the dispersion of the data is reasonable is as follows. First, from the range of the data, estimate u using Eq. (5.6.2). Second, divide by the median thus
5.6.
USEFUL INEFFICIENT STATISTICS
805
obtaining an estimate of the fractional standard deviation (F.S.D.) in the observed distribution, that is F.S.D. =
(5)
N
obs
range di (median)
(5.6.10)
This F.S.D. is dimensionless, regardless of whether the data are in terms of counts xi,scaled counts ui, time intervals ti, or counting rates Ai. Third, estimate a representative value of Z,the number of scale-of-1 (dimensionless) counts in a typical individual measurement. By Eq. (5.2.13) the expect,ed F.S.D. from a Poisson distribution of events is l / d ? . Fourth, compare the F.S.D. from Eq. (4.6.10) with 1 / 4 2 . If the ratio lies outside the domain (5.6.11) the dispersion is probably unusual ( P 2 0.98 or P 0.02). The statistical justification for this simple comparison test of F.S.D. and l/& can be expressed analytically as follows : For a Poisson distribution, with F = n = 1 degrees of freedom,
As an illustration consider the data of Table IV. The range of Ai is 17.895 - 17.733 = 0.162 counts/sec, and the median value is 17.796 counts/sec. Then the observed fractional standard deviation is
F.S.D.
0.162 'v
t/T (17.796)
=
o.ao34, or 0.34%.
The total number or scale-of-400 counts in each measurement is approximately fi N 225, hence there are about 400fi = Z = lo6 ecale-of-1 counts in each measurement. Then the expected fractional standard deviation is 1
7;7s
=
0.0032, or 0.32%.
Then F.S.D. 0.0034 1 / 4 $ - 0.0032 -Iu-
=
1.05
806
5. STATISTICAL
FLUCTUATIONS
and the data may be considered as a reasonable sample of a Poisson distribution. Indeed, using Eq. (5.6.12) we would have
x2 II (n - 1) and, with F
= n
-1
=
[El‘ =
6(1.05)2 = 6.6
1/d2
6, Fig. 6 gives P = 0.4.
5.6.5. Examples
As an illustration, analyze the data for Geiger-Muller A , from Table 111. 2 ‘v median = 224
(39)2 = - 1 (range)2 = -6 5.8. n median 7 224
x 2 ‘v n
Then F
=
~
n - 1 = 6, and P
‘v
0.5. Using Eq. (4.6.12) &s an estimate of
x2, we would have x2 =
(n - 1)
(median) [range/-\/; l/-\/median
n (median)
which is the same, in this case, as Eq. (5.6.4). These results are compared with those obtained from the corresponding efficient statistics in the following table. The results are seen t o be compatible. z Efficient statistics Inefficient statistics
227 224
01
XP
P
4.9 5.6
4.4 5.8
0.6 0.5
0
12.8 14.8
When analyzed in the same way, the data from Table I11 on counter X give the results shown in the tabulation below.
Efficient statistics Inefficient statistics
z
0
05
X’
I’
244 244
5.3 5.7
2.2 2.3
0.58 0.67
0.98
0.99
These inefficient statistics are so simple that they should be used routinely for the preliminary appraisal of all laboratory data.
APPENDIX 1 * 1. Evaluation of Measurement 1.1. General Rules
In a concise expression of the results of the measurement of a physical quantity, three pieces of information should be given: a number, a numerical statement of reliability, and an appropriate set of units. The number is generally an estimate expressed in a finite set of digits (the exceptions are numbers which are exact by arbitrary definition, or mathematical constants such as the base of natural logarithms) reflecting the limited accuracy of physical measurement. The statement of reliability is usually written as plus or minus one, or at most two digits in units of the last digit of the number, together with a sufficient explanation to allow interpretation. In particular, one should state how many measurements were employed in the determination of the number and of its reliability. As will be seen below, this is of great value in the critical comparison of the results of different experiments, and in their combination with results of previous work. The number of digits that can be read is indicated by the smallest scale division, the least count of an instrument. Usually, one additional digit can be estimated between scale divisions. No more-and no lessdigits should be recorded than can be read reproducibly. To remove ambiguities, a standard form may be used for the recording of data: the decimal point is put just after the first nonzero digit, and the number is multiplied by the appropriate power of ten. Every digit is then understood to be significant. The statement of reliability, or the statement of the magnitude of error, is automatically indicated by the number of significant digits. The final result will generally have one more significant digit than the individual readings. This procedure implies that one should not round off readings. Any round-off increases the error. I n the course of computation, round-off may be inevitable. A brief discussion of errors so introduced, with further references, is given in Section 7. For the estimation of the best value of the desired quantity and of the significance of the result statistical techniques are used. The terms “best” and “significant” should be understood in a technical sense: e.g., “best” and “significant” according to some statistical criterion. The criteria applied depend on assumptions which may or may not be true: attention should be paid to their validity. In the following, only a prescription of the techniques can be given. For this reason, a word of warning is in order:
* Appendix 1 is by Sidney Reed.
807
808
APPENDIX
1
these techniques, properly used, can improve the understanding of the results and the judgment of their worth-but they are not a substitute for thought.’ It should be emphasized that work in certain fields, e.g., cosmic ray or high energy physics, requires more complete attention to statistical techniques in the planning and interpretation of experiments than can be discussed here.2-3
2. Errors 2.1. Systematic Errors, Accuracy Statements about reliability of a measurement require assessment of the accuracy and of the precision of the work. Lack of accuracy is considered to be due mainly to what long usage has termed Systematic errors. In general, systematic errors are definite functions of experimental method, instruments, or environmental conditions. If detected they can usually be corrected for. Sometimes a single correction will be adequate for the entire work and can be applied a t the end. Constant, or slowly varying systematic errors are hard to detect. The crucial test is the comparison of measurements of the same quantity obtained from different experiments, using different principles. 2.2. Accidental Errors, Precision Precision implies close reproducibility of the results of successive individual measurements. It is assumed that, in general, there is a variation from measurement to measurement. This scatter of data is usually considered due to accidental errors; it is imagined that the experiment is aimed at a constant quantity, superimposed on which there is a random sum of small effects independent of each other and of the quantity itself which are responsible for the variation of the results. Absence of variation is not necessarily an indication of precision; it may be due simply to an excessively large least count of the instrument used. For further references, see E. B. Wilson, Jr., “An Introduction to Scientific Research.” McGraw-Hill, New York, 1952; H. Cramer, “Elements of the Theory of Probability and Its Applications.” Wiley, New York, 1955; J. Cameron, in “Fundamental Formulas of Physics” (D. Menrel, ed.), Chapter 2. Prentice-Hall, New York, 1955; L. Parratt, Probability and Experimental Errors in Science, Wiley, N.Y., 1961. * See, for example, L. JLnossy, “Cosmic Rays.” Oxford Univ. Press, London and New York, 1953. * M. Annis, W. Cheston, and H. Primakoff, Revs. Modern Phys. 26, 818 (1953); J. Orear, Univ. of California Radiation Lab. Rept. UCRL-8417 (L958).
APPENDIX
1
809
3. Statistical Methods To analyze accidental errors, the actual data are imagined to be a random selection, one for each measurement, of values from a large reference distribution which could be generated by infinite repetition of the experiment. In statistical terms, this is a finite sample from a “parent distribution” (p.d.). For reasons of mathematical convenience, it is usual to assume that the p.d. can be approximated satisfactorily by an analytic function (p.d.f.) having two or three parameters. A finite data sample permits at most the assignment of odds to the values of the p.d.f. parameters which represent the best value and the significance of the measurement. In most cases, a reasonable, explicit assumption of a definite form of the p.d.f. is desirable. Which form should be taken depends on a preliminary assessment of the probabilistic features of the experiment. If the errors are accidental in the sense described in Section 2.2 above, a normal (see Section 3.1) distribution function (n.d.f.) is appropriate. If the experiment is directly concerned with probabilistic phenomena, e.g., counting experiments in nuclear physics, the Poisson or some other discrete probability distribution function may be chosen. 1,2.3 For the problem of estimation of the best value alone one does not need to assume any particular p.d.f.; a systematic estimation using least squares can be made.4 A sharp quantitative statement of the statistical significance of a difference between two “best” estimates of the same quantity cannot be made, however, without assuming a definite form for the p.d.f.
3.1. M e a n Value and Variance The fraction of readings d N ( x ) / N drawn from the p.d.f. j(x) lying in dx is the range between x and x
+
dN(x)/N
=
f ( x ) dz.
(3.1)
The function f ( x ) is normalized: J f ( x ) dz = 1. The average of any function g(x), denoted by , is defined by = J g ( x ) f ( z )dx. The range of integration may, for mathematical convenience, extend in both directions to infinity. Of special importance are the average of x called the mean < X > = J x f ( x )dx (3.2) 4
E. R. Cohen, Revs. Modern Phys. 26, 709 (1953).
810
APPENDIX
1
and the average of (x - < x > ) ~ called , the dispersion or variance of x U’(X>= J(x - )”~(z)dx.
(3.3)
The square root of the variance, u(x) is called standard deviation or sometimes standard error. I t is a measure of the spread of the data and thus of the precision. An important example of a p.d.f., often assumed to apply to accidental errors, is the Gaussian or normal distribution (n.d.f.) : fl(x) =
[&ru2(z)]-’exp[ - (z - <x>)*/w(x)]
(3.4)
characterized by two parameters, the mean < x > and the variance uz(x). N measurements xi allow the formation of the sample mean N
(3.5)
and the sample variance (3.6)
The mean has the property of being the value of a parameter a which minimizes Zz, (xi - a)2. On the grounds of consistency, one expects that in some sense 3 converges to < x > and s to u as N -+ co .* For computation, it is useful to subtract a constant A of the order of size of xi, so that N
(3.7)
and
3.2. Statistical Control of Measurements The use of any p.d. implies that the data may be regarded as drawn at random from it. There are statistical tests for this,’ but in the case of data scatter because of accidental errors, a rough “control chart” can assist in detecting systematic departures which are functions of time. Such a chart may be made by plotting, on the abscissa, the order (in time) of the reading, and on the ordinate, the reading itself. If there is previous information on the scatter of the data using the same instrument under similar conditions, so that u(x) is known, one can, at least
* This is so in the technical sense of convergence in probability; see, e.g., Cramer, reference 1.
APPENDIX
1
81 1
tentatively, draw lines on the chart at z k 34 which should, if the data are in control, bracket practically all the points. It is quite valuable to have such a chart associated with a precise instrument. If no previous information is available, one should take a number of points, draw lines at z k 3s and continue for a few more readings in order to see whether the additional data fall between these lines. If it appears that randomness is a fair assumption, one can use the function “chisquare”6 to test the overall fit of an assumed p.d.f. The chi square, x2, function can be defined as*
xz = sum of
(observed values - values expected from p.d.f.)2 value expected from p.d.f.
I
and is tabulated as function of the number of degrees of freedom. Here the number of degrees of freedom equals the number of terms in the sum minus the quantity: one plus the number of p.d.f. parameters which must be estimated from the data. I n the case of a n.d.f., this is the number of terms minus three. It is generally necessary to group the observed data and the corresponding values from the p.d.f. into cells.6 For moderate numbers of readings, say 20 or so, x 2 will only show marked discrepancies between the data and the proposed p.d.f. The x 2 tables give the probability that tabulated values of x2 would be exceeded by those computed from a random sample from the assumed p.d.f.
4. Direct Measurements It is useful to distinguish between direct measurements, such as can be made of length, time, or electrical current; and indirect measurements, in which the quantity in question can be calculated from measurement of other quantities. In the latter case the law of connection between the quantities measured and sought may also be in question. In such a case, one has first to decide whether the proposed relation holds for any values of the quantities (establishment of the law of connection), and then to make as good an estimate as possible of the quantity desired.? In the case * An alternative definition has, instead of the expected value, the p.d.f. variance in the denominator. 6 See any standard statistical tables, e.g., R. A. Fisher and G. Yates, “Statistical Tables,” Oliver & Boyd, Edinburgh and London, 1953;C. D. Hodgman, ed., “Handbook of Chemistry and Physics.” Chemical Rubber Publ., Cleveland. (New editions of the latter volume are published frequently.) 6W. G. Cochran, Ann Math. Statistics 23, 315 (1952);also Parratt, reference 1. 7See Wilson or Parratt, reference 1; Annis et al., reference 3; and Cohen el al., reference 12.
812
APPENDIX
1
of direct measurements only the latter problem needs to be solved. This simple situation will be discussed first.* There are several cases, depending on what information is available at the start.
4.1. Errors of Direct Measurements If one has information at the start of the experiment regarding the variance of readings of the measuring instrument under similar conditions, the following procedure can be employed: One can draw up a control chart, using the previous u(x) together with the mean Z of a short preliminary run. If subsequent readings appear to be in statistical control, i.e., if the points fall between the lines a t z k 3a, one can terminate the process a t a definite number of readings which depends on the precision desired. One can then say that the most likely value of < x > is given by the mean 2, and that the reliability of this estimate is such that the probability is one-half that the interval 3 - 0 . 6 7 ~ ~ / 4 I % < x > I Z 0.67u/dN contains <x > . The precision increases with N in the sense that the interval having a definite probability of containing < x > narrows proportional to N-’/2.* In this case, the interval length is sharply defined (for fixed N and probability) and if the results are quoted as
+
2
L-
e;
N measurements
the meaning of this statement is as stated above. Frequently the only information available at the start is that provided by the data itself. If the data seem t o be in statistical control, one can make statements about the probability of bracketing the p.d.f. mean which differ from those possible when u(x) is known. The levels of probability now depend on the number of data points in the sample and the intervals bracketing < x > can now vary in length from sample to sample. The type of statement that can be made for this case is that the best estimate of < x > is 2 and that the probability is 1 - P that the interval between
(4.1) will include, on the average of many samples of size N , the p.d.f. mean <x>. The function t ( P , f )is called Student’s or Fisher’s t and is tabulated
* An interval of this type is called a conjidence interval. It should be distinguished from a tolerance interval which will contain a definite fraction of the population, e.g., a single observation. See, e.g., reference 1; or N. Arley and K. Buch, “Probability and Statistics,” p. 168. Wiley, New York, 1950. A valuable, readable discussion is given by W. E. Deming and R. T. Birge, Revs. Modern Phys. 6, 119 (1934).
APPENDIX
1
813
giving probability levels P that a value should be found at least as large as that tabulated, as function of the number of degrees of freedom f, which in this case is N - 1 . 6 ~ 9 Equation (4.1) indicates that the precision increases, i.e., that the interval narrows as N increases. From this it might be inferred that with repetition of the experiment the precision of the mean would improve indefinitely. That would be so under ideal circumstances but in actual experiments the probability of occurrence of systematic errors increases with the length of the investigation. Another limitation is due to the least count of the instruments used. All the statements above depend on the approximation of a continuous parent distribution function. This approximation improves with the ratio of 0 to the least count w. (w I &T is usually considered reasonable; see Eisenhart et ~ 1 . ’ ~If ) w is large, and the readings come out constant, statistical methods cannot be applied. If there are random errors in the data, one can improve the precision by repetition, but a very large number of measurements are required if u = w.* In order to be fairly sure the accidental errors are in a state of statistical control, a certain number of measurements is necessary (how many depends largely on the investigation, but the number must be fairly large). Once such a state has been reached, further measurements can be added to refine the precision of the mean, but on the hypothesis that each added measurement can be joined with the preceding ones. For this reason a continuously running control chart is very useful. 4.2 Rejection of Data
A closely related problem is the rejection of data. Often some criterion for doing so is given, based on analysis of the data alone. This is not recommended. A measured point should not be excluded on statistical grounds alone. If a control chart as suggested in the previous paragraph is used, and if a point or set of points seem out of control, the experimental conditions should be carefully re-examined, and if no assignable causes can be found, the point should be remeasured after settings have been remade, etc. This presumes that the analysis is made in the course of the
* If the least count is large, attention must be paid t o providing a random character for the data. For example, an improvement in the precision of a measurement of length can be obtained by putting the ends of an object at random between the least count marks of a scale rather than always setting one end to coincide with one of the markers. 9 R. A. Fisher and F. Yates, “Statistical Tables.’, Oliver & Boyd, Edinburgh and London, 1953. 10 C. Eisenhart, M. W. Hastay, and W. A. Wallis, eds., [‘Techniques of Statistical Analysis.” McGraw-Hill, New York, 1947.
814
APPENDIX
1
experiment itself, which is the procedure to be strongly recommended. Unless assignable causes can be found, the suspected data should be retained. In any case a clear statement of the situation is imperative, and a procedure once adopted should be followed consistently. l 1
4.3. Significance of Results Of particular importance is the comparison of the best estimates from two experiments using different physical principles to measure the same quantity. Under the assumption that both are under statistical control, i.e., their data may be regarded as drawn, at random, from normal p.d.f.’s and that these p.d.f.’s have different means but the same variance uZ, one can proceed as follows: If nl and n2 are the numbers of measurements, 2l and ZZtheir sample means, and s12and sz2 their sample variances, the expression
+
should have Student’s t distribution with n1 n2 - 2 degrees of freedom, If the value of 1 from Eq. (4.2) corresponds to a low level of probability for the appropriate number of degrees of freedom, the probable existence of a systematic difference between 21 and 22, as opposed to an accidental or chance difference, is indicated. In many cases, the results from these experiments will have unequal precision, so that they cannot be assumed to be samples drawn from populations having one and the same variance. This situation is discussed by Fisher and Y a t e ~ . ~
5. Indirect Measurement Errors of indirect measurement occur when the desired quantities are obtained from those measured with the aid of equations. These equations may be the expression of an established physical relation, or may have to be established by means of the data itself. 5.1 Propagation of Errors
A simple case of indirect errors is the propagation or compounding of absolute errors. The desired quantity z is related to the directly measured different quantities dl), d 2 ) ,. . . dm) by a known function, x = f(z(” .
.
. 29).
A critical discussion is given by N. Arley, Kgl. Danske Videnskab. Selskab., Mat.fus. Medd. 18, No. 3 (1949). 1‘
APPENDIX
815
1
Using Taylor’s theorem to expand around the true values zo, obtains
zO(~),
one
and retaining only the Squaring, averaging over all d.f.’s for the di), lowest order terms, one has
where the correlation coefficient pii is defined as
The derivatives are evaluated at z O ( ~ ) or at the nearest approximation thereto. The approximation resulting from neglecting the higher order terms may be poor, for example if any of the higher derivatives, e.g., multiplied by the kth moment < (di)- z O ( ~ ) ) ~ > is comparable with the lowest order terms. If the errors in the d i ) are independent, the equation has the form usually given u”2)
=
<(x -
z0)2>
=
c (&)’
UZ(dj)).
(5.3)
One can assign a measure of error u to x on the basis of preassigned error values in the d j ) using the above formulas. The interpretation of the statistical significance of the error in z , however, requires some knowledge of the distribution of the errors. In an error analysis of this type, the data should be carefully examined for independence. If the data are obtained from a least squares fit of all before the data jointly, for example, or if in the computation of the di) insertion into the formula some common error-contributing components are used, e.g., one of the physical constants, the correlation coefficient pij will in general not be zero.12The most accurate determination of the l2 E. R. Cohen, J. W. M. DuMond, T. W. Layton, and J. S. Rollett, Revs. Modern Phys. 27,363 (1955);E.R. Cohen and J. W. M. DuMond, in “Handbuch der PhysikEncyclopedia of Physics” (S. Flugge, ed.), Vol. 35. Springer, Berlin, 1957.
816
APPENDIX
1
physical constants themselves requires a knowledge of the correlation coefficients in addition to that of the standard errors. Another practical consequence refers to the design of the experiment. is that in which The best distribution of errors among the quantities di) all the errors are equal. As a consequence, effort should not be spent on further refining the measurements expected to be most accurate, but on those expected to be least accurate in order to approach the optimum condition. A third consequence occurs in the transformation of weights that must be made when transforming a variable in order to bring an equation into a more tractable form for the determination of its parameters by least squares fitting. For example,* y = abz is often transformed into a straight bz for the determination of a and b by line by setting y’ = In y = In a least squares procedures. If the error in z is assumed to be negligible and if that in y is constant, each value of y l does not have equal weight but must be weighted according to
+
W(y’)
= (.“(y’))-‘
=
or, more generally
Yl
=
fb);
W(Y7 =
[(;)z rJz(u)]-l
(5.4)
[($
(5.5)
rJ2(Y)]-1.
5.1.1, Least Squares. Assume a set of data y is generated by changing values of a variable 5. In the general case, both z and y are subject to errors. We assume a certain functional relation between y and z, and are attempting to estimate values of its parameters and to estimate how well it fits the data. A rather clear case is that in which there is some physical reason for believing that the functional relation has a definite form. It is important to have some idea of the over-all behavior of the data, in order not to waste time trying to fit the wrong type of function. The data should be plotted first. It may be that to start arbitrarily with a definite kind of function, e.g., a polynomial, is really a mistake and the data would be much better fitted by a sum of exponentials, for example. In order to interpret a data plot, some acquaintance with the behavior of simple functions is he1pf~l.l~ There are two general classes of least squares approximation functions, one represented by the polynomial functions, and the other by periodic functions. A special case of polynomia1 function, the linear or straight line relation, is important enough to deserve discussion. Is Graphs of simple functions are given by A. C. Worthing and J. Gattner, “Trerttment of Experimental Data.” Wiley, New York, 1943. *Other examples are given by Parratt, reference 1.
APPENDIX
1
817
The usual treatment, and the one given below, is that wherein the errors in y are assumed to be much greater than those in x, so that x may be taken as exact. Even so, the errors in y may vary with x, and this may be known azpriorior not. If the errors in both x and ZJ are comparable, a partial analysis is available. l 4 If there is assumed to be a linear relation between an exact, but unknown, Y and z, and if all the measurements are given equal weight, this relation could be written y=a+px whereas the approximation to this relation is =
a
+ bx.
With given data yi, a and b are determined by the least squares’ condition be minimized with respect to a and b. that the difference Zi(yi This leads to
For equispaoed z data, use of the explicit sums of Z%,x; and Z~-_,xi2 in the above formulas give 6 b = n(n2 - 1) [(Yn - y d ( n - 1)
+ (yn-1 - yz)(n - 3) + - . .]
(5.7)
this can be written in the easily memorizable form, suggested by Birge16a
+
+
b = (Y7L - Y l ) b - 1) (Yn-1- ZJ2)(n- 3) (n - ‘ 1 ) 2 (n - 3 ) 2 (n
The intercept is
+
+
(yn-2
5)2
- y3)(n
+..
- 5)
+
*
.
-
*
(5.8)
Formulas for the variance and statistical significance of a and b are given, e.g., by Cameron’ or Kendall.I6 If, assuming negligible error in z, a polynomial of the nth order f(x) is 14 A. Wald, Ann. Math.Statistics 11, 284 (1940); J. Neyman and E. M. Scott, ibid. 22, 352 (1941);23, 135 (1953). 16 M. G. Kendall, “The Advanced Theory of Statistics,” Vol. 2. Griffin, London, 1948.
818
APPENDIX
1
considered to be appropriate to fit the data I/, e.g., (5.10)
the least squares procedure is to determine the polynomial coefficients ak to minimize N
Q
=
2
n
Wj
j=1
(yj -
1akzj')'. k=O
(The number of data points N > n) where wj is the weighting factor of yj, usually taken as the inverse of the variance a2(yj), wj = (r2((yj))-'. This gives the sequence of linear equations N
n
The minimal value of Q is then
Assuming the yj are normally distributed an estimate of goodness of fit is given by computing Q2jn. If the deviations from the assumed function would be due to chance fluctuations, Q2/nshould behave as x2 with N - n - 1 degrees of freedom. The statistical significance of the coefficient a, can be estimated by forming the ratio
F, = ( N
- n - 1)(QgE1)- Q'"! )/&"?) min
min
(5.12)
which under the assumptions mentioned should have Fisher's F distributionl for one degree of freedom in the numerator and N - n - 1 degrees in the denominator. Standard tables6v9give probability levels that values at least as large as those tabulated could be found by chance, as function of the degrees of freedom in numerator and denominator. Even if it appears that a coefficient an is not statistically significant, the fit may not yet be satisfactory according to the x2 criterion; if the fit is not satisfactory one should proceed to the next higher power, n 1. Formulas for the variance of the coefficients a, assuming these are normally distributed, are given, e.g., by Cameron' and in standard statistical texts.16
+
APPENDIX
1
819
The use of orthogonal polynomials offers considerable advantage in least squares procedures. These are discussed, e.g., by Birge16a and in standard texts.
6. Preliminary Estimation Under certain assumptions, it is possibIe to use Fisher’s method of Maximum Likelihood’ to make preliminary estimates of the numbers of measurements needed to achieve a desired precision, or to be able to establish agreement with a hypothesis, given in terms of a theoretical formula for the distribution of measurements. This is discussed, e.g., by Orear.3
7. Errors of Computation Recent progress in automatic computation has given much incentive to the discussion of numerical analysis and errors of computation, particularly the errors of solving large sets of linear equations. Introductory discussions with extensive further references are given by Hildebrand.16 Errors of linear equation solving are treated by Dwyer. l7 The publication “Mathematical Tables and Other Aids to Computation” (MTAC) contains useful material, including lists of recent tables. l9 Apart from gross errors, or mistakes, the errors of computation can be classed into: (1) the propagation of errors initially present; (2) the generation of errors in detailed steps of the computation, now commonly called round-off errors; and (3) the type of error due to cutting off, a t a finite number of steps, a computation involving an infinite limiting process or asymptotic series, called truncation error. Although these types of error combine in complicated ways, fortunately the over-all error of the computation can be estimated by a simple addition. I n extensive computations, the distribution of errors has to be obtained from statistical considerations in order to make a practical over-all estimate. l 6 There are some accepted rules for good computational design. The procedure should allow the computer to find and get rid of mistakes, to R. T. Birge, Revs. Modern Phys. 19, 298 (1947). F. B. Hildebrand, “Introduction to Numerical Analysis.” McGraw-Hill, New York, 1956. A useful index is given in the appendix of this text, as well as an extensive bibliography. 17 P. S. Dwyer, “Linear Computations.” Wiley, New York, 1951. 18 Mathematical Tables and Other Aids to Computation (MTAC). National Research Council, Washington, D.C.; published as a quarterly since 1943. 1 9 A. J. Fletcher and others, “Index of Mathematical Tables.” McGraw-Hill, New York, 1946. 15s 18
820
APPENDIX
1
estimate bounds for other errors, and to check the final result. There should be a minimum number of individual operations, including look-ups, resetting of scales, and arithmetic operations. Approximate operations, such as division and root extraction, should be eliminated or at least put off as far as possible towards the end of the calculation. An assessment should be made of the number of digits required a t each step to maintain the desired overall accuracy. Regarding calculational aids of moderate accuracy, a few general statements can be made. For multiplication and division, tables of logarithms to 4, 5, 7, and 10 decimal places are available. A calculating machine which multiplies and divides is usually more convenient. Tables of powers of 2, sin x, cos x, exp x and other special functions are also available. A comprehensive index to 1946 has been published,lg and more recent lists can be found in MTAC.'* The ordinary ten inch slide rule has a relative accuracy of about 0.3% per operation of multiplication or division, and can only handle numbers having three significant figures. The relative accuracy, but not the convenience, of a slide rule improves roughly proportional to the length of scale. Automatic desk calculating machines are often so designed that a fixed number of decimal places can be carried throughout. In such a case the number having the largest number of decimal places should fix the position of the machine. I n calculation with numbers having a definite number of significant figures, one additional digit should be carried. It is not recommended that numbers be systematically rounded off in order to eliminate doubtful digits during the course of a calculation. Doing so generally increases the error. There are conventional rules of rounding off followed in many tables and in circumstances where rounding off is inevitable and under the control of the computer. These are given here so that an estimate may be made of the errors thereby introduced. To round off a number, in the conventional sense, to n digits in the decimal system, the nth digit is 1st digit is greater than 5, and left unaltered if increased by 1 if the n 1st digit is less than 5. If the n 1st digit is equal to 5, the nth the n is left unaltered if even. If odd, it is increased by 1. The rounded-off number is said to have n significant digits or figures, and the error of the rounded-off number is not larger than one half unit in the nth digit. If two numbers are rounded off to the same set of n significant figures, their difference does not exceed one unit in the nth place from the true value x rounded off to n significant figures. In multiplication, raising to powers, or division, the relative errors and the number of significant figures are important, but not the decimal place. In addition or subtraction the absolute values of errors and the location of the decimal point do matter.
+
+
+
APPENDIX
2
82 1
The subtraction of nearly equal numbers, causing a loss in significant figures, should be avoided, or made explicit if possible. Numerical analysis of the arithmetic operationst gives the following results: If x* represents a number x rounded off to n significant figures, the error in the n t h significant digit of zmwith m = any real number, is less than 10”[(1 5 - 11. If y* is also rounded off to n significant figures, the product x*y* differs from the true value xy by less than 4 in units of the n t h digit. When handling an (algebraic) sum of numbers having the same number of significant figures, but different decimal positions, a good rule is to round off the ones with more decimal positions to one place beyond th at of those with the least decimal positions. If a n auxiliary table of entry differences is given to facilitate interpolation, the difterence between rounded off values (such as those usually given in tables) has less “round-off” error than the difference between entries before these are rounded 0ff.t
+ -
Appendix 2. Kinematics* The purpose of this chapter is to present a summary of the kinematics of particle reactions. I n particular, we shall discuss the following topics: (1) the equations for the Lorentz transformation from the laboratory system to the center-of mass system; (2) the kinematic criterion for the ident’ification of reaction products based on the maximum possible angle a t which a particle of a given mass can be emitted in the laboratory system; (3) the kinematics of two-body decay; (4) the thresholds for associated production of unstable particles ( K mesons and hyperons); ( 5 ) the relativistic two-body and three-body phase space factors. 2.1. Equations for the lorentr Transformation from the laboratory System to Center-of-Mass System; The Jacobian for Particle Reactions
I n a collision process, we have initially two particles ( a and b) whose momenta and total energies in the laboratory system will be denoted by pa, p b , and E,, Eb, respectively. The total momentum and energy in the
t See Eisenhartlo for a brief discussion of errors of interpolation; see also reference 16. * Appendix
2 is by R. M. Sternheimer.
822
APPENDIX
2
laboratory system will be called po and Eo,respectively, so th a t we have
As a result of the reaction between the initial particles ( a and b ) , we haven particles, which will be labeled i = 1 , 2 , . . . , n. The momenta and total energies of these final particles in the laboratory system will be denoted by pi and Ei, respectively. The center-of-mass system for the collision is defined to be th a t coordinate system (moving at a uniform velocity V with respect to the laboratory system) for which the total combined momentum of all of the particles is zero. Thus if pa and pb are the c.m. momenta of the initial particles a and b, and if p, denotes the c.m. momentum of the ith final particle, we have n
Here and in the following, the abbreviation c.m. is used for center-ofmass. We also note that the symbols for all quantities pertaining to the c.m. system will be barred. The velocity V of the c.m. system with respect to the laboratory system is given by V = po/Eo. (4) Thus the direction of motion of the c.m. system is along the direction of the total laboratory momentum pa, which will be taken as the x-axis. I n Eq. (4) and in the following, we assume that the units have been so chosen that the velocity of light c = 1. In particular, we can assume that energies are expressed in Bev, moment>ain units of Bev/c, and velocities in units of c. We define yv by the equation: yv
G
(1 -
vyi2.
(5)
We shall now verify that, with V as given by Eq. (4), the total momentum in the c.m. system is zero. For this purpose, we write down the equations for the Lorentz transformation from the laboratory system to the c.m. system: pi2 = Y v ( p i + - VEJ (6) pit = pit (7) E o. = Y v ( E i - v p i z ) (8) where p i , is the x component of the momentum pi, and pit is the transverse momentum (component of pi at right angles t o V) in the laboratory
APPENDIX
2
823
system; piz and sir are the corresponding components of the cam.momentum pi. Equation (6) gives: n
n
n
where we have used Eq. (4) for V ,and the conservation of momentum and energy in the laboratory system. We aIso have: n
n
2
pit
=
i=l
1
pit =
o
i=l
where the last equality follows from the fact that the z-axis is along the direction of PO, so that the component of po along any direction perpendicular to the x axis must be zero. Thus we have proved that n
2pi=0 i=l
where the sum extends over the final particles i = 1, 2, . . . n. Of course, the same property holds for the initial momenta, by the law of conservation of momentum (pa Pa = 0). The total energy of all of the particles in the c.m. system will be denoted by 80.From Eqs. (4), (5), and (8), one finds
+
n
Eo
=
2 8,
i=l
= YV
n
n
i= 1
i=l
( 2 Ei - V
pi,) = ~ v ( E o- Vpo) = E O / Y V . (12)
An alternative method of deriving this result is to note that the total momentum po and the total energy EOform a four-vector, with E O as the fourth (timelike) component. For such a vector, the square of the length 1 is invariant (constant) under a Lorentz transformation. I n the c.m. system, the total momentum POis zero, so that we have -12
= Eo2 -
p02 =
-
Eo2.
(13)
Upon using Eq. (4), one obtains 8 0 2
=
Eo2(1
- V2)
(14)
which is equivalent to Eq. (12). The c.m. angle between the direction of the ith particle and the 2 axis
824
APPENDIX
2
(direction of V) will be called &. Thus Piz and pit are given by
The equations for the Lorentz transformation from the c.m. system to the laboratory system are as follows:
The laboratory angle of emission ei (with respect to the x-axis) is given by O i = tan-'(pdpi,)
(20)
where pi, and pit are obtained from Eqs. (17) and (18). PI
FIG. 1. Diagram showing momenta in the center-of-mass system for a reaction in which three particles are produced. The momenta of the initial particles are ii,, and fib; the momenta of the reaction products are @ I , PI, and fia.
Figure 1 shows the momenta in the c.m. system for a collision in which 3 particles are produced. In this figure, pa and p b are the c.m. momenta of the two initial particles (Pa = P b ) . The final particles have momenta pl, pz, and p3 (Zi",, pi = 0). The x-axis is along the original direction of motion of particle a, i.e., along pa. Particle b is assumed to be initially at rest in the laboratory system (pa = 0). For particle 1, the momentum components pl2 and p I t are shown.
APPENDIX
2
825
The Jacobian for Particle Reactions. For calculations of the cross section of particle reactions, it is useful to obtain the expression for the Jacobian J for the transformation from the c.m. system to the laboratory system. The Jacobian is the quantity by which the cross section in the c.m. system must be multiplied in order to obtain the corresponding cross section in the laboratory system. Two cases can be distinguished: (1) If there are only two particles in the final state (n = 2), the system is determined kinematically if one of the angles 8i (8, or 8 2 ) is given. (2) If there are more than two particles (n > 2), the system is not determined completely by giving the angle 8i of any one final particle, and a different expression applies for the Jacobian. For a two-particle reaction, we have
da/d0
=
Jl(da/dH)
(21)
where da/dfl and da/dH are the differential cross sections in the laboratory system and the c.m. system, respectively; J 1 is the appropriate Jacobian. For a reaction involving more than two particles, we have
d2a/dE d0
=
Jz(d2a/dB d o )
(22)
where d2a/dE d0 and d 2 c / d 8 dH are the differential cross sections (with respect to both energy and angle) in the laboratory system and the c.m. system respectively, Jz is the Jacobian. We shall first consider the case of only two particles. The Jacobian J 1 is then given by J1 = (d0/dH)-' (23) where dH = sin 8[d8l = Id cos 81 and dQ = sin Ode = Id cos 01 are the elements of solid angle in the c.m. system, and the laboratory system, respectively, for the particle considered. Here and in the following, we omit the subscript i [see Eqs. (15)-(20)], with the understanding that the formula for J 1 always refers to a particular final particle. From Eqs. (15)-(20), one finds cos
where p and
A
e
=
A - ~ C OfiS+ p )
(24)
are defined by
p =
B
v/z,
=I
+ 2p cos 8 +
p2
- V2 sin2 8.
(25) (26)
Here B is the velocity of the particle in the c.m. system. Upon differ-
826
APPENDIX
2
entiating both sides of Eq. (24) with respect to cos 8, one obtains
an - 11 + p cos el _ ail - y v 2 A a ~ 2 Hence from Eq. (23) ,TI
=
+
yv2Aa’211 p cos
el-’.
Equation (28) gives J 1 in terms of the center-of-mass quantities fi and However, in some applications, it is more convenient to obtain Jl from the velocity v and angle 8 of the particle in the laboratory system. In order to derive the expression for J 1 in terms of v and 8, we define the quantities p and A as follows:
e.
p = v/v A = 1 - 2p cos 8
+ p 2 - V2 sin2 8.
(29) (30)
From Eqs. (6)-(€9, one obtains cos
e = A - ~ ~ ~ 8c o-sp ) .
(31)
Upon expressing 8 in terms of v and e, one finds
p = P~--1/2(i- vvcos
el.
Substitution of Eqs. (31) and (32) in Eq. (26) for
A
(32)
A gives
=I/(~~~A).
(33)
Finally, upon inserting Eqs. (31)-(33) into Eq. (28), one obtains
J~ = y v - 2 ~ - 1 q i - cos 81-1
(34)
which gives J 1 in terms of the laboratory quantities v and e for the case that only two particles are present in the final state. We note that in Eq. (27), we have evaluateddOldil, and have then taken its reciprocal to obtain J1. This derivative is particularly simple, since the energy E does not depend on the angle 8 in the c.m. system. We could also have obtained dil/dQ = J1 directly. However, since E is a function of 0 in the laboratory system, dil/dQ is given by
where ail/an and ail/aE are the partial derivatives for fixed E and e, respectively. Equations (28) and (34) give the relation between the cross section du/dQ for a two-particle process in the laboratory system and the cor-
APPENDIX
2
827
responding cross section du/dn in the c.m. system [see Eq. (21)]. The equations for J 1 can also be used for the two-body decay of unstable strange particles ( K mesons or hyperons). JI gives the relative number of decays per unit solid angle in the laboratory system, assuming isotropic decay in the rest system of the particle. In this case, 6 is the laboratory angle of one of the decay products with respect to the direction of the primary particle. For the case that more than two particles are present in the final state, J is given by the following determinant I
ail
aQ I
As for the two-body case, it is convenient to obtain J Z by evaluating its reciprocal:
1 an
m/aQ is given by Eq.
an I
(27). For the other derivatives, one obtains from
Eqs. (19) and (24) :
Upon inserting Eqs. (27) and (38)-(40) into Eq. (37), one obtains
which gives J z in terms of the center-of-mass quantities Z, and 8. From Eqs. (18) and (20), one can also show that Eq. (41) is equivalent to:
J z = sin 8/sin e
=
p/p.
(42)
The result sin J/sin fl has been given by Marshak.’ In order to obtain J Z in terms of the laboratory quantities v and 8, we substitute Eq. (33) into 1
R. E. Marshak, “Meson Physics,” p. 300. McGraw-Hill, New York, 1952.
828
APPENDIX
2
Eq. (41), getting J2
= yy1A-1/2.
(43)
Equations (41)-(43) give the relation between the differential cross section in the laboratory system d2a/dE dQ and the corresponding cross section in the c.m. system, d 2 a / d 8 d n [see Eq. (22)J.The equations for JZcan also be used to obtain the laboratory angle and energy distribution of secondary particles arising from the three-body decay of an unstable particle (e.g., K meson). Thus if d 2 n / d 8 d n is the probability of decay per unit energy and angle in the rest system for one of the decay products, the corresponding distribution in the laboratory system d2n/dE dQ is given by 2n d2n -d- J2-e (44) dE dQ dE dQ I n particular, for a n isotropic distribution in the rest system of the unstable particle, d 2 n / d 8 d o is independent of the angle t7 and is a function of 8 only. We summarize the preceding results: the Jacobian J is given by Eqs. (28) and (34) for the case of two particles in the final state, and by Eqs. (41)-(43) for the case of more than two particles. In these equations, p , 6, p, and A are defined by Eqs. (25), (26), (29), and (30), respectively. 2.2. Maximum Angle Criterion for the Identification of Outcoming Particles in Fundamental Collisions
For nucleon-nucleon and pion-nucleon collisions in which mesons are produced, it can be shown2 that the angle e of the recoil nucleon cannot exceed a maximum value emwhich depends on the energy of the incident nucleon or pion and on the number of mesons produced. Thus if the angle 0 for a particle observed, for example, in a cloud chamber or bubble chamber, is larger than the value of Om for the recoil nucleon, i t can be concluded that the particle cannot be a nucleon and must therefore be a pion or a K meson. There is a maximum possible angle 0, for any emitted particle (e.g., for pions) at energies sufficiently near threshold. However, for pions, em becomes 180" a t an energy not far above threshold, when the c.m. velocity ij of the emitted pion exceeds the velocity V of the c.m. system with respect to the laboratory system. In order to obtain the expression for Oml we note that the c.m. velocity tl of the particle considered will have its maximum value when the other particles (2, 3, . . . n) all move with the same velocity2 and in the direction opposite to Thus particles 2, 3, . . . n are equivalent R. M. Sternheimer, Phys. Rev. 93, 642 (1954).
2
829
irni
(45 1
APPENDIX
to a single particle of mass M , given by
M =
i=2
where mi is the mass of the ith final particle. From the definition of the c.m. system, the momentum of M is -plm, where plm is the (maximum) momentum of 1. Thus plmcan be obtained from the equation:
(pi,
+ m12)1/2+ (pl, + M2)'/2
= 2 0
(46)
where EOis the total energy in the c.m. system. Upon solving Eq. (46) for plm,one finds plm = B'/2/(2E') (47) where B is defined b y
+
B = [(b, M ) 2 -
rn12][(Eo -
- m12].
(48)
Hence the maximum possible velocity D1, of particle 1 is given b y
nl,
=
+
[~/(4~3'2rn~2~ ) ] 1 / 2 .
(49)
From Eq. (20), one finds for the laboratory angle O1 of the particle:
el
where is the angle of emission of the particle in the c.m. system. The maximum value of el is obtained from the condition
d tan Ol/d&
=
0
which gives cos
el
=
-nlm/V.
Upon inserting Eq. (52) into Eq. (50), one obtains the maximum angle
elm: where nlm is given b y Eq. (49). Figures 2 and 3 show the curves of elmfor recoil nucleons from nucleonnucleon and pion-nucleon collisions, when n = 1, 2, or 3 pions are produced. The curves are plotted as a function of the kinetic energy of the incident nucleon or pion in the laboratory system. For nucleon-nucleon collisions, Oln, increases rather slowly with the incident nucleon energy
830
APPENDIX
2 I
I
-
-
-
I
0
0.5
I .o I.5 2 .o 2.5 INCIDENT NUCLEON’ENERGY T~ (IN Bevl
I
3.0
FIQ.2. Maximum possible laboratory angle el,,, of the recoil nucleon in a nucleonnucleon collision in which n ( = 1, 2, or 3) pions are produced, as a function of the laboratory kinetic energy T N of the incident nucleon.
APPENDIX
2
83 1
TN beyond 3 Bev. Thus at TN = 6.0 Bev, BIm is 77.2" for n = 1, 66.0" for n = 3, and 56.8" for production of n = 5 pions. It may be noted that Eq. (53) is completely general, and can also be used, for example, to obtain the maximum angle of hyperons or K mesons in associated production. The existence of the maximum velocity 81, implies that the laboratory momentum p l of the particle cannot exceed a certain maximum value (to be denoted by pl,) which is a function of el. In order to obtain plm,we note that by virtue of Eq. (8) we have Elm
+
= ~v[(pL,
- V p l m cos ell
m12)1/2
(54)
where Elmis the maximum energy in the c.m. system. From Eq. (49) one finds that 81, is given by 8 1 ,
=
+
( 4 8 , ~ r n ~ 2 B)1/2/(2170).
(55)
Upon solving Eq. (54) for PI,, one obtains
Thus if the measured momentum of a particle exceeds the value of plmfor recoil nucleons and K mesons at the angle el, the particle can be identified with certainty as a pion. Equation (56) can also be used to obtain the laboratory momentum of a particle in a two-body reaction as a function of the laboratory angle 81. In this case, Elmis the c.m. total energy of the particle as obtained from Eqs. (48) and (55). However, in Eq. (48), M is now the mass of the other particle resulting from the collision (i.e., A4 = m2),ml is the mass of the particle considered, and Bo is the total energy in the c.m. system.
2.3. Kinematics of Two-Body Decay For the study of the two-body decay of unstable strange particles, it is useful t o obtain the distribution of the energies El and E2 of the decay products in the laboratory system. We have
El
=
+
~ ~ ( 8T'P1 1 cos $1)
(57)
where El and pl are the total energy and momentum of the decay particle (secondary) in the rest system of the primary; is the angle of emission in the rest system, T' is the velocity of the primary, and y v = (1 - V2)-1/2. [The rest system is that in which the decaying particle (primary) is at
el
832
APPENDIX
2
rest. This coordinate system is completely analogous to the center-of-mass system considered previously. Thus the total momentum of the decay products in the rest system is zero, in similarity to the c.m. system for LL particle reaction.] The maximum and minimum values of the laboratory total energy El are given by
The number of decays dn/dEl per unit energy interval dE1 is dn __---. -
dE1
dn d cos
d cos Sl dE1
el
(59)
On the assumption that the decay of the primary particle is isotropic in its rest system, dn/dcos& is constant (= &). Furthermore, from Eq. (57), one finds
One thus obtains
Hence the energy distribution dn/dEl is constant (i.e., independent of E l ) between E1,min and The distribution of the angle a between the two decay particles in the laboratory system has been obtained by Sternheimer.s One finds for the relative number of decays per interval da:
where C is defined as:
C = +(mo2- mI2 - m2 ).
(63)
Here mo and po are the mass and momentum of the decaying particle, mi and E; are the mass and laboratory total energy of the decay products (i = 1, 2); p1 is the momentum of either particle in the rest system. It may be noted that a is related to El and E 2 by the equation: cos a
=
ElE2 - C (E l 2- m12)1/z(E22 - m22)1/2
R. M. Sternheimer, Phys. Rev. 98, 205 (1955).
(64)
APPENDIX
833
2
The application of the kinematics of two-body decay t o the interpretation of cloud chamber pictures of the V particles ( K mesons and hyperons) has been described by Bridge4 and by Thompson.6 The relation between the energy and angular distribution of primaries produced in a reaction and the energy and angular distribution of secondaries resulting from a two-body decay of the primaries has been discussed by Sternheimer.6
2.4. Threshold Energies for Associated Production of Unstable Particles In this sect,ion, we obtain the energy thresholds for associated production of unstable particles in nucleon-nucleon, pion-nucleon, and y-nucleon collisions. The following basic reactions will be mainly considered :
+ + + + + + +
N + N-l Y K + N N N - Kf K2N ?r+N+Y+K ?r N-+ K f KN y + N - + Y + K y N-+ K f KN
+ + +
where N = nucleon, Y = hyperon. We will consider both the case where the target nucleon is at rest and the case where it moves towards the incident particle with a kinetic energy of 25 MeV. As explained below, the latter assumption gives an approximate effective threshold for associated production in a collision with a nucleus. The laboratory kinetic energies of the initial particles will be called T , and Tb; their masses are denoted by ma and mb. I n order t o obtain the expression for the threshold value of T,, we write the equations for th e total energy Eo and momentum pa in the laboratory system:
+
Eo = ma -k mb T , 4-Tb P O = (Ta2 2m,T,)ll2 - (Ti,’
+
+ 2mbTb)’‘2.
(65) (66)
In Eq. (66), it has been assumed that the target particle (usually a nucleon) moves backward in the laboratory system, i.e., towards the incident particle. The required total energy Bo in the c.m. system, which is determined by the combined mass of the reaction products, is given by 3 0
= Eo(1 - V2)”’
(67)
4 H. S. Bridge, in “Progress in Cosmic Ray Physics” (J. G. Wilson, ed.), Vol. 111, Chap. 11, pp. 209-218. Interscience, New York, 1956. 5R. W. Thompson, in “Progress in Cosmic Ray Physics” (J. G. Wilson, ed.), Vol. 111, Chap. 111, pp. 266-280. Interscience, New York, 1956. 8 R. M. Sternheimer, Phys. Rev. 99, 277 (1955).
834
APPENDIX
2
where V = PO/&. From Eqs. (65)-(67), one obtains E02
-I- 2[m,Tt, -I- mbTa -I- TaTb -k ( T G 2 2maTa)"2(Tb2 2mbTb)1/2].(68)
= (ma
Upon solving Eq. (68) for T,, one finds
[GEb - 2mamb2- pb(G2- 4ma2mb2)1/2]/(2mb2) (69) where G is defined by G = Eo2 - ma2- mb2.
T,
=
In Eq. (69), p b and Eb are the momentum and total energy of the target particle. For a target particle at rest, Eq. (69) reduces to the well-known formula :' T , = [Eoz- (ma mb)21/(2mb). (71)
+
In both (70) and (71), 80is, of course, equal to the sum of the masses of the final reaction products of (1)-(VI). Thresholds for unstable particle production by both nucleons and pions have been previously obtained.8 We have recalculated these results, using somewhat more recent mass values for the various particles. The values used weregJO(in Bev/c2); r:0.1395;K:0.4935;N:0.938;A': 1.115; 2+:1.189; 2-:1.197; %-:1.321; and 2:1.615, where 2 is the hyperon observed by Eisenberg," with Q value of -6 MeV. It has been assumed that this particle decays into a A' and a K meson and that it has strangeness12 S = - 3 . The results are given in Table I. This table is divided into 3 parts. Parts A, B, and C give the thresholds for production by nucleons, pions, and y-rays, respectively, for the reactions listed. In each case, we give two results: ( 1 ) for target nucleon at rest (Tb = 0 ) ;(2) for target nucleon of 25 Mev moving towards the incident particle (Tb = 25 Mev). The reason for choosing Tb = 25 Mev is that the nucleons in the nucleus have kinetic energies of the order of 25 Mev or less. The maximum energy is ~ 2 Mev 5 and is often referred to as the Fermi energy. For a target 7 See, for example, R. E. Marshak, "Meson Physics," p. 73. McGraw-Hill, New York, 1952. 8 R. M. Sternheimer and D. H. Wilkinson, Brookhaven Cosmotron Internal Report RMS-DHW 1 (1955). 9 A. M. Shapiro, Revs. Modern Phy8. 28, 164 (1956). 'OR.Budde, M. Chretien, J. Leitner, N. P. Samios, M. Schwartz, and J. Steinberger, Phys. Rev. 108, 1827 (1956). 11Y. Eisenberg, Phys. Rev. 96, 541 (1954). See also W. F. Fry, J. Schneps, and M. S. Swami, ibid. 97, 1189 (1955). I* M. Gell-Mann, Phys. Rev. 99, 833 (1953); T. Nakano and K. Nishijima, Progr. Theoret. Phys. (Kyoto) 10, 581 (1963); A. Pais, Phy8iCa 19, 869 (1953).
APPENDIX
835
2
nucleon with the maximum energy (-25 Mev) and moving towards the incident particle, the threshold for the reaction has the lowest value. It is seen that the thresholds for Tb = 25 Mev are markedly lower than those for Tb = 0, so that the thresholds for production in a nucleus are appreciably decreased as compared to production on a free proton (liquid hydrogen target). TABLE I. Thresholds for Unstable Particle Production (inBev) Reaction (A) N + N - + A " + K + N N -+ Z+ K N N
+
N N N N
+ + + N Z- + K + N +N-+ K + + K - + 2 N + N - + E + 2K + N 4-N - + Z + 3 K + N -+
(B) H + N - + A " a N - + Zf a N 4 2H N + K+ H N -+ E: H N-+ Z
+ + + + +
+K
+K +K + K- + N
+ 2K + 3K
(0 Y + N + A " + K
+N + y +N
+K 2- + K y + N - + K+ + K - + N y
Z+
+
TS= 0
Tb
=
25 Mev
1.581 1.784 1.807 2.493 3.740 6.796
1.105 1.263 1.281 1.818 2.800 5.218
0.760 0.890 0.904 1.356 2.221 4.489
0.578 0.680 0.692 1.050 1.736 3.537
0.910 1.040 1.054 1.506
0.723 0.826 0.837 1.196
The strange particles are generally produced by means of incident nucleons or pions. However, recently both at the Cornell and Cal. Tech. electron synchrotrons, K particles have been made by means of an energetic ?-ray beam (bremsstrahlung from the electron beam). The thresholds for production by incident y-rays are listed in part C of Table I.
EXAMPLES: As an application of the equations discussed above, we consider the reaction : p p - + x+ K+ n (VW
+
+ +
for incident protons of kinetic energy T , = 3.0 Bev. For the kinetic energy of the target proton, we assume the two possibilities considered above: (1) Tb = 0 ; (2) Tb = 25 Mev (moving backward in the laboratory system). We want t o obtain the maximum possible c.m. velocity iiz,,,, of
836
APPENDIX
2
the 2+ particle, and the corresponding laboratory energy of the 2+, if it has been emitted with the maximum velocity at c.m. angles = 0' and 60'. In the latter case, we also obtain the laboratory angle e2 of the Z+. Table I1 gives the values of the various kinematic quantities. The quantities in the laboratory system pertaining to 82 = 0" and 8, = 60" are labeled by the superscripts (1) and (a), respectively. First, the total momentum p o and the total energy Eo are obtained, together with the TABLE 11. Kinematic Quantities for the Reaction p
+p
3
Z+
+ K+ + n
for 3.0-Bev Incident Protons
PO (Bev/c) E O (Bev)
v
?"
E D(Bev) B (Bev4) o2.m
Ex,.,, (Bev) P Z , ~(Bev/c)
E',L' (Bev) E g ) (Bev) P!?~ (Bev/c) p!$,'f (Bev/c)
$g)
ezSm
3.825 4.876 0.7845 1.612 3.025 20.69 0.5344 1.407 0.752 3.219 2.744 2.385 0.651 15.3" 30.0"
3.607 4.901 0.7360 1.477 3.318 45.27 0.6488 1.563 1.014 3.411 2.860 2.448 0.878 19.7" 51.7'
velocity ' I of the c.m. system and the corresponding value of yv [Eqs. (1)-(5)]. From these results, one can deduce the total energy Eo in the c.m. system. For the 2+ to have its maximum c.m. velocity B Z , ~ ,it is necessary that the K+ particle and neutron move with the same velocity and in the same direction (opposite to tz,,). The sum of the masses of K+ and n is: M = 1.433 Bev. Upon inserting this value in Eq. (48), one , Ex,,,, from Eqs. (47), (49), and obtains B , and hence f i ~ , ~P, Z . ~and (55). If the 2+ is produced at & = 0", the laboratory total energy is EL1) = 3.219 Bev for Tb = 0 and E$) = 3.411 Bev for Tb = 25 Mev (corresponding to kinetic energies T$' = 2.030 Bev and 2.222 Bev, respectively). For the case 82 = 60', we have obtained EL2) together with the components of the momentum &,: and p;:: from Eqs. (17)-(19). From the values of and p;::, one deduces that the laboratory angle OL2' is 15.3' for Tb = 0 and 19.7' for T b = 25 MeV. The last row of Table I1 gives the maximum possible laboratory angle [see Eq. (53)].
837
2
APPENDIX
As a second example, we have considered the reaction T-
+p
4
A'
+ K"
(VIII)
at an incident pion kinetic energy of 1.0 Bev. It is assumed that the K O meson is produced at a c.m. angle eR = 60°, and we wish to obtain the energy and angle of the K" in the laboratory system, and the Jacobian J K at this angle. The intermediate quantities and the final results are listed in Table 111, both for Tb = 0 and Tb = 25 MeV. We note that the TABLE 111. Kinematic Quantities for K" Mesons Produced at e ' ~= 60° by the Reaction Tp -+ A" K" for 1.0-Bev Incident Pions
+
Quantity
PO (Bev/c) Eo (Bev) V 7-v
Eo (Bev) B (Bev4)
fi3 E K (Bev) PK
(Bev/c)
E K (Bev) P K . ~(Bev/c)
PK,, (Bev/c) OK
p_K AK JK
+
Ta
=0
1.131 2.078 0.5443 1.192 1.743 1.196 0.5365 0.585 0.314 0.799 0.567 0.272 25.6" I .015 2.823 4.471
Tb = 25 Mev 0.913 2.103 0.4341 1.110 1.895 3.217 0.6922 0.684 0.473 0.873 0.592 0.410 34.7" 0.627 1.879 2.416
c.m. velocity D K can be obtained by means of Eq. (48)for B, in which m l = VZK = 0.4935 Bev and M = m~ = 1.115 Bev. The values of E K ,OR, and J E in Table I11 pertain to 8 K = 60". An extensive discussion of the kinematics of A meson production by incident nucleons and y-rays (photoproduction) on nucleons has been given by Marshak.13 The effect of the Fermi motion on the threshold for pion production in nuclei14 and the absolute threshold for pion production16 are also discussed in Marshak's book.' Recently, LeipuneP has calculated extensive tables of the kinematics of two-body reactions and two-body decays. The two-body decays conR. E. Marshak, "Meson Physics," pp. 3 and 41. McGraw-Hill, New York, 1952. W. McMillan and E. Teller, Phys. Rev. 73, 1 (1947). 16 W. H. Barkas, Phys. Rev. 76, 1109 (1949). l6 L. B. Leipuner, "Relativistic Two-Body Kinematics," Brookhaven Cosmotron Internal Report LBL-2 (1958). l3
838
APPENDIX
2
+
+
sidered in this work include the following: 0' + n+ n-; A' --t p n-; 2- + n r - ; Z0 + ha y ; and Z+ Ao n-. Among the 14 two-body reactions investigated, five are of the type of reactions (111) and (V) p 4A' K + ; T+ p 3 2+ Kf. Four additional above, e.g., y reactions involve the O2 particle (a long-lived species of the neutral K meson which can exhibit strangeness12 S = - l), e.g., O2 p A' p; 02 p + E" Kf. Leipuner has also carried out calculations for the elastic K nucleon scattering. The remaining 4 reactions do not involve the strange particles. These reactions are: y p 4p r'; y p 4n n+; r* p + ~ * p ; and p p + d +A+. For each case, for a given incident laboratory energy (for a reaction) or kinetic energy of the parent particle (for a two-body decay), Leipuner's tables" give the following quantities as a function of the c.m. angle of the heavier of the two outgoing particles: the kinetic energies TI,Tz,the momenta p l , p 2 , the velocities vl, v2, and the angles el, 02 of the two outgoing particles (all quantities in the laboratory system). Moreover, the tables list the c.m. momentum PI ( = p z ) of either secondary particle, the velocity V of the c.m. system, the corresponding yv = (1 - V2)-lI2, and the c.m.'velocities gl and g2 of the two secondary (outgoing) particles. For the exothermic reactions involving a threshold, the threshold energy and threshold momentum are also listed. Far the two-body reactions, the tables generally extend from incident energy T, = 50 Mev (or from threshold) to T, = 2.5 Bev, at intervals of 50 MeV. For the particle decays, the range of the calculations extends from p = 50 Mev/c (momentum of the parent particle) to p = 2.5 Bev/c, at intervals of 50 Mev/c. These tables were calculated on an IBM-704 Computer.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
el
2.5. Phase Space Factors for Two- and Three-Particle Final States It may be convenient to list here the expressions for the relativistic two-body and three-body phase space factors. These phase space factors F2 and F J are related to the density of final states per unit total energy 8, in the c.m. system. F1 and F, have the significance that if the matrix element for the formation of the particles in the final state were constant (i.e., independent of energy and angle), then F 2 and Fs would give the relative probability of various final states. For more than 3 particles in the final state, it has not been possible t o obtain an analytic expression for the phase space factor." The expression for Fz is given by
17
M. M. Block, Phy.9. Reu. 101, 796 (1956); and private communication.
APPENDIX
2
839
where El and Bz are the c.m. total energies of the two particles, 30is the total energy in the c.m. system (3, = 8, BZ),and p1 is the c.m. momentum of either particle. In order t o derive Eq. (72), we note that F 2 is the density of final states (p) per unit solid angle (in the c.m. system), and is given by
+
We have
so that
Upon inserting Eq. (75) into Eq. (73), one obtains Eq. (72) for F P . The relativistic three-body phase space factor F a has been obtained by Block,” and is given by
where D is defined by
Here pl, PI, B,, and d o l are, respectively, the c.m. momentum, kinetic energy, total energy, and solid angle of the particle considered; m2 and ma are the masses of the other two particles, 8 0 is the total energy in the c.m. system. The following derivation of Eq. (76) has been obtained by Block.“ It follows essentially the same lines as that given in reference 1 for the nonrelativistic case.lS The density ‘of final states is
where pz and dQz are the c.m. momentum and solid angle of particle 2. The c.m. momentum and total energy of particle 3 will be denoted by R. E. Marshak, “Meson Physics,” p. 81. McGraw-Hill, New York, 1952.
840
APPENDIX
2
p 3 and 8,, respectively. We have
+
p2
-pl
Upon taking the direction of
pa2 =
p3
=
(79)
-p1.
as the polar axis, we can write
- 152 cos
&)z
( ~ 1
+
~
2
sin2 2 ez
(80)
where ez is the angle between pz and -pl. Differentiation of both sides of Eq. (80) gives Padp3 = - ? j l p z d ( C O S 82). . (81) For fixed p1 and p2, the differential d 8 0 in the denominator of Eq. (78) is equal to dEa, which is given by
d83
dpa/E3.
=
(82)
From Eqs. (81) and (82), one obtains d(cos
82)
-
d8o
8 , can be written as 8 3
=
--. 8
3
p1pz
w - (m22+
p22)1/2
where W is the energy available to particles 2 and 3:
w=
8 0
- 8,.
(85)
Upon inserting Eqs. (83) and (84) into Eq. (78) one obtains
dp
=
27rp1dpl d%
/
Pl,l&¶X
P9.min
[W - (m2
+
p22)112]lj2
dpz
(86)
where the factor 2a comes from the integration over the azimuthal angle of particle 2; and p 2 , m i n are the maximum and minimum values of p2, which are obtained for & = 0’ and 180°, respectively. The corresponding values of p 3 are p2 - p1 and p 2 p1, respectively. Thus Ij 2,max and p2,min are obtained by solving the equation
+
where D is given by (77). In Eq. (86), we note that p1 dpl thus obtains
=
8, d 8 1 . One
APPENDIX
where
and have
E2,mox
p2,min.We
E2,min are the values
2
of
8,corresponding
Upon inserting Eqs. (88) and (90) into Eq. (89), one obtains Eq. (76) for the three-body phase space factor. For a reaction with 3 final particles, in which the matrix element for the formation of particle 1 is Hl(pI, gl) (after integration over the variables pertaining t o particles 2 and 3), the center-of-mass differential cross section is given b y d2a 27r IH1\2Fs (91) dF1da, ft uv where ijl is the relative velocity of the initial particles in the c.m. system. The total phase space available t o the three particles, analogous to 47rFz for the two-body case, is given by
where is the maximum possible c.m. kinetic energy of particle 1 [see Eq. (55)]. Examples : As an illustration of the three-body phase space factor F S , we have considered three examples, which are presented in Figs. 4-6. Figure 4 shows the phase space factor F for the K+ mesons from the reactions: p p
+ p - - t Ao + K+ + p + p + Z+ + K+ + n
for a n incident proton kinetic energy T , = 3.0 Bev, and a target proton at rest in the laboratory system. In this case, in Eq. (76), m2 and m3 are the masses of the A' and proton for reaction (IX), and the Zf and neutron masses for reaction (X). F is plotted against the c.m. kinetic energy FK of the K meson. If the matrix element for the K meson production were constant as a function of PK,then F would give directly the energy distribution of the K mesons in the c.m. system. I n Fig. 4, and also in Figs. 5 and 6, the phase space factor has been normalized as follows:
/o''"FdP
=
1
(93)
842
APPENDIX
2
C.MS. ENERGY OF K MESON ( I N MeV) FIG.4. The relativistic three-body phase space factor F for the K+ meson from the
+
+
+
+
+
+
p -+ A" K + p and p p --t 2+ K+ n,for an incident proton reactions: p kinetic energy T, = 3.0 Bev and a target proton at rest in the laboratory system. F is plotted against the c.m. kinetic energy of the K + meson.
9
I
I
I
I
I
I
I
I
LL1 0I0
I
I
1
F (N)
50
100
150
200 250 CMS. ENERGY OF NUCLEON OR PION ( I N MeV)
300
FIG.5. The relativistic three-body phase space factor F for the pion and the nucleons p .+ p n ?r+, for an incident proton kinetic energy from the reaction: p T p = 1.0 Bev and a target proton a t rest in the laboratory system. The abscissa represents the kinetic energy of the pion or nucleon in the c.m. system.
+
+ +
APPENDIX
2
843
where f',,,is the maximum possible value of the c.m. kinetic energy T', and is expressed in Bev. Figure 5 shows the phase space factor F for the pion and the nucleons from the reaction: p p -+ p 7E n+ (XI)
+
+ +
for an incident proton kinetic energy T, = 1.0 Bev and a target proton at rest in the laboratory system. The pion phase space factor F(r+) is obtained from Eq. (76)by taking m2 = m s = mN, whereas the nucleon phase space factor F ( N ) is obtained by taking m2 = mN and m3 = m,,
ENERGY OF p MESON OR ELECTRON ( I N MeV)
FIG.6. The relativistic three-body phase space factor F for the p meson from the K$ decay, and the electron from the K,f, decay. The abscissa represents t h e kinetic energy of the p meson or electron in the rest system of the K particle.
where mN and m, are the nucIeon and the pion mass, respectively. It may be noted that according t o the Fermi statistical theory of pion production,l@the energy distribution of the pions and nucleons is given by the corresponding phase space factors F(T+) and F ( N ) , the assumption being made that the matrix element for pion production is constant aa a function of the pion and nucleon kinetic energies. Figure 6 shows the phase space factors for the p meson from the K:* decay : K:3-+p*+no+ v (XI11 E. Fermi, Progr. Theoret. Phya. (Kyoto) 6, 570 (1950); Phy8. Rev. 92, 452 (1953); SS, 1435 (1954). l9
844
APPENDIX
2
and for the electron from the K t 3 decay:
K:3
-+
e*
+ + 7r0
V.
(XIII)
The K:3 and K:3 are relatively infrequent modes of decay of the K meson, which generally decays into two particles. (The most frequent modes of decay are: Ks2 + 2 ~ and , K$ -+ ,u* F.) For K:3 and K:3J we have: BO= mKJm2 = mz, ma = 0 in Eq. (76)) where mg and mr0are the K and T O masses, respectively. The phase space factor F is shown as a function of the p meson or electron kinetic energy in the rest system of the K particle. Phase space calculations with applications to Fermi's statistical theory of meson produ~tion'~ have been carried out by several a ~ t h o r s . ' 7 , ~ ~ - ~ 6
+
ACBNOWLED~MENT
I wish to thank Dr. Luke C. L. Yuan for several helpful suggestions concerning this chapter. C. N. Yang and R. Christian, Brookhaven Cosmotron Internal Report (1953). J. V. Lepore and R. N. Stuart, Phys. Rev. 94, 1724 (1954). zz R. H. Milburn, Revs. Modern Phys. 27, 1 (1955). * 3 J. V. Lepore and M. Neuman, Phys. Rev. 98, 1484 (1955). 24 M. M. Block, E. M. Harth, and R. M. Sternheimer, Phys. Reu. 100, 324 (1955). 26 G. E. Fialho, Phys. Rev. 106, 328 (1957). 41
Appendix 3. Properties of Elementary Particles and Particle Resonance States TABLE I. Intrinsic Properties of Elementary Particles@*b ~
Class Photon Leptons
AntiSpin Isotopic Particle particle (unitsh) spin Z
I,
Strangeness S
Mass
K+
Mass (Mev) 0 0 0 0.510976 4 0.000007 105.655 0.010 135.00 f 0 . 0 5 139.59 f 0.05 493.9 -1- 0.2
0 0 0 1 206.77 264.20 273.18 966.6
KO
497.8 4 0 . 6
974.2
Nucleons
p n
Hyperons
Ao
938.213 k 0.01 939.507 f 0.01 1115.36 k 0.14
1836.12 1838.65 2182.80
Stable (1.013 f 0.029) X 103 (2.51 k 0.09) X 10-10
Zf
1189.4
0.2
2327.7
(0.81 +0.06) -0.05
zo
1191.5 k 0 . 5
2331.8
<0.1
z-
1196.0 f 0 . 3
2340.6
(1.61 +o'lo) -0.09
ZO
1311 f 8
2566
-1.5
z-
1318.4 f 1 . 2
2580.2
(1.3:;)
Y V. VF-
eP-
Mesons
ro K+
(me)
Mean life (sec) Stable Stablec Stablec Stable (2.212 & 0.002) x 10-6 (1.9 rt 0.6) X 10-18 (2.55 -1- 0.03) x 10-8 (1.224 f 0.013) x 10-8 (KiO:(1.00 -1- 0.04) x 10-10 \KP: (6.1 +1.6 -l.l) x
x
x
10-10
10-10 X 1O-lo
X
x
10-lo
~~
a From W. H. Barkas and A. H. Rosenfeld, Proc. Ann. Rochester Conf. High-Energy Nuclear Phys. 10,877 (1960); G. A. Snow and M. M. Shapiro, Revs. Modern Phys. 33, 231 (1961). * For I . and S, the first and second numbers in parentheses pertain, respectively, to the particle and the antiparticle listed onthe same line. e G. Danby, J. M. Gaillard, K. Goulianos, L. M. Lederman, N. Mistry, M. Schwartz, and J. Steinberger, Phys. Reu. Letters 9,36 (1962).
846
APPENDIX
3
TABLE 11. Decay Modes, Branching Ratios, and Q Values for Mesons and Hyperons0
Particle
Decay modes P+
P+
a+
K++
+
Y
+ ~(Kfi2) + r'(Kr2)
+ *++ + a' + e+ + + a 0 ( K e g ) + + rD(Kfis) a+ + aa+
T-(T)
a+
*'(TI)
Y
p+
Y
+ + aT *+ + a- + a' + + e*
?yo
z++
0
b
{Y,F]
a0
a0
&(MeV) Branching ratio (%) 33.93 139.08 135.00 133.98 388.2 219.3 75.1 84.3 358.4 253.2 218.6 227.8 357.7
99.99 (1.21 f 0.07) -99 -1 58 f 2 25.6 f 2 5.7 f 0.3 1.7 f 0.3 5.1 f 0.8 3.9 f 0.5 69 f 3 31 f 3
357.7
48.4 f 5b
83.6 92.8
x
10-2
13.4 f 1.8 ?
37.56 64 f 3 40.85 36 k 3 0.10 f 0.03" 176'64 0.10 f 0.03 116.2 50 f 2 110.3 50 f 2 251.2 -0.5 76.1 100 116.9 100 63.4 -100 60.6 -100
From G.A. Snow and M. M. Shapiro, Revs. Modern Phys. 33, 231 (1961). D. Luers, I. S. Miltra, W. J. Willis, and S. 8.Yamamoto, Phys. Rev.Letters 7 , 255
(1961). c R. P. Ely et al., Proceedings of the 1962 International Conference on High-Energy Physics at CERN, p. 445.
TABLE 111. Properties of Particle Resonance Statesa Class Nucleon isobars (S = O,B = 1)
Pion resonances ( S = 0 , B = 0)
State N3
*
N1* NT* Nj* Nj** 7
Mass Width r (Mev) (Mev) 1220 1510 1690 1670 1900
200
550
<12
Decay reaction
Q
Na*+N+r
90
N:+
N +r N:-+ N + 2 r
?
Ispin Spin and parity
~]-+z++r-+rO
++ 8 0
I
Q+
I-
$(?)
? ?
Q
Production processes
O-(G=+l)
+N-,
+N + r + N +2?r
r++n+p+t~
(25%) or I]
+ neutrals
1-(G
=
-1)
K- + p +
1-(G
=
-1)
p p
(75%) W
P
785
750
<20
~130
w + A+
p-+
+ r-
2r
+TO
0
1
1-(G = +1)
A
T+
+
p0
?r-
+
885
Hyperon isobars ( S = -1,B = 1)
40
? ?
K*+K+r
K*-+K+?r
Y1*
1385
~ 5 0 Y1*4 A + r YI*-+Z + r (<5%)
1
(8 + ?)
r-
K-
P-
P+
+
K*(E*)
7L
+p +P + P-+ + P P +P + nr
T+
nances [S = + l ( - l ) , B = 01
+7
+ p + w + + r+ p w + 2r+ + 2rr++n+o+p +p + +
a-
K ( K ) meson reso-
{ ++ NN ++r 2
N;,
+p+ +p
P
K*
4
+ z:
K*- + p
K-+p+Yi*+r KzO+ p + y:-* + r K-+d+Y, +P rp + Y1* K
+
+
~
TABLE 111. Properties of Particle Resonance States. (Continued) Class Hyperon isobars (S= -1,B=l) (Continued)
State
Yo* Y,**
1405 1520
Decay reaction
~ 4 0 Yo*+Z + r 16 Y,**-+ff+N(+)
Y,**+ z + r (9) Y:* 4 A 2r (4)
+
(Y2*) Y,*** Hyperon isobarb (S = -2, B = 1)
Mass Width 1’ (Mev) (Mev)
$*
(1550) 1815 1530
? >120
<35
-
-
E*-+ E
t r
I spin
Spin and parity
0 0
Q+
KK-
?
r-
?
Production processes
+p
4
+ p-+
Yo* .$ r+
+ r-
Y,**
0
(2) 0 +(?)
rg? ?
+ p-+ K + + Y:+ p Y:** R- + p + x*- + K+ K - + p--+ + KO K-
4
a From R. H. Dalitz, “Three Lecturee on Elementary Particle Resonances,” Brookhaven National Laboratory Report BNL-735 (T-264), pp. W 9 1 , December, 1961. For each class of resonances, the baryon number B and the strangeness S are listed in the first column of the table. For the pion resonances, the G parity quantum number is also given after the spin and parity. The fractions in parentheses listed after some of the decay reactions indicate the probable values of the corresponding branching ratios. For the nucleon and hyperon isobars, the corresponding antiparticle stat.es (anti-isobars) have not been listed. These antiparticle states have opposite values for B and S to the corresponding particle states shown in the table. bG. M. Pjerrou et al., Phys. Rev. Letters 9, 114 (1962); L. Bertanca et al., Phys. Rev. Letters 9, 180 (1962).
Author Index Numbers in parentheses are footnote numbers. They are inserted to indicate that the reference to an author’s work is cited with a footnote number and that his name does not appear on that page. Allred, J. C., 419, 423, 426, 427(106), 428, 451, 506 A l m h , O., 160 Aamodt, L. C., 98 Alseevsky, N. E., 732 Aamodt, R. L., 494, 501 Abragam, A., 106, 123, 124, 126(68), 127, Alvager, T., 332 142, 143(42), 145, 146(42), 147(42), Alvarea, L. W., 588, 635 148, 151, 171(6), 172(3), 173, 178, Amaldi, E., 449, 747 Amaldi, U., Jr., 260, 269(184), 272(184) 191, 192(64), 200 Abraham, M., 96, 102, 103(3, 15), 124, Amble, E., 98 Ambler, E., 164(5, 17, 24, 25, 32, 43), 165 181(6), 182 (43, 45), 166, 167, 168, 186, 188, 189 Adair, R. K., 368, 369(1), 374, 375, 383, (47), 190(47), 194, 200, 202, 203 385(1), 387(28), 388 (104), 204(104), 211(69), 212(69) Adams, G. D., 626 Ammar, R. G., 758 Adams, J. B., 665, 668 Anderson, A. C., 176 Aeppli, H., 142, 146 Anderson, C. D., 345 Aglintsev, K. K., 540 Anderson, G. A., 618 Agnew, H. M., 566, 567(35), 568(35) Anderson, H. L., 111, 556, 557(1), 560(1) Agnew, L. E., Jr., 747 Anderson, W. A., 122 Ahn, S. H., 453 Anderson, W. K., 603 Ahrens. L. H., 302 Andrew, E. R., 106 Aiello, W. P., 384 Angstrom, K., 283 Aitken, I).K., 649 Annis, M., 808, 809(3), 811 Ajzenberg, F., 354, 357(6) Ajzenberg-Selove, F., 352, 354, 359(1a), Apalin, V. F., 252 Appel, H., 269, 272(210) 365, 366(1a), 441 Arbman, E., 732 Alberghini, J. E., 254 Arfken, G. B., 134 Albers, J. R., 235, 238, 254 Argo, H. V., 441, 442(139) Albert, R. D., 392, 526 Alburger, D. E., 443, 446(147), 536, 578 Arley, N., 812, 814 Armenteros, R., 345, 752 (74, 75), 579 Alder, I<., 142, 143, 145, 146, 148, 208, Armstrong, A. H., 372,378(13), 427(106), 428, 442(13), 450, 451, 452 359 Alders-Schonberg, H., 142, 143, 145(41), Arnold, J. T., 111, 122 Arnold, W. R., 430 146(41), 148(41) Arm, R. G., 134, 135 Aldrich, L. T., 302 Alikhanov, A. I., 231, 233, 256(80, 88), Arthur, J. S., 431, 479(115) Arutyunyan, V. M., 251 257(88) Asaro, F., 153 Allan, D. L., 450 Ascoli, G., 219 Allen, A. J., 431, 479(115) Ashby, N., 228 Allen, B., 343, 371, 373(4) Aston, F. W., 1 Allen, R. C., 407, 474, 475(204) k t r o m , B., 284 Allen, W. D., 410 Attix, F. H., 539, 541(40) Allison, R. W., 584, 587(9) 849
A
850
AUTHOR INDEX
Avakyan, R. O., 228, 230(77) Aves, R. V., 561 Awschalom, M., 296 Axel, P., 333
B Bach, D. R., 434 Backenstoss, G., 244, 246(147) Backenstoss, G. K., 297 Backus, J., 583 Bacon, G. E., 607 Bacon, R. H., 320 Backstrom, G., 578(72), 579 Bainbridge, K. T., 1, 3, 4, 223, 228(52), 239(52), 277, 281, 339, 731 Bair, J. K., 375, 474 Baker, J. M., 179, 181(11, 16), 182 Baker, W. B., 558 Balabina, G. V., 732 Baldinger, E., 309, 311(6), 314(6), 319 (61, 475 Ball, W. P., 407 Bame, S. J., Jr., 441, 449 Band, I. M., 153, 161(7) Banerjee, H., 235, 238, 239(115) Barber, R. C., 4 Bardeen, J., 124, 126(68), 172(2), 173 Bargmann, V., 223, 225(59) Barkas, W. H., 747, 837, 845 Barker, E. F., 20 Barker, K. H., 345 Barnes. C. A., 262, 333 Barnes, C. B., 188 Barnes, R. G., 60 Baroni, G., 747 Barschall, H. H., 363, 371, 372(10), 373, 374, 375, 376, 377, 383, 387 (28), 388, 397(15), 403, 448(15), 449, 452, 474, 475 Bartholomew, R. M., 292 Bartlett, J., 251 Barton, M. Q., 624, 657, 663, 676(4) Bashandy, E., 318, 322(26) Bashkin, S., 374 Bastien, P., 758 Bateman, H., 765 Battat, M. E., 460 Bauer, R. W., 164(7, 12, 14), 166, 201, 202(101), 206(101), 262 Baxter, P., 37
Bay, Z., 309, 311(5), 321(5), 322(5), 323(5), 539, 540(38) Bayatyan, G. L., 228, 230(77) Bayhurst., B. P., 290, 573 Beach, L. A., 537 Beard, D. B., 272 Beattie, J. W., 541, 542(42) Beauchamp, R. K., 384 Becker, R. L., 254, 368, 369(1), 371, 372, 376, 385(1), 388(1) Beghian, L. E., 311 Beheian, L. E., 557 Behman, G. A., 580, 651 Behrend, H. J., 269 Bell, P. R., 157, 526 Bell, R. E., 137, 311, 313(17), 314(17), 322(12), 323(32), 336, 338(61), 408 Beltrametti, E. G., 260, 272 Benczer-Koller, N., 244, 246(142), 323 Bender, P. L., 122 Bender, R. S., 431, 433, 479(115) Bennett, W. H., 588 Benson, J. L., 6, 8 Bent, R. D., 443, 446(146, 148), 447(148) Berestechij, V. B., 254 Berger, D. W., 440 Berger, M. J., 445, 517 Berlin, T. H., 345, 656 Berman, A. I., 547 Bernardini, M., 260, 269(184), 272(184) Bernstein, S., 164(10, 36), 165(52), 166, 167, 169, 176(20) Berthelot, A., 296 Bethe, H. A., 235, 258,339,353,363,366, 381, 395, 399, 400(80), 401(80), 402(80), 403(80), 407 Beun, J. A., 164(15), 166, 201 Beyster, J. R., 399, 400(80), 401(80), 402(80), 403(80), 407, 474 Bhalla, C. P., 210 Bhanot, V. B., 8, 9(25) Bichsel, H., 262 Biedenharn, L. C., 129, 130, 131, 134, 135(4, 7), 138, 142, 145, 162, 194(1) Bielesch, A., 560 Bienlein, H., 228, 230(75), 252, 255(75), 256(75), 257(75) Bieri, R., 16, 18(5) Bincer, A. M., 209, 210(126), 241 Birge, R. T., 812, 819 Birge, R. W., 747
AUTHOR INDEX
Birk, M., 327 Birkhoff, R. D., 228 Bishop, G. R., 139, 141(35), 164(9, 181, 165(50), 166, 167, 201 Bisi, A., 260 Biswas, S., 37, 39, 40, 41(1) Bivins, R., 207 Blair, J. M., 429 Blair, J. S., 479 Blaise, J., 55, 58 Blake, B., 254, 258 Blaton, J., 339 Blatt, J. M., 353, 362, 363, 366 Blaugrund, A. E., 330, 331(47), 332 Bleaney, B., 86, 101, 102, 103(18), 128, 164(26), 166, 170(4), 173, 176, 179, 181(5, 11, l6), 182, 184, 198(29), 201 Bleuler, E., 408, 479 Blewett, J. P., 223, 583, 624, 630, 638, 645, 653, 662, 663, 679, 751 Blin-Stoyle, R. J., 168, 198(10) Bloch, F., 104, 107, 108(2), 116, 124, 126(68), 171(5), 173 Block, M. M., 838, 844 Block, R., 385 Blocker, W., 544, 545(50) Bloembergen, N., 104, 107(1), 108(1) Bloom, M., 123 Bloom, S. D., 269, 271(211), 272(211), 327, 328(35), 329(35), 330 Blosser, H. G., 408, 685 Bobone, R., 254 Bockelman, C. K., 355, 356, 374, 375, 383, 387(28), 388, 440 Bodenstedt, E., 142 Bobel, G., 235 Bockmann, K., 241 Boehm, F., 260, 262, 267, 269(181), 271 (203) Bohm, D., 646, 653 Bohr, A., 359 Bohr, N., 337 Bolef, D. I., 120 Bollinger, L. M., 390, 393, 395(58), 396(58) Bondelid, R. O., 427(106), 428 Bonilla, C. F., 602 Bonner, T. W., 362, 371, 375, 376, 403, 443, 446(146) Booth, R., 407 Boreli, F., 452
85 I
Born, M., 395 Bostrom, C. O., 525(22), 526, 527(22), 528(22), 534(22) Bothe, W., 203, 276, 285, 288, 290(42) Boueher, R. R., 556, 557(7), 561 Bouches, R., 277 Bouchiat, C., 220 Bowers, K. D., 179, 181(5), 182, 199(36) Bowman, H. R., 562 Bowman, W., 239 Boyd, A. W., 292 Boyer, K., 430, 477 Boyle, E. J., 601 Bradley, G. E., 164(39), 167 Bradner, H., 758 Brady, F. P., 430 Braunstein, R., 33 Bray, P. J., 60 Breazeale, W. M., 614 Breen, R. J., 556, 557(6), 558, 561 Breit, G., 59, 235, 422 Breitenberger, E., 150, 151 Bret, G. C., 317 Bretscher, M. M., 709 Brickwedde, F. G., 189 Bridge, H. S., 833 Briggs, G. H., 438 Brill, T, 618 Brink, G. O., 75 Briscoe, W., 430 Britten, R. J., 696 Brobeck, W. M., 645 Brolley, J. E., Jr., 339, 344, 348, 372, 373 ( l l ) , 417, 427, 428, 449, 450, 588 Bromley, D. A., 480, 483, 709 Brooks, F. D., 484 Brosi, A. R., 228, 230, 255(78), 256(78), 257(78), 259(78) Brovetto, P., 124, 126(68), 260, 269(184), 272 Brown, A. B., 10, 341,344, 433,434(123), 436(123), 438(123), 439(123), 476 (123) Brown, F., 292 Brown, G., 449, 450(161) Brown, K. L., 487, 491, 547, 644 Brown, R., 108 Browne, C. P., 355, 356, 433(125), 434, 435(125), 440 Brownell, G. L., 539 Bruck, H., 663
852
AUTHOR INDEX
Brueckner, K. A., 345 Brugger, R. M., 371 Bruner, J. A,, 709 Brussard, P. J., 206 Brussel, M. K., 363, 481 Budnick, D., 269 Buch, K., 812 Budde, R., 834 Buechner, W. W., 355, 356, 433(125), 434, 435(125), 585 Buhring, W., 233, 256(89) Bullock, M. L., 428 Bunbury, D. StP., 333, 336(58, 59), 337 (59), 338(59) Burcham, W. E., 354, 365 Burde, J., 337, 338(66) Bureau, A. J., 284 Burgess, J., 108 Burgov, N. A., 537, 538(34) Burkhardt, J. L., 468 Burris, W. R., 520, 524(20), 527(20) Burrows, H. B., 428 Butler, C. C., 345 Butler, J. P., 298 Butler, J. W., 362 Button, J., 758 Byers, D. H., 277
C Cabell, M. J., 298 Cacho, C. F. M., 164(44), 167 Cachon, A., 345 Caldwell, D. O., 480 Calogero, F., 220, 223(43) Camac, M., 631, 740(17) Cameron, A. G. W., 462, 463, 468, 469 (183, 199), 470, 485(199) Cameron, J. R., 476, 808, 812(1), 817, 818, 819(1) Campion, P. J., 577 Carassi, L. M., 223 Carlson, R. O., 29 Carlson, B. C., 366, 696 Carroll, E. E., Jr., 484 Carson, A., 563, 564(31) Carter, D. S., 408 Carter, R. E., 399, 400(80), 401(80), 402 (SO), 403(80), 407(80) Carter, R. S., 460, 472(179), 475(179) Carver, J. G., 392
Case, K. M., 223 Casimir, H. B. G., 48, 50, 119 Castagnoli, C., 747 Casto, C. C., 568 Cavallo, L., 575, 576(57) Cavanagh, P. E., 231, 232(79), 255(79), 256(79), 257(79) Chadwick, J., 276, 293 Chamberlain, O., 223, 232(53), 502, 747 Chaminade, R., 296 Chao, C. Y., 433, 436(122), 443(122), 444 (122) Chaplin, G. B. B., 137 Charlton, E. E., 626 Charpak, G., 224, 225, 227(63, 64) Chase, C. T., 422 Chase, R. L., 314 Cheka, J. S., 228 Chernick, J., 611 Cheston, W., 808, 809(3), 811(3) Chidley, B. G., 309 Childs, 0. W., 618 Childs, W. J., 434 Chinowsky, W., 758 Chodorow, M., 643 Chretien, M., 834 Christian, R., 844 Christofilos, N., 693, 696(2) Christy, R. F., 363, 448 Chupp, W. W., 747 Church, E. L., 151, 152, 153(1) Cini, G.; 124, 126(68) Claesson, A., 235 Clark, D. J., 685 Clarke, R. L., 408 Clement, J. D., 452 Clement, J. R., 189 Cobble, J. W., 577 Coburn, H. C., 138 Cochran, D. R. F., 274 Cochran, R. C., 558, 560(14) Cochran, R. G., 614 Cochran, W. G., 811 Cochrane, W., 490 Cockcroft, J. D., 580 Coester, F., 131, 162, 194(2) Coffinberry, A. S., 561 Cohen, B. I., 329 Cohen, B. L., 404, 408, 480, 632 Cohen, E. R., 117,339,809, 811(12), 815 Cohen, L. D., 485
853
AUTHOR INDEX
Cohen, S. G., 337, 338(66) Cohen, V. W., 59, 77, 102, 103(10) Colby, L. J., Jr., 577 Cole, F. T., 680 Cole, T. E., 618 Coleman, C. F., 135, 139, 141(36), 142(36), 231, 232(79), 255(79), 256 (79), 257(79) Coles, D. K., 26, 27(15) Collington, D. J., 116 Collins, R. L., 111 Collins, T. L., 33, 298 Colombo, S., 319 Combrisson, J., 123, 124, 126(68) Compton, D. M. J., 273 Condon, E. U., 46, 47(5) Connally, R. E., 526 Conner, Jerry P., 373, 383, 452 Connors, D. R., 464, 468(192), 469(192) Cook, C. F., 362, 375 Cook, L. J., 385 Cook, V., 759 Cooke, A. H., 181(2, 7, 14), 182, 189 Cool, R. L., 674, 699 Coombes, C. A., 748, 750(3), 752 Coon, J. H., 377, 382 Cooper, D. I., 709 Coor, T., 374, 377(22), 378(22), 383(22), 385(22), 388(22) Cork, B., 709, 748, 750(3), 752, 759 Corson, D. R., 547, 548(58) Cottini, C., 314, 315(18), 319(18) Courant, E. D., 423, 653, 663, 665, 692, 693, 694, 696(1, 5), 719, 726(1), 751 Cowan, C. L., Jr., 296, 297, 408 Cox, J. A. M., 195 Cox, R. T., 422 Cramer, H., 808, 812(l), 819(1) Cranberg, L., 327, 329, 372, 378(13), 384 397, 442(13), 448(78), 456, 457(171) Crane, H. R., 216, 224(16), 225(15), 227(16), 581, 656 Crane, W. W. T., 562 Craven, J. H., 289, 575 Crewe, A. V., 648 Critchfield, C. L., 419 Cross, W. G., 731, 740(18) Culligan, G., 260, 311 Culvahouse, J. W., 124, 128, 164(34), 167, 200, 209 Cunningham, J. E., 601
Cuperman, S., 233, 256(90), 257(90) Curie, M., 282 Curran, S. C., 298 Curtis, M. L., 569, 570 Curtis, R. B., 195 Curtiss, L. F., 556, 563, 564(31) Cutkosky, R. E., 261
D Dabbs, J. W. T., 164(10,36), 165(52, 61, 62), 166, 167, 169, 176(20), 182 (19, 21), 202, 204, 207(106, 107) Dagley, P., 164(11, 23), 166, 209 Dahl, O., 758 Dahlberg, D. A., 390, 393, 395(58), 396(58) Dailey, B. P., 121 Dalitz, R. H., 848 Damerow, R. A., 8 Damm, C. C., 5, 6, 7(18, 19) Danby, G., 845 d’Andlau, C. A., 752 D’Angelo, N., 295 Daniel, H., 284, 288, 298 Daniels, J. M., 164(9, 19, 22, 26, 31), 166, 167, 181(4), 182, 187, 190(50), 199, 201, 213 Danilyan, G. V., 537, 538(34) Danos, M., 24 Dar, Y., 330, 331(47), 332(47) Darby, E. K., 296 Darby, P. F., 588 Darden, S., 388, 389(54), 391(54) Darden, S. E., 384, 388, 472 Darwin, C. G., 219 Das, T. P., 106, 121 Daudel, P., 277 Daudel, R., 277 Daunt, J. G., 164(27), 166,170(8),173,188 Davis, L., 77 Davis, R. C., 526 Davis, R. W., 382, 403, 405 Davisson, C. M., 157 Day, E. A., 635 Day, R. B., 378, 385, 397, 445, 448(78), 459, 460, 485 Dayhoff, E. S., 61 Dayton, I. E., 419, 707 de Benedetti, S., 137, 141, 149(39), 272, 309, 311, 312(7)
854
AUTHOR INDEX
de Groot, S. R., 162,195,200,219,223(33) Dehmelt, H. G., 104, 111, 120(3, 32), 171 (a), 173, 191 Deichsel, H., 252 DeJuren, J., 385 de Klerk, D., 186, 189(49), 190(48, 49) Delaney, C. F. G., 291, 301 Dellis, A. N., 116 Demidov, A. M., 534 Deming, W. E., 812 Dempster, A. J., 1, 3 Dennison, D. M., 20,656, 732 DePasqualli, G., 85, 244, 246(141) De-Shalit, A., 233 DeShong, J. A., 618 Detoeuf, J.-F., 296 DeTroyer, A., 566 Deutsch, J. P., 260, 269(187) Deutsch, M., 129, 139, 141(32), 148(32), 164(7, 12, 14), 166, 201, 202(101), 206(101), 239, 262, 430 Devons, S., 129, 131, 134, 135(6), 137, 138(6), 142, 143(6), 145, 148, 150, 151, 162, 194(3), 333, 335(57), 336, 337(59), 338(59) de Waard, H., 146, 284, 337 DeWire, J. W., 537, 545, 546(53), 547, 548(58), 549(53), 553(53) Diambrini, G., 537 Dicke, R. H., 277 Dickinson, W. C., 121, 122, 421 Dickson, J. J., 618 Diddens, A. N., 164(4, 15, 28), 166, 201, 209(102), 216, 269117) Diebel, R. N., 526 Dilworth, C., 37, 39, 41(2) Dismuke, M. M., 157 Dismuke, N., 157 Diven, B. C., 408, 409, 449 Dixon, W. R., 466, 560 Dobrinin, Yu. P., 293 Dobrokhotov, E. I., 296 Dobrowoleki, W., 102, 103(11, 12, 13) Dodder, D. C., 420, 421 Dodson, R. W., 566, 567, 568 Doggett, J. A., 445, 517 Dohne, C. F., 141, 148(40) Dolbilkin, B. S., 537, 538(34) Dolphin, G. S., 541 Domen, S. R., 541, 542, 543(46), 547(46), 548(46), 550(46), 551(46), 553(46)
Donelian, K. O., 614 Dorain, P. B., 102, 103(17) Doran, G. A,, 663 Douglas, R. A., 369, 467, 469(197), 470 (197), 485(197) Dove, R. B., 188 Draper, J. E., 330, 525(22), 526, 527(22), 528(22), 534(22) Dreicer, H., 29 Drever, R. W. P., 298 Driscoll, R. L., 122 Drummond, J. E., 18 Dubroniv, A. V., 732 Duckworth, H. E., 4, 9 Duffield, R. B., 619 Duffus, H. J., 181(2), 182 Dumond, J. W. M., 117,339,811(12),815 Dunham, J. L., 30 Dunning, J. R., 394 Dunoyer, L., 59 Dupree, T. H., 468 Durand, H., 164(9, 25), 166, 200 Durieux, M., 189 Dwyer, P. S., 819 Dyson, F., 619 Dyson, F. J., 237, 238(110) Dehelepov, B. S., 157, 354
E Eakins, G. W., 298 Eastwood, T. A., 298 Eberhard, P., 758 Eby, F. S., 428, 429 Edlund, M., 591, 596 Edmonds, A. R., 183,207(41) Edmonde, D. T., 181(14), 182 Edvarson, K., 578(69), 579 Edwards, P. D., 542, 547(44), 548(44), 550(44) Eggler, C., 563 Eichelberger, J. F., 569, 570(44) Eisberg, R. M., 428, 430, 479 Eisenberg, Y., 834 Eisenbud, L., 422 Eisenhart, C., 813 Eisenetein, J. C.,184,194,211(69), 212(69) Eider, F., 707 Ekspong, A. G., 747 Elbek, B., 434, 440 Elioff, T., 747
AUTHOR INDEX
Eliseiev, G. P., 231, 233, 256(80, 88), 257 (88)
Ellett, W. H., 526 Elliot, J. O., 472 Elliott, J. P., 359 Elliott,, L. G., 336, 338(61), 563, 564(29), 566 Ellis, C. D., 276 Ellis, R. E., 428 Ely, R., 431, 479(115), 846 Endt, P. M., 354 Engelke, Charles, 388 Eppling, F. J., 476 Ershler, B. V., 231, 256(80) Esbach, J. R., 95, 98 Estermann, I., 59 Etherington, H., 591, 592 Evans, R. D., 339, 351, 359, 360, 761, 769, 781, 792 Everhart, G. G., 403 Everling, F., 9, 16, 17(1), 18(5), 339, 340 (1l), 342( 11) Eversdijk Smulders, M. C., 165(58, 59), 167 Ewald, H., 34 Ewan, G. T., 244, 246(143), 247(143)
F Fabricand, B. P., 29 Fabricand, B. P., 463, 470(186) Fagg, L. W., 139, 214, 264(2), 265(2), 266(2) Faissner, H., 214 Falk, C. E., 374, 376, 377(22), 378(22), 383(22), 385(22), 388(22) Falkoff, D. L., 221, 222(47), 235(47) Fano, U., 131, 138(10), 162, 194(5), 195, 197, 204(78), 207(5), 221, 222(46, 49), 234(46, 49), 235,238,265(46,49) Farkas, I., 241 Farley, F. J. M., 224, 225, 227(63, 64) Farwell, G. W., 479, 480 Faust, W. R., 141, 148(40) Fay, H., 41 Feather, N., 504 Fechter, H. R., 501 Fechtig, H., 299 Fedchenko, E. D., 481 Feher, 6, 92, 102(ld), 103(ld), 124, 126, 127, 128, 129(73), 172(1), 173
855
Fehrentz, D., 288 Feingold, A. M., 151, 460 Feld, B. T., 293, 381, 383(38), 390, 404, 556 Fellberbaum, J., 594 Febner, G., 251, 252 Felthauser, H. E., 382 Ferentz, M., 134, 135, 208 Ferguson, A. J., 134 Ferguson, A. T. G., 327 Fermi, E., 65, 751, 843 Fernald, R. A., 329 Ferrero, F., 464 Ferroni, S., 124, 126(68), 260, 269(184), 272 Feshbach, H., 381, 383(38) Feynman, R. P., 215, 217, 221(12) Fialho, G. E., 844 Fields, R., 374, 461, 563, 564(24) Fields, R. E., 368, 369(1), 385(1), 388(1) Fierz, M., 215 Figuera, A. S., 537 Filimnov, S. I., 732 Findley, D. E., 419, 482 Findley, R. W., 137, 309, 312(7) Fireman, E. L., 296, 297 Fischbeck, H., 290 Fischer, G. E., 428 Fischer, J., 494 Fisher, R. A., 811, 813, 814, 818(5) Fitch, V., 494 Fleischmann, R., 252 Fleming, D. M., 539, 540(39), 541(39) Fleming, M. R., 561 Flerov, G. N., 297 Flerov, N. N., 405 Fletcher, A. J., 819, 820(19) Fletcher, P. C., 98 Flynn, K. F., 291 Foldy, L., 646, 653 Foley, H. H., 61 Ford, G. W., 241,258(129), 261, 263(129) Ford, K. W., 479 Fowler, J. L., 339, 344, 348, 372, 373(11), 449,475, 588 Fowler, T. K., 329 Fowler, W. A., 10, 12(8), 19,341,344(12), 431, 432(116), 433, 434, 436(122, 123), 438, 439(116, 123), 442, 443, 444(122), 476(123), 477(133), 478 (133)
856
AUTHOR INDEX
Fowler, W. B., 747 Fradkin, D. M., 217, 220(27), 223(27) Frank, N. H., 653 Frank, S. G. F., 260 Frankel, S., 138, 151, 263, 460 Franklin, J., 251 Frana, W., 265 Franzen, W., 309, 311(6), 314(6), 319(6) Franainetti, C., 747 Frauenfelder, H., 85, 129, 134, 135(3), 137, 138(3), 142, 143, 146, 148, 149, 151,200,201(91), 244,246(141, 144), 247(144), 248(144), 249(144), 254, 273, 274, 297 Freeman, A. J., 170(13), 173 Freeman, Joan M., 397, 448(79) Freier, G., 373, 387(16), 388(16) Fremlin, J. H., 297 French, J. B., 353 Fretter, W. B., 36, 752 Frey, W. F., 157 Friedland, S. S., 483 Friedlander, G., 573, 699 Friesen, E. W., 36 Frisch, D. H., 385, 709 Frisch, 0. R., 563, 564(30), 566(30) Frisch, R. D., 59 Froissart, M., 220, 223(42) Froman, D. K., 427(106), 428 Fronsdal, C., 220, 235, 236 Fry, W. F., 834 Frye, G. M., Jr., 452 Fulbright, H. W., 296 Fulk, M., 373, 387(16), 388(16) Fuller, C. S., 92, 102(ld), 103(ld), 128 Fuller, E. G., 515, 517, 518(9), 543, 551 (48) Fuller, G. H., 44, 58(4) Fulta, S. C., 485, 511, 514(6) Furlan, G., 241
G Gaerttner, E. R., 392 Gaillard, J. M., 845 Galanina, N. D., 393 Galbraith, W., 748, 750(3) Gallaher, L. J., 164(15), 166 Galonsky, A. I., 228, 230, 255(78), 256 (78), 257(78), 259(78), 476 Galster, S., 260, 261, 271
Gammel, J. H., 452 Gard, G. A., 231, 232(79), 255(79), 256 (79), 257(79) Garrett, C. G. B., 186, 190(46) Garwin, R. L., 224, 225,227(63, 64), 311, 504 Gatti, E., 309, 314, 315(18), 319 Gattner, J., 816 Gavrilov, B. I., 463 Geiger, J. S., 244, 246(143), 247(143) Geiger, K. W., 560 Gell-Mann, M., 834 Gentner, W., 299 Geoffrian, C., 228 George, E. C., 37, 39(1), 40(1), 41(1) Gerdau, E., 142 Gere, E. A,, 92, 102(ld), 103(ld), 126 (68), 128 Gerhart, J. B., 244, 246(146), 262, 359 Gerholm, T. R., 137, 142, 146, 158 Gerlach, W., 59 Germain, P., 668 Gerstenkorn, S., 55 Geschwind, S., 22, 23(6, 7, 8), 24, 25, 32(7), 34 Geshkenbein, B. V., 254 Ghiorso, A., 297 Ghoshal, S. N., 408 Giauque, W. F., 189 Gibbons, J. H., 370, 371, 375(3), 376(3), 378(3), 385, 386(3), 392 Gibbs, R. C., 134 Gibson, H. F., 532(27), 533 Gibson, W. M., 441 Giese, C. F., 6, 8 Gillet, V., 200 Gilliam, 0. R., 98, 102, 103(10) Gimmi, F., 138, 143(21) Gittelman, B., 262 Gittings, H. T., 403 Glaser, D. A., 746 Glasoe, G. N., 330 Glasser, R. G., 37, 39, 40 Glasstone, S., 591, 596 Glendenin, L. E., 291 Glover, K. M., 570 Gluckstern, R. L., 228,229,230(74), 235, 255(74), 256(74), 257(74), 259 Glueckauf, E., 566 Godel, J. B., 618 Godlove, T. F., 392
857
AUTHOR INDEX Goldberg, E., 476 Goldberg, M. D., 396 Goldberger, M. L., 381, 383(38) Goldemberg, J., 463, 464(189), 471(189), 472( 189) Goldfarb, L. J. B., 129, 131, 134, 135(6), 137, 138(6), 142, 143(6), 145, 148, 150, 151, 162, 194(3, 7) Goldhaber, G., 747, 752, 758 Goldhaber, M., 139, 141(33), 148(33), 260, 269(180), 277, 281(9), 293, 297 Goldhaber, S., 747, 752, 758 Goldring, G., 143, 146, 147, 327, 330, 331(47), 332, 333, 335(57) Goldsack, S. J., 37, 39(2), 41(2) Goldschmidt, G., 164(22), 166 Goldstein, H., 381, 383(38) Good, M. L., 751 Good, R. H., Jr., 217, 220(27), 223(27), 225(61) Good, W. E., 26, 27(15) Good, W. M., 329, 392 Goodings, D. A., 170(12), 173 Goodman, A., 333 Goodman, C., 446, 460 Goodman, L. S., 74, 77 Goodwin, P. N., 539, 540(39), 541(39) Gore, E. K., 290 Gordon, J. P., 29 Gordy, W., 23, 98 Gorodetzky, S., 311 Gorter, C. J., 105, 164(3, 6, 15), 165, 166, 170(1, 5, 9), 173, 176, 201, 202, 203 (105) Gottstein, K., 41 Goudsmit, P. F. A., 160 Goudsmit, S., 49 Could, H. L. B., 247 Goulianos, K., 845 Gove, H. E., 363, 430 Gove, N. B., 354 Grace, M. A., 164(8, 11, 18, 19, 23, 25, 26, 29, 33, 42, 44), 166, 167, 168, 182(17), 186, 198, 200, 201, 209, 327 Graham, R. L., 244, 246(143), 247(143), 311, 322(12) Grainger, L., 603 Graves, A. C., 566, 567(35), 568(35) Graves, E. R., 377, 403, 405, 458, 460 Gray, T. P., 195, 198 Green, G. K., 653, 665, 668
Green, R. E., 311, 313(17), 314(17) Green, T. A., 152, 153 Green, T. S., 434 Greenberg, J. S., 228, 229, 230(745, 255 (74), 256(74), 257(74), 259 Gregory, J. W. G., 648 Gregory, M., 164(11), 166 Grenacs, L., 260, 269(187) Grenchik, R., 428, 429(109), 487 Griffing, D. F., 164(20, 21), 166 Grishuk, G. I., 254 Grismore, R., 330 Grismore, W., 316 Grodzins, L., 205, 214, 228(1), 252(1), 255(1), 256(1), 260, 262, 263(1), 264(1), 265(1), 269(180) Groshev, L. V., 534 Giinther, C., 142 Giirsey, F., 233 Giithner, K., 228, 230(75), 252, 255(75), 256(75), 257(75) Gunnink, R., 577 Gunst, S. B., 269 Gunther-Mohr, G. R., 22,23(6), 27,34(6) Gutowsky, H. S., 111
H Haag, J. N., 164(41), 167, 182(10) Haddad, E., 449 Haddock, R. P., 747 Hafner, E. M., 374, 377(22), 378(22), 383(22), 385(22), 388(22) Hagedorn, F. B., 438, 439, 477(133), 478 (133) Hahn, E. L., 106, 111, 121, 123 Hahn, F. S., 329 Hain, K., 41 Halbach, K., 108 Halban, H., 164(19, 22, 25, 26), 166, 168, 200,201, 557 Halford, D., 128 Hall, G. R., 102, 103(18), 286, 570 Hall, R. N., 583 Halliday, D., 339 Halpern, I., 594 Halpern, J., 463, 464(187), 465(187) Hamer, E. E., 618 Hamermesh, B., 461, 464(181), 466 Hamermesh, M., 304 Hamilton, D. R., 59,77,139,141(34), 148 (34)
858
AUTHOR INDEX
Hamilton, J., 215, 221(11) Hamilton, J. H., 157 Hammer, C. L., 284 Hamher, S. H., 165(62), 167, 182(21), 202, 207(107) Hand, L. N., 712 Handley, T. H., 408 Hanna, S. S., 139, 214, 263, 264(2), 265 (2), 266(2), 335, 338 Hans, H. J., 376 Hans, H. S., 472 Hansen, L. F., 752 Hansen, N. E., 485, 511, 514(6) Hansen, P. G., 263 Hansen, W. W., 104, 107(2), 108(2) Hanson, A. O., 85, 244, 246(141), 371, 384(9), 406, 419, 441(84), 449, 464, 556, 563, 566, 567(35), 568(35) Hardy, J. D., 20 Harmsen, D. M., 244,245(145), 246(145), 247(145), 248(145), 249(145) Harrer, J. M., 618 Harrick, N. J., 60 Harris, S. M., 148 Harris, S. P., 566 Harrison, F. B., 296, 408 Harth, E. M., 844 Hartley, W. H., 463, 464(190), 466, 467 (190) Hartogh, C. D., 200 Harvey, B. G., 337 Harvey, John A., 385, 386(46), 396, 407 (46), 410, 430, 460(46) Harvey, R. A., 541, 542(42) Haslam, R. N. H., 462, 467, 469(183, 197), 470(197), 485(197) Hastay, M. W., 813 Hatcher, C. R., 485, 511, 514(6) Hauser, I., 134 Hausman, H. J., 431, 479(115) Havens, W. W., Jr., 390,392,394,395(56) Hawkings, R. C., 289, 292, 575 Haworth, L. J., 392 Hayden, R. J., 297 Hayes, F. N., 409 Hayward, E., 515, 517, 518(9), 543, 551 (48) Hayward, R. W., 164(5, 17, 24, 32), 165 (45), 166, 167, 202, 203(104), 204 (104), 577 Heard, Harry G., 496
Heckman, H. H., 747 Hedgran, A., 157, 578(75), 579 Heer, C. V., 188 Heer, E., 138, 142, 143, 145(41), 146(41, 50), 148(41, 46, 50), 200, 201(93) Heinberg, M., 264 Heine, V., 170(12), 173 Heinrichs, O., 557 Heintee, J., 233, 256(87, 89), 290 Heitler, W., 139, 141(38), 215, 221(7) Helmholz, L., 566, 567(35), 568(35) Hemmendinger, A., 408, 409(91), 441, 442 (139) Henderson, C., 649 Henderson, G. H., 438 Henderson, W. J., 541, 542(42) Henkel, R. L., 372, 374, 378, 381(19), 407, 410, 442(13) Henri, V. P., 309, 311(5), 321(5), 322(5), 323(5) Henry, K. M., 558, 560(14) Herb, R. G., 363, 384, 476(42), 585 Hereford, F. L., 139 Hereward, J., 333 Herta, M. R., 556, 557(6), 558, 561 Hereberg, G., 20, 33(2, 3) Herzog, B., 123 Hess, V. F., 283 Hess, W. N., 558, 560(16) Hester, R. E., 484 Heyd, J. W., 569, 570(44) Heydenburg, N. P., 359, 445, 446, 447 (156), 448 Hibdon, C. T., 375, 376(23) Hickey, F. C., 290 Hiebert, R. D., 417, 418(96) Higginbotham, W. A., 314, 317 Higgins, G. H., 562 Higgs, R. W., 459 Hildebrand, F. B., 819 Hildebrand, R. H., 385 Hill, J. S., 164(11, 23), 166, 209 Hill, R. D., 164(20), 166 Hill, R. W., 476 Ilillger, R. E., 95, 98 Hillman, P., 374, 385(20) Hilton, H. H., 334 Hincks, E. P., 566 Hine, G. J., 539 Hine, M. G. N., 665, 668 Hintenberger, H. H., 4, 7(11)
a59
AUTHOR INDEX
Hintz, N. M., 481 Hipple, J. A., 116, 122 Hirschberg, L., 37, 39(2), 41(2) Hisdal, E., 511 Hoang, T., 759 Hobson, J. P., 74, 77, 78 Hockney, R. W., 434 Hofstadter, R., 310, 443, 445(145), 531 Hogg, B. G., 9 Holder, B. E., 119 Holland, R. E., 304, 329, 375, 376(28) Hollander, J., 157 Hollander, J. M., 153, 278, 354, 562, 572, 579 Holm, K., 244, 245(145), 246(145), 247 (145), 248(145), 249(145) Holmgren, H. I)., 428, 429 Holt, J. R., 260 Honig, A., 27, 33(17) Hoover, J. I., 141, 148(40) Hopkins, J. C., 244, 246(146), 262 Hoppes, D. D., 164(5, 17, 24, 32), 165 (45, 46), 166, 167, 202, 203(104), 204 (104, Horen, D., 157 Hornyak, W. F., 357, 374, 376, 377(22), 378(22), 383(22), 385(22), 388(22), 443, 445(144), 466 Horseley, R., 285 Horsley, R. J., 462, 469(182) Horwitz, N., 748 Hough, P. V. C., 434 Hoyle, M. B., 134 Hubbard, E. L., 642, 709 Hubbs, J. C., 74, 75, 76, 77, 78, 102, 103 (19) Huber, O., 151 Huber, P., 475 Hudson, A. M., 427(106), 428 Hudson, R. P., 164(5, 17,24, 32, 43), 165 (43), 166, 167, 168, 181(3), 182, 186, 187, 189(47), 190(47, 51), 200, 202, 203(104), 204(104) Huffman, J. R., 611 Hufford, D. L., 566, 567, 568(35, 39) Hughes, D. J., 363,385,386(46), 390,392, 393, 394(57), 395(57), 396, 407(46), 460, 472(179), 475(179), 563 Hughes, H. K., 28 Hughes, I. S., 139 Hughes, R. H., 25
Hughes, V. W., 228, 229, 230(74), 255 (74), 256(74), 257(74), 259 Huiskamp, W. J., 164(3, 4, 6, 13, 28), 165 (13, 59), 166, 167, 168, 201, 202, 203 (105), 209(102), 216, 269(17) Huizenga, J. R., 160, 571(51), 572 Hull, J. R., 169, 175(23) Hull, M. H., Jr., 235 Hull, R. A., 169, 175(23) Hultberg, S., 157, 160 Hurlimann, T., 450 Hurst, D. G., 618 Hurst, R., 286 Hussain, T., 557 Huster, E., 290 Hutchison, C. A., Jr., 102, 103(17), 128 HUUS,T., 359, 446 Hyams, B. D., 244, 246(147), 297
I Ignatiew, K. G., 393 Igo, G., 428, 430, 479 Inghram, M. G., 297 Inglis, D. R., 359, 365 Innes, G. S., 541 Isenor, N. R., 4 Ivanter, I. G., 237
J Jack, W., 484 Jackson, H. G., 298 Jackson, H. L., 476 Jackson, J. D., 254 Jacob, M., 162 Jacobsohn, B. A., 108, 110(15), 111(15) Jacobson, J. C., 333 Jacquinot, P., 54 Janecke, J., 284 JiLnossy, L., 808 Jansen, J. F. W., 160 Jarmie, N., 363, 430, 449(114) Jarvis, G. A., 441, 442(139) Jastram, P. S., 164(27), 166 Jauch, J. M., 131, 162, 179, 194(2), 200 (39), 212(39), 215, 221(8) Jeffries, C. D., 96, 102, 103(3, 11, 12, 13, 15), 116, 124, 126(68), 127, 168, 171 (7), 173, 181(6, 12), 182(14), 192 (14), 194(14), 202(14)
860
AUTHOR INDEX
Jen, C. K., 98 Jensen, E. N., 298 Jensen, J. H. D., 216, 217(22), 220(22), 359 Jentschke, W., 676 Jesse, W. P., 500 Johansson, B., 332 Johansson, F., 43 Johns, H. E., 467, 469(197), 470(197), 485 (197) Johnson, C. E., 164(8, 9, 18, 25, 29, 33, 40, 41, 42, 44), 165(47, 50, 55, 60), 166, 167, 181(8), 182(10, 17), 186, 200, 201 Johnson, C. H., 375, 441, 452, 475 Johnson, E. G., 3, 7(9) Johnson, V. R., 371, 441(7) Johnson, W. H., Jr., 8, 9(25), 33 Johnson, W. R., 237 Johnstone, C. W., 327 Jones, G. A., 147, 336 Jones, K. W., 369 Jones, R. J., 633 Jones, R. V., 102, 103(11, 12, 13) Jordan, E. B., 3 Jergensen, M. H., 322, 323(32) Jost, R., 285 Judd, D. L., 732 Jung, H., 284 Jupiter, C. P., 485, 511, 514(6)
K Kaeser, R. S., 165(45), 167, 181(3), 182, 187, 190(51) Kalbfleisch, G. R., 758 Kalbitzer, S., 299 Kalkstein, M. I., 296 Kandiah, K., 137 Kane, J. V., 138 Kanner, H., 309, 311(5), 321(5), 322(5), 323(5) Kanner, M. H., 375, 475 Kantz, A., 531 Kaplan, D., 123 Kaplan, I., 339, 611 Kaplan, M., 164(2), 165 Karmohapatro, S. B., 737 Karplus, R., 24, 108 Kastler, A., 171(1), 173, 191 Kato, W. Y., 390, 394(57), 395(57)
Katoh, T., 579, 580 Katz, L., 462,463,464(189), 467,468,469 (183, 197, 199), 470, 471(189), 472 (189), 485(197, 199), 512 Kauw, G., 292 Kedzie, R. W., 96, 102, 103(3, 15), 124, 181(6), 182 Keefe, D., 759 Keepin, G. R., 594, 595(5) Kegel, G. H. R., 311 Keil, E., 533 Kelley, G. G., 526 Kellogg, J. M. B., 60 Kelly, E. L., 683, 686, 709 Kendall, M. G., 817, 818 Kennedy, J. M., 134 Kennedy, J. W., 573 Kenney, R. W., 544, 545(50) Kepler, R. G., 752 Kerimov, B. K., 235, 237, 251 Kerlee, D. D., 479 Kerst, D. W., 542, 547(44), 548(44), 550 (44), 626, 629, 681, 687, 731, 735, 739(16) Kerth, L. T., 747, 759 Ketelle, B. H., 228, 230,255(78), 256(78). 257(78), 259(78) Keuper, J. P., 139 Khoe, T. K., 219, 220(35), 223(35), 225 (35) Khutsishvili, G. R., 164(30), 167, 168 Kiehn, R. M., 460 Kiesling, J. D., 751 Kimball, C., 461, 464(181), 466 King, J. G., 61, 72, 76(16) King, R. W., 354 Kingdon, K. H., 71 Kington, J. D., 474 Kinsey, B. B., 481 Kirn, F. S., 532(27), 533 Kisslinger, L. S., 153, 162 Kister, J. M., 407 Kitchen, Summer W., 501 Kittel, C., 124, 126(68), 243 Kjellman, J., 310 Klein, M. J., 178 Klein, M. P., 119 Klema, E. D., 150, 151 Klochov, D. S., 297 Knable, N., 385 Knight, W. D., 111, 120(29), 122
861
AUTHOR INDEX
Knipper, A., 311 Knipper, A. C., 164(18, 44), 166, 167, 201 Knoebel, H. W., 111 Knop, G., 244, 246(147) Fo, R., 568 Koch, H. W., 485, 509, 510(1), 516, 517 (10, 11), 518, 519(17), 520, 521(17), 522(18), 523(18), 524(1), 528, 626 Kockum, J., 517 Konig, L. A,, 4, 7(11), 9, 17(1), 339, 340 ( I l ) , 342(11) Korner, H. J., 142 Koester, L. J., Jr., 297, 552 Kofoed-Hansen, O., 239 Kogan, A. V., 164(1), 165 Kohler, D., 334 Kohman, T. P., 287, 302 Kolbenstvedt, *H., 238, 239(118) Kolomenskij, A. A., 668 Komar, A. P., 544, 545(51), 546(51), 550 (51), 553(51) Konopinski, E. J., 216 Kopfermann, H., 46, 47, 48, 50(7), 121 Koppe, H., 258 Korringa, J., 124, 126(68), 171(4), 173 Koski, H. G., 60 Koster, G., 121 Koster, G. F., 194 Koute, S. L., 619 Kramer, G., 241 Kramers, H. A., 105, 178 Kraushaar, J. J., 139, 141(33), 148(33), 151, 277 Kresnin, A. A., 235, 241 Krienan, F., 647 Kristiansson, K., 41, 42, 43 Krohn, V. E., 142, 143, 146(44), 148 Kronig, R. de L., 105 Kriiger, H., 111, 120(32) Kruglov, S. P., 542,544,545(51), 546(51), 547(45), 548(45), 550(51), 553(51) Kruse, H. W., 297 Kruse, T. H., 443, 446(148), 447(148) Kulenkampff, H., 534 Kul’kov, V. D., 164(1), 165 Kune, W. E., 428 Kuperman, S., 233 Kurti, N., 164(8, 11, 19, 22, 25, 26, 29), 166, 168, 169, 170(2), 173, 176, 186, 200, 201, 209 KuBEer, I., 285
Kusch, P., 60, 61 Kusourov, G. I., 732 Kutikov, I. E., 252, 293
L Ladage, A., 228, 230(76), 256(76) Ladenburg, R., 36 Lamb, W. E., 121 Lamb, W. E., Jr., 61, 395 Lamb, W. A. S., 484 Lambertson, G. R., 748, 750(3), 752 Lampi, E. E., 373, 387(16), 388(16) Landau, L., 339 Lane, A. M., 353, 359 Lander, R. L., 747 Lang, H. J., 384 Langer, L. M., 290, 296 Langham, W. H., 569 Langmuir, I., 71 Langsdorf, A. J., Jr., 375, 376(23) Langsdorf, A. S., Jr., 339, 343, 373 Lannutti, J. E., 758 Laubenstein, M. J., 371, 441(7) Laughlin, J. S., 541, 542(42) Laukien, G., 44, 58(3) Lauritsen, C. C., 10, 12(8), 19, 333, 338, 341,344(12), 431,432(116), 433,434, 436(122,123), 438,439,442,443,444 (122), 476(123), 477(133), 478(133), 581 Lauritsen, T., 337, 338, 352, 354, 357(6), 358, 365, 434, 442, 443, 445 Lawrence, E. O., 630 Lawson, J. D., 665 Lawson, J. S., Jr., 148, 151, 297 Layton, T. W., 117, 811(12), 815 Lazar, N. H., 408, 526, 527 Lazarenko, V. R., 296 Laeareva, L. E., 463, 537, 538(34) LeBlanc, M. A. R., 164(31), 167 LeBlanc, W. H., 539, 540(39), 541(39) LeCouteur, K. J., 648 Lederman, L. M., 225, 845 Lee, C. A., 29 Lee, T. D., 216 Lefevre, H. W., 317 Lehmann, P., 147, 148 Leighton, R. B., 345 Leininger, R. F., 277 Leipuner, L. B., 837
862
AUTHOR INDEX
Leiss, J. E., 485, 511, 543, 551(49) Leith, C. E., 385 Leitner, J., 834 Lemmer, H. R., 164(8, 18, 25), 165(50), 166, 167, 200, 201 Lemonick, A., 139, 141(34), 148(34) Leonard, B. P., Jr., 472 Lepore, J. V., 844 LBvBque, A., 147 Levi, C., 296 Levi, M. W., 200 Levin, J. S., 327, 384, 390, 394(57), 395 (57), 456, 457(171) Levine, C. A., 297 Levine, N., 85, 149, 151, 244, 246(141) Levine, S. H., 433 Lew, H., 71, 72, 77 Lewin, W. H. G., 159 Lewis, G. M., 291, 298 Lewis, H. R., 254 Lewis, I. A. D., 316 Lewis, R. R., 195 Li, C. W., 12(8), 19, 431, 448 Libby, W. F., 296, 301, 302(104) Lifshitz, E., 339 Likely, J. G., 430 Lillie, Alan B., 452 Lind, D. A., 459, 460(173) Lindenbaum, S. J., 662 Lindqvist, T., 146, 269 Lindsay, G. R., 333, 335(57) Lindstrom, G., 578(70, 71, 73, 75), 579 Lipkin, H. J., 233, 244, 246(144), 247 (144), 248(144), 249(144), 254 Lipman, N. H., 311 Lipnik, P., 260, 269(187) Lipps, F. W., 196, 197, 265 Lipworth, E., 77 Listengarten, M., 152, 153(1) Little, R. N., Jr., 472 Little, W. A., 176 Littler, D. J., 566, 567(38) Livesey, L. C., 618 Livingood, J. J., 624, 629, 657, 662 Livingston, M. S., 223, 339,430,583, 624, 630, 633, 638,645,662, 663,664,676, 679, 692, 693(1), 696(1), 726(2) Livingston, R., 98, 111 Llewelyn, P. M., 102, 103(18), 128 Lloyd, S. P., 133 Loeffler, F. J., 546, 547(55), 549(55)
Lonsjo, O., 376, 403 Lofgren, E. J., 747 Logan, J. K., 189 Lokan, K. H., 512 Loper, 0. J., 440 Lorrain, P., 582 Louisell, W. H., 216, 225(15) Lounsbury, M., 292 Louvegnies, M., 55 Lovberg, R. H., 481 Lovejoy, C. A., 165(48, 54), 167, 182(13) Luckey, D., 714 Luebke, E. A., 392 Liiders, G., 665 Luers, D., 846 Lukashevich, I. I., 252 Lukyanov, S. Yu., 296 Lundby, A., 269 Lutsenko, D. L., 534 Lyman, E. M., 419 Lynch, F. J., 304 Lynch, G. R., 758 Lyon, W. S., 408, 569 Lyubimov, V. A., 231, 233, 254, 256(80), 256(88), 257(88)
M McAulay, I. R., 291, 301 McCarthy, J. A., 296 McClelland, C. L., 446 McClure, R. E., 111 McCord, R. V., 295 McCrary, J. H., 376, 446 McDaniel, B. D., 443, 536, 547, 548(58) McDonald, J. E., 221, 222(47), 234(47) McElhinney, J., 541 McEllistrem, M. T., 369, 433 MacFarland, C. E., 709 MacFarlane, M. H., 353 Macfarlane, R. D., 287 McGinnis, C. L., 354 McGowan, F. K., 139, 141(37), 149(37), 150, 151, 162, 448 MacGregor, M. H., 407 McGruer, J. N., 419, 433, 482 McIntyre, J. A., 310, 443, 445(145) Mack, J. E., 44, 58(2) McKensie, J. M., 483 MacKenzie, R. D., 244, 246(143), 247 (143)
AUTHOR INDEX McKibben, J. L., 406, 441(84) McKim, F. R., 181(14), 182 Macklin, R. L., 371, 408 McMaster, W. H., 138, 197,221,222(48), 222(51), 234(48), 234(51), 235(51), 237(51), 238(51), 239(51), 241(51), 263(51, 122), 265(51) McMillan, E. M., 385, 636, 645(26), 651 McMillan, W. G., 277, 837 McNeill, K. G., 464 Macq, P. C., 260, 269(187) McReynolds, A., 619 McVoy, K. W., 235, 237(101), 238, 239 (113) Maeder, D., 445 Maglic, B. C., 758 Magurno, B. A,, 363 Mahl, H., 290 Malmfors, K. G., 310 Malone, D. P., 228, 229, 230(74), 255 (74), 256(74), 257(74), 259 Malvano, R., 464 Mandel, M., 27, 33(17) Manfredini, A., 747 Manley, J. H., 392 Mann, A. K., 463, 464, 517 Mann, L. G., 269, 271(211), 272(211) Mann, M. M., 618 Mann, W. B., 289, 539, 540(38), 575, 576 Manning, G., 333, 336(58, 59), 337(59), 338(59) Manquenouillc, R., 311 Margolis, B., 251 M a r y i n , G., 288 Marin, P. C., 244, 246(147), 297 Marion, J. B., 343, 371, 373(4), 514 Mark, H., 441 Marklund, I., 269 Marrus, R., 102, 103(19) Marshak, H., 165(57), 167, 385 Marshak, R. E., 827, 834, 837, 839 Marshall, J., 494, 502 Marshall, L., 751 Marshall, W., 170(10), 173 Martin, H. C., 409 Martin, H. J., 433 Massey, H. S. W., 216, 249(13), 250(13), 252(13) Matsuda, H., 4 Mattauch, J., 3, 4, 7(7), 14, 16, 18
863
Mattauch, J. H. E., 9, 17(1), 339, 340 ( l l ) , 342(11) Maximon, L. C., 235, 237 May, A. N., 566 May, M. M., 139 Mayer, J. W., 483 Mayer, M. G., 359 Mayer-Kuckuk, T., 292 Medveczky, L. L., 558 Meissner, K. W., 53, 54 Meister, H. J., 223, 225(62) Melkonian, E., 390, 394(56), 395(56) Mellet, R., 277 Mendlowitz, H., 223 Menes, M., 120 Merrit, W. F., 289, 292, 575 Merzbacher, E., 217, 222(31) Metropolis, N., 207 Metzger, F., 139, 141(32), 148(32) Meyer, D. I., 746 Meyer, H., 181(8, 9), 182 Meyer, St., 276, 283 Meyers, L. H., 111 Michel, L., 217, 220(25), 223, 225(59) Middleton, R., 434 Miedema, A. R., 164(3, 4, 6), 165(58), 166, 167, 176, 202, 203(105) Mihailovi, M. V., 285 Mikaelyan, L. A., 231,232, 252, 256(81), 257(81) Milburn, R. H., 844 Miller, D. W., 375, 387(28, 49), 388, 431, 432(119), 433, 434(119), 438(119) Miller, J., 148, 485 Miller, L. C., 293 Miller, S. C., 239 Miller, W., 532(27), 533, 534 Miller, W. C., 462, 463(182), 466(182), 468( 182) Miller, W. F., 517 Millman, S., 59, 60 Mills, R. L., 385 Miltra, I. S., 846 Miskel, J. A., 269, 271(211), 272(211) Mistry, N., 845 Mizobuchi, A., 147 Mladjenovic, M., 157 Moak, C. D., 583, 587(8) Mobley, R. A., 385 Mobley, R. C., 329, 330(40), Moch, R., 296
864
AUTHOR INDEX
Mossbauer, R. L., 273 Mohr, C. B. O., 251 Molina, E. C., 770 Moljk, A., 298 Monahan, J. E., 339, 343(8), 373 Monk, G. S., 701 Montague, B. W., 751 Montalbetti, R., 462, 463, 464(189), 469 (183), 471(189), 472 Montgomery, C. G., 788 Montgomery, D. D., 788 Mooring, F. P., 374 Moran, H. S., 517 Moreland, P. E., Jr., 4 Morita, A., 209 Morita, M., 195, 210 Morita, R. S., 195, 209, 210 Morrison, G. C., 449, 450(161) Morrison, P., 353, 363, 366 Morton, W. T., 449, 450(161) Mosrkowski, S. A., 278 Mott, N. F., 216, 249(13), 250(13), 252 (13) Mottelson, B., 359 Mots, J. W., 509,510(1), 520,524(1), 532 (27), 533, 534 Moyer, B. J., 501 Moser, F. S., 438 Mozley, R. F., 707 Muehlhause, C. O., 330, 375 Miihlschlegel, B., 258 Miiller, R., 445 Miiller-Warmuth, W., 4 Munnich, K. O., 302 Muirhead, H., 449, 450(161) Mukhtarov, A. I., 241 Muller, T., 224, 225, 227(63, 64f Mullin, C. J., 241, 258(129), 263(129) Murray, J. J., 748, 758 Murray, R. B., 558, 559 Mutchler, G. S., 164(7), 166 Muxart, R., 277 Myers, I. T., 539, 540(39), 541(39) Myers, 0. E., 570
N Nadzhafov, I. M., 235 Nagel, B., 238 Nagle, D. E., 77, 274
Nagy, K., 241 Nakano, T., 834 Nakasima, R., 354 Naqvi, S. I. H., 526 Nathan, O., 263 Nathans, R., 463, 464(187), 465(187) Navarro, Q. O., 165(56),167, 182(15) Neal, R. B., 644 Neamtan, S. M., 263 Neher, H. V., 781 Neidigh, R. V., 480 Neiler, J. H., 392 Nelson, D. F., 225, 251, 252 Nereson, N., 388, 389(54), 391(54) Nethercot, A. H., Jr., 22, 28, 30(5), 32(5, 181, 33(5) Neuert, H., 557 Neuman, M., 844 Newman, Eugene, 408 Newson, H. W., 385 Newton, J. O., 327 Newton, T. D., 322, 323(30) Neyman, J., 817 Nicodemus, D. B., 382 Nielsen, C. E., 663 Nielsen, J. M., 526 Nielsen, K. O., 153, 434 Nielsen, 0. B., 153 Nier, A. O., 3, 6, 7(8, 9, lo), 8, 9(25), 33, 34 Nierenberg, W. A., 61, 74, 75, 76, 78, 102, 103(19), 145 Nierhaus, R., 292 Nijgh, G. J., 153, 157, 160, 208, 570, 571 (50), 578(50), 579 Nikitin, L. P., 164(1), 165 Nikitin, S. J., 393 Nikolayev, F. A., 537, 538(34) Nasson, A., 310 Nilsson, S. G., 153 Nishijima, K., 834 Nobles, R. A., 407 Norberg, R. E., 123 Northrop, J. A., 430 Notley, M. J. F., 561 Novey, T. B., 142, 143, 145(41), 146(41, 48, 50), 148(41, 48,50), 200, 210(93), 262 462,463(182), 466,469(182) Noyes, J. Nozawa, M., 579, 580
c.,
AUTHOR INDEX
0 Oakley, D. C., 545, 546(52), 549(52) Ogata, K., 4 Ohkawa, T., 679, 687(87) Okazaki, A., 388 Okorokow, W. W., 393 OIesen, M. C., 434 Olsen, H., 235, 237, 238, 239(118) Olson, R. A., 474, 475(204) Olsson, I., 301 Olt, R. G., 569, 570(44) O’Neal, R. D., 563 O’Neill, G. K., 687 Orear, J., 808, 819 Ornstein, L. Th. M., 160 Orton, J. W., 179, 182(37) Ostermukhova, G. P., 540 Ott, I). G., 409 Overhauser, A., 124, 126(68), 171(3), 173, 191 Owen, G. E., 345 Owen, J., 102, 103(14), 179, 182(36), 199 (36)
P Packard, M., 104, 107(2), 108(2), 123 Page, L. A., 214, 217(3), 228(3), 239(3), 245(3), 262(3), 263(120), 264(3), 269 Pais, A., 834 Pake, G. E., 106, 111 Palevsky, H., 390, 393, 394(57), 395(57), 428, 429(109), 487 Palfrey, T. R., 546, 547(55), 549(55) Palmer, R. R., 390, 393(58), 395(58), 396 (58) Paneth, F. A., 566 Panofsky, W. K. H., 491, 506, 544, 545 (501, 644, 712, 750, 751 Panussi, Ph., 284 Papineau, L., 296 Park, E. C., 285 Park, J. G., 102, 103(16) Parker, G. W., 165(61, 62), 167, 182(19, 21), 202, 207(106, 107) Parker, S., 501 Parkinson, W. C., 316, 330, 434 Passatore, G., 221, 222(50), 234(50), 258 Pate, B. D., 288, 289, 290(46), 568, 569, 572,575, 576
865
Patro, A. P., 269 Paul, E. B., 408 Paul, W., 288 Pauli, W., 44, 217, 222(30) Pearce, R. M., 296 Peierls, R., 285 Peker, L. K., 354 Pelekhov, V. I., 534 Penfold, A. S., 485, 511 Pennington, E. M., 9 Peressutti, G., 241 Perez y Jorbtt, J., 139, 141(35), 164(9, 18), 165(50), 166, 167, 201 Perkins, D. H., 747 Perkins, R. B., 472, 474, 475(204) Perkins, R. W., 526 Perl, M. L., 746 Perlman, I., 153, 278 Perov, I. S., 241 Perry, C. L., 157 Perry, J. E., Jr., 372, 378(12), 385, 441, 449 Persico, E., 228 Peshkov, V. P., 188 Petch, H. E., 311, 322(12) Peters, B., 37, 39(1), 40(1), 41(1), 752 Peterson, J. M., 385, 477 Peterson, J. R., 747 Peterson, R. E., 375, 383 Peterson, R. W., 434 Peterson, V., 494, 501 Petree, B., 374 Petterson, B. G., 158, 269 Pettus, W. G., 252 Peyrou, C.,37 Philips, W. R., 147 Phillips, D. D., 403, 427(106), 428 Phillips, J. A., 430 Phillips, R., 494, 501 Phillips, R. P., 750 Phipps, T. E., 60 Piccioni, O., 661, 699 Pick, R., 147 Pidd, R. W., 216, 224(16), 225(15), 227 (16), 252 Pieper, G. F., 428 Pignatello, M., 319 Pines, D., 124, 126(68), 172(2), 173 Pipkin, F. M., 77, 124, 128, 139, 141(34), 134(34), 164(34, 38, 39), 167, 202, 209
866
AUTHOR INDEX
Pixley, R. E., 484 Pjerrou, G. M., 847 Placaek, G., 381, 395 Pleasonton, F., 295 Podolanski, J., 345 Pohlit, W., 542, 547(47), 551(47) Poole, M. J., 453 Popov, V. S., 223 Poppema, 0. J., 164(16), 166, 201 Porschen, W., 292 Porter, R. A., 277 Portis, A. M., 107, 129 Poss, H. L., 376, 377, 385(34) Post, R. F., 317, 643 Postma, H., 164(6, 13), 165(13, 57, 58, 59), 166, 167, 176, 202, 203(105) Potter, R. M., 427(106), 428, 566, 567 (35), 568(35) Pound, R. V., 104, 107(1), 108(1), 111, 113, 120(29), 124, 142, 143(42), 145, 146(42), 147(42), 148, 151, 170(7), 173, 183, 200 Povelites, J. G., 566, 567(35), 568(35) Powell, C. F., 428 Powell, J. L., 384, 452, 476(42) Powell, W. M., 747 Pratt, R. H., 235, 238(92) Preston, R. S., 263 Preston, W. M., 428 Prestwood, R.J., 290, 573 Price, P. C., 150 Primakoff, H., 295, 808, 809(3), 811(3) Proctor, W. G., 111, 120, 122, 123, 127 Prokofiev, Yu. A., 293 Prud’Homme, J. T., 472 Prudkovsky, G. P., 732 Pruitt, J. S., 542, 543, 547(46, 47), 548 (46), 550(46), 551(46, 47, 491, 553 (46) Pryce, M. H. L., 102, 103(18), 178, 179, 181(20), 182, 184 Purcell, E. M., 104, 107(1), 108, 111, 113 f l ) , 228 Pushkin, E. V., 228, 230(77) Putmsn, J. L., 577 Putnam, T. M., 417, 427(109), 428, 449 Pytte, A., 261
Q Quisenberry, K. S., 3, 6, 7(10), 8,’33, 34
R Rabi, I. I., 29 Rabi, I. I., 59, 60 Raboy, S., 142, 143,146(44), 148(48), 526 Racah, G., 162, 194(5), 207(5) Raczka, A., 241 Racaka, R., 241 Rsdeloff, J., 142 Radford, H. E., 181(3), 182, 187, 190(51) Radkewich, I. A., 394 Rainwater, L. J., 390, 392, 394, 395(56) Ramsey, N. F., 29,60,61,86,98(lb), 102 (lb), 103(lb), 106, 121, 374, 385(20) Ranken, W. A., 446 Rasetti, F., 276 Rasmussen, J. D., 278 Rasmussen, J. O., 153, 165(48, 49), 167, 182(13), 202 Rasmussen, S. W., 427(109), 428 Rasmussen, V. K., 338, 357, 431, 432 (119), 434(119), 438(119), 443, 445 (144) Rausch, W., 290 Rawitscher, G. H., 251 Reardon, W. A., 339, 343(8), 373 Redfield, A. G., 107, 111 Rees, J. R., 479, 480, 481(217) Reibel, K., 517 Reilley, E. M., 431, 479(115) Reines, F., 296, 297, 408 Reinov, N. M., 164(1), 165 Reiter, R. A., 309 Renzetti, N. A., 59 Retherford, R. C., 61 Reynolds, C. A., 165(57), 167 Reynolds, H. L., 408 Reynolds, J. B., 77 Reynolds, J. H., 297 Reynolds, S. A., 569 Ribe, F. L., 451 Richards, H. T., 369, 371, 441(7) Richardson, J. G., 480 Richardson, J. R., 645, 686 Richert, R., 311 Richings, H. J., 311 Ridgway, S. L., 699 Ridley, B. W., 231, 232(79), 255(79), 256 (791, 257(79) Ries, R. R., 8 Riealer, W., 292
867
AUTHOR INDEX
Rispoli, B., 537 Ritchie, R. H., 228 Robert, J., 277, 282 Roberts, A., 111, 651 Roberts, J. H., 453, 556, 557(8) Roberts, L. D., 164(10, 36), 165(52, 61, 62), 166, 167, 169, 176(20), 182(19, 21), 202, 204, 207(106, 107) Roberts, T. R., 3, 7(8), 189 Robertson, J. C., 484 Robinson, C. S., 626 Robinson, E. S., 427(106), 428 Robinson, F. N. H., 164(8, 19, 22, 26), 166, 176, 181(4), 182, 187, 190(50), 200, 201 Robinson, G., 19 Robinson, H. P., 570 Robinson, W. A., 120 Robson, J. M., 293 Rockwood, C. C., 318 Rogers, B. S., 409 Rogers, F. T., 228 Rogers, J., 267, 271(203) Rohrlich, F., 215, 221(8) Rollett, J. S., 117, 811(12), 815 Romanowski, T. A., 309 Rose, M. E., 129, 134, 135(4), 138, 142, 145, 148, 150, 151, 152, 153, 155, 157, 161, 162, 170(6), 173, 176, 179, 194, 200(39), 206(4), 210, 212(39), 217, Z20(26), 223(26), 251, 254, 258, 272, 446 Rose, P. H., 588 Rosen, L., 388, 397, 417, 419, 426, 427 (106), 428, 448(78), 449, 451, 452, 453, 458, 506 Rosen, S. P., 295 Rosenblum, B., 20, 23(8), 28, 30(5), 32, 33(5) Rosenblum, E. S., 732 Rosendorf, R., 44 Rosenfeld, A. H., 845 Rosentsveig, L. N., 241 Rosenzweig, N., 134, 135, 208 Ross, R., 748 Rossi, A., 85, 149, 151,244, 246(141, 144), 247 (144), 248(144), 249 (144) Rossi, B., 407, 410 Rossi, B. B., 288, 375 Rotblat, J., 428 Rotenberg, M., 207
Rothem, T., 233 Roulston, K. I., 526 Rourke, F. M., 298, 571(51), 572 Rozentsveig, L. N., 235 Rozics, J. D., 237 Rubin, A. G., 441 Rubin, S., 431, 432(116), 434(116), 438, 439(116) Rudik, A. P., 254 Runnalls, 0. J. C., 556, 557(7), 561 Russell, B., 374, 461, 563, 564(24) Russell, J. T., 317 Rutherford, E., 276, 281, 438 Rutledge, A. R., 134
5 Sachs, D., 374, 461, 563, 564(24) Sadauskis, J., 500 Sadykhov, F. S., 237 Sailor, V. L., 165(57), 167 Saito, N., 302 Sala, O., 363, 374 Salant, E. O., 377, 385(34) Salinger, G. L., 176 Salmi, E. W., 407 Salomons-Grobben, N., 160 Salpeter, E. E., 108, 235 Samios, N. P., 834 Samoilov, B. N., 164(35), 165(35), 167, 170(11), 173, 184, 186(44) Sampson, M. B., 431, 432(119), 433, 434 (119), 438(119), 480, 481(217) Sanders, J. E., 410 Sanders, J. H., 116 Sanders, L. G., 557 Sands, M., 624 Sandweiss, J., 747 Sapp, R. C., 164(27), 166, 200 Sarkar, S., 235, 241 Satchler, G. R., 195, 198 Sawter, C. V., 209 Sawyer, G. A., 430 Schardt, A. W., 327, 333, 443 Scharenberg, R. P., 143, 146, 147, 311 Schawlow, A. L., 20, 23(4), 24(4), 25(4), 26, 27(15), 30(4), 31, 32(4), 33(4). 34(4), 86, 97(la), 99(la) Schecter, L., 428 Scheer, M., 534 Scherb, F., 385
868
AUTHOR INDEX
Scherrer, P., 138, 142, 143, 146(43, 50), 148(46, 50) Schiff, L. I., 317, 436, 468, 511, 534 Schlein, P., 758 Schmid, L. P., 169, 175(24) Schmidt, F. H., 244, 246(146), 262 Schmidt-Rohr, U., 284 Schmitt, H. W., 371 Schnell, W., 751 Schneps, J., 834 Schoen, A., 273 Schooley, J. F., 164(40), 165(49, 55), 167, 182(13), 194, 200, 202, 211(69), 212 (69) Schopper, H., 205, 260, 261, 264, 265 (200), 268(200), 269, 270, 271(200), 272(200) Schrack, R. A., 543, 551(49) Schrank, G., 419 Schriifer, E., 534 Schuhl, C., 485 Schulta, H. L., 392 Schumann, R. P., 298, 571(51), 572 Schupp, A. A., 216,224(16), 225,227(16) Schupp, G., 298 Schwarta, M., 834, 845 Schwartz, R. B., 363 Schwarzer, D., 297 Schwarzschild, A,, 244, 246(142) Schwebel, A., 575, 576(55) Schweidler, E., 276 Schwinger, J., 24, 630 Scobie, J., 298 Scolman, T. T., 3, 6(10), 7(10), 8, 33, 34 Scott, B. F., 567, 568(39) Scott, E. M., 817 Scott, G. G., 243 Scott, J. M. C., 241 Scott, M. B., 419 Scurlock, R. G., 164(18, 29, 33, 42, 44), 165(47, 60), 166, 167, 182(17), 186, 201 Seaborg, G. T., 278, 297, 354, 562, 572, 579 Seagondollar, L. W., 375 Seagrave, J. D., 363, 374, 381(19), 430, 443, 449(114), 451 Sears, B. J., 134 SegrB, E., 276,277,281, 563,564(27), 747 Seidl, F. G. P., 390,394(57), 395(57), 566
Seliger, H. H., 203, 288, 289, 539, 540 (38), 575, 576, 577 Sens, J. C., 224, 225, 227(63, 64) Septier, A., 709 Serber, R., 10, 626, 731, 735, 739(16) Serra, A., 537 Severiens, J. C., 164(4), 166, 335, 338 Seward, F. D., 485, 511, 514(6) Sewell, D. C., 385 Seyerlein, J., 534 Shafer, R. E., 511, 514(6) Shapiro, A. M., 834 Shapiro, M. M., 845, 846 Shapiro, P., 459 Sharma, R. C., 206 Sharp, W. T., 134 Sheffield, J. C., 571(51), 572 Sheldon, J., 561 Sher, R., 594 Sherman, N., 251, 252(149, 150) Sherr, R., 359, 385, 448 Shiren, N. S., 643 Shirley, D. A., 164(2, 40, 41), 165(48, 49, 51, 54, 55, 56), 167, 181(12), 182(10, 13, 15), 200, 202 Shoemaker, F. C., 659, 696, 707 Shore, F. J., 165(57), 167 Shortley, G. H., 46, 47(5) Shrader, E. F., 317 Shrader, F. S., 329 Shugart, H. A., 74, 76 Shull, F. B., 709, 732 Siegbahn, K., 157, 239, 358, 359, 431, 578(69, 73), 579, 732 Siekman, J. E., 164(16), 166 Siekman, J. G., 337 Silsbee, H. B., 60, 74, 78 Silverman, A., 676 Simmons, B. E., 429 Simms, P. C., 311, 323 Simon, A., 179,194,200(39), 211(66), 212 (39) Simon, F. E., 164(26), 166, 169, 170(2,3), 173, 201 Simon, Sir Francis, 176 Simons, D. G., 164(7), 166 Simpson, 0. C., 59 Simpson, R. E., 164(39), 167 Sinclair, D., 139 Singletary, J. B., 453 Sippel, R. F., 443, 446(146)
AUTHOR INDEX
Sjostrand, N. G., 390,393,394(57), 395(57) Skarsgard, H. M., 408 Sklyarevskii, V. V., 164(35), 165(35), 167, 170(11), 173, 184, 186(44) Skobkin, V. S., 297 Sliitis, H., 290 Sleeper, H. P., Jr., 619 Slichter, C. P., 124, 126(68), 172(2), 173 Sliv, L. A,, 152, 153, 161 Smales, A. A., 302 Smaller, B., 108 Smart, J. S., 169, 175(24) Smirnov, G. V., 252 Smith, B., 285 Smith, C. C., 577 Smith, D. B., 289, 290(48), 363, 430, 449 (114), 575 Smith, F. M., 747 Smith, J., 74, 77(22) Smith, J. R., 474 Smith, Lloyd, 635, 642 Smith, L. G., 5, 6, 7, 8(21) Smith, R. K., 372, 378(12), 449 Smith, W. V., 23 Snell, A. H., 293, 295 Snow, G., 374, 377(22), 378(22), 383(22), 385(22), 388(22) Snow, G. A., 377, 385(34), 845, 846 Snow, W. J., 517 Snowdon, S. C., 363, 376, 472 Snyder, C. W., 10, 341, 344(12), 431, 432 (116), 433, 434(116, 123), 436(123), 438(123), 439(116, 123), 476(123) Snyder, H., 10 Snyder, H. S., 663, 665, 692, 693(1), 694, 696(1, 5), 719, 726(1) Sodickson, L., 239 Softky, S. D., 333 Sokolov, I. A., 164(1), 165 Sokolowsky, W. W., 394 Solomon, I., 123 Soltan, A., 581 Sommer, H., 116 Sommerfeld, A., 235 Sosnovsky, A. N., 293 Sosnowski, R., 231 Sowter, C. V., 164(11, 23, 33), 166, 167 Sperduto, A., 355, 356 Spinrad, B. I., 618 Spivak, P. E., 231, 232, 252, 256(81), 257 (81), 293
869
Spohr, D. A., 176 Squires, E. J., 214 Stafne, M. J., 396 Stahl, R. H., 374, 385(20) Stanford, C. P., 164(10), 16 167 Stannard, F. R., 37 Stapleton, H. J., 181(12), 182 Stapp, H. P., 217, 220(24) Starfelt, N., 517, 520, 522(18), 523(18) Statz, H., 194 Staub, H., 407, 410, 475 Staub, H. H., 282, 288, 375 Stavinsky, V. S., 463 Stebbins, A. K., 408 Stech, B., 208, 220, 262 Steenberg, N. R., 139, 198, 199, 200, 206 Steenland, M. J., 164(3, 4, 15, 28), 165, 166, 168, 176, 186, 190(48), 198(12), 201, 202, 203(105), 209(102), 216, 269(17) Steffen, K. G., 557 Steffen, R. M., 129, 133, 142, 143, 145, 146, 148, 151, 200, 201(92), 254, 269, 270, 272(209) Stehle, P., 241 Steinberger, J., 834, 845 Steiner, H. M., 747 Steinwedel, H., 288 Stel’makh, M. F., 164(1), 165 Stelson, P. H., 139, 141(37), 149(37), 162, 428, 448 Stepanov, E. P., 164(35), 165(35), 167, 170(11), 173, 184, 186(44) Stephens, F. S., 153 Stephens, W. E., 450, 463, 464(190), 466 (190), 467(190), 484, 485 Stephenson, J., 239 Stephenson, T. E., 164(10), 166 Stern, O., 59 Sternheimer, R. M., 121, 694, 696, 699, 703(4), 705(4), 706, 712, 713(14), 732, 735, 736(3, 4), 737(3), 738(3), 739, 740(4), 741(4), 752, 828, 833, 833, 834, 844 Stevens, K. W. H., 86, 101, 179, 182(35) Stevenson, M. L., 758 Stewart, L., 452, 453, 560 Stierlin, U., 244, 246(147) Stitch, M. L., 27, 33(17) Stokes, R. H., 430
870
AUTHOR INDEX
Stolovy, A., 164(37), 165(53), 167, 169 Stone, It. 8., 619 Stone, T., 481 Stora, R., 220, 223(42) Storey, R. S., 484 Stork, D. H., 747, 758 Storrs, C. L., 385 Stovall, E. J., Jr., 427(106), 428, 430 Strandberg, M. W. P., 23, 24(9), 29, 32 (91, 95, 98 Stratton, T. F., 429 Strauch, Karl, 462, 463(184), 467(184), 468 Strauss, N. G., 318 Strom, P. O., 74 Strominger, D., 278, 354, 562, 572, 579 Stroot, J. P., 269 Stroth, J. E., 244, 246(146) Strube, G., 142 Strumski, C. J., 709 Stuart, R. N., 844 Stuart, R. V., 429 Stubbs, T., 758 Stump, R., 277 Suchorutchkin, 5. I., 393 Suess, H. E., 302 Sugarman, R. M., 317 Sugimoto, K., 147 Sunderland, R. J., 74 Sunyar, A. W., 260, 262, 269(180) Sutton, R. B., 309 Svartholm, N., 431, 732 Svelto, V., 309, 319(4) Swami, M. S., 834 Swank, R. K., 428, 429(109), 487 Swanson, 309 Swanson, W. P., 546, 551(56) Swartz, C. E., 497, 498 Sydoriak, S. G., 189 Symon, K. R., 678, 681(86), 683(86)
T Taconia, K. W., 188 Tallmadge, F. K., 419, 426, 427(106), 428, 506 Talmi, I., 366 Talyzin, V. M., 405 Tamas, G., 485 Tanttila, W. H., 120, 123 Tanner, N., 484
Tasehek, R. F., 371, 384(9), 397, 409, 441, 442(139), 448(78), 449,474, 475 (204), 566, 567(35), 568(35) Tassie, L. J., 233, 251 Tate, R. E., 561 Tauber, H., 302 Tautfest, G. W., 438, 487, 491, 501, 506, 546, 547(55), 549(55) Tavernier, G. C., 566 Taylor, A. E., 385 Taylor, H. L., 376, 403 Taylor, J. B., 71 Taylor, R. D., 274 Taylor, R. T., 164(19, 42, 44), 165(47, 60), 166, 167, 182(17), 186 Taylor, T., 619 Telegdi, V. L., 147,223,224,225(59),227 (63), 333, 336(55) Teller, E., 837 Temmer, G. M., 164(43), 165(43), 167, 200, 263, 359,445,446,447(156), 448 Templeton, D. H., 164(41), 167, 182(10) Tendam, D. J., 408, 479 Teng, L. C., 219, 220(35), 223(35), 225 (35), 648 Terentiev, V. V., 297 Ternonick, A., 77 Terrell, J., 408, 409 Thaxton, H. M., 422 Theilacker, J. S., 603 Theis, W. R., 239, 241, 263(121) Thirion, J., 333, 336(55) Thode, H. G., 285 Thomas, G. E., 390,393,395(58), 396(58) Thomas, H. A., 116, 122 Thomas, L. H., 683 Thomas, R. G., 337, 358, 384, 445 Thompson, R. W., 345, 346, 347, 833 Thompson, T. J., 618 Thomson, G. P., 490 Thomson, J. J., 1 Thornton, R. L., 477, 647 Thun, J. E., 158 Ticho, H. K., 755, 758 Titterton, E. W., 464 Tobailem, J., 277, 281, 282 Tochilin, E., 562 Tolansky, S., 54 Tolhoek, H. A., 162, 164(3, 6, 28), 165, 166, 168, 195, 196, 197, 198(12), 200, 201, 202, 203(105), 204(76), 206, 209
87 1
AUTHOR INDEX
(102), 216, 217(14), 219, 223(14, 331, 249(14), 250(14), 252(14), 255(14), 256(14), 264(14), 265, 269(17) Toller, A. L., 385 Tollestrup, A. V., 19, 433, 436(122), 443 (122), 444(122), 445 Tomkins, F., 58 Toppel, B. J., 443, 446(147) Toptygin, I. N., 258 Torrey, H. C., 104, 107(1), 108(1), 111, 113(1) Towle, J. H., 333, 336(59), 337(59), 339 (59) Townes, C. H., 20, 22, 23(4, 6, 8), 24, 25 (4), 27, 28(5), 29, 30, 31, 32(4, 5), 33(4, 5, 17), 34(4, 6), 86, 97(la), 99 (la), 121, 192 Trail, C. C., 441, 526 Trambarulo, R. F., 23 Treacy, P. B., 147 Treibwasser, S., 61 Trenam, R. S., 181(5), 182 Tretihkov, E. F., 254 Tribuno, C., 464 Trigger, K. R., 116 Tripp, R., 748 Trischka, J. W., 28, 33 Trump, J. G., 585 Trumpy, G., 216, 269(18) Tsai, Y. S., 241 Tuberfield, K. C., 116 Tuck, J. L., 430, 648 Turner, C. M., 327,328(35), 329(35), 587 Turner, J. F., 231, 232(79), 255(79), 256 (79), 257(79) Twiss, R. Q., 653 Teara, C., 485, 511, 514(6)
U tZberall, H., 220, 235, 236 Ullman, J. D., 244, 246(144), 247(144), 248(144), 249(144) Untermyer, s.,618
v Valuev, B. N., 463 Van de Graaff, R. J., 584, 585, 587(11), 588 Van der Leun, C., 354
Van der Vlugt, N. J., 176 van Dijk, H., 189 Vanetsian, R. A., 481 Van Hoomissen, J. E., 462, 463(182), 466 (182), 469(182) van Lieshout, R., 157, 160, 208, 354, 570, 571(50), 578(50), 579 Van Loef, J. J., 459, 460(173) van Nooijen, B., 159 Van Patter, D. M., 12, 371 Van Rossum, L., 747 Van Vleck, J. H., 211, 212(131) van Wageningen, R., 164(16), 166 Varian, R., 123 Vegors, S. H., 333 Veksler, V. I., 636, 645(27), 648, 649, 651, 751 Verbinski, V. V., 450 Vincent, D. H., 291 Vincent, L. D., 472 Vise, J. B., 244, 246(142) Vishnevskii, M. E., 228, 230(77), 254 Vishop, G. R., 164(22), 166 Visotskii, G. L., 235 Visscher, W. M., 274 Vitale, S., 260, 272 Vladimirski, V. V., 394, 668 von Friesen, S., 41, 42, 43(8) von Herrmann, P., 428 Von Hipple, A. R., 105 von Issendorff, H., 228, 230(75), 252, 255 (75), 256(75), 257(75) von Schweidler, E., 283 Voshage, H., 4
W Wadey, W. G., 392 Waggoner, M. A., 153 Wagner, L. R., 302 Walchli, H. E., 122 Wald, A., 817 Waldman, B., 462, 463(182), 466(182), 468( 182) Waldmann, L., 16, 18(5) Waldorf, W. F., 479 Walker, R. L., 443, 536, 545, 546(52), 549(52), 563, 564(30), 566(30), 567 (35), 568(35) Wall, N. S., 479, 480 Wallis, W. A., 813
872
AUTHOR INDEX
Walt, M., 388, 397, 407, 448(78), 474 Walter, F. J., 204 Walter, M., 151 Walter, R. I., 102, 103(10) Walters, M. C., 297 Walton, E. T. S., 580 Walton, R. B., 388, 452, 472, 474, 475 (204) Wang, T. C., 111 Wangsness, R. K., 107, 108, 110(15), 111 (15) Wanlesa, R. K., 285 Wanlass, S. D., 345 Wapstra, A. H., 9, 17(1), 153, 154, 157, 158,159,160,208,260,269(181)339, 340(11), 342(11), 570, 571(50), 578 (50, 73), 579 Ward, A., 484 Ward, A. G., 618 Ward, I. M., 102, 103(14) Warren, R. E., 384, 476(42) Watson, M., 758 Watson, R. E., 170(13), 173 Wattenberg, A., 374, 461, 556, 563, 564 (23, 24) Way, K., 134, 354 Weaver, H. E., 111, 114 Weaver, R. S., 322 Webb, T. S., 438, 439, 477, 478(133) Weber, W., 327 Wegener, H., 228, 230(75), 252, 255(75), 256, 257(75) Wegner, H. E., 330, 430, 479, 480 Weills, J. T., 618 Weinberg, A. M., 591, 611, 618 Weinstein, R., 239 Weiss, M. M., 102, 103(10) Weisskopf, V. F., 278, 353, 362, 363, 366, 381, 383(38) Wells, F. H., 316 Welton, T., 251 Welton, T. A., 194 Wende, H., 4, 7(11) Weneser, J., 152, 153(1) Wenny, D. H., 247 Wenzel, W. A., 432, 751, 752, 759 Werle, J., 220 Wessel, G., 71, 77 West, H. I., Jr., 151 West, J. M., 618 Westenbarger, G. A., 165(51), 167
Westendorp, W. F., 626, 629 Wetherill, G. W., 302 Wexler, S., 74, 77 Whaling, W., 12(8), 19, 371, 431, 432 Wheatley, J. C., 164(20,21,28), 166, 176, 201, 209(102), 216, 269(17) Wheeler, J. A., 36 White, F. A., 571(51), 572 White, H. E., 46, 48 White, H. S., 747 White, M. G., 428, 630, 659 White, R. L., 26, 27(15) White Grodstein, G., 157 Whitehead, M., 747 Whitley, S., 181(7), 182, 189 Whitmore, B. G., 558 Whittaker, E., 19 Whittle, C., 164(15), 166 Wick, G. C., 31, 162, 382 Wideroe, R., 623, 634(1) Wiedenbeck, M. L., 134, 135 Wiegand, C., 277, 502, 747 Wiegand, C. E., 281 Wiggins, J. S., 483 Wightman, A,, 139 Wightman, A. S., 220 Wigner, E., 215 Wigner, E. P., 363, 591 Wilcox, R. M., 239 Wiles, D. R., 285 Wilhelmi, Z., 231 Wilkinson, D. H., 139, 336, 365, 834 Willard, H. B., 228, 230, 255(78), 256 (78), 257(78), 259(78), 375, 474 Williams, D., 111 Williams, E. J., 422 Williams, J. H., 371, 373,384(9), 387(16), 388(16), 419,426, 427(109), 428, 449, 481, 506, 566, 567(35), 568(35) Williams, W. S.C., 217 Williamson, R. M., 385 Willis, E. H., 302 Willis, W. J., 846 Wilson, E. B., Jr., 25, 808, 811, 812(1), 819(1) Wilson, E. D., 277, 281 Wilson, R. R., 545, 546(54), 547, 548(58), 653 Wimett, T. F., 594, 595(5) Winhold, E. J., 450, 463, 464(190), 466 (190), 467(190), 468
873
AUTHOR INDEX
Winsberg, Lester, 501 Winter, R. G., 296 Wintersteiger, V., 445 Winther, A., 208, 359 Wojtkowska, J., 231 Wolf, G., 291 Wolf, W. P., 181(2, 7, 14), 182, 189 Wolfenstein, L., 214 Wolff, M. M., 484 Wolfson, Y., 327 Wong, E., 102, 103(17) Wood, D. E., 453 Wood, E., 385 Woodbury, H. H., 445 Wooten, J. K., 207 Worcester, J. L., 75, 76, 102, 103(19) Worthing, A. C., 816 Worthington, H. R., 419, 482 Wrage, E. G., 299 Wright, D. U., Jr., 558 Wright, G. T., 484 Wu, C. S., 164(24, 32), 166, 167, 202, 203 (104), 204(104), 216, 244, 246(142), 260, 261(188), 323, 359, 394 Wyckoff, H. O., 532(27), 533, 534, 539, 540, 541(40) Wyckoff, J. M., 516,517(10, ll), 518,519 (17), 520(10), 521(17), 528 Wyld, H. W., 254
X Xuong, N. H., 758
Y Yaffe, L., 288, 289, 290(46), 568, 569, 572, 575, 576
Yamamoto, S. S.,846 Yang, C. N., 216, 844 Yates, G., 811, 813, 814, 818(5) Yeater, M. L., 392 Yekutielli, C., 44 Yergin, P. F., 463, 470(186) Yntema, J. L., 428 Yokosawa, A,, 232, 584, 587(10) Yoshizawa, Y., 579, 580 Ypsilantis, T., 747 Yu, F. C., 122 Yuan, L. C. L., 376, 377, 385(34)
z Zacharias, J. R., 59, 60, 61, 72, 76(16), 77 Zahringer, J., 299 Zajec, E., 709 Zappa, L., 260 Zatsepina, G. N., 463 Zeiger, H., 29 Zeitler, E., 533 Zel’dovich, I. B., 235 Zendle, B., 541 Zerby, C. D., 517 Zichichi, A., 224, 225, 227(63, 64) Ziegler, B., 528 Zimmerman, J. R., 111 Zimmermann, R. L., 709 Zinn, W. H., 618 Zinov’era, K. N., 188 Ziof. T.. 759 Zdbel, W., 142, 146 Zucker, A., 408, 633 Ziinti, W., 151 ZupanEiE, C., 446
Subject Index A Absorber-shift method, 336-337 Absorption in a cavity containing a paramagnetic sample, 88 cells, gas, 24-28 in a gas-filbd waveguide, 87-88 magnetic resonance, 106-111 self-, with @-raysource, 290 Accelerators, 623 ff d.c., 580 ff fixed-field alternating-gradient, 678686 radial sector, 678-680 spiral sector, 680-681 linear, 634-645 electron, 642-645 proton, 635-642 pulsed, as neutron source, 392-393 Accuracy of mass spectroscope mass values, 7-9 of measurement, 808 Activation cro ss-section, neutron, 407411 Adiabatic slow passage, 108-1 10 Age determinations, 298-302 Airy formula, 52 Alignment electric hyperfine structure, 183-184 of magnetic hyperfine structure, 176183 nuclear, 163, 174 Alpha particle counters, proportional, 286-288 sources neutron, 556-562 radioactive, 567-572 (a,n)neutron sources, 560 ?-radiation from, 561-562 Analyzers, magnetic, reaction product, 431-440 Angle criterion, maximum, for outcoming particle identification, 828-831 mean scattered, charged particle in emulsion, 37-38
polarization, 225 time rate of change, 225 ff solid, of charged particle beams, 506508 Annihilation -in-flight in polarized target, 262-263 positron, 238-239, 262, 485, 511-514 of slow positrons, 263-264 Antiproton beam, 752-755 Aperture effective, 710-711 finite, correction, 247-248 Asymmetry function &O), 250-252 ratio, polarimeter, 141 Attenuation, coefficient, photon total, 518-520, 529-531 AU'~*decay scheme, 159
B Back scattering in Faraday cup, 490-491 Be(a,n) reaction, 556-560 Be9 bombardment with deuterons, 357358 Beam(s) absolute charged particle, measurement by counting, 494-495 atomic, deflection due to magnetic field, 69 colliding, 686-689 high momentum, using counters as detectors, 752-755 monitoring based on induction electrodes, 495-498 neutron, from reactors, 572-580 parallel, of charged particles, 485 ff polarized, 216-223 separators, 747-760 degrader type, 747-748 electromagnetic, 748-750 nuclear interaction, 751-752 radiofrequency, 750-751 size, finite, correction for, 419 transport systems, 691-760 examples, 752-760 874
SUBJECT INDEX
Bending, beam, 692-747 B1“energy level diagram, 361 Beta-particle back scattering, 288 counters, 80 sources in nuclear orientation, 203 radioactive, 572-580 Betatron, 626-630 Bliabha scattering, 242 Block theory of magnetic resonance absorption, 106-1 11 Branching ratios, 297-298 for mesons and hyperons, 846 Breit-Rabi diagram for CslSz,64 Bremsstrahlung, 235-237, 509-51 1 Bridge balanced microwave, with superheterodyne detection, 26-27 circuit for d.c. comparison of capacit ances, 488-490 Bubble chamber measurement of mass, 35-37 separated beams for, 755-759
C Calibration of radioactive neutron sources, 565-567 Calorimeter differential, 282-283 X-ray, 541-543 Camera, multiplate, for measuring differential cross-section, 426-428 “Centroid shift” method, 321-322 Cerenkov monitors, 502 spectrometer, 53 1 Cesium beam decay, 67-68 Chain reaction, fission, 591 Channeling of atomic beams, 76 Chi-square estimate of, 804-806 function, 811 test, Pearson’s, 786-789 Chronotron, 316-317 Circuits, scaling, 795-799 constant number of counts, 796-797 constant time interval, 795-796 varying time and counts, 797-799 Clocks, electronic, 308-320
875
Cloud chamber measurement of mass, 35-37 Cockroft-Walton accelerator, 581-584 Coincidence(s) accidental, rate of, 292 delayed-, measurement of, 316 measurement analysis, 320-327 method t o measure specific activities, 291 rate, radiation, 132, 136-137 Collection of radioactive atom beams, 76-78 Collisions between counter-circulating beams, 686-689 Compton polarimeter, 139-142 scattering from polarized ferromagnets, 264 Condensers, cylindrical and spherical, 228-231 Control rods and drives, reactor, 603 Conversion coefficients correlation function, 152-162 from p and y ray intensities, 154-157 for mixture of two multipolarities, 160 special methods, 157-160 Y-shell, 152 data, use of, 160-162 Converter balanced-, measurement of photon total beam, 544 time-to-height, 312, 313 ?-ray, 156 Coolant, reactor, 602 Cooling magnetic, 186-190 very low temperature, 175-176 Copper activation measurement of relative photon beam, 546-547 Correlation angular, 129-151 determination of spins, 131-137 of conversion electrons, 137-138 directional, between two successive y-rays, 133-135 experimental tests, 148-149 function , 150-15 1 conversion coefficients, 152-162 theoretical directional, 132-136 y-y, procedure in, 149-150
876
SUBJECT INDEX
Coulomb energy difference between two nuclei, 363-364 excitation by passing charged particles, 446-448 scattering, 233 Counters, particle, 78-80 alpha proportional, 286-288 beta ray 47~,288-290 BF,, 375, 393 cerenkov differential high pressure, 758-760 characteristics, 503 gas ionizing, 308 gas recoil, 375-376 Geiger-Muller, 442 nonpolarizable, 793 polarizable, 792-793 radioactivity, 283-284 scintillation, 307, 376 NaI(Tl), 445 telescope geometry, 503-505 solid semiconductor ionization, 308 telescopes, 441 relative monitor, 502 Counting-rate mean, 799-800 meters, 799-802 average of continuous readings, 801802 average of 12 independent readings, 800-801 Counting, single-channel, effects of resolving time, 792-794 Coupling interionic, effect on nuclear orientation, 199-200 quadrupole, 147-148 Cross-section capture, neutron, 407-411 determination by magnetic reactionparticle analyzers, 434 ff diffwent ial elastic, 472-482 interaction, 411-472 evaluation of errors, 424-426 magnetic analyzers, 431-440 sources of charged particles, 413416 thick target, 436-439 thin target, 435-436
for neutron-induced reactions, 448460 for elastic scattering of photons by nuclei, 517-518 for electron-electron scattering, 241243 for 7-induced reactions, 461472 for Mott scattering, 250 for production of thin-target bremsstrahlung, 520-524 total for Compton effect, 267 interaction, 366-396 back-angle, 370 monoenergetic fast neutrons, 366388 using polyenergetic neutrons, 388396 measurements with fast neutrons, 385-388 with slow neutrons, 394-396 Crystal growth from aqueous solut,ion, 184-186 Curie formula for nuclear magnetization, 107 Current integrator circuit, 418 Cyclotron, 630-634 azimuthally varying field, 683-686 electron, 648-651 “frequency-modulated,” 645-651 “sector-focused,” 678-686
D Data rejection, 813-814 statistical characterization, 771-780 Dating of large ages, potassium-argon method, 300 radiocarbon, 301-302 Decay direct observation of, 280-285 double beta, 293-297 of free neutron, 293-295 lifetime, radioactive, 275-279 modes for mesons and hyperons, 846 number of per angular interval, 832 per unit energy interval, 832
877
SUBJECT INDEX
radioactive, 359-360 law of, 766 sources of photons, 514-515 sequence, intermediate state of, reorientation effects, 199-200 two-body, kinetics of, 831-833 Deltamax, 247 Density matrix describing polarization state, 221-222 Depolarization, 248-249, 257-259 Detection crystal video, 24-25 modulation signal, 90-92 particle, of atomic beams, 71-85 of photodisintegration processes, 462 ff sensitivity in microwave spectrometer, 89 Detector (9) Compton y-ray, sensitivit.y of, 204-206 used in differential cross-section measurements, 426 for elastic-scattering experiments, 477479 for electronic timing, 306-308 energy-sensitive, 413-430 neutron, 375-376 particle, solid state, 482-483 recoil-proton, 441, 453 for slow neutron measurement, 393 Deviation, standard, 774-776, 803, 810 fractional, 805 Diffraction grating, 53 Discharge tube, hollow cathode, 51-52 Disintegration rate, radioactive a-source, 568-570 Dispersion index binomial, 790 Poisson, 788-791 law of mass spectrometer, 5-6 Distributions binomial, 761-762 composite, 780-786 frequency, 761-771 Gaussian, 810 interval, 767-771 multinomial, 762-763 normal, 763-764, 810 parent, 809 Poisson, 764-767
Dop p 1er broadening, 50 of cross-section resonance, 395 shift measurement of recoil nuclei stopping, 336-337 Drift tube structure, proton linear accelerator, 639-640
E Electron(s) conversion, directional correlation of, 137-138 cyclotron, 648-651 -electron scattering detection, 244 ff polarization dependence, 239-249 linear accelerator, 642-645 motion in electromagnetic field, 222233 nuclear double resonance, 127-129 polarization, 214-274 conversion, 253-255 detection, 239-264 longitudinal, 256 non-relativistic, 219 relativistic, 219-220 transverse, 252-256 polarized target, 243 a t rest, polarization of, 217-219 synchrotron, 653-656, 676-678 Emission angular distribution a-particle, 206-207 pparticle, 209-210 y-ray, 207-209 charged particle from charged particle induced reactions, 413-440 neutral particle from charged particle induced reaction, 440-448 secondary electron from Faraday cups, 491 monitor, 501 Emulsion measurement of nucleon mass, 37-44 measurement of scattered particle flux, 505-506 ENDOR, 127-129 Energies end point, 359
878
SUBJECT INDEX
threshold for unstable particle production, 833-838 Energy of annihilation photons, 512 binding, 1, 2 due t o electric quadrupoIe moment, 48 equivalent of a mass of nuclear magnitude, 339 interaction, nuclear quadrupole, 97 kinetic, of cyclotron particles, 631 levels nuclear, from reaction energies, 352366 rotational, of symmetric-top molecule, 20-21 lost per revolution, betatron, 630 due t o magnetic moment, 46-47 nuclear binding, change during fission, 592-593 reaction, of nuclear reaction, 339-352 threshold, 340-341, 346-349 endoergic reaction, 346-347 X-ray total beam, 537-553 Entropy, molar, 212-214 Equilibrium, radioactive, 275-276 Errors accidental, 808 of computation, 819-821 of direct measurements, 812-813 probable, 778 propagation, 783-784 in indirect measurements, 814-816 round-off, 819 standard, 776-778, 803, 810 systematic, 808 truncation, 819 Excitation energies, determination, 354363 Exposure-dose, 537, 539-540
F Factorial, 763 Fnraday cup, 416-417, 486-494 Field index, magnet, 731 magnetic, nuclear magnetic resonance measurement of, 122-123 First moment method, 321 Fisher’s t function, 812-813
Fission cross-section, 407-411 neutron sources, 562 Fit, goodness of, tests for, 786-792 Flight-distance, direct, measurement of lifetime, 335-336 Fluctuations, statistical, in nuclear processes, 761-806 Flux average primary neutron, at saatterer, 454 charged particle, 485-507 decrease rate, 367 densities of particles scattered from a target, 502-507 at detector of doubly scattered neutrons, 382 distribution, H FB reactor, 601 fast neutron, absolute measurement of, 440-44 1 Fluxmeters, 434-435 Focusing action of magnetic lens, 696-699 alternating-gradient, 663-664, 693 beam, 692-747 double, 692 ff of two-magnet system, 704-706 equations for wedge-shaped magnets, 731-747 properties of two-magnet system, 701712 strong, 693 ff with electrostatic lenses, 70&709 system with momentum analysis, 742746 vertical, by fringing field of wedgeshaped magnet, 732-734 Foil, magnetized ferromagnetic, polarized electrons from, 243 Forces, betatron, 735 Frequency absorption, pure rotational, 21 distributions, 761-771 of operation, proton linear accelerator, 638 resonance, chemical shift of, 121 revolution, synchrocyclotron, 646 shift correlation with mass change, 29-35 isotopic, 21-22 Fuel elements, reactor, 601
879
SUBJECT INDEX
G Gain, intensity due t o magnet system, 710 due to rectangular quadrupole, 712-713 Gamma ray counters, 78-80 decay of nuclear states, 357-359 emission, 442-448 from neutron-induced reaction, 459460 flux measurement, 527-528 -induced reaction cross-sections, 461472 radioactive sources, 572-580 transmission through polarized absorber, 267-269 Gases, microwave spectroscopy in, 93100 Geiger-Miiller counter, 442 Generator, electrostatic, 584-590 for elastic scattering measurements, 476 g-factor, 224 Ground state, 352 Gyromagnetic ratio of intermediate state in a y-y cascade, 144 nuclear, 122-123 angular correlation determination, 142-147 proton, 122-123
H Half-life, 275, 278 Hazards, nuclear reactor, 605-606 Heat flow in very low temperature cooling, 175-176 influx during magnetic cooling, 182183 production rate due t o radioactivity, 286 Heating effect of radioactive substance, 187-188 local, of gas target along incident beam path, 423 Helicity electron, 215, 219 determination using bremsstrahlung, 259-262
of y-rays by Mossbauer effect, 273-274 by transmission, 267-269 persistence, 215, 223 positron, from annihilation-in-flight, 262 transfer efficiency, 234 polarization measurement depending on, 259-262 Hgle7 decay scheme, 159 Hgeos decay scheme, 159 Hornyak detector, 466 Hs(d,n)Hea reaction, 371 HJ(d,n)He4reaction, 372 HJ(p,n)Hes reaction, 371
I Image aberrations, mass spectrometer, 7 Inscattering correction of neutron cross-section, 378-382, 394 multiple, 382-383 Integrator, fast, 497 Intensity, photon beam, 508 Interaction between electrons and photons, 233-235 Interferometer, Fabry-Perot, 53-54 Intervals confidence, 785-786 for standard deviation of a normal distribution, 791-792 distribution, 767-771 3-fold, 768-771 involving s random events, 768-771 selection, “overlap” method, 310-311 between successive random events, 767-768 tolerance, 812 Ion(s) paramagnetic, in solid state, spin Hamiltonian for, 178-179 retention on metal surfaces, 77-78 Ionization chamber, 280-282 high energy, calibration, 547-553 low energy free-air, 539-541 as relative beam monitor, 499-501 -range measurement of mass in cloud chamber, 36-37
880
SUBJECT INDEX
Irradiation facilities, reactor, 608-61 1 Isochromat, 468 Isotope production and preparation, 80-85 shift, 45-46
Li’(p,n)Be’ reaction, 369-370 back-angle total cross-section, 385, 386 Lund photometric method, 41-43
M
J Jacobian for particle reachions, 825-838 Jeffries method of nuclear orientation, 192
K Kinematics of particle reactions, 821-844 two-body decay, 831-833 Klein-Nishina formula for Compton scattering, 139
L Larmor frequency, 105, 143, 227 Least count, instrument, 807 Least’ squares method, 16-18, 816-819 Lens defocusing, matrix for, 721-723 equation, magnet system, 696-719 focusing, matrix for, 719-720 high-current focusing, 713-714 magnetic defocusing action, 699-701 focal length, 696, 699 focusing action, 696-699 Life, mean, from counting rate, 320 excited state, 303 Lifetime deiermination, 275-338 long, 275-302 short, 302-338 excited state, 353 Light scintillation, collection of, 307 sources atomic beam, 52-53 for hyperfine structure measurement, 50-53 Limiter circuits, 317-318 Linewidth of microwave gas spectra, 2324
Magnet ( 8 ) alternating gradient synchrotron, 665 Cambridge electron accelerator, 677 design electron synchrotron, 654 proton synchrotron, 657-659 geometry, AVF cyclotron, 679 ff for nuclear and quadrupole resonance, 112 pole shaping, betatron, 628 quadrupole, 692-694 comparison with wedge-shaped magnets, 746-747 focaI properties, 756 with rectangular aperture, 712-713 system focal length, 726-731 lens equation, 696-719 position of focal planes, 726-731 “strong focusing,” matrix method of calculation, 7 19-73 1 three-, matrices for, 724-726 two-, matrices for, 723-724 wedge-shaped focusing equations, 731-747 focusing with uniform field, 740-742 horizontal focusing, 734-735 for n < 1, 736-738 for n > 1, 738 vertical defocusing for n < 0, 738740 vertical focusing by fringing field, 732-734 for n > 0, 735-736 Magnification lateral, double focusing magnet system, 709, 729-730 longitudinal, magnet system, 730, 740 wedge-shaped magnet, 739 Mass(es) atomic from microwave spectra, 20-35 from nuclear reaction and disintegration energies, 9-20
881
SUBJECT INDEX
comparison, doublet method, 5-7 determination of nuclei and particles, 1-44 difference ration, 33 excess, I 1 atomic, 342 number, 1-2 reduced, 45 unit atomic, 1, 9 relative nuclidic, 1, 9 Mean values, 772-773, 803, 809-810 difference of two, 784-785 Measurement evaluation, 807-821 statistical control of, 810-81 1 Median value, 773 Microwave spectra, atomic mass determinations from, 20-35 spectroscopy, 23-29, 85-103 in gases, 93-100 high resolution, 92-93 paramagnetic resonance, 95-96, 100103 Modal value, 773 Modulation in microwave spectroscopy of gases, 90-92 Microtron, 648-65 1 Molecule centrifugal distortion due to rotation, 30 symmetric top, 20-21 MZller scattering, 242 Moment (s) evaluation of, 49-50 of inertia, molecule, contribution of molecular electrons to, 31-32 nuclear, determination, 44-214 indirect methods, 129-147 microwave method, 85-103 spectroscopic methods, 44-129 nuclear quadrupole, 99 quadrupole, determination, 120-121, 147-148 Momentum acceptance focusing system with, 742-743 image width due to, 716-719 dispersion at image, 714-716 wedge-shaped magnet, 739
-ionization measurement in cloud chamber, 35-36 -range measurement in cloud chamber, 36 Monitoring, induced radioactivity, 501 Monitors, charged particle, primary, 486-498 secondary, 499-507 Monochromator, neutron crystal, 392 Mossbauer effect, 273-274 Mott scattering, 249-259 Multiplication, neutron, 405-407
N Neutron background extraneous, from beam direction, 377-378 room scattered, 376-377 cross-section differential elastic, 472-475 nonelastic, 397-411 total, transmission measurement, 367 ff emission, 440-442 flux, nonexponentiak attenuation, 383385, 394-395 generated per nonelastic collision, 405 ff induced reactions, leading to emission, 448-460 charged particle, 449452 y-ray, 459-460 neutron, 452-459 proton, 450-451 moderating material, 602 prompt fission, energy distribution, 594-595 -proton scattering total cross-section, 385, 386 sensing instrumentation, 604-605 sources monoenergetic, 369-373 radioactive, 555-567 Slow, 390-392 threshold measurement, 360-362 Noise power, crystal, 89 Nuclei mirror, 365 oriented, production of, 168-184
882
SUBJECT INDEX
dynamic methods, 171, 190-194 static methods, 168 ff transient methods, 172, 190-194 Wigner, 365 Nucleon mass measurement in emulsions, 37-44 constant sagitta method, 37-41 photometric method, 41-43 Nucleus ground state configuration, 352 initial recoil velocity, 333
0 Orbits, particle, in FFAG accelerator, 681-682 Orientation, nuclear, 162-214 degree of, 194-201 dynamic, 124-127 ferromagnetics and antiferromagnetics in, 184 Overhauser method, 191-192 paramagnetic salts for, 181-182, 184186 Oscillations, “betatron,” 628, 648, 735
P Packing fraction, 2 Pair production, polarization in, 237-238 spectrometer measurement of phot,on beam energy, 546 Parity determination, 44-214 angular correlation, 137-142 indirect methods, 129-147 spectroscopic methods, 44-129 Particles, elementary, intrinsic properties, 845 “Peak matching” technique, 6 Phase acceptance, 637 factors for two- and three-particle final states, 838-844 stability alternating gradient synchrotron, 666-668 electron cyclotron, 652-653 linear accelerator, 636-638 synchrocyclotron, 646-647
Photodisintegration, 461-472 using bremsstrahlung, 468-472 Photoelectric effect, polarization in, 238 Photomultiplier tubes a t low temper,ature, 203-204 Photon beam energy absolute high-energy, 541-546 relative, 546-547 beam sources, 509-515 bremsstrahlung, 509-511 monoenergetic of fixed energy, 514515 positron annihilation-in-flight, 511514 -difference method, 469-472 flux, differential, 508-553 polarization, 196, 197, 214-274 detection, 264-274 Photoneutron sources, 562-565 Photopeak area, 156 Plot, parametric, 135 Po-a-Be neutron source, 556-559 Poisson distribution, 764767 index of dispersion. 789-791 statistics applications, 792-802 Polarimeter Compton, 264-273 backward scattering, 272 Beard-Rose method, 272-273 forward scattering, 269-272 theoretical background, 264-267 transmission method, 267-288 y-ray, 138-142 Polarization in crossed fields, 227 degree of incident y-ray beam, 141 -direction correlations, 138 of electrons, 214-274 external field, 169-176, 225 linear of -prays, 138-142 in longitudinal fields, 225-226 of magnetic hyperfine structure, 176183 measurement, 201-206 nuclear, 163, 174 of assembly of nuclei, 169 photon, 196, 197, 214-274 in transverse electric field, 226 in transverse magnetic field, 226-227
SUBJECT INDEX
Positron annihilation-in-flight, 238-239, 262, 485, 511-514 cross-section per MeV, 512 with polarized electrons, 262-264 decay in magnetic fields, 264 Potentiometer, slide-back, 486 ff Power, photon beam, 508 Precession of atomic beam, 70-71 Precision vs accuracy, 779-780 of measurement, 808 Probabilities, combined, 780-781 Processes, independent random, superposition of several, 781-783 Proton gyromagnetic ration, 122-123 linear accelerator, 635-642 from neutron-induced reactions, 45045 1 synchrotron, 656-663, 668-676 Pulse duration of rf accelerator, 329 -generator circuit, basic, 498
Q &-equation, 341, 343 inelastic scat,tering, 350 relativist,ic, 344 &-surface representation of Thompson, 346 “Quanta, equivalent,” 508 Quantameter, 544-546 Quantum number m,magnetic, 147 “ z component,” 353 Q value(s) for mesons and hyperons, 846 negatron decay, 351 nuclear reaction, 9 ff, 339-352, 354-357 positron decay, 351-352
R Ra-a-Be neutron source, 558-560 Radionuclei, short-lived, investigation by atomic beams, 58-85 Range-energy relation of particle in emulsion, 37
883
Reaction cycIes, 14-16 endoergic, 340 exoergic, 340 nuclear, reaction energy of, 339-352 product masses, 429-430 Reactivity, reactor, 596 ff Reactor engineering, 601-605 fuel handling, 605 graphite moderated, 611-614 heavy water enriched uranium, 618-619 natural uranium, 619 high flux beam, 620-622 isotope, 622 kinetic behavior, 595 ff light water moderated pool, 614-618 neutron source, 590-622 tank-type light water, 618 thermal column, 609-610 water boiler, 611 zirconium hydride enriched uranium, 619-620 Recoil(s) methods, nuclear, for lifetime measurement, 333-338 proton, in nuclear emulsion, 455 Rectifiers, cascade, 581-584 Resolution finite angular, correction for, 419-420 finite energy, correction for, 247-248 of mass spectroscope, 4 Resolving power, theoretical, of diffraction grating, 53 Resonance curve saturation, 109 electron nuclear double, 127-129 magnetic corrections in, 121-122 quadrupole splitting of, 119-120 molecular beam electric, 28-29 nuclear magnetic and quadrupole, 104129 experimental apparatus, 111-115 quadrupole, 119-121 states, particle, 847-848 Response function for coincidence scintillation spectrometer, 529, 530
884
SUBJECT INDEX
for single-crystal scintillation spectrometer, 516-517 Roentgen below 3 Mev, 539 Rubidium isotopes, 81-85
5 Salts, paramagnetic, for nuclear orientation, 181-182, 184-186 Sample(s) mean, 810 scattering, for transmission total crosssection, 373-374 Scalar regularizing action, 771 Scattering diffraction, of neutrons, 381 multiple in absorbers, charged particle loss due to, 420-422 of reaction products in gas target, 422-423 in spherical shell, 400, 401-403 plural and multiple in Mott scattering target, 256-257 of protons and deuterons from B10, 355-356 -range measurement of mass in cloud chamber, 37 Scintillation duration, 307 monitors, 501 spectrometer measurement of photon total beam energy, 543-544 Scintillators, large liquid, 408-409 Shower method measurement of photon total beam energy, 544-546 Shutter methods, electronic, for lifetime measurement, 327-333 microwave cavity, 330-331 radiofrequency deflecting, 327-329 Signal-to-noise ratio for nuclear induction apparatus, 112-1 13 Sign conventions in polarization, 222-223 Slit-edge penetration by charged particles, 423424 Sml*z fission product, 599 Sources artificial particle, 580-690 low energy, 580-622 medium and high energy, 623
neutron nuclear reactor, 590-622 radioactive, 555-567 photon beam, 509-515 radioactive, 555-580 a-particle, 567-572 &particle, 572-580 7-particle, 461-462, 572-580 Specific activity method to measure halflife, 279, 286-292 Spectra, optical, hyperfine structure in, 44 ff Spectrograph magnetic, 433-43 Spectrometer anticoincidence scintillation, 524-528 coincidence scintillation, 528-531 Compton, 533-534 gas maser, 29 magnetic, 431-433, 531-537 mass, double focusing, 2-3 millimeter wave, 28 nuclear induction, 113-115 pair, 534-537 three-crystal, 446, 447 Walker and McDaniel, 443 photon, 515-537 single-crystal scintillation, 516-524 Stark-modulation, 25-26 high temperature, 27 total absorption, 515-531 Spectroscope hyperfine structure, 53-55 microwave paramagnetic resonance, 95-96, 100-103 Spectroscopy mass, 1-9 microwave, 23-29, 85-103 optical, 44-58 ultraviolet, 44-58 Sphere transmission measurement of total nonelastic cross-section, 397-405 Spin determination, 44-214 indirect methods, 129-147 microwave method, 85-103 spectroscopic method, 44-129 echoes, 123 isotopic, 353 measurement in liquids, 117-119 transformers, 227-233 Of U "', 55-56
885
SUBJECT INDEX
Splitting, electric hyperfine, 178, 183 Stark effect broadening, 51 modulation, 90 States electron polarization, 216-217 mixed, 218 isobaric, 363-366 nuclear, formed after radioactive 8-decay, 304 Stat ist ics methods, 809-81 1 Poisson, 792-802 useful inefficient, 803-806 Stokes parameter for photon and elect,ron beams, 221-222 Storage rings, magnetic, 687-689 Structure, hyperfine effects of nuclear moments and spins on, 46-49 electric, 183-184 magnetic, 176-183 in optical spectra, 44 ff spectroscope, 53-55 Student’s t function, 812-813 Supermendur, 247 Switches, series, 311-314 Synchrocyclotron, 645-651 Synchrometer, mass, 5 Synchrotron alternating-gradient, 663-678, 693-694 electron, 676-678 proton, 668-676 constant-gradient, 651-663 electron, 653-656 proton, 656-663 “racetrack,” 656
Tensors, statistical, 194, 197 expansion in powers of 1/T, 210-214 Thermometer, nuclear, 190 Threshold, neutron detector, 404 measurement, 360-362 Time coincidence resolving, 325 -of-flight delay curve, antiproton, 754 measurement of neutron reaction cross-section, 456-457 interval sorter, 314 resolution, 324 ff resolving effects on coincidence and anticoincidence circuits, 794-795 effects on single-channel counting, 792-794 transit, in scintillation counter, 308 two-source method, 793-794 Timing, electronic, of decay, 306-333 Tl*OO decay scheme, 156 Track width, mean, 42-43 Trajectories, particle, in strong focusing magnet, 694-696 Transfer efficiency, 234 polarization, 233-239 Transformation, Lorentz, from laboratory t o c.m. system, 821-824 Transformers polarization, 227-233 spin, 227-233 Transitions, “forbidden,” microwave magnetic field induced, 124-125 Transmission, y-ray, law of, 766
U
T u236
Target (8) -detector geometry in scattering and reaction experiments, 412-413 in elastic scattering experiments, 479 for neutron cross-section measurement, 372-373 proton synchrotron, 661 Tbl69 hyperfine structure, 57-58 Temperature, very low, measurement, 189-190
nuclear properties, 594 spin, 55-56 V Van de Graaff generators, 584-590 tandem, 588-590 Van Vleck-Wesskopf equation, 87 Variance, 773-774, 809-810 sample, 774
886
SUBJECT INDEX
Vector, polarization, 218 Voltage deflection, on condenser plates, 228229 -mukiplying circuit, 581-584
W Waveguide gas absorpt.ion cell, 24-25 gas-filled, absorption, 87-88 iris-loaded, 642-643 “Whole number rule,” 1 Wick’s limit, 382 Wien filter, 231-232 “Wobble stretching,” 32-33
Y Yield(s) of E2 y-rays, 447 from F lB ( p ,a ) y reaction, 443-444 of y-rays per incident charged particle per unit solid angle, 443 nuclear reaction, 362-363 photodisintegration, per roentgen, 469-470 reactor, 593
2 Zeeman modulation, 90