Nuclear Physics (Methods of Experimental Physics)

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Methods of Expe rim en taI Physics VOLUME 5

NUCLEAR PHYSICS PART A

METHODS OF

EXPERIMENTAL PHYSICS: L. Marton, Editor-in-Chief Claire Marton, Assistant Editor

1. Classical Methods, 1959 Edited by lmmanuel Estermann

2. Electronic Methods Edited by E. Bleuler and R. 0. Haxby 3. Molecular Physics Edited by Dudley Williams

4. Atomic and Electron Physics Edited by Vernon W. Hughes and Howard L. Schultz 5. Nuclear Physics (in two parts), 1961 Edited by Luke C. L. Yuan and Chien-Shiung Wu 6. Solid State Physics (in two parts), 1959 Edited by K. Lark-Horovitz and Vivian A. Johnson

Volume 5

Nuclear Physics Edited by

LUKE C. L. YUAN Brookhaven National laborafory Upton, New York

CHIEN-SHIUNG WU Columbia University New York, New York

PART A

1961

ACADEMIC PRESS

@

New York and London

Copyright @ 1961, by

ACADEMIC PRESS INC. ALL RIQHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM BY PHOTOSTATl MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS

ACADEMIC PRESS INC. 111 FIFTH AVENUE NEWYORK3, N. Y.

United Kingdm Edition Published by ACADEMIC PRESS INC. (LONDON)LTD. 17 OLD QUEEN STREET, LONDONS.W. 1

Library of Congress Catalog Card Number 61-17860 PRINTED IN THE UNITED STATES OF AMERICA

CONTRIBUTORS TO VOLUME 5, PART A D. E. ALBURGER, Brookhaven National Laboratory, Upton, N e w York M. BLAU,Institut fur Radiumforschung, Vienna, Austria J. E. BROLLEY, JR.,Los Alamos Scientific 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 Xtate Radiations, fnc., Culver City, California

T. R. GERHOLM, Institute of Physics, University of Uppsala, Uppsala, Sweden W. W. HAVENS,Pupin Physics Laboratory, Columbia University, New York, New York R. HOFSTADTER, Physics Department, Stanford University, Standford, California D. J. HUGHES, Brookhaven National Laboratory, Upton, N e w York* S. J. LINDENBAUM, Brookhaven National Laboratory, Upton, New York G. C. MORRISON, Atomic Energy Establishment, Harwell, Berkshire, England G. D. O'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

*

Deceased. V

vi

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

FOREWORD TO VOLUME 5A After a longer delay than originally expected, I a m able to present here the next volume in the series of Methods of Experimental Physics: the first part of the “Nuclear Physics” Methods. Much thought .and work went into this volume and I am in the best position to appreciate all the effort of my fellow editors who devoted so much time in preparing this particular volume. The aims of the publication did not change. The reception of the earlier volumes had proven th a t it was really useful to concentrate on “a concise, well illustrated presentation of the most important methods, or general principles, needed by the experimenter, complete with basic references for further reading. Indication of limitations of both applicability and accuracy is a n important part of the presentation. Information about the interpretation of experiments, about the evaluation of errors, and about the validity of approximations should also be given. The book should not be merely a description of laboratory techniques, nor should i t be a catalog of instruments.” I n these troubled times, when furthering of scientific education is so important, we hope th a t these volumes can be of real help as well to the educator, as to the research worker. At the time of writing my foreword, two other volumes are well under way. The manuscripts of the companion volume of the present one, Volume 5B, are accumulating rapidly and printing should follow this one within a few months. In an even more advanced stage is our Volume 3, “Molecular Physics,” which, under the very valuable leadership of Professor Dudley Williams, promises to become a perfect companion t o the already existing volumes. At the writing of this foreword, Volume 3 is just entering the page-proof stage. I would like to report also on further development. It has been suggested that, in order to enhance the usefulness of this collection to the graduate student, we supplement the planned six volumes with a seventh devoted to nothing but problems. I n discussing this idea with a number of my colleagues I found favorable reaction and a t present I a m investigating how t o organize such a volume. It is a pleasant duty to thank again all those who have devoted SO much time and work to the preparation of this volume. I n the first place come the volume editors whose intelligent and devoted handling of the material is beyond praise. One shouldn’t, however, forget the authors. They have been most accommodating and I join the volume editors in expressing my appreciation. The publishers deserve our gratitude for

vii

viii

FOREWORD TO VOLUME

5A

being very patient and very helpful during this longer delay than we expected. As in the past, Mrs. Claire Marton has been most helpful in handling many of the problems of the editorial office. To all these people go my heartfelt thanks. L. MARTON Washington, D. C. June, 19fl

PREFACE TO VOLUME 5A The field of experimental nuclear physics has in the last two decades, experienced a tremendous growth of activity in all its branches. The difficulty in performing nuclear physics experiments is also greatly multiplied with the increasing complexities of the problems involved. There are, at present, many articles and books which give excellent reviews on basic principles and details of techniques of various detectors, methods and specific topics in nuclear physics. But it is often hard to obtain comprehensive information on the principal methods and their relative merits for the measurement of a specific physical quantity in the field of nuclear physics. This knowledge is especially desirable when one wishes to make a choice among the various methods on the basis of their feasibility, the accuracies attainable, and the limitations in their application under specific conditions. For any specific method of measurement, the comprehensive procedure of converting the experimental data into the desired physical quantities including the necessary corrections involved is often not explicitly mentioned in the literature. It is the intention of the present volume to try to meet some of the requirements mentioned above. All possible methods that deal with the measurement of each particular physical quantity are grouped together so as to achieve a more coherent presentation. Furthermore, the scope of this book is not limited to the usual treatment of low energy nuclear physics only, but it comprises both the high and low energy regions. We hope that this volume will serve as an informative source and as a reference book for physicists in general and, in particular, as an instructive and useful guide for all those who are interested in doing research in this field. Every effort has been made to obtain leading experts in each field to prepare contributions on the specific topics involved so that their intimate knowledge and experience can be shared. Owing to the comprehensive coverage of this book and to the enthusiastic response of a large number of contributors who treated their subject matter so thoroughly, it was found necessary to divide the work into two volumes rather than to publish a single volume a8 originally planned. For this reason and because an unusually large number of contributors have been involved, there has been some unavoidable delay in the completion of this book. ix

X

PREFACE TO VOLUME 5 A

We wish to take this opportunity to express our deepest appreciation and thanks to all the contributors for their understanding and cooperation, to the publisher and to Dr. L. Marton, the Editor-in-Chiefl for their invaluable help and continuous encouragement. CHIEN-SHIUNG Wu Columbia University LUKEC. L. YUAN Brookhaven National Laboratory August 14, 1961

CONTENTS, VOLUME 5, PART A CONTRIBUTORS TO VOLUME 5, PART A. . . . . . . . . . . . . .

v

FOREWORD TO VOLUME 5A . . . . . . . . . . . . . . . . . . vii PREFACE TO VOLUME 5A . . . . . . . . . . . . . . . . . . . CONTRIBUTORS TO VOLUME 5, PARTB. . . . . CONTENTS, VOLUME 5, PART B. . . . . . . .

ix

. . . . . . . . . xv . . . . . . xvii ,

.

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 E / 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 . . . . . . . .

.

.

. . . . .

. . . .

. . .

. .

1.2. Ionization Chambers. . . . . . . . . . . . . . . . . by ROBERTW. WILLIAMS 1.2.1. General Considerations. . . . . . . . . . . . . 1.2.2. Pulse Formation. . . . . . . . . . . . . . . . 1.2.3. Quantitative Operation and Some Practical Considerations . . . . . . . . . . . . . . . . . 1.2.4. Amount of Ionization Liberated. . . . . . . . . 1.2.5. Noise: Practical Limit of Energy Loss Measurable. 1.2.6. Some Types of Pulse Ionization Chambers . . . . 1.2.7. Current Ionization Chambers and Integrating Chambers. . . . . . . . . . . . . . . . . .

1 1

4 44 55 56 73 76 89 89 95 100 103 105 107 109

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 1.4. Scintillation Counters and Luminescent Chambers . . . 120 by GEORGET. REYNOLDS and F. REINES 1.4.1. Scintillation Counters. . . . . . . . . . . . . . 120 1.4.2. Solid Luminescent Chambers . . . . . . . . . . 159

xi

xii

CONTENTS. VOLUME

5.

PART A

1.5. cerenkov Counters . . . . . . . . . . . . . . . . . and LUKEC. L. YUAN by S. J . LINDENBAUM 1.5.1. Introduction . . . . . . . . . . . . . . . . . 1.5.2. Focusing cerenkov Counters . . . . . . . . . . 1.5.3. Nonfocusing Counters . . . . . . . . . . . . . 1.5.4. Total Shower Absorption Cerenkov Counters for Photons and Electrons . . . . . . . . . . . . . 1.5.5. Other Applications . . . . . . . . . . . . . . .

162 162 168 186 189 191

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 . . . . . . . . . . by M . BLAU 1.7.1. Introduction . . . . . . . . . . . . 1.7.2. Sensitivity of Nuclear Emulsions . . . . 1.7.3. Processing of Nuclear Emulsions . . . . 1.7.4. Optical Equipment and Microscopes . . 1.7.5. Range of Particles in Nuclear Emulsions 1.7.6. Ionization Measurements in Emulsions . 1.7.7. Ionization Parameters . . . . . . . . 1.7.8. Photoelectric Method . . . . . . . . .

. . . . . 208

. . . .

. . . .

. . . .

. . . .

. . . . . . . .

. . . . . . . .

. 208 . 210 . 216 . 224 . 226 . 240 . 245 . 264

1.8. Special Detectors . . . . . . . . . . . . . . . . . . 265 1.8.1. The Semiconductor Detector . . . . . . . . . . 265 by S. S. FRIEDLAND and F. P. ZIEMBA 1.8.2. Spark Chambers . . . . . . . . . . . . . . . . 281 by BRUCECORK

.

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 graphic Emulsions . . . . . . . . . by M . BLAU

. . . . 289

. . . .

289

. . . . 289

. . . . 293 Photo-

. . . . 298

CONTENTS, VOLUME

5,

PART A

...

XI11

2.1.2. Principal Methods of Measuring Nuclear Size. . . 307 by ROBERTHOFSTADTER 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 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. WILLIAMS 2.2.1.2.2. Scintillation Spectrometry of Charged Particles 41 1 by G. D. O’KELLEY 2.2.1.2.3. Measurement of Range and Energy with Cloud Chambers and 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 LUKEC. 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

xiv

CONTENTS, VOLUME

5,

PART A

2.2.2.4. Determination of Momentum and Energy of Neutrons with He3 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 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 y-Ray Energy by Absorption. . 671 by ROBLEYD. ZVANS 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

SUBJECTINDEX . . . . . . . . . . . . . . . . . . . . . . .

718

CONTRIBUTORS TO VOLUME 5, PART B E. AMBLER,Low Temperature Section, National Bureau of Standards, Washington, D.C. F. AJZENBERO-SELOVE, Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania M. BLAU,Institut fur Radiumforschung, Vienna, Austria M. H. BLEWETT, Brookhaven National Laboratory, Upton, New Yorlc 0. CHAMBERLAIN, Department of Physics, University of California, Berkeley, California B. CORK,Lawrence Radiation Laboratory, University of California, Berkeley, California H. DANIEL, M a x Planck Institute for Nuclear Physics, Heidelberg, Germany M. DEUTSCH, Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts H. E. DUCKWORTH, Department of Physics, McMaster University, Hamilton, Ontario, Canada R. D. EVANS,Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts G. FEHER,Department of Solid State Physics, University of California, L a Jotla, California H. FRAUENFELDER, Department of Physics, University of Illinois, Urbana, Illinois W. B. FRETTER, Department of Physics, University of California, Berkeley, California W. GENTNER,M a x Planck Institute for Nuclear Physics, Heidelberg, Germany S. GESCHWIND, Bell Laboratories, Murray Hill, New Jersey J. G. HIRSCHBERG, Project Matterhorn, Princeton University, Princeton, New Jersey J. C. HUBBS,E-H Research Laboratories, Inc., Oakland, California C. D. JEFFRIES, Department of Physics, University of California, Berkeley, California J. K. JEN, Applied Physics Laboratory, The Johns Hopkins University, Silver Spring, Maryland H. W. KOCH,Betatron Laboratories, National Bureau of Standards, Washington, D. C. xv

xvi

CONTRIBUTORS TO VOLUME

5,

PART B

H. KOUTS,Brookhaven National Laboratory, Upton, N e w York D. W. MILLER,Department of Physics, Indiana University, Bloomington, Indiana W. A. NIERENBERG, Department of Physics, University of California, Berkeley, California G. D. O’KELLEY, Oak Ridge National Laboratory, Oak Ridge, Tennessee L. ROSEN,Department of Physics, Los Alamos Laboratory, Los Alamos, New Mexico R. M. STERNHEIMER, Brookhaven National Laboratory, Upton, New York A. N. WAPSTRA,Institute voor Kernphysisch Onderzoek, Amsterdam, Holland W. WHALING, Department of Physics, California Institute of Technology, Pasadena, California

CONTENTS, VOLUME 5, PART B 2.3. Determination of Mass of Nuclei and of Individual Particles 2.3.1. Mass Spectroscopy by H. E. DUCKWORTH 2.3.2. Nuclear Disintegration Energies and Reaction Values by W. WHALING 2.3.3. Microwave Method by S. GESCHWIND 2.3.4. Cloud Chambers by W. B. FRETTER 2.3.5. Photographic Emulsions by M. BLAU 2.4. Determination of Spin, Parity and Nuclear Moments 2.4.1. Spectroscopic Methods 2.4.1.1. Optical and Ultra-Violet Spectroscopy by J. G. HIRSCHBERG 2.4.1.2. Atomic Beam by W. NIERENBERG and J. C. HUBBS 2.4.1.3. MICROWAVE METHOD by J. K. JEN 2.4.1.4. Nuclear Magnetic and Quadrupole Resonances by C. D. JEFFRIES and G. FEHER 2.4.2. Indirect Methods 2.4.2.1. Angular Correlation by H. FRAUENFELDER 2.4.2.2. Conversion Coefficients by A. N. WAPSTRA 2.4.2.3. Oriented Nuclei by E. AMBLER 2.5. Determination of Polarization of Electrons and Circular Polarized Photons by H. FRAUENFELDER

xvii

xviii

CONTENTS, VOLUME

5,

PART B

2.6. Determination of Life-Time 2.6.1. Long Life-Time by W. GENTNER and H. DANIEL 2.6.2. Short Life-Time by M. DEUTSCH 2.7. Determination of Nuclear Reactions 2.7.1.Determination of the Q-Value for Nuclear Reactions 2.7.2.Determination of Nuclear Energy Levels from Reaction Energies by F. AJZENBERG-SELOVE 2.7.3.Total Interaction Cross Sections 2.7.4. Differential Interaction Cross Sections 2.7.5.Elastic Cross Sections 2.7.6.Inelastic Cross"Sections 2.7.7.Nuclear Production Cross Sections by L. ROSENand D. W. MILLER 2.8. Determination of Flux Densities 2.8.1.Flux of Charged Particles by 0. CHAMBERLAIN 2.8.2.Flux of Photons by €1. W. KOCH 3. Sources of Nuclear Particles and Radiations

3.1. Natural Sources-Radioactivity by G. D. O'KELLEY

3.2. Artificial Sources 3.2.1.Low Energy Sources 3.2.1,l.Cascade Transformer 3.2.1.2.Van de Graaff by H. BLEWETT 3.2.1.3.Nuclear Reactor 3.2.1.4.Neutron Sources by H. KOUTS 3.2.2.Medium and High Energy Sources 3.2.2.1. Linear Accelerator 3.2.2.2.Cyclotron 3.2.2.3.Betatron

CONTENTS, VOLUME

5,

PART B

3.2.2.4. Synchrotron 3.2.2.5. Synchrocyclotron 3.2.2.6. Proton Synchrotron 3.2.2.7. Alternate Gradient Synchrotron 3.2.2.8. Fixed Field Alternate Gradient Synchrotron by H. BLEWETT

4. Beam Transport System by R. STERNHEIMER and B. CORK 5. Statistics by R. C’ VANS Appendix 1. System of Units 2. Kinematics by R. STERNHEIMER

AUTHORINDEX

SUBJECTINDEX

xix

This Page Intentionally Left Blank

1. FUNDAMENTAL PRINCIPLES AND METHODS OF PARTICLE DETECTION

1 .l. Interaction of Radiation with Matter*

7

1.1 .l. Introduction In this chapter, we shall discuss the various processes which take place when charged particles and y radiation pass through matter. For any type of charged particle (proton, meson, electron, etc.), there will be a loss of energy as the particle traverses the material, due to the excitation and ionization of the atoms of the medium close to the path of the particle. The loss of energy per cm of path, dE/dx, is generally referred to as the ionization loss. In Section 1.1.2, we give a simplified derivation of the theoretical expression for dE/dx, the well-known Bethe-Bloch formula, including a discussion of the density effect which becomes important at high energies. The ionization loss of a fast charged particle is frequently used as a means of identifying the particle, by observing its track in a cloud chamber, bubbIe chamber, or in photographic emulsion. The ionization loss dE/dx is a function onIy of the veIocity v of the particle (for a given charge), so that a simultaneous measurement of dE/dx and of the momentum p enables one to determine the mass m of the particle. The ionization loss can also be used to determine approximately the energy of the particle, if its identity has been established by other methods. A further important property of the ionization process is that the energy w required to form an ion pair in a gas is approximately independent of the energy and the charge of the incident particle, so that when a particle is stopped in a gas, a measurement of the total number of ion pairs enables one to obtain the energy of the incident particle, provided that the value of w for the stopping gas is known. This property has been widely used in the operation of ionization chambers. 1 In Section 1.1.2, expressions for dE/dx are given for various cases, together with a discussion of the fluctuations of the ionization loss (Landau effect). The recent experiments on the ionization loss of relativistic charged particles will be discussed in some detail. For particles heavier than electrons (e.g., protons, K , T , or p mesons), the ionization loss dE/dx is the most important mechanism of energy loss. As a result, a particle with a given incident kinetic energy T will have a quite well-defined range R, which depends on T, on the mass m and on the

t See also, Vol. 4, B, Parts 6, 7,and 8.

1See also this volume, Chapter

1.2.

* Chapter 1.1 is by R. M. Sternheimer. 1

2

1.

PARTICLE DETECTION

charge z of the particle, as well as on the stopping substance. The relation between R and T is known as the range-energy relation. Tables of the range-energy relation for protons of energies T , = 2 Mev to 100 Bev have been recently calculated by Sternheimer for the following materials : Be, C, All Cu, Pb, and air. These range-energy relations differ from the results of Aron et a1.2 in two respects: (1) the density effect correction is included a t the higher energies ( T , 2 2 Bev); (2) recent values of the mean excitation potential I (which enters into the Bethe-Bloch formula) have been used, which are somewhat higher than the value I = 11.52 ev employed by Aron et al. The tables of the range-energy reIations are given in Section 1.1.3, together with a table of the values of dE/dx which were used in the calculation of R ( T ) . Section 1.1.3 also includes a brief discussion of the range straggling. Section 1.1.4 gives various formulas pertaining to the scattering of heavy particles (heavier than electrons) by atoms. When electrons pass through matter, they lose energy by ionization in the same manner as any charged particle (see Section 1.1.2). However, in addition, a high-energy electron will produce electromagnetic radiation (bremsstrahlung) in the field of the atomic nuclei.* For electrons above the critical energy E , (e.g., 47 Mev for All 6.9 Mev for Pb), the energy loss due to radiation exceeds the ionization loss, and constitutes the predominant mechanism for the slowing down process. The y quanta from the bremsstrahlung can create electron-positron pairs, which in tu r n can produce additional y rays. The resulting electromagnetic cascade is called a shower and has been widely observed in cloud-chamber pictures both with incident electrons and y rays. Th e theoretical expressions for the bremsstrahlung and a discussion of shower production are presented in Section 1.1.5. The multiple scattering of charged particles is considered briefly in Section 1.1.6. The penetration of y rays through matter is characterized by a n absorption coefficient r which determines the exponential attenuation of the y ray beam. The processes which contribute to r are the photoelectric effect, the Compton scattering, and the pair production. A summary of the theoretical expressions for these three processes is given in Section 1.1.7. The discussion of Sections 1.1.4-1.1.7 follows closely the review article

* See also Vol. 4 , A, Section

1.5.2. R. M. Sternheimer, Phys. Rev. 116, 137 (1959). 2 W. A. Aron, B. G. Hoffman, and F. C. Williams, University of California Radiation Laboratory Report UCRL-121 (1 951); Atomic Energy Commission Report AECU-663 1

(1951).

1.1.

INTERACTION O F RADIATION W I T H MATTER

3

by Bethe and Ashkin3 on the “Passage of Radiations through Matter.” I n 1956, in order to solve certain difficulties connected with the decay of the strange particles (particularly the K meson), Lee and Yang4 discussed the consequences of a possible nonconservation of parity in the weak interactions (beta decay, strange particle decay, a- and p-meson decay). They suggested a number of experiments to test this hypothesis. These experimentss7 were performed soon after the publication of their paper, and have shown very clearly that parity is not conserved in the weak (decay) interactions, in contrast to the strong interactions which conserve parity to a high accuracy. An important consequence of parity nonconservation is th at the electrons (or positrons) from the beta decay of unpolarized nuclei should be strongly longitudinally polarized, i.e., the electron spin should be aligned predominantly antiparallel to the electron direction of motion, while for positron decays, the positron spin should be aligned predominantly parallel to the positron direction of motion. The magnitude of the polarization P is predicted to be v/c in each case, where v is the velocity of the particle (electron or positron). Thus for relativistic electrons or positrons, P should be essentially 100%. It should be noted that the prediction that P = v / c follows only from a particularly simple theory of parity nonconservation, namely the two-component theory of the neutrino. I n a separate article,*we have given a discussion of the proposals of Lee and Yang4 concerning parity nonconservation in weak interactions. This article also contains a description of the crucial experiments of Wu et d 6on the beta decay of oriented nuclei (Co60), and of Garwin and co-workersO on the polarization of the p+ from a+ decay, which together with the work of Friedman and Telegdi,7 were the first experiments that demonstrated the violation of parity conservation in weak interactions. We have also summarized*the two-component theory of the neutrino, which was proposed independently by Lee and Yang,g Landau,lo and Salam.” A large number of experiments have been performed to establish the longitudinal polarization of the electrons and positrons from beta decay. 3 H. A. Bethe and J. Ashkin, Passage of radiations through matter. In “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 1, p. 166. Wiley, New York, 1953. 4 T. D. Lee and C . N. Yang, Phys. Rev. 104, 254 (1956). 6 C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys. Rev. 106, 1413 (1957). 6 R. L. Garwin, L. M. Lederman, and M. Weinrich, P h p . Rev. 106, 1415 (1957). 7 J. I. Friedman and V. L. Telegdi, Phys. Rev. 106, 1681 (1957). 8 R. M. Sternheimer, Advances in Electronics and Electron Phys. 11, 31 (1959). 9 T. D. Lee and C . N. Yang, Phys. Rev. 106, 1671 (1957). 10 L. D. Landau, Nuclear Phys. 8, 127 (1957). 11 A. Salam, Nuovo n’mento 1 101 6,299 (1957).

4

1. PARTICLE

DETECTION

These investigations involve a variety of methods to determine the longitudinal polarization : scattering of the polarized electrons on nuclei (Mott scattering) ; scattering on polarized electrons (ferromagnetic 3d electrons of iron in a magnetic field), which is often referred to as Mdler scattering; circular polarization of the bremsstrahlung emitted by the polarized electrons; and annihilation of the polarized positrons in various materials. The experiments have in turn led to important developments of the theories presented in Sections 1.1.4 and 1.1.5 on the scattering and interaction of electrons in matter. These new theoretical results, as well as a review of the experiments on the longitudinal polarization, are presented in the latter part of the article on parity nonconservation." 1.1.2. The Ionization Loss dE/dx of Charged Particles 1.1.2.1. The Bethe-Bloch Formula. The theoretical expression for dE/dx is based on the Bethe-Bloch formula, which has been derived from the work of Bohr,12Bethe,13 Bloch,I4 and others. The Bethe-Bloch formula for particles heavier than electrons is given by

dx

-2p-6-

u

]

(1.1.1)

where n = number of electrons per cma in the stopping substance, m = electron mass, p = v/c, where v = velocity of the particle, z = charge of the particle, I = mean excitation potential of the atoms of the substance, Wmsx= maximum energy transfer from the incident particle to the atomic electrons, 6 is the correction for the density effect, which is due to the polarization of the medium, as will be discussed below, and U is a term due to the nonparticipation of the inner shells ( K ,L, .) for very low velocities of the incident particle. This term is generally called the shell correction term, and will be discussed below [see Eq. (1.1.34)l. The maximum energy transfer W,, is given by

..

Wmsx= 2mv2/(1 - p 2 )

( I .1.2)

for energies E << (mi2/2m)c2, where mi is the mass of the incident particle. Throughout this chapter, m (without subscript) denotes the mass of the electron. The Bethe-Bloch formula (1.1.1) is obtained in the following manner. The eIectromagnetic field of the passing particle will excite the

* Other aspects of electron polarization are discussed in Vol. IV, A, Chapter 3 5; this volume, Chapter 2.5. l2 N. Bohr, Phil. Mag. [6] 26, 10 (1913); [6] SO, 581 (1915). l a H. A. Bethe, Ann. Physik [7] 6, 325 (1930). l4 F. Bloch, 2.Physik 81, 363 (1933).

1.1.

5

INTERACTION O F RADIATION WITH MATTER

atomic electrons from their initial ground state to higher excited states, either discrete states or states in the continuum corresponding to ionization. For close collisions, with small values of the impact parameter b ( b 10-8 cm), tho at,omic electrons can he considered as free, i.e., the atomic binding forces are negligible compared to the field of the passing particle during the collision. Small values of b are associated with large energy transfers W . (W = l/b2.) The following approximate derivation of Eq. (1.1.1) follows closely that given by Fermi.16 The atomic electron is assumed to be essentially a t rest before the collision, i.e., its velocity veI is assumed to be small compared t o the velocity v of the incident particle. The impulse I* given t o the electron in the direction perpendicular to the path of th e passing particle is given by

-

(1.1.3) where F , is the perpendicular component of the force acting on the atomic electron. The factor (ze2/b2)in Eq. (1.1.3) gives the order of magnitude of FI,while the factor ( b / v ) is of the order of the collision time. An exact calculation shows th at II is actuaIly equal to 2ze2/bv. The longitudinal part of the impulse, Ill, is zero. I , is equal t o the momentum pel acquired by the electron. The corresponding energy acquired (which is lost by the passing partide) is given b y (1.1.4) The number of electrons per cm of path in the range of impact parameters db is 2 m b db. Hence the energy loss per cm t o electrons in from b t o b the range (b, b db) is

+

+

-dE(b)

$):(

=

-~

(27rnb db) =

4nnz2e4db -. mu2 b ~

( 1.1.5)

The complete rate of energy loss d E / d x is obtained by integrating Ey. (1.1.5) between the limits bmin and b,,,, which are the minimum and maximum values of b for which the above treatment of the energy transfer to the atomic electrons is valid. Approximate values of bmin and bmsxwill be obtained below. One thus finds - - d=E-

dx

l6

4?mz2e4 mv2

bmsx In -* bmin

(1.1.6)

E. Fermi, “Nuclear Physics,’’ p. 27. Univ. of Chicago Press, Chicago, 1950.

6

1.

PARTICLE DETECTION

The upper limit b, arises from the fact that if the time of collision T is large compared to the period of revolution of the atomic electrons in their respective Bohr orbits, the passing particle does not lose any appreciable energy to the electrons, since the perturbation of the electron motion is then essentially an adiabatic process. I n the nonrelativistic region, the time of collision r is -b/v. At relativistic energies, the region of space in which the perpendicular component El of the electric field (and hence the impulse I J is large, is contracted by a factor (1 - p2)-1'2where p = v/c. Hence the time T during which a particular electron experiences the field El is also decreased by the factor (1 - p2)-lI2, so th a t we have 7

N

(b/v)(l -

p2)liZ.

(1.1.7)

If denotes the mean frequency of excitation of the electrons, b,, is determined by (1.1.8) 1/Y g (brn"JV)(1 - p 2 ) l i z whence (1.1.Y) b,,, g ( v / F ) ( l - p - 1 ' 2 . The lower limit bminarises from the limitation of the classical treatment presented above. Thus if the de Broglie wavelength X is larger than the impact parameter b, the above classical considerations will lose validity. I n this connection, X must be taken as the wavelength of the atomic electron in the center-of-mass system, which approximately coincides with the rest system of the incident particle (for particles heavier than electrons). Since we assume that vel <
=

h(l - P2)1/2/(mv)

( 1.1.10)

where h is Planck's constant h divided by 2 ~Upon . inserting Eqs. (1.1.9) and (1.1.10) into Eq. (1.1.6), one finds (1.1.11)

I n the logarithm of Eq. ( l . l . l l ) , f i Y is 1 / ( 2 ~ times ) the mean excitation potential of the atom (denoted previously by I ) . Thus the argument of the logarithm is 27rmv2/[1(1 - p2)]. Actually, an accurate treatment of the limits b,,, and bmi, gives

- _dE _ _-_4?mx2e4 _ In -. dx

mu2

2mv2 1(1 - p2)

( 1.1.12)

Equation (1.1.12) is ident,ical with the Bethe-Bloch formula ( l . l . l ) ,

1.1.

INTERACTION O F RADIATION WITH MATTER

7

except for the relativistic term -2p2 and the terms due to the density effect and the shell corrections ( - 6 - U ) in the square bracket of (1.1.1). The equivalence of the logarithms of Eqs. (1.1.1) and (1.1.12) can be easily verified by substituting Eq. (1.1.2) for W,,, in Eq. (1.1.1). 1.1.2.2. Dependence of the Ionization Loss on the Particle Velocity. The Density Effect. We will now discuss the genera1 behavior of Eq. (1.1.1) for dE/dx as a function of the particle velocity u. At low energies, -dE/dx decreases rapidly with increasing v. This decrease is essentially due to the fact that with increasing u , the collision times 7 decreases, resulting in a decrease of the excitation of the atomic electrons. The momentum transfer pel is proportional to 7 or to l / v . The resulting energy transfer equals p$/2m, and is therefore proportional to 1/u2, as is seen from Eq. (1.1.1). I n the relativistic region (v = c), the factor 1/v2 becomes nearly constant, and the velocity dependence is mainly determined by the behavior of the square bracket of Eq. (1.1.1). In view of Eq. (1,1.2), the logarithm of (1.1.1) can be written

L

=

2

111

2mv2 1(1 - p")'

(1.1.13)

The denominator (1 - p2) results in a logarithmic increase of dE/dx with increasing energy, which is called the relativistic rise. This increase takes place after dE/dx reaches a minimum a t u = 0 . 9 6 ~(see Figs. 1 and 2). As is seen from Eq. (1.1.1)) a part of this rise is due to the distant collisions [the factor (1 - p2)-' in the logarithm of (l)],and a part is due to the close collisions; this part results from the increase of Wm, with increasing energy. As is shown by Eq. (1.1,9), the increase due to the distant collisions arises directly from the relativistic increase of b,,. It is in this connection th at the density effect (represented by the term 6) becomes important. Thus with increasing energy, the cylindrical region around the path of the particle where the atoms are excited will increase However, the atoms close t o the path of the particle in radius (= Lax). will produce a polarization which reduces the electric field acting on the electrons at larger distances, thus diminishing th e energy loss to these electrons. The existence of this effect was first suggested b y Swann16 in 1938. It has been calIed density effect, since it depends strongly on the density of the medium, in as much as the polarization is proportional to n, the number of electrons per cm3. The density effect correction was first evaluated by Fermi" under simplified assumptions. Further calculations 16

'7

W. F. G. S w a m , J . Franklin Znst. 226, 598 (1938). E. Fermi, Phys. Rev. 66, 1242 (1939); 67, 485 (1940).

a

1.

PARTICLE DETECTION

of this effect have been carried out by Halpern and Ha11,18Wick,lg Sternheimer,20s21BudinilZ2and others. I n the extreme relativistic region, the density effect correction 6 is given by 6

= -

ln(1 - p2) - ln(12/h2vp2) - 1

(1.1.14)

where v p is the plasma frequency of the electrons and is given by yP =

(ne2/i..m)1/2.

The term - ln(1 - 02) in 6 cancels the term due to (1 - p2) in the logarithm of Eq. (1.1.1). Thus the part of the relativistic rise due to the broadening of the field region is completely canceled by the density effect in the limit of very large energies. There remains the part of the rise due to the term In W,.,, i.e., due to the energy transfer in close collisions. W,,, increases as (1 - p2)-’ for not too high energies [Eq. (1.1.2)]. At very high energies [E > (mi2/2m)c2], W,,, increases approximately a s (1 - p2)-1/2. Indeed, Vmax approaches the value E - (mi2/2m)c2,a s is shown by the following general formula due to Bhabha.23

where E is the total energy of the particle (including the rest mass). Thus in the extreme relativistic region, the logarithm of (1.1.1) has a term - ln(1 - p2)3/2,whereas 6 has a term - ln(1 - p2), so that the relativistic rise is only one-third as large as it would be without the density effect. 1.1.2.3. Energy Loss due to eerenkov Radiation. It should be noted that the relativistic rise includes the energy loss due to Cerenkov radiation. The Cerenkov loss is given by the formula of Frank and T t ~ m m . ~ ~ (1.1.16) where the integral extends over the frequencies v for which pn > 1, and where n(v)is the index of refraction of the medium. The Cerenkov loss is 0. Halpern and H. Hall, Phys. Rev. 67, 459 (1940); 73, 477 (1948). G. C . Wick, Ricerca sci. 11, 273 (1940); 12, 858 (1941); Nuovo cimento [9] 1, 302 (1943). 20R. M. Sternheimer, Phys. Rev. 88, 851 (1952); 91, 256 (1953); 93, 351, 1434 (1954). 2 1 R. M. Sternheimer, Phys. Reu. 103, 511 (1956). 22 P. Budini, Nuovo cimento [9] 10, 236 (1953). 23 H. J. Bhabha, Proc. Roy. SOC. A164, 257 (1937). Z4 I. Frank and I. Tamm, Compt. rend. acad. sci. U.R.S.S. 14, 109 (1937). l8

l9

1.1.

INTERACTION OF RADIATION WITH MATTER

9

zero at low energies, and increases to a small saturation value in the region of the relativistic r i ~ e . ~ " The ~ ' magnitude of (1.1.16) is in all cases small compared to the magnitude of the relativistic rise. This result arises from the fact that there is a large number of absorption lines and cont i n ~ a , ~corresponding ~-~~ to excitation of the electrons in the K , L, M . . . shells, except for the very light atoms (H, He) where, however, there is still a wide spectrum of absorption frequencies corresponding to the continuum above the discrete spectrum for excitation from the 1s shell. As a result, the expression for the atomic polarizability C Y ( V ) contains a large number of terms, one term for each absorption frequency (discrete line spectrum or excitation to the continuum). For such a behavior of C Y ( V ) , the index of refraction %(.) is less than 1 over a considerable region of V . As a result the region of integration of (1.1.16) is considerably restricted, and actually the only region which makes an important contribution to (dE/dx)c is the region of v below the first absorption limit, i.e., effectively the optical and near-ultraviolet region, as has been shown by Sterr~heimer.~~ It should also be noted that the width of the spectral lines gives rise to a strong absorption of the cerenkov radiation for values of v close to the frequencies of the atomic transitions, thus resulting in a further reduction of the Cerenkov energy loss. For condensed materials, - (l/p)(dE/dx)c is of the orderz0 of lop3 Mev/g cm-2 and hence completely negligible compared to the total ionization loss (> 1 Mev/g cm-2).31 For gases, the cerenkov loss is somewhat more important,29of the order of 0.1 Mev/g for Hz and He, and 0.01 Mev/g cm-2 for gases with medium and large Z . Even for H,, the cerenkov loss accounts for only -15% of the relativistic rise. Comprehensive treatments of the stopping power problems in dense materials, including the Cerenkov radiation, have been recently given by fan^,^^ Budini and Taff Bra,3 5 and Tidman. 3 4 The Bethe-Bloch formula, Eq. ( l . l . l ) , includes the cerenkov loss. l7 Thus, in order to obtain the energy -(dE/dx)d deposited close to the path of the particle, it is necessary in principle to subtract - (dE/dx)c. A. Bohr, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 24, N o . 19 (1948). H. Messel and D. M. Ritson, Phil. Mag. [7] 41, 1129 (1950). 17 M. Schoenberg, Nuovo cimento [9] 8,159 (1951); 9,210,372 (1952); M. Huybrechts and M. Schoenberg, ibid. 9, 764 (1952). 28 P. Rudini, Phys. Rev. 89, 1147 (1953). 99 R. M. Sternheimer, Phys. Rev. 89, 1148 (1953); 91, 256 (1953); 93, 1434 (1954). 80 G. N. Fowler and G. M. D. B. Jones, Proc. Phys. SOC.(London) A66, 597 (1953). 81 See also J. R. Allen, Phys. Reu. 93, 353 (1954). 84 U. Fano, Phys. Reu. 103, 1202 (1956). 88 P. Budini and L. Taffara, Nuouo cimenlo [lo] 4, 23 (1956). *4 D. A. Tidman, Nuclear Phys. 2, 289 (1956); 4, 257 (1957). 26 24

10

1.

PARTICLE DETECTION

We have (1.1.17) However, as pointed out above, (dE/dx) c is generally negligible compared to the relativistic rise of dE/dx, except for gases of low 2 (H2, He). While the energy loss due to Cerenkov radiation is very small compared to the total ionization loss, the Cerenkov effecta6has received an important application in the design of Cerenkov counters," which are based on the property that the Cerenkov radiation is absent unless the velocity of the particle exceeds a critical value vc determined by v,no/c

=

(1.1.18)

1

where no is the index of refraction of the radiating substance (usually a liquid) in the optical region. The Cerenkov counter is used as a velocity selector, and together with a momentum measurement, it enables one to identify charged particles by placing an upper or lower limit on the mass. 1.1.2.4. Evaluation of dE/dx. The M e a n Excitation Potential 1. Equation (1.1.1) can be written in the following form: Pdx:

P2

B

P + In Wmax,Mev + 0.69 + 2 In mic - 2p2 - 6 - U -

(1.1.19) where p is the density of the medium in g/cm3, so that -(l/p)(dE/dz) gives the energy loss in Mev/g cm-2; A and 3 are defined by:

A = 21rnx2e4/(mc2p) 3 = 1n[mc2(1OE ev)/P].

(1.1.19a) (1.1.19b)

I n Eq. (1.1.19), Wmax,Mev is the value of Wmax[Eqs. (1.1.2), (1.1.15)] in MeV. In order to obtain - (l/p)(dE/dz) in Mev/g cm-2, one must take A as: 0.1536(Z/A0)x~,where 2 = atomic number, A0 = atomic weight of stopping substance. The following expression for 6 has been obtained by Sternheimer :20,21 6 = 4.60GX 6 = 4.606X

+ C + U ( X I- X)' +C

(Xo < X < X i ) (1.1.20) ( X > Xl) (1.1.20a)

where X = loglo(p/m,c), X Oand X I are particular values of X which depend on the substance. X Ois the value of X below which 6 is zero; XI is the value of X above which the high-energy expression, Eq. (1.1.20a), applies.

* See also this volume,

Chapter 1.5. A review of the applications of the eerenkov effect has been given by J. V. Jelley, Progr. in Nuclear Phys. S, 84 (1953). as

1.1.

11

INTERACTION O F RADIATION WITH MATTER

In the region X > XI, the ionization loss becomes independent of the excitation potential I , as has been first shown by Fermi.I7 I n Eqs. (1.1.20) and (1.1.20a), a, s, wd C are constants which depend on the substance and on the value chosen for I . C is defined as

C = -2 ln(I/hv,)

-

1.

(1.1.21)

The mean excitation potential I has been the subject of numerous investigations. I n 1933, Bloch14showed that, on the basis of the ThomasFermi model, I should be proportional t o the atomic number: I = kZ,but he could not determine theoretically the value of the proportionality constant lc, which therefore has to be obtained from experiment. An early determination by Bethe36of I for air from the range-energy relation of (Y particles gave I,, = 80.5 ev. I n 1940, Wilson3? obtained a value for aluminuni,IA1= 150 ev. Both of these results giveI/Z 11.5 ev. I n 1951, Bakker and Segr&38 measured dE/dx for 340-Mev protons in a number of materials, and obtained values of I of the order of -92 - 1 0 2 ev for heavy elements. Measurements of the ranges of 340-Mev protons by Mather and Segr&39 led to similar values of I . For Al, Mather and Segr83g = 148 i -3 ev, in good agreement with Wilson’s37earlier result. found IA1 On the other hand, Sachs and Richardson40 from a determination of the absolute stopping power for 18-Mev protons in A1 obtained a substjailtially namely I , = 168 k 3 ev. For heavy elements Sachs higher value for IAl, and Richardson obtained I ,- 1 4 2 - 182 ev, which is also considerably higher than the results of Bakker and Segr&38From measurements of the range in A1 of protons of various energies from 35 to 120 MeV, Bloembergen and van Heerden4l deduced a value I A 1 = 162 & 5 ev. A similar measurement for 18-Mev protons in A1 by Hubbard and M a ~ K e n z i e ~ ~ gave I A I = 170 ev.43 I n 1955, C a l d ~ e 1 1recalculated ~~ the values of I from the data of Sachs and Richardson40 with the inclusion of the low-energy shell corrections [ C , and CL, see Eq. (1.1.34)]. Th e resulting values of I are somewhat smaller than those originally obtained by Sachs and Richardson, but are still considerably above the Bakker-Segr& values. Thus for Al, C a l d ~ e 1 1 ~ ~ found = 163 ev, and for the heavy elements, I,- 132 - 1 4 2 ev. M. S. Livingston and H. A. Bethe, Revs. Modern Phys. 9, 261 (1937).

R. R. Wilson, Phys. Rev. 60, 749 (1941).

** C . J. Bakker and E. SegrB, Phys. Rev. 81,489 (1951). R. L. Mather and E. SegrB, Phys. Rev. 84, 191 (1951). D. C . Sachs and R. J . Richardson, Phys. Rev. 83,834 (1951); 89, 1163 (1953) 41 N. Bloembergen and P. J. van Heerden, Phys. Rev. 83, 561 (1951). 4 * E. L. Hubbard and K. R. MacKenzie, Phys. Rev. 86, 107 (1952). 43 See also D. H. Simmons, Proc. Phys. Sac. (London)A66, 454 (1952). 4 4 D. 0. Caldwell, Phys. Rev. 100, 291 (1955). 40

12

1.

PARTICLE DETECTION

Caldwell also showed that the various experiments are generally consistent with I values which are independent of the velocity of the incident particle. 4 6 This result was important, since it had bee%previously believed that I might be velocity-dependent, in order to reconcile data from different experiments. 4 6 Recently, two accurate determinations of I have been made from measurements of the range and stopping power of low-energy protons ( s 2 0 Mev). From measurements of the range of protons of various energies from 6-18 MeV, Bichsel et aL4’ have obtained the following I values for Be, Al, Cu, Ag, and Au: I B e = 63.4 f 0.5 ev, I A I = 166.5 k 1 ev, Icu = 375.6 k 20 ev, I A g = 585 k 40 ev, and I A u = 1037 k 100 ev. The result for Be confirms an earlier determination by Madsen and V e n k a t e ~ w a r l uwho , ~ ~ obtained IBe= 64 f 5 ev. The large value of I / Z for Be (1/Z i X 16) had been previously predicted by A. B ~ h (who r ~ ob~ tained I = 60 ev) on the basis of polarization effects caused by the presence of the two conduction electrons per atom. For Al,Cu, Ag, and Au, 13 ev, which is somewhat smaller the I values of Bichsel et al. give I / Z than Caldwell’s results,44but is considerably higher than the values of I / Z obtained by Bakker and Segr&.@ The other recent determination of I values has been made by Burkig and M a ~ K e n z i eThese . ~ ~ authors measured the stopping powers relative to aIuminum of a number of metals of 19.8-Mev protons. The resulting values The values of I of I are based on the value I A I = 166.5 ev of Bichsel et of Burkig and MacKenzie for Be, Cu, Ag, Au, and P b are: IBe= 64 ev, Icu = 366 ev, I,, = 587 ev, IAu = 997 ev, and I,, = 1070 ev. These values of I are in good agreement with the results of Bichsel et aL4’ In 1959, Zrelov and stole to^^^" measured the range R in copper of 660-Mev protons from the Dubna synchrocyclotron. These authors have obtained a value R = 257.6 f 1.2 gm/cm2, which leads to a calculated mean excitation potential ICu= 305 f 10 ev ( I / Z = 10.5 k 0.3 ev). This value of Icuis appreciably smaller than the values obtained by Bichsel et al.47and by Burkig and M a c K e n ~ i eat~ ~lower energies (6-20 Mev). Zrelov and stole to^^^^ have also determined the stopping power relative to copper for H, Be, C, Fe, Cd, and W for 635-Mev protons. For H, Be, and

-

See also W. Brandt, Phys. Rev. 104,691 (1956);111,1042(1958);112,1624(1958). M. Scharff, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 27, No. 15 (1953). 47 H. Bichsel, R. F. Mozley, and W. A. Aron, Phys. Rev. 106, 1788 (1957). 48 C. B. Madsen and P. Venkateswarlu, Phys. Rev. 74, 648 (1948). 49 V. C.Burkig and K. R. MacKenzie, Phys. Rev. 106,848 (1957). 4OeV. P. Zrelov and G. D. Stoletov, Zhur. Eksptl. i Teoret. Fiz. 86, 658 (1959): [translation: Soviet Phys. J E T P 9,461 (1959)l. 45

teJ. Lindhard and

1.1.

INTERACTION OF RADIATION WITH MATTER

13

C, the resulting values of I / Z are -14-15 ev (IH = 15, I B e = 61 f 6, I0 = 85 f 8 ev). The value of I B e is in good agreement with the results of earlier experiment^.^'-^^ On the other hand, for Cd and W, the values of I / Z are 9.8 and 9.2 ev, respectively, indicating that for heavy elements, the value of I / Z may be appreciably lower than the results (-13 ev) of references 47 and 49. The values of the constants a, s, and C for the density effect correction 6 which are given in reference 20 were based on the Bakker-Segr838values of I , whereas the results for 6 of reference 21 were obtained by means of . ~ ~denote these values of I by I1 and Iz, the higher I values of C a l d ~ e 1 1We respectively. The values of 6 for any intermediate 1 value, I = I O (e.g., that of Bichsel et aZ.47),can be obtained by logarithmic interpolation, as follows. Let 61 and 8 2 denote the values of 6 pertaining to I I and 12, respectively.20.21Then 60 appropriate to I = I0 is given by 60 =

where q is defined by

761

+ (1 -

7162

(1.1.22) (1.1.23)

Of course, Eqs. (1.1.22) and (1.1.23) apply also, within reasonable limits, if I0 is outside the range (IJZ), i.e., for I0 < I1 or I 0 > Iz. When the stopping material is a compound containing several atomic species, the stopping powers of the individual elements are approximately additive (Bragg’s rule). Thus Eq. (1.1.1) still holds for the compound, provided that the mean excitation potential I in this equation is replaced by the following average potential 1: (1.1.24) with (1.1.25) Here n, is the number of atoms of the ith type in the compound, with atomic number Z, and excitation potential Ii; fi of Eq. (1.1.25) is the oscillator strength of the atomic electrons belonging to the ith species. The density effect correction 6 of Eq. (1.1.1) must be replaced by 8 defined bv (1.1.26) i

where 6i is the value of 6 for the ith constituent.

14

1.

PARTICLE DETECTION

In order to obtain - (l/p)(dE/dz) in Mev/g cm-2, the constant A of Eq. (1.1.19a) must be taken as follows:

(1.1.27)

where Ai is the atomic weight of the i t h element. Extensive experiments have been carried out by Thompsons0t o verify the additivity of the stopping powers for a number of organic compounds (containing C, H, N, 0, and Cl). I n these experiments, the 340-Mev protons from the Berkeley cyclotron were slowed down to 200 MeV. It was found that the stopping powers of the constituent elements in the compound are additive to within 1%. Small deviations (less than 1%) from additivity were observed; these were attributed to the influence of the chemical binding. Thompson also obtained values of I for the elements C, H, N, and 0, using liquid targets for H, N, and 0. 1.1.2.5. The Restricted Ionization Loss, - (l/p)(dE/dz)~,. Equation (1.1.1) or (1.1.19) gives the average energy loss of the charged particle. These expressions include the possibility of large energy transfers, up to the maximum value W,,, [Eqs. (1.1.2), (1.1.15)]. I n certain applications, one is, however, interested in the restricted energy loss with energy This is true in particular for transfers less than a certain fixed value WO. the grain count in nuclear emulsion* and for the drop count of tracks in cloud chambers. t For the grain count in emulsion, the relevant quantity is the ionization loss with energy transfers less than W O 5 kev,26because larger energy transfers generally result in the development of neighboring grains not directly in line with the track, so that the grain count along the track is no longer proportional to the complete ionization loss. A similar phenomenon takes place for cloud chamber tracks. For energy transfers which are larger than -1 kev, a cluster of drops is formed, so that the drop count along the track is not proportional to the complete ionization loss. The restricted energy loss, with maximum energy transfer W o is , given by

-

* See also this volume,

Chapter 1.8.

t See also this volume, Chapters 1.6 and

1.7. T. Thompson, University of California Radiation Laboratory Report UCRL-1910 (1952). See also T. Westermark, Phys. Rev. 93, 835 (1954). 60

1.1.

15

INTERACTION OF RADIATION WITH MATTER

or in terms of the constants A and B

P

p)

dx wo

=

$ [B + 0.69 + 2 In P + In -

wO,,,

- /Y

-6-u

mic

1

(1.1.29) where W0,Mevis the value of W Oin MeV. Whereas the average ionization loss - ( l / p ) ( d E / d z ) continues to rise the indefinitely with increasing energy (due to the increase of W,,,), restricted energy loss - ( l / p ) (dE/dz) W , levels off to a constant value at high energies, which is generally referred to as the Fermi plateau. This I I1111111

I11111111

I

11111111

I11111111

I Ill1

72

fi MESON MOMENTUM (IN Mev/c)

FIG.1. The ionization loss of p mesons i n 02 as a function of the p-meson momentum. The solid curve represents the restricted energy loss, - ( l / p ) ( d E / d x ) w ,with W o= 1 kev, as obtained from Eq. (1.1.28). The crosses represent the experimental data of Ghosh et aL61The theoretical curve and the experimental drop count (number of ion pairs per cm) have been normalized a t the minimum of ionization.

result arises from the fact that the logarithmic term in 6 exactly cancels the effect of the (1 - ,@) denominator in Eq. (1.1.28), as has been discussed above. An example of the relativistic increase of - (l/p) (dE/dz)w,, and the Fermi plateau at very high energies is presented in Fig. 1, which shows the restricted energy loss of p mesons in oxygen (at normal pressure) for an assumed value Wo = 1 kev. A measure of the relativistic increase is given by the ratio R = Jplat/Jmin, where JP1,, and Jmi,are the values of - (l/p)(dE/dz)~,in the plateau region and at the minimum of ionization 61 S. K. Ghosh, G. M. D. B. Jones, and J. G. Wilson, Proc. Phys. SOC.(London) A66, 68 (1952); A67, 331 (1954).

16

1.

PARTICLE DETECTION

(v = 0.96c), respectively. As is seen from Fig. 1, for 0 2 , with W O= 1 kev, = 1.08 Mev/g cm-2, so that R = 1.51. we have Jplat= 1.63, Jmin Figure 1 also s h o w the data of Ghosh et aLblobtained from the drop count in an expansion cloud chamber filled with oxygen a t normal pressure (see Section 1.1.2.9). The theoretical curve and the experimental data have been normalized at the minimum of ionization (Jmi.= 44 ion pairs/cm). The equivalent number of ion pairs is indicated on the right-hand scale.

'*4"

- 1.2

z 2

1.11 10'

' " t l l l l l

"111111'

'

1

111111

' ' I J I I I '

lo3 lo" 10' lo6 p MESON MOMENTUM (IN Mev/c)

' " ' 1 ' 1 '

10'

FIG.2. The ionization loss of p mesons in He as a function of the p-meson momentum. The solid curve represents the restricted energy loss, -(l/p)(dE/dx) wO1with W O= 1 kev. The dashed curve shows the energy deposited along the track, -(l/p)(dE/dx)d, after subtraction of the estimated energy escape due to Cerenkov radiation, - (I/p) ( d E / d s ) c [see Eq. (1.1.17)]. The flat part of the curves a t very high momenta ( >lo5 Mev/c) is often referred to as the Fermi plateau.

The theoretical curve is in good agreement with the data, except a t the highest momenta of the experiment ( p , 2 10 Bev/c), where the data give an indication that the relativistic rise may be somewhat smaller than predicted by the theory. However, the uncertainties of the measurements make it impossible to decide at present whether there is a real discrepancy. The Cerenkov loss ( d E / d z ) c may also be partly responsible for the apparent disagreement, since it reduces the energy deposited along the track (dE/dz)d. However, - (l/p)(dE/dx)c is expected to be quite small for oxygen (-0.02 Mev/g ern+ in the region of the Fermi plateau29). Figure 2 shows the relativistic rise in helium at normal pressure. In this = 1.22, Jplst = 1.79 Mev/g so that R = 1.47. We have case, Jmin made an estimate of the cerenkov energy ~ o s sand , ~ ~the dashed curve of

1.1.

17

INTERACTION OF RADIATION WITH MATTER

Fig. 2 showsthe resultant energydepositedalong the track - (l/p) (dE/dz)d. The difference between the solid and the dashed curves represents the cerenkov loss, - (l/p)(dE/dz)~[see Eq. (1.1.17)]. The relativistic rise of - (l/p) (dE/dz)w o in gases has been observed in a large number of experiments. A summary of these experimental investigations is given in Section 1.1.2.9. 1.1.2.6. The Most Probable Ionization Loss Eprob. Fluctuations of the Energy Loss. As has been shown by Williams,62Landau,63and others, the ionization loss in a thin absorber is subject to appreciable fluctuations, because of the statistical nature of the ionization process. The energy loss in a thin absorber has a considerable spread about the most probable value Eprob. This spread is often referred t o as the Landau effect, since Landau was the first to calculate the expected distribution of the energy losses. Further contributions to this problem have been made by S ~ r n o nand ,~~ by Blunck and L e i ~ e g a n gThe . ~ ~ distribution is asymmetric, with a long tail on the side of high-energy losses which is due to the infrequent collisions with very large energy transfers which result in a relatively large angle scattering of the incident particle. The full width of the Landau distribution at half-maximum is of the order of 20% of Eprob for typical cases. From Landau’s theory,63one obtains the following expression for eproh for a thin absorber (of thickness t g/cm2) : Eprob

=

~

mv2p -

- P2

+ 0.37 - 6 - U (1.1.30)

Equation (1.1.30) can be written as follows:

B

P At + 1.06 + 2 In mic - + In

As an example, a thickness of 6.97 g/cm2 of Be gives a most probable loss = 10 Mev for 3.0-Bev protons. The Landau distribution for this case is shown in Fig. 3. This figure shows the rapid rise of the probability P ( E ) on the side of low-energy losses, and the long tail on the side of large energy losses. The maximum energy transfer of a 3-Bev proton to a single electron, Wmx = 17 MeV, is indicated on the abscissa for comparison with eprob

E. J. Williams, Proc. Roy. SOC.A126, 420 (1929). D.Landau, J. Phys. U.S.S.R. 8, 201 (1944). 6 4 K. R. Symon, quoted by B. Rossi, in “High-Energy Particles,” p. 32. PrenticeHall, New York, 1952. 6 s 0. Rlunck and S. Leisegang, Z.Physin‘k 128, 500 (1950). 61

sSL.

18

1.

PARTICLE DETECTION

the values of Epr& and e~~ (average energy loss). can, of course, be ob= eP(e) de. The tained by integrating over the distribution, i.e., values of the loss E for which the distribution has half its maximum value 1 = a 1 - 8 ~€ ~ 0 ~9.13 ~ Mev and E Z = 11.20 MeV. The fractional spread (€2 - t l ) / ~ p r o= h 0.21 is a measure of the width of the distribution. The ratio ( € 2 - eprob)/(Eprob - el) is 1.38, which is a measure of the skewness of the distribution. The average loss in the same thickness of Be is: eAv = 6.97 X 1.593 = 11.10 MeV, where 1.593 Mev/g cm-2 is - ( l / p ) ( d E / d x ) (see Table 111). The difference between and Eprob is also

lo"

-

t

0.35

n

0.30 -

7

-

gr

W

=0.25 -

--zu0.20 -I

-

a

0.15 -

-

i m

0

0.10

-

0.05-

-

0- L

7

ENERGY LOSS 6 (IN M e V )

FIG.3. The Landau distribution of energy losses E for 3-Bev protons traversing a thickness 6.97 gm/cm2 of Be, for which €pr& = 10 Mev, EA" = 11.10 Mev, and W,, = 17 Mev.

an indirect measure of the importance of the infrequent large energy transfers. In similarity to the restricted energy loss, the most probable loss eprob also levels off to a constant value (Fermi plateau) at very high energies. This result is, of course, due to the fact that the close collisions (which would result in an unlimited increase of dE/dx) do not contribute to eprob, but only to the tail of the Landau distribution. A summary of the experimental determination of Eprob in thin absorbers will be given in Section 1.1.2.9. 1.1.2.7. Ionization Loss of Electrons. Equations (1.1.2.8) and (1.1.2.9) for (dE/dz)w, and Eqs. (1.1.30) and (1.1.31) for eprab are valid for any type of charged particle: electron, meson, etc. These expressions do not iiivolve the close collisions which differentiate slightly between

1.1. INTERACTION

O F RADIATION W I T H MATTER

19

electrons and particles heavier than electrons. On the other hand, Eqs. (1.1.1) and (1.1.19) for the average energy loss are applicable only for particles heavier than electrons. These expressions include a term due to close collisions. For electrons, this term is somewhat different, and the average ionization loss is given by:56

where T , is the kinetic energy of the electron. The factor $T, in the logarithm represents the effective maximum energy transfer W,,,. The reason for this result is that the maximum possible energy transfer from the incident electron t o the atomic electron is T,. However, since the two electrons are indistinguishable, one can call the incident electron after the collision that which has the highest energy. Since this energy is 2 $ T e , the effective maximum energy transfer is $T,. Aside from the replacement of W,, by +Te,the square bracket of Eq. (1.1.32) for electrons differs from that for heavy particles [Eq. (1.1.1)] by an amount: A = (17/8) - In 2 = 1.43

(1.1.33)

a t very high energies ( p = 1). In (1.1.33) the term In 2 is due t o the fact atomic electron) that the reduced mass of the system (indident particle is +m for a n incident electron, as compared to = m for a heavy p a r t i ~ l e . ~ ’ The term 17/8 is due t o the difference between the cross sections for close collisions of electrons as compared t o heavy particles. The effect of A on dE/dx is relatively small ( 5 10%) since the value of the square bracket of (1.1.32) is generally -20. 1.1.2.8. The Shell Correction Term U. We shall nbw discuss the correction U for the nonparticipation of the K , L, . . . electrons a t low energies of the incident particle. This correction has been introduced b y Bethe.36 U is given by

+

(1.1.34) where CKand CLare the K and L shell corrections, respectively. CKand C L are negligible at high energies, and become appreciable only when the velocity v of the particlgis decreased to a value of the order of the velocity 66

H. A. Bethe, in “Handbuch der Physik” (H. Geiger and K. Scheel, eds.), Vol. 24,

p. 273. Springer, Berlin, 1933; C. Mgller, Ann. Physik [5] 14, 531 (1932). 67 H. A. Bethe and J. Ashkin, i n “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 1, p. 253. Wiley, New York, 1953.

20

1.

PARTICLE DETECTION

of the atomic electrons in the K and L shells, respectively. Thus the shell corrections will enter at a somewhat higher energy for heavy elements than for light elements. As an example, for Cu, 2 C ~ / 2is less than 0.05 for proton energies above T , = 65 MeV, corresponding to v, = 0.35~.CLbecomes appreciable only at still lower energies. For Cu, ~ C L / Z < 0.05 for T, > 11 Mev (v, > 0.15~).Detailed calculations of CK and CL have been carried out by Walske.68 The corrections for the M , N , and higher shells of heavy atoms are generally negligible, except at very low energies (T, 5 1 MeV). ,, where the present theory becomes unreliable for other reasons (capture and loss of electrons by the incident particle, see Section 1.1.2.10).

I

I

I

I

I

I

I

I

I

%

sz Q

tE

1.8 1.6

1.4

1.2 1.0’ I02

I

’ 1 1 1 1 1 1 1

lo3 p

’

‘ ‘ * l t 1 l ’

lo4

’

lo5

’

lo6 ’

11111111

lo7

MESON KINETIC ENERGY (IN MeV)

FIG.4. The average ionization loss of p mesons in Be, Al, Cu, Ag, and Au, as a function of the p-meson kinetic energy [Eq.(1.1.19)].

1.1.2.9. Example: - (l/p)(dE/dx) for /I Mesons in Various Materials. Experimental Verification of the Bethe-Bloch Formula at Relativistic Energies. In summary, Eqs. (1.1.1), (1.1.19) and (1.128)-(1.1.32) givethe

expressions for the ionization loss for 4 different possibilities: (1) average energy loss of particles heavier than eIectrons; (2) average energy loss of electrons; (3) restricted energy loss (energy transfers less than a fixed value W O;) (4) most probable loss in a thin absorber. As an example of the behavior of dE/dx as a function of energy, Figs. 4 and 5 show curves of -(l/p)(dE/dx) versus kinetic energy T, for /I mesons in various solids and gases. In calculating the curves of Fig. 4 for the solids, we used for the excitation potential I of each substance the 68

M. C. Walske, Phys. Rev. 88, 1283 (1952);101,940 (1956).

1.1.

INTERACTfON OF RADfATfON WfTH MATTER

21

average of the I values determined by Bichsel et aL4' and by Burkig and M a c K e n ~ i e .The ~ ~ resulting I values are: IBe= 64 ev, Id = 166 ev, Icu= 371 ev, IA== 586 ev, and I , = 1,017 ev. For Fig. 5, we used the I values given in reference 21 for Hz, He, and air: I H 2= 19 ev, I H e= 44 ev16Q and Isir= 94 ev (the last corresponding to I = 132 ev). For Ar and Xe, the values of I / Z were obtained by interpolation from the above I values = 230 ev, and ZXe = 684 ev. for Cu, Ag, and Au. This gives: IAr 4 and 5 show the ionization minimum (for T , 200-300 MeV) Figures and the relativistic rise at higher energies. The value of - (l/p) (dE/dz)a t the minimum decreases with increasing 2, on account of the increase of I

-

p MESON KINETIC ENERGY (IN MeV)

FIG.5. The average ionization loss of p mesons in Hz, He, air, Ar, and Xe, as a function of the p-meson kinetic energy [Eq. (1.1.19)]. The curves for He, air, Ar, and Xe pertain to normal pressure.

in the denominator of the logarithm of Eq. (1.1.1). For H,, three curves are presented corresponding to different pressures. On account of the density effect, the ionization loss decreases with increasing pressure a t very high energies ( T , 2 10 Bev). It may be noted that, in contrast to Figs. 1 and 2 for - ( l / p ) ( d E/d z)w0,which show a plateau at high energies, the curves of Figs. 4 and 5 have an unlimited logarithmic increase, which is, of course, due to the fact that they represent the average energy loss, including all possible energy transfers up to Wmax[cf. discussion following Eq. (1.1.29)]. Figures 4 and 5 also apply for protons, provided that the numbers on the abscissa are multiplied by the factor mp/m,, where m pand m,, are the 69 E. J WilIiams, Proc. Cambridge Phit. Soc. 33, 179 (1937).

22

1.

PARTICLE DETECTION

proton and the p-meson mass, respectively. [ d E / d x for protons of energy T, is equal to dE/dx for p mesons of energy T,(m,/m,).] The Bethe-Bloch formula has been verified in numerous experimental investigations. A summary of the low-energy work on the ionization loss d E / d x and on the range-energy relation is to be found in the articles of Bethe and A ~ h k i n Allison ,~ and Warshaw,60and Taylor.61The experimental studies at relativistic energies are discussed in the review articles of Price62and of Uehling.'j3 Here we shall present a summary of some of the experiments performed at relativistic energies to verify the existence of the relativistic rise and of the Fermi plateau (due to the density effect). An outline of the main features of some of the experimental investigations on the relativistic rise and the density effect is given in Table I. The most extensive experiments on the energy loss in gases a t relativistic energies have been made by means of expansion cloud chambers,* by obtaining the drop count along the tracks of the particles. The momentum is determined by measuring the curvature in the magnetic field of the cloud chamber. One of the earliest experiments of this type is that of Ghosh and co-workerssl who measured the restricted energy loss (W O= 1 kev) of 1.1 mesons in 0 2 a t normal pressure. They observed a rise from 44 drops per cm to 60 drops per cm in going from the minimum ionization at p , = 0.4 Bev/c to the Fermi plateau starting at p , = 20 Bev/c. From their determinations of the drop count at p , = 7, 15, and 30 Bev/c, it is clear that the energy loss - (dE/dx)w-,does not increase indefinitely with increasing p,, but instead levels off to a saturation value. This result provides a direct confirmation of the existence of the density effect in gases a t high energies (see Fig. 1). Aside from the work of Ghosh et u Z . , ~ ~there have been several other cloud-chamber determinations of the relativistic rise, although these have generally somewhat poorer momentum determinations. Carter and W h i t t e m ~ r eperformed ~~ measurements in a helium-filled chamber, both for p mesons and electrons, and obtained evidence that high-energy electrons ionize more heavily than minimum, which would confirm the relativistic rise. These authors also obtained direct evidence for the relativistic rise by comparing the ionization due to p mesons with momenta between 70 and 250 Mev/c (14.7 f 0.35 droplets/mm on the photographic film) with the ionization for a group with p , > 1500 Mev/c

* See also in this volume, Chapter 1.6. S. K. Allison and S. D. Warshaw, Revs. Modern Phys. 26, 779 (1953). A. E. Taylor, Repfs. Progr. in Phys. 16, 49 (1952). 6a B. T. Price, Repis. Progr. in Phys. 18, 52 (1955). 83 E. A. Uehling, Ann. Rev. Nuclear Sci. 4, 315 (1954). G4 R. S. Carter and W. L. Whittemore, Phys. Rev. 87, 494 (1952). 60

1.1. INTERACTION

23

O F RADIATION WITH MATTER

TABLEI. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss For each particular method of determination, the experiments are listed in chronological order. For additional details and a more complete list of references, see text (Section 1.1.2.9).

Author

Method of determination

Corson and Cloud chamber (drop count) Brodeae (1938)

Sen Gupta67

Cloud chamber

(1940)

Hazen66(1945)

Cloud chamber

Haywardas (1947) Cloud chamber

Carter and Whittemore'4

Cloud chamber

(1952)

Ghosh et aL61

Cloud chamber

(1954)

Kepler et aLa9 (1958)

Cloud chamber

Type of particle, energy range, and material traversed

Results

Electrons (0.3-60 MeV) Observation of miniin cloud chamber mum of ionization filled with Nzat 1.5 (at T , 2 MeV) and atmos pressure. relativistic rise for electrons. Electrons (2-500 MeV) Observation of minimum of ionization and relativistic rise for electrons. Electrons in air. Two Observation of relativistic rise (of -40% energy groups: 1.42.1 Mev and 30-240 between two energy groups) in agreement MeV. with Bethe-Bloch formula. Electrons in He. High-energy electrons ( T , > 100 MeV) have 1.4 times minimum ionization, in agreement with theory. Increase of ionization p mesons in He (at between the two pressure P = 98 cm momentum groups Hg). Two momenin good agreement tum groups: p = 70with theory. 250 Mev/c, and p > 1500 Mev/c. p mesons in 0, (at nor- Observation of relativmal pressure); p = istic increase of ionization and levelling 0.3-30 Bev/c. off to Fermi plateau above 6 Bev/c in reasonable agreement with calculations including the density effect. p mesons with p / p c = Observation of relativistic rise and Fermi 3-80, and electrons plateau: in good with p/pc = 50agreement with 2000, in following

-

24

1.

PARTICLE DETECTION

TABLE I. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss (Continued) ~~

Author

~

Type of particle, energy range, and material traversed

Method of determination

gases: He (1.3 atmos); Ar (0.2 atmos) ; Ar-He mixture (each a t 0.2 atmos); Xe-He mixture (each at 0.1 atmos) p mesons (0.3-70 Bev/c) in mixture of argon (P = 774 mm Hg) and ethylene (P = 46 mm Hg). p mesons in Ne.

Results theory for He; for other cases, calculated rise is somewhat larger than experimental values.

.

Parry et al.1° (1953)

Eyeions et

al.7'

(1955)

Proportional counter

Proportional counter

Palmatier, et aZ.I2 (1955)

Proportional counter

p

mesons (0.2-15 Bev/c) in argon at pressures from 2 to 40 atmos.

Lanou and Kraybi11728 (1959)

Proportional counter

p

mesons (3.3-140 Bev/c) in a mixture of 95% He and 5% CO2 a t a total pressure of 2.7 atmos.

Barbersl (1955)

Ionization chamber

Electrons (1-35 MeV) in Hi, He, and N2 (normal pressure).

Observation of relativistic rise and Fermi plateau in good agreement with theory. Observation of relativistic rise and Fermi plateau in good agreement with theory. Decrease of the relativistic rise with increasing pressure, in good agreement with calculations of Sternheimer20 on the density effect. Observation of relativistic rise in He and saturation of the most probable ionization loss a t p/m,c 200 (Fermi plateau). In this region, the ionization loss is 1.28 & 0.04 times minimum. Observations in good agreement with calculated relativistic rise for Nz, but experimental increase of ionization somewhat smaller than calculated increase of IdE/drl for Hzand

2

1.1. INTERACTION

O F RADIATION WITH MATTER

25

TABLEI. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss (Continued)

Author

Barbers2 (1956)

Method of determination

Ionization chamber

Type of particle, energy range, and material traversed

Electrons (1-35 MeV) in H2 and He a t 1 and 10 atmos pressure.

Results He, possibly clue to production of Cerenkov radiation which does not contribute t o ionization.20 Observations in good agreement with calculated relativistic rise. At 10 atmos, reduction of ionization due to density effect is observed for T, 10 Mev in HZ and for T. >, 18 Mev in He. The reductions at 35 Mev are in reasonable agreement with calculations of Sternheimer.20 Observed relativistic rise between 1.7 Mev (minimum) and 9.0 Mev in reasonable agreement with calculations. Relativistic rise of 1.17 k 0.03 between sea level spectrum and underground spectrum, in good agreement with calculations using density effect correction.

2

Herefords7 (1948)

Low-pressure counter

Shamos and Hudes88 (1951)

Low-pressure counter

McClureS6 (1953)

Pickup and Vo.yvodics9 (1950)

Electrons (0.2-9.0 MeV) in H 2 ( P = 7 cm Hg).

Cosmic-ray p mesons at sea level (average momentum = 3.5 Bev/e) and under 140 feet of rock (average momentum = 48 Bev/c); primary specific ionization in H2 filled counter ( P = 2.0 cm Hg). Observations in good Electrons (0.2-1.6 Low-pressure agreement with MeV) in He, He, Ne, counter Bethe’s theory of and Ar. primary specific ionization. p-decay electrons and First observation of Nuclear relativistic fi mesons -10% relativistic emu1sion rise of grain count in (grain count) in plate exposed to

26

1.

PARTICLE DETECTION

TABLE I. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss (Continued) ~

Author

Method of determination

Type of particle, energy range, and material traversed

Results

emulsion between sea-level cosmic-ray T/m& 3 and spectrum; and highT/mor2 20, in reaenergy electrons, n sonable agreement mesons, and protons with theoretical prein plate exposed t o dictions. cosmic rays at high altitudes. Blob count increases Electrons (5 Mev-5 Nuclear Bev. emulsion -5% between 5 Mev and 15 MeV, then re(blob count) mains constant to 25 Bev; relativistic rise is smaller than value predicted by theory (14%). Grain count increases Nuclear n mesons (200 Mev/c-3 by -8% between Bev/c). emu1sion 500 and 1500 Mev/c, (grain count) in reasonable agreement with calculations of Budini.z* Ratio R of plateau to Electrons (y > lo), Nuclear minimum blob emulsion T mesons (y < loo), count, R = 1.14 k (blob count) and protons (y < 0.03, in good agree10). ment with theoretical value 1.14;slow rise of grain count until saturation is reached for > 100, in good agreement with calculations of Sternheimer.20 Nuclear Electrons from p decay Relativistic increase of emulsion (average energy = 14% (between mini(blob count) mum and plateau 34 MeV) and nionization), and slow mesons (31-230 Mevl. rate of rise, in good agreement with theory. Ratio R of plateau t o Nuclear T mesons (109.1 Mev) minimum grain emulsion from Krz decay, and count, R = 1.133. (blob count) p mesons (152.7

N

Morrishgo (1952)

Daniel et aLS4 (1952)

Stiller and Shapiros2 (1953)

Fleming and Lord93 (1953)

Alexander and JohnstongTa (1957)

1.1.

27

INTERACTION O F RADIATION WITH MATTER

TABLE I. Summary of Some of the Experimental Investigations on the Relativistic Rise and t h e Density Effect for the Ionization Loss (Continued)

Author

Method of determination

Type of particle, energy range, and material traversed MeV) from K,? decay.

JongejansSTb (1960)

Nuclear emuIsion (blob count)

Beam pions (5.2-5.7 Bcv/c) from Berkeley Bevatron; secondary pions produced by beam pions (1.8 < y 5 3.5); electron pairs (65 < y < 1100).

Whittemore and Street104 (1949)

Crystal counter (pulse-height distribution)

Cosmic-ray p mesons in AgCl crystal. Two energy groups: T, = 0.3 Bev (minimum ionization), and T, > 1.6 Bev.

Bowen and Roser1°6 (1952)

Scin tillator (pulse height distribution)

Cosmic-ray p mesons (30 Mev-3 Bev) in anthracene crystal.

Hudson and Hofstadterllo (1952)

Scintillator

Cosmic-ray p mesons in NaI (Tl) crystal ( p r > 225 Mev/c).

Baskin and Winckler (1953)

Scintillator

Cosmic-ray p mesons (80-2200 MeV) in xylene solution (with terphenyl).

O6

Results The authors have obtained an accurate calibration curve for grain count versus p p c for the region 0.5 < p < 0.95. Ratio R of plateau t o minimum grain count, R = 1.129 f 0.010. The relativistic rise of the grain count is slow, with an appreciable increase (-4%) taking place between y = 40 and y 1000 (plateau). Observed relativistic increase between the two energy groups is in agreement with the prediction of the Bethe-Bloch formula, including density effect correction. No detectable relativistic rise of most probable energy loss eprob, in good agreement with theory including the density effect. Observed pulse-height distribution in good agreement with calculations including the density effect. No relativistic rise is observed, in agreement with calculat i o n 9 including the density effect.

-

28

1.

PARTICLE DETECTION

TABLE I. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss (Continued) ~~~

Author Bowen111 (1954)

Millar et (1958)

uZ.109

Paul and Reich"4 (1950)

Goldwasser, MiIIs, and Hanson 119 (1952)

Goldwasser et diZ1 (1955)

Method of determination

~~

Type of particle, energy range, and material traversed

Results

Relativistic increase in Accelerator-produced reasonable agreeT- mesons (60-220 ment with calculaMev), p- mesons tions; rise to plateau (245 Mev), and cosvalue may be somemic-ray p mesons (4 what faster than pregroups with average dicted by theory [obenergies: 0.37, 0.76, served rise: (10.9 i 1.47, and 5.23 Bev); 1.0)% at T, = NaI(T1) crystal. 50m,c*]. Scintillator Cosmic-ray p mesons Observed value of €i,rob (two energies: T p = (0.30 Bev)/e,,,b (2.2 0.30 Bev and 2.2 Bev) = 1.016 2c Bev); liquid scintil0.005, in good agreelation counter filled ment with BetheBloch formula, inwith triethylbenzene (plus tercluding density efphenyl). fect correction. Energy loss in Electrons (2.8 and 4.7 Observed mean energy loss is in better agreethin sample Mev) in samples ment with calculated (-0.3 gm/cma) of Be, C, H20, Fe, and value if density efPb. fect correction is included. Electrons (9.6 and 15.7 Observed most probaEnergy loss ble energy loss (-1 MeV) in thin samples (-1 gm/cm*) of Mev) is in good Be, polystyrene, Al, agreement with Cu, and Au. Landau formula, including correction for density effect. Energy loss Electrons (15.7 MeV) Direct observation of in thin samples of the density effect by comparing energy Teflon and Kel-F, and in the correloss in solid and gasesponding gases (same ous samples of the chemical composisame substance. The tion) : perfluororeduction of the ionization loss in the cyclobutane and chlorotrifluorosolid samples is in good agreement with ethylene. calculations of the density effect.

Scintillator

1.1.

INTERACTION O F RADIATION W I T H MATTER

29

TABLEI. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss (Continued)

Author Hudson121 (1957)

Method of determination Energy loss

Type of particle, energy range, and material traversed

Results

Electrons (150 MeV) in Observed energy loss thin targets (-2.5 eprob in good agreement with Landau gm/cma) of Li, Be, C, and Al. formuIa including the density effect correction.

(17.9 f 0.25 droplets/mm). The theoretically predicted value for this high-energy group of p mesons is 18.5 droplets/mm, in good agreement with the experimental result. Hazene6 verified the relativistic rise for electrons in air. Even earlier experiments by Corson and Brodese and b y Sen GuptaG7gave convincing evidence for the relativistic rise b y using cosmic-ray electrons. More recently, Hayward68 showed that high-energy is the minielectrons have a n ionization of 1.4Jni, in helium, where Jmin mum ionization. 9 measured the relativistic rise I n a recent experiment, Kepler et ~ 1 . 6 have of the ionization loss of p mesons and electrons in He, Ar, and Xe, by obtaining the drop count in a n expansion cloud chamber. Measurements were made for He a t -1.3 atmos, for Ar a t -0.2 atmos, for a n Ar-He mixture, each a t -0.2 atmos, and for a Xe-He mixture, each a t -0.1 atmos. I n each case, the cloud chamber contained alcohol and water vapor a t a partial pressure of -5 cm Hg. The p-meson momenta extend from minimum ionization (p/m,,c = 3) to p/m,c = 80. The electron momenta extend from p/mc = 50 to z 2 0 0 0 . Thus the entire region of the relativistic rise is covered in these measurements, including the Fermi plateau, which starts a t p/mc E! 1000. For the helium experiment, the theoryzkz2is in very good agreement with the data, if Williams’ valuesg of the excitation potential I for He is used (IH, = 44 ev;69 I for gas mixture = 49.4 ev69). For the argon and the argon-plus-helium experiments, the calculated rise is larger by a t least one standard deviation than W. E. Haaen, Phys. Rev. 67, 269 (1945). D. R. Corson and R. B. Brode, Phys. Rev. 63, 773 (1938). 67 R. L. Sen Gupta, Nature 146,65 (1940); Proc. NatE. Znst. SCi. fndiu 9,295 (1943). 88 E. Hayward, Phys. Rev. 72, 937 (1947). 60 R. G. Kepler, C. A. d’Andlau, W. B. Fretter, and L. F. Hansen, Nuovo cimento [lo] 7, 71 (1958). See also A. Rousset, A. Lagarrigue, P. Musset, P. Rancon, and X. Sauteron, Nuovo cimento [lo] 14, 365 (1959). 66

68

30

1.

PARTICLE DETECTION

the experimental values. Thus for the Ar-He mixture, a t p/mc % 1700,the is 1.59 f 0.04, whereas the calculated resultz0is observed value of J/Jmin is 1.64. Here J is the observed ionization loss, - (l/p)(dE/dx)w,, and Jmin the value of J at the minimum of ionization. In this experiment, the effective maximum energy transfer W O(determined by the size of a blob = 40 drops) was 700-1000 ev. For the Xe-He mixture, the discrepancy is appreciably larger. For p/mc E 1000, the experimental J/Jmin = 1.58 -t 0.05, whereas the values calculated from the theories of BudiniZ2and SternheimerZ0are 1.78 and 1.75, respectively. On the whole, it appears that the experimental points for 4, AF-He, and Xe-He lie on a curve which increases less rapidly with increasing momentum than the theoretical curve obtained from the expression for - (l/p) (dE/dz)w, [Eq. (1.1.28)]. Kepler et ~ 1 have . given ~ ~ various possible reasons for this discrepancy for the heavy gases, in particular: (1) a variation of the maximum energy transfer W Owith atomic number 2, due to the 2 dependence of the binding energy of the struck electron; (2) the ratio of the energy loss to excitation and to ionization may depend on the velocity v and on 2, in such a manner as to decrease the slope of the curve of - ( l / p ) ( d E / d z ) w , versus p / p c at high momenta beyond the ionization minimum; (3) shielding effects of the inner electron shells in the heavy elements are important at very high energies. However, these shielding effects are taken into account, a t least in first approximation, by the density effect term 6 in Eq. (1.1.28). As a check on the Xe-He experiment, as Kepler et al.69 have determined the number of drops per cm at Jmin 28.9 k 0.6 under well-controlled conditions. This experimental value can be compared with the theoretical predictions: 31.8 0.4 drops/cm for the Bakker-SegrP excitation potentialszoI , and 29.6 ? 0.4drops/cm for the higher C a l d ~ e 1 values 1 ~ ~ ofz1I . It is seen that the experimental value is in good agreement with Caldwell's results, which are also favored by several other ionization loss experiments (see Section 1.1.2.4). Important experiments on the ionization loss a t relativistic energies have been carried out with proportional counters. I n these experiments, the Landau distribution is measured, from which, of course, one obtains the most probable loss Ep& The most accurate determinations are those of Parry et aL7Oon the ionization loss eprobof p mesons in argon, and those of Eyeions et d."who obtained Eprob for p mesons in a neon-filled counter. In both cases, a relativistic rise of -50% was found, and the leveling off to

+

70 J. K. Parry, H. D. Rathgeber, and J. L. Rouse, PTOC. Phys. Soc. (London) A66, 541 (1953). 7l D. A. Eyeions, B. G. Owen, B. T. Price, and J. G. Wilson, PTOC. Phys. Soc. (London) A68, 793 (1955).

1.1.

INTERACTION OF RADIATION WITH MATTER

31

the Fermi plateau was clearly observed. These two experiments used a cosmic-ray magnetic spectrometer, in which the high-energy particles are passed through a strong magnetic field and the resulting deflection is measured in a hodoscope array of Geiger counters placed below the proportional counter. Palmatier and c o - w o r k e r ~have ~ ~ investigated the relativistic rise and the Fermi plateau of Eprob for p mesons in a counter filled with argon a t various pressures up to 40 atmospheres. These authors have directly verified the dependence of Eprob on the pressure, i.e., the increase of the density effect with increasing pressure,20 and the resulting decrease of cprob,plat/Eproh,min, the ratio of the plateau to the minimum value of eprob. The calculated values of €,,rob are in reasonable agreement with these experimental results. Lanou and Kraybil1728have recently carried out a-similar investigation using p mesons of momenta 3.3-140 Bev/c in a proportional counter filled with a mixture of 95% He and 5% COZ at a total pressure of 2.7 atmospheres. These authors have observed the relativistic rise of cprobin helium, and have found that the rise saturates at momenta p/m,c 2 200 due to the density effect. In the region of the Fermi plateau, the most probable ionization loss is 1.28 k 0.04 times the value a t the minimum, in agreement with the calculations of Sternheimer.20,21 Several other experiments with proportional counters demonstrate the relativistic rise, but were not accurate enough to establish the existence of the Fermi plateau. Among these studies, we may mention the experiments of Kupperian and Palmatier,73Becker et aZ.,74 Price et aZ.,76and Eliseiev et aZ.76 Several experimenters have investigated the width A of the Landau distribution as a function of the value of A t / ( p 2 1 ) ,which enters as a parameter in Landau’s theory. West77found that, although the percentage width 100A/Eprob decreases with increasing At/(P21), as required by Landau’s theory, the value of A/cprob is larger than Landau’s result by a factor of -2. This discrepancy for the width of theidistribution can be reE. D . Palmatier, J. T. Meers, and C. M. Askey, Phys. Rev. 97, 486 (1955). R. E. Lanou and H. L. Kraybill, Phys. Rev. 113, 657 (1959). 73 J. E. Kupperian and E. D. Palmatier, Phys. Rev. 91, 1186 (1953). T 4 J. Becker, P. Chanson, E. Nageotte, P. Treille, B. T. Price, and P. Rothwell, Proc. Phys. SOC.(London)A66, 437 (1952). 75 R. T. Price, D. West, J. Becker, P Chanson, E. Nageotte, and P. Treille, Proc. Phys. soc. (London)A66, 167 (1953). 7 6 G. P. Eliseiev, V. K. Kosmachevsky, and V. A. Lubimov, Doklady Akad. Nauk. S.S.S.R. 90, 995 (1953); (English translation: NSF-tr-163, Dept. of Commerce, Washington, D.C.) 77 D. West, Proc. Phys. SOC.(London)A66, 306 (1953). 71

788

32

1.

PARTICLE DETECTION

moved by improvements in the Landau theory which have been discussed by fan^^^ and H i n e ~ . ~ ~ Igo and co-workerssOhave measured the distribution of energy losses of 31.5-Mev protons in a $inch proportional counter filled with an Ar-C02 mixture (96 % Ar, 4 % C02). The pulse-height distribution was in reasonable agreement with the Landau distribution, although slightly wider in the region of the tail for large energy losses. Barbers1.s2has measured the specific ionization of electrons in Hz, He, and Nz in a n ionization chamber. The electrons were obtained from the Stanford linear accelerator and had energies ranging from 1 to 35 MeV. A collimated beam of electrons was sent through an ionization chamber into a Faraday cup, so that the ratio of the collected ionic charge to the charge collected in the Faraday cup is proportional to the specific ionization. In Barber's first experiment,S' H2, He, and Nz at atmospheric pressure were used. Under these conditions, one does not expect any density effect correction, since the density effect sets in above 35 Mev for gases a t normal pressure.20 A t minimum ionization, the number of ion pairs per cm (probable specific ionization) was 7.56 f 0.09,6.15 k 0.08, and 53.2 f 0.7 in H2, He, and N2, respectively (at normal temperature and pressure). These results were compared with the theoretical expression for the ~ , (l.l.28)] with W o= 17.4 kev restricted energy loss, - ( l / p ) ( d E / d ~ ) [Eq. for Hz, 16.4 kev for He, and 70 kev for Nz,as determined from the size of the ionization chamber and the experimental conditions. Barbers1 thus obtained the following values for w, the average energy required to produce an ion pair: 37.8 k 0.7, 44.5 k 0.9, and 34.82",; ev for Hz, He, and N'L,respectively. These results are in reasonable agreement with ,~~ and H ~ r s t , ~ ~ the values of w obtained by Jesse and S a d a u s k i ~Bortner and Bakker and S e g r P (see Section 1.1.2.12). The total number of ion pairs per cm a t the ionization minimum as obtained from the average energy loss, - (l/p)(dE/ds) [Eq. (1.1.1)] without any limitation on the maximum energy transfer, was found to be: 9.19 f 0.18,7.55 k 0.16, and 61.62::; for the three gases. The relativistic increase of the ionization from minimum (at -1.7 MeV) to 35 Mev is 1.17 for H2,1.20 for He, and 1.24 for Nz. For N P , the calculated increase of the ionization loss agrees within 1% with the observed rise, but for Hz and He, the predicted increase is U. Fano, Phys. Rev. 92, 328 (1953). K. C. Hines, Phys. Rev. 97, 1725 (1955). G. J. Igo, D. D. Clark, and R. M. Eisberg, Phys. Rev. 89, 879 (1953). *1 W. C. Barber, Phys. Rev. 97, 1071 (1955). 82 W. C. Barber, Phys. Rev. 103, 1281 (1956). W. P. Jesse and J. Sadauskis, Phy8. Rev. 90, 1120 (1953). R 4 T. E. Bortner and G. S. Hurst, Phys. Rev. 90, 160 (1953). 79

1.1.

INTERACTION O F RADIATION WITH MATTER

33

somewhat higher than the observed value, assuming that the energy loss per ion pair w is independent of the electron energy. In particular, for Hz, the deviation between the calculated and the observed values would correspond to an increase of (3.3 i-0.7)% in w as the electron energy is increased to 35 MeV. It is possible that the lowering of the rate of rise of the ionizatJionis due t o the production of c e r p k o v radiation29which is not reabsorbed t'o form ions in the gas. In Barber's second experiment,82 the specific ionization of electrons in H2 and He was measured at 1 and 10 atmospheres pressure. At 10 atomspheres, a sizable density effect is expected for both gases a t 35 MeV, whereas at 1 atmosphere the density effect correction is negligible. The experimental setup was essentially the same as in the first experiments1 (ionization chamber; Faraday cup). The theory is in good agreement with the experimental results which show that at 10 atmospheres, the specific ionization J [Eq. (1.1.28)] a t first increases above the minimum, from -2 to 10-15 MeV, but levels off above 10 Mev for H2 and 18 Mev for He. For Hz at 10 atmospheres, the value of J ( 3 5 Mev)/Jmi, is 1.12 upon inclusion of 6 (density effect), as compared to the experimental value: J / J m i ,= 1.11. (The calculated value of Jmin is 3.39 Mev/g cm-2.) Without the density effect, the calculated J / J m i ,would be 1.20. Thus the data provide a good confirmation of the existence of the density effect for gases at high pressure. Upon taking into account the experimental uncertainties, BarberB2finds that the ionization loss is decreased by (8 f 1 ) % by the density effect, as compared to the theoretical reductionz0of 6.5 %. Similar agreement is obtained for the measurements in He at 10 atmospheres, where the observed decrease of the ionization J at 35 Mev is (3.5 f 1.3) %, as compared to the calculated valuez0of 3%. Low-pressure Geiger counters have also been used to determine the relativistic rise of the ionization loss in gases. In a low-pressure counter, one attempts to measure the total number of ionizing collisions:

N = /owm"xP(W)dW where P(W) dW is the probability of a collision with energy transfer between Wand W dW. By contrast, a proportional counter measurement gives the total energy deposited: & = J ~ " " " P ( W ) W dW. The fluctuations in & are largely due to the presence of the large energy transfers W of the order of W,., which are weighted by the factor W in the integrand for &. For N , on the other hand, there is no factor Win the integrand, so that the large energy transfers are weighted much less heavily than for &, and the fluctuations in N are correspondingly reduced. The condition under which the Geiger counter measures N rather than & is that the incident

+

34

1.

PARTICLE DETECTION

particles make on the average less than one ionizing collision in traversing the counter. N is proportional to the primary specific ionization J,, of the incident p a r t i ~ l e . ~ ~ Several experiments have been performed using low-pressure Geiger counters. The most extensive recent measurements for electrons in the 0.2-1.6 Mev energy rangepre those of McC1ureS6who obtained the primary specific ionization J,, of electrons in this energy range for Hz, He, Ne, and Ar. For Hz, the results could be fitted to the theoretical curve of J,, versus p/mc obtained by Bethe.86Somewhat earlier, H e r e f ~ r d , using ~’ counter measurements, obtained evidence for the relativistic rise of the ionization loss of electrons by measuring J,, for electrons in hydrogen in the range from 0.2 to 9.0 MeV. Evidence for the relativistic rise for p mesons in hydrogen has been obtained by Shamos and Hudes.88 The relativistic rise of the ionization loss in photographic emulsion has been the subject of numerous investigations. The first definite evidence for a -10% rise in the grain count in emulsion was obtained by Pickup and VoyvodicS9in 1950. The minimum value of the grain count G is obtained for a ratio T/moc2 3 of the kinetic energy T to the rest energy mgc2 of the particle. The rise starts at T/moc2 N 3 and continues until the plateau value G,,,, is reached for T/moc2 10-100. The precise value of T/moc2at which the plateau is reached has been the subject of some controversy, with some experiments favoring a rapid rate of rise of G to GPIsta t T / m d 10, while others give evidence of a more gradual rise for which the plateau is reached only for T/moc2 50-100. In the experiments, following a suggestion of Morrish,Soone obtains generally the blob count rather than the grain count. Here a blob is defined as either a single grain or a group of overlapping grains which cannot be resolved. It was found that the blob count is considerably more independent of the observer than the grain count.91 In the region between the minimum and the plateau, the blob count is proportional to the grain count.92For comparison of the theory with the grain or blob count observations, one must calculate the restricted energy loss [Eq. (1.1.2S)l with maximum energy transfer W O 5 kev. The reason is that for energy transfers W >, 5 kev, the delta-ray will have a large enough range to traverse one or more addi-

-

-

-

-

-

G. W. McClure, Phys. Rev. 90, 796 (1953). H. A. Bethe, in “Handbuch der Physik” (H. Geiger and K. Scheel, eds.), Vol. 24, p. 515. Springer, Berlin, 1933. *’ F.L. Hereford, Phys. Rev. 73, 982 (1947);74, 574 (1948). M. H. Shamos and I. Hudes, Phys. Rev. 84, 1056 (1951). 88 E. Pickup and L. Voyvodic, Phys. Rev. 80, 89 (1950). A. H. Morrish, Phil. Mug. [71 43, 533 (1952). 91 See also L. Jauneau and F. Hug-Bousser, J . phys. radium 13, 465 (1952). 92 B. Stiller and M. M. Shapiro, Phye. Rev. 92, 735 (1953). 86

88

1.1.

35

INTERACTION OF RADIATION WITH MATTER

tional grains not directly in line with the path of the particle. If only the grains along the track are counted, or if a blob count is made, in which all of the overlapping grains due to an energetic 6 ray are counted as a single unit, then the observed count will be essentially unaffected by the presence of the high-energy 6 rays, so that a cutoff at WO 5 kev is indicated, as was first pointed out by Messel and Ritson.26The resulting values of - (l/p)(dE/dx)wo as obtained by Sternheimer20n21 give a relativistic rise of 14% which saturates slowly and does not level off until T/moc2 100. The magnitude of the increase and the gradual character of the relativistic rise are in good agreement with the observations of Stiller and Shapirog2 using cosmic rays, and those of Fleming and Lordg3using cosmic-ray electrons and accelerator-produced 7-mesons. On the other hand, Budini’s calculations22*2s give a more rapid rate of rise, with GpIstbeing reached for T/moc2 between 10 and 40, depending on the specific assumptions made about the widths of the spectroscopic lines of the Ag and Br atoms of the emulsion. Budini’s calculations are in reasonable agreement with the results of Daniel et al. 9 4 which indicate a more rapid rate of rise than those of references 92 and 93. Data on the ionization loss in emulsion have also been obtained by McDiarmidlg6Michaelis and Violet,g6Morrishlg7and others. Alexander and Johnstong7ahave obtained a very accurate calibration curve for the grain density as a function of ppc for ?r and p mesons. In this work, the authors used T and p mesons of constant and precisely known energy from the K,2 and Kp2decays of K partides at rest. The calibration curve extends from p = 0.5 to0.95, corresponding to 1 < g* < 3, where g* is the grain density normalized to the minimum of ionization:

-

-

g* = (dE/dX)w,/[(dE/dX) Wolrnin.

I n the range 1 < g* < 1.6, the accuracy of the calibration curve for g* is estimated to be better than 1%. For the ratio of plateau to minimum grain count, the authors have obtained g,*l, = 1.133. Recently, J ~ n g e j a n s ~ has ’ ~ measured the relativistic rise of the grain density in Ilford G5 emulsion, using pion tracks of energy -5.4 Bev from the Berkeley Bevatron, and secondaries produced by the pions in the emulsion [pions stopping in the emulsion, with y between 1.8 and 2.5; J. R. Fleming and J. J. Lord, Phys. Rev. 92, 511 (1953). R. R. Daniel, J. H. Davies, J. H. Mulvey, and D. H. Perkins, Phil. Mag. [7]43, 753 (1952). 9 5 I. B. McDiarmid, Phys. Rev. 84, 851 (1951). 06 R. P. Michaelis and C. E. Violet, Phys. Rev. 90, 723 (1953). $7 A. H. Morrish, Phys. Rev. 91, 423 (1953). 978 G. Alexander and R. W. H. Johnston, Nuovo cimenfo [lo] 6, 363 (1957). wb B. Jongejans, Nuovo cimenlo [lo] 16, 625 (1960). 98 94

36

1.

PARTICLE DETECTION

other secondary pions; electron pairs with y between 65 and 1100, where y = (T/moc2) I]. The momentum of the tracks was determined from multiple scattering measurements. For the ratio GPlat/Gmin,a value 1.129 k 0.010 was obtained, in good agreement with the results of Stiller and Shapirog2 (1.14 2 0.03), Fleming and Lordg3 (1.14 f O.Ol), and Alexander and Johnston97a (1.133 f 0.008). The rise of the grain density G to the plateau value was found to be slow, with a n appreciable increase 1000. This result is in good agreetaking place between y = 40 and y ment with the calculations of Sternheimer,20*21 and with the experiments of references 92 and 93. A calibration curve for g” versus (y - 1) is given in the paper of J0ngejans.97~This curve was calculated using a value of the excitation potential I = 501 ev for AgBr. In comparing the theory with these data for emulsion, it must be borne in mind that one would expect large fluctuations of the ionization loss in each grain, since the most probable energy loss Eprcb for a minimum ionizing particle in a 0.2 p AgBr grain is only -50 ev. On the other hand, the threshold value 7 for the energy deposit below which the grain is not exposed is of the order of several hundreds of volts. Thus the effect of Landau-type fluctuations on the grain count is expected to be quite important, as was first pointed out by BarkasI9*and subsequently by Brown.99I n view of these difficulties and limitations of the theory, the detailed quantitative agreement which has been obtained for emulsion is surprisingly good. In connection with the emulsion measurements on electron-positron pairs produced by y rays of very high energy ( E , 2 10 Bev), Perkins’** observed that there is a reduction of the ionization loss J below the value of twice minimum ionization (2Jmi,) due to the interference between the electromagnetic fields of the positron and electron. With increasing distance x from the origin of the pair, the ionization J varies form 0 to 2Jni,, as a result of the increase of the distance d between the positron and electron. Several authorslo’ have treated theoretically the problem of the ionization loss of an electron-positron pair, along the lines of Fermi’s cal~ulation’~ of the ionization loss of a single particle. The theory gives the dependence of the ionization J on the distance d. The asymptotic value 2Jmi. is attained when d becomes large compared to c/(2rvP)

+

-

W. H. Barkas, in “Colloque Bur la sensibilite des cristaur et des emulsions photographiques,” Paris, September, 1951; see also W. H. Barkas and D. M. Young, University of California, Radiation Laboratory Report URCL-2579, revised (1954). 99 L. M. Brown, Phys. Rev. 90, 95 (1953). loo D. H. Perkins, Phil. Mag. [7] 46, 1146 (1955). 1olA. E. Cudakov, Izvest. Akad. N a u k S.S.S.R. 19, 650 (1955); I. Mito and H. Ezawa, P r o p . Theoret. Phys. (Kyoto) 18, 437 (1957); G. Yekutieli, Nuovo cimento [lo] 6, 1381 (1957).

1.1.

INTERACTION OF RADIATION WITH MATTER

37

( = 0.51 X lodEcm for emulsion). Upon using the theoretical dependence of J on d, the measurements of J ( x ) enable one to obtain the value of the opening angle of the pair 0 ( = d/x), from which in turn one can estimate the energy E , of the parent y ray.lo2The results obtained by this methodLo3 are in reasonable agreement with the conventional determination of the y-ray energy from the subsequent development of the shower of electrons and y rays (see Section 1.1.5.2). Besides the experiments on photographic emulsion, crystal counters and scintillators* have also been used to observe the relativistic increase of the ionization loss of p mesons in condensed materials. The first of these experiments was carried out by Whittemore and Streeti0*in 1949, using a silver chloride crystal. These authors compared the ionization pulses produced by p mesons of range > 112 cm of P b (T, > 1.6 Bev) with those produced by minimum ionization p mesons (T, = 0.3 Bev) , and found a definite relativistic increase. The results were in agreement with the predictions of the Bethe-Bloch formula including the density effect correction. Experiments with p mesons passing through an anthracene scintillator have been performed by Bowen and Roser,lo6who obtained no detectable relativistic increase of the ionization loss Eprob above the minimum value. This result is in agreement with the theoretical predictions, since the density effect sets in at a relatively low energy (near the ionization minimum) for low atomic number, and is large enough to prevent any rise of €pro,, from occurring. Similar results were obtained by Baskin and W i n ~ k l e r using , ~ ~ ~a ~liquid ~ ~ ~scintillator of low Z (xylene). In these experiments, it is assumed that the light output of the scintillator is proportional to the energy deposited by the incident particle ( p meson). This assumption has been verified by Chou,lo8who showed that the response of most scintillators is nearly linear up to 3Jmin-4Jmi,. Millar, et al. lo9 have exposed a large-area liquid scintillation counter to cosmic-ray p mesons. The counter was filled with triethyl-benzene (plus terphenyl). The most probable loss Eprob and the Landau distribution were obtained both for T, = 0.30 Bev and T , = 2.2 Bev. The value of fprob

* See also in this volume, Chapter 1.4. A. Borsellino, Phys. Rev. 89, 1023 (1953). W. Wolter and M. Miesowice, Nuovo cimento [lo] 4, 648 (1956). lo4 W. L. Whitternore and J. C. Street, Phys. Rev. 76, 1786 (1949). 106T.Bowen and F. X. Roser, Phys. Rev. 86, 992 (1952). 108 R. Baskin and J. R. Winckler, Phys. Rev. 92, 464 (1953). 107 See also A. G. Meshkovskii and V. A. Shebanov, Doklady Akad. Nauk S.S.S.R. 83, 233 (1952). 108 C. N. Chou, Phys. Rev. 87, 903 (1952). 109 C. H. Millar, E. P. Hincks, and G. C. Hanna, Can. J . Phys. 36,54 (1958). 102

108

38

1.

PARTICLE DETECTION

at 0.30 Bev is higher by (1.6 & 0.5) % than the value a t 2.2 Bev, in good agreement with Eq. (1.1.30) including the density effect correction 6. The prediction of the Landau theory for the width of the pulse-height distribution (18 % at half-maximum) is in reasonable agreement with the observed width (20.5% at half-maximum in the central area of the counter) when the width due to the counter resolution function (8%) is taken into account. Hudson and Hofstadterl'O have exposed a thallium-activated sodium iodide crystal [NaI(TI)] t o the cosmic-ray p-meson spectrum and have found that the resulting observed pulse-height distribution is in much better agreement with the theoretical distribution obtained upon inclusion of the density effect correction 6 (as calculated from the paper of Halpern and Hall's) than with a theoretical distribution obtained by setting 6 = 0. In each case, the theoretical curve was obtained by folding the Landau straggling distribution6awith the cosmic-ray p-meson spectrum. In a later investigation, Bowen'" used a NaI(T1) crystal to measure the energy loss of T- and p mesons of selected energies or energy groups. The T- mesons were produced by the Chicago 450-Mev cyclotron and had well-defined energies extending from 61 to 222 MeV. In addition, 245-Mev p- mesons arising from the decay of 227-Mev T- mesons were used. Moreover, four energy groups of the cosmic-ray p-meson spectrum were studied. These groups were separated in energy by using various thicknesses of iron absorber. The average energies of the p mesons in the four groups were: T, = 368,755,1470, and 5230 MeV. The energies of the T- from the cyclotron were: 61, 85, 118, 163, and 222 MeV. At each of these energies, the most probable loss Eprob was obtained from the observed pulse distribution. Bowen thus obtained values of epr& as a function of T/moc2. The theoretical prediction20321 for 6prab versus Tlmoc2 is in reasonable agreement with these data. Thus from the calculations one obtains an 8.2% increase (relative to minimum ionization) at T,, = 50mpc2,and an asymptotic value of the rise (at very high energies) of 11.4%. The experimental value is 10.9 5 1.0% at T, = 50m,c2. This result may indicate that eprob rises to the plateau value somewhat more rapidly than predicted by the theory. It may be noted that the reason why there is a relativistic rise of Eprob for NaI but none for anthracene or xylene is that with increasing 2, the density effect correction 6 sets in a t higher energies, thereby leading to a relativistic rise before the energy loss eprob saturates due to the onset of 6. The density effect has been extensively studied by observing the energy Hudson and R. Hofstadter, Phys. Rev. 88, 589 (1952). T. Bowen, Phys. Rev. 96, 754 (1954).

11oA. l11

1.1.

INTERACTION OF RADIATION WITH MATTER

39

loss of electrons in passing through thin foils. The straggling of the energy loss in thin foils was already clearly demonstrated in 1928 by the work of White and Millington,"2 as well as that of Madgwick.'13 More recently, Paul and Reich114measured the energy loss of 2.8-Mev and 4.7-Mev electrons in foils of Be, C, Fe, and Pb. Chen and Warshaw116showed that eprob for electrons with energies T , < 2 Mev is correctly given by Landau's theory.63 However, from their data they were unable to discriminate between the Landau distributionKaof energy losses, and the (somewhat wider) distribution of Blunck and Leisegang.66On the other hand, in the experiments of Kalil and Birkhoff,'ls an accurate comparison could be made with the Blunck-Leisegang distribution, and it was found that while for the heavy elements (e.g., Pb) this distribution is in essential agreement with the observations, for the light elements (e.g., Be) the predicted width of the distribution at half-maximum is too small by a factor of -1.8. However, the discrepancy for light elements was not observed in a more recent experiment by Hungerford and Birkhoff."' ~ ~ good ~ agreement with the BlunckKageyama et ~ 1 also. obtained Leisegang distribution for foils of Al, Cu, In, and Pb. Goldwasser, Mills, and Hansonllg have measured the energy loss of 15.7-Mev electrons in passing through thin samples of Be, polystyrene, Al, Cu, and Au. With the exception of Au, they found that good agreement for eproo could be obtained by using the asymptotic value of the density effect correction [Eq. (1.1.2Oa)l. For the case of Au, the expression for 6 for intermediate energies, Eq. (1.1.20), must be used, as was pointed out by Warner and Rohrlich.120The energy loss distributions of Goldwasser et aZ.l19 are in essential agreement with those predicted from the Landau theory.6a Goldwasser, Mills, and RobillardlZ1have obtained a direct demonstration of the density effect, by measuring the energy loss of 15.7-Mev electrons in (solid) Teflon and Kel-F, and then in the gases corresponding to Teflon and Kel-F (i.e., gases having the same chemical composition). It was found that the difference between the values of Cprob P. White, and G. Millington, Proc. Roy. SOC.A120, 701 (1928). E. Madgwick, Proc. Cambridge Phil. SOC.23, 970 (1927). 114 W. Paul and H. Reich, 2.Physik 127, 429 (1950). 116 J. J. L. Chen and S. D. Warshaw, Phys. Rev. 84, 355 (1951). 116 F. Kalil and R. D. Birkhoff, Phys. Rev. 91, 505 (1953). 117E. T. Hungerford and R. D. Birkhoff, Phys. Rev. 96, 6 (1954). 118 S. Kageyama, K. Nishimura, and Y. Onai, J . Phys. Sor. Japan 8, 682 (1953); Kageyama, S., and Nishimura, K., J . Phys. SOC.Japan 7 , 292 (1952). 119 E. L. Goldwasser, F. E. Mills, and A. 0. Hanson, Phys. Rev. 88, 1137 (1952). 120 C. Warner and F. Rohrlich, Phys. Rev. 93,406 (1954). lZ1 E.L. Goldwasser, F. E. Mills, and T. R. Robillard, Phys. Rev. 98, 1763 (1955); see also A. M. Hudson, Phys. Rev. 106, 1 (1957). 112

11*

40

1.

PARTICLE DETECTION

in the solid and the gaseous phases is given by the predicted density effect correction 6. The ionization loss is rapidly becoming an important tool in bubble chamber investigations. The bubble count (number of bubbles per cm of path) is a function only of the velocity of the particle and the temperature . the ~ first ~ to~make .a systematic study of of the liquid. Glaser et ~ 1 were the bubble count as a function of the velocity of the particle, by using secondary protons and T+ mesons of momenta between 0.53 and 1.60 Bev/c from the Brookhaven Cosmotron. They found that the bubble density b is approximately proportional to 1/P2. This indicates that the bubble formation is proportional to the number of slow 6 rays (secondary elect,rons). The number of 6 rays per gm/cmZ is given by

where El' is the lower limit and Ez' is the upper limit of the energies of the 6 rays considered in ns; El' and E l are in electron volts. I n similarity to the grain count in emulsion or the drop count for cloud-chamber tracks, E 2 is taken as the energy of a S ray that has a long enough range to extend to a visible distance from the track of the incident (primary) particle. One thus obtains E2' = 50 kev. El' is taken as -3 times the mean excitation potential I of the atoms of the liquid, so that the 6 rays with energy E1' can be treated as free electrons during the collision. Thus El' is a t most a few kev, and therefore 126 is not very sensitive to the precise value of E i , since 1/Ez/ << l/El'. From their experimental data, Glaser et have obtained the following empirical expression for the bubble count b : b = (A/p2) B(T) bubbles/cm (1.1.36)

+

where A = 9.2 5 0.2 bubbles/cm for protons in propane, and B(T)is a function only of the temperature T. For propane, B C 8 at 56"C, =11 at 57"C, and B rises rapidly above 57°C to a value of =38 at 59.5"C. A is constant between 55°C and 59.5"C1but decreases rapidly below 55OC. (At 50°C no tracks are visible.) By using fast comparison tracks of known velocity, the velocity can be determined to an accuracy of 5% for proton tracks 10 cm long. ' ~ ~ suggested that instead of In a more recent paper, Willis et ~ 1 . have counting the number of bubbles along the track, the distribution of 122

D. A. Glaser, D. C. Rahm, and C. Dodd, Phys. Rev. 102, 1653 (1956);see also

G. A. Askarian, Zhur. Eksptl. i Teorel. Fiz. 30, 610 (1956);[translation: Soviet Phys. JETP 3, 4 (1056)l. 123 W.J. Willis, E. C. Fowler, and D. C. Rahm,Phys. Rev. 108, 1046 (1957).

1.1.

INTERACTION OF RADIATION WITH MATTER

41

lengths of the spaces between bubbles should be obtained. This method was found to be more accurate than the direct bubble count, because it avoids errors in the bubble counting method which are due to limited optical resolution of the images of neighboring bubbles and possible bubble coalescence. This procedure for bubble tracks is somewhat analogous to the blob count for nuclear emulsion tracks, since an aggregate of neighboring or overlapping bubbles (“blobs”) is effectively treated as a single unit. The bubble count obtained from the distribution of spacings can be fitted by the formula: b = C ( T ) / p 2 ,where C ( T )is a function only of the temperature. Willis et al. point out that a similar dependence to the 1/p2dependence of n6 is obtained by considering the restricted energy loss with maximum energy transfer WO= 70 kev, above which the 6 rays are assumed t o form separate tracks not in line with the track of the incident particle. Equation (1.1.28) for - (l/p) ( d E / d x ) w , gives a n approximate dependence l/p1,8a(in the velocity range of interest) which is experimentally indistinguishable from l/p2. Blinov et in a n experiment similar to th a t of Willis et ~1.~123 also obtain a l/p2 dependence for the bubble density as a function of velocity. 1.1.2.10. Capture and Loss of Electrons at Very Low Energies. For very low energies, the charged particle may capture an electron. Subsequently the electron may be lost again. This process is very complicated, and a complete theoretical treatment does not exist a t present. For a review of the literature on this problem, the reader is referred t o the review articles of Allison and Warshaw,60 and Bethe and Ashkin.3 A thorough discussion of the problems involved has been given by Bohr.126 From the Thomas-Fermi model, BohrlZ6has obtained the following expression for the capture cross section uc:

-

~ T U H ~ Z ~ ’ ~ ( V O /(VV > )> ~ Vo) (1.1.37) where v o = e2/h is the velocity of an electron in the first Bohr orbit of hydrogen (of radius aH),and v is the velocity of the incident particle. Equation (1.1.37) holds for heavy eIements, which have several atomic electrons with velocities vel larger than v. For light elements, where this condition is not fulfilled, Brinkman and K r a m e r P have derived the following formula for u c : UC

uc =

(2’~,/5)a,2zz(v,/v)’2

which is expected to hold for ( v / v o )

(1.1.38)

2 10.

124 G. A. Blinov, Iu. S. Krestnikov, and M. F. Lomanov, Zhur. Eksptl. i Teoret. Fiz. 31, 762 (1956); [translation: Soviet Phys. JETP 4 , 661 (1957)l. 126 N. Bohr, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 18, No. 8 (1948). 126 H. C. Brinkman and H. A. Kramers, Koninkl. Ned. Akad. Wetenschap., Proc. 33,

973 (1930).

42

1. PARTICLE DETECTION

The theory of the loss of electrons by a charged particle (ion) is less complicated. The cross section for loss al is of the order of r u H z S10-16 cm2 for v V O , i.e., for protons of -25 kev. uz falls off rather slowly with increasing v. The capture and loss cross sections of protons are equal (a, = u l ) for air at -20 kev and for hydrogen a t -50 kev. Above this energy, we have ae < a. For intermediate 2 values, BohrlZ6has obtained the followingestimate of a1: a1 7raH222/3(421). (1.1.39)

-

-

Thus for protons, q / a Cvaries as Z 1 / 8 T 6 for / 2 medium 2. The ratio is nearly independent of 2 and increases rapidly with energy. In agreement with this result, it is found experimentally that at predominates rapidly over ac as the energy is increased above the value (-25 kev) for which ac = a ~ . The processes of capture and loss of electrons are very important for the energy loss and range of fission fragment^.^ 1.1.2.11. The Stopping Power at Very Low Energies. For very low energies, where the velocity of the particle is less than the velocity of the atomic electrons ( T , 5 25 kev for protons), Fermi and Teller127have obtained the following expression for the energy loss: (1.1.40) where Ry is the Rydberg unit, and urn is the maximum velocity of the electrons of the substance if the latter are regarded as constituting a where TL is the number degenerate Fermi gas. Thus urn = (3~~/8?r)'/~(h/m), of electrons per ~ m Equation . ~ (1.1.40) shows that the energy loss increases with increasing v in this region, in contrast to the decrease with increasing v at higher energies ( = l / v 2 ) . Experimentally, good evidence has been obtained for the increase of the stopping power with increasing velocity a t low energies. Warshawl28 has made careful measurements of dE/dx for protons in Be, Al, Cu, Ag, and Au in the energy range from 50 to 400 kev. For all cases, he obtained a maximum of d E / d x in the neighborhood of T, = 100 kev. In the region below the maximum, an extrapolation of Warshaw's results (from -50 to -25 kev) could be well fitted by the Fermi-Teller formula. The maximum of the ionization loss dE/dx, to be denoted by J,, has the value 640 Mev/gmcm-2 for Be, where it occurs a t TP.,= 75 kev. For Al, J, = 440 Mev/gm cm+ and T,,, = 72 kev. For Cu, Ag, and Au, J , = 230, 140, and 100 Mev/gm cm-2, respectively. The corresponding values of T,,, are 140, 160, and 160 kev, respectively. 127

la*

E. Fermi and E. Teller, Phys. Rev. 73, 399 (1947). S. D.Warshaw, Phys. Rev. 76, 1759 (1949).

1.1.

INTERACTION OF RADIATION WITH MATTER

-

43

For somewhat higher energies (T, 400 kev), above T,,,, BohrlZ6has given an approximate theory based on the Thomas-Fermi model of the atom, and has obtained the following expression for the energy loss: dE-- 1 6 ~ n Z l ' ~ h e ~ -dx mu

(1.1.41)

In this region, -dE/dx goes as l / u , instead of the l / u 2 dependence which prevails at somewhat higher energies. Warshaw l Z 8has also obtained reasonable agreement of Eq. (1.1.41) with stopping power data for Cu, Ag, and Au in the range from T, = 350 to 550 kev. For a more detailed discussion of the stopping power measurements at low energies ( T , 2 2 MeV) the reader is referred to the review article of Allison and Warshaw.60 1.1.2.12. The Energy w Required to Produce an Ion Pair in a Gas. When a heavy charged particle passes through a medium, it excites and ionizes the atoms of the material. The ion pairs which are formed by direct action of the particle in the immediate vicinity of its path are called the primary ions. The most energetic of these primary ions, called delta rays, may travel a considerable distance before being themselves stopped by the medium. I n the slowing down process, the delta rays produce additional ions called secondary ions. The sum of the primary and secondary ionization constitutes the total ionization produced by the passing charged particle. It has been found experimentally that the energy w required to produce an ion pair is approximately independent of the energy and charge of the incident particle. Moreover, w does not vary appreciably for different gases, all values being of the order of 25-35 ev/ion pair. Typical values of w, obtained by Jesse and S a d a ~ k i s are ' ~ ~ as follows: 36.3 ev for H2, 42.3 ev for He, 35.0 ev for N2, 26.4 ev for Ar, 34.0 ev for air, and 32.9 ev for COz. Fano130has proposed a theory which explains both the fact that w is independent of the energy and charge of the incident particle, and also the smallness of the variation of w with the atomic number 2. The constancy of w as a function of E has been widely used in ionization chambers for the determination of the energy of particles. Thus if a particle is stopped in the gas of an ionization chamber, its initial energy is proportional to the total number of ions produced, which can be electronically measured by means of a linear amplifier. It is necessary to know the value of w for the gas in the ionization chamber; w can be determined by measuring the number of ions produced by a particle whose lZ9 W. P. Jesse and J. Sadaukis, Phys. Rev. 107, 766 (1957); see also T. E. Bortner and G. S. Hurst, Phys. Rev. 95, 1236 (1954); R. H. Frost and C. E. Nielsen, Phys. Rev. 91, 864 (1953). u0U. Fano, Phys. Rev. 70, 44 (1946); 72, 26 (1947).

44

1.

PARTICLE DETECTION

energy is known by other means (e.g., natural a particle or particle originating from an exothermic nuclear reaction). 1.1.3. Range-Energy Relations The mean range R of a particle of kinetic energy TI is given by (1.1.42) where -(l/p)(dE/dx) is the average energy loss as obtained from Eq. (1.1.1) or (1.1.19). In this section, we shall restrict ourselves to particles heavier than electrons, since the range of high-energy electrons is determined by the bremsstrahlung and shower production rather than the ionization loss, as will be discussed in Section 1.1.5 [see Eqs. (1.1.81) and (1.1.82)]. 1.1.3.1. Summary of Range-Energy Relations. Range energy relations have been obtained by several authors. I n 1937, Livingston and Bethe36 published range-energy relations for protons, deuterons, and a particles in air. In obtaining these results, various experimental data on the ranges of natural a particles were used. The ranges of a particles and protons are related by (1.1.43) R,(T,) = 1.0072 R,(3.971 T,) - 0.20 cm where R,(T,) is the proton range for an energy T p ( E*T,), and R,(T,) is the a-particle range for an energy T,; 1.0072 = za2(mp/ma);3.971 = (m,/m,) [Eq. (1.1.44)], and the constant term131-0.20 cm is due to the capture and loss of electrons at low energies which has a somewhat different effect on a particles and protons. The proton range-energy relation of Livingston and Bethe extends up to 15 MeV. In 1947, Smith1S2obtained range-energy relations for protons up to 10 Bev, both for air and aluminum. For air, Smith used the same value of I as Livingston and Bethe:36 Iair = 80.5 ev; for Al, Wilson’s value3’ l a l = 150 ev was used. Somewhat later, Aron el aL2 calculated proton range-energy relations for a number of metals and gases, up to 10 Bev, using a value of I = 11.52 ev, which was essentially derived from Wilson’s result for Al. The calculations of Aron et al. as well as those of Smith neglect the density effect, which becomes important for proton energies T, above -2 Bev. The tables of Aron et al. have been extended by Rich and made^.'^^ A summary of the range measurements at various energies has been given by Bethe and A s h k h 3 P. M. S. Blackett and L. Lees, Pmc. Roy. SOC.A134, 658 (1932). J. H. Smith, Phys. Rev. 71,32 (1947). l a 3M. Rich and R. Madey, University of California Radiation Laboratory Report UCRL-2301 (1954). l31

182

1.1.

INTERACTION OF RADIATION WITH MATTER

45

There have been several determinations of the range-energy relation for nuclear emulsion. In 1953, V i g n e r ~ n 'obtained ~~ a range-energy relation, based on older data, particularly those of R ~ t b l a t , and ' ~ ~ Cuer and Jung. l3-5 Vigneron's results were later extended by Barkas and Young.137 Calculations of the range-energy relation for high energies, including the density effect correction in dE/dx, have been carried out by Baroni et ~ 1 . Friedlander, l ~ ~ Keefe, and M e n ~ n , ' ~have ~ " made a comparison of the ranges in emulsion and in aluminum for protons of energies 87, 118, and 146 MeV. Recently, Barkas and his c o - w o r k e r ~have ~ ~ ~ made very extensive measurements of the ranges in Ilford G5 emulsion, taking into account the effect of the water content of the emulsion on the rangeenergy relation. The water content determines the density of the emulsion. Barkas140 has calculated a new and very accurate range-energy relation for Ilford G5 emulsion for a "standard density" of 3.815 gm/cm3, and has given the correction which must be applied to ranges measured under nonstandard conditions to obtain the corresponding ranges for the standard density (and hence the energy of the particle). In obtaining the values of d E / d x used in calculating the range [Eq. (1.1.42)], Barkas has included both the shell correction U at low energies and the density effect correction 6 a t high energies. The mean excitation potential I was used as an adjustable parameter, to be determined so as to give the best fit of R ( T ) t o the available range measurements. I n this manner, a value I = 331 f 6 ev was obtained, which gives an average I / Z = 12.1 & 0.2 ev for the elements of emulsion (excluding the hydrogen). This value of I/2 is in good agreement with the recent results of Bichsel el aL4' and of Burkig and M a c K e n ~ i e . ~ ~ 1.1.3.2. Calculations of the Range-Energy Relations of Protons for 6 Substances. As mentioned above, the range-energy tables of Aron et aL2 do not take into account the density effect correction 6. Moreover, these tables were calculated for an excitation potential I = 11.52 ev, which is somewhat lower than the most recent value, I 12.5-132 ev, as obtained

-

l a 4 L.

Vigneron, J. phys. radium 14, 145 (1953) J. Rotblat, Nature 167, 550 (1951). 136 P. Cuer and J. J. Jung, Sci. et ind. phof. 22, 401 (1951). 137 W. H. Barkas and D. M. Young, University of California Radiation Laboratory Report UCRL-2579, revised (1954). 188 G. Baroni, C. Castagnoli, G. Cortini, C. Franzinetti, and A. Manfredini, Report BS-9, Istituto di Fisica dell'Universit8, Rome, 1954. la*s M. W. Friedlander, D. Keefe, and M. G . K. Menon, Nuovo cimenfo [lo] 6, 461 (1957). 189 W. H. Barkas, P. H. Barrett, P. Cuer, H. H. Heckman, F. M. Smith, and H. K. Ticho, Phys. Rev. 102, 583 (1956); Nuovo cimento [lo] 8, 185 (1958). $40 W. H. Barkas, Nuovo cimento [lo] 8, 201 (1958). la6

46

1.

PARTICLE DETECTION

from references 47 and 49. Sternheimer' has carried out calculations to determine new range-energy relations for some of the commonly used materials, using the higher I values and including the density effect correction. Range-energy relations have been obtained for 6 substances : Be, C, Al, Cu, Pb, and air, for proton energies T , from 2 Mev to 100 Bev. The reason for choosing 100 Bev as the upper limit of the tables is th a t 10 Bev, upon this enables one to obtain pmeson ranges u p to T, applying a small correction for the fact th at the maximum energy transfer W,,, becomes slightly dependent on the mass ratio m,/m a t the highest energies considered. This correction will be given below. The ranges obtained in this work' are -1 to -9% higher for T , = 10 Bev than t,hose of Aron et aL2 and of Smith.132 The largest differences occur for Be (9.2%) and C (6.4%). Details of the calculations of the ranges are given in reference 1. The values of - ( l / p ) ( d E / d z ) were calculated from Eq. (1.1.1). The mean excitation potentials I were obtained from the work of C a l d ~ e 1 1 , ~ ~ . ~ ~ following I values were Bichsel et U Z . , ~ ~and Burkig and M a ~ K e n z i eThe used: IBe = 64 ev, Ic = 78 ev, Iair= 94 ev, IAl= 166 ev, Icu = 371 ev, and Ipb= 1070 ev. The density effect correction 6 was evaluated from the calculations of Sternheimer.20*21 The K and L shell corrections C K and CL a t low energies were obtained from Walske's Table I1

-

TABLE 11. Values of the Constants Used t o Obtain the Ionization Loss' Z is the mean excitation potential. A and B are the constants appearing in Eq. (1.1.19). C, a, s, X O ,and XI enter into the expression for the density effect correction 6 [Eqs. (1.1.20)and (1.1.2Oa)l. Material Z (ev) A

Be C A1 Cu

Pb Air

64 78 166 371 1070 94

(s) B

0.0681 0.0768 0.0740 0.0701 0.0608 0.0768

18.64 18.25 16.73 15.13 13.01 17.89

-C

2.83 3.18 4.25 4.71 6.73 10.70

a

S

0.413 0.509 0.110 0.118 0.0542 0.126

2.82 2.67 3.34 3.38 3.52 3.72

xo

XI

-0.10 -0.05 0.05 0.20 0.40 1.87

2 2 3 3 4 4

gives the values of the constants which were used in the calculation of the ionization loss [Eq. (1.1.19)] and the density effect correction 6 [Eqs. (1.1.20) and (1.1.2Oa)l. Table I11 gives the values of - (l/p)(dE/dx) for protons. The resulting range-energy relations are presented in Table IV. Recently Sternheimer140ahas derived an expression for the rangeenergy relation R(T,) for protons as a function of the mean excitation 140s

R. M. Sternheimer, Phys. Rev. 118. 1045 (1960).

1.1. I N T E R A C T I O N

O F RADIATION W I T H MATTER

47

TABLE 111. Values of the Ionization Loss -(l/p)(dE/dz) (in Mev/gm crn-l) for Protons in Be, C, Al, Cu, Pb, and Air1

2 3 4 5 6 7 8 9 10 12 14 16 18 20 22.5 25 27.5 30 35 40 45 50 55 60 65 70 75 80 90 100 110 120 130 140 150 160 180 200 225 250 275 300 325 350 375

131.9 97.45 78.06 65.59 56.69 50.15 45.03 40.99 37.63 32.44 28.62 25.65 23.30 21.38 19.41 17.80 16.47 15.34 13.53 12.15 11.05 10.15 9.412 8.788 8.254 7.791 7.385 7.026 6.424 5.933 5.527 5.187 4.896 4.644 4.424 4.232 3.908 3.647 3.384 3.173 3.000 2.853 2.730 2.625 2.534

140.6 104.4 83.97 70.74 61.29 54.28 48.81 44.47 40.87 35.29 31.17 27.96 25.42 23 34 21.21 19.46 18.01 16.79 14.82 13.32 12.12 11.14 10.33 9.645 9.062 8.556 8.112 7.719 7.061 6.526 6.079 5.706 5.388 5.112 4.872 4.659 4.304 4.016 3.728 3.497 3.307 3.148 3.013 2.896 2.797

A1

cu

Pb

Air

110.8 83.16 67.44 57.19 49.84 44.38 40.09 36.67 33.80 29.35 26.04 23.45 21.39 19.70 17.95 16.52 15.32 14.31 12.67 11.41 10.41 9.584 8.902 8.325 7.831 7.402 7.026 6.693 6.132 5.674 5.292 4.973 4.700 4.464 4.258 4.077 3.768 3.522 3.272 3.072 2.908 2.771 2.655 2.555 2.469

78.93 61.83 51.27 44.08 38.73 34.71 31.50 28.94 26.77 23.38 20.83 18.82 17.22 15.91 14.54 13.42 12.48 11.68 10.38 9.383 8.584 7.925 7.378 6.914 6.514 6.167 5.861 5.590 5.133 4.760 4.449 4.187 3.961 3.767 3.594 3.445 3.192 2.989 2.783 2.616 2.480 2.366 2.268 2.185 2.112

41.14 34.62 29.85 26.36 23.65 21.54 19.81 18.40 17.18 15.23 13.73 12.52 11.54 10.73 9.874 9.163 8.564 8.050 7.203 6.548 6.020 5.581 5.213 4,900 4.629 4.391 4.181 3.996 3.682 3.424 3.209 3.027 2.870 2.734 2.616 2.511 2.333 2.189 2.042 1.924 1.828 1.747 1.678 1.619 1.568

134.0 99.86 80.53 68.00 58.99 52.32 47.11 42.96 39.51 34.15 30.20 27.10 24.66 22.66 20.61 18.93 17.53 16.35 14.44 12.98 11.82 10.87 10.09 9.420 8.852 8.360 7.928 7.546 6.904 6.382 5.950 5.587 5.276 5.007 4.773 4.567 4.221 3.942 3.660 3.434 3.248 3.093 3.961 2.848 2.751

48

1.

PARTICLE DETECTION

TABLE 111. Values of the Ionization Loss - (l/p) ( d E / d x ) (in Mev/gm crn-)) for Protons in Be, C, Al, Cu, Pb, and Air1 (Continued)

400 450 500 550 600 700 800 900 1000 1250 1500 1750 2000 2250 2500 2750 3000 3500 4000 4500 5000 6000 7000 8000 9000 10,000 12,500 15,000 17,500 20,000 22,500 25,000 27,500 30,000 40,000 50,000 60,000 70,000 80,000 90,000 100,000

2.453 2.321 2.215 2.129 2.059 1.950 1.871 1.812 1.767 1.692 1.649 1.623 1.608 1.599 1.595 1.593 1.593 1.597 1.604 1.612 1.621 1.638 1.655 1.670 1.685 1.699 1.728 1.753 1.774 1.792 1.808 1.822 1.835 1.847 1.886 1.915 1.939 1.959 1.976 1,991 2.005

2.709 2.563 2.448 2.355 2.278 2.159 2.074 2.009 1.960 1.879 1.833 1.806 1.791 1.782 1.778 1.777 1.778 1.784 1.793 1.802 1.813 1.834 1.854 1.873 1.890 1.905 1.939 1.968 1.993 2.014 2.033 2.050 2.065 2.077 2.122 2.156 2.183 2.206 2.225 2.242 2.257

2.392 2.268 2.169 2.090 2.022 1.921 13 4 9 1.795 1.754 1.687 1.649 1.629 1.618 1.613 1.611 1.613 1.615 1.624 1.635 1.647 1.659 1.682 1.704 1.724 1.743 1.759 1.796 1.827 1.853 1.876 1.895 1.913 1.929 1.944 1.991 2.027 2.056 2.080 2.100 2.118 2.134

2.049 1.945 1.863 1.795 1.741 1.658 1.598 1.555 1.522 1.471 1.443 1.429 1.422 1.420 1.422 1.425 1.429 1.440 1.452 1.465 1.478 1.502 1.524 1.544 1.562 1.579 1.615 1.645 1.671 1.693 1.712 1.729 1.745 1.759 1.804 1.839 1.866 1.890 1.909 1.926 1.941

1.523 1.448 1.390 1.343 1.305 1.246 1.205 1.175 1.153 1.120 1.104 1.099 1.099 1.102 1.108 1.114 1.121 1.135 1.150 1.164 1.178 1.204 1.227 1.248 1.267 1.284 1.321 1.351 1.377 1.399 1.418 1.436 1.451 1.465 1.511 1.546 1.574 1.597 1.616 1.633 1.648

2.666 2.524 2.413 2.323 2.249 2.136 2.055 1.995 1.950 1.877 1.838 1.819 1.809 1.806 1.808 1.812 1.818 1.834 1.851 1.870 1.889 1.924 1.958 1.989 2.017 2.044 2.102 2.151 2.194 2.232 2.265 2.296 2.323 2.348 2.433 2.499 2.552 2.597 2.631 2.661 2.687

1.1.

INTERACTION O F RADIATION W I T H MATTER

49

TULE IV. Range-Energy Relations for Protons in Be, C, Al, Cu, Pb, and Air1 The range R is given in grn cm-z.

TP (MeV) 2 3 4 5 6 7 8 9 10 12 14 16 18 20 22.5 25 27.5 30 35 40 45 50 55 60 65 70 75 80 90 100 110 120 130 140 150 160 180 200 225 250 275 300 325 350 375

Be

C

A1

cu

Pb

Air

0.0091 0.0180 0.0296 0.0436 0.0601 0.0789 0.0999 0.1232 0.1487 0.2061 0.2719 0.3459 0.4278 0.5175 0.6404 0.7750 0.9212 1.079 1.426 1.817 2.249 2.722 3.234 3.784 4.371 4.995 5.655 6.349 7.840 9.461 11.21 13.08 15.06 17.16 19.37 21.68 26.61 31.91 39.03 46.67 54.78 63.33 72.30 81.64 91.34

0.0084 0.0168 0.0275 0,0406 0.0558 0.0732 0.0926 0.1141 0.1376 0.1904 0.2508 0.3187 0.3937 0.4759 0.5884 0.7116 0.8452 0.9891 1.307 1.663 2.057 2.488 2.954 3.456 3.991 4.559 5.160 5.792 7.148 8.623 10.21 11.91 13.72 15.62 17.63 19.73 24.20 29.02 35.49 42.42 49.77 57.53 65.65 74.12 82.91

0.0115 0.0221 0.0355 0.0517 0.0704 0.0917 0.1155 0.1416 0.1700 0.2337 0.3062 0.3872 0.4766 0.5742 0.7073 0.8526 1.010 1.179 1.551 1.967 2.427 2.928 3.469 4.051 4.670 5.327 6.021 6.750 8.313 10.01 11.84 13.79 15.86 18.04 20.34 22.74 27.85 33.34 40.72 48.61 56.98 65.79 75.02 84.62 94.58

0,0190 0.0335 0.0513 0.0724 0.0967 0.1240 0.1542 0.1874 0.2234 0.3035 0.3943 0.4954 0.6066 0.7276 0.8922 1.071 1.265 1.472 1.927 2.434 2.992 3.599 4.253 4.954 5.699 6.488 7.321 8.195 10.06 12.09 14.27 16.58 19.04 21.63 24.35 27.19 33.23 39.71 48.39 57.66 67.49 77.82 88.61 99.85 111.5

0.0410 0.0676 0.0988 0.1345 0.1746 0.2190 0.2674 0.3198 0.3761 0.5000 0.6385 0.7912 0.9576 1.138 1.381 1.644 1.926 2.229 2,885 3.614 4.411 5.275 6.202 7.192 8.243 9.352 10.52 11.74 14.35 17.17 20.19 23.40 26.80 30.37 34.11 38.02 46.29 55.14 66.98 79.61 92.95 107.0 121.6 136.7 152.4

0.0087 0.0175 0.0287 0.0423 0.0581 0.0761 0.0963 0.1185 0.1428 0.1974 0.2598 0.3299 0.4073 0.4920 0.6078 0.7346 0.8720 1.020 1.346 1.712 2.116 2.557 3.035 3.549 4.097 4.678 5.293 5.940 7.327 8.835 10.46 12.20 14.04 15.99 18.03 20.17 24.73 29.64 36.23 43.29 50.79 58.68 66.95 75.56 84.50

50

1.

P AR T I C L E DETECTION

TABLE IV. Range-Energy Relations for Protons in Be, C, Al, Cu, Pb, and Air1 (Continued)

400 450 500 550 600 700 800 900 1000 1250 1500 1750 2000 2250 2500 2750 3000 3500 4000 4500 5000 6000 7000 8000 9000 10,000 12,500 15,000 17,500 20,000 22,500 25 ,000 27 ,500 30,000 40 ,000 50,000 60,000 70,000 80,000 90,000 100,000

101.4 122.3 144.4 167.5 191.3 241.3 293.7 348.1 404.0 548.9 698.8 851.7 1007 1163 1319 1476 1633 1946 2259 2570 2879 3493 4100 4702 5298 5889 7347 8784 10,202 11,604 12,993 14,370 15,737 17,095 22,450 27,711 32 ,899 38,030 43,112 48,152 53 , 158

91.99 111.0 131.0 151.8 173.4 218.6 265.9 314.9 365.3 495.8 630.7 768.2 907.3 1047 1188 1328 1469 1750 2029 2308 2584 3133 3675 4212 4743 5270 6570 7850 9112 10,359 11,595 12,820 14,036 15,243 20,003 24,677 29,286 33,843 38,356 42 ,833 47,278

104.9 126.4 148.9 172.4 196.7 247.6 300.7 355.6 412.0 557.7 707.7 860.4 1014 1169 1324 1479 1634 1943 2250 2555 2857 3456 4046 4629 5206 5777 7183 8563 9922 11,262 12 ,588 13,901 15,202 16 ,494 21,574 26,550 31 ,448 36,284 41,067 45,807 50,509

123.5 148.6 174.9 202.2 230.5 289.5 350.9 414.4 479.4 646.8 818.7 992.9 1168 1344 1520 1696 1871 2220 2566 2908 3248 3919 4580 5232 5876 6512 8077 9610 11,117 12,604 14,072 15,525 16,964 18,391 24,002 29,491 34 ,888 40,214 45,477 50,692 55,863

Pb

Air

168.6 202.3 237.6 274.2 312.0 390.5 472.2 556.3 642.2 862.7 1088 1315 1543 1770 1996 222 1 2445 2888 3326 3758 4185 5024 5847 6655 7450 8234 10,153 12,023 13,856 15,657 17,432 19,184 20,915 22,629 29,344 35,883 42,290 48,596 54,820 60 ,975 67,070

93.73 113.0 133.3 154.4 176.3 222.0 269.8 319.2 370.0 500.9 635.7 772.5 910.3 1049 1187 1325 1463 1737 2008 2277 2543 3067 3583 4089 4589 5081 6287 7462 8612 9742 10,853 11,950 13,032 14,102 18,282 22 ,336 26,295 30,177 34,002 37,781 41,519

1.1.

INTERACTION O F RADIATION W I T H MATTER

51

potential I . The expression for R(T,) is obtained by an interpolation of the previously calculated range-energy relations’ for Be, Al, Cu, and P b (see Table IV). The result for R(T,) is accurate to within 1% for values of I between 60 and 1100 ev. The expression for R(T,) can be used to calculate the range-energy relation for any substance, provided a n appropriate value of I is assumed. 1.1.3.3. Range-Energy Relations for Particles other than Protons. The Correction Factor Fi. The range R for any other particle i (heavier than an electron) with energy Tican be obtained from the proton ranges of Table IV by means of the relation

(tl)

R;(Ti) = - - R, zi2

)

3 Ti F i

(mi

( 1.1.44)

where z, is the charge of the particle, mi is its mass, mp = proton mass, and R,[(mp/m,) Ti] is the proton range for the appropriate energy (m,/m,) Ti. I n Eq. (1.1.44), the factor Fi corrects for the slight dependence of the maximum energy transfer W,,, on mi a t very high energies. Thus Wmax for 1.1, T , and K mesons is slightly smaller than for protons with the same value of yi = Ei/mic2, where Ei is the total energy (including rest mass) of the particle. Hence - (l/p) (dE/dz) is decreased and the range Riis slightly increased for mesons (Fi > 1). From Eqs. (1.1.1) and (1.1.15), one finds that the change of - (l/p)(dE/dz) is given by

Values of F, for p mesons are given in Table V. These values were obtained’ by numerical integration of Eq. (1.1.42) with - (l/p) (dE/dz) calculated from the appropriate W,,, for p mesons. Table V shows that the correction for 1.1 mesons is very small ( F , - 1 5 0.01), and that the values of F , are practically independent of 2, being nearly the same for Be and Pb. For 7r and K mesons, the corrections F , and F K are not tabulated, since one will not generally be interested in the ranges of these particles for y i 2 5 , in view of the large probability that they will interact before coming t o the end of the range. Actually for a given yi, the corrections are even smaller than for 1.1 mesons. Thus for Pb, F , = 1.0095 for Y~ = 100, and F K = 1.0017 for Y K = 100. It should be noted that a t very high energies [E >> (mi2/m)c2], spindependent effects on the energy loss in close collisions will be present,141 which are not included in the Bethe-Bloch formula. 1.1.3.4. Range Straggling. When a beam of particles loses energy by 141 See, for example, B. Rossi, York, 1952.

“

High-Energy Particles,” p. 14. Prentice-Hall, New

1.

52

PARTICLE DETECTION

TABLE V. Values of the Factor F, Which Enters into the Expression for the p-Meson Range R,, at Very High Energies [Eq. (1.1.44)11 F,, is given for Be and Pb, as a function of 7, = E,/m,,ea.

4 6

8 10 15 20 25 30 40 50 60 70 80 90 100

1.0013 1.0017 1.0021 1.0025 1.0032 1.0039 1.0047 1.0054 1.0066 1 -0077 1.0088 1.0098 1.0107 1.0116 1.0125

1.0010 1.0014 1.0017 1,0020 1.0027 1.0034 1,0041 1.0047 1.0058 1.0068 1.0079 1.0089 1.0098 1.0107 1.0115

ionization, all of the particles do not come to the end of their range and stop after traversing the same thickness of material. Instead there is a distribution of the ranges due to the statistical nature of the ionization loss process. This distribution is a Gaussian. The probability p(R) dR of a particle of well-defined initial energy T ohaving a range between R and R 4-dR is given by

p(R)dR

= __ 1 cYT1'2

[

exp - ( R

a~0)2] dR

(1.1.46)

where CY'

E

2(R - RQ);,= 2Jp(R)(R- Ro)'dR.

(1.1.47)

In Eqs. (1.1.46) and (1.1.47), Ro is the mean range142obtained above [Eq. (1.1.42)]by integration over the average energy loss - (l/p)(dE/dx). An approximate equation for ( R - Ro);, has been obtained by Bohr.12 For sufficiently large initial velocities v of the particle (2mv2 2 I K , where IK is the ionization potential of the K shell), Bohr's formula gives

loTo ( d E / d ~ )dT -~

( R - Ro);, = 4re4z2NZ

(1.1.48)

where T Ois the initial energy of the particle, N = number of stopping atoms per ema, and Z = atomic number of stopping material. I n practice, it is difficult to obtain directly the value of ( R - RO);, from the observed distribution of ranges. Instead one obtains the number142 See

also H. W. Lewis, Phys. Rev. 86, 20 (1952); U. Fano, ibid. 92, 328 (1953).

1.1.

INTERACTION O F RADIATION W I T H MATTER

53

FIG.6. Schematic number-distance curve, showing RQ, Rextrand S (cf. Bethe and Ashkin, reference 3, p. 246, Fig. 15b).

distance curve by plotting the number of particles which survive as a function of the thickness traversed. An example of such a curve is shown in Fig. 6. From the Gaussian of Eq. (1.1.46), one finds that the curve of N has half its maximum value, N = +No, at R = Ro,the mean range of the particles which are assumed t o be initially homogeneous in energy. Moreover, a t R = Ro,the theoretical curve of N versus R has its maximum slope, - (1/ad2).By drawing a tangent to the N versus R curve a t its steepest point ( R = Ro)and obtaining the intersection of the tangent with the R axis (see Fig. 6), one finds the extrapolated range, Re,,,, which is given by (1.1.49) Re,,= = RO TT 1 112a.

+

The difference Rextr- ROis defined3 as the straggling parameter S. Thus

S2 = h a 2 = +(R - Ro);,

(1.1.50)

and the distribution function p(R)dR [see Eq. (1.1.46)] can be written as f o l l o ~ s : ~ 1 (1J.51) p(R)dR = - exp ( R - R d 2 ]dR. 2s

[ (&)

The value of ( R - Ro);, obtained from the measured S by means of Eq. (1.1.50) can be compared with the theoretical expression, Eq. (1.1.48). Good agreement has been obtained in several experiments. As an example, the calculated percentage straggling of protons in 100S/Ro decreases slowly from 2.29 for RO= 5 cm, to 2.13 a t 149 H. A. Bethe and J. Ashkin, in “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 1, p. 244. Wiley, New York, 1953.

54

1.

PARTICLE DETECTION

10 cm, 1.78 at 100 cm, 1.57 a t lo3 cm, 1.31 a t lo4cm, and 0.97 a t lo5em. Millburn and S ~ h e c t e r 'have ~ ~ found experimentally that SIRo varies very slowly with 2. Thus the value of SIR0 relative t o copper is 0.90 for Be, 0.95 for Al, 1.02 for Ag, and 1.06 for Pb. The range straggling in emulsion has been thoroughly investigated by Barkas et U Z . , ' ~ ~ who used protons and T+, T - , and p+ mesons, These authors have found that the Bohr formula [Eq. (1.1.48)] gives reasonable agreement with their data. They have also discussed various additional straggling effects which are present in range measurements with nuclear emulsion. Values of the range straggling for six substances (Be, C, Al, Cu, Pb, and air) have been recently calculated by Sternheimer.'45a In these calculations, the following expression for ( R - Ro)i,was used:

Equation (1.1.48a) differs from the Bohr formula [Eq. (1.1.48)] by the inclusion of the following factors in the integrand: (1) the factor (1 - &P2)/ (1 - pz), which is a relativistic correction that was first derived b y Lindhard and S ~ h a r f f (2) ; ~ ~the factor [l (2m/mi)y]-' which is derived in reference 145a, and which becomes important only for very high energies (y >, mi/2m); (3) the correction factor K which takes into account the effects of binding on the atomic electrons a t low velocities of the incident particle [v 5 (IK/2m)1'2].The correction K is similar to the binding effect corrections C K and CL which enter into the Bethe-Bloch formula [Eq. (1.1.34)]. K becomes 1 for sufficiently high energies ( T , 2 100 Mev for Al; 400 Mev for Pb). The expression for K has been obtained by Bethe.3v36 / R ~ The percentage range straggling t = 100[(R - R O ) ~ " ~decreases with increasing energy until a minimum is reached for T,/mLc2 2.5, which is in the same region as the minimum of the ionization loss d E / d x . Beyond the minimum, e increases with energy, as a result of the effect of the factor (1 - p2)-' in the integrand of (1.1.48a). We note th a t c as defined above is related to S by: t = 100(2/~)"~S/Ro.As a n example, for p mesons in Cu, e, decreases from 3.94 a t T , = 10 Mev to a minimum of 2.69 a t 280 MeV, and then increases to 3.07 at T, = 1 Bev, 4.07 at 3 Bev, and 5.74 a t 10 Bev. It is that B is almost independent of 2, showing only a small increase in going from Be to Pb (at a constant

+

-

144 G. P. Millburn and L. Schecter, University of California Radiation Laboratory Report UCRL-2234, revised (1953). l P sW. H. Barkas, F. M. Smith, and W. Birnbaum, Phys. Reu. 98, 605 (1955). 145B R. M.Sternheimer, Phye. Rev. 117, 485 (1960).

1.1.

INTERACTION O F RADIATION W I T H MATTER

55

energy T). As an example, for 300-Mev protons, Q,(Z)/~,(CU) is 0.885 for Be, 0.895 for C, 0.941 for Al, and 1.113 for Pb.

1.1.4. Scattering of Heavy Particles by Atoms For nonrelativistic velocities, the Rutherford formula for the elastic scattering of heavy particles by nuclei of charge Z e is given b y d@Ll(B.) =

2?re422Z2sin e de 16T2s i n 4 ( p )

(1.1.52)

where d% is the differential cross section, e is the scattering angle, ze is the charge of the particle, and T is its nonrelativistic kinetic energy. Equation (1.1.52) can also be written (1.1.53) where TYevis T in units MeV. For the scattering of identical particles of spin the M ~ t t formula l ~ ~ gives for the cross section:

a%(@)=

2T24e4 cos f? sin T2

e de 1 -~ [sin4

L

-

1

e +X e

1 cos sin2 e cos2 e

+ (protons, electrons),

tz

In tan2 e)].

(1.1.54)

The last term in the square bracket (involving ti) arises from the quantummechanical exchange phenomena which are a consequence of the identity of the incident particle and the scatterer. I n the inelastic collisions of heavy charged particles with atoms, it is of interest t o obtain the energy distribution of the secondary electrons (6 rays). The angle of ejection $ in the laboratory system is related as follows t o the energy W of the secondary electron:

W

=

(1.1.55)

2 mu2 cos2 $

where m = electron mass, v = velocity of incident heavy particle. The maximum value of $ is 90’ in which case W = 0. The cross section for ejection of a 6 ray with energy between W and W dW is

+

(1.1.56) 148 N. F. Mott, Proc. Roy. Soc. Al26, 259 (1930); see also N. F. Mott and H. S. W . Massey, “The Theory of Atomic Collisions,” 2nd ed. Oxford Univ. Press, London and New York, 1949.

56

1.

PARTICLE DETECTION

The cross section for finding a 6 ray between $ and $

+ d$ is (1.1.57)

For relativistic energies, BhabhaZ3has shown that the collision cross section for a particle with spin 0 is given by ( 1.1.58)

where W,,, is the maximum possible energy transfer to the atomic electrons [Eq. (1.1.15)]. Of course, for p+ 0, Eq. (1.1.58) reduces to the nonrelativistic expression (1.1.56). It may also be remarked that for energy transfers W << W-,, Eq. (1.1.58) gives the same result for d@ as Eq. (1.1.56). In this important case, there are no relativistic corrections to the Rutherford formula. 1.1.5. Passage of Electrons through Matter

Electrons passing through matter lose energy by ionization and excitation of the atomic electrons of the medium, in the same manner as heavy particles. The expression for the ionization loss of electrons has been discussed in Section 1.1.2.7 [Eq. (1.1.32)]. However, in addition, for highenergy electrons, there is the possibility of radiation in the field of the nucleus (bremmsstrahlung). Above a certain energy, called the critical energy E,, the radiation loss predominates over the ionization loss, whereas for E < E,, the ionization loss is the dominant mechanism of energy loss. 1.1.5.1. Radiation by Electrons (Brernsstrahlung). The energy distribution and total cross section for the bremsstrahlung depend strongly on whether the field of the nucleus is effectively screened by the atomic electrons. Thus if the effective impact parameter b is small compared to the atomic radius a U,Z-'/~(aa = Bohr radius), there is essentially no screening, whereas for b >> a, the screening is virtually complete. The relevant parameter for the screening is E, defined by

-

(1.1.59) where Eois the initial total energy of the electron, E is its final energy, and hv (= Eo - E ) is the energy of the radiated quantum. For 5 >> 1, there is no screening, and the energy distribution of the radiation, as obtained

1.1.

INTERACTION O F RADIATION WITH MATTER

by Bethe and Heitler,14' is given by

@(Eo,v) dv

= 4 Z 2 r 0 2d-v ~

137 v

[ 1 + )(':

-

"1

- --

3 Eo

[In

("") - i] mc2hv

57

(1.1.60)

where T O = e2/mc2is the classical electron radius. For 5 = 0, we have complete screening, and @(EO,v)is given by

Equations (1.1.60) and (1.1.61) give only the probability of radiation in the field of the nucleus. There is also some radiation in the field of the atomic electrons. This contribution *el is of the order of 1/Z of the nuclear contribution. For the case of complete screening, Wheeler and Lamb148 have derived the following expression for the energy distribution : 4zr02 dv @,~(Eo,zJ) d v = __ 137 Y

([

1+

( : J 2

_--

3 Eo (1.1.62)

The bremsstrahlung in the field of the atomic electrons, in the limit of no screening, has been investigated by Borsellin0,'~9 Votruba,160Rohrlich, 161 Nemirovsky,162and Watson.lK3 Figure 7 shows the energy distribution of the photons from the bremsstrahlung as a function of the photon energy hv divided by the kinetic energy To of the incident electron. The curves pertain to various values of To/mc2 and were taken from the work of Bethe and Heitler.I4' These curves pertain to lead and include the effect of screening. It may be noted that the function plotted is the frequency distribution of the energy radiated hv@[in units mc25,see Eq. (1.1.66)]. This distribution approaches a constant value as v + 0, in contrast to the probability distribution 147 H. A. Bethe and W. Heitler, Proc. Roy. SOC. A146, 83 (1934); for a discussion of the screening, see also W. Heitler, "The Quantum Theory of Radiation," 2nd ed., p. 168. Oxford Univ. Press, London and New York, 1944. 14* J. A. Wheeler and W. E. Lamb, Phys. Rev. 66, 858 (1939). 149 A. Borsellino, Nuovo cimento [9] 4, 112 (1947); Rev. univ. nuc. Tucumdn Ser. A . Mat. y $8. ledrica A6, 7 (1947). 160 V . Votruba, Phys. Rev. 73, 1468 (1948). l61F. Rohrlich, Phys. Rev. 96, 657 (1954); see also J. M. Jauch and F. Rohrlich, I' The Theory of Photons and Electrons," p. 249. Addison-Wesley, Cambridge, Massachusetts, 1955; F. Rohrlich and J. Joseph, Phys. Rev. 100, 1241 (1955); J. Joseph and F. Rohrlich, Revs. Modern Phys. 30, 354 (1958). 162 P. Nemirovsky, J. Phys. U.S.S.R. 11, 94 (1947). 16 3 K. M. Watson, Phys. Rev. 72, 1060 (1947).

58

1.

PARTICLE DETECTION

itself, which behaves as 1 / v for small frequencies v. Figure 7 shows th a t for small To, the energy distribution hv4, decreases uniformly as hv is inhv9 decreases less rapidly near creased from 0 t o TO.With increasing TO, Y = 0, with the main decrease occurring close to the maximum value hv = TO.For any finite T O 9 , becomes 0 a t hv = TO.T h e value of hv4, at 20mc2h). v = 0 is independent of To (hva

hv/To

FIG. 7. Energy distribution of bremsstrahlung emitted by a fast electron. These curves were taken from the work of H. A. Bethe and W. Heitler [Proc. Roy. SOC.A146, 83 (1934)l; see also Bethe and Ashkin, reference 3, p. 270, Fig. 19. The curves pertain to lead, and include the effect of screening.

The energy loss of the electrons by radiation is given by

-

=

f$)rad

N

1”

hv4,(EoJv)d v

(1.1.63)

where vo = Eo/h, and N = number of atoms per cm3. Equations (1.1.60) and (1.1.61) show that (P(E0,v)is roughly proportional t o 1 / v for small frequencies v, so that the integrand hvQ(Eo,v)does not have any strong dependence on v, and actually becomes almost independent of Eo and v/To for sufficiently high energies ( T O2 100mc2)(see Fig. 7). As a result, - (dEo/dz),,d is approximately proportional to the primary energy Eo. Thus we write.

where

9.a

rs)

=

NEoQ,,,

rad

is the integral of (1.1.63) divided by Eo.

( 1.1.64)

1.1.

59

INTERACTION O F RADIATION WITH MATTER

For mc2<< Eo << 137rn~~Z-"~, screening can be neglected, and one finds (1.1.65)

where I is defined by @ -=

1)r02 =

137

Z(Z

+ 1)5.79 X lo-@ cm2.

(1.1.66)

one obtains: For the case of complete screening (Eo>> 137rn~~Z-'/~), = I ' [ 4 ln(183Z-1/3)

where

I' E Z(Z

+ $1

( 1.1.67)

+ {)5.79 X 10-z8 cm2

(1.1.68)

and where { is the correction for the contribution of the atomic elect r o n ~ ; {' ~is~of the order of 1.2-1.4. The distance Xo over which the electron has its energy decreased by a factor e is called the radiation length. Thus XOis defined by

l/Xo = 4NI' ln(183Z-1'3).

( 1.1.69)

For large energies, we have [from Eq. (1.1.67)]: (1.1.70)

where b = 1/[18 1n(183Z-1/3)]is very small compared to unity (b = 0.012 for air, 0.015 for Pb). Table VI gives values of the radiation length XO for various materials. TABLE VI. Values of .the Critical Energy E, and Radiation Length XOfor Various Substances This table is taken from Bethe and Ashkin, reference 3, p. 266. Substance

E , (Mev)

X O(gm/cm2)

340 220 103 87 77 47 34.5 24 21.5 6.9

58 85 42.5 38 34.2 23.9 19.4 13.8 12.8 5.8 36.5 35.9

~~

Hydrogen Helium Carbon Nitrogen Oxygen Aluminum argon Iron Copper Lead Air Water

~

83 93

60

1.

PARTICLE DETECTION

Bethe and Heitler14’ have given the following approximate formula for the critical energy: E, S 1600mc2/Z (1.1.71) from which they have obtained the following expression for the ratio of the radiation loss to the collision loss:

-

(dEo/dz)r,fi - EoZ (dEo/dz),,11 - 1 6 0 0 ~’~ ~ ’

(1.1.72)

It should be noted that Eqs. (1.1.71) and (1.1.72) are very approximate, as can be seen by comparing the values of E, calculated from Eq. (1.1.71) with the actual values given in Table VI. With increasing energy of the electron, the radiation becomes increasingly peaked forward. Aside from a factor which depends slowly on E O and hv, the angular distribution of the radiation du/dQ is determined by:154

(1.1.73) where 0 is the angle of emission of the radiation and B is a constant. Thus the average angle of emission is given by

<@>

(1.1.74)

mc2/Eo

which becomes very small with increasing Eo. Recently tfberal1155has investigated the bremsstrahlung produced by fast electrons in single crystals. He has shown that interference phenomena are expected to occur which can enhance the radiation and markedly change the y-ray spectrum. A similar effect for pair production (see Section 1.1.7.3) is also discussed by Uberall. The crystal effect is small at low energies, and sets in for q noh/a, where q is the momentum transfer to the target atoms, a is the lattice constant, and no is of the order of 2 or 3. This condition corresponds to an electron energy To 200 Mev 1 Bev for pair production. The for bremsstrahlung, and y energy hv interference effect is confined to angles 00 of order 0 0 5 (137Z-’I3) X (mca/Eo) between the primary beam and the line of atoms participating in the interference. I n a second paper, tfberal1165has discussed the polarization of bremsstrahlung emitted from a monocrystalline target. The polarization P is all), where ul and ~ 1 are 1 the cross secdefined as: P = (aL - q ) / ( a l tionsjor producing radiation polarized perpendicular and parallel, respec-

-

-

+

lS4 A. Sommerfeld, “Atombau und Spektrallinien,” Vol. 2, p. 551. Vieweg, Braunschweig, Germany, 1939. lKK H. Vberall, Phys. Rev. 103, 1055 (1956); 107,223 (1957).

1.1.

INTERACTION O F RADIATION WITH MATTER

61

tively, to the production plane (formed by the incoming electron momentum PO and the emitted y-ray momentum k). He has shown that, in typical cases, P is increased by a factor of -1.5 above the value obtained when an amorphous target is used. Moreover, there is a net polarization with respect to the plane formed by the incident direction (PO) and the crystal axis. Thus for a Cu crystal, at T = O", with an incident electron energy EO= 600 MeV, for 0 0 = 20 X rad, the polarization POof the entire bremsstrahlung cone is E0.15 for z 3 kv/Eofrom 0 to 0.2. Between 2 = 0.2 and 0.5, PO decreases slowly to 0. Here eo is the angle between the primary direction and the crystal axis. For Oo = 5 X 10-8 rad, Po is 0.31 a t x = 0, and decreases rapidly with increasing x, becoming negative at z = 0.19, with minimum value PO= -0.11 at x = 0.33. Thus by using an appropriate angle 00, it may be possible to obtain partially polarized y radiation of sufficient intensity to perform high-energy polarization experiments. 1.I .5.2. Shower Production. As the eIectron proceeds through the material, it will create a shower, which is produced as follows. The electron loses energy by bremsstrahlung, producing a high-energy y ray. The y ray in turn can produce an electron and positron by pair production (see Section 1.1.7.3below). The pair in turn can radiate energy by bremsstrahlung, thereby producing photons, which can then create more pairs. In this way a cascade of photons and high-energy e+ and e- is produced, which is called a shower. The number of e+ and e- present increases at first with increasing thickness t, then attains a maximum a t a certain thickness t,, and decreases for larger t. I n this connection, as was mentioned above, it is convenient to define a critical energy E, by the condition that for EO = E,, the radiation loss, Eq. (1.1.701, is equal to the energy loss by ionization. Values of E , for various materials are given in Table VI. Figure 8 shows the expected number of electrons n as a function of the thickness t in radiation lengths Xo. I n the figure, loglo n is plotted against t for 4 different values of the total energy E Oof the primary electron, which is given in units of the critical energy E,. These curves were taken from the work of Rossi and Greisen.lS8Figure 8 shows that for a shower withiiincident energy E O = 100E,, n increases from n = 1 at t = 0 to a maximum n, G 10 a t t = t, 4, and thereafter decreases to n = 1 at t g 12, and is negligible for t >, 12. The thickness t, at which the maximum is reached, and the value n, at the maximum both increase with increasing energy EOof the primary electron. Thus for E O = 104E,, we have n, 1000, t, E 9. I n this case, n becomes negligible only for t 2 30. Many authors have treated analytically the complicated mathe166

B. Rossi and K. Greisen, Revs. Modern Phys. 13, 240 (1941).

1.

62

PARTICLE DETECTION

matical problems involved in shower production. A review of these calculations is given in the book by Ro ~ si.'~ ' Wilson168has recently treated the problem of the shower development by a Monte Carlo method, in which a large number of electrons are followed through the material, with a statistical (probability) determination of the bremsstrahlung and pair production processes in each particular case history." Neglecting scattering, by means of an approximate ((

5

4

3 C

2 2

s

I

0

-I

0

t

FIG.8. The number n of electrons in a shower as a function of the thickness traversed of B. Rossi and K . Greisen [Revs. Modern Phys. 13, 240 (1941)l. t in radiation lengths. These curves were taken from the work

theoretical model of shower production, Wilson finds for the mean range r (in units Xo) of an electron of initial energy Eo r

=

ln 2 In

(-E ,EoIn 2 + 1).

(1.1.75)

The distribution of ranges around r is approximately Gaussian, and the root mean square straggling s (in units X , ) is given by (1.1.76) Wilson has shown from his Monte Carlo calculations that for a n incident electron, one is more likely to find 1, 3, 5 . . . than 2, 4,6, . . . electrons and positrons a t a given thickness in the shower, since electrons lS7

B. Rossi, " High-Energy Particles," Chapter 5. Prentice-Hall, New York, 1952. R. R. Wilson, Phys. Rev. 84, 100 (1951); 86, 261 (1952).

1.1.

INTERACTION O F RADIATION W I T H MATTER

G3

and positrons are formed in pairs. On the other hand, if the shower is initiated by a y ray, one is more likely to find 2 , 4 , 6 . . , than 1 , 3 , 5 . . . electrons and positrons a t any thickness in the shower, Wilson’s shower curves as obtained b y the Monte Carlo method are more spread out than those of the general (analytic) shower theory, i.e., the shower penetrates to a greater thickness t than according to the general t,heory. One of the reasons for this difference is that Wilson’s calculations take into account the fact that the low-energy y rays have a relatively long mean free path (see Section 1.1.7.4). 1.1.5.3. Production of Secondary Electrons by Electrons. Scattering of Electrons by Electrons and Nuclei. The cross section for ejection of secondary electrons (6 rays) by an electron passing through matter is given by146

d@e(T.W) ., I

=

( + ( T -1W)’ - W ( T 1- W )cos [

re4 - dW wi1

T

ln

(‘+)I} (1.1.77)

where T is the kinetic energy of the primary electron, and W is the energy transfer, i.e., the kinetic energy of the secondary electron. In Eq. (1.1.77), the second and third (cosine) terms in the curly bracket are exchange terms. For small W , these terms become negligible, and the resulting cross section [re4/(TWZ)]dW is the same as that for primary heavy particles (e.g., protons) [see Eq. (1.1.56)]. For relativistic energies of the incident electron, the electron-electron scattering cross section has been obtained by Mdler,L69and is given by

1

+

( T - W ) z+ ( T

+ mcz)2

(1.1.78)

For T < mc2, Eq. (1.1.78) reduces to the nonrelativistic Mott formula, Eq. (1.1.77), in which the cosine factor in the last term is ~ 1unless , W7 is very close to either 0 or T. I n his calculations,169MGller summed over both directions of polarization of the two electrons, so that Eq. (1.1.78) represents the average cross section for unpolarized incident electrons. However, recently in connection with the experiments on parity n o n co n ~e r v a tio n ~in-~beta decay, it has become of interest to evaluate the cross sections for polarized electrons scattered by polarized electrons, in particular for the case that the two spin directions are parallel or antiparallel to each other, and along 159 C. Mdler, Ann. Physik [5] 14, 531 (1932).

64

1.

PARTICLE DETECTION

the relative direction of motion of the electrons. This problem arises because it has been shown from the two-component theory of the neutrino,*ll that the electrons and positrons from beta decay are expected to be longitudinally polarized10.16't-162 (i.e., with average spin direction along the direction of motion). The value of the polarization P is predicted to be: P = T v/c, where v = velocity of p particle, and the minus sign applies to electrons, while the plus sign applies to positrons. Thus high-energy electrons from p decay (v c) are almost 100% polarized, with the spin pointing opposite to the direction of motion. A method of determining the longitudinal polarization consists in scattering the &decay electrons on the electrons in a ferromagnetic sample of material. As is well known, if a strong magnetic field is applied to an iron sample, the 3d electrons of iron will be polarized in the direction of the applied field.* Thus if a magnetic field is applied along a direction parallel or antiparallel t o the direction of the incident p electrons, one can obtain the polarization P of the incident beam from a comparison of the c o u n h g rates of scattered electrons a t a particular angle, for the two field directions. This result arises from the fact that the cross sections for parallel spin directions (bp and for antiparallel spin directions & are appreciably different from each other, for all values of the incident energy, provided that the energy transfer W is sufficiently large (WIT 2 0.2, where T is the kinetic energy of the incident electron). The first calculation of the spin-dependent cross sections r#Jp and (ba was carried out by Bincer,163and we shall here briefly summarize his results. For the differential cross sections in the center-of-mass system of the two electrons (to be abbreviated as c.m. system), one obtains

-

d(bp =

eddn [2 C O S ~t7 2~284sin4 t7

dr#Ja=

e4dQ [I -2 ~ ~ sin4 8 4 t7

--

+p(3

G + C O S ~t7)

COS~

+

B4(1

+

s)]

COS~

(1.1.79)

+

G + B2(2 + 3 C O S ~t7 - C O S ~8)

COS~

+ 84(5 - 4

G + C O S ~t7)]

COS~

(1.1.80)

where 8 is the c.m. scattering angle, B is the c.m. velocity of either electron (in units of c), is the total c.m. energy of either electron, and d o is the element of solid angle in the c.m. system. We have S2 = (y - l ) / ( y l),

+

* See also Vol. 4, A, Chapter

3.5. J. D.Jackson, S. B. Treiman, and H. W. Wyld, Phys. Rev. 106, 517 (1957). Iel L. Wolfenstein and L. A. Page, BUZZ. Am. Phys. Sac. [Z] 2, 190 (1957). 162 R. B. Curtis and R. R. Lewis, Phys. Rev. 107, 543 (1957). A. M. Bincer, Phys. Rev. 107,1434 (1957);see also G.W. Ford and C. J. Mullin, Phys. Rev. 108, 477 (1957). l60

1.1.

65

INTERACTION O F RADIATION WITH MATTER

where y = E/mc2,with E = total laboratory energy of incident electron. Upon defining x = cos' 8, Eqs. (1.1.79) and (1.1.80) can be rewritten as follows : d+P

(1.1.81) @a

( 1.1.82)

We note that cos

e = 1 - 2w

(1.1.83)

where w is the fractional energy transfer, w = W/T. Thus after integrating over the azimuthal angle, one obtains

J dn

=

27r sin ede

=

-2nd(cos #)

=

47r dw.

(1.1.84)

I n view of Eq. (1.1.83), we have: x = (1 - 2 ~ ) Upon ~ . substituting these results in Eqs. (1.1.81) and (1.1.82),one finds for the average differential cross section per unit w (for unpolariaed electrons) :

x [ y y i - 2~

+ 3 ~ -2 2w3 +

w4)

- (27

- i ) ( w - 2w3

+

W4)l.

(1.1.85)

Finally, upon using the relations: T = (y - 1)mc2, and W = wT, one can easilyshow that Eq. (1.1.85) is equivalent to Eq. (1.1.78) ford@(T,W). From Eqs. (1.1.81) and (1,1.82), one obtains

+ +

&, -- ~ ' ( 1 &

+

62 x2) - 2 y ( l - Z) 1 - 5' 87' - 2y(4 - 52 2') 4 - 62 22"

+ +

+

( 1.1.86)

Figure 9 shows the curves of as a function of w, for y = 1,3, and co , as obtained by B i n ~ e r . 'It~ ~ is seen that decreases rapidly with increasing w, independently of y. The minimum value is attained for w = 0.5 (0 = go"), and is given by (Y - 1)' = 4(2y2 - 2y

(1.1.87)

+ 1)

which becomes 0 for y 4 1 (nonrelativistic energies) and

+ for y

4

co

.

66

1.

PARTICLE DETECTION

The dependence of the electron-electron scattering cross section on the relative directions of polarization has been used in a few experiments t o determine the longitudinal polarization of electrons from B decay. 164,18i, The arrangement of the experiment of Frauenfelder et ~ 1 . lis~ shown ~ I .3

.8

.6

0, +o

.4

.2

0 0

.I

.2

.3

A

.5

W

FIG.9. The ratio &/&, for electron-electron scattering, as a function of the relative kinetic energy t,ransfer w.This figure is taken from the work of A. M. Bincer [Phys. Rev. 107, 1434 (1957), Fig. 11, and is reprinted with the permission of the author and the Editor of the Physical Review.

schematically in Fig. 10. The scattered electrons are recorded in coincidence by the counters 61 and Cz.The counting rates are compared for opposite directions of the magnetizing current around the Deltamax scattering foil. The scattering angle 0 is usually so chosen that it corre164 H. Frauenfelder, A. 0. Hanson, N. Levine, A. Rossi, and G. De Pasquali, Phys. Rev. 107, 643 (1957). lo5 N. Benczer-Koller, A. Schwarzschild, J. B. Vise, and C. S. Wu, Phys. Rev. 109,

85 (1958).

1.1.

67

INTERACTION O F RADIATION W I T H MATTER

sponds approximately to the maximum relative energy transfer w = 0.5, for which has its smallest value, as discussed above. For w = 0.5, the two final electrons have equal energies in the laboratory system and are emitted symmetrically with respect to the incident direction a t a n angle 8, given by sin2 8, = 2/(7 3). (1.1.88)

+

A description of the experiments on the M3ller scattering of P-decay electrons, as well as a more complete discussion of the theory, can be found in reference 8.

COLLIMATOR

& SOURCE

FIG.10. Schematic view of the experimental arrangement of Frauenfelder et a1.ln4 used to demonstrate the longitudinal polarization of electrons from the fl decay of P32 and Pr144,by means of the Mflller (electron-electron) scattering.

For the scattering of relativistic electrons b y nuclei, McKinley and FeshbachlGB have obtained the following expression :

I

+ ZsP - s i n ( p ) [ l - sin(+0)] . 137

(1.139)

This expression applies provided th at 2/137 is not too large, Le., not for the heaviest nuclei. For P + 1, Eq. (1.1.89) may be rewritten as follows:167 Ze2 cos2(&8) sin4((B8) -{I+--

-)

d @ = ( 2E

] (2ssin

a 2 sin(;O)[l - sin(@)] 137

cos2 (Be)

0 dB)

(1.1.90)

where E is the total laboratory energy of the electron. We note th a t both in (1.1.89) and (1.1.90), 0 is the angle of scattering in the center-of-mass system. W. A. McKinley and H. Feshbach, Phys. Rev. 74, 1759 (1948). R. Hofstadter, Revs. Modern Phys. 28, 214 (1956).

lB8

68

1.

PARTICLE DETECTION

In connection with the longitudinal polarization of the electrons from beta decay,loJao-lezthe Mott scatteringleaof the electrons from a heavy nucleus has also been used to detect the po1arization.t I n this type of experiment, the longitudinal polarization of the electrons is first transformed into a transverse polarization, for instance, by deflecting the electrons through -90" by means of an electrostatic field. A system of crossed electric and magnetic fields may also be used, t o take advantage of the fact that the focusing condition for the electrons can then be made identical with the condition for turning the spin through 90". After the particles have thus acquired a substantial amount of transverse polarization (spin d perpendicular to momentum p), they are scattered through an angle 0 in the plane perpendicular to the (d,p) plane, which we assume to be horizontal, and the up-down asymmetry of the scattered intensity is observed. That is, the intensity of the electrons scattered through an angle 8 in the upward direction is different from the intensity of the electrons scattered through the same angle 8 in the downward direction. As was first shown by Mottlesin 1929, the asymmetry in the scattering of transversely polarized electrons is largest for heavy elements and for large scattering angles (6 90"-150"). For a beam with transverse polarization P , the ratio R of the scattered intensities in both azimuthal directions perpendicular to the (d,p) plane (i.e., upward and downward in the example discussed above) is given by

-

(1.1.91)

where s(0)is a function, first calculated by Mott,le8which depends on the atomic number of the scatterer, the incident electron energy, and the angle of scattering 8. The most complete recent calculation of S(0) has been carried out by Sherman,169who has tabulated s(e)at intervals of 15" for various values of the electron velocity p, for three elements: mercury (2 = 80) ; cadmium (2 = 48);and aluminum (2 = 13). s(0) is given by

- p*)1'2 s(e) = 2PE(1 [F(B)G*(B)+ F*(e)G(0)] sin e(da/dQ)

(1.1.92)

where 5 = 2/(137p), X is the de Broglie wavelength, P(0) and G ( 0 ) are the

t See also Vol. 4, A, Section 3.5.1. 168 N. F. Mott, PTOC. Roy. Soc. Al24,425 (1929);A136,429 (1932);see also the discussion in N. F. Mott and H. S. W. Massey, "The Theory of Atomic Collisions," 2nd ed., pp. 74-85. Oxford Univ. Press, London and New York, 1949. lEg N. Sherman, Phys. Rev. 103, 1601 (1956).

1.1.

69

I N T E R A C T I O N O F RADIATION WITH MATTER

regular and irregular solutions, respectively, of the Schroedinger equation for the electron in the field of the nucleus. These functions are in general complex, and the asterisk denotes the complex conjugate. In Eq. (1.1.92) du/dQ denotes the differential cross section for an unpolarized beam of electrons, which is given by

+ ]GI2 sec2(+0)].

du/dQ = X2[t2(1- pz) ]PIz csc2(+0)

(1.1.93)

has also tabuIn addition to the asymmetry function i3(0), Sherman*eB lated the values of the real and imaginary parts of F and G, as well as the

FIQ.11. The asymmetry factor -S(O) for mercury (2 = 80) as a function of the scattering angle e. The values of S(O) were obtained from the results of Sherman.leg The curves for electron velocities j3 = 0.2, 0.4, 0.6, 0.8, and 0.9 correspond to electron energies T, = 10.5, 46.6, 128, 341,and 661 kev, respectively.

differential cross section du/dQ. As mentioned above, Ii3(0)l is largest for heavy elements and large values of 0. Figure 11 shows the curves of s(0) versus 0 for 2 = 80 and for various velocities p. x(e)is zero for 0 = 0" and 180" for all energies, and a t B = 1for all angles 0. Several experiments have 0.6 (T, 130 kev), been carried out at 0 = 90" using electrons with p for which lS(90")1 has its maximum value. It may be noted that for p = 0.6, IS(0)l increases from 0.271 a t 90" to 0.424 a t 120" and 0.418 at 135'. Nevertheless, it has been found desirable t o work a t -90" because of the rapid decrease of the cross section du/dQ with increasing angle.

-

-

70

1.

PARTICLE DETECTION

FIQ.12. The ratio 9 = (du/dn)/(duz/dn)for mercury (2 = 80) as a function of the scattering angle 0. The solid curves of 7 were obtained from the results of Sherrnan.lBg The dashed curve of 9 for fi = 0.6 was calculated from the formula of McKinley and FeshbachlG6[Eq.(1.1.89)].

Figure 12 shows the values of the ratio q for mercury, as obtained by Sherman,169 where q is defined by (1.1.94) where duR/dQ is the Rutherford cross section, which is given by the factor outside the curly bracket of Eq. (1.1.89): (1.1.95) The dashed curve in Fig. 12 shows the values of q predicted by the formula of McKinley and FeshbachIG6for = 0.6, i.e., the curly bracket of Eq. (1.1.89). It is seen th at the actual values of q differ appreciably from the McKinley-Feshbach result, as would be expected in view of the large 2 value (2/137 = 0.58). Among the earlier determinations of S(O), we may mention the calculations of Mott, 1 6 * Bartlett and Watson,17o Bartlett and Welton,'" and Mohr and T a ~ s i e . " ~ J. H. Bartlett and R. E. Watson, Proc. Am. Acad. A d s Sci. 74, 53 (1940). J. H. Bartlett and T. A. Welton, Phys. Rev. 69, 281 (1941). l72 C. B. 0. Mohr and L. J. Tassie, Proc. Phys. Soc. (London) A67, 711 (1954); C. B. 0.Mohr, Proc. Roy. Soc. A182, 189 (1943). 170

1.1.

INTERACTION O F RADIATION WITH MATTER

71

It should be pointed out that the function S(0) was originally introduced by Mott"j8in connection with the double scattering of a n initially unpolarized beam of electrons. I n this case, the polarization P after a single scattering through an angle O1 is given by S(B1),and the direction of the spin d after the scattering is perpendicular to the plane of the scattering. After a second scattering, the relative intensity of the beam as a function of the angle cp between the first and second planes of scattering is given by I(el,ez,cp) = 1

+ s(e1)s(e2)co~ (o

(1.1.96)

where 0 2 is the angle of the second scattering. Among the more recent double scattering experiments which have attempted to verify the theoretical values of S(0),we may cite the works of Shull et u Z . , ' ~ ~ Ryu et u Z . , ' ~ ~ and Louisell et A review of these investigations has been given by Tolhoek. 176 The Mott scattering has been used in several e x p e r i m e n t ~ ' ~ ~on - ~the 8~ longitudinal polarization of p-decay electrons, and has shown th a t the polarization agrees with the predicted value, P = v/c, within the experimental errors.8 The same conclusion was obtained from the experiments using the MIdller scattering.'64s165 1.1 -5.4. Range of p Rays in Matter. In some cases, a crude value of the energy of a beam of homogeneous p rays is obtained by measuring the socalled practical range R, in some material, such as aluminum. The practical range is obtained by extrapolating the straight-line (maximum slope) part of the graph of transmission versus thickness traversed, and taking into account the background (cf. Fig. 6). For p rays in aluminum, Katz and P e n f ~ l dhave l ~ ~ given the following expressions for R, as a funcC. G.Shull, C. T. Chase, and F. E. Myers, Phys. Re*. 63, 29 (1943). N. Ryu, K. Hashimoto, and I. Nonaka, J . Phys. Soc. Japan 8, 575 (1953). l76 W. H. Louisell, R. W. Pidd, and H. R. Crane, Phys. Rev. 94, 7 (1954). 178 H. A. Tolhoek, Revs. Modern Phys. 28, 277 (1956). 177 H. Frauenfelder, R.Bobone, E. von Goeler, N. Levine, H. R. Lewis, R. N. Peacock, A. Rossi, and G. De Pasquali, Pkys. Rev. 106, 386 (1957). 178 H. De Waard and 0. J. Poppema, Physica 23, 597 (1957). 179 P. E. Cavanagh, J. F. Turner, C. F. Coleman, G. A. Gard, and B. W. Ridley, Phil. Mag. [8]2, 1105 (1957). l80 A. I. Alikhanov, G. P. Eliseiev, V. A. Lubimov, and B. V. Ershler, Nuclear Phys. 6, 588 (1958). 181 A. de-Shalit, S. Kuperman, H. J. Lipkin, and T. Rothem, Phys. Rev. 107, 1459 (1957). 182 H. J. Lipkin, S. Kuperman, T. Rothem, and A. de-Shalit, Phys. Rev. 109, 223 (1958). 183 L. Katz and A. S. Penfold, Revs. Modern Phys. 24, 28 (1952). l73

17*

72

1.

PARTICLE DETECTION

tion of the incident electron energy TO(in MeV) :

R,

=

412Tonmg/cm2

n = 1.265 - 0.0954 In TO

(1.1.97)

for 0.01 d To =< 2.5 MeV, and

R,

=

530To - 106 mg/cm2

(1.1.98)

for 2.5 5 T O6 20 MeV.

FIG.13. The practical range R P of low-energy electrons in aluminum as a function of the electron kinetic energy To.This curve was calculated from the range-energy relation given by L. Katz and A. S. Penfold [Revs. Modern Phys. 24, 28 (1952)]. See also Eqs. (1.1.97) and (1.1.98) of text.

Figure 13 shows a plot of Eqs. (1.1.97) and (1.1.98). This curve is in good agreement with the experimental data on the maximum range of electrons from natural beta emitters. Among the earlier works on the range of low-energy /3 rays, we may mention those of Marshall and Ward, l S 4 FeatherJ1S5Flammersfeld, l 8 6 Bleuler and Zunti, l*7 Glendenin, and Hereford and Swann.lE9 lS4

J. Marshall and A. G. Ward, Cun. J. Research A16, 39 (1937).

N. Feather, Proc. Cambridge Phil. SOC.34, 599 (1938). A. Flammersfeld, Nuturwissenschaften 33, 280 (1946). lg7 E. Bleuler and W. Ziinti, Helv. Phys. Actu 19, 137, 375 (1946); 20, 195 (1947). la*L. E. Glendenin, Nucleonics 2, 12 (1948). lag F. L. Hereford and C. P. Swann, Phys. Rev. 78, 727 (1950). lS6

I**

1.1.

INTERACTION OF RADIATION W I T H MATTER

73

1.1.6. Multiple Scattering of Charged Particles When a charged particle penetrates a thick absorber, it undergoes a large number of small-angle Coulomb scatterings. This process, which is called multiple scattering, was first treated quantitatively by Williams.l90 In addition, the particle may undergo a small number of relatively largeangle scatterings, for which the probability can be directly obtained from the Rutherford scattering formula. We shall here be concerned with the multiple scattering only. The resultant distribution of the space angle 6 between the incoming and outgoing directions of the particle is given by (1.1.99)

where

<e2> is the mean square value of

(2)

<e2>

= 2OI2In - =

8 and is given by1g1

el2 In [ 4 ~ Z ~ / % ~($)2]iVt

(1.1.100)

Here Omin is the minimum angle of scattering in a single encounter, Omin Ei X/a, where X is the de Broglie wavelength of the particle and a is the radius of the atom, a a& 1‘3. In Eq. ( l . l . l O O ) , t is the thickness of material traversed, z is the charge of the particle, m is the electron mass (regardless of the type of scattered particle: proton, meson, etc.), N is the number of atoms of absorber per cm3, and el is that angle for which there is, on the average, only one collision with 8 > 01 throughout the absorber. el2 is given by e12 = 4 T ~ ~1)9e4t/(p42. ( ~ (1.1.101)

-

+

Equation (1.1.100) can be written as follows: <e2>

=

0*1572(2 4- 1)z2t ln[1.13 X 104Z4/3z2A-1tp-2](1.1.102) A (PO)

where pv is in MeV, t is in gm cm-2, and A is the atomic weight in gramsThe expression preceding the logarithm in (1.1.102) is el2. Rossi and GreisenlS8have given a somewhat different formula for < e2> which has been frequently used in experimental applications. This expression is given by (1.1.103) E. J. Williams, Proc. Roy. SOC.A169, 531 (1939);Phys. Rev. 68,292 (1940). H. A. Bethe and J. Ashkin, in “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. I, p. 285. Wiley, New York, 1953. 1**

191

74

1.

PARTICLE DETECTION

where Xo is the radiation length in the material [cf. Eq. (l.l.G9) and Table VI], and E, is a constant energy given by

E,

=

( 4 X~ 137)1%c2

=

21.2 MeV.

( 1.1.104)

As was shown by Bethe and Ashkin,lgl Eq. (1.1.103) applies only for relatively large thicknesses t > to, where t o is given by

to

=

6.7(137/2)2A113gm cm-2.

(1.1.105)

For Pb, to = 110 gm cm-2, while for C, to = 8000 gm cm-2. For small thicknesses, Eq. (1.1.103) overestimates the mean square multiple scattering angle. Thus for 3-Bev protons and samples of thickness t = 10 gm cm-2 of C, Cu, and Pb, < 0 2 > u2 = 0.123", 0.248', 0.388' from Eq. (1.1.102) for C, Cu, and Pb, respectively, whereas the corresponding values from Eq. (1.1.103) are: <82>1'2 = 0.159", 0.289", and 0.429', respectively. The factor F by which Eq. (1.1.103) differs from (1.1.102) is: F = 1.29, 1.17, and 1.11 for C, Cu, and Pb, respectively (for t = 10 gm cm-2). It is often useful to consider the projected angles 8, and 01/, i.e., the projections of the angle 8 on the zy plane perpendicular to the direction of motion of the particle. The distribution of the 8, values is a Gaussian: (1.1.106) where

< eZ2> is the mean square value of <e,2>

=

;<e2>

8, and is given by

(1.1.107)

with < 8 2 > given by Eq. (1.1.102). Thus the denominator of the exponent is the same ( = <e2>) for both P(8) and P,(Q [Eqs. (1.1.99) and do, for the projected angle 8, (1.1.106)]. Of course, the distribution P1/(02/) has the same form as P,(8,) do,. The distribution of the lateral displacement r of the particles has been determined by Ferrni,Ig2and is given by (1.1.108) where t' = t/X, is the thickness traversed in units of radiation lengths X O . More elaborate theories of the multiple scattering have been developed Snyder and Scott,Ig6 by Goudsinit and S a ~ n d e r s o n , ~ ~ ~ E. Fermi, quoted by B. Rossi and K. Greisen, Revs. Modern Phys. 13,265 (1941). S. Goudsmit and J. L. Saunderson, Phys. Rev. 67, 24 (1940). ls4 G. MoliBre, 2. Naturforsch. Sa, 78 (1948). 195 W. Paul and H. Steinwedel, in "Beta- and Gamma-Ray Spectroscopy" (K. Siegbahn, ed.), p. 1. Interscience, New York, 1955. 196 H. S. Snyder and W. T. Scott, Phys. Rev. 76, 220 (1949). 192

193

1.1.

INTERACTION OF RADIATION WITH MATTER

75

Lewis,lg7and Bethe. l g 8 These theories treat more accurately the transition from the small-angle region of multiple scattering to the large-angle region where single scattering predominates. This transition region is sometimes called the region of plural scattering. I n a recent investigation, Nigam and c o - w ~ r k e r s have ' ~ ~ made a critical study of the MoliBre theorylS4of multiple scattering, and have obtained a consistent treatment of the scattering of a charged particle by the field of an atom, up to the second Born approximation. I n this work, Nigam et al. have used the expression of Dalitz2""for the scattering cross section of a relativistic particle of spin in a screened atomic field, for which the potential is: V = -(Ze2/r)exp(--r), where K is a n arbitrary constant, and r is the distance from the nucleus. It was found that the deviation of the complete expression for the "screening angle " 8, from the value given by the first Born approximation is considerably smaller than was obtained by MoliBre. Moreover, the expression for the distribution function P(e) contains additional terms of order xZ/137, which were not obtained by Moliitre. Nigam et al. have carried out calculations of the distribution function P(0) for the case of 15.6-Mev electrons scattered by Au and Be, in order t o The compare their theory with the experimental results of Hanson el aLZo1 theoretical results are in good agreement with the data. The two cases considered correspond to electrons of average energy 15.7 Mev scattered by a gold foil of thickness t = 37.2 mg/cm2, and 15.2-Mev electrons scattered by a beryllium sample of thickness t = 491.3 mg/cm2. The experimental distributionsof Hanson etal. havea llewidth O,(exp) = 3.78" for Au and 4.33" for Be. Here 0, denotes the angle (measured from the direction of the incident beam) a t which P(0) has fallen off t o l/e of its value a t e = 0". e, is thus given by < 02> l i 2 for a Gaussian distribution [Eq. (1.1.100)]. It may be noted, however, th a t the actual multiple scattering d i s t r i b ~ t i o n ~ ~deviates ~ - ' ~ ~ somewhat from a Gaussian a t all angles. I n particular, at large angles (0 >, 28,), the actual P(0) lies above the Gaussian of Eq. (l.l.lOO), and slowly approaches the single-scattering cross section (which decreases only as m e - * ) . For comparison with the values of e,(exp), the theory of Nigam et a l l g 9gives 0, = 3.80" for Au, and 4.35" for Be, in very good agreement with the data. On the other hand, MoliBre's theorylg4gives e, = 3.83" for Au, and 4.56" for Be. The result

+

H. W. Lewis, Phys. Rev. '78, 526 (1950). 1**H.A. Bethe, Phys. Rev. 89, 1256 (1953). 199 B. P. Nigam, M. K. Sundaresan, and T. Y. Wu, Phys. Rev. 116, 491 (1959). zoo R. H. Dalitz, Proc. Roy. Soe. A206, 509 (1951). 201 A. 0. Hanson, L. H. Lanzl, E. M. Lyman, and M. B. Scott, Phys. Rev. 84, 634 (1951).

76

1. PARTICLE DETECTION

for Be is thus too large by 5%. It may be noted that from the simple expression of Bethe and Ashkin given above [Eq. (1.1.102)], one obtains 0, = 3.94" for Au and 4.33" for Be. The value for Au is too large by 476, while the result for Be agrees exactly with B,(exp). On the other hand, the formula of Rossi and Greisen [Eq. (1.1.103)] would give the values 5.82" for Au and 6.56"for Be, which are both considerably larger than O,(exp). The multiple scattering has been frequently used for a crude measurement of pv for charged particles in nuclear emulsion.202If the track of the particle is subdivided into sections (cells) of length t , the average angle between successive sections is given by (1 .l.109)

where ,6 = v/c, t is measured in microns, pv is in MeV, A~ is in degrees, and K(t,,6) is a slowly varying function of t and p. The theoretical value194*196 of K is between 23 and 24, which is in satisfactory agreement with the experimental results both of the Bristol groupzo3and of C o r ~ o n , * ~ ~ namely K = 25.1 f 0.6 for P = 1 and t = 100. 1.1.7. Penetration of Gamma Rays For y rays passing through matter, there is an exponential attenuation such that the intensity I ( z ) after traversing a thickness 2 is given by

I ( z ) = I(O)exp( - Nuz)

=

I(O)exp( -m)

(1.1.110)

where I(0) is the incident intensity (at 5 = 0), N is the number of atoms of absorber per cm3,u is the total cross section for absorption or scattering of the y rays, and r = Nu(cm-') is the absorption coefficient of the radiation. There are three processes which contribute to 6: (1) the photoelectric effect, which consists of the ionization of atomic electrons by the incident photon. (2) The Compton scattering of the photons by the atomic electrons. I n this process, the atomic electrons can generally be considered as free, and the energy transfer to the electron is a function of the scattering angle O of the ? ray and its initial energy hvo. The energy transfer is determined in a straightforward manner from conservation of momentum and energy. (3) The production of an electron-positron pair in the field of a nucleus. P. H. Fowler, Phi.?. Mag. [7] 41, 169 (1950). Gottstein, M. G. K. Menon, J. H. Mulvey, C. O'Ceallaigh, and 0. Rochat, Phil. Mag. [7] 42, 708 (1951). go4 D. R. Corson, Phys. Rev. 80, 303 (1950); 84, 605 (1951). Zo2

2osK.

1.1.

77

INTERACTION OF RADIATION WITH MATTER

The photoelectric effect predominates at low y energieszo6(hv < 0.05 Mev for All hv < 0.5 Mev for Pb). The Compton effect gives the main contribution at intermediate energies (0.05 < hv < 16 Mev for Al; 0.5 < hv < 4.8 Mev for Pb). The pair production predominates a t high energies (hv > 16 Mev for Al; hv > 4.8Mev for Pb). We shall now consider separately each of these three processes. For a general review of the subject of the interaction of y rays with matter, the reader is referred to the review articles of Bethe and Ashkinj3 Davisson and Evans,206and D a v i s ~ o n . ~ ~ ’ 1.1.7.1. Photoelectric Effect. For energies far above the K absorption edge and in the nonrelativistic range (hv << mc2) the cross section for the photoelectric effect from the K shell is given byzo8 @,,hot.K

(1.1.111)

=

where 40 is the Thomson scattering cross section:

40 =

sT3 (x)’ = 6.651 X mc2

cm2.

( 1.1.112)

In obtaining Eq. (1.1.111), the Born approximation was used, the wave function of the outgoing electron being taken as a plane wave. This procedure is not justified when the kinetic energy of the ejected electron is of the same order or less than the binding energy of the K electrons.2n9 Thus for relatively small y energies hv [i.e., for hv/(Z2Ry) 1 - 10, is considerably smaller than would be where R y = Rydberg unit], aPhat,~ given by Eq. (1.1.111). The ratio of the actual @ ) p h o t , ~to the value of the expression (1.1.111) is 0.12 for hv = ZZRy, 0.43 for hv = 10Z2Ry, and 0.90 for hv = 1000Z2Ry. For very high y energies, hv >> m d , relativistic effects become important. This problem has been treated by SauterlZ1OHulme,211 and others. The following formula obtained by Hall is valid in the limit hv >> mc2, and includes the effect of the Coulomb field of the nucleus on

-

2 0 6 See, for example, W. Heitler, “The Quantum Theory of Radiation,” 2nd ed., p. 216, Fig. 21. Oxford Univ. Press, London and New York, 1944. 206 C. M. Davisson and R. D. Evans, Revs. Modern Phys. 24, 79 (1952). 207 C. M. Davisson, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 24. Interscience, New York, 1955. 208 H. Hall, Revs. Modern Phys. 8, 358 (1936). a** M. Stobbe, Ann. Physik [5] 7, 661 (1930). Z1O F. Sauter, Ann. Physik [5] 9, 217 (1931); 11, 454 (1931). 211 H. R. Hulme, Proc. Roy. Soc. A133, 381 (1931). P1z H. Hall, Phys. Rev. 46, 620 (1934); 84, 167 (1951).

78

1.

PARTICLE DETECTION

the outgoing electron, which becomes appreciable for heavy elements : @phot.K

3 Z6damc2 2 1374 hv

= - - -exp[ -TCY

+ 2 d ( 1 - In

CY)]

( 1.1.113)

where CY = 2/137. Equation (1.1.113) shows that a p h o t , R decreases quite slowly with increasing v (only as v-l) in the relativistic region, as compared to the k 7 1 2 decrease at nonrelativistic y energies [Eq. (1.1.111)]. Aside from the approximate calculations mentioned above, which are based in part on the Born approximation, and on the use of plane-wave or nonrelativistic wave functions, Hulme et ~ 1 . have ~ ~ 3 carried out exact calculations for the photoelectric effect from the K shell, using the appropriate Dirac wave functions in the field of the nucleus. The calculations of Hulme et al. were carried out for two y-ray energies, hv = 0.354 and 1.13 MeV, and for three values of 2 : 26, 50, and 84. These results have been extensively used to check the validity of various approximation formulas and to obtain smooth curves of versus hv in the intermediate energy region (hv 1 Mev). I n obtaining the photoelectric absorption coefficient, one must include the contribution of the absorption from the L, M , . shells. Latyshev214 has made direct measurements of the photoelectrons ejected from the K and L shells of P b and Ta, for the ThC’ y rays (hv = 2.62 Mev). ,,, have Detailed calculations of the total photoelectric cross section ,,@ been carried out by White1215who used the results of StobbelZo9 Sauterj210 and Hulme et aL213According to White,21bthe ratio [ of the total photoelectric cross section @phot to the K shell contribution aphot.K is -1.15 for heavy elements. White has obtained E for various values of 2, for two y-ray energies: (1) a t the K absorption edge; (2) for mc2/hv = 1.5 (i.e., hv = 0.340 Mev). At the K edge, [ = 1.02 for 2 = 6, 1.11 for 2 = 29, 1.14 for 2 = 50, and 1.167 for 2 = 92. For hv = 0.340 MeV, [ = 1.01 for 2 = 6, 1.07 for 2 = 29, 1.10 for 2 = 50, and 1.138 for 2 = 92. Figure 14 shows the plot of loglo(@ph,t/&,)versus hv for C, All Cu, Sn, and Pb, as obtained from the results of White. For C and All and decreases approxifor the heavier elements a t low photon energies, aphot mately as Y - ” ~ , as expected from Eq. (1.1.111). On the other hand, for Pb is proportional to v-1 [see at high energies, between 5 and 50 MeV, aphot Eq. (1.1.113)]. White’s calculations include the effect of the L and M

-

. .

Z13

H. R. Hulme, J. McDougall, R. A. Buckingham, and R. H. Fowler, Proe. Roy.

Sac. A149, 131 (1935).

G. D. Latyshev, Revs. Modern Phys. 19,132 (1947). G. R. White, Natl. Bur. Standards Rept. 1003 (1952); see also Appendix I by C. M. Davisson, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 857. Interscience, New York, 1955. 214 216

1.1.

INTERACTION O F RADIATION W I T H MATTER

79

shells, which becomes the sole contribution to the photoelectric effect a t low energies below the K absorption edge. Thus for Sn, the break in the cm2 curve of @)phot at 29.25 kev is due t o the K edge [@,hot = 1.05 X = 8.58 x on low-frequency side of K edge ( L and M shells only); cm2 on high-frequency side ( K L M shells contribute)]. Similarly, for Pb, the break a t 88.2 kev is due to the K edge, while the discontinuity in the region of 15 kev is due to the L absorption edges (LI, LII, ,5111). Below 13.07 kev (LIrledge), only the M , N , and 0 shells contribute to the photoelectric effect.

+ +

PHOTON ENERGY h v ( I N MeV)

FIG.14. The cross section *,,hot for photoelectric absorption for C, Al, Cu, Sn, and Pb, as a function of the photon energy hv. These curves were obtained from the results of White.216

The theoretical results presented above are in fairly good agreement with various experiments on the photoelectric e f f e ~ t . ~ ~ ~ s ~ ~ ~ , ~ ~ ~ 1.1.7.2. Compton Scattering. The Compton scattering consists of the scattering of y-rays b y atomic electrons which can be considered as free (no atomic binding forces) for sufficiently high y-ray energies. From the laws of conservation of energy and momentum, one finds th a t the frequency v of the scattered quantum is given by v =

yo

1

+ (hvo/mc2)(1 - cos e)

(1.1.114)

where vo is the frequency of the incident quantum, and 0 is the angle of H. A. Bethe and J. Ashkin, in “Experimental Nuclear Physics” (E. Segr6, ed.), Vol. 1, p. 304. Wiley, New York, 1953.

1.

80

PARTICLE DETECTION

FIG.15. Schematic diagram of Compton effect, showing notation used in the text: hvo and hv are the energies of the incident and the scattered quanta, respectively;

T

is the kinetic energy of the recoil electron.

scattering of the y ray. The notation used is shown in Fig. 15. The kinetic energy T of the electron is given by

T

=

h(vo -

V)

=

2 m ~ ~ ( h vCOS' o ) ~cp

(hvo

+ mc2)2- (hvo)2cos2

cp

( 1.1.115)

where (a is the angle between the directions of the outgoing electron and the incident y. The angles cp of the electron and 6 of the y ray are related as follows: tan

cp = Y Os c:;':

0 = (mcT$hv)

cot(&@.

(1.1.116)

The energy of the scattered photon decreases with increasing 6. The minimum value, attained for 6 = 180", is given by (1.1.117) The maximum possible angle of the recoil electron is cp = go", in which case the energy of the electron is T = 0, while the scattered y ray continues with its initial energy (hv = hvo) in the forward direction. The differential cross section for Compton scattering was first obtained by Klein and NishinaZ17in 1929. The Klein-Nishina formula gives

) 817

0. Klein and Y . Nishina, 2.Physik 62, 853 (1929).

(1.1.118)

1.1.

81

INTERACTION OF RADIATION W I T H MATTER

where ro = e2/mc2, k = hv, ko = hvo, and d@c is the cross section for scattering of the y ray through an angle e into the solid angle dQ. Upon substituting Eq. (1.1.114) for k = hv, one obtains

(1.1.119) where y 3 ko/mc2. Equation (1.1.119) gives the cross section as a function of the angle 8. For small values of y, the distribution follows the 1 cos2 0 law characteristic of classical electromagnetic theory. As y increases, the distribution becomes increasingly peaked forward, as is generally the case for any high-energy process. The differential cross section as a function of the energy k is given by

+

d@c =

[

?rro2mc2dk 1 + J2;( kko

- 2(y Y+2 1)

+

(1

+ 2r)k y2ko

1

I k~ r2k

(1.1.120)

with

1

1

+

5 -k 5 1 27 - ko

(1.1.121)

Detailed calculations of various quantities and spectra pertaining to the Compton scattering have been carried out by Figure 16 shows the spectrum of the scattered quanta [Eq. (1.1.120)] for incident y energies hvo = 0.5, 1, 2, and 3 MeV. The minimum value hv,i. [Eqs. (1.1.117), (1.1.121)] increases slowly with increasing hvo. Thus hvmin= 0.169 Mev for hvo = 0.5 MeV, and hvmi, = 0.236 Mev for hvo = 3 MeV. (The asymptotic value in the limit v o 4 co is m c 2 / 2 = 0.255 Mev.) It is seen from Fig. 16 that, as hv is increased above hvmin,d @ ~ / d ( h vfirst ) decreases t o a minimum value, and then increases uniformly up to hv = hvo. The minimum of the cross section becomes increasingly more shallow as the primary energy hvo is increased. The total Compton scattering cross section @C is given by

-

(1.1.122) where 40is the Thomson cross section [Eq. (1.1.112)]. For small y (y

<< l),

A. T. Nelms, Nail. Bur. Standards Circ. 642 (1953). See also R. Latter and H. Kahn, “Gamma-Ray Absorption Coefficients.” Published by The Rand Corporation, Santa Monica, California, 1949; G. Allen, Natl. Advisory Comm. Aeronaut. Tech. Notes 2026 (1950). zl$

82

1.

PARTICLE DETECTION

Eq. (1.1.122) gives 9 c Ei 4

0 0 - 2Y)

(1.1.122a)

whereas for very large y (y >> l), one obtains (1.1.122b) Equations (1.1.122) show that the Compton cross section decreases uniformly with increasing energy of the y quantum. Figure 17 shows a , was taken from Bethe and Ashkin, referplot of % / 4 0 versus h v ~ which ence 3, p. 322. For comparison, we have also shown the photoelectric

PHOTON ENERGY hv (IN MeV)

FIG.16. Spectrum of Compton scattered quanta, as obtained from Eq. (1.1.120), for incident photon energies hvo = 0.5, 1, 2, and 3 MeV.

cross section divided by 2,in the same units 40 [i.e., @phot/z+O]. The reason by 2 is that a p h o t pertains to the photoelectric effect for for dividing aphat the entire atom, so th at @.phot/Z represents the photoeffect per atomic electron and is therefore the quantity t o be compared with 9 c (Compton scattering per electron). We note th at the energy hvo E h ; ~for which +phot/Z = 9~increases with increasing 2. Thus hih = 0.02, 0.05, 0.13, and 0.53 Mev for C, All Cu, and Pb, respectively. As hvo is increased above h h , the photoelectric effect rapidly becomes unimportant compared to the Compton scattering as a source of y-ray attenuation. It should be noted th at the expression for @c [Eq. (1.1.122)] no longer applies for very low photon energies, where the binding of the atomic electrons must be taken into account. I n this case, the incoherent (Compton) scattering will be reduced, both because of the binding of the atomic

1.1.

INTERACTION O F RADIATION WITH MATTER

83

electrons, and because of the effect of the exclusion principle in preventing transitions to occupied atomic levels. On the other hand, there will also be a substantial amount of coherent scattering from the atom as a whole, so that the total scattering cross section will generally be larger than the value Z@cwhich would be calculated from Eq. (1.1.122). For a discussion of these effects, the reader is referred to the review article of Davisson.*07 The Klein-Nishina formula has been tested in various experiments, and has been shown to be in good agreement with the experimental dat~.206,207.216

and the photoelectric cross section per FIG.17. The Compton total cross section electron, iP,hot/Z, for C, Al,Cu, and Pb, as functions of the incident photon energy hvo. The curves of GphOt/Z were obtained from the results of White.216

1.1.7.3. P a i r Production. The theory of the pair production by y rays is closely related to the theory of the bremsstrahlung by a high-energy electron. The general formula obtained by Bethe and Heitler147using the Born approximation is very complicated and will not be given here. However, the formula simplifies considerably if the energies of both positron and electron are not too high so that screening can be neglected, i.e., if ( 1.I . 123)

where E+ and E- are the total energies of the positron and electron, respectively, and k = hv is the energy of the incident photon. If in addition to Eq. (1.1.123) , all energies involved are large compared to mc2,the

1. PARTICLE

84

DETECTION

energy distribution of the positrons (or electrons) is given by Q(E+) dE+

=

45 dE+

ka

kmc2 ( I. 1.124)

where 5 is defined by

5

(Z2/137)ra2

(1.1.124a)

with T O = e2/mc2 (classical radius of the electron = 2.82 X cm). As is also true for the general Bethe-Heitler formula, Eq. (1.1.124) gives a symmetric energy distribution for the positron and electron. Actually for small velocities v+ and v- of the pair, and for large 2, the Coulomb effect (which is neglected in the Born approximation) becomes important and results in a somewhat asymmetric distribution favoring higher energy positrons. For very large energies E+, E-, the screening is complete (4 = 0 ) , and Q(E+)is given by Q(E+) dE+

=

45dE+

[( +

~-

k3

+ Em2+ +E+E-)ln(183Z-1’s) - +E+E-]. (1.1.125)

Figure 18 shows the energy distribution of the pair particles (positrons or electrons) as obtained from the calculations of Bethe and Ashkin (reference 3, p. 328). For hv up to 10mc2,the curves do not include screening and are valid for all elements; for higher photon energies] the calculations of Bethe and Ashkin were done for Pb and include the effect of screening. It is seen that for small values of hv, the energy distributions are generally quite flat between the minimum and maximum values, T+,mi,= 0 and T+,,,, = k - 2mc2, where T+ is the kinetic energy of the positron. For k/mc2 5 30, the distribution has a broad maximum at T+ = +T+,mx= +lc - mc2.For larger k/mc2, Q(E+) has a broad minimum at +T+,m.xand two subsidiary maxima on each side of the minimum, which implies that for large k/mc2, either the positron or the electron tends to carry off most of the energy of the y ray. For any finite k/mc2, = k - 2mc2. the distribution is zero a t the two ends, T+,mi,= 0 and T+,msx Figure 18 shows that the distributions are symmetrical with respect to T+ = $T++,., = +lc - me2. This is a consequence of the use of the Born approximation for Q(E+), which gives identical spectra for the positron and electron. In analogy with the bremsstrahlung in the field of the atomic electrons] which has been discussed above [Eqs. (1.1.62), (1.1.68)], there is also the possibility of pair production in the field of the atomic electrons. Wheeler

1.1.

85

INTERACTION O F RADIATION WITH MATTER

and Lamb14*have shown that, for complete screening, the electronic pair production is a fraction [/Z of the production in the field of the is the same quantity that appears in the nucleus, where [ (-1.2-1.4) formula for the bremsstrahlung [Eq. (l.l.68)l. Thus, for the case of complete screening, the total @(E+)dE+ (including the electronic con[) in the definition of b tribution) is obtained by replacing Z2 by Z(Z [Eq. (1.1.124a)I. For low energies, where screening can be neglected (hv 5 20 Mev), the pair production in the field of the atomic electrons 15* Rohrlich,'j' Nemirovhas been calculated by B o r s e i l i n ~ , 'Votruba, ~~ sky,L62and Watson.153

+

10

a

NE N I Lb : I

- 4

'? -

w"

a

2

'0

0.1

0.2

0.3

0.4 (E;

0.5

0.6

07

Q0

0.9

I

mc2)/~v-2mc2)

FIG.18. Energy distribution of the positron (or electron) in an electron pair as a function of the positron kinetic energy for various energies hv of the incident y ray. These curves were taken from the calculations of Bethe and Ashkin (reference 3, p. 328, Fig. 38). For hv 5 IOrnc~,the curves do not include screening and are valid for all elements. For higher photon energies, the curves pertain to Pb and include the effect of screening.

The angular distribution of the pair electrons becomes increasingly forward with increasing energy of the primary quantum, in analogy with the bremsstrahlung distribution. In particular, for high y energies k, the average angle between the incident y ray and the direction of motion of the electron (or positron) is given by <9>

Emmc2/E

(1.1.126)

where E is the energy of the electron (or positron). Equation (1.1.126) is completely analogous to Eq. (1.1.74) for the bremsstrahlung. The pair production in monocrystalline targets has been discussed by ubera11.166

86

1.

PARTICLE DETECTION

The total cross section for pair production cally for two limiting cases:

aPaiF can be obtained analyti-

' / ~ screening), Eq. (1.1.124) is valid, (1) For me2 << hv << 1 3 7 r n ~ ~ Z - (no and one finds (1.1.127) ~ screening), integration of Eq. (2) For h v >> 1 3 7 m ~ ~ Z - "(complete (1.1.125) gives = $[? ln(183Z-1'3) - A]. (1.1.128) For intermediate values of hv, the total cross section @pair must be obtained by numerical integration. Figure 19 shows the resulting curves of 14

12 10 @'pair

v

@

8 6

4 2 n "I

2

5

10

20

100 200 hv/mc2

500

2000

l0,OOO

FIG.19. The total cross section for pair production Qlpair as a function of the 7-ray energy, for air and Pb, and for the hypothetical case of no screening. These curves were taken from the calculations of Bethe and Ashkin (reference 3, p. 338, Fig. 41).

for the (hypothetical) case of no screening, and for air and Pb (including screening), as obtained by Bethe and Ashkin (reference 3, p. 338). It is seen that for large energies (hv/mc2 2 50), the values of for air and P b fall below the curve for no screening, and slowly approach the asymptotic value [Eq. (1.1.128)] which is 14.1 for air and 11.6 for Pb. As mentioned above, the Bethe-Heitler theory based on the Born approximation cannot be expected to give accurate results for high Z and low energies of the positron or electron, since the wave functions for these particles will then be appreciably distorted by the Coulomb field

1.1.

87

INTERACTION O F RADIATION WITH MATTER

of the nucleus. Jaeger and Hulme220have calculated the pair production cross sections at two photon energies (hv = 3mc2 and 5.2mc2), using the exact Dirac wave functions for the pair particles. At hv = 3mc2, they obtained results for 2 = 50, 65, and 82; at hv = 5.2mc2, the calculations were carried out for 2 = 82 only. For the worst case, 2 = 82, hv = 3mc2, the Born approximation cross section is too low by a factor of 2 (0.34 X cm2, as compared to 0.67 X cm2 from the exact calculation). For the other 2 values, and hv = 3mc2, the Bethe-Heitler result is also too small, but the deviation decreases rapidly with decreasing 2 or increasing hv. Thus for hv = 5.2mc2, 2 = 82, the error is only 16%. Recent experimental results a t high energies (210 Mev)221-227 have shown that the measured pair production cross sections are appreciably lower than the Bethe-Heitler calculated values, and that the deviation is proportional to Z2. Thus Lawson,222from his measurements at 88 MeV, concluded that the ratio of the experimental to the theoretical cross section can be approximately represented by

~

~= 1 -~1.5 x 1~ 0-522. /

a

(1.1.129) ~

In view of these results, Bethe et al.2zshave carried out an accurate calculation of the pair production, in which the Born approximation was not used. They have found that the correction to the Born approximation result of Bethe and Heitler147is a reduction of the cross section proportional to Z2, as is indicated by the experimental data. Upon applying this correction, Bethe et a1.228have obtained excellent agreement with the measurements of LawsonZP2 at 88 MeV, and those of DeWire et aLZz4a t 280 MeV. It may be noted that the correction to the Born approximation is a reduction of the cross section a t high energies, as compared to an increase of the cross section at low energies.220 The correction goes through 6 Mev.226n228 zero at hv 1.1.7.4. Total Absorption Cross Section for y Rays. The total cross section u for the removal of a y-ray photon from the incident beam is given by u z= a p h o t 2a.c @'pair. ( 1.1.130) Z z O J . C. Jaeger and H. R. Hulme, Proc. Roy. SOC.A163, 443 (1936); J. C. Jaeger,

-

+

+

Nature 137, 781 (1936). 2 2 1 C. D. Adams, Phys. Rev. 74, 1707 (1948). a22 J. L.Lawson, Phys. Rev. 76, 433 (1949). 223 R. L. Walker, Phys. Rev. 76, 527 (1949). 224 J. W. DeWire, A. Ashkin, and L. A. Beach, Phys. Rev. 83, 505 (1951). 2zaC. R.Emigh, Phys. Rev. 86, 1028 (1952). z*BE. S. Rosenblum, E. F. Shrader, and R. M. Warner, Phys. Rev. 88, 612 (1952). 227 A. I. Berman, Phys. Rev. 90, 210 (1953). 228 H.A. Bethe and L. C. Maximon, Phys. Rev. 93, 768 (1954); H. Davies, H.A. Bethe, and L. C. Maximon, ibid. 93, 788 (1954).

~

88

1.

PARTICLE DETECTION

The complete absorption coefficient c equals Nu.Figure 20 shows the mass absorption coefficient T/P (in units cmZ/gm) for Al, Cu, and Pb. For Pb, we have presented the separate contributions to r / p due to the photoelectric effect, Compton effect, and pair production (dashed curves). The values of r / p were obtained from the tables of White.216It is seen from Fig. 20 that, as a function of frequency, T has a minimum, which 4 for Pb. occurs at hv ? 20 Mev for Al, .=8 Mev for Cu, and ~ 3 . Mev For Pb, the minimum lies in the region where the Compton effect is predominant. For lower v, the photoelectric effect predominates, and the with increasing v is responsible for the decrease of rapid decrease of apbot ale c

$'

0.16

E

0

-$g

0.14

0.12

I-

0.10

2

'a

0.08

0.06

t 0.04 a 0

2 O.O2 0

0.1

0.5

0.2

I

2

5

10

20

50

100

PHOTON ENERGY h Y (IN MeV)

FIG.20. The mass absorption coefficient T/P for Al, Cu, and Pb, as a function of the photon energy hv. For Pb, the separate contributions of the photoelectric effect, Compton effect, and pair production are shown by the dashed curve8. The curves of r / p shown in this figure were obtained from the tables of White.216

the total absorption coefficient c. At frequencies somewhat above the position of the minimum, the pair production becomes the main effect, and is responsible for the rapid increase of with increasing v, until a t very high energies (hv 5 Bev), T approaches a constant value as a result of the saturation of due to screening. The minimum of r implies that y rays of energies of the order of 5-20 Mev have a relatively long mean free path in matter. The values of ~ / at p the minimum for Al, Cu, and Pb, are as follows, according to the data of White: 0.0217 cm2/gm for Al; 0.0306 cm2/gm for Cu, and 0.041 cm2/gm for Pb. The corresponding values of the maximum mean free path are: Amsx = 46.1 gm/cm2 for A1 (at hv = 20 Mev); ,A, = 32.7 gm/cm2 for Cu (at hv = 8 Mev); X, = 24.4 gm/cm2 for P b (at hv = 3.4 Mev).

-

1.2.

IONIZATION CHAMBERS

89

C ~ l g a t has e~~ carried ~ out measurements of the total y-ray absorption cross section a for y energies of 0.411, 0.664, 1.33, 2.62 Mev using radioactive sources, and 4.47,6.13, 17.6 Mev using y rays from nuclear reactions. The measurements were made for a variety of absorbers (polyethylene, C, Al, Cu, Sn, Pt, Pb, Bi, and U). I n general, the theory is in good agreement with these data, when account is taken of a small correction due to Rayleigh in addition to the three principal effects: the photoelectric effect, Compton scattering, and pair production in the field of the nucleus and the atomic electrons. The present theory is in reasonable agreement with measurements of the absorption coefficient at various energies up to 280 M ~ v . ~ ~ ~ , ~

1.2. Ionization Chambers* 1.2.1. General considerations’ An ionization chamber is a device which measures the amount of ionization created by charged particles passing through a gas. The basic processes of ionization of gases by charged particles have been discussed in Chapter 1.1.7 If an electric field be maintained in the gas by a pair of electrodes, the positive and negative ions will drift apart, inducing charges on the electrodes which can be detected as a voltage pulse. Or if a steady flux of particles enter the chamber one can measure the average current caused by the ionization. The latter application will be specifically considered in Section 1.2.7; the other sections of this chapter, however, will be principally devoted to the ionization chamber as a detector of single particles and therefore as a pulse instrument. 1.2.1.1. Essentials of a Pulse Ionization Chamber. Figure 1 shows schematically the essential parts of this very simple device. One of the electrodes, the “collector” (a misnomer, as we shall see), is designed to have a low capacity both to the other electrode and to ground, so that a very small charge will still give a measurable potential change. The small amount of charge is characteristic: if the particle loses 1 Mev in collisions S. A. Colgate, Phys. Rev. 87, 592 (1952). W. Franz, Z. Physik 98, 314 (1935); P. Debye, Physik. Z. 31, 419 (1930). t See also Vol. 2, Chapter 4.1 and Vol. 4, B, Chapter 7.5 and Section 9.2.3. 1 General references for Sections 1.2 and 1.3 are: D. H. Wilkinson, “Ionization Chambers and Counters.” Cambridge Univ. Press, London and New York, 1950; B. B. Rossi and H. Staub, “Ionization Chambers and Counters.” McGraw-Hill, New York, 1949; S. C. Curran and J. D. Craggs, “Counting Tubes.” Academic Press, New York, 1949; see also Vol. 4, A, Section 2.1.5. **o

*an

-

* Chapter

1.2 is by Robert W. Williams.

90

1.

PARTICLE DETECTION

with the gas, the number N of electrons released will be about 30,000, or coulomb. The passage of a particle creates the ionization, for all practical purposes, instantaneously. The positive ions then drift toward the negative electrode with a velocity of the order of (1cm/sec) X [(760 mm Hg)/p X [ E / ( l volt/cm)] or in a typical case 0.001 cm/psec. The electrons, if they do not suffer attachment and thereby become heavy negative ions, will drift toward the positive electrode with velocities, under comparable conditions, of 1-5 cm/psec. For definiteness assume, as is usually the case, that the collector is the anode. The collector potential is lowered both by the motion of electrons toward it and by the motion of positive ions away from it. It is instructive to calculate explicitly the potential change in a highly idealized case; for

a charge of 5 X

'\ \

I

I

= -k

FIG. 1. The essentials of a pulse ionization chamber (schematic). The dotted line illustrates the path of an ionizing particle whose passage leaves pairs of ions in the gas of the chamber.

example, insulated long cylindrical electrodes with cylindrical sheets of positive and negative charge, Q+ = Q- = Q formed a t radius r1 (Fig. 2). Let the initial potential difference between inner and outer electrodes be Vo(b)- V O ( Uwhere ), V,(r) is the potential at any point in the chamber before the charge sheets have been moved apart, and assume that the negative sheet of charge collapses uniformly toward the central electrode and the positive one expands toward the outer electrode. When the two charge distributions are at r- and r+ respectively we find from elementary calculation that the potential has changed by an amount which is independent of the magnitude of the initial potential, and which can be written (1.2.1)

This result, that the potential change of the electrode is proportional t o the fraction of the total potential drop through which the charge has

1.2.

IONIZATION CHAMBERS

91

moved, is not restricted to this special geometry; it will be discussed in more detail in Section 1.2.2. We use it here to note that the total voltage pulse, if all the ions are ultimately collected, is Q/C as expected, and that the part of this pulse corresponding to electron motion occurs orders of magnitude more rapidly than the part due to ion motion. It is therefore important t o know under what circumstances the electrons will remain free as they drift through the gas, and what mechanisms may prevent complete collection of all ions.

FIG.2. Idealized cylindrical ionization chamber.

1.2.1.2. Behavior of Ions and Electrons in Gases. The positive ions which are formed upon passage of a charged particle through the gas remain nearly in thermal equilibrium with the gas. The presence of the electric field causes them to drift toward the cathode, but the increase in kinetic energy is very small and they rapidly reach a terminal mean velocity for which the energy gained from the electric field is dissipated in molecular collisions. This “drift velocity’’ would be expected to depend on the ratio of field strength to mean free path, and indeed it is observed experimentally t o be, in the ionization-chamber range, a linear function of E / p : 760E (1.2.2) w = Ko-

P,

with E in volts per centimeter and p in mm Hg, K Oranges from about 8 cm/sec for Ne, t o 2.5 cm/sec for Ar, t o 1 cm/sec for Xe.2 Negative ions 2 A. M. Tyndall, “The Mobility of Positive Ions in Gases.” Cambridge Univ. Press, London and New York, 1938.

92

1.

PARTICLE DETECTION

(not electrons) have about the same drift velocities. A simple theory of drift velocity8 yields the expression

_e _X -E muP where is the mean free path and u the ion’s speed, assumed the same for all ions. Since practical limitations (Section 1.2.3) usually restrict the average value of E / p to not more than -1 volt/cm/mm Hg, the minimum time required for ions to cross even a small ionization chamber, say 1 cm, will be -0.2 msec, and in most cases it will be considerably longer. In nearly all cases where ionization chambers are used to count individual particles only the fast portion of the pulse, due to the motion of the electrons, is utilized. The electrons which are released in the initial ionization prove to remain free, in most gases, until they impinge on an electrode (or other surface). The exceptions are electronegative gases which have an appreciable probability to form negative ions by electron attachment. For 0 2 , the most dangerous offender, the probability of attachment, per collision, is 10V to Water vapor has a similarly large attachment probability, and Clz, NH,, N20, HzS, SO?, NO, and HC1 are all known to be bad. In a nonattaching gas-Ar, N2, CH4 are among the commonly used ones-the electrons continue to drift toward the anode, but acquire from the field a kinetic energy many times their thermal energy, because the mechanisms of energy transfer to the gas are relatively inefficient. The denoted by the ratio of mean kinetic energy to thermal energy (#KT), “agitation energy” 7,is typically of the order of 100 in a noble gas, where elastic collision and atomic excitation are the only available energytransfer mechanisms; it is down by an order of magnitude in diatomic gases, and is not much greater than 1 in polyatomic gases. Table I gives some values5 for the drift velocity and agitation energy of electrons in various gases. Argon is a particularly important gas; it is convenient and is widely used. Its first excitation level is very high, 11.5 volts, and in consequence the electron agitation energy is large. A small amount of polyatomic impurity gas lowers the agitation energy greatly; Rossi and Staub’ find w

=

8 13. B. Rossi and H. Staub, “Ionization Chambers and Counters,” p. 6. McGrawHill, New York, 1949. D. H. Wilkinson, “Ionization Chambers and Counters,” p. 41. Cambridge Univ. Press, London and New York, 1950. 6Based on R. H. Healey and J. W. Reed, “The Behavior of Slow Electrons in Gases,” Amalgamated Wireless, Ltd., Sydney, Australia, 1941; and tables in Wilkineon,‘ and Rossi and Staub,s

1.2.

93

IONIZATION CHAMBERS

about a factor of ten for 10% COZ at E / p = 1. This has the effect of increasing the drift velocity, for two reasons: the electron speed is lowered [see Eq. (1.2.2)], and because of the Ramsauer resonance effect on the cross section of noble gas atoms for electrons the mean free path proves to increase, in the region of interest (which is from -10 ev to -1 ev). Table I includes the drift velocity in 5 % and 10% CO,; at E / p = 1 the increase over pure Ar is a factor of ten; there is a comparable increase when the mixture is compared to pure C02, because the mean free path of -1 ev electrons in argon is so great. TABLEI. Drift Velocity w and Ratio of Agitation Energy t o Thermal Energy 9 for Electrons in Various Gases a t Room Temperature; Principally from Healey and Reed5 Values are approximate and in noble gases are strongly impurity sensitive.

E/p

He Ne Ar HI

N2

coo

0.95 Ar 0.05 COz 0 . 0 9 Ar 0 . 1 coz

=

0.2 v/cm/mm Hg

w (cm/piiec)

tl

0.5 0.5 0.3 0.4 0.4 0.1

62 120 2.7 6.5 1.5

11

E / p = 1 v/cm/mm Hg

---

w (cm/Mec) 0.9 1.5 0.5 1 .o 0.8 0.55

3.3

4.3

0.9

5.3

9

53 216 285 9.3 21.5 1.5

R. H. Healey and J. W. Reed, “The Behavior of Slow Electrons in Gases.” Amalgamated Wireless, Ltd., Sydney, Australia, 1941. 0

The greater drift velocity of Ar-C02 mixtures is often of great practical value in obtaining a faster pulse, and these mixtures are widely used. Unpurified tank argon alone will give rise to an electron drift velocity considerably greater than that of Table I, and is satisfactory for many applications where E / p is large (-1) and where accurate pulse height is not essential. In high-pressure cylindrical-geometry chambers, where E / p at the outer electrode is usually quite low, the Ar-C02 mixture may show some attachmentJ6and pure Ar will give better performance. There are two additional complications in the motions of ions or elec6 Under ordinary conditions COO has negligible electron attachment. Experience with cosmic-ray ionization chambers [H. S. Bridge, W. E. Hazen, B. B. Rossi, and R. W. Williams, Phys. Rev. 74, 1083 (1948)Jindicates that a t very low E / p values (-0.01), “pure” Ar has distinctly less attachment than Ar-COZ mixtures.

94

1.

PARTICLE DETECTION

trons in gases which can lead to reduction in pulse height; we now consider the first of these, diffusion. The center of gravity of a group of ions which is liberated at a point in an electric field will have a displacement proportional to the time, while the mean distance of the ions from the center of gravity will increase as the square root of the time. The importance of diffusion can be measured by the ratio of the latter distance to the former one. This ratio can be calculated from kinetic theory; the mean free path cancels out, and one has, for a gas at room temperature.’ diffusion distance drift distance

=

o.18

d+

(1.2.3)

where q is the agitation energy and V the voltage difference between the point of release of the ions and the final position of the center of gravity. For heavy ions 7 = 1 and the effects of diffusion will generally be small. For electrons 7 may be -100 in noble gases, and diffusion may be important; electrons may diffuse back to the cathode, or out beyond the boundaries of the apparatus. The “ C 0 2 effect ” may be utilized to reduce r] and therefore decrease the electron diffusion. The second effect is recombination, the neutralization of positive and negative ions before they are collected. This is a complex subject, considered in detail by Wilkin~on,~ and we shall only summarize the principal results. The recombination coefficient a is defined by writing the rate of disappearance of ions, when n+ positive ions and n- negative ions per cubic centimeter are present, as an+n-,8 with a depending on agitation energy, and, of course, on the nature of the negative ions; a is cm3/sec for heavy negative ions, and cm3/sec for electrons. In air chambers or other chambers where attachment is more or less complete, recombination can be a serious cause of pulse loss, particularly in those current chambers which contain a large density of ionization (Section 1.2.7). Free-electron chambers are better both because of the smaller value of 01 and the shorter time which the electrons spend in the gas. The over-all improvement factor is -lo7; therefore free-electron chambers do not suffer from recombination effects under any circumstances ordinarily realized. Ionization chambers used as monitors of the direct beams of pulsed machines (synchrotrons, etc.) present special problems. They will be considered in Section 2.7.1. 7 D. H. Wilkinson, “Ionization Chambers and Counters,” p. 37. Cambridge Univ. Press, London and New York, 1950. * This assumes that an ion, when created, does not recombine preferentially with the other member of the pair. The assumption is surely correct if the negative ions are electrons, b u t it might be expected to break down in very high-pressure air.

1.2.

95

IONIZATION CHAMBERS

1.2.2. Pulse Formation Quantitative use of the ionization chamber as a particle detector requires consideration of shape and magnitude of the voltage pulse caused by release of ionization. We at first calculate the potential change of the collecting electrode due to the motion of a single charge, assuming that the electrode is essentially insulated (i.e., referring to Fig. 1, that the time constant RC of the grid resistor and the capacity of the collector is large compared to the collection time of an ion). The result which was obtained for a special case in Section 1.2.1 can be obtained for arbitrary two-electrode geometry under the assumption (always true in practice, even in a Geiger counter) that the charge released in the gas is small compared to the charges residing on the electrodes which give rise to the initial potential difference Vb Va. Consider an electron of charge -e at point r l in the gas. The potential at rl will consist of two parts: the potential Vo(rl) due to the charges on the electrodes, which is, by our assumption, essentially independent of any charges which may be present in the gas, and V8(rl), the potential due to other ionization which may be present in the gas. Then the electrostatic energy of the electron consists of two independent parts, the spacecharge energy and the energy in the field of the electrodes, and we may consider the latter ~ e p a r a t e l y The . ~ energy of the system of electrodes, plus the electron, is

-

+BqiVi = i[-eV,(r,)

-k

&ova0

- QoVao].

where V , and v b are the electrode potentials, and the subscript 0 refers to initial values. As the electron drifts from rl to r 2 the work done on it by the field must be extracted from this system; QO and VbO must remain fixed, but Val the collector potential, can change by some amount AV. Thus we have -e[Vo(rz) - Vo(r1)l

=

+[-eVo(rd

+ eV&d +

QO

AVI (1.2.4)

which is the desired result. 9 The space charge has no direct effect on the size of the pulse due to one electron but it may affect the velocity of the electron and therefore the pulse shape, and secondary effects of the electron. In pulse ionization chambers these questions are unimportant because the space charge is so small, but they are crucial in Geiger counter operation (Section 1.3.2).

1.

96

PARTICLE DETECTION

Two examples will serve to illustrate the behavior of pulse ionization chambers utilizing electron collection (the pulse shape of “slow” chambers is not usually interesting). First the parallel-plate chamber of separation d, ion-pair formed at X;O from the positive electrode, at t = 0. AV will have a rapid contribution from the motion of the electron which has drift velocity w-, and a much slower contribution from the positive ion (drift velocity w+). -e A V = - (C =

w-t

+ wft )

-e

(7w+t + 2)

-e C

finally.

--

until w-t until w+t

= =

xo

d - xo

Figure 3 illustrates this pulse using drift velocities characteristic of argon. To take advantage of the fast electron-collection pulse one must eliminate the effects of the slow positive ions by incorporating a low-frequency rejection network in the collector circuit of the chamber or, more commonly, in the amplifier-e.g., a short time-constant T in a resistancecapacitance coupling stage such that RC = r << t+, the collection time of the ions. The rapidly-rising (electron-collection) part of the pulse will not be much affected so long as r >> t- where t- is the collection time of the electrons, but the positive-ion pulse will be reduced by roughly r/t+. For a given form of input pulse the detailed shape of the pulse after passing through the low-frequency rejection network can be obtained by standard transient-response analysis-Wilkinsonl gives several examples. The dotted line in Fig. 3 illustrates the effect of a time-constant r = 5t(10% pulse-height loss). An approximation which is sufficient for some purposes is to assume that the voltage rise due to the electrons is undistorted, but that the voltage then returns t o zero with the time constant r, and with no positive ion contribution. A network which gives a more nearly square-topped pulse is illustrated in Fig. 4;it consists of a shorted delay line in series with a resistance equal to its characteristic impedance. It is less convenient than the RC, and causes a 50% amplitude loss, so that it is usually used only when there is some reason to require a good pulse shape. The pulse-shaping action of this network is illustrated in Fig. 5. The second example of the voltage pulse in an electron-collection chamber is that of cylindrical geometry: a small central electrode, of radius a, is surrounded by a concentric cylinder of radius b. This is a simple, lowcapacity arrangement, and its chief virtue is that the electron-collection

1.2.

IONIZATION CHAMBERS

97

C 3Q.C d C

nv t

, I \'\ \

'.. -

FIQ.3. Idealized voltage pulse from a parallel-plate ionization chamber. An ion pair is released at distance Xa from the negative electrode. The time scale would be approximately right if, for example, d were 2 cm, V were 1000 volts, and P were 1 atmos. The dashed line indicates the effect of a five-microsecond time constant.

0-jD.L.

FIQ.4. Shorted-delay-line pulse-shaping network. Rk is equal to the characteristic impedance of the delay-line.

t

-AV

FIQ.5. Effect of delay-line pulse shaping on a typical pulse from a cylindrical ionization chamber; 270 is the round-trip time of the line, and must be greater than t-, the electron collection time, if t h e pulse is to have a flat top.

98

1.

PARTICLE DETECTION

pulse is reasonably independent of the point at which the electron is released, since most of the potential drop occurs near the central wire, Figure 6 shows the electron pulse in a chamber with b/a = 120, calculated for a single electron [curves (a) and (a’)]; for uniform ionization in the chamber [curves (b) and (b‘)]; and for uniform ionization along a straight line passing through the axis of the chamber [curve (c)]. The pulse shape,

FIG.6. Electron pulse in a cylindrical ionization chamber: 1- is the drift time from outer to inner electrode; curves (a) and (a’) are for a single electron, (b) and (b’) for a uniform distribution, and (c) for a linear distribution.

AV(t)/AVfina~,is calculated from Eq. (1.2.1), with elapsed time related to drift velocity by

t=

J:&

and with two assumptions for w: constant (solid curves) and proportional to (E)”q(dotted curves). Inspection of curve (a) shows that a considerable portion of the chamber volume gives rise to pulses of nearly maximum height. It is easy to show that uniform ionization gives a pulse which is a fraction f of the total-charge-collection pulse, f = b2/(b2 - u2)

This is 0.90 for b/a

=

120.

- 1/[2 ln(b/u)]

1.2.

IONIZATION CHAMBERS

99

The parallel-plate chamber can be modified so that it gives an electron pulse independent of the point of production,1° by adding a third electrode. A grid at a fixed potential, near the collecting electrode, will shield the collecting electrode from the rest of the chamber (Fig. 7) so that only that portion of the electron’s travel which takes place between the grid and the collector will cause a pulse to be induced. The electrostatic situation near the edge of an electrode will in general be complicated-either the field will be quite distorted and therefore the effective volume, and expected pulse shape, somewhat uncertain, or the field shape can be maintained by “guard electrodes”-extra electrodes, Grid, Useful Volume

-

Collector

h

FIG. 7. Schematic diagram of a gridded chamber. Electrons originating in the shaded area will give pulses of nearly uniform height as they pass between the grid and the collector.

held at the average potential of the collecting electrode, which maintain the symmetry of the field beyond the edge of the collecting electrode. The pulse induced by an electron near the edge of the collector is more complicated in this case; detailed calculations in simple cases are given by Rossi and S t a ~ b . ~ In many applications of ionization chambers one is interested in average current or in total amount of charge collected (Section 1.2.7). For these chambers (as well as for “slow” pulse chambers which respond to the motion of ions) the details of the eIectron pulse are unimportant. The guard electrode helps t o define accurately the volume of gas from which ionization is collected. The role of the guard electrode in preventing leakage current is discussed in Section 1.2.7. 10 For the detailed theory of this device, Bee D. H. Witkinson, “Ionization Chambers and Counters,” p. 74. Cambridge Univ. Press, London and New York, 1950.

100

1.

PARTICLE DETECTION

1.2.3. Quantitative Operation and Some Practical Considerations 1.2.3.1. Attachment, Diffusion, Recombination. Those basic phenomena of the passage of electricity through gases which affect the operation of ionization chambers are discussed in Section 1.2.1. Their importance may range from very little in the case of large E / p and nonabsolute pulseheight requirements (e.g., a low-pressure parallel-plate chamber used as a counter) to considerable in the opposite extremes (e.g., a high-pressure cylindrical chamber used for proton recoil pulse-height spectrum work). The maximum value of E / p which can be used is determined by the condition that there should be no gas multiplication (Section 1.3.1) even in the region where E / p is largest. For cylindrical or spherical chambers this means that E / p will necessarily be low in the region near the outer electrode; and in general the difficulties mentioned are associated with low E / p . Even with parallel-plate chambers the inconvenience of working with high voltages will often put a practical limit on the available E / p . Electron attachment can be eliminated by sufficiently rigorous exclusion of electronegative gases, of which O2 and H20 are the most frequent offenders. Noble gases can be purified very effectively by circulation over hot calcium c h i p ~ . ~Purified Jl argon in a clean metal-and-glass chamber with soldered seals has been found to remain free from attachment for years, in an application in which the minimum E / p was about 0.01 v/cm/mm Hg. However, a chamber containing volatile material (e.g., rubber gaskets) must be purified frequently if quantitative performance under low E / p conditions is to be maintained. The Ar-C02 mixture previously referred to can also be purified with hot ~ a l c i u r nand , ~ is free of attachment under most conditions. However, there are indications12 that at very low E / p this mixture, unlike pure argon, shows noticeable attachment. Gases which cannot be purified by such drastic methods, e.g., BF3 (which in pure form does not have serious attachment), must be prepared with great care. Graves and Fromanla describe a suitable technique for preparing BF3for ionization chambers. Recombination can be shown to be negligible14under nearly any circumstances in chambers in which no attachment takes place. This subject will therefore be treated in Section 1.2.7, in connection with current chambers. L. Colli and U. Facchini, Rev. Sci. Instr. 23, 39 (1952). H. S. Bridge, W. E. Hazen, B. B. Rossi, and R. W. Williams, Phys. Rev. 74,1083 (1948). l 3 A. C. Graves and D. K. Froman, “Miscellaneous Physical and Chemical Techniques of the Los Alamos Project,” p. 154. McGraw-Hill, New York, 1952. l4 D. H. Wilkinson, “Ionization Chambers and Counters,” Chapter 111. Cambridge Univ. Press, London and New York, 1950. 11

la

1.2.

101

IONIZATION CHAMBERS

Diffusion effects are important only for electrons under conditions of large agitation energy q [Eq. (1.2.3)].A typical situation in which diffusion may be important occurs when ionization is released adjacent to the wall of the chamber, as when a noncollimated alpha-particle source is incorporated in the negative electrode as a calibrating standard. It can be assumed that electrons which diffuse back t o the negative electrode are lost, and with the help of Eq. (1.2.3) an estimate of the pulse loss from this effect can be made. Addition of a polyatomic gas, with consequent 10 r

0

0.5

1.0

15

2 .o

2.5

E (ev) FIG.8. Attachment probability h upon collision of an electron with an oxygen molecule, as a function of electron energy E. From Wilkinson,6 by permission.

reduction of 7, is of course desirable for applications where diffusion must be minimized. An example of a complete diffusion calculation is worked out by Rossi and Staub.16 1.2.3.2. Checks for Quantitative Operation. Spurious Effects. The most commonly used test for proper ionization-chamber operation is the examination of the pulse height, from some reproducible source, as a function of collecting voltage. If a chamber exhibits a good “plateau”region in which pulse height is independent of voltage-it is usually considered free from the defects we have outlined. However, examination of the dependence of electron attachment coefficient on electron energy, Fig. 8, shows that in oxygen, at least, there is a region in which increasing electron energy causes an increase in attachment coefficient, which might l6 B. B. Rossi and H. Staub, “Ionization Chambers and Counters,” p. 27. McGrawHill, New York, 1949.

102

1.

PARTICLE DETECTION

compensate the decreased number of collisions which electrons suffer at higher energies. Also, the drift velocity (in impure argon, for example) l6 may not increase with E. The existence of a plateau is therefore not a sufficient indication, in a pulse chamber, that all electrons are being collected. A source of alpha particles of known energy can be used to release a known amount of ionization in the chamber (Section 1.2.4), say Qo. The pulse height due to the electron motion should then be

Vo = (fQo/C,)G where f is the (average) fraction of total voltage drop through which the electrons move, C, the collector capacity, and G the amplifier gain. A quantitative check system based on this principle is outlined by Bridge and associates.6 A more elaborate method of checking, which does not depend on knowing the capacity of the chamber, has been used by Hazen and collaborators.I6 They provide a polonium source which remains at the potential of the negative electrode but can be moved toward the collecting electrode, reducing the degree of attachment by reducing the path length through which the electrons move. Constant pulse height as the source is moved in is a reliable check in this case. A still more elaborate method, using a pulsed X-ray source and measuring the fraction of total current carried by electrons, is described by Rossi and Staub.” In the design and construction of pulse ionization chambers a reasonable care must be taken to avoid spurious pulses from high-voltage leakage or breakdown or electrical pickup. Any good insulators can be used (in contrast to the current chambers described in Section 1.2.7, which require very high quality insulators) ; in particular, glass-Kovar seals are very useful. In most applications it is possible to provide a grounded conductor (guard electrode) which separates the high-voltage insulator from the collector insulator, thereby greatly reducing the dificulties caused by leakage across the high-voltage insulator. It is sometimes convenient to have the collector at high voltage, and to connect it to the amplifier through a coupling condenser. The condenser must then be selected very carefully. Ceramic condensers seem to be the most satisfactory. The signal obtained from a pulse chamber is often of the order of a millivolt or less. Obviously the collector and the amplifier input must be completely shielded. Ordinarily a double shield (i.e., a grounded case See F. E. Driggers, Phys. Rev. 87, 1080 (1952). B. B. Rossi and H. Staub, “Ionization Chambers and Counters,” p. 58. McGrawHill, New York, 1949. l8

l7

1.2.

IONIZATION CHAMBERS

103

outside the high-voltage electrode) is necessary; occasionally one must take special precautions-such as connecting all grounds together only a t the first tube of the amplifier-to prevent electrical pickup. The use of a short time-constant in the amplifier eliminates most ac and microphonic difficulties, although in low-level work it is often necessary t o operate the preamplifier filaments on dc.

1.2.4. Amount of Ionization liberated The basic process by which a fast charged particle loses energy in a gas have been discussed in Chapter 1.1,* where it is pointed out th a t the total energy loss of a particle of charge Z and velocity @ is, within certain limitations, just proportional to Z 2 times a function of @.Experimentally it is found that a given energy loss gives rise t o a number of ion pairs that depends on the gas, but is approximately (to at worst 10%) independent of the nature and speed of the particle. One can understand in a qualitative way why this should be so: the primary energy-loss event results either in excitation of the gas molecule, or in ionization. I n the latter case the electron may be ejected with considerable energy, but if so it will itself undergo further excitation or ionization collisions, 80 that the energy ultimately either goes into ionization or into excitation (whence it is dissipated in collisions or escapes as radiation.I8 The partition of energy between ionization and excitation depends mainly on the behavior of rather slow electrons even though the primary particle may be of very high energy. The energy loss corresponding to the formation of one ion pair W proves t o be a few times the ionization potential. T he constancy of W means th at the energy of a particle which stops in an ionization chamber can be measured by measuring the quantity of ionization released, or more generally the energy lost in the chamber by particles passing through is directly proportional to the ionization. This is an important and much-used property of the ionization chamber, and for quantitative work i t is clearly necessary to have accurate empirical data ~ ' provided a large amount on W . The work of Jesse and S a d a u k i ~ ' ~ -has of information on W ;it extends previous work, and where a cross-check

* See also Vol. 4, A, Parts 1 and 4. Xenon and, to a lesser extent, krypton and argon give off a considerable fraction of this energy as "scintillation" light in the visible and ultraviolet region. See R. A. Nobles, Rev. Sci. Znstr. 27, 280 (1956);C. Figgler and C. M. Huddleston, Phys. Rev. 96, 600 (1954);A. Sayres and C. S. Wu, Rev. Sci. Instr. 28, 758 (1957).Such noble gas scintillations are discussed in Chapter 1.4. 18 W. P. Jesse and J. Sadaukis, Phvs. Rev. 97, 1668 (1955);100, 1755 (1955);W.P. Jesse, H. Forstat, and J. Sadaukis, ibid. 77,782 (1950). 20 W.P. Jesse and J. Sadaukis, Phys. Rev. 102, 389 (1956). 21 W.P. Jesse and J. Sadaukis, Phys. Rev. 107, 766 (1957). 16

104

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PARTICLE DETECTION

is possible it agrees with other contemporary data; the relative accuracy between different gases is a few tenths of a per cent, and the absolute accuracy about 1%. Their principal results are summarized in Table 11, TABLE 11. Average Energy, in ev, to Make an Ion Pair W for Beta Particles, Po210Alpha Particles, and Low-Energy Alpha Particles, in Pure Gases* Gas He Ne Ar Kr Xe Hz

Air Nz 0 2

COZ CaHi CzHe CHI CzHz

Mean W,g for beta particles

W , for PoZlO alpha particles

W , for alpha particles of 1.2 Mev

42.3 36.6 (26.4) 24.2 22.0 36.3 34.0 35.0 30.9 32.9 26.2 24.8 27.3 25.9

42.7 36.8 26.4 24.1 21.9 36.3 35.5 36.6 32.5 34.5 28.0 26.6 29.2 27.5

42.4 37.4 (26.4) 24.1

37.1 38.1 36.3 29.8 28.5 31 .O 29.0

W. P. Jesse and J. Sadaukis, Phys. Rev. 102, 389 (1956); 107, 766 (1957).

where values of W are listed as the energy loss in electron volts corresponding to one ion pair. The table is actually based on the assumption that in argon W is a true constant for different particles; this assumption is strongly supported in reference 21. The constancy of W in argon for alpha particles of varying velocities has been checkedlS for a range of 1 to 9 MeV. There is evidence from a study of Po210 recoil nuclei20 that these extremely slow and heavy particles have a W ,in argon, about four times that of alpha particles. However, fission fragments already exhibit “normal” behaviorzz-they have a W of 36 ev in air, and presumably would show the standard W in argon. Bakker and SegrtP found W for 340-Mev protons to be 35.3 for Hz and 33.6 Nz,3% lower than the W,gvalues of Table 11. BarberZ4studied the specificionization of high-energy electrons (1 to 34 Mev). By assuming the validity of the theoretical energy-loss expression he found, for elec22 D. H. Wilkinson, “Ionization Chambers and Counters,” p. 21. Cambridge Univ. Press, London and New York, 1950. [See D. West, Can. J. Research A26, 115 (1948); N. D. Lassen, Phys. Rev. 70, 577 (1946).] ** C. J. Bakker and E. Segr6, Phys. Rev. 81, 489 (1951). 24 W. C. Barber, Phys. Rev. 97, 1071 (1955).

1.2.

IONIZATION CHAMBERS

105

trons of -2 MeV, W values of 37.8 0.7 in Hz, 44.5 t- 0.9 in He, and 34.8 5-0.9 in Nz. At higher energies the Nz value remains constant but the Hz value increased by about 3% a t 34 MeV, presumably because Cerenkov radiation begins to carry off some of the energy. The general conclusion to be drawn is that in pure noble gases, and in hydrogen, W is independent of the energy and nature of the particle, to about 1%, over a wide range. I n air and other complex gases one can expect a lessened ionization efficiency a t high specific ionizations, i.e., an increase in W which can be of the order of 10%. However, it has been found’g that in the noble gases of very high ionization potential, notably helium, minute admixtures of impurity reduce the value of W very decidedly: for example, 100 parts per million of Ar in He has a 25% effect. Presumably this is caused by ionizing collisions of metastable He atoms with argon atoms (the first excited state of He is higher than the ionization potential of Ar). Argon should be free of this difficulty, since it has a much lower ionization potential. Statistical fluctuations in the actual number of N of ions formed in a chamber around the average number of fl = A E / W (where energy AE is lost in the chamber) present two quite different problems. If the particle loses all its energy, fluctuations in the rate of energy loss will be correlated, since the total energy is fixed, and the standard deviation in N will be it has been calculated to be about two-thirds of this.25 less than 4%; But if the energy lost in the chamber, AE, is only a small fraction of E , the fluctuations in AE will depend on the number of primary collisions N , and on the energy given to the delta ray in each collision; the standard deviation will be considerably greater than The problem is more characteristic of proportional counters than of ionization chambers, since the former must be used when AE is small; discussion is therefore postponed to Section 1.3.1.

dz.

1.2.5. Noise: Practical Limit of Energy Loss Measurable

The smallest charge which can be detected as a pulse in an ionization chamber is limited by the intrinsic noise of the first stage of the amplifier. Amplifier noise as it affects the sensitivity of ionization chambers is discussed by ElmoreZ* and Gille~pie.~’ * Depending on conditions (fast or slow rise, large or small chamber capacity), different sources of noise may become the predominant source of noise charge. The optimum signal-to-

* See also Vol. 2, Chapter 12.5. U. Fano, Phys. Rev. 72, 26 (1947). W. C. Elmore, Nucleonics 2, (3), 16 (1948). 27 A. B. Gillespie, “Signal, Noise, and Resolution in Nuclear Counter Amplifiers.” Pergamon, New York, 1953. 26

26

106

1.

PARTICLE DETECTION

noise situation proves to be one in which rise time is comparable with pulse duration. Elmore shows that in a typical short-pulse-duration chamber, if C,,, is the chamber capacity in micromicrofarads, R, the “equivalent shot noise resistance” in ohms, and T the pulse duration in microseconds, the most probable noise charge in electron charges is Q/e

-

CPpfdR./7

for the best signal-to-noise ratio. The equivalent resistance R, is defined so that 4kTR, = 2eI,/gm2 where k is the Boltzmann constant, and I , and gm are the plate current and transconductance of the first tube of the amplifier. A typical value for R, is 10000, for 7 , 10 psec, and for C , 30 ppf, so that Q / e 300 ion pairs equivalent noise. This would correspond to about 9000 ev of energy loss in the chamber. Of course fluctuations of three or four times the most probable noise occur frequently, and the minimum charge which can be detected reliably corresponds to about 5 times this, or an energy loss of 50,000 ev. This is somewhat better than is usually achieved in practice, although Wilkinson28cites some experience indicating that it may be attainable. Improvement by further lengthening T is not very effective, even if speed of response can be sacrificed, since grid resistor noise, independent of T , becomes important. For T not restricted, Elmore finds for the optimum case the most probable noise is

-

Q / e = 735[C,,t(R,/R,)11’2 where R, is the effective grid resistance (the grid resistor in parallel with the equivalent noise resistance of grid current). For a high gm tube such as the 6AK5, R, is limited by grid current, so one is limited to a threshold sensitivity which proves to be two or three times better than that of the short-pulse limit. The pulse duration and rise time, for a typical case, might be 50 Msec. The selection of amplifier rise time and clipping time depends not only, or even principally, on the noise problem, but on the particular application in hand-the speed of counting, necessity for accurate timing, shape of ionization chamber pulses, quantitative preservation of pulse height, etc. If accurate reproduction of pulse shape is not important, signal-to-noise and pulse-height reproduction can both be improved by using a rise time and clipping time which are equal, and somewhat longer than the rise time of the slowest chamber pulse. For a detailed discussion of several specific cases, see Wilkinson,’ Chapter 4. x8 D. H. Willcinson, “Ionization Chambers and Counters,” p. 142. Cambridge Univ. Press, London and New York, 1950.

1.2. IONIZATION

CHAMBERS

107

1.2.6. Some Types of Pulse ionization Chambers 1.2.6.1. Alpha-Particle Chambers. The typical energy release of a radioactive or induced alpha emission is several MeV, so it is clear from the foregoing that the ionization chamber is very well suited to the detection of these particles, or measurement of their energy. Typical ranges of these alpha particles in air at one atmosphere are a few centimeters, so that it is easy to contain the entire path of the particle in the chamber. Absolute counting of alpha particles is usually accomplished by placing a thin deposit containing the alpha-active material on the negative electrode of the chamber. Ideally this constitutes a ‘ I 27r” counter (assuming the electrode is plane), so that the fractionf of all disintegrations counted is +. However, the finite thickness t of the deposit will introduce a correction; the fraction which escapes can easily be shown to be $(l - t/2R), where R is the range of the particle in the material; since there will be which is the least energy an escaping alpha can have some energy Emin and still be counted, because a nonlinear discriminator of some sort must be set to reject the unwanted small pulses due to noise, etc., the range is the range of a particle of energy Emin, and the fraction reduced by R(Emin), becomes f = +{ 1 - t/2[R - R(EmiJ]}. A second correction is required to take into account the backscattering of alpha particles which start into the material in a direction away from the gas, but are deflected into the counting volume by multiple scattering. Rossi and S t a ~ give b ~ ~ numerical results of a calculation of this effect for various materials and energies. Typical values would be, for an alpha particle of 3.68 cm range, an increase in f of 8 % for gold or 2 % for aluminum. Measurement of alpha-particle energy by means of ionization chambers is considered in Section 2.2.1.2. 1.2.6.2. Proton Recoil Detectors. Neutrons can be studied by observing the ionization of recoil protons from n - p collisions in the counter gas, if it is hydrogenous, or in a hydrogenous foil or lining of the chamber. A proton recoil at an angle 0 with respect to the direction of a neutron of energy EOwill have an energy of EOcos2 e (nonrelativistic). I n the energy range from a few hundred kilovolts to a few Mev the recoils can be stopped in the gas (a 5-Mev proton has a range of 34 cm in air at 1 atmosphere) and the recoil chamber can be used as a measure of the energy and absolute flux of neutrons; this application is discussed in Section 2.2.2.1. As a method for counting neutrons on a relative basis the recoil ioniza29 B. B. Rossi and H. Staub, ‘‘Ionization Chambers and Counters,” p. 127 McGrawHill, New York, 1949.

108

1.

PARTICLE DETECTION

tion chamber is usually less satisfactory than the proportional counter (Section 1.3.1) because the continuous energy distribution of the protons leads to many undetectable recoils. A large number of specific designs of proton recoil detectors are discussed in references 3 and 4. More recent work is discussed by Johnson and Trai130 and by Berenson and S h ~ r m a n . 3 ~ 1.2.6.3. Boron Trifluoride Chambers and Fission Chambers. Neutron: of all energies can be detected by the energy released in an ionization chamber by a nuclear disintegration; it is sometimes convenient to detect high-energy protons this way also. For neutrons the most widely used reaction is Bl0(n,a) Li7, which releases 2.34 Mev for thermal neutrons, and of course more for fast neutrons. The cross section for this reaction is very large, so that even though natural boron contains only 19% BO ' it leads to relatively high efficiency counters. Among the gaseous compounds of boron BFt is the most stable and satisfactory. It is reasonably free from electron attachment when pure, but the commercial gas often contains impurities which are difficult to remove and which lead to attachment. A method of preparing the pure gas is described by Graves and Froman. l3 Boron enriched in Bl0is available from suppliers of stable isotopes. BO ' has a cross section for slow neutrons which varies as l / v up to energies in the kilovolt range. Thus it lends itself to absolute measurements of neutron density n (rather than flux nv) in the low-energy range, since the disintegration rate is proportional to nvu, and therefore, to n, since u l/v. An ionization chamber filled with BF3provides a very stable method for a relative measurement of neutron flux, since it can be arranged so that the majority of the pulses are the same height (by using cylindrical geometry or a gridded chamber) and the pulse height is essentially independent of the applied voltage. However, the pulses are small and for most applications a proportional counter (Section 1.3.1) is more convenient. For measuring an integrated flux or time-average flux a current chamber (Section 1.2.7) filled with BF3 is very satisfactory. Another useful reaction for neutron measurement is the fission of heavy nuclei. Fission releases nearly 200 MeV; the fragments have a range of about 2 cm in air, and ionize most heavily at the start of their range. Very thin deposits of fissionable material are necessary to obtain the full energy of the fragment, but the energy release is so large that for counting this is not very critical. UZ3& has a large slow-neutron fission cross section, and competes in

-

30 31

C . H. Johnson and C. C. Trail, Rev. Sci. fnstr. 27, 468 (1956). R. E. Berenson and M. B. Shurman, Rev. Sci. Instr. 20, 1 (1958).

1.2.

IONIZATION CHAMBERS

109

efficiency with boron. U233and Pu239 are also slow-neutron fissionable, but have higher alpha activities. However, none of these substances is generally available. At higher neutron energies several heavy elements show convenient fission “thresholds”; some of these are listed in Table 111. Above bismuth TABLE 111. Approximate Thresholds of Various Heavy Substances for Fission by Neutronsn Nucleus

Threshold

NpZ3’

0.4Mev 0.5 Mev 1.1 Mev 1.3 Mev 60 Mev

Pa232 U238

Th234 Bi2o9

*From B. T. Feld, in. “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 2, p. 347. Wiley, New York, 1953.

all of these elements are alpha-active. Rapid electron collection and a short resolving time are necessary to prevent “pile up”-imitation of a large fission pulse by two or more alpha pulses.

1.2.7. Current Ionization Chambers and Integrating Chambers* Historically the earliest use of ionization chambers was not as singleparticle detectors but as meters for the average rate of ionization occurring in a gas, and such current-or charge-meters still have very wide utility; for example, in detecting and measuring radioactivity, in radiological health measurements, in cosmic-ray intensity studies, and in beammonitoring at particle accelerators. In general the problems of current chambers are quite different from those of pulse chambers and they will not be discussed here in The current t o be measured is usually very small and the use of good insulators is essential. Polystyrene, amber, quartz, and Teflon are satisfactory. Guard electrodes to prevent a direct leakage path between high-voltage electrode and collector (Section 1.2.3.) are essential in this application. Care should be taken th a t ionization cannot collect on an insulating surface. The second principal problem of current chambers is recombination. Since the recombination rate of ions is proportional to the square of the ionization density, recombination affects the linearity as well as the

* See also Vol. 4,A, Section 2.1.5. For more information on current chambers see D. H. Wilkinson, “Ionization Chambers and Counters,” Chapter 5. Cambridge Univ. Press, London and New York, 1950. 32

110

1.

PARTICLE DETECTION

absolute accuracy of ionization chambers. I n many applications reproducibility and accuracy are the essential qualities demanded of the current chamber. A pure, nonattaching gas is essential unless the E / p values are everywhere high. A small amount of attachment, which may be quite acceptable in a pulse chamber, will often lead to recombination and therefore to nonquantitative, nonlinear effects in a current chamber. Recombination is particularly serious in chambers used with pulsed accelerators, since the ion density is high during the pulse even though the average ion current may be low. A well-designed ionization chamber can have a time-constant of weeks, and its accuracy is usually limited by the precision of measurement of current or amount of charge. The transient response of a current chamber is of course quite different depending on whether or not attachment is taking place. I n a free-electron chamber the part of the current carried by electrons will have a response in the microsecond region. Rossi and S t a ~ describe b ~ ~ a chamber which was used to detect changes in gamma-ray flux occurring in less than 1 psec.

1.3. Gas-Filled Counters* 1.3.1. Gas Multiplication; Proportional Counterst

I n the discussion of the behavior of electrons in gases (Section 1.2.1) it has been assumed that the electrons released by the initial ionizing particles do not create further ionization after they have been slowed down to the mean agitation energy. They continually gain energy from the electric field, at, an average rate weE,where w is the drift velocity, but it has been assumed that they lost this energy in elastic collisions or excitation collisions with gas molecules. If the field strength is sufficiently great, however, some electrons will acquire an agitation energy greater than the ionization‘potential of the gas molecules, and new ionization will be formed. If-the average rate of ionizing collisions is a per centimeter, the average number of electrons after I centimeters, per original electron, would be n = eaz. This effect is called gas mu2tipZication.f By its use the total amount of ionization from a given initial act can be increased, thereby increasing the available signal. It is, of course, an undesirable 33 B. B. Rossi and H. Staub, “Ionization Chambers and Counters,” p. 106. McGrawHill, New York, 1949. t See also Vol. 4, A, Section 2.1.4. $ See also Vol. 2, Chapter 4.1.

* Chapter

1.3 is by Robert W. Williams.

1.3. GAS-FILLED

COUNTERS

111

effect in an ionization chamber, where its presence destroys the quantitative relationship between the energy lost in the chamber and the amount of ionization released. A counter which is designed to use gas multiplication as a n amplifying device is called a proportional counter. It will normally have a very smallradius positive electrode, so that the region of high field and therefore of gas multiplication will be confined to a small volume near the electrode. The most frequently used geometry is cylindrical, with concentric electrodes of radii a and b, and with a << b. We will discuss only this case; others which are found useful, such as a small loop of wire inside a sphere, or a grid of parallel wires inside a disk, are fairly obvious extensions. Since the region of multiplication is confined to a small volume near the central (positive) electrode, nearly all the electrons released by the particle to be detected will traverse the entire region of multiplication and therefore will have an average multiplication, which we designate by M , and which is independent of the part of the chamber in which the ionization was released. The signal will therefore be proportional to the initial ionization, provided that the space-charge near the wire remains small enough that the field distribution is substantially unaltered (otherwise the multiplication probability will be reduced). The space-charge limitation means that the largest usable M , in a given counter, will depend on the amount of ionization released by the particle traversing the counter, and will be larger for a small initial ionization. An actual calculation of gas multiplication is not very useful, since it depends on cross sections which are not well known, and may be impuritysensitive. The most practical scheme for designing counters is to extrapolate from empirical data with the help of scaling laws. Assuming that all multiplication occurs near the central electrode, and that photoelectric processes can be ignored, one can show’ that the multiplication M , a t voltage V Oand pressure P , will have the functional form =f

(&,Pa).

(1.3.1)

Multiplication depends sensitively on the nature of the gas, and it is clear (Section 1.2.1) that pure simple gases, especially noble gases, will have a relatively low threshold for multiplication, because energy-loss mechanisms are inefficient. However, there are two disadvantages to pure noble gases which often makes it advisable to go to a mixture or a complex gas: the multiplication is a very sensitive function of voltage; and the 1 B. B. Rossi and H. Staub, “Ionization Chambers and Counters,” pp. 78-86. McGrsw-Hill, New York, 1949. Reprinted by permission of the U.S. Atomic Energy Commission.

112

1.

PARTICLE DETECTION

energetic photons from excited atoms may eject photoelectrons from the cathode (or from other molecules) which will start new electron “avalanches.” One thus approaches the “breakdown” threshold, which is reached when the expected number of new avalanches per avalanche is 1000

too M

10

200

600

1000

1400

1800

2200

2600 3000

FIG. 1. Gas multiplication M versus voltage, for a counter with wire diameter 2a = 0.010 in., cylinder diameter 2b = 0.87 in. (A) Tank hydrogen, 99.97% pure, with pressures of 10 (A,) and 55 (Ae) cm Hg. (B) Methane, 85% pure, with pressures of 10 (B1) and 40 (BZ) cm Hg. (C) Tank argon, 99.6% pure, with pressures of 10 (Ct) and 40 (C,) cm Hg. (D) Mixture of 90% hydrogen and 10% methane, with pressures of 10 (D,) and 40 (Dz) cm Hg. (From Rossi and Staub,‘ by permission.)

greater than one. Breakdown counters (Geiger counters, etc.) are considered in Section 1.3.2. Data from the Los Alamos studies on proportional counters’ are reproduced in Figs. 1 to 6, where experimentally observed values of multiplica-

1.3.

GAS-FILLED

COUNTERS

113

tion M are plotted against voltage. These curves together with Eq. (1.3.1) will facilitate the design of proportional counters of various shapes. The time response of a proportional counter can be understood by reference to the ionization-chamber equation, Eq. (1.2.4) and its accompanying discussion. The mean free path of electrons in gases at one atmos-

FIQ.2. Tank argon, 99.6% pure. Wire diameter 0.001 in.; cylinder diameter 1.56 in. Gas multiplication M versus voltage for a pressure of 6.8 atmos. (From Rossi and Staub,' by permission.)

phere is of the order of cm, and the gas multiplication becomes important only in the last few mean free paths, so that the extra electrons of the avalanche are liberated very close to the central wire, usually within considerably less than one radius. The fraction of potential through which they fall is

where 6 is the distance from the ionizing collisions to the wire. The motion of the electrons toward the wire therefore causes only a small fraction of the total pulse. The bulk of the pulse begins t o rise very steeply as the

114

1.

PARTICLE DETECTION

VOLTS

FIG. 3. Spectroscopic nitrogen. Wire diameter 2a = 0.001 in.; cylinder diameter 1.56 in. Gas multiplication M versus voltage for pressures from 0.79 to 4.25 atmos. (From Rossi and Staub,' by permission.)

VOLTS

FIG.4. Boron trifluoride. Gas multiplication M versus voltage. (A) Wire diameter 2a = 0.010 in.; cylinder diameter 2a = 1.50 in.; pressure p = 10 cm Hg. (3)Wire diameter 2a = 0.001 in.; cylinder diameter, 1.56 in.; pressure p = 80.4 cm Hg. (From Rossi and Staub,' by permission.)

1.3.

115

GAS-FILLED COUNTERS

positive ions which are formed near the wire start to move out of the very strong field there. Figure 7 illustrates the pulse schematically: the period of delay, which may be from 0.1 psec a t low pressure to 1 psec or more for a high-pressure counter, the steep rise as multiplication frees the positive ions, the gradual lessening of ion drift velocities as they move outward. A sharply differentiated proportional counter signal will give a very 1000-,

--

I

[

I

I

,

I

I

I

-4

-

-

-

300

600

900

1200

IS00

VOLTS

FIG.5 . Mixture of 98% argon and 2 % Con. Wire diameter 2a = 0.010 in.; cylinder diameter 2b = 0.87 in. Gas multiplication M versus voltage for pressures of 10 and 40 cm Hg. (From Rossi and Staub,’ by permission.)

short pulse (perhaps 0.2 psec) but in coincidence work the effect of the delay may be more important than pulse length. The usefulness of the proportional counter as a quantitative instrument depends on the observed behavior of the multiplication-its linearity, constancy throughout the chamber, fluctuations, and reliability. According to the Los Alamos studies’ the “proportional” action for different parts of the counter is excellent provided there is negligible attachment. E / p is usually quite high in a proportional counter, so this difficulty is only serious in a gas like BF3, which often has impurities showing some attachment. However, even BF, counters may be satisfactory from this

1. PARTICLE

116 ‘Oo0

DETECTION

!

E

3

1700 2100 2500 2900 3300 3700

4100 4500

VOLTS FIQ.6. Mixture of 90% argon and 10% COi. Wire diameter 2a = 0.005 in.; cylinder diameter 2b = 1.56 in. Gas multiplication M versus voltage for pressures of 1.13, 2.15, and 3.5 atmos. (From Rossi and Staub,’ by permission.)

t FIG.7. Schematic representation of a proportional-counter pulse. In practice the slow portion of the pulse would usually be eliminated by an RC or other low-frequency rejec,tionnetwork.

1.3.

GAS-FILLED

COUNTERS

117

point of view, when only neutron detection and not the energy of the disintegration is of interest. The linearity of response is limited by space-charge considerations and therefore by total ionization. For a proton or alpha-particle a gas multiplication of 100 is usually safe, while for a beta ray the multiplication can be made correspondingly higher. Thus Curran et aL2 have used proportional counters with multiplication as high as lo4 to measure very weak beta rays down to a few hundred electron volts, and even t o detect single photoelectrons from ultraviolet light. The statistics of the multiplication process have been worked out by Snyder3for a simplified model. The result is, for large gas multiplication, that the standard deviation is larger by a factor of d2 than th a t expected from the statistics of the initial ionization process if the latter is Poisson. Th a t is, if N is the average initial ionization caused b y the primary particle, the standard deviation in the relative output, AV/V, would be expected to be d T N for an ionization chamber, but is

AV/V = d 2 / N

(1.3.2)

for a proportional counter. Experimentally2the fluctuations seem to be a little bit smaller than this. Such fluctuations are, therefore, sufficiently small that the proportional counter can be used as a reliable instrument for obtaining information on individual events. The fluctuations in initial ionization have here been assumed t o be Poisson, that is, equal to dw. However, they can be much larger (because of knock-on processes) for a fast particle which does not lose, on the average, a large amount of energy in the counter. Such “Landau” fluctuations are thoroughly treated by Rossi4* The general reliability and stability of proportional counters which contain a t least some polyatomic gas is very good. Obviously the high-voltage supply must be well stabilized and the gas composition must remain constant; under these conditions they are very satisfactory. The factor of 100 or so in pulse height, over the ionization chamber, brings the signal from a heavily ionizing particle u p t o a level th a t is very convenient electronically, and brings the signal from a relativistic particle into the detectable range. Many of the functions of proportional counters can be performed more efficiently or a t higher resolution in time with scintillation counters (Section 1.4.1.9) ; however, proportional counters require less *Also refer to Section 1.1.2.6 of this volume.

1s.C. Curran, J. Angus, and A. L. Cockroft, Phil. Mag. [7] 40, 36, 53, 929 (1949). SH. 9. Snyder, Phys. Rev. 72, 181(A) (1947). B. B. Rossi, “High-Energy Particles,” p. 29. Prentice-Hall, New York, 1952.

118

1.

PARTICLE DETECTION

total energy loss, and are more nearly linear when used with heavily ionizing particles. Proportional counters are available from some commercial radioactivity-equipment suppliers, with BF, and other fillings, in various cylindrical sizes and as thin-window counters.

1.3.2. Geiger Counters and Other Breakdown Counters In the preceding section we considered the possibility that a sufficiently large electron-initiated avalanche in a counter with a high-field region might produce, in addition to the n electrons of the avalanche, a photoelectron at the cathode or in the low-field region of the counter. If the probability of doing this is y per avalanche electron, the chance of starting a second avalanche is y n , and the expectation value of the total number of electrons produced from one original electron by the series of avalanches is

M

=n

+ y n 2 + y2n3+ . . .

which sums to M = n/(l - yn) if y n is less than 1; but which diverges if yn is greater than 1. The latter is the phenomenon of breakdown, and leads to a counter which responds to each electron introduced into it with a signal which is limited only by some other mechanism. The Geiger counter is a breakdown counter in which the discharge is eventually stopped because the positive-ion space charge that accumulates around the anode modifies the field to the point where yn < 1. As the electrons are collected, leaving the slowly moving positive ions in place, the potential of the anode is nearly unchanged (see preceding section), the charge on the anode is reduced, and one has instead a positive spacecharge “sheath” around the anode. The field is thus smoothed out, becoming weaker near the wire, and gas multiplication is reduced. A simple-gas counter rekindles itself , however, and will exhibit a relaxation phenomenon indefinitely unless electrically “quenched.” The source of the new discharge, which takes place after the positive-ion ‘‘ sheath” of space charge has moved away from the anode and the field is partially restored, has been shown to be the ejection of electrons from the cathode by the positive ions as they collide with it. It is easy to make a counter which does not have this defect (and therefore does not require quenching), and nearly all counters now used are made this way. One adds to the filling gas, normally argon, a polyatomic gas or vapor such as alcohol, ethylene, or petroleum ether. The ionization potential of the polyatomic molecule is less than that of argon, and the Ar+ ions transfer their charge to the complex molecules. If the molecular gas represents about 10% of the total pressure, the probability of charge transfer is essentially unity.

1.3.

GAS-FILLED COUNTERS

119

The energy available in neutralizing the complex ion at the cathode is now dissipated in molecular dissociation, with the result that the probability of releasing an electron from the cathode proves to be negligible. Of course this means that the complex gas is consumed as the counter operates, and the counter has a finite life. The counter with a complex vapor added is called self-quenching, and we restrict ourselves to this type in the general discussion that follows. A very common type of counter is cylindrical, about 1 in. in diameter, with a 0.001-in. diameter central wire and a filling of argon (9 cm Hg) and alcohol (1 cm Hg). Such counters typically have an operating region of about 1000-1200 volts over which the size of all pulses, for a given voltage, is the same independent of initial ionization, and over which the counting rate rises by 3 to 5% (because of increase of effective counting volume and increase of spurious pulses, presumably) ; this is the so-called plateau. I n the self-quenching counter the photoelectrons which maintain the discharge are released from the vapor molecules and not from the cathode -the vapor is fairly opaque to the ultraviolet photons in question, and the photon mean free path is small compared t o the counter radius. The discharge therefore spreads down the central wire with a speed characteristic of the mean free path of a photon divided by the time necessary for an avalanche to develop. This is fairly slow, about 10 cm per microsecond. As with the proportional counter, the pulse is mainly due to positive-ion motion, but the important part of that motion takes place more quickly than the discharge-spreading time, so that the observed pulse rise is more or less linear for the first microsecond or two (depending, obviously, on the length of the counter and on whether the discharge starts in the middle or at one end). A 12-in. counter with a low-capacity load, operated 50 volts above threshold will give the order of 5 volts in a microsecond or two; the rate of rise falls off quickly from then on. The counter does not recover sufficiently to give another full-sized pulse until 100-150 psec have elapsed. Since about lo9 ions are released per count, all resulting in decomposition of the vapor molecules, the counter’s characteristics change with use, the threshold rises and eventually it becomes unusable, either because of too much tar or not enough alcohol; the useful life usually is between lo8 and lo9 counts. Halogen-filled counters do not suffer from this defect, but they exhibit electron capture, with the obvious concomitant disadvantage of unpredictable time delays. Electronically quenched simple-gas counters are of course a possibility; but if one is going to that much trouble it may be better to use high-multiplication proportional counters. The principal advantages of the Geiger counter are the simplicity of its associated equipment (because of its large pulse), and its high effi-

120

1.

PARTICLE DETECTION

ciency. The efficiency for a minimum-ionizing charged particle passing through the gas, averaged over the area, is more than 99%; ordinarily the inactive volume represented by the finite thickness of the walls is a more serious effect. The efficiency of a brass-walled Geiger counter for gamma rays is roughly, in per cent, equal to the energy of the gamma ray in MeV, in the vicinity of 1 MeV. Wilkinson6 gives a large amount of information on Geiger counters, and discusses some examples of nonstandard counters (odd shapes, etc.). Although the Geiger counter is the only breakdown counter presently in wide use, it should be mentioned that uniform-field breakdown counters have been successfully operated.6 A novel breakdown counter of quite different sort is described by Conversi and Gozzini.7 They have found that glass tubes filled with neon, when placed in a sufficiently strong electric field, will break down and emit a flash of light when ionization is released in them. Applications appear to be restricted t o tracing the paths of high-energy particles.

1.4. Scintillation Counters and Luminescent Chambers* 1.4.1. Scintillation Counters 1.4.1 .l. Infroduction. The field of scintillation counting has developed mainly in the past twelve years, and has for some eight years been one of the principal techniques for particle detection. By direct or secondary processes it is now possible to detect every known type of elementary particle by this means. The instrumentation has developed from a 1 cmS crystal mounted on a 931-A photomultiplier, to the use of large tanks,? crystals or plastics as indicated in Fig. 1, coupled with photomultipliers with cathodes 2, 5, or 16 in. in some cases.$ Scintillators exist in several forms: crystals, liquids, plastic solids, and gases. I n all cases, the phenomenon depends on the fact that the suitable ‘ ‘ f l ~ o r sgive ’ ~ off pulses of light when a charged particle passes through D. H. Wilkinson, “Ionization Chambers and Counters,” Chapter 7. Cambridge Univ. Press, London and New York, 1950. J. W. Keuffel, Rev. Sci. Znstr. 20, 202 (1949). M.Conversi and A. Gozzini, Nuovo cimento [lo] 2, 189 (1955). t Refer to Section 1.4.1.10 of this volume. 3 See also Vol. 2, Section 11.1.3.

__

* All of Chapter 1.4 is by George T. Reynolds, except Section 1.4.1.10 which is by F. Reines.

1.4.

SCINTILLATION

COUNTERS

121

them.* This light is directed t o a photomultiplier cathode where it ejects electrons that are then accelerated and multiplied in the dynode structure of the tube. The charge finally collected a t the anode is recorded by suitable circuits. Among the materials found suitable in crystal form are inorganic and organic substances (see Table I). The liquids and plastics are, so far, organic. There are several important distinctions between the characteristics of organic and inorganic scintillators, including life-

FIG.1. Two examples of large tank scintillators. Tanks of the general design features have been made up to four feet in dimension.

times, linearity of energy response, temperature effects, etc., but the basic difference appears to be due to the fact that the process by means of which light is emitted from an inorganic crystal is due primarily to the crystal structure, whereas organic substances exhibit luminescence by virtue of molecular properties. 1.4.1.2. Scintillation Process in Organic and Inorganic Substances. A discussion of the scintillation process in organic substances has been given by Brooks.’ Briefly, the phenomena involved are a s follows, with reference to Fig. 2. A charged particle passing through a scintillator

* See also Vol. 6, B, Part 11. IF. D. Brooks, in “Progress in Nuclear Physics” (0. R. Frisch, ed.), Vol. 5, pp. 252-313. Pergamon, New York, 1956.

1.

122

PAltTICLE DETECTION

TOLE I. Scintillator Characteristics Inorganic crystal scintillators Relative Decay time kmSx pulse height (sea) (A)

Scintillator Sodium iodide (thallium activated) Cesium iodide (thallium activated) Cesium iodide (cooled to 77°K) Cadmium tungstate Lithium iodide (europium activated)

Scintillator Anthracene p,p'-Quaterphenyl trans-Stilbene Diphenylacetylene Terphenyl

Solvent Toluene

p-Xylene PCH

2.1

0.25 X

0.6

lo-'

Density (@;m/cc)

4100

3.7

651

1.5 X 10-6

4500

4.5

62 1

c

4500

4.5

621

6 X 2 X 10-6

5300 4400

7.8 4.1

1325 446

-5.0 0.2 0.7

Organic crystal scintillators Amax Density Relative Decay time pulse height (mrsec) (A) (gm/cc) 1.00 0.85 0.60 0.45 0.40

Primary solute (gm/l)

32 7 6.4 5.4 5.0

4400 4200 4100 3900 4000

1.25 1.16 1.18 1.23

mp ("C)

a//3

217 318 124 62.5 213

0.1

Liquid scintillators Relative Secondary pulse Decay solute height to time in Am,, Density (gm/l) anthracene mpsec (A) (gm/cc) a/@

Diphenylanthracene (10) BBO (1) POPOP (1.5) PRD (8) a-NPO (3) DEAMC (1.2) PPO (4) P-TP (9)

PBD (10) PBD (10) P-TP (5) P-TP (3) PPO (12)

mp (OC)

POPOP (0. I) a-NPO (0.I) DPH (0.1) BPOP (0.2) DPH (0.01)

0.85 0.70 0.67 0.63 0.63 0.59 0.54 0.48 0.66 0.59 0.51 0.77 0.68 0.50 0.48 0.33

0.87 0.09 <2.8

-

<3.0 -2.2

<3.0

8.0

4300 3700 4200 3800 3550

1.4.

123

SCINTILLATION COUNTERS

TABLE I. Scintillator Characteristics (Continued) ~

Solvent

~~

Primary solute (gm/l)

Plastic sczntillators Relative Secondary pulse Decay solute height to time in (gm/l) anthracene mpsec

Polystyrene TP (36) TPB (16) PVT TP (30)

0.28 0.38 0.33 0.44 0.47 0.50 0.52 0.45 0.52 0.54

CY-NPO(1) POPOP (1) BBO (1) DPS (1) PBD (30) (20) (20) BBO (10)

BBO (1) DPS

53.0 4.6 4.0

(A)

Density (gm/cc) CY/B

3550 0 . 9 4800

5 3 . 0 -3800

0.10 0.08

0.10

0.44

Explanations: TP: p-terphenyl; TBP: tetraphenylbutadiene; PPO: 2,5 diphenyloxazole; NPO: 2-( l-naphtyl)-5-phenyloxazole; PBD : 2 phenyl 5-(4-biphenylyl)1,3,4 oxadiazole; POPOP: 1,4, bis 2-(5-phenyloxa~olyl)-benzene;BBO: 2,5,-di-(4biphenyly1)-oxazole; DPS: diphenylstilbene.

Be!

W

03

(0)

I

(b)

i

(C)

i

(d)

I

(el

FIG.2. Energy levels in an organic molecule and luminescence processes. (a) Excitation. (b) Internal degradation. (c) and (d) Internal conversion. (e) Fluorescence.

124

1.

PARTICLE DETECTION

looses energy ionizing, exciting, * and perhaps dissociating the molecules in a column near its trajectory, to an extent depending on the rate of energy loss, and therefore the velocity, of the particle. In the case of excitation, the processes are those of section (a) of the figure, where the electronic levels of the molecule are indicated. This excitation can be transferred to other molecules or dissipated in nonradiative processes, such as transitions among vibrational levels (heat). The present understanding of the process is that transitions from higher excited levels to the la eve1 occur rapidly (10-l2 second) so that the situation of section (c) of the figure is that from which the radiative transitions of section (e) or the nonradiative (quenching) transitions of section (d) occur. Since transitions to 10 levels occur quickly, many aspects of the process are independent of the excitation mechanism, and much has been learned from the study of photoluminescence. However, although the molecular quantum efficiencies of some successful scintillators are as high as 90%, the energy conversion factor or scintillation efficiency in charged particle detection is the order of 4% for anthracene to 10% or somewhat more for certain inorganic crystals. A great deal has been learned about the luminescence process in organic scintillators by the study of liquid and solid solutions, and these results have aided in the description of the process in crystals. The mechanism by means of which the energy is transferred from the molecules which are initially affected to those which finally radiate is not comp.etely understood, but it appears that the main process is one in which collisions are responsible for the transfer, either by exciton transfer or by a dipole resonance interaction such as that suggested by Forster2 until the energy is quenched or radiated. It appears that even the molecules of crystals are not in permanent strong attraction, and that here also collisions play an essential role. The detected spectrum in a practical scintillator depends on the emission and absorption spectra of the molecule, as well as the size of the sample. The effect of environment is indicated by the case of anthracene, where the absorption spectrum shifts to longer wavelengths by approximately 250 A as the material is changed from vapor to dilute solution to solid. The closer collisions possible in the liquid state result in increased self-quenching, reducing the quantum efficiency and the lifetime. The role of radiative transfer has been indicated as minor, being most significant in plastic scintillators where it accounts for, a t most, 20 % of the energy transfer.a

* Refer to Chapter 1.1 of this volume. 2 8

T. Forster, Ann. Phusik [6]2, 55 (1948). R. K. Swank and W. L. Buck, Phys. Rev. 91,927 (1953).

1.4.

SCINTILLATION

COUNTERS

125

The nature of the phenomenon in inorganic crystals is discussed by Swank,4 Birks, and Curran. * The successful inorganic scintillators from ionic crystals, and in particular, the alkali halides, notably NaI with thallium impurity, are of interest. In such crystals, charged particles may raise electrons into the conduction bands or into excited levels. The electron and the hole it leaves in the filled band move rapidly throughout the crystal (as an “exciton”) until captured by an imperfection, giving up energy in the form of vibrational transfer, or until captured by an impurity. A suitable impurity will become excited and then radiate as a scintillator. The alkali halides with thallium added as the impurity have high transparency to their own radiation. The best known are listed in Table I. The thallium impurity concentration does not seem to be critical between approximately 0.1 and 0.2%. Other inorganic scintillators, including the alkaline earth tungstates are also listed in the table. Most of these crystals are available commercially. 1.4.1.3. Scintillator Characteristics. 1.4.1.3.1. GENERAL PROPERTIES OF SCINTILLATORS. The most commonly used inorganic crystal for highenergy particle detection is NaI(T1). The organic crystal usually taken as standard is anthracene, although it is more difficult to make this reproducible than is stilbene. Anthracene has been taken as about onehalf as sensitive as NaI(T1) on the basis of pulse height) but the work of Sangster’ indicates that it may be relatively better than this. The comparison of various scintillators in the literature often leads to inconsistencies because of the fact that the spectral output of the scintillator and that of the photomultiplier are not exactly matched. Reflector and container characteristics can markedly influence comparative results. Removal of oxygen from the organic liquid scintillators by bubbling argon or nitrogen, has been found to make a significant increase (approximately 25%) in pulse height in certain cases. A useful quantity in the description of fluorescence phenomena is the quantum efficiency number of quanta emitted = number of quanta absorbed‘ The “practical” q of a scintillation counter may be less than the molecular q due to reabsorption. Since reabsorption depends on crystal size, it is a practical consideration.

* Reference is made again to Vol. 6, B, Part

11.

R. K. Swank, Ann. Rev. Nuclear Sn’. 4, 111-140 (1954). 6 J. B. Birks, “Scintillation Counters.” McGraw-Hill, New York, 1953. 8 S. C. Curran, “Luminescence and the Scintillation Counter.” Academic Press, New York, 1953. R. C. Sangster, J . Chem. Phys. 24, 670 (1956).

1,

126

PARTICLE DETECTION

The scintillation efficiency is defined as energy emitted per fluorescence energy lost by charged particle per fluorescence' ~~

The role of reabsorption can be seen as follows, with reference to Fig. 3. Let E be the energy lost in the scintillator by a charged particle, w the Emission Spectrum

Absorption Spectrum

FIG. 3. Schematic representation of the absorption and emission spectra of a scintillator.

energy necessary to produce a fluorescence excitation, and qo the molecular quantum efficiency. The number of photons in a scintillation is given by

N =

E

-w qo.

(1.4.1)

If the ratio of area b to a in Fig. 4 is denoted by K, then of the N photons emitted, K N escape, (1 - K ) N are reabsorbed, and qO(1 - K)N are reemitted. Thus finally N o = NK[1 - (1 - K)qo]-' escape, so that (1.4.2)

is the practical q. The effect of reabsorption is also to lengthen the lifetime. A typical result for anthracene is q o = 0.94 and q = 0.80. Since direct measurements on anthracene indicate that the energy lost by a charged particle in producing one observable photon is about 65 electron volts (corresponding to q) this result indicates that the number of electron

1.4.

127

SCINTILLATION COUNTERS

volts required to release L photon in the initial encounter is about GO electron volts. The scintillation efficiency is determined as follows. For a photon of wavelength 4400 A, hv = 2.8 volts. Therefore the efficiency is (2.3/60) = 4 or 5%. Measurements on stilbene, terphenyl, diphenylacetylene, and plastic scintillat ons indicate a performance about 0.5 to 0.6 of anthracene and

c 0

E-

MeV/mg /cm2 of Anthrocene

dx

FIG.4. Energy response of an anthracene scintillator.

so about 100 to 120 volts per photon. The best solutions require about 80 volts per photon. Thus, in an application where a minimum ionizing particle passes through a 1 cm thick plastic scintillator, and the light collection efficiency is lo%, the number of photoelectrons from the cathode of a photomultiplier of cathode efficiency 10% is determined to be -lo' X 0.10 X 0.10 120

165.

Calculations similar to this for particular arrangements afford the means for estimating pulse heights and resolutions that can be achieved, according to principles discussed by Morton.8 8

G. A. Morton, RCA Revs. 10, 525 (1949).

128

1.

PARTICLE DETECTION

1.4.1.3.2. ENERGY RESPONSE.The energy response of. organic and inorganic scintillators differs markedly in the degree of linearity. By energy response is meant the light output as a function of energy loss of the particle. The early work of Jentschke and his groupg has been generally confirmed. Although not strictly proportional to energy loss, the light output from inorganic crystals [NaI(Tl) is the standard example] is very nearly proportional to the energy loss down to -1 Mev for protons and -15 Mev for a’s. For p’s, protons, and a’s of the same energy loss a t these low energies the response of sodium iodide is roughly 1 : 1:0 . 6 Recent work in this area is discussed by Brolley and Ribe.’” On the other hand, the organic scintillators are not nearly as proportional. In fact, an early description of this lack of proportionality was given in the so called a l p ratio-the ratio of pulse height (i.e., light output) per unit energy loss for 5 Mev a’s and relativistic p’s. For organic liquids, plastics, and crystals this ratio lies between 0.08 and 0.10. This lack of uniformity has generally been attributed to “damage” suffered by the molecule when large energy transfers, due to high rates of energy loss by the particle, are involved, It can be shown’’ that without specifying any details as to the nature of the damage, the rate of luminescence can be given by -dL_ - A(dE/dx) (1.4.3) dx 1 B(dE/dk)

+

where dE/dx is the energy loss, and A and B are constants of the material related to energy transfer and molecular “damage.” Brooks’ has reviewed more refined attempts to describe this nonuniformity, including those based on bimolecular processes. Results of comparing data with theory are shown in Fig. 4 where curve (a) is based on Eq. (1.4.3) and curve (b) based on a consideration of the role of bimolecular processes. At high values of d E / d x surface effects may be important.12 Because of the way that the measurements are made these might have been the cause of the apparent flattening of the curve in the region of the a points. With this in mind, curve (b) might be favored. There are still several factors that have not been investigated, including the effect of the extent of the ionized column cross section, and dependence on charge multiplicity. For example, curve (b) is consistent with results of fission products compared to a response. Further, the detailed nature of the distribution of C. J. Taylor, W. K. Jentschke, M. E. Remley, F. S. Eby, and P. G. Kruger, Phys Rev. 84, 1034 (1951); also F. S. Eby and W. K. Jentschke, ibid. 96, 911 (1954). l o J. E. Brolley, Jr. and F. L. Ribe, Phys. Rev. 98, 1112 (1955). l 1 J. B. Birks, Proc. Phys. SOC.(London) A64, 874 (1951). 12 E. Mateosian and L. C. L. Yuan, Phys. Rev. 90, 868 (1953).

1.4.

129

SCINTILLATION COUNTERS

the energy transfer (6 rays, etc.) might result in more “damage” locally caused by a’s than protons and therefore effects of a different nature. 1.4.1.3.3. LIFETIMES.The lifetimes of inorganic scintillators are generally longer than those of the organic scintillators. Table I gives some of the lifetimes for the common scintillators. I n the case of the organic substances, the study of lifetimes adds much to the understanding of the processes involved. For example, an impurity that tends to quench will reduce the lifetime; if there is significant optical reabsorption the lifetime will be increased; WrightI3 has shown tha t surface effects can also change the lifetime. Generally, an exponential decay time is indicated by experiment. Since most decay times are measured with photomultipliers followed by suitable circuitry, it is important to keep the usual response time consideration in mind. It can be shown4 th at the pulse-height maximum from such a system is given by (1.4.4) where N is the total number of excitations; q is the quantum efficiency of the cathode; G is the multiplication factor of the photomultiplier; g is the fraction of photons collected; and y = ( R C / r ) , the circuit constant over the decay time of the fluorescence. Thus, if y is large, the pulse height does not depend on r ; otherwise there is a dependence. This has served as a basis for finding long components after initial short components in organic and NaI crystals.* Also, since most organic scintillators have lifetimes of the order of 10-9 to lo-* second, the spread in transit times for the photomultiplier is a n important limitation, sometimes amounting to as much as 2 x second. Recently developed photomultipliers have succeeded in reducing this time spread t o the millimicrosecond region. I4 There are three general methods for measuring lifetimes. 1. I n the notation above, the current from the photomultiplier is given by 1 = NgqGef(t) where f ( t ) is the fraction decaying a t a given time, i.e.,

Z

=

Zof(t)

Thus, the voltage a t the input is ‘v = V,,’(e-’/RC - e - t / 7 )

= (ZO/T)e-?

= V,,’e-t/r[et/RC+t/T

-

Vo’e-t/r

1-

-

11 - 11

MT+RC)I/RCT

(1.4.5)

* On pulse-height determination, see also Vol. 2, Section 9.6.1. G. T. Wright, Proc. Phys. Soc. (London) A68, 241 (1955). Proc. 6th Scintillation Counter Symposium, IRE Trans. on Nuclear Sci. NS-6, No. 3 (1958); E. H. Thorndike and W. J. Shlaer, Rev. Sci. Znstr. 30, 838 (1959). 13 14

130

1.

PARTICLE DETECTION

so that for RC small,

The procedure is to connect the photomultiplier directly to the plates of a scope. If RC is small, the anode voltage is proportional to the anode current so that a measurable exponential is obtained. This method has been used by Post who pulsed the photomultiplier high voltage, and by Swank, using pulsed X-rays. 2. Leibson has determined lifetimes using a pulsed X-ray source, modulated by diffraction from an oscillating quartz crystal (frequency u). Then the light output is

F(t)

=

qJ0

4 1

+

0272

sin(ut -

e)

(1.4.7)

where J Ois the initial intensity and tan e = wr. Thus r can be determined from intensity decrease or phase. 3. A further method utilizes a shorted delay line and diode rectifier giving a waveform of the sort:

Thus, changing the delay line changes the pulse recorded by a following slow amplifier, from which the lifetimes can be deduced. Work on lifetimes originally reported by Wright have been continued a t Harwell, relating to the difference in decay times of certain organic materials depending on whether the excitation was by P’s or a’s. The basic difference seems to lie in the effect of the specific ionization. Work by Owen14indicates that although all the scintillators examined have nearly TABLE 11. Decay-Time Dependence on Excitation Fastest component showing proton/electron difference Phosphor Stilbene Anthracene Quaterphenyl Toluene solution Borate-xylene solution

Fast comp. decay time in mpsec 6.2 33 4.5

12.8 4.2

Decay time Intensity ratios in mpsec Ple

370 370 350 200 200

1.8 2.1 2.1 1.8 1.9

a Ratio of amplitude of these components under proton and electron excitation for pulses of equal peak height. This component is estimated to contain approximately 10% of the total energy for electron excitation.

1.4.

131

SCINTILLATION COUNTERS

the same long time decay component, the relative amount of light output found in the long component depends on the nature of the excitation. This is summarized in Table 11, for results using a PoBe source (4.5-Mev y’s and 12-Mev neutrons, which give approximately equal pulse heights), From these results it appears possible to distinguish between excitations due to a’s, neutrons or protons on the one hand, and y’s or 6’s on the other, by comparing the ratios of peak heights to total charge in the photomultiplier output pulses.

p- Terphenyl

p-p‘ Quaterphenyl

34uo Wavelength

-

4600

5

)O

i;

FIG.5(a). Emission spectra of scintillator solutions in toluene (from Brooks).

I n considering 1.4.1.3.4. SPECIALDISTRIBUTION OF SCINTILLATION. the spectral distribution of the scintillator output one must be concerned with how well the emission is matched to the photomultiplier used. Some typical scintillator emission curves are indicated in Figs. 5(a) and (b). A. Crystals. The details of the emission spectra of crystals depends on their thickness, as mentioned in earlier discussions of reabsorption. The peak responses recorded for organic and inorganic scintillators are given in Table I in the column labeled Am*=. B. Liquids. Here the wavelengths are the order of 200 or 300 8 shorter than crystals (thus note the evidence for interaction effects in the solid state) but considerable success has been achieved by use of wavelength shifters (for example diphenylthexatriene and aNPO) as indicated in

132

1.

PARTICLE DETECTION

Table I. Note that DPH added to terphenyl gives ,A, at 4500 (10 mg/ a t 4100 (20 mg/liter). liter) and aNPO added to terphenyl gives ,A, at C. Plastics. As Table I shows, terphenyl in polystyrene gives ,A, about 3500 b and the addition of TPB as a shifter changes this to about 4500 b. TPB in Polystyrene or polyvinyltoluene has Amax a t about 4500 A.

Wovelength

FIG.5(b). Emission spectra of solid solutions in polystyrene (from Brooks).

These spectra determine the nature of the cathode response required of the photomultiplier. Commercial photomultiplier responses vary considerably, but fairly typical responses are given in Fig. 6.* The spectra also determine the nature of the reflecting surface required, as discussed in the next paragraphs. 1.4.1.4. Liquid Scintillators. Five gm/liter of terphenylI6 in toluene gives a response of the order of 0.45 that of anthracene. Development of other solvents and solutes to which suitable wavelength shifters have been added have improved this response up to nearly 0.8. A good review of the liquid scintillation counter field will be found in the report of the Liquid Scintillation Counter Symposium held a t Northwestern University in August, 1957, to be published by Pergamon Press. *See also Vol. 2, Section 11.1.3.2. l 6 G. T. Reynolds, F. B. Harrison, and G. Salvini, Phys. Rev. 78, 488 (1950).

1.4.

SCINTILLATION

133

COUNTERS

The scintillation process in liquids is more complex than in crystals, but correspondingly more can be learned b y varying the parameters. Using the concepts discussed earlier concerning the nature of the scintillation process in organic crystals we can describe the process in a liquid solution as follows. The solvent initially receives most of the energy given up by the charged particle to be detected. I n the course of collisions, L

1111111111111 3 4 5 6 7 8

.5:

s4

u

Lin

3 4 5 6 7 8 Wave’ength angstroms x s9

3 4 5 6 7 8

3 4 5 6 7 8 Multi-alkali

SII

FIG.6. Spectral response of various photomultiplier cathodes.

some of this energy is transferred to the solute, or fluor of the solution. Evidently the characteristics of the successful solvent must include ability t o keep from quenching the energy for a time long enough for a transfer to the solute. The energy levels of the solute and solvent must be related so that this transfer can occur. Finally, the solute must have a high probability for radiating the energy as light a t a wavelength suitable for detection by standard photomultipliers. The processes involved are indicated in Table 111,in which c is the concentration of solute. I n these TABLE 111. Energy Transfer Processes Description

Process

S

+ hvi+

S*

s*+ s + s* + s+ s + s or S*+ S s* + s+ s + s* hY*

+

S* + F - + F* S F* + F h~a F*+S+F+S or F * + F F* + F + F F S* F - , S F

+

+

+

+

Excitation of solvent Fluorescence of solvent (small)

Reaction probability

kl

Self-quenching of solvent Energy transfer (solvent) Energy transfer to solute Fluorescence of solute

kz

Internal quenching of solute Self-quenching of solute Quenching of solvent by solute (small)

kb Ck6 Ck7

134

1.

PARTICLE DETECTION

terms the quantum efficiency for the solution is of the form (1.4.8)

where

ka

m=k4

+ kti

and the quantum efficiency for the solvent is neglected. In these equations, the k’s are reaction probabilities and have the following significance: k l , fluorescence of the solvent; ckg, energy tranfer to the solute; h, fluorescence of the solute; k6, internal quenching of the solute; cks, self-quenching of the solute. This shows that the response as a function of concentration rises to a maximum and then falls off. The form of the equation given above can be compared with those given by Furst and Kallmann :18 (1.4.9)

Figure 7 shows some results of studies of relative pulse heights as a function of concentration. The maximum response is seen to be very broad. Some “typical” results indicate that the ratio of transfer to quenching in solvent is about 0.75; the ratio of self-quenching of the solute to emission plus internal quenching of solute is about 0.02; and that internal quenching of solute is small compared to emission. Certain results are known concerning the effects of “secondary solutes” and “ secondary solvents.” a. The secondary solvent is an intermediate in the transfer process. It may be useful in the “loaded” scintillators (discussed later). The secondary solvent must be soluble and also have good energy transfer to and from the solute and solvent. Napthalene appears to be best (300 gm/liter), but not when used with terphenyl. b. Secondary solutes are wavelength shif ters and will be discussed later. POPOP, diphenylthexatriene, etc., used in concentrations of about 1 gm/liter are in this category. Possibly nonradiative transfers are involved. The energy response of liquids is very similar to that of crystals. The a/@ ratio generally decreases as the concentration decreases. l8

M. Furst and H. Kallmann, Phya. Rev. 86, 816 (1952).

1.4.

135

SCINTILLATION COUNTERS

1.4.1.5. Plastic Scintillators. Plastic scintillators (i.e., solid solutions) were developed in 1950 and 1951 by Schorr and Tornsy, and Koski. Much fine recent work has been done by Swank. Common plastics have been terphenyl in polystyrene and terphenyl in polyvinyl-toluene, with wavelength shifters similar to those used in liquids. Tetraphenyl butadiene is also used in place of terphenyl and most recent combinations

0.8

0.6

B- Excitation

0.4

RCA-6342 MgO Reflector

0.2

0

4

8

12

16

20

24

Grams of fluor per liter of solution

FIG.7. Light output as a function of concentration in a typical solvent. (All solutions argon-saturated.)

are shown in Table I. The concentration of solute in plastic scintillators is about 5 to 10 times that in liquids. The energy dependence in plastics is very similar to that in liquids. The concentration effects in plastics are similar to those in liquids, but the work of Swank and Buck3 is particularly useful here in that it shows that radiative transfer is relatively more important in plastics, due t o the better scintillation properties of the solvent (polystyrene as compared to toluene) and the strong absorption of the solute in this region. Even so, the relat.ve importance of radiative transfer is only about 20% a t most.

136

1.

PARTICLE DETECTION

Various methods of polymerizing the solute are used, including: 1. mixing solute in molten polystyrene solvent; 2. polymerizing a monomer solution by means of a catalyst (benzoyl peroxide -55 %) ; 3. polymerizing the monomer without the catalyst.

Method No. 2 probably gives best results, but No. 3 is used most for ease of handling. A high-temperature (146°C) process results in a shorter time, lower average molecular weights and better response by -lo%, compared to lower temperatures (126°C). A good plastic appears to be : 20 grams of 2 phenyl 5-(4-biphenylyl)-1,3,4 oxadiazole (PBD) in 1000 grams of polyvinyltoluene (PVT) with 1 gram of p , p’ diphenylstilbene (DPS) as a wavelength shifter. This gives a pulse-height response equal to 0.54 that of anthracene. Certain commercial plastics are available, some apparently exceeding this performance. The effect of concentration on performance of plastics is similar to those found in the case of liquids. 1.4.1.6. light Pipes and Reflectors for Scintillation Counters. A crystal scintillation counter is normally placed in a metal container. I n the case of the hygroscopic alkali halides this container must also seal the crystal from moisture and, of course, in all cases the box must be lighttight. Efforts to determine whether specular or diffuse reflecting surfaces provide the best light collection have not shown any striking differences but it is generally believed that diffuse reflecting surfaces are best. In the case of crystal and plastic scintillators this diffuse surface can be easily applied by “smoking” on MgOz or painting with certain commercial preparations such as Tygon. In the case of liquids the problem is more difficult if it is desired to have the liquid in direct contact with the reflector. In this case special procedures to apply titanium dioxide in lacquer to an etched container face could be followed. Containers for liquid scintillators can be made out of aluminum with thin transparent windows of alite. If the sealing is with Neoprene gaskets it is necessary to boil the Neoprene in toluene for several days to remove impurities. Using this procedure, counters have been constructed that were usable over many months. If an all transparent container is required, it can be fabricated out of Lucite, but then phenylcyclohexane must be used as the solvent of the solution. Acetic acid has proved to be a suitable cement in this case. When the setup is such that the photomultiplier cannot be put in direct contact with the scintillator, Lucite light pipes have been used. Care must be taken in the selection of the cement used to attach the

1.4.

SCINTILLATION

COUNTERS

137

light pipe t o the scintillator face or window in order to avoid crazing. A commercial product designated as R 313 has proved useful here. If the photomultiplier is immersed directly in the liquid it is necessary to run the cathode near ground potential to avoid deterioration of the cathode. If a light pipe link is used, optical coupling with the photomultiplier face is generally made b y the use of some high viscosity silicone. 1.4.1.7. Loaded Scintillators. The usefulness of scintillators can be enhanced for some applications by the process of "loading" the scintillator with selected heavy elements. Neutron detection experiments have suggested loading with boron, cadmium, etc. Some success has been achieved by using :

+ +

1. triethylbenzene methyl borate [BIO ( n p )reaction]; 2. triethylbenzene cadmium octoate [Cd113(n,r>l; or toluene cadmium propionate 3. toluene samarium or gadolinium propionate [Sm149( n , ~ ) ] .

+

+

}

Generally however, quenching sets in before much of the desired substance is in solution, so that concentrations have until recently been restricted to approximately one percent. Recently studies b y Kallmann" and Swank have shown that intermediate solvents such a s naphthalene and biphenyl can be used to extend the amounts and types of compounds that can be successfully dissolved without quenching. Recent work by Hyman16 has resulted in plastic scintillators containing u p to 5% b y weight of lead, with a response that is about 50% that of a n unloaded plastic scintillator. 1.4.1.8. Noble Element Scintillators. In the past several years efforts to prepare gaseous scintillation counters have resulted in certain successes in the use of the noble gases. These counters have provided properties of interest in speed, large light output, linearity, simplicity and flexibility in Z and density. Work has been done by Northrup and noble^,^^-^* Eggler and H u d d l e ~ t o n , 'and ~ Sayres and Wu120among others. Early results were difficult to correlate until the importance of eliminating impurities was fully realized. Other factors that must be taken into account are the size of the container and t,he effect of wavelength shifters. Consistent and successful gas scintillation counters have been reported by Sayres and Wu20 constructed along the lines shown in Fig. 8. The gases used were helium, argon, krypton, and xenon. Since the gas is H. Kallmann, I R E Trans. on Nuclear Sci. NS-3, No. 4 (1956). J. A. Northrup and R. Nobles, IRE Trans. on Nuclear Sci. NS-3, No. 4 (1956). 19 C. Eggler and C. M. Huddleston, IRE Trans. on Nuclear Sci. NS-3, No. 4 (1956). 10 A. Sayres and C. S. Wu, Rev. Sci. Znstr. 28, 758 (1957); W. R. Bennett, Jr. and C. S. Wu, Bull. Am. Phys. SOC.[2] 2, No. 1 (1957). 17

18

138

1.

PARTICLE DETECTION

susceptible to impurities it is necessary to take special precautions in construction. Teflon is used as gasket material; metal components and gaskets are carefully baked out. I n addition a gas circulating pump and calcium purifier were also incorporated. With proper precautions and initial purification Sayres and Wu found no measurable deterioration in performance in their counters over a period of five days. Most of the light coming from the noble element scintillators lies in the ultraviolet so that wavelength shifters are necessary. Some of the ,Q.

phenyl,

as ut

u234

I' alpha source

1

0

I in.

m

u Scale

FIG.8. A typical gas scintillator. (From Sayres and WulZoRev. Sci. Instr. 28,759 (1957), Fig. 3.)

early work was confused by the quenching effects of certain wavelength shifters. The most successful method for wavelength shifting has been to deposit thin layers of quaterphenyl or diphenylstilbene (30 to 50 pgm/ emz) on the walls of the container and adjoining photomultiplier face. Difficulties in the interpretation of the role of nitrogen as a wavelength shifter have been removed by the work of Bennett and WuZowhich indicates that the observed spectra, rise times, etc., when nitrogen is present are explained on the basis of collision phenomena rather than photoexcitation. Using a chamber of the sort shown in Fig. 8, Sayres and Wu have made systematic observations using helium, argon, krypton, and xenon as the scintillators count Po210a particles. In each case the pressure was adjusted t o be as high or higher than that needed to stop the a particles in the chamber volume. The results of these tests show that pulse heights reach

1.4.

SCINTILLATION

139

COUNTERS

a maximum when the particle is stopped in the chamber, and that the resolution also improves with increasing pressure. Resolutions (in per cent peak width) of 5 % to 10% were obtained. The relative performances as well as the effect of wavelength shifting is shown in Table IV. TABLE IV. Performance of Noble Element Scintillators Glass phototube (6292)

Gas (optimum pressure) Xenon (6 psi) Krypton (8 psi) Argon (10 psi) Helium (45 psi) [Noise/background]

Without quaterphenyl

With quaterphenyl

6 <3 <3

105 50 15 38 3

9 3

Quartz phototube (K 1306) Without quaterphenyl

With quaterphenyl

88

145 83 28 40

11 14 3

4

The linearity of these scintillators has been demonstrated for charged particles having high rates of energy loss, including fission fragments. Nobles2‘ using xenon gas and protons, deuterons and a particles from 2-5 Mev found that pulse height versus energy yielded a straight line, but with an intercept at 0.5 MeV. Other work indicates linear response among similar particles, but some “inverse saturation” effects. It is possible that some of these discrepancies are the result of spurious rather than inherent effects. There are conflicting reports in the literature concerning the lifetimes of the scintillation processes in the noble elements, but it seems clear that these counters are “fast.” In gases there appears to be an inverse pressure dependence’* so that the lifetimes are approximately 75 mpsec at 25-cm pressure, 30 mpsec a t 50-cm pressure, and 14 mpsec a t 100-cm pressure. The light output relative to NaI(T1) is also the subject of conflicting reports. The work of Northrup14 indicates that pulse heights from argon, krypton, and xenon are only a little less than those from NaI(T1). This is consistent with the work of Sayres and Wu, where comparisons showed gas pulse heights about one-half those of CsI. Several special applications of gas counters, or counters containing gas mixtures have been suggested and tried. Sayres and Wu constructed a slow neutron detector by placing a metallic boron film (20 pgm/cm2, 97% B’O) in a xenon gas counter. The resulting spectrum of pulse heights when slow neutrons irradiated the counter showed two peaks in the ratio 21

R. A. Nobles, Rev. Sci. Instr. 27, 280 (1956).

140

1.

PARTICLE DETECTION

7:4 as expected for the Li and a particles from the reaction Bl0 (%,a)Li7. I n a similar way, a fission counter was constructed by placing a thin U2a5 disk in a xenon chamber and irradiating with slow neutrons. Figure 9 shows the fission spectrum obtained.

-a's

1500 61.4 Mev

N

1000 -

500

-

FIG.9. The fission spectrum of U*36in 29 psi of xenon, using the K-1306 phototube and quaterphenyl wavelength shifter.

Some very interesting results have been obtained for binary mixtures by Northrup et aZ.,14particularly with regard to the possibility of neutron detectors utilizing He3. Sayres and Wu have also investigated a mixture of 90% He and 10% xenon and found that, a t proper pressures, the response is as good as that of a pure xenon counter. This counter also serves to show the good discrimination properties possible for ionizing particles and y rays. Results of Po2l0a and radium y tests are given in Table V. Scintillations have also been detected from the liquid and solid states of some of the noble elements. The work of NorthrupI4 shows that liquid and solid xenon, krypton, and argon give pulses the order of one-half or better than those obtained from NaI(T1). The techniques are very similar to those involved in the work on gas scintillators. Liquids and solids appear to have scintillation decay times under 10 mpsec.'*

I .4.

141

SCINTILLATION COUNTERS

The great interest in He4 and He3 in neutron scattering and detectioii has resulted in an investigation of liquid helium as a scintillator by Thorndike and Shlaer14 and Wu and her collaborators.22Using a! particles Wu et a1.22found that, under optimum conditions, relating from P3, TABLE V. Results of Poz1"a and Radium Scintillator

Pulse height of a source

y

Tests Peak effect of y source

-

CsI (1 mm thick) Anthracene (3 m m thick) He 10% Xe (60 psi total) 16292 tube with quaterphenyl j

+

160 22 I05

75 77 5

to the light collection aiid range of particles the signal is well above the background, and energy resolution is very satisfact,ory (-20%). In the system used by these workers, the pulse-height ratio between liquid helium aiid CsI (at room t,emperature) is roughly 1:2. This same work indicated that He vapor also scintillates satisfactorily. In conclusion, it is possible to say that techniques have beeii developed that allow the practical use of the noble elements as scintillators in the gaseous arid liquid states; and that certain of t,hem are also good scintillators a s solids. 1.4.1.9. General Applications of Scintillation Counters. Certain special applications of scintillation counter techniques are discussed elsewhere in this book. It is appropriate a t this point to describe some general applications of the technique. For some experiments scintillation counters have B marked advantage over the visual techiiiques of cloud chamber, diffusion chamber, bubble chamber, and emulsion, due to t.heir fast, t,iming aspect. This advantage might be exploited in a lifet,ime det,ermination, such as the T+ lifetime measurements of Jakobsen et al.?' or t,he Kf meson lifetime determination of Motley and I ? i t ~ h . ~ In* l i f e t h e experiments scintillators are arranged to give a fast t,imiiig pulse when the parent particle enters and does not leave the absorber, and another pulse when the decay product leaves the absorber. With current techniques and organic scintillators, timing resolutions of several milliniicroseconds are possible. Similar techniques, involving large area liquid scintillators have been used t80 measure capture h i e s of p - mesniis ill the t,ens of millimicrosecond 22

23 2'

C. S. \Vu, private coiiiinunication. M. J. Jskohsen, A. G. Schula, and J. Steinberger, P h p . Rev. 81, 894 (1951). H.. Motley a n d V. Fitch, Ph?p. Rev. 106, 265 (1952.

142

1.

PARTICLE DETECTION

range.25 The fast timing and good geometry aspects of scintillation counters have also been used in time of flight applications such as the antiproton work at Berkeley.28Here it was possible to distinguish between R mesons and antiprotons of the same momentum (1.19 Bevlc) by their time of flight over a 40-ft base line. The times were 40 and 51 mpsec respectively. Another area where the fast time resolution of scintillators is an advantage is in certain scattering experiments such as those of Yuan and Lindenbaum2’ and Cronin et aL28 In these applications the sacrifice in precise spatial resolution is offset by the advantage of being able to record many events in a single pulse of the beam of the high-energy particle accelerator. From the above remarks, it is obvious that a very great advantage would result if the tracks of particles could be simultaneously viewed, as in chambers and emulsions, and timed, as in counters. This combination of properties is discussed in Section 1.4.2.

1.4.1 .lo. l a r g e Scintillation Counters.* 1.4.1.10.1. GENERAL CHARACLarge volume liquid2Yscintillation detectors, arbitrarily defined as >+ cubic meter, are useful in the efficient detection of neutrons and gamma rays as well as charged particles. I n the case of neutral radiations this characteristic stems from the detector size which is by definition

TERISTICS.

2K J. W. Keuffel, F. B. Harrison, T. N. K. Godfrey, and Geo. T. Reynolds, Phys. Rev. 87, 942 (1952). 28 0. Chamberlain, E. Segrh, C. Wiegand, and T. Ypsilantis, Phys. Rev. 100, 947 (1955). 27L. C. L. Yuan and S. J. Lindenbaum, Proc. 4th Conf. on High Energy Nuclear Physics, Rochester, New York, pp. 98-100 (1954). 2 * J. W. Cronin, R. Cool, and A. Abashian, Phys. Rev. 107, 1121 (1957). 29 Large volume detectors using plastic scintillators have been employed by the Los Alamos Bomb Test Division in the detection of gamma rays. A total absorption spectrometer using a large single crystal of NaI, 9a in. diameter by 9 in. high is described by R. C. Davis, P. R. Bell, G. C. Kelly, and N. H. Lazar, in the Proc. 5th Scintillation Counter Symposium, February, 1956, Washington, D.C., published in I R E Trans. on Nuclear Sci. NS-8,82 (1956). We restrict our discussion to liquid detectors because of the relative ease vis-a-vis plastic with which it is possible to incorporate other substances such as neutron absorbers into liquids. Single crystals are limited in size although i t may be that a transparent slurry of smaller crystals in a bath of optically matching liquid could be devised to overcome this limitation.

-

* Section

1.4.1.10 is by F. Reines.

1.4. SCINTILLATION

COUNTERS

143

large or a t least comparable with the absorption mean free path for these radiations in the scintillator. Charged particles can in some cases also be studied more effectively by such detectors because of their more complete absorption of the particle energy and decay particles, if any. I n addition t o particle containment, large detectors have been constructed which have fair energy and time resolution, and neutron capturing elements such as boron and cadmium have been incorporated into the scintillator to provide a distiiictive signal on neutron capture. A limited amount of spatial resolution can be achieved by the use of several tanks or of light shields dividing a common liquid container. The characteristics resulting from large size have been found useful in a variety of problems ~ ~ a s the requiring very high (>75%) neutron detection e f f i ~ i e n c ysuch measurement of neutron multiplicity from the fission process,31 in the measurement of large-volume low-intensity gamma sources such as the human body,S2 and in free neutrino studies which required a large hydrogeneous target and good positron and neutron detection efficiencies.* Apart from size, which introduces special problems of construction and transparency of the liquid to scintillation light, large volume scintillation detectors are in principle much the same as the more conventional small detectors. A particle enters the liquid and causes the emission of scintillation photons which traverse the liquid to its boundary and either strike a photomultiplier face or container wall and are absorbed or are reflected back into the liquid. The scintillation light,, diffusing in this manner, is eventually absorbed either by the liquid, the walls, or the photocathodes. Light striking the photocathode is converted with an efficiency which can be a s high as 10% t o photoelectrons, and the electrical pulse thus produced, amplified b y the photomultiplier tube itself, and followed by external amplification has the characteristics of amplitude, shape, and time of occurrence. These characteristics are used to study the primary physical event under consideration. 1.4.1.10.2. ENERGYLoss IN LIQUID SCINTILLATORS. The energy deposited in the scintillator in the first place depends on the particle involved, its energy, and of course on the size of the detector. The scintillation efficiency is also a function of the particle involved and in some instances, its energy as well. Discussions of 'the energy loss of various particles, i.e., gamma rays * See Section 2.2.4 of this volume. F. Reines, C . L. Cowan, F. B. Harrison, and 1). 8. Carter, Rev. Sci. Znstr. 26,

:Io

1061 (1954). 41 €3. C. Diven, H. C. Martin, R. F. Taschek, and J. Terrell, Phys. Rev. 101, 101'2 (1956). 32 E. C . Anderson, I R E Trans. on Nuclear Sci. NS-3, 93 (1956).

144

1.

PARTICLE DETECTION

electrons, protons, 1nesoiis,33~3*arid iieutroiisS6 caii be fouiid in lnaiiy places.* We note here the fact that in the hydrocarbons comprising liquid scintillators, the energy loss of gamma rays is, for eiiergies less thaii 10 Mev, primarily by mealis of the Comptoii eEect and helice takes place over ail extended region. Figure 10 shows the total mean free path 70

60

I

50

u

I

I-

:

40

W

w

a

LL

0

I

2

3

4

5

6

7

E

10

9

GAMMA ENERGY (MEV)

FIG. 10. Total gamma-ray lnean free path versus energy in toluene (C,H8, p gm/cm3).

=

0.87

x(cm) versus energy for a gamma ray of energy E (MeV), in toluene (C,H,, p = 0.87 gm/cm3)).Since the iiieaii free path is inversely proportional to density this curve can be simply scaled to other organic solvents. Because the Comptoii effect results only in a partial energy loss, the gamma-ray absorption process requires several collisions. The absorption

* Refer to Section 1.1 2 of this volume. R. Latter and H. Kahn, (;anma ray absorption coefficients. Rand Report &I70 (1949); see the article by H. A. Bethe and .J. Ashkin: Passage of radiation through niatter, i n “Experiinentnl Xurlear I’hysics” (E. SegrP, ed.), Vol. I, pp. 166-357. Wiley, Nrw York, 1953. 3 4 M. Itich and R. Mady, Range Energy Tal~les.UCItL-2301 (1954). 3 5 Bec the article by B. Feld: The neutron, i n "Experimental Nuclcni P ~ ~ R I v ~ ’ ’ (E. SegrB, ed.), Vol. 11, pp. 208-586. Wiley, New York, 1953. 33

length may be roughly ehtimated by random walk considerations modified by the correlation in direction of the incident arid scattered gamma ray: e.g. for a 1-Mcv ganima t>hcCompton mean free path is 17 cm implying absorption lengths a h i t 35 cm. The Comptmi recoil electrons are for these energies readily :~l~sorJwci by the scintillator with a n energy loss I

1

I

I

I

I

4

5

6

7

8

9

I

I

I

1

30

a

e II -

aQ

10

5

0

I

2

3

1 0 1 1

1 2 1 3 1 4

NEUTRON ENERGY ( M E W 1710.

11. Neutron, proton cwlhsion incan free path in toluene.

of about 1.6 Mev/cm so that the energy deposition of a gamma ray is given essentially by the Comptori process. Neutrons give up their energy to the scintillator largely by elastic c~ollisionswith protons. The process of nentroii slowing down arid diffusion has been extensively and it is known that the distance travelled by a fast neutron prior to its thermalieation is of the order of the mean free path for the first collision. Figure 11 gives the mean free path for n,p collisions in toluene. The slowing down process is quite rapid (-2 X

146

1.

PARTICLE DETECTION

sec) so that the sequence of proton recoils involved occurs within the resolving time, 2 X lo-’ sec, of more or less conventional electronics. Under these circumstances the neutron slowing down pulses “pile up” giving one pulse. Because of the nonlinearity of the scintillator response to protons and the various combinations of recoil energy loss possible for a neutron, the sum pulse varies. Consequently, there is no unique scintillation response for a given neutron energy.36The role of the other major scintillator constituent, carbon, in slowing down the neutron is small because of the relatively great mass of the carbon nucleus and also because the neutron collision cross section presented by the scintillator protons is somewhat larger than that of scintillator carbon. Nevertheless, the neutron loses an average of 14% of its energy in each collision with a carbon nucleus and since the carbon recoil nuclei are so inefficiently signaled by the scintillator, this fact introduces additional nonlinearity into the response to neutron energy. The light output of a liquid scintillator (5 gm/liter terphenyl in toluene) for electrons and protons as determined by Harrison87is shown in Fig. 12. As an example of the photon yields to be expected from liquid scintillators we quote the absolute photon yield determined by Post38for terphenyl (8 gm/liter) in toluene, 150 ev/photon. 1.4.1.10.3. DESIGN CONSIDERATIONS. As outlined in Section 1.4.1.10.1, the response of the detector to a primary event is a consequence of several factors : 1. the energy deposited in the scintillator and the fraction of this energy which appears as scintillation light, 2. the transparency of the liquid to its own scintillation; 3. the reflectivity of the container walls and the fraction of the wall area covered by photocathodes; 4. the photoelectric efficiency of the photocathodes and the electrical characteristics of the photomultipliers, photomultiplier ganging circuits, and amplifiers. This lack of uniqueness could be eliminated in principle by the use of a system fast enough to observe the individual proton recoils. Indeed, since one is here concerned with correcting for a nonlinearity, it need not be made with precision. Thus far such corrections have not been attempted although the multiplicity of recoiLs has been observed by D. W. Mueller’s group a t Los Alamos (private communication), and was actually used to discriminate between neutrons and gammas by F. D. Brooks, in “Liquid Scintillation Counting” (C. G. Bell and F. N. Hayes, eds.), pp. 268-269. Pergamon, New York, 1958. F. B. Harrison, Nucleonice 10(16), 40 (1952). 88 R. F. Post, Phys. Rev. 79, 735 (1950), as interpreted by S. C. Curran in hie book entitled “Luminescence and the Scintillation Counter.” Academic Press, New York, 1963.

1.4.

147

SCINTILLATION COUNTERS

The fraction F of scintillation light which eventually reaches a photocathode can be related to the mean transmission over the light path between reflections t of the scintillation light by the liquid, the reflectivity I2

I

I

I

I

I

I

I

I

I

10

_I W

a

0

‘I,s

ELECTRONS

* U a

a

t

m

R

a

-.

6

%

P W

I $

4

_I

2

a

0

1

2

3

4

5

6

7

8

’

9

1

0

PARTICLE ENERGY (MEV)

FIG.12. Scintillation light output for electrons and protons versus particle energy in a 5 gm/liter terphenyl toluene solution (Harrison37).

of the wall r, and the fraction f of the wall surface uniformly covered by photocathodes by the simple formula:* F

=

tf/[l - tr(1 - f)].

(1.4.10)

The mean transmission t is in general a complicated function of the mean free path for scattering A, and absorption A. of the light by the liquid and the detector size and shape. For t,he simple case of a spherical detector and a liquid which has a scattering mean free path much greater than the detector diameter, t is given by the relationship A,

t = - (1 d

- e-”Aa).

(1.4.11)

* A more precise formula would replace t in the numerator by to, where fa is the light transmission, averaged over points uniformly distributed throughout the scintillator, to the wall.

148

1.

PARTICLE DETECTION

As a n example, suppose d = 1 meter, A, = 2 meters, r = 0.9, f = 0.05. Then t = 0.787 and the fraction of scintillation light collected F = 0.121. The enhancement of light collection caused by the reflectivity of the container walls is seen in this example, to be a factor of 2.4. The reflectivity of a given wall coating, scintillator combination can be measured by using a light source centrally placed in a sphere which is so small that t = 1.0. A uniform distribution of holes around the sphere would enable the light to emerge and appropriate filters can be used around the light sensing element, to match scintillation light. Small optical mockups have been used to measure the uniformity of light collection in this limiting, t = 1, case.39 The total mean free path for scattering plus absorption can be measured for the scintillating liquid by means of a Beckman spectrophotometer, provided a standard is available for comparison. As a standard we have used reagent grade toluene measured in a six foot long box with a modulated light source at one end to provide an ac signal, a photoelectric detector a t a variable distance from the source, and light baffles between for collimation. The total mean free At path was measured as the departure from inverse quar re,^" i.e., I(r,At) = Ioe-r/X1/4~r2.

(1.4.12)

With this arrangement we have found reagent grade toluene to have a total mean free path of 5 meters. Thus far no measurements have been made which enable a direct and accurate determination of the absorption mean free path A, for a given scintillator. In principle, given the total mean free path ',A ( = A'; )';A and a light, source which mocks up the scintillation light, an experiment the result of which depends on the ratio X,/At can be done. For example, a centrally placed light source surrounded by a spherical volume of liquid with photoelectric detectors on a spherical black wall represents a calculable system. A drawback is the which is required in order to make a useful rather large radius (-A,) measurement. Because of the complicated dependence of the various light collection factors on the geometry, the eventual optimum design and resultant, characteristics for any given application is probably better determined from studies of complete detectors than from an a priori synthesis. 1.4.1.10.3.1. Scintillator Transparency and ReJlecting Paint. The problem of scintillator transparency4I has been met, by standard chemical

+

H. W. Kruse and F. Reines, unpublished. C. L. Cowan, Jr., F. B. Harrison, A. D. McGuire, and F. Reines (unpublished). 4 * For more details see A. R. Ronzio, C. L. Cowan, Jr., and F. Reines, Rev. Sci. Instr. 29, 146 (1958). 3a *O

1.4.

SCINTILLATION COUNTERS

149

means in the case of triethylbenzene (TEB, C2Hy, p = 0.87 gm/cm3). Reagent grade toluene with a total mean free path for s5attering and absorption of about 5 meters at a wavelength of 4200 A is generally acceptable without further treatment. Purified triethylbenzene has a mean free path of from 5 to 10 meters. Purification of the crude T E B is accomplished by digestion wit,h sodium methylate (about one pound per hundred gallons of TEB) for four hours followed by fractional distillation in which a narrow cut is t,aken at. the constant boiling point range. The liquid is best st,ored in glass or metal cont,ainers painted wit8hEpon base enamel because of it,s tendency, especially in t,he case of toluene, to leach impurities from t,he container walls. Storage under an inert atmosphere such as argon is advisable in order to avoid the lowering of scintillation efficiency due to the act.ion of dissolved oxygen. An essential ingredient in any large scint’illator is the so called wavelength shifter which absorbs the primary scintillation light and re-emits it a t a lower frequency which is: (a) less readily absorbed by the scintillat,ing solution; (b) more easily reflected by available inert,, adhesive container coatings;42(c) more efficient,ly detected by the photomultiplier tubes. A popular shifter is 1’0P0P43 which moves the maximum in the emitted spectrum from the region of 3800 to 4200 A. The 5-meter mean free path quot,ed above is in purified solvent and does not apply to t,he case of an actual scintillator in which e.g. 3 gm/liter of terphenyl and 0.3 gm/liter of POPOP have been added t,o the solvent. For a scintillating solution the total mean free path drops to about one meter, presumably due to fluorescent or reradiative scattering. Judging from the actual performance of large detectors, itJ seems that only a small part, of the mean free path reduction is due to increased absorption by t,he terphenyl and POPOP. 1.4.1.10.3.2. Uniformity of Light Collection. If it is desired to employ the detector in any manner which requires energy resolution then it is clearly necessary t o be able to make the amount of light collected by the photocathodes dependent only on the number of photons emitted. In view of the fact that the optical t#rarismission of the liquids and the reflectivity of the container are imperfect, this aim is accomplished in an approximate sense by surrounding the liquid with a multitude of photomultiplier tubes. The fraction of the liquid surface area covered by photoin the largest detectors (-1.5 meterss) constructed at cathode is Los Alamos. Numbers from 45 2-in. h b e s to 110 5-in. tubes have been

+

4 2 Plasite paint with TiO:! pigment has been found to be suitable coating material. Ohtained from Wisconsin Protective Coat.ing Go., Green Bay, Wisconsin. F. N. Hayes, D. G. Ott, and V. N. Kerr, Nucleonics 14, 42 (1956). See also Table I, Section 1.4.1.3.1 of this volnmc.

1.

150

PARTICLE DETECTION

used to help obtain uniformity of light collection. Figure 13 shows three arrangements designed to this end. All the tubes are usually connected in parallel although it is on occasion useful to divide them into two interleaved banks which are connected in prompt coincidence to disSCINTILLATING

ISOLATION LlQUlO TO /OBTAIN MATCH OPTICAL

.TRANSPARENT WINDOW

x-

x--- - SCINTILLATING LlOUlO

PHOTOMULTIPLIER TUBES DISTRIBUTED AROUND CYLINDRICAL WALL

\

I

X X X

SCINTILLATING LlOUlD

X X

X

X X X

\ r c ) RECTANGULAR

1

X X X

PARALLELOPIPED

FIQ. 13. Photomultiplier tube arrangements designed for uniformity of light collection.

criminate against tube noise. In arrangement (a) light from events which occur over an appreciable distance compared with the photomultiplier tube dimensions is collected in a uniform manner. In (b) the photomultipliers are isolated from the scintillating liquid by a nonscintillating, optically matching so that short-range particles cannot deposit their energy in the near vicinity of a photomultiplier cathode. Arrange-

'* Cerenkov radiation will of course reRult in some light from energetic particles passing through this region.

1.4.

SCINTILLATION COUNTERS

151

ment (c) is a variant of (b) which has been found useful in work where a large, unobstructed area was necessary. The light collection characteristic of a uniformly distributed source was measured in an optical mockupa9 of a rectangular detector 9 X 4+ X 2 ft to have a half-width a t halfmaximum of 7%. The over-all light collection was estimated as %. Given this figure, 1 Mev deposited in the scintillator, a conversion to photons of 1 photon/l50 ev deposited, and 1 photoelectron emitted by the photocathode per 10 incident photons, we conclude that 170 photoelectrons are produced per Mev absorbed by the liquid. This figure implies a statistical uncertainty of k 4 1 7 0 or ?8% due to fluctuations in the number of photoelectrons. The over-all uncertainty for 1 Mev deposited in this example is therefore conservatively taken as k 15%. 1.4.1.10.3.3. Photomultiplier Selection and Circuitry. * Very large detectors built thus far have employed 2-in. RCA and Dumont and 5-in. Dumont (K-l198), photomultiplier tubes. Sixteen-inch tubes which are just becoming available in quantity have a wide variation of response across the photocathode and hence, assuming only a few are used, introduce nonuniformity in the light collection of the system. As mentioned, photomul4iplier tubes have been used in gangs with as many as 110 5-in. tubes connected in parallel. Under these circumstances, uniformity of light collection requires that all the tubes have the same gain. Since tubes differ from each other in gain for a fixed voltage distribution along the dynodes, it is necessary to adjust the voltage in each case so as to equalize the gains of all tubes in the gang. This is done by observing the response to a source such as Csla7 placed in a reproducible fashion near a NaI crystal which is viewed by the photomultiplier tube under adjustment. The tubes are similar enough to require only the selection of an appropriate load resistor. Figure 14 shows a resistance network on one tube. Figures 15 indicate how a number of tubes can be connected in parallel. Also shown in Fig. 15(a) is a simple switch which enables a given tube to be thrown out of the gang. This switch is useful in checking the performance of individual tubes in situ, especially in the location of tubes which have become noisy, In addition to equalizing gains it is also of importance to select tubes of acceptable noise level because despite the large capacitance of the ganging network even one noisy tube can result in variable and unacceptable backgrounds. Occasionally problems arise in which even the best collection of tubes results in unacceptable levels due to tube noise. In such cases it has been found that tube noise can be greatly reduced by the expedient of dividing the tubes into two interleaved bunks axid requiring a coincidence between the two banks. The price one pays for this reduction in background is in a smaller light collec*See also Vol. 2, Section 11.1.3.

152

1. PAltTICLE DETECTION GAIN BALANCE RESISTOR

H.V.

-

2 MEG I

I

I

1

G-

SCREEN (13)

CATHODE (14) SWITCH

FIG. 14. Voltage divider network used on a 5-in. 1)umont K-1198 photoInultiplier tube.

t,iori and hence in decreased energy resolution per bank. I n c,ases where the energy resolution required in making a coiiicidence is not unduly restrictive it is possible t,o add the signal from the two banks through iso1ztt)ion networks (i.e., separate preamplifiers) and regain the resolution lost in t,he division. A problem occasionally eucountered is in the electrical oscillations which sometimes result., in the ganging “yoke.” Parasitic resistors c:m be added t,o suppress such ringing as show11 in Icig. 15(b). The variatiou of

1.4.

I53

SC'INTILLATION ('OUNTElIS

FIG, 15(:1).Ganging yokc for a largc number of 5-in. photomultiplier t rrhes. 93"COAXIAL SIGNAL LEAD ( T O PREAMPLIFIER) COAXIAL HIGH VOLTAGE LEAD

SIGNAL YOKE

FIG.15(b). Schematic drawing of ganging yokc for a large number of 5-in. photoinultiplier tubes.

154

1.

PARTICLE DETECTION

photomultiplier response due to ambient magnetic fields can also be a cause for concern. In general the use of mu metal shields is cumbersome, expensive, and in many cases impractical because of the restriction imposed on light collection. One solution to this problem is to build the 18

17 16

15

t

14

13

z

3 12

>

a a

11

g

10

E U

-

9

W

G

8

0

7

&

z F

2 3

6

8

5

0

20

30

40

50

60

70

80

PULSE HEIGHT (VOLTS)

FIG. 16. Peaked spectrum due t o cosmic rays which pass through the giant slab detector of Fig. 5. The liquid depth, 56 cm, corresponds to a peak energy of -90 MeV. These data were taken under about 200 f t of rock at an altitude of 7300 f t and correspond t o a muon rate through the detector of 13/sec. (Cowan and Reines, unpubished, 1957.)

detector of steel which will shield the tubes, another is to recalibrate the system a t each position of use simply foregoing the loss of resolution due to the magnetic field effects. 1.4.1.10.4. CALIBRATION AND USES. Two general calibration schemes

1.4.

155

SCINTILLATION COUNTERS

are available for use with large detectors: t,he first makes use of cosmic radiation, the second of radioactive sources. Minimum ionizing cosmicray particles, for example, deposit an amount of energy which is proportional to the track length in the scintillator. In consequence a pulse-height spectrum due to cosmic rays penetrating the detector has a peak which may be associated with the mean energy loss of minimum ionizing cosmic rays. A typical “through peak” is shown in Fig. 16. Once the through peak is located, the detector can be replaced by a standardized pulser which is then adjusted to give an output voltage equal to the through B

I

I

I

1

I

I

I

PULSE HEIGHT ( M E W

Fro. 17(a). Mu-meson decay electron spectrum seen with 75-crn cylindrical detector. Data were taken at 7300 ft above sca level and 40,000 counts were recorded in 36 hours. (Reines et al.so)

peak value. Energy gates can then be set using the pulser. An independent energy calibration can also be obtained, along with a check of the system used to study events in delayed coincidence, by employing the phenomenon of muon decay, p*-+ 8’ Yv+. (1.4.13)

+ +

In this instance, a delayed coincidence is required electronically with a fist pulse of, say 20-40 Mev energy followed within 10 psec by a second pulse of energy > 15 MeV. The energy spectrum of these second pulses is due t o the decay electron. Fortunately, backgrounds are small, and despite considerable distortion of the decay electron spectrum due to bremmstrahlung losses and edge effects, the end point (53 MeV) is sharp. Figure 17(a) shows such a decay spectrum measured with the 75-cm cylin-

1

156

1.

I’AHTICLE

DETECTION

drical detector. Figure 17(b) shows the associated time interval or decay time spectrum. These calibration schemes give the energy response for a distributed source: the variation of response across the detector can be investigated by means of aperture detectors used in coincidence to gate the detector under study. The distributed or localized response can also be determined by means of radioactive sources such as the posit,ron I

I

I

I

I 8

I

10

v)

Iz

3

0 0

10

z

IC

I

I

I

2

4

6

I, \ 10

12

MICROSECONDS

FIG.17(b). Half-life measurement of the 8-decay of cosmic fi mesons stopping in the detector used as a check on the time calibration of the apparatus. The entry of the meson yielded a ‘ I first pulse ” and the decay electron the “second pulse.’’ This measurement was made in conjunction with the energy calibration using the decay electron spectrum end-point. (Reines et ~ 1 1 . 3 ~ )

emitter Cu‘j4which can be dissolved as a salt into the scintillator and the resultant spectrum observed. If the Cu64is encapsulated, only the annihilation radiation enters the detector, giving gamma rays of unique energy for measurement. These calibrat,ioii techniques incidentally indicat,e some of the uses which can be made of large volume liquid scintillation detectors. Another degree of freedom which can be incorporated into the detector is a sensitivity to neutrons which are fast or are produced in association with a

1.4.

SCINTILLATION COUNTEItS

157

charged particle. In this case the delayed coinviderice technique mentioned above in connection with muon decay can be employed, the first pulse being due to the associated rharged particle or a recoil proton, the second to the capture of the moderated neutron by the scintillator solution. A good neutron capturer is cadmium which has a large capture cross section (5300 barns at thermal energies for the natural isotopic mixture) and on the average, four capture gamma rays with energy totalling 9 MeV. The use of energetic gamma ray.; helps discriminate

K >

0 20

1.0 c W

a

0.16

zI-

0 12

0 t W

K

t a m 4

0 5 2

a a

0 08

w

a

u

0 04

r a a

0

0

0 CAPTURE TIME t ( p SEC)

Fro. 18. Keutron capture versus time spectrum seen with 75-cm cylindrical detector = 0.003%.These measurements mere made using cosmic-ray neutrons: 9 X lo4 neutrons were rounted in 20 hours. The solid lines represent theoretical values ohtained by means of a Monte Carlo calculation. (Iteines et ~ 1 . ~ ~ ) CY

against background. Cadmium-bearing compounds such as Cd propionate and more recently Cd octoate have been used with some success.41 Figure 18 shows a typicd neutron capture versus time curve for the 75-cm cylindrical detector with a Cd to H at,omic ratio, a = 0.0032. I n this case the neutrons were fast, arising from various cosmic-ray events such as p- cnpt'ure and stars in 90 kgm of P b placed on top of the detector. 4 5 The kinds of electronic circuitry employed with large detectors are indicated schematically in Fig. 19. Shown are the positive high-voltage supply (h.v.) (required t o maintain the photocathode a t ground potential so as to prevent degeneration due to high-voltage gradients at the photocathode), preamplifiers, amplifiers, coincidence circuits, scalers, and 45 In some experiments this cosmic-ray neutron background can be quite troublesome. A partial remedy is to construct the detector of light elements and avoid using heavy elements close to the detector as part of the shielding.

158

1.

.PARTICLE DETECTION

pulse-height analyzer. * Photographic records of oscilloscope traces triggered by appropriate pulses from coincidence circuits are sometimes employed to assist in the identification of the signals and the elimination of noise and background events. As an example of how this circuitry is used, consider the case of muon decay as measured with a large liquid scintillation detector. A pulse corresponding to an entering muon of 20-40 Mev passes through circuit I and registers as a pulse on the scalar. In addition a pulse is sent to coincidence unit I1 making it sensitive for, say, 10 psec. If a pulse in the energy range 15-60 Mev passes through circuit I1 during this 1 0 ~ s e c

‘ I l -

SWITCH

FIG.19. Schematic of electronics associated with a large liquid scintillation detector to measure delayed coincidences, e.g., muon decay.

it registers a delayed coincidence on the scaler and, a t the same time triggers the pulse-height analyzer gating circuit. The second pulse is then analyzed by the pulse-height analyzer. If it is desired to analyze the first pulse, the delay line can be used to store it pending the electronic decision that an appropriate delayed coincidence has occurred. In addition to this sequence, the time interval between the two pulses which comprise the delayed coincidence is measured by a time delay analyzer triggered by the two pulses. The second pulse scaler reads the rate,at which single pulses occur in the energy range 15-60 MeV. The example just given is only meant to be indicative of the kind of use to which such detectors can be put. The reader is referred to the literature for more details. a0--aa-46 I n general, however, it should be noted that the field of large scintillation counters is still a new one and in many *On these circuit elements consult also Vol. 2, Chapters 6.1, 6.2, Sections 7.2.2, 9.1.1, and Chapter 9.6. 4 4 F. Reines, in “Liquid Scintillation Counting” (C. G. Bell and F. N. Hayes, eds.), pp. 246-257. Pergamon, New York, 1958.

1.4.

SCINTILLATION COUNTEItS

159

cases the answers to questions associated with specific uses must be found by t,he experiment,alist.*

1.4.2. Solid Luminescent Chambers Almost as soon as the nature and usefulness of scintillation counters became apparent, the possibility of “seeing” the path of a charged par-

FIG. 20. (a) Sample scintillator filaments. (b) A filament scintillation counter. comparison solid scintillation counter.

(c) A

ticle in the scintillator was discussed. Early reports, later borne out by publication^,^^ of Russian work date back to 1954. This work involved the photography of tracks of protons a t two times minimum ionization in CsI crystals, with the aid of an image intensifier tube.48 I n this arrangement, involving as it does a solid CsI crystal, the depth of focus

* The author acknowledges collaboration with Dr. C. L. Cowan, Jr. on the problems associated with the development of large volume liquid scintillation detectors. 47E.K. Zavoiakii, M. M. Butslov, A. G . Plakhov, and G. E. Smolkin, J . Nuclear Energy 4,340 (1957). B. R. Linden and P. A. Snell, Proc. IRE (Insl. Radio Engrs.) 46, 513 (1957).

problem is severe in the optical link coupling the crystal with the image intensifier. This is particularly true since, as will he discussed later, the light output of a scintillator is very low compared with photographable intensities. Recently a technique has been d c t v e l ~ p e dfor ~ ~the ~ ~ preparation ~ of plastic scintillators in the form of long, thin filaments as shown in Fig. 20. Such filaments are now available commercially. 61 The filaments are

FIG. 21. Crossed filament array with simulated stereo views light piped from rear to front faces.

arranged in rows, stacked alteriiatively a t right angles as shown in Vig. 2 1, to furnish the stereoviewing necessary for three-dimensional reconstruction of the particles’ path. Each of the two orthogonal sets of filaments is viewed separately by image intensifiers. The major advantage of such a system is that since the filaments act as light pipes for the scintillation output, only those filaments traversed by the particle actually put out light, and this light is piped to the end of the respective filaments, so that the optics problems are restricted to a plane source. Thus the coupling to the image tubes can be made directly. Filament arrays with individual diameters from 0.5 to 1.0 mm have been prepared. G. T. Reynolds and P. E. Condon, Rev. Sci. Instr. 28, 1098 (1957). G. T. Reynolds, N?ccleonies 16 (6), 60 (1958). 51 Available froin Pilot Cherniral Corp., Watertown, Connrcticrit, and Xuclear Enterprises, Edinburgh, Srotland. 48

1.4.

SCINTILLATION C O U N T E R S

161

It has been s h o ~ 1 that 1 ~ ~for ~ a 1-mm diameter filament traversed by a minimum ionizing particle, approximately 16,000 photons/cm’ result. This shows that recording on fast film requires an image intensifier with a gain of lo5. Figure 22 shows a track of a minimum ionizing p meson

FIG.2 2 . A minimum ionizing p-meyon track 1 in. long obtained with a scintillation chamber niadc up of &5-1nm diameter fi1ament.s 1 i in. long.

1 in. long obtained in n chamber made up of ~.5-111mdiameter filaments. Six stages of image intensification prereded an image orthicon tube and the track was photographed on the face of 3, kinescope. The Russian reports imply n tube of that gain, and there is every evidence that similar tubes will he available commercially to Western scieiitists i l l t.he near future.

162

1.

PARTICLE DETECTION

A solid scintillator detector composed of plastic scintillator filaments has the advantage of simultaneous fast timing (approximately 3 X sec) and good space resolution, allowing the detection of relatively rare events in fluxes 1000 to 10,000times those possible with bubble chambers. With the image tube? gating techniques available, the scintillation chamber can be triggered after the event, similar to cloud-chamber operations. There are no moving parts; the nuclear composition is simple (carbon and hydrogen) and loading with selected 2 material can be easily accomplished by placing thin sheets between the rows of filaments. Following a suggestion of Kalibjian,62Jones and Per16a have applied the idea of regenerative feedback to the problem of viewing a CsI crystal. Although lack of precise registry of successive images in practice prevents simple application of the regenerative idea, forced registration, or alternatively, alternate cycling of two image tubes in the regenerative chain, offer promising approaches for this general idea. Several commercial laboratories are currently engaged in the development of channeled image intensifiers in which secondary electron cascade (and possible subsequent photon internal regeneration) paths are restricted to small cross-section channels.

1.5. cerenkov Counters* 1.5.1. Introduction

cerenkov counters have recently been playing an increasingly important role in the detection of high-energy particles, especially in experiments performed in particle accelerators in the multi-Bev range. Not only do these Cerenkov detectors prove to be extremely useful in many counter experiments but they can also be employed in conjunction with bubble and scintillation chambers to select and identify high-energy particles as desired. $

t See also Vol. 2, Section 11.2.3. R. Kalibjian, UCRL 4 i 3 2 (1956). L. W. Jones aud M. L. Perl, Rev. Sci. Instr. 29, 441 (1958). $ Regarding the Cerenkov effect see also Vol. 4, A, Section 1.5.3.

6*

65

-

*Chapter 1.5 is by S. J. Lindenbaum and Luke C. 1. Yuan.

1.5. EERENKOV

163

COUNTERS

A cerenkov counter can be constructed from any relatively transparent optical medium which possesses an index of refraction sufficiently greater than 1 in the region of the visible spectrum and its neighborhood. When a charged particle of velocity v(cm/sec) travels in a medium of index of refraction n such that v > (c/n)-i.e., when tJheparticle velocity exceeds the velocity of light in the medium Cerenkov radiation (first observed by Cerenkov) is 'The Cerenkov photons are radiated with uniform probability along the elements of conical surfaces of angle 6 relative to the direction of motion of the particle, where 0 is given by2nS ( I .5.1)

and p = the ratio of the particle velocity to the velocity of light in vacuum. n(v) = the optical index of refraction of the medium at the frequency v of the emitted photon. The instantaneous apex of the cone passes through the position (macroscopic) of the particle. The Cerenkov radiation is polarized such that the electric vector lies in the plane formed by the photon direction and the direction of motion of the particle. The intensity of cerenkov radiation per unit length per unit frequency interval is then given by -d2N =-

dx dv

sin2 e

= 2Tz2 -sin2 0

137c

(1.5.2)

where d 2 N / d x dv is the number of photons emitted per cm of path per unit frequency interval, v is the frequency of emitted photons, e is the electron charge, Z is the ratio of the magnitude of the charge of the moving particle to the electronic charge, c is the velocity of light in vacuum in cm/sec, and h is Planck's constant.

Figure 1 summarizes the relevant features of the Cerenkov radiation. 1P. A. cerenkov, Compt. rend. acad. sci. U.R.S.S. 2, 451 (1954);Phys. Rev. 62, 378 (1937). * I. Frank and I. Tamrn, Compt. rend. aead. sci. U.R.S.S. 14, 109 (1937). 3 G. B. Collins and V. G . Reiling, Phys. Rev. 64,499 (1938). 4 H.0.Wyckoff and J. E. Henderson, Phys. Rat. 64, 1 (1938). 6 J. Marshall, Ann. Rev. N d e a r Sci. 4 , 141 (1954);CERN Sumposium, ffenevo, Proc. 2, 62 (1956).

164

1.

PARTICLE DETECTION

Figure 2 depicts the relationship between index of refraction and velocity for a series of Cereiikov angles. Figure 3 depicts the variation of Cerenkov angle with p for various fixed indices of refraction corresponding to some of the commonly available values. For most practical cases the index of refraction is relatively constant over the visible spectrum which is contained in a frequency

CHARGED

CTnRY

FIG. 1 . Relevant features of Cerenkov radiation. (Instant

1.28 1.26 1.24 1.22 1.20 1.18 1.16 n 1.14 1.12 1.10 1.08 1.06 1.04 1.02 I.00.

I \

\,,

I

=

Instantaneous.)

I

I

,975

1.00

-

BOO

1

,825

,050 .075

,900

.925

,950

B

FIG.2. Index of refraction

a8

a function of velocity

for a series of Cerenkov angles.

interval -3.5 X 10'" cycles/sec. Furthermore, practical Cerenkov counters use photons only in the visible and near ultraviolet regions of the spectrum. For a particle with 2 = 1 traveling in a medium of coilstant, index of refraction, the number of photons in the visible spectrum generated per em of path length ( d N / d z )is found by evaluating (1.5.2) to be

I

x

500 sin2 8

(1.5.3)

1.5.

165

EERENKOV C O U N T E R S

where I = number of photons generated in the visible spectrum per cm of path. Commercial photomultipliers in general use a t present, have equiva lent photocathode efficiencies of -0.05 to 0.10 electron per photon over a frequency interval approximately equal to the visible frequency interval. Therefore if all the generated Cerenkov light is collected without absorption or other loss and a conservative average photocathode efficiency of -0.05 is assumed for the photomultipliers, one obtains as the

50”-

40’-

0 60

070

0 80

0 90

10

B FIG.3. Variation of Cerenkov angle with B for various fixed indices of refraction.

resultant electrical signal, S(photoelectrons/cm), generated at the photomultiplier cathode : (phototdtrons

= 25 sin2 8.

(1.5.4)

The maxipum Cerenkov signal will obviously be obtained for any medium when the particle velocity approaches c, i.e., when 0 3 1. This will be the case, for example, for a relativistic electron 2 10 MeV. If the particle traverses wat>er (n = 1.33) one can evaluate (1.5.4) and one finds S 10 photoelectrons/cm. For Lucite, another commonly 14 photoelect,rons/t:m. For Pb loaded used medium, ( r ~= 1.5) arid 8 16 photoelectrons/cm. glass TL = 1.7 and S On the other hand a minimum ionizing particle in plastic scintillator would lose about, 1.5 Mev/cm and generate -6300 photons/Mev. *

-

-

-

* Other organic phosphors such as anthracene, stilbene, and diphenyl acetylene are either generally equal or superior to plastic scintillator in photon yield. The numerical evaluation is based on data listed in “Handhuch der Physik-Encyclopedia of Physics” (S. Fliigge, ed.), Vol. XLV, p. 145 (Nuclear Instrumcntation 11). Springer, Berlin, 1959.

1.

166

PARTICLE DETECTION

This would yield -475 photoelectronsjcm if all the light were collected. Hence for a relativistic ionizing particle of @ -+ 1 the ratio of scintillator photoelectrons/cerenkov photoelectrons n. 33 for Lucite and = 44 for water. In many practical cases the ionization loss in plastic scintillator would be several times minimum with a corresponding proportional* increase in photoelectrons/cm generated in the scintillator, whereas also in many practical cases @ < 1 which reduces the Cerenkov signal. Hence one can say that generally speaking the ratio of photoelectrons generated by charged particles in plastic scintillator/cm to that generated by Cerenkov radiation in optical media/cm is greater than -30 to 50, i.e., S(scintillator)/S(Cerenkov) > 30 to 50. From the foregoing it is obvious that Cerenkov counters would only be employed when one wishes to make use of the special characteristics of this radiation. The main use of Cerenkov counters is to restrict the velocity range of the particles counted. This can be done in the following ways. 1. Detection of cerenkov light sets a threshold for /3, i.e., @ > l/n. 2. Measurement of the angular range of the Cerenkov cone determines the generating particle velocity to lie in a range < @ < @ 2 where 81 and @ z are determined by the index of refraction of the medium and the details of the measuring system.

One might remark at this point that with the best of modern techniques signals of -5 to 10 photoelectrons on the average can be utilized to count with an efficiency approaching 100%. Average signals of even 2-3 photoelectrons have been used to count with maderate efficiency.6sCHence although the Cerenkov signal levels are much lower than scintillation signal levels it has proven quite feasible to construct many types of useful Cerenkov counters. There are two general categories of Cerenkov counters: focusing or angular selection counters and nonfocusing count,ers. In the focusing type, a system for focusing the photons in the Cerenkov cone on the detector photomultiplier or photomultipliers is included. This type allows a selection of a range of angles O1 < 8 < e2 of the Cerenkov light, The nonfocusing type most commonly used merely att$empts to collect as

* Saturation effectsfor very high specific ionization cases are neglected in the above treatment. S. J. Lindenbaum and A. Pevsner, Rev. Sci. Znstr. 26, 285 (1954). 6a Using modern high gain photomultipliers such as the RCA-6810A, the authors have found it quite easy even without further amplificationto attain efficiencieswhich correspond to counting all cases where one or more photoelectrons are generated i.e. eff. = 1 e-< where 7t is the mean number of photoelectrons generated.

-

1.5.

167

~ E R E N K O V COUNTERS

many cerenkov photons as possible onto the photomultipliers more or less independent of their cone angle. However, there is a special class of nonfocusing counters which select an angular range of cerenkov light by making use of the properties of internal reflection and the appropriate use of absorbing coatings of black paint. Both types and combinations of them have been extensively employed. Probably one of the first working Cerenkov counters was built by D i ~ k e It . ~ mas the focusing counter schematically depicted in Fig. 4. A design proposal for the type of counter Dicke used was previously made by Getting.s Dicke employed the 20-Rlev electrons from a betatron to test his counters. A 20-Mev electron t,raveling parallel to the axis generates Cerenkov radiation (as shown) which is internally reflected

EERENKOV RAY INCIDENT PARTICLE- TRAJECTORY

PHOTO M ULT IPL IE R

FIG.4. Cerenkov counter designed by Dicke.

by t he rod and cone until it leaves the base of the cone and is focused by the lens a s shown on the 1P28 photocathode. A fast particle of different velocity such that 0 differed sufficiently would not be focused a t the photocathode and hence could be discriminated against. Although Dicke probably detected Cereiikov light he was not able at the time to rule out all other possibilities. Jelleyg later achieved success with a nonfocusing water cerenkov counter shown in Fig. 5. The cone of Cerenkov light generated in the water is relected by the silver-coated glass cylindrical container, and the light enters the photocathode of the photomultiplier at the bottom directly, and also from the back side of the photocathode by reflection from the MgO cone. The black paint on the outside of the tube was removed t o allow this. Jelley showed quite clearly that the cosmic-ray counts were due to cerenkov light since he painted black the end wall of the glass container opposite the photomultiplier (i.e., the top end in the diagram) and by a R. H. Dicke, Phys. Rev. 71, 737 (1947). I. A. Getting, Phys. Rev. 71, 123 (1947). J. V. Jelley, PTOC. Phys. SOC.(London) A M , 82 (1951).

1.

I G8

PARTICLE DETE("PI0N

roincidencc method selected part,icles moving downward. He then showetl tha t rotating the counter by 180" so that the photomultiplier was on top, caused the counts in the photomultiplier to be reduced to a small fraction of their former value. This demonstrated conclusively that most of thc counts were due to light which was dirrcted forward, and cerenkov radiation is the only known possibility. GLASS END PLATE WITH BLACK PAPER

DISTILLED WATER

CONTAINER SILVERE D ON THE OUTSIDE

LIGHT TIGHT ENVELOPE

'

LIGHT GATHERING CONE COATED WITH M g 0

PHOTOMULTIPLIER

AMPLIFIER

FIG.5. Nonfocusing water Cerenkov coiinter designed hy Jelley.

Several general reviews6,l0--lla have been written on Cerenkov counters. I n the present article representative types of the most generally useful types of counters will he discussed without necessarily including all reported counters. "

1.5.2. Focusing Cerenkov Counters

Basically a focusing Cerenkov counter consists of three elements: a radiator, a focusing system including in some cases an output coupler, and a photomultiplier or series of photomultipliers. 1.5.2.1. Radiator. This is the opticd medium in xvhich the Cerenkov radiation is generated. The radiator is generally designed so as to main10 Cerenkov and other fast countcr terhniqurs. P E R N Symposium, Geneva, Proc 2, 61-103 (1956). 11 J . V. Jelley, "Cerenkov Radiation." Pergnmon, Kew York, 1958 1 1 D. ~ Blanc, " DPtecteurs de particulrs," pp. 170-187. Masson, Paris, 1950

1.5.

6ERENKOV COUNTERS

169

tail1 a defiiiite relatioil between the cone angle of Cerenkov light a i d the direction of the particle in order to allow a measurement of this angle. A common type of radiat,or consists basically of a cylinder of solid optical medium such as Lucite or glass. The bases and cylindrical surface are optically polished. The charged particles are incident on one base of the cylinder in a direction parallel t o the axis as shown in Fig. 6. If a fast charged particle does not scatter or interact, or appreciably slow down in such a radiator, the unique angle 0 of the cerenkov photons relative to the cone axis is maintained regardless of the number of reflections from the cylindrical surfare.

FIG.6 . Cercmkov rays in a radiator.

This can easily be seen since the photon momentum p originally has a component pll = p cos 8 along the cone axis and a (*omponelitp , = p sin 8 perpendicular to the cone axis. The perpendicular component can be further broken down into pI = ps p. where p+ is tangent to the cylindrical surface a t the point of contact hut perpendicular to the axis, and p. is perpendicular to the surface and the axis. lcor a specular reflection a t the cylindrical surfwe pi1 and ps are obviously unaffected, p, is reversed iii direction from an outward going normal to the surface to an inward going normal to the surfave without change of magnitude. Hence i t is obvious that the angle 8 relative to the axis is maintained and also that the minimum distance of appiwch to the axis for a skew ray is maintained. If @ < I, the critical angle i h exceeded a t the cylindrical surface and all light is internally reflected. When the photon reaches the exit base surface of the cylinder and enters the air, its angle to the axih is changed due to refraction from 0 tn 8 , where

+

( I .5.5)

If sin 8 < 1 the ray is transmitted (at least partially) to the air. However, if sin F) I the ray is interirally reflecte,! and is trapped.

>

170

1.

PARTICLE DETECTION

Since P 5 1 ; it follows that: sin 0 5 -\/nz - 1. Hence, if n 5 -\/2 there is always some transmission out of the cylinder. However, for even common optical media such as glass or Lucite, if ---t 1 entrapmeiit results. Hence in these cases an output coupling section is necessary to allow escape of the light from the radiator. In the Getting-Dicke counter (Fig. 4) the flared cone of the radiator cylinder serves this purpose. Another type of radiator for use with liquid optical media is a cylindrical container with specularly reflecting walls which is filled with a transparent liquid and is fitted with a thin glass window at the exit end to allow escape of the Cerenkov radiation. The principle of operation is similar to the Lucite or glass rod described above, except for the fact that specular reflection a t the boundary replaces the internal reflection. If a thin polished glass cylinder is used as the cylindrical container internal reflection can still be employed. One should remark that for p = 1 the internal reflection will even, in the case of the solid cylindrical radiator (Lucite), be complete only for particles exactly parallel to the axis, and hence a reflecting coating or a slight outward taper toward the front face of the cylinder to insure internal reflection may be in order if the beam divergence is appreciable. In many cases the base of the radiator where the charged particles enter is coated with black paint to absorb Cerenkov radiation of particles proceeding in the wrong direction. Gas radiators have also been employed although not to the extent of liquid or solid radiators. Gas is used for very fast particles where /3 -+ 1 and it is desired to employ a low index of refraction medium to set a high p threshold or to improve the velocity revolution de/d,B. A major advantage in a gas Cerenkov counter is the feasibility of varying the index of refraction n by simply varying the pressure of the gas and, to a lesser extent, by varying the temperature. Thus charged particles of a desired velocity or momentum can be easily selected during the course of an experiment by providing the appropriate pressure of the gas in the counter. 1.5.2.2. Focusing System. The focusing systems generally make use of n series of cylindrically or spherically symmetrical surfaces around the axis parallel to the designed-for direction of motion of the incident particle. Following the work of Jelley, Marshall6employed the focusing counter system shown in Fig. 7. The Lucite radiator is joined to a Lucite heniispherical lens which has a focal length equal to twice its radius of curvature, Therefore there is a sharp focus a t 3 radii for rays froin the radiator coplanar with the axis of the system. A cylindrical mirror constructed of glass tubing which is aluminized on the inside is inserted such that the rays strike its surface at a distance -1.5 radii from t,he radiator end so

171

h

D

172

1.

PARTICLE: D E T E C T I O N

that a sharp focus is made a t the axis of the lens. The hemispherical lens serves as a coupler to efficiently remove the photons from the radiator and avoid the total internal reflection for very fast particles ( p + I ) . For rays skew to the axis the focusing does not work due to conservation of angular momentum, and has been shown5 t o lead to an image diameter; sin e D>_nd--(1.5.6) sin 8’ where D is the diameter of the image and the equality holds under ideal conditions; n is the index of refraction; d is the diameter of the radiator; fl is the C‘erenkov cone angle and 8’ is the half cone angle of the rays which form the image. CYLINDRICAL MIRROR

>

BAFFLE

PHOTOMULTIPLIER

Lll!LL-

FIG.8. Schematic of the most commonly applied t,ype of focusing counter.

At the position of the image a photomultiplier is used a s a detector which converts the incident photon into an electronic pulse. A diaphragm can be used a t the image to limit the acceptance circle. Diaphragms can also be used in other parts of the system. Marshall has described5 variations of his counter in one of which the hemispherical lens is split by a light shield, and two plane mirrors are placed colinear with the axis to form two images on two photomultipliers such that stray eerenkov light? produced in the lens cannot lead to a roincideiice but the desired light from tjhe radiator does. Also a two photomultiplier coincidence eliminates the counts due t o a particle directly striking the photomultiplier and greatly reduces phototube noise counts. A schematic of the most commonly applied type of focusing COUIIter6~12J3 is shown in Fig. 8. It uses the cylindrical radiator and cylindrical mirror but not the hemispherical coupler. The radiator can be a solid polished cylinder of an optical medium such as Lucite, glass or quartz, or a polished glass

’*

S. J. Liudenliaiim and I,. C. I,. Yuan, (’ERN Symposic~m,OPrwiw, Z’io(. 2, (23 (1956). l 3 0 Chamhrrlain and C Wicgand, (‘ERN Syi:iposzum, Geneva, Proc 2, 6 3 (IR5ti)

1.5.

~ E R E N K O V COUNTERS

173

cylindrical shell container filled with a liquid which acts as the radiator. One can also use as a radiator a metal cylindrical shell which is filled with liquid or gas and contains a polished aluminum cylindrical inner wall, or a separate polished aluminum cylindrical mirror and an exit window a t one end with plane surfaces perpendicular to the axis to allow the photons to escape. The cerenkov light will escape from the end of the radiator only if eo is small enough. Let us denote the index of refraction of the generating medium by no, that of the glass by n,, and that of the air by n,. Also denote the angle of the Cerenkov light to the axis by 80, OQ, and 8,. Then due to the relations no sin eo = n, sin $# = n, sin 8, and the geometries used, the angle to the axis ea of the cerenkov light escaping into the air is given by sin 8, = 2 sin na

eo c-. no sin eo

and, provided 0, is real, is independent of the index of refraction of the glass exit window. 0, is real provided

Since n, = 1 this reduces to (no2 - l/p2) 5 1 < pno. Thus if Cerenkov light is generated in the radiator these inequalities are always satisfied for no I z/% For no > 4 3 , P must not be too large to satisfy these inequalities. The cerenkov photons of cone angle 8, are then reflected from the cylindrical mirror as shown in Fig. 8. The position of the image along the axis is determined by the Cerenkov cone angle 8. Provided that the diameter of the cylindrical mirror D, is 2 3 times the diameter of the radiator d, the image is not much affected by the optical aberrations of the focusing system. The magnification of the system is approximately one and the image is a circle of diameter equal to the effective radiator diameter. However, it can be shown that the effective angular uncertainty of acceptance of such a system is proportional to d/D,. Hence high-resolution counters require a large D,/d. In many practical cases, two or more photomultipliers, generally off the axis, are employed in coincidence to effectively eliminate counts due to tube noise coincidences, direct excitations of the photomultipliers by a charged particle, and stray light coming through a part of the baffle system due to a particle of wrong velocity proceeding through the radiator in the wrong direction. Such a particle may be due to a scattering or inelastic interaction of the incident particle, or due to background

174

1.

PARTICLE DETECTION

particles. Possible ways of splitting the light are indicated in Figs. 9 and 10. One should remark here that in the case of liquid or gas radiators, cerenkov light will be generated in the solid transparent exit window. u

0 2 4 6 8

INCH

HALF TRANSPARENT MIR

LINED WITH ALUMINIZED POLYSTYRENE

FIG.9. High-velocity resolution gas counter designed by Lindenbaum and Yuan.

/ / /

/

/ /

-

\

\

PHOTOMULTIPLIER

\

/

\ \

'

/

/

PLANE MIRROR

\ \

\

I--.---\'

I

'

I--/

t

----

ZERENKOV RADIATOR

ED - BLACKEN BAFFLE

FIG.10. Velocity selecting counter designed by Chamberlain, SegrB, Wiegand, and Ypsilantes.

For p + 1, if the exit window, as is generally the case, is made of glass, quartz, or other transparent optical media of n > 4 2 , the light generated by a particle parallel to the axis will be trapped in the window and can be absorbed by blackening the outer boundary of the window.

1.5.

EERENKOV

COUNTERS

175

However, a sufficiently slow particle will produce light in the window of the same cone angle as the desired signal and hence will get through the optical system. But the number of photons involved will be smaller than those of the desired signal by approximately the ratio of the window thickness to the radiator length. Hence by designing a large ratio of radiator length to window thickness, they can be discriminated against. Particles proceeding along some directions in the window may also generate an appreciable signal some of which gets through the baffle system. However, in general this can also be discriminated against by pulse height and light splitting with a double coincidence requirement. Furthermore, a directional requirement can be made by requiring one more coincidence after the particle passes through the focusing counter. 1.5.2.3. Resolution. The velocity resolving power of a focusing cerenkov counter can be expressed in terms of the partial derivative ae/ap which from Eq. (1.5.1) is

ae -ap

1 p2n

sin 0'

(1.5.7)

For counters designed for high energy resolution of relativistic particles

p -+1 and n = 1. Hence, since the variation of p with energy is very slow, large values of ae/ap are required to obtain good energy resolution. But a6 ap

N

-. 1

sin 6

(1.5.8)

-

Hence small values of 6 are required for high resolution. However, intensity const sin2 6. Therefore : ae 1 const - W v N W s1n dintensity

(1.5.9)

Hence an index of refraction must be used such that 6 - 0 in order to obtain a good resolution. However, it is also obvious from Eq. (1.5.9) that 6 must be sufficiently larger than zero to allow an adequate number of photons to be generated. Gases under variable pressure and temperature are the practical sources of these low index optical media. Figure 11 shows the index of refraction of one of the commonly used gases as a function of pressure a t several temperatures. If it is desired to cover a wide range of index of refraction from near unity to -1.2 to 1.3, then a gas with a critical temperature near or above room temperature is desirable. This allows liquefaction and the resultant high values of index of refractions to be obtained with moderate pressures, at temperatures which are not excessively high or low.

1.

176 1.20 1.18

PARTICLE DETECTION

-

co2

25.05%

321)80 40°C

1.16

1.14 c

49.71'

4

x

1.1 2

2

1.10

5

1.08

n

1.06 1.04

1.02 1.00 PRESSURE -(ATOM.)

FIG.11. Index of refraction of COZ gas as a function of pressure at several temperatures.

1.5.2.4. Practical limitations. Let us now consider the various practical limits to the resolution of a focusing cerenkov counter. These fall into three categories : (a) the finite width (AO) of the cerenkov angular cone radiated relative to the instantaneous position of the particle; (b) the deviations of the particle trajectory tangent from the optical system axis direction, and various geometrical and optical factors contained in the resolution; (c) characteristics of the incident beam of particles and the effects resulting from their interaction with the cerenkov counter itself. I n category (a) we have the following effects. 1. Difraction-The Cerenkov cone angle has a width A6 due to diffraction which essentially depends on the length of path in the radiator over which the coherence conditions are unchanged. Although it has been ~ h o w n ' ~that - ~ ~ the emission of individual photons do not affect this coherence, large enough Coulomb scattering and nuclear shadow scattering do. Lil43lSand Dedrickle have shown th at the characteristic distance which determines the diffraction width of a Cerenkov cone is much greater than l4 16

l6

Yin Yuan Li, Phys. Rev. 80, 104 (1950). Yin Yuan Li, Phys. Rev. 82, 891 (1952). K. G . Dedrick, Phys. Rev. 87, 891 (1952).

1.5.

~ E R E N K O V COUNTERS

177

the mean distance between emission of successive photons but much smaller than the total path length in the radiator material. In practical cases the diffraction width is generally negligible. 2. Dispersion-Since the index of refraction n is a function of frequency Y , it follows from Eq. (1.5.1) that there will be a dispersion width A0, introduced. The dispersion width Ad, can be estimated as follows:

ae

AB, = - A n an

(visible) =

An pn2 sin 0'

(1.5.10)

For a relativistic particle in Lucite A0, = 0.8", in fused quartz A0, = 0.6" in water AO, = 0.5'. It is obvious from Eq. (1.5.10) that At?, can become large a t small angles. In this connection it is interesting to compare the behavior of the ratio of the dispersion width AO, to the velocity resolving power a0/ap as a function of angle 8. Using the foregoing we find

that the above ratio is independent of angle. Hence, the increasing dispersion width a t small angles is accompanied by a proportional increase in velocity resolving power a t small angles. Therefore, it is generally desirable to go to small angles for increased resolution, since beam angular divergence, and Coulomb and shadow scattering widths are more or less independent of angle and generally larger than dispersion widths. In category (b) we have the following effects. 1. Scattering-Even for a particle originally parallel to the optic axis of the system, both Coulomb and nuclear shadow scattering change the direction and position of the trajectory in the counter. Since the Cerenkov photons are radiated at a polar angle 0 to the trajectory direction] the scattering of the trajectory leads to a distribution in 0 when all the photons radiated from the various parts of the trajectory are considered together. Light media of low atomic number such as water or Lucite or especially gas minimize both optical and Coulomb scattering. 2. Optical resolution-Any practical optical system has a characteristic angular resolution due to the finite size of the object, the inherent resolution limits of the optical system and optical aberrations. For the optical system shown in Fig. 8, the angular resolution can be defined as l/A0, where A 0 is the width of the range of cone angles leaving the exit window of the radiator which are transmitted with greater than

178

1.

PARTICLE DETECTION

half the peak intensity by the optical system to the photomultipliers comprising the detectors. D,/d. Hence it is For a fixed angle 8, the optical resolution l / A e obvious that a large enough mirror to radiator diameter ratio is required for high optical resolution. Practical counters used by the authors and others use a D,/d ratio -3 to 10. For these values of D,/d spherical and other optical aberrations of the system have little additional effect on the resolution. In category (c) (beam characteristics and effects of interactions) we have the following effects. 1. Ionization loss in the radiator-The ionization loss leads to a systematic decrease in p as the particle progresses through the radiator. This of course leads to a decreasing C'erenkov cone angle which gives an energy loss width term A8dr,dz to the cone angular spread. This effect is only important for thick radiators and lower energy particles. Marshall6 has shown how the use of tapered outward toward the front radiators can correct for this efiect. Another indirect effect of the ionization loss on the Cerenkov radiation is the production of 6 rays of sufficient velocity to themselves produce Cerenkov radiation. This latter effect, of course, leads to a type of general background light. However, this effect is generally not a serious background limitation in practical Cerenkov counter applications. 2. Nuclear interactionsThe special cases of Coulomb and nuclear shadow scattering which lead to small changes of particle direction have been previously considered. In addition, one can also have inelastic nuclear interactions which change the direction, energy, and type of particle as well as adding new particles. These interactions generally terminate the Cerenkov radiation pattern of the original particle a t the point of interaction, but also in many cases supply new Cerenkov light emitted by the products of the interaction which acts as a background. Actually background-producing particles can enter the counter from any point of its surface and both directly emit C'erenkov light and also indirectly via products of nuclear interactions which they induce. It is again desirable to use light media as radiators to reduce the number of nuclear interactions. 3. Beam characteristicsA practical beam of particles even if momentum analyzed so as to be nearly monochromatic in energy has both an energy spread, and an angular spread. These two effects obviously lead to a spread AObeamin the cerenkov cone radiated which limits the practical velocity resolution of the counter. 4. Magnetic J i e l d s T h e presence of strong stray magnetic fields can cause a curvature of the radiating charged particles path and hence impart a distribution to the Cerenkov cone angle relative to the axis of

-

1.5. EERENKOV

COUNTERS

179

the system. This effect is not important in most practical cases. Stray electric fields can also in principle modify the cerenkov cone angle but the fields usually encountered are too weak. 1.5.2.5. Photomultipliers.* The photomultiplier characteristics most useful for application to Cerenkov counters are the following. 1. End window semitransparent photocathode type of large enough cathode area to efficiently cover the image in a focusing type counter and collect as many photons as possible. For nonfocusing counters the large area end window type are also desirable for highest detection efficiency. 2. A high efficiency for converting photons to photoelectrons over as wide as possible a section of the visible and ultraviolet spectrum as is transmitted by the radiator. A peak of conversion efficiency in the blue or ultraviolet is in general desirable. 3. As high a gain as feasible to reduce the need of electronic amplification of the small Cerenkov pulses. 4. A good signal-to-noise ratio. 5. Preferably a small spread of transit time from photocathode to first dynode structure) and a small time spread in the photomultiplier structure, in order to take advantage of the very short time spread in the cerenkov pulses. The authors) experience has been that the fourteen-stage 56AVP (Philips) and the 6810A and 7264(RCA) represent a reasonable compromise with the above requirements for general purpose use. There are many other phototubes manufactured by Dumont, EMI, RCA and others which are more suitable for particular cases. Some of these are described in the various references given for individual counters. I n particular the 5-in. and 16%. diameter RCA and Dumont phototubes are useful for large counters. 1.5.2.6. Some Practical Focusing Counters. Figures 9 and 10 show various practical focusing counters of the type depicted schematically in Fig. 8. Figure 9 shows a type of high-velocity resolution counter with both a liquid and a gaseous radiator which was first constructed in 1952 by the authors and tested at the Brookhaven Cosmotron.lEs The gas generally employed is C O z which can be varied continuously in index of refraction over the range 1.004 to 1.21 by varying the pressure over the range 0 to 200 atmospheres and the temperature over the range 25" to 50°C (see Fig. 11). *See also Vol. 2, Section 11.1.3. Recent modifications include an anti-coincidence channel to improve background rejection when K mesons are detected in the presence of a large r-meson background and these changes are not shown. S. Ozaki a.nd J. Russel have collaborated with the authors in these modifications. 16*

180

1.

PARTICLE DETECTION

At 25"C, a pressure of approximately 75 atmospheres liquefies the CO, which then has an index of refraction -1.2. Hence when the counter is set for a particular angle of detection, the index of refraction and hence the velocity interval accepted can be changed a t will. For the counter shown, the mean value of P accepted can be varied from = 0.83 to /3 = 1.00 with a velocity resolution AD = ,0.005. Since the cerenkov cone angular range is not changed, the photon intensity and geometrical resolution properties are constant as the index of refraction is changed. Hence also the efficiency of detection is approximately constant. When used a t detection angles of -10" the efficiency is 290%. The liquid radiator can be used with Minnesota Mining & Mfg. Co. fluorochemical 0-75 with a variable temperature to cover the range n = 1.26 to 1.31. Water, sugar water, and then various standard liquids can be used to cover the range n = 1.33 to 1.7. A list of the index of refraction of some solid and liquid substances is shown in Table I. TABLE I. Index of Refraction of Solid and Liquid ~

~~~

Index of refraction Material

nd

Solid (at 18°C) Fused quartz Polymethyl methacrylate (Lucite) Quarts Polystyrene Glass (ordinary Crown) (light fiint) (dense flint)

1.458 1.489 1.493 1.5443 1.592 1.517 1.580 1.655

Liquid (at -20°C) Fluorochemical FC-75 Water Paraldehyde Carbon tetrachloride Toluene Benzene Chlorobenzene Carbon-di-sulfide

1.276 1.333 1.405 1.46 1.497 1.501 1.525 1.630

-

~~~

Reciprocal dispersion V = (na - l ) / ( w - nJ 65 49 70 30 60 42 34

56 49 30 31

The small gaps that exist are easily filled by using the movement of the photographic bellows to change the angle of detection. With a particular liquid the mean value of fi accepted can be varied over a considerable range by changing the distance between the light splitter and the radiator via the bellows. Although the efficiency can in principle change

1.5.

EERENKOV COUNTERS

181

as the angle of detection is changed this effect is small since theefficiency can be made close to 100%. The velocity resolution however does change somewhat in a calculable way with angle. The major use of such a counter is as a mass spectrometer wherein small changes of resolution are not too important. A velocity selecting counter of this general type was employed by Chamberlain et ~ 1 . 'in~ their discovery of the antiproton at Berkeley. Figure 10 shows a diagram of their counter. STAINLESS CYLINDER 4 " 1.D x 8" LONG I" QUARTZ

WINDOW

FRONT SURFACE

TWO ELEMENT LUCITE LENS

RING APERTURE PHOTOMULTIPLIER C7170 (RCA)

FIG.12. Focusing counter using gaseous or liquid fluorochemical 0-75 designed by Baldwin et al.

Using a fused quartz solid radiator and a cylindrical mirror arrangement as shown in Fig. 10 they were able to attain a velocity resolution such that when AD 0.03 the counting rate dropped to -3% of the peak value. This counter was used as an element in a counter telescope to select antiprotons from negative pions in a momentum analyzed beam. A gas focusing counter using gaseous or liquid fluorochemical 0-75 (normally liquid) at elevated temperatures and high pressures has been designed and used by an M.I.T. group, Baldwin et ul.18 A diagram of this counter is shown in Fig. 12. By varying the temperature up to -255°C

-

l7 0. Chamberlain, E. SegrB, C. Wiegand, and T. Ypsilantis, Phys. Rev. 100, 947 (1955). . E. Baldwin, D. Caldwell, S. Hamilton, L. Osborne, and D. Ritson, Scintillation Counter Conference, Washington, D.C., January, 1958; D. Hill, private communication.

182

1.

PARTICLE DETECTION

and the pressure over the range 1-20 atmospheres the index of refraction can be varied from near unity to -1.28. The optical system employed essentially consists of a lens system with Schmitt correction which focuses light of a particular Cerenkov polar cone angle relative to the lens axis into a narrow ring located approximately one focal length behind the lens. For small angles the ring diameter is proportional to the Cerenkov cone angle and hence a small annular disc opening before the photomultiplier restricts the photons accepted to a small Cerenkov cone angular interval.

FIG.13. Focusing liquid counter employing a spherical mirror, designed by Huq and Hutchinson.

One advantage of this system is that for obtaining high-velocity resolution with large diameter radiators the necessity for a large cylindrical mirror can be avoided. However the alignment of the lens system must be rather carefully performed. This counter has been used in the detection of K+ mesons in positive analyzed momentum beams a t the Bevatron. The velocity resolution is such that the counting rate dropped by a factor of 20 when Afl 0.006. Another variation of the ring type of focusing counter employing a spherical mirrorIg instead of a series of spherical lenses is shown in Fig. 13. This counter is particularly suited for liquid radiators. The liquid radiator fills the space between a spherical mirror through which the beam enters and a plane mirror face through which the beam leaves. Both mirrors are front silvered. After reflection from the plane

-

1s

M. Huq and G. W. Hutchinson, Nuclear Instr. and Methods 4, 30 (1959).

1.5. EERENKOV

COUNTERS

183

mirror the Cerenkov cone is focused into a ring defined by an annular stop of inner radius equal to 4.25 cm and external radius of 4.5 cm. The focused ring diameter is an increasing function of Cerenkov angle. The counter described was used in a 900-Mev proton beam and exhibited an over-all angular resolution of for a cone angle -35'. This corresponded to an energy resolution -54%. A counter similar in principle using ethylene gas as the radiator has been designed by von Dardel and his co-workers*Oat CERN t o be used for experiments with the 25-Bev alternate gradient synchrotron beam. Operating a t a gas pressure of up to 70 atmospheres (at room temperature) they were able to separate antiprotons from k- and T- up to 16 Bevlc momentum. A drawing of this counter is shown in Fig. 14a. A counter designed by the authors and their co-workers for particle separation at the Brookhaven 32 Bev Alternate Gradient Synchrotron is shown in Fig. 14b. The major differences from the CERN counter is the use of COS gas and the addition of an anti-coincidence channel which collects the n-meson light when K-meson or anti-proton light is tuned into the signal channel. This technique greatly reduces the background level. One should remark a t this point that cerenkov radiators and optical systems of the focusing type with film or other integrating detectors2I have been employed to measure the velocity distribution of particles in a beam. Since we are concerned here with cerenkov counters we shall not describe these. There is a special class of velocity interval selecting counters which do not use focusing. The lower velocity limit is set by the threshold and the upper velocity limit by the internal reflection and subsequent absorption of light with a cone angle greater than ec where ec is that angle for which total internal reflection occurs. A counter of this type designed by Fitch and Motleyzz is shown in Fig. 15. The velocity range selected is 0.65 < p < 0.78. From the discussion in Section 1.5.2.2 it is clear that n L .\/z is required in order for total internal reflection to occur at the exit face. This is the major limitation of this kind of velocity interval selector. One is restricted to a low threshold velocity and also one has a much poorer resolution than obtainable in the focusing type of counter. However the simplicity of the device is of course an advantage for those cases where it can be used. Another variation of velocity interval selection can be obtained by

*+'

G. von Dardel, private communication (1960). R. L. Mather, Phys. Rev. 84, 181 (1951); J. V. Jelley, AERE NP/R 1770 Atomic Energy Research Establishment, Harwell (1955), unpublished. 22 V. Fitch and R. Motley, Phys. Rev. 101, 496 (1956). 20

21

FIG.14a. Focusing gas counter designed by von Dardel et al.

signal light; (F), special collecting mirror for Cerenkov anti-coincidence light; ( G ) , reflecting mirror for directing light into photomultipliers; (a), RCA type 6810A or Amperex 56UVP photomultiplier; (I), 4 RCA type 6810A or Amperex 56UVP photomultiplier connected in parallel.

186

1.

PARTICLE DETECTION

using black paint or other arrangements to eliminate the largest cerenkov cone angles corresponding to the highest velocities and of course a threshold velocity is set by the index of refraction. A counter of this type has been designed by Hughes, Palevsky, and co-workers23 for use in detecting high-energy neutrons. The counter consists of a high-pressure COZ cylinder which allows a variable index of refraction to set a variable threshold velocity; the cylinder is painted with black paint so that only small angle light will escape. BLACK PAINT

FIQ.15. Velocity interval selecting counter designed by Fitch.

1.5.3. Nonfocusing Counters The counter described in Fig. 5 constructed by Jelley represents one general type of nonfocusing counter, namely what can be. called the “end on type.” This ‘counter is mainly suitable for use as a last element in a telescope, as the thickness of material in the counter and the presence of the phototube in the beam do not make it convenient as in one of several counter elements in a telescope. A relatively thin transmission type nonfocusing cerenkov counter has been designed by Lindenbaum and Pevsner.s In this counter two 23

D. Hughes, H. Palevsky, and co-workers, private communication.

1.5. EERENKOV

COUNTERS

187

5819 RCA photomultiplier tubes faced the sides of an aluminum liquid container of 3 in. X 3 in. cross section and 14 in. thick with 31-mil walls. Liquids of various indices of refraction were used as cerenkov radiators. The ends of the phototubes were immersed directly in the liquids so that the semitransparent photocathode was covered by liquid. O-ring seals around the tubes were used to make a tight seal. Aluminum foil lined the inside of the container except for the ends of the phototubes. This counter was used as an element in a counter telescope in an 87-Mev negative pion beam which had been momentum analyzed. Differential range curves taken in this beam with turpentine (n = 1.475), ethylene glycol (n = 1.427), and water (n = 1.33) all exhibited the usual T - and pmeson peaks appropriately shifted to correspond to a velocity threshold a t that cerenkov angle which provided -2 photoelectrons at the photomultipliers. The only observable background was small and due to accidental coincidences. The absolute counting efficiency obtained was greater than 90% relative to filling the counter with a scintillator. There was no evidence for any background counts due to scintillation or other noncerenkov counts. It has been the general experience with Cerenkov counters that, except for substances that tend to scintillate, the cerenkov effect is large compared to general light background due to other sources even for nonfocusing counters. A number of improved versions of this type of counter using 6810A 14-stage RCA photomultipliers have been designed and employed by the authors at the Brookhaven Cosmotron for the past few years. A typical example is shown in Fig. 16. Counting efficiencies of -95-98 % have been attained for relativistic particles. The counter output was amplified and limited with one distributed 140 mc amplifier with 18 db gain. The output of the amplifier was fed to one grid of a 6BN6 dual grid tube coincidence circuit. Another counter element or series of elements were put in coincidence with the Cerenkov counter via the other grid. The inside of the counter is coated with white reflecting paint to diffuse the light or in some cases coated with aluminum foil. A gas counter of this type designed to use CO:, mainly is shown in Fig. 17. The mirrors increase the efficiency of light collection at very small angles. The pressure can be varied from one to 200 atmospheres. The temperature can be regulated from 0" to 100°C. This allows the index of refraction to be varied continuously from 1.004 to 1.21. The liquid version of this counter (Fig. 16) allows the index of refraction to vary with suitable liquids over the range 1.33 to 1.7. The index of refraction of fluorochemical o-75 can be varied over the range 1.26 to

188

1.

PARTICLE DETECTION

1.32 by varying the temperature. Hence except for some small gaps most of the index of refraction range 1.004 to 1.7 can be attained by these counters. A series of these transmission type counters were employed by the authors24 a t the Brookhaven Cosmotron in an investigation of positive SCALE 1:2

LIT€ WINDOW

DIRECTION OF BEAM

-

-

-

-

-

FIG.17. Nonfocusing gas counter designed by Lindenhaum and Yuan.

K-meson production in positive proton collisions. After momentum selection via magnetic deflection, a velocity interval is selected by requiring a coincidence in a counter with a threshold p1 and a n anticoincidence in a counter with a threshold pz where pz > pl. Hence the velocity range selected is represented by: p1 < p < p2. 2 4 S. J. Lindenbaum and L. C. L. Yuan, Phys. Rev. 106, 1931 (1957).

1.5. EERENKOV

COUNTERS

189

Such a coincidence anticoincidence pair then acts as a mass spectrometer in a momentum analyzed beam, and can be employed t o detect only K + or another mass component in the beam. To insure a high efficiency in the anticoincidence, 2 or 3 counters are used. As a matter of fact the major practical use of the focusing type of Cerenkov counter is also to act as an element which by selecting a velocity interval in a momentum analyzed beam a t a high energy accelerator selects a particular mass of particles. Another major advantage of Cerenkov counters is that they do not detect background due to low velocity particles (i.e., below their threshold). Hence the accidental counting rates are reduced and jamming is avoided. Heiberg and Marshallz6and also Porter26have reported using a fluorescent material additive to a water Cerenkov counter so that the violet and ultraviolet components of the radiation can be transformed into a nondirectional light of more usable wavelength for the photomultiplier. Gains of less than a factor of two have been realized in certain cases with this technique. Atkinson and Perez-Mendez have reportedz7a gas Cerenkov threshold device for discriminating against inelastically scattered pions in a negative pion momentum analyzed beam.

1.5.4. Total Shower Absorption eerenkov Counters for Photons and Electrons*

Kantz and Hofstadter first suggested28 the principle of using a total absorption cerenkov counter to measure the energy of a photon or electron of energy >,100 MeV. The basic idea is that a block of a relatively clear optical medium of short radiation length such as lead loaded glass, with dimensions equal to many radiation lengths is used as a shower producing medium for a photon or electron entering near its center. If the block is large enough, the mean total path length of electrons and photons is approximately linearly related to the energy of the incident photon or electron. Since for electrons of energy 2 several Mev the mean number of Cerenkov photons emitted per unit path length is independent of the energy, the mean number of Cerenkov photons emitted in the counter is also a linear function of the incident energy. If the side walls of the block are reflecting for the shower photons, and the front end is

* Refer to Section 2.2.3.7. 26

E. Heiberg and J. Marshall, Rev. Sci. Znslr. 27, 618 (1956).

** N. Porter, Nuowo cimento [lo] 6, 526

(1957). H. Atkinson and V. Perez-Mendez, Rev. Sci. Znstr. SO, 864 (1959). 28 A. Kantz and R. Hofstadter, Nucleonics 12, (3), 36 (1954); R. Hofstadter, CERN Symposium, Geneva, Proc. 2, 75 (1956). 2’J.

190

1.

PARTICLE DETECTION

optically coupled directly to one or a series of large photocathode photomultipliers, a sizeable fraction of the photons generated will strike the photocathodes. A fraction of the light emitted will be reabsorbed before reaching the photocathodes. Both this reabsorption and the fraction of photons collected on the photocathode will be only slightly dependent on the incident energy over the energy range for which a well-designed counter is useful. Hence the mean number of photoelectrons generated at the photomultiplier photocathodes will be approximately a linear function of the energy of the incident photon or electron. Known energy electron beams can be used to calibrate the counter. Obviously there will be appreciable fluctuations from the mean number of photoelectrons for individual showers caused by the same energy electron. These fluctuations arise from the stochastic nature of the shower, the partial loss of electrons from the counter even in large counters, the fluctuations in Cerenkov photon emission, photon collection, and photoelectron production. The over-a11 effect of the fluctuations can be expressed in terms of the per cent energy resolution. Although different definitions have been employed, a convenient one is the full width at half-maximum of the counter response to monoenergetic electrons divided by the energy. The most commonly used type of total shower absorption Cerenkov counters employ Pb-loaded glasses or heavy crystals which are nearly colorless. A typical glass of this type is manufactured by the Corning Glass Co. It has a radiation length of 1-in., a density of 3.9, an index of refraction of 1.65, and a critical energy of 16 MeV. The critical energy is that energy at which ionization loss equals radiation loss. The Schott glass works in Germany makes two varieties of Pb-loaded glass suitable for total absorption counters. The lighter one, type SF-1, is clear white, contains 62% PbO, has a density of 4.44, and a radiation length of 2.0 cm. Counters of this type have been designed and utilized by Kantz and Hofstadter12*C a s ~ e l s , ~ ~ Brabant et u Z . , ~ Swartz13* ~ Filosofo and Y a m a g a t ~Koller , ~ ~ and S a c h ~and , ~ ~others. Jester30 employed a 12-in. diameter Corning glass cylinder 14 in. long (2 optically coupled 7-in. cylinders). Four 5-in. diameter Dumont 6364 photomultipliers were placed with their cathodes in optical contact with J. M. Camels, CERN Symposium, Geneva, Proc. 2, 74 (1956). M. H. L. Jester, Univ. of California Radiation Laboratory, Report No. 2990 (1957). 91 J. M. Brabant, B. J. Moyer, and R. Wallace, Rev. Sci. Znstr. 28, 421 (1957). C. Swartz, I R E Trans. on Nuclear Sci. NS-8, 65 (1956). 33 I. Filosofo and T. Yamagata, CERN Symposium, Geneva, Proc. 2, 85 (1956). 34 E. L. Koller and A. M. Sachs, Phys. Rev. 116, 760 (1959). 29

30

1.5.

~ E R E N K O V COUNTERS

191

one end of the cylinders using Dow-Corning Silicone No. 200 as the optical coupling. He has reported obtaining a linear response from 50 to 200 Mev with a resolution of -45%. An improved version of Jester’s and its linear response until counter was developed by Brabant et -1.5 Bev was demonstrated. A typical counter of this type designed by Swartz a t Brookhaven is shown in Fig. 18. The observed resolution was better than 30% for 400-Mev electrons.

FIG.18. Typical shower detector designed by Swartz.

Another obvious way to obtain a linear relation with electron or photon energy is to use a total shower absorption scintillation counter. Various versions of this type have been reported10*11.31utilizing NaI at low energies ( 5100 Mev), ordinary and heavy liquid scintillators, and combinations of liquid or plastic scintillators sandwiched between Pb plates to reduce the radiation length. In general the cerenkov type appears more likely to provide a compact, high resolution, low background, higher absolute energy accuracy, counter for the several hundred Mev to several Bev region. 1.5.5. Other Applications Several useful applications of cerenkov counters which have not been discussed are the following.

192

1.

PARTICLE DETECTION

1. As a directional device-All focusing counters are highly directional. Furthermore front to back discrimination is easily attained with nonfocusing counters by painting the back end of a cylindrical radiator black. Wincklerss used this method to measure albedo in the atmosphere. Various other methods of obtaining directional sensitivity are also possible. 2. Since the number of Cerenkov photons is proportional to Z2 for a known velocity particle, various charged particles can be separated by pulse height. There is some advantage over scintillators in that the broad Landau ionization loss distribution does not exist. Of course the statistical fluctuations of the number of photons and photoelectrons are larger in the Cerenkov case due to the smaller numbers. Nevertheless one can probably do better in many cases with a cerenkov counter than a scintillator. 3. A s a source of very fast light pulses for very short resolution time of flight work-A particle traveling along the axis of a radiating cylinder like that in Fig. 1 will produce a light pulse a t the front face of the cylinder of width in time equal to:

(-p

At8o,, = 1 v

1

e - 1)

COS~

(1.5.10)

where 1 is the cylinder length in cm, v is the particle velocity in cm/sec, and 0 is the Cerenkov cone angle. The cause of this time width is due to the fact that photons emitted in the interior of the radiator arrive later a t the exit than those emitted at the exit face. This is due to two reasons: (a) the velocity of light is less than the particle velocity; (b) the particle travels along the axis while the photons travel a t angle 8 to the axis. For a relativistic particle p + 1 (1.5.11) For a Lucite radiator of 2411. length At

-

sec.

For a gaseous cerenkov radiator of 6411. length operated a t a 10" Cerenkov angle with p 4 1, At

-

2X

J. R. Winckler, Phys. Rev. 86, 1034 (1952).

sec.

1.5. EERENKOV

193

COUNTERS

An optical system of the type shown in Fig. 8 will in principle approximately preserve the value of At for a photocathode placed in the image plane. In this respect one should note that an additional path length of 1.2 in. second and a n additional path length of in air gives a delay of -1O-lO 0.12 in. gives a delay of -10-l1 second. Hence great care must be taken to maintain a pulse width of to 10-l1. Of course there is no existing production type photomultiplier of end type photocathode of sizeable area which will allow one to transform these short time width light pulses into electrical pulses without appreciable lengthening. The best photomultipliers presently available would lead to a width of one to several millimicroseconds sec) for a n instantaneous light sec pulse. Although developmental types may provide widths of -1O-lO in the near future. Techniques employing rf gating of the first dynode for short intervals to reduce timing errors to -10-10 sec have also been considered. I n this connection one should note that the other general and older method for measuring velocity is by electronically measuring the time of flight between two counters. Presently available photomultipliers and ordinary coincidence and chronotron techniques allow a time of flight measurement of a n accuracy close to 10-lo second, when one demands counting each individual particle with a moderate efficiency. Therefore for a typical relativistic particle ( p 4 1) timed over say 10 f t , which is a typical telescope distance, we would have A@ 0.1. If the distance were increased to -50 f t we would have Ap 0.01. With a 50-ft distance and an accuracy of timing of -1O-lO sec we would have A 0 0.001. Of course short lifetime particles cannot be effectively counted over such large distances. I n order to generate pulses with sharp enough timing to maintain 10-10 sec coincidence resolution, entails a series of severe problems. Also changes of signal path length of -1 in. would bring the counters out of coincidence. Gas cerenkov counters of the focussing type can in th e future be expected with proper design and precautions t o reach velocity resolutions of AD < 0.001. This would allow one, for example, in a 20 Bev/c beam a t the Brookhaven AGS or CERN strong focusing 25-Bev proton accelerators, t o separate antiprotons from all other known particles. * From the foregoing one can conclude th at cerenkov counters appear most promising in providing highest velocity resolution. Furthermore they can be used with existing photomultipliers and electronic techniques

-

* Refer to Section 2.2.1.3.

- -

194

1.

PARTICLE DETECTION

most conveniently, and do not require large time of flight distances. Finally their direct velocity selectivity makes them extremely useful in reducing background and pileup problems.

1.6. Cloud Chambers and Bubble Chambers* Cloud chambers and bubble chambers are used to make visible the paths of high-speed charged particles. In cloud chambers, the track of the particle is formed when a supersaturated vapor condenses preferentially on the ions formed by the charged particle as it passes through a gas. Droplets formed on the ions grow large enough so that, with the proper illumination, they are visible and can be photographed. Bubble chambers operate quite differently. The path of the particle is delineated by the bubbles formed when a charged particle passes through a superheated liquid. Energy deposited along the track by the ionizing particle creates locally heated centers around which bubbles of vapor start to grow. When these bubbles reach a suitable size they, too, may be illuminated and photographed. The most important basic difference to be noticed between cloud chambers and bubble chambers is that the former operate with gases, and the latter with liquids. There are many other differences, aside from technical operating problems, and they will be discussed in a section concerned with the advantages and limitations of each method. We will now treat only the fundamental principles involved in the operation of these devices.

1.6.1. Cloud Chambers The supersaturation of vapor needed to form droplets on ions may be obtained in two ways: (1) by the rapid expansion of a volume of gas containing the vapor (expansion cloud chamber) ; and, (2) by the diffusion of vapor from a warm region where it is not supersaturated to a cold region where it is supersaturated (diffusion cloud chamber). The technical problems of construction and operation of these two types of cloud chambers are quite different. However, once supersaturation is obtained, by whatever method, the formation of the droplets proceeds according to well-known thermodynamic principles. *Chapter 1.6 is by W. B. Fretter.

1.6.

CLOUD CHAMBERS AND BUBBLE CHAMBERS

195

The theory of the formation of liquid droplets from a supersaturated vapor, treated by J. J. Thomson’ in 1888, and developed by various other physicists is summarized by J. G. Wilson in his excellent treatise on cloud chambers,2 and by Das Gupta and Ghosh3 in their review article. More recent developments in cloud-chamber technique are described in a report4 of a conference on cloud chambers. Liquid droplets may form in a supersaturated vapor on nuclei present in the gas or, if the supersaturation is sufficiently high, spontaneously on microscopic fluctuations in density in the vapor. The latter process determines the upper limit of supersaturation desirable to attain in an expansion type cloud chamber, but is not of use in track formation. The nuclei present in cloud chambers, upon which droplets form at lower supersaturations include ions, both those of the track and others formed in the chamber, foreign suspended particles such as dust, chemical compounds which may act as condensation nuclei, and re-evaporation nuclei. The latter are produced by the evaporation of large droplets to a point where further evaporation ceases. Of all these condensation nuclei, the only ones wanted to exist in the chamber a t the time of production of supersaturation and subsequent photography are the first, those ions produced by the passage of the charged particles. All others must be cleared from the chamber in various ways. 1. Unwanted ions are removed by an electrostatic clearing field. 2. Dust particles and re-evaporation nuclei are removed by production of supersaturation successively, in the case of expansion chambers, and continuously, in the case of diffusion chambers, until the nuclei are carried to the bottom of the chamber, where they adhere to the wall. The condensation of vapor on ions involves in the first approximation the dielectric constant of the liquid and the external medium, the surface tension of the liquid, the molecular weight of the vapor, and the degree of supersaturation of the vapor. The theory is given by Wilson2 and only the broad outlines will be given here. The effect of the charge is to modify the surface energy of an incipient droplet in such a way as to permit it to grow by the condensation of molecules out of the vapor. If the supersaturation, that is the ratio of the vapor pressure existing ( p ) to the 1 J. J. Thomson, “Applications of Dynamics in Physics and Chemistry.’’ Macmillan, London, 1888. 2 J. G. Wilson, “The Principles of Cloud-Chamber Technique.” Cambridge Univ. Press, London and New York, 1951. a N. N. Das Gupta and S. K. Ghosh, Revs. Modern Phys. 18, 225 (1946). 4 ‘‘Report of the Conference on Recent Developments in Cloud-Chamber and Associated Techniques, March, 1955’’ (N. Morris and M. J. B. Duff, eds.), University College, London, 1956.

196

1.

PARTICLE DETECTION

equilibrium vapor pressure (PO)at the temperature after the expansion is sufficiently high, charged droplets will grow and continue to grow until other limitations occur. The value of the supersaturation necessary for drop formation on ions depends on the nature of the vapor and the sign of the ion. The latter fact indicates that, in the initial stages of formation of the drop, the polar nature of some of the vapors used plays an important role. For example, water vapor condenses preferentially on negative ions, and higher supersaturation is needed for condensation on positive ions, while for ethyl

EXPANSION RATIO

FIQ. 1. Positive and negative ion thresholds, 70% ethanol and 30% water, in a cloud chamber filled with oxygen. Curve at left is for positive ions; curve at right is for negative ions.

alcohol, the opposite effect occurs, as is shown6 in Fig. 1. The value of the supersaturation required2 varies from p / p o = 4.14 in water to 1.94 in ethyl alcohol, to name two commonly used vapors. Mixtures of alcohol and water are also sometimes used, in which case the value of p / p ~may be as low as 1.62. The rate of growth of drops from a supersaturated vapor determines the length of time required for a drop to reach visible (or photographable) size. Rapid growth makes possible short photographic delay times, thus minimizing distortion effects due to motion of the gas, and it also is desirable in producing large droplets which fall quickly to the bottom of the chamber, leaving no residue of re-evaporation nuclei. The rate of drop growth has been studied by Hazen" and by Barrett and Germain.' C. E. Nielsen, Ph.D. Thesis, University of California, Berkeley, California, 1941. (1942). '0. E. Barrett and L. S. Germain, Reu. Sci. Instr. 18, 84 (1947).

* W. E. Hazen, Rev. Sci. Znstr. 13, 247

1.6.

CLOUD CHAMBERS AND BUBBLE CHAMBERS

197

Theoretically, the drop growth is determined by the diffusion of vapor toward the growing drop and by the conduction of heat away from the growing drop. Practically, it has been founda that the heat conductivity of the gas is the predominant factor in ordinary operation of cloud chambers near atmospheric pressure. The rate of growth of droplets in xenon, which has a very low heat conductivity, is very small, but if a n equal pressure of helium, with high heat conductivity, is added to the xenon, the drop growth is nearly as rapid as it is in pure helium. Kepler et al. also found th at the rate of drop growth is not very dependent on the gas pressure in the range 0.2 atmos to 1.4 atmos, indicating th a t diffusion processes are not limiting the growth. In practice, the time required for a drop to reach visible size is between 50 and 250 milliseconds, depending on the observational conditions. The diameter of the drops in a cloud chamber, a t the instant of photography, is of the order of 10W cm. 1.6.1.1. Expansion Chamber. Supersaturation is produced by a rapid, nearly adiabatic expansion of the mixture of gas and vapor. The drop in temperature during such an expansion is given by T1/T2= ( V Z / Z J I ) Y - ~ where y = c,/c,, the ratio of specific heats a t constant pressure and dependconstant volume of the gas mixtures, or Tl/T2 = (pl/pz)(~-l)’u, ing on whether the expansion is volume-defined or pressure-defined. Here T lis the initial (absolute) temperature, T 2 the final temperature, v 1 the initial volume, and v 2 the final volume, with p l and pz the corresponding pressures. The change in temperature is clearly dependent on y, and the large value of y for monatomic gases makes them desirable for use in cloud chambers. Most cloud chambers in current use are volume-defined, generally by mechanical means, but pressure-defined cloud chambers, with only a thin rubber diaphragm separating the pressure vessels, are sometimes used, especially in cloud chambers containing metal pIates. There are .many factors to consider in the design of a n expansion cloud chamber, aside from the purely mechanical ones, and those connected directly with the experimental setup. Some of these factors are discussed briefly a s follows. 1.6.1.1.1. PRESSURE AND TYPE OF GAS. Cloud chambers may be operated a t widely varying pressures, from a fraction of a n atmosphere, to 50 atmospheres. At low pressure, the vapor becomes a n appreciable fraction of the total amount of gas present, changes the value of y, and provides an appreciable amount of ionization. I n the range of pressures from 0.1 atmos to 2 atmos, the operation of a cloud chamber is not 8 R. G. Kepler, C. A. d’Andlau, W. B. Fretter, and L. F. Hansen, Nuovo cimento 1101 7, 71 (1958).

198

1.

PARTICLE DETECTION

difficult. At higher pressures, the chamber becomes more difficult to clear of old droplets, and the time necessary to wait between expansions increases as the operating pressure increases. Scattering of the passing particle by high-density gas reduces the accuracy of curvature measurements if the chamber is placed in a magnetic field. On the other hand, the sensitive time of high-pressure chambers is long compared with chambers operated near atmospheric pressure. A good discussion of the properties of high-pressure chambers has been given by Burh0p.O When the type of gas is not specified by the experiment to be done, noble gases are preferred because of the lower expansion ratio required. Argon is most commonly used, or a mixture of half argon and half helium which gives easily visible tracks and rapidly growing droplets. 1.6.1.1.2. USE WITH MAGNETIC FIELD.Very often information on the momentum of the tracks passing through a cloud chamber is required, and the cloud chamber must be placed in a magnetic field. One factor to be considered here is the design of the chamber and its expansion mechanism to make the best use of the field. See Fig. 2. It is desirable to make the expansion mechanism occupy the least space possible, and often most of it can be placed outside the magnet, with a rod or tube leading through the iron to compress or expand the chamber. The magnetic field should be as uniform as possible to avoid large corrections in the momentum, and accurate fiducial marks should be provided in the chamber to serve as points of reference. Generally the larger the magnetic field, the better, since the spurious curvatures produced by scattering and by motion of the gas are relatively less import,ant if the field is large. Wilson2 summarizes the relative importance of these two types of error for various lengths of track and magnetic field values. The term “maximum detectable momentum” is often used as an index of performance of a chamber. This is the particle momentum for which the true curvature is equal to the uncertainty in curvature, and for a chamber where tracks of length about 40 cm can be measured in a field of lo4 gauss, the maximum detectable momentum under very good conditions may be as high as 50 Bev/c. 1.6.1.1.3. USE WITH PLATES. Since the stopping power of gas is so low, and the probability of nuclear interaction in a typical cloud chamber is very small, it sometimes is desirable to place sheets of heavier material in the chamber, leaving gas spaces between, in which the tracks may be seen. The minimum distance between such plates is about 3 in., and they must be coated with reflecting materials to increase the light scattered by the droplets. Such multiplate chambers have proved valuable in investigations of nuclear reactions, and the short-lived unstable E. H. S. Burhop, Nuowo dmento [9] 11, Suppl. No. 2 (1954).

1.6.

199

CLOUD CHAMBERS AND BUBBLE CHAMBERS

particles produced in these reactions. The range of particles can be measured if the particle stops in one of the plates, and scattering in the plates can also be determined. If y rays traverse the chamber, the probability of pair production may be increased if plates are introduced, and

A I

REAR

C

\ i? APPROX

FRONT WINDOW

,//Y,///; .// B

,/.v

//+

6

1

SCALE IN INCHES

2

3

4

FRONT

FIG.2. Cloud chamber designed for use in a magnetic field. The back plate of the cloud chamber moves to produce the expansion. (A) Vertical section parallel to front. (B) Vertical section parallel to side. (C) Horizontal section.

details of nuclear reactions can be observed. Multiplate chambers do not operate well in regions of high background unless sufficient shielding is available. A multiplate chamber has been operated near the Bevatron'O with a strong pulsed electric field between the plates, which reduced the ion background to the point where counter control may be used. 1oR. W. Birge, H. W. J. Courant, R. E. Lanou, and M. N. Whitehead, Univ. of California Radiation Laboratory Report UCRL-3890 (1957).

200

1.

PARTICLE DETECTION

1.6.1.1.4. COUNTERCONTROL.Expansion cloud chambers may be operated a t random, with a repetitive cycle accelerator, or with counter control. Normally counter control is used for cosmic-ray experiments. Here a particle passes through the chamber and associated counters, giving a signal which triggers the expansion of the chambers.* Since the speed of expansion is of the order of 10 milliseconds, counter controlled tracks are broadened by diffusion to widths of about one or two millimeters. If the chamber is triggered before the particles pass through, as with an accelerator, the tracks are much sharper, easier to photograph, and to measure. If unusual events are to be observed, however, it is sometimes advantageous to use counter control even a t an accelerator. CONTROL. No cloud chamber operates con1.6.1.1.5. TEMPERATURE sistently without adequate temperature control, and if accurate momentum measurements are required, extreme precautions must be taken to avoid certain types of temperature gradients. The order of magnitude of temperature control required may be 10.1"C for most applications, and +O.Ol"C for accurate momentum measurements. Normally a temperature gradient of about O.Ol"C/cm is maintained from top to bottom of an expansion cloud chamber to provide stability of the gas. It is also good practice to measure the temperature and the temperature differences around a cloud chamber as routine operating procedure. The speed of expansion of a cloud 1.6.1.1.6. SPEEDOF EXPANSION. chamber is usually an important factor only in the case of counter control, when diffusion of the ions before they are immobilized by the forming drops may make the track too wide for accurate measurement. Cloud chambers can be made to complete their expansion in as little as 0.004 set," but expansion times of 10 to 20 milliseconds are more commonly used. The width of a track is given by2

X

=

4.68(D7)'I2

where X is the width which contains 90% of the drop images, D is the diffusion coefficient in cm2sec-1, and 7 is the expansion time. Tracks about 1 mm wide are obtained with expansion times of 14 milliseconds in air a t NTP. The speed of expansion is increased if the moving parts are of low mass; however, if the speed is too great and gas must move at speeds near the speed of sound, undesirable shock wave effects occur which often can completely spoil the operation of the chamber. In cloud chambers expanded by a moving piston, provision should always be made to catch and damp the motion of the piston at the end of its stroke.

* See also Vol. 2, Part 8. R. V. Adams, C. D. Anderson, P. E. Lloyd, R. R. Rau, and R. C. Saxena, Revs. Modern Phys. 20, 334 (1948). 11

1.6.

CLOUD CHAMBERS AND BUBBLE CHAMBERS

20 1

1.6.1.1.7. RECYCLING TIME.A conventional expansion cloud chamber requires a t least one minute to prepare for each expansion. This time is occupied by slow, clearing expansions, waiting for the motion of the gas to cease, and the vapor to diffuse back through the gas. Such chambers cannot operate a t the repetition frequency of a pulsed accelerator, and in this respect are far inferior to bubble chambers, which can recycle every few seconds. Various attempts have been made12 to shorten the resetting time by quick recompression, overcompression, etc., but it seem difficult to operate on much less than a one minute cycle. Under these circumstances the operating characteristics are quite different from those at longer times, and the chamber must be adjusted to take these differences into account. 1.6.1.1.8. PHOTOGRAPHY AND ILLUMINATION. Although the illumination problems of diffusion and expansion cloud chambers have some common features, the illumination of an expansion chamber is sometimes easier because it may be possible to illuminate from the rear and use the large amount of forward-scattered light. Illumination a t right angles to the direction of viewing gives about 100 times less light than illumination from the rear but in many cases, because of mechanical reasons, right angle illumination is necessary. The design of the illumination system depends on the degree of detail required in the tracks. If individual drops must be photographed, the requirements are stringent. Flash-tube light sources are now universally used. A brief discussion of recent techniques of photography and illumination is given in the paper by flretter;I2 and Wilson2 discusses fully the photographic problems involved in drop photography. 1.6.1.2. Diffusion Chambers. Supersaturation in a diffusion cloud chamber is produced by the diffusion of a vapor from a warm region where supersaturation does not exist into a cold region where supersaturation occurs. The diffusion cloud chamber is a continuouslg sensitive instrument. The region of supersaturation is necessarily horizontal because it is maintained by temperature gradients in a gas, where stability in the gravitational field occurs when the thermal gradient is vertical. The thickness of the sensitive region depends on the thermal conditions, but cannot be made more than two or three inches. A review article on diffusion cloud chambers, covering the theory of operation and the techniques of experimental use was published by Snowden,l3 and more recent developments were reviewed by Fretter. l 2 In several ways, the design factors for diffusion cloud chambers are similar to those for expansion cloud chambers. A magnetic field is often I* 18

W. B. Fretter, Ann. Reu. Nuclear Sci. 6, 145 (1955). M . Snowden, Prop. in Nuclear Phys. 3, 1 (1953).

202

1.

PARTICLE DETECTION

required, and the chamber, together with the cooling arrangement must then be designed to fit in a magnet. The illumination and photography present similar problems, but the photographic background in a diffusion chamber may consist of a black-dyed liquid which gives excellent contrast with the brightly illuminated track. It is considerably more difficult to utilize plates of heavy material, as in a multiplate expansion chamber because of thermal problems involved, and although some attempts have been made along these lines, the use of plates in a diffusion chamber has not become an important device. There are, however, certain unique design factors in diffusion cloud chambers. 1.6.1.2.1. PRESSURE AND TYPE OF GAS. Diffusion cloud chambers operate satisfactorily with air and argon at atmospheric pressure. Methyl or ethyl alcohol are commonly used as vapor. Dry ice (solid CO,) usually provides the cooling for the bottom of the chamber, either directly by contact, or by circulation of a liquid such as acetone over dry ice and through cooling tubes on the bottom of the chamber. A low-pressure chamber has been constructed for use with helium by Choyke and Nie1~en.l~ In this chamber the bottom was cooled with liquid air and the chamber operated in the pressure range of 75 cm to 15 cm Hg. The temperature of the top had to be maintained a t less than -20°C to provide mass stability. Such a chamber might be used, for example, for the observation of low-energy electrons whose range would be small in an atmospheric pressure chamber. The most' useful diffusion cloud chamber for high-energy nuclear research has been the high-pressure hydrogen chamber. Shutt16has shown that the light gases such as hydrogen, deuterium, and helium are unsuitable for use in diffusion chambers near atmospheric pressure, but work well a t pressures of the order of 25 atmos. Thus a desirable increase in density is obtained along with proper operation. Such chambers have been widely used in connection with accelerators and until the advent of the hydrogen bubble chamber provided the only means of observing directly interactions of fast particles with protons and deuterons. Although the technical problems of operating a t 25 atmos pressure are substantial, the high-pressure hydrogen diffusion chamber is an important instrument in nuclear physics. 1.6.1.2.2. USE WITH ACCELERATORS. The diffusion cloud chamber, being continuously sensitive, is adaptable to the rapid cycling of pulsed accelerators. For such operation, the recharging of the condenser bank supplying energy to the flash tubes illuminating the chamber must be done at a rapid rate. More basic problems are those of background W. J. Choyke and C. E. Nielsen, Rev. Sci. Insts. 2S, 207 (1952). 1sR.

P. Shutt, Rev. Sci. Instr. 22, 730 (1951).

1.6. CLOUD

CHAMBERS AND BUBBLE CHAMBERS

203

produced in the chamber during the acceleration cycle, and the ion load supportable by the chamber during the pulse. The first of these is handled by proper shielding of the chamber and insertion of the target late in the acceleration cycle. The ion load allowable in the chamber is limited by the diffusion rate of the vapor into the region depleted by formation of tracks during the previous exposure. If the cycling time is not less than five seconds, the chamber will usually recover adequately. 1.6.2. Bubble Chambers

It has long been known that liquids may be heated above the boiling point, without actually boiling. Such superheated liquids are unstable and erupt into boiling after short periods of time. Boiling may start, that is bubbles may form, at surfaces or a t nucleation centers within the liquid. D. A. Glaser was the first to conceive the idea that nucleation centers within the liquid might be created by deposit of energy by passing charged particles, and to see that such a process could be used to detect fast-moving charged particles. The bubble chamber can be thought of as the inverse of a cloud chamber, with a gas bubble forming in a superheated liquid instead of a liquid drop forming in a supersaturated gas. The first bubble chambers were constructed so that the only nucleation centers were provided by the ionizing particle. They were made entirely of glass and were thus limited in size. Later experiments showed that gasketed chambers could operate satisfactorily if the expansion conditions were properly controlled. Development of this technique has been rapid and bubble chambers of large size are in operation or under construction. Many different types of liquids have been used, for example, liquid helium, liquid hydrogen, organic liquids, liquid xenon, and certain other inorganic liquids. Although the general principles of the operation of a bubble cloud chamber are known, there is as yet no satisfactory theory which predicts, for example, the degree of superheat required or the number of bubbles formed as a function of energy loss. It is found experimentally that bubble chambers operate with pressure'8 about two-thirds of the critical pressure and the temperature about two-thirds of the way from the normal boiling temperature to the critical temperature. Some examples of pressure and temperature are given in Table I. The liquid in a bubble chamber is superheated by a sudden reduction of pressure. After the track forms and the photograph is taken, the pressure is increased to the initial value, the bubbles collapse and the chamber is ready for another expansion. The great advantage of the bubble chamber over the expansion cloud chamber is that all this can take place in a few 16

D. A. Glaaer and D. C. Rahm, Phys. Rev. 97, 474 (1955).

204

1.

PARTICLE DETECTION

TABLE I. Operating Conditions of Typical Bubble Chamber Materials For the methyl iodide-propane chamber the ratiation length is 7 cm; for liquid xenon it is 3.1 cm. Operating pressure (psi)

Operating temperature

Density gm/cm3

Hydrogen Heliums Xenonb Propane

70 15 300 315

28°K 4°K - 19°C 58°C

Isopentane Methyl iodide-propane WFs*'

350 450 426

157°C 125°C 149°C

0.07 0.07 2.3 0.4 (0.078 gm/cm* of H) 0.5 1.3 2.42

Substance

~

0

---

~

W. M. Fairbank, E. M. Harth, M. E. Blevins, and G. G. Slaughter, Phys. Rev.

100, 971 (1955). * J. L. Brown, D. A. Glaser, and M. L. Perl, Phys. Rev. 102, 586 (1956).

seconds, making the bubble chamber match a pulsed accelerator in its duty cycle. A chamber described by Glaser and Rahm16 filled with isopentane, became fully sensitive to minimum ionizing particles 3.5 milliseconds after the expansion was initiated, and remained sensitive for about 10 milliseconds. Photographs must be taken within this interval. In liquid hydrogen bubble chambers" the bubbles grow much more slowly, and delay times of the order of 50 milliseconds are required. Although the exact process of nucleation of bubbles is still not under~ ~ of stood, the rate of growth of the bubbles can be e ~ p l a i n e d ' *in~terms the heat flow in the liquid. In liquids of high thermal conductivity the rate of growth of bubbles is expected to be large, and the experimental results give close agreement with the theory. The nucleation process itself has been discussed16,20.21 in connection with measurements on bubble density. Deposit of a substantial amount of energy, such as might occur when a delta ray is made, seems to be necessary. That the bubble density varies with velocity in the same way delta rays do supports this idea. Another theoretical approach to this problem has been made by Askar'ian,22who finds an expression giving the specific number of bubbles. 17 D. Parmentier, Jr. and A. J. Schwemin, Rev. Sci. Znstr. 26, 954 (1955); also D. E. Nagle, R. H. Hildebrand, and R. J. Plano, ibid. 27, 203 (1956). 18 H. K. Forster and N. Zuber, J . Appl. Phys. 26, 474 (1954). 19 M. S. Plesset and S. A. Zwick, J . Appl. Phys. 26, 493 (1954). 20 D. A. Glaser, D. C. Rahm, and C. Dodd, Phys. Rev. 102, 1653 (1956). 11 G. A. Blinov, I. S. Krestnikov, and M. F. Lomanov, Soviet Phys. J E T P 4, 661 (1957). 22 G. A. Askar'ian, Soviet Phys. J E T P 4, 761 (1957).

1.6.

CLOUD CHAMBERS AND BUBBLE CHAMBERS

205

1.6.2.1. Bubble Chambers. Design Factors. 1.6.2.1.1. TYPEOF LIQUID. The choice of liquid depends primarily on the nature of the experiment to be done. Liquid hydrogen has the advantage of presenting a purely protonic target to the incident particle, but its low density makes it necessary to construct rather large chambers to have appreciable probability for interaction, and the low temperature involved creates cryogenic problems. A source of liquid hydrogen must be a t hand. Organic liquids such a s pentane or propane present a mixture of nuclei as targets, complicating the analysis of the pictures, but their density is much greater. The density of free protons is about the same in liquid hydrogen as in propane. Organic liquids must be heated above room temperature to produce the required superheat. Neither liquid hydrogen nor organic liquids are very efficient in materializing photons. Liquid xenon, liquid SnC14, and liquid WFs23 may be used as h i g h 4 materials with good detection efficiency for photons, but the cost of sufficient xenon to fill a reasonable size bubble chamber is very high. Another type of bubble chamber24 is th a t in which a gas is dissolved in a liquid under pressure. When the pressure is released the gas forms bubbles. The liquids used are usually organic liquids; thus the matter of choice of liquid is the same a s for a n ordinary organic liquid bubble chamber. 1.6.2.1.2. CONTROL OF TEMPERATURE. In order to ensure reproducible conditions, particularly for bubble counting, the temperature of a bubble chamber must be controlled to about 0.1"C. This implies accurate thermostatting and consideration of temperature gradients. Rapid recycling causes heating of the liquid, and compensation for this must be provided. Generally the production and maintenance of the proper temperature conditions presents a major problem in bubble chamber design. 1.6.2.1.3. MAGNETIC FIELD. It is often desirable to immerse the bubble chamber in a magnetic field so th at measurements of momentum may be made. See Fig. 3. For liquid hydrogen the multiple Coulomb scattering is not important compared to the deflection in the magnetic field a t 10 kilogauss or higher. Scattering is more serious in organic and other heavier liquids. Since single scattering can often be detected by visual inspection of the track with fields of 20 kilogauss measurements of momentum accurate to 10% may be made in bubble chamber filled with organic liquids. I n most cases it is desirable to design chamber and magnet together because of the many interlocking mechanical and thermal problems. 23 J. H. Mullins, E. D. Alyea., L. R. Gallagher, J. K. Chang, and J. M. Teem, Bull. Am. Phys. Soe. 2, 175 (1957). 2 4 P. E. Arzan and A. Gigli, Nuovo czmento [lo] 3, 1171 (1956); 4,953 (1956); see also B. Hahn and J. Fischer, Rev. Sci. Instr. 28, 656 (1957).

206

1.

PARTICLE DETECTION

1.6.2.1.4. ILLUMINATION. The first photographs of tracks in a bubble chamber were taken with bright-field illumination. I n this system a bright diffuse source of light is placed back of the chamber and light is scattered out of the beam by the bubbles. Thus the bubbles appear dark against a bright background. The scattering angle can be quite small, of the order

FIG.3. Large liquid hydrogen bubble chamber and associated magnet.

of one degree. Dark field illumination has also been developed, similar to the " straight-through" illumination in cloud chambers. This may be done by a series of plastic strips placed a t an angle, illuminated by a flash tube. 1.6.2.1.5. PHOTOGRAPHY AND REPROJECTION. Photography of a small bubble chamber presents no serious problems, and reprojection of the photographs is similar to reprojection of cloud-chamber photographs. For a large bubble chamber, however, the optical problems may be formidable.

1.6.

CLOUD CHAMBERS AND BUBBLE CHAMBERS

207

The index of refraction of the liquid is not negligible and the glass through which the photograph is taken may be quite thick. Thus the images are displaced, and for a large chamber the displacements may not be linear with distance off axis. The analysis of the pictures is very complicated unless suitable optical elements are introduced. 1.6.2.1.6. SENSITIVETIME, COUNTERCONTROL.Measurements of sensitive time in a pentane chamber have been made by Glaser, who found that the chamber was sensitive for 10 milliseconds. Further experiments on the nucleation centers in this chamber showed that the lifetime of such centers was never more than 1 millisecond and was usually less. Since several milliseconds are required to perform the expansion, counter controlled expansions seem to be impossible, at least with present techniques. Thus for cosmic ray studies where countercontrolled expansion are often required, bubble chambers have not been very useful. Some attemptsz5have been made to recycle bubble chambers a t a very high rate and have therefore a large fraction of the time during which the chamber was sensitive.'Photographs of the chamber would be taken only when an interesting event is detected by a system of counters, analyzed for events occuring during the sensitive time. However, it seems likely that the principal use of bubble chambers will be with accelerators, to which they are ideally adapted. 1.6.2.1.7. SAFETY. The dangers inherent in operation of bubble chambers are so great that all possible precautions against accident must be taken. Hernandez et aLZ6described the safety measures taken in the construction of hydrogen bubble chambers, and tests on explosions of hydrogen gas. The safety precautions to be taken with bubble chambers containing organic liquids must be also carefully considered when the chamber is large, since such liquids are necessarily hot and at high pressure, and an explosion would be disastrous. 26 E. V. Kurnetsov, M. F. Lomanov, G . A. Blinov, and Chuan Chen-Niang, Soviet Phys. JETP 6, 773 (1957). 2 6 H. P. Hernandez, J. W. Mark, and R. D. Watt, Rev. Sci. Instr. 28, 528 (1957).

1.7. Photographic Emulsions* 1.7.1. Introduction The use of in the field of nuclear physics dates as far back as the discovery of radioactivity, the latter being first observed by photographic methods. A new era in the use of photographic emulsions was initiated when Kinoshita8 and Reinganumg were able to identify a trajectories-rows of developed silver grains-marking the passage of an a particle through emulsions. After Rutherford’s discovery of the disintegration of light elements by a particles, there arose a definite need for sensitive tools to detect and measure protons emitted in these disintegrations. Since only a few sensitive instruments were available a t this time, experiments with photographic emulsions were initiated. The trajectories of slow protons were first detected in 1925;‘O in the following years the tracks of faster protons-up to about 50 MeV-were observed, due to the subsequent improvements of the quality of emulsions, processing techniques, and thickness of emulsion layers. The grain density in proton tracks was smaller than in a tracks of equal velocity, and it was soon definitely established that the grain density in tracks is a function of the specific ionization loss which a particle suffers in the penetration of matter. The earliest experiments were concerned with particles emitted in the disintegration of nuclei by a particles of radioactive origin. Attempts were made to determine the yields and angular and energy distribution of disintegration products in these reactions. The low intensity of radiation, available from radioactive sources seriously limited the accuracy of the 1

a

M. M. Shapiro, Revs. Modern Phys. 13, 240 (1941). P. Demers, Can. J . Research A26, 223 (1947). H. Yagoda, “Radioactive Measurements with Nuclear Emulsions.” Wiley, New

York, 1949. J. Rotblat, P r o p . i n Nuclear Phys. 1, 37 (1950). A. Beiser, Revs. Modern Phys. 24, 273 (1952). OL. J. Vigneron, J . phys. radium 14, 121 (1953). Y.Goldschmidt-Clermont, Ann. Rev. Nuclear Sci. 3, 141 (1953). L.Voyvodic, i n “Progress in Cosmic Ray Physics” (J. G. Wilson, ed.), Vol. 11, p. 219. North Holland Publ., Amsterdam, 1953. Ib M. M. Shapiro, in “Handbuch der Physik-Encyclopedia of Physics” (S. Flugge, ed.), Vol. 45, p. 342. Springer, Berlin, 1958. ?OC. F. Powell, P. H. Fowler, and D. H. Perkins, “The Study of Elementary Particles by the Photographic Method,” Pergamon Press, London, 1959. S. Kinoshita, Proc. Roy. SOC.A83, 432 (1910). 9.M. Reinganum, Phys. 2. 12, 1076 (1911). l o M. Blau, J . Phys. 34, 285 (1925).

* Chapter 1.7 is by M. Blau. 208

1.7.

PHOTOGRAPHIC EMULSIONS

209

measurements. Emulsions were exposed to cosmic radiation on high mountains and in balloon flights, leading to the discovery of fast neutrons in cosmic radiation simultaneously with cloud chamber experiments. In these exposures multiple disintegration of emulsion nuclei by cosmic radiation-stars-were observed for the first time. l1 With the availability of collimated proton-deuteron- and a-particle beams from accelerators, it became possible to correlate the residual range of particles in emulsions and the grain density in tracks with mass, charge, and energy of the incident particles; these calibration tracks were then used for the energy determination of particle tracks emitted in di~integrations.l~-'~ The investigations were greatly enhanced by the availability of new emulsion types, containing higher concentration of silver halides, with which much denser and therefore better defined tracks could be obtained. These emulsions were manufactured first by Ilford and later also by Kodak and Eastman-Kodak. However, the photographic method up to 1948 was limited to the detection of particles with velocities p 5 0.4; the then available emulsions were not sensitive enough to record particles of higher velocity and therefore smaller specific ionization. I n 1948 Kodak, Ltd., in England, and soon afterward Eastman-Kodak and Ilford were successful in manufacturing the so-called electron sensitive or minimum ionization emulsions with which all charged particles, regardless of velocity, can be recorded. Another shortcoming of the older emulsion techniques was overcome by a new processing technique16-the so-called temperature methodwith which plates with up to 1 mm emulsion thickness can be developed. Before the invention of this procedure, the maximum thickness which could be developed within a reasonable length of time was 200 microns. The development of plates with emulsions thicker than 1 mm, although possible, is very lengthy; it is also difficult to obtain uniform development and to avoid distortion. However, since many experiments in the highenergy range require thicker emulsion layers, the manufacture of stripped emulsions or pellicles must be considered a very great improvement in emulsion techniques. Tightly compressed stacks of these emulsion sheets are exposed to the radiation and later developed separately. Various marking systems have been devised which make it possible to follow the particle trajectories through adjacent sheets. This increases the measural 1 M. Blau and H. Wambacher, Sitzber. Akad. Wiss. Wien, Math.-naturw. Kl. Abt. Zla 146,623 (1937). l2 W. Heitler, C. F. Powell, and C. E. F. Fertel, Nature 144,283 (1939). I3T. R. Wilkins, J . Appl. Phys. 11, 35 (1940). l 4 J. Chadwick, A. N. May, C. F. Powell, and T. C. Pickavance, Proc. Roy. SOC. A183, 1 (1944). l 6 C. C. Dilworth, C. P. S. Occhialini, and R. M. Payne, Nature 162, 102 (1948).

210

1.

PARTICLE DETECTION

ble path length of high-energy particles and therefore allows a greater number of measurements on a single track to be made. Consequently the statistical error in the measurement is diminished. The greater observable path length is of special importance in the investigation of interaction and decay events because a greater number of events becomes observable. Another milestone in the development of the photographic method is the introduction of a measuring technique with which multiple scattering in particle trajectories can be determined.16~~7 This technique is indispensable for mass measurements of particles in the relativistic energy range; for lower energy particles the results of scattering measurements supply a valuable complement to ionization and range measurements. Perhaps the greatest triumph of the photographic method is the discovery of unstable particles. In 1947 Perkins18 discovered the negative ?r meson and shortly afterward Lattes et ~ 1 . found ~ 9 the positive counterpart. Since then a great number of unstable particles-heavy mesons and hyperons-have been detected and their properties investigated through work in nuclear emulsions. The first heavy meson, unambiguously defined by its decay, was the 7 meson, discovered by Brown et a1.2Qin nuclear emulsions. The contribution of nuclear emulsion work in the field of strange particles could be adequately described only together with the development of particle physics. The recent improvements in mass and energy measurements are primarily the result of these experiments. So far the above discussions have mainly dealt with field of high-energy particles. The method has also been successfully applied in the field of slow neutrons, or photo-disintegrations, and in problems connected with fission. In most of these experiments the emulsions are loaded (impregnated) with small amounts of the element under investigation. There is, furthermore, a large field of application for nuclear emulsions in problems of biochemistry, biophysics, medicine, and mineralogy.

1.7.2. Sensitivity of Nuclear Emulsions The process of latent image formation for particles in nuclear emulsions is essentially the same as the case of light in ordinary emulsions. The fact that one observes rows of silver grains in the former case, and not in light exposures, may be explained by the larger energy of the particles and by the different mechanism of energy dissipation. The general theory of P. H. Fowler, Phil. Mag. 171 41, 169 (1950).

Y.Goldschmidt-Clermont, Nuovo cimento [91 7, 331 (1950). D.H.Perkins, Ndure 169, 126 (1947). l9

C. M. G. Lattes, G . P. S. Occhialini, and C. F. Powell, Nature 160, 486 (1947).

ao

R. H. Brown, U. Camerini, P. H. Fowler, H. Muirhead, C. F. Powell, and D. M.

Ritson, Nature 163,82 (1949).

1.7.

PHOTOGRAPHIC EMULSIONS

21 1

latent image formation is a solid state problem and will not be discussed in detail here. I n a recent articlez1the latest experimental and theoretical data were reviewed on which the theory of the latent image formation is based. Electrons and positive holes are liberated through irradiation and move independently through the crystal. The authors consider as a first step in the latent image formation the creation of a “pre-image speck,” which is a combination of an electron and a silver ion, absorbed near a dislocation site. The pre-image speck is unstable and decays in a fraction of seconds, if not, another electron and silver ion is deposited at the same dislocation site. The pre-image is now converted into the “ sub-image,” a neutral complex Agz. The sub-image has a longer lifetime and can be developed, but only with strong developers or large induction periods. The next step then is the absorption of another silver ion and the subsequent neutralization by a photoelectron. Experimental investigations make it plausible that the neutral aggregate Ag, can be considered as the origin of the stable latent image. Since silver in contact with silver halide acquires positive charge, it is likely that Ag3 will combine with a silver ion to form a stable tetrahedral combination Ag,. Ag, due to its positive charge, is now easily reduced by the developer. It is believed that in exposures of extreme short duration predominantly sub-images are formed, since recombination phenomena prevent the formation of stable Ag, complexes. Fast particles traverse a silver halide grain in about 10-14 sec, a time interval which is exceedingly short in comparison to the small mobility of ions. Therefore most of the ionization energy of fast particles will be spent in the formation of sub-images, which are less effective than stable latent images in the subsequent developing procedures. That explains why, in spite of the relatively high ionization power of fast charged particles, special types of emulsions are necessary for the detection of particles. Problems connected with the sensitivity of nuclear emulsions, i.e. , the maximum particle energy which can be detected and the maximum grain density in tracks of particles of given properties, have been well studied.1~22~23 A few a ~ t h o r s ~ *attempted ~ 4 , ~ ~ to describe the relation between a grain density and specific ionization loss by semiempirical mathematical expressions. However, in later experiments with particles of higher energy and in emulsions of higher sensitivity it was found that 21

T. W. Mitchell and N. F. Mott, Phil. Mug. [8] 2, 1149 (1957).

** C. M. G. Lattes, P. H. Fowler, and P. Cuer, PTOC.Phys. Soc. (London)69,883 (1947). J. H. Webb, Phgs. Rev. 74, 511 (1948). M. Blau, Phys. Rev. 76, 279 (1948). a5L. van Rossum, J . phys. radium 10, 402 (1949). 25

a4

212

1.

PARTICLE DETECTION

these relations are valid only within a limited energy range. More detailed information about the phase of the theoretical and experimental investigations prior t o the use of electron sensitive emulsions, may be found in * These articles also contain discussions of the first the literat~re.26~-~~ experiments in electron-sensitive emulsions and the earlier investigations on stopping power and the range energy relation in emulsions. Electron-sensitive emulsions are able to detect all charged particles, no matter what their energy is, thereby opening a completely new area of nuclear physics to emulsion research. It became necessary to re-examine and to revise the methods of earlier investigations in order to adapt the techniques to the new problems. In particular, the study of grain density as a function of specific ionization or energy loss has been resumed during the last years. As a result it has been found necessary to introduce certain changes due t o theoretical considerations and because of practical reasons connected with the new measuring techniques. At very high energies the grain density decreases slowly approximately proportional to the ionization loss until a minimum value is reached a t energies of about three times the rest mass of the particle. The grain density starts then to rise again slowly for still higher energies (relativistic increase)a1up to the so-called plateau value, which is about 10% higher than the minimum grain density. Grain density is not only a function of energy loss, but depends also upon emulsion sensitivity and development conditions. However, it has been found that the ratio g/gminor g/gpl is nearly independent of development and changes in emulsion sensitivity; where g is the grain density in the track element under investigation and gminand gpl are the grain densities a t minimum ionization and plateau value. For slower or multiply charged particles the relationship between specific energy loss and grain density becomes more complicated. With increasing ionization loss the grain density in particle tracks tends to reach a saturation value which is due to the limited number of grains per unit length. The saturation value depends strongly on development (size of grains) and upon the emulsion sensitivity.

* See also Vol. 4, A, Section 2.1.7. J. W. Mitchell, ed., “Fundamental Mechanism of Photographic Sensitivity.” Butterworths, London, 1951. 2 6 P. H. Fowler and D. H. Perkins, in reference 25a, p. 340. R. Morand and L. van Rossum, in reference 25a, p. 317. ** R. W. Berriman, in reference 25a, p. 272. ISL. Vigneron and M. Boggardt, in reference 25a, p. 265. J. Rotblat and C. T. Tay, in reference 25a, p. 331. a1 E. Pickup and L. Voyvodic, Phys. Rev. SO, 98, 251 (1950). Also refer to Section 1.7.6 of this volume. 258

1.7.

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213

I n a later chapter we will discuss theoretical and semiempirical equations, governing the correlation between specific energy loss and grain density or related parameters. These relations are extremely important in problems of particle identification. It can be considered as a general statement that quantitative results in nuclear emulsions can be obtained only if appropriate calibration methods are employed. Therefore, it is understandable that, for emulsion experiments, as for any other measuring technique, based on calibration methods, technical details and reproducibility considerations will play an important role. We will return to this topic later; here, only the more technical aspects of emulsion properties, sensitivity requirements, and processing conditions will be treated. The chief purposes of emulsion experiments are: (a) the identification of particles by mass and charge measurements; (b) the determination of particle energies; (c) the investigation of lifetime and decay characteristics of unstable particles; (d) the study of scattering, interaction, and production cross sections; and (e) the detailed study of the nature and energy a s well as angular distribution of the particles emitted in these events. In many problems the emulsion serves only as the detector of particles from an external source, while in others it is utilized as a reaction chamber in which the particles interact with nuclei of the emulsion itself, or with additional nuclei introduced into the emulsion for the specific purpose of the experiment. Such experiments can be carried through successfully only if: (1) the emulsion sensitivity is sufficient for the detection of particles in the energy interval under consideration; (2) the discrimination among trajectories of particles with different properties is satisfactory, i.e., the difference in grain density is appreciable; (3) the emulsion thickness is large enough to observe, on the average, appreciable segments of the trajectory; and (4) the geometrical relations prevailing a t exposure are well reproduced in the developed emulsion. The latter depends on the processing conditions. Item (3) depends upon the size of the emulsion as well as the thickness of the emulsion layer, if glass-backed plates are used, or on the number of sheets within the emulsion stack; (1) and (2) depend mainly on emulsion properties but can be changed slightly through development conditions. The simultaneous attainment of the highest sensitivity and the best discrimination properties is not always possible. I n emulsions of high sensitivity the number of developed grains tends to reach the saturation value for particles of relatively high kinetic energy (about 30 MeV for protons). Thus, a further increase in ionization energy does not essentially change the grain density in the trajectory. Therefore it is very fortunate

214

1.

PARTICLE DETECTION

that various types of emulsions are available, permitting a choice appropriate to the problem at hand. The various types of emulsions and their specific properties have been discussed in great detai1;1-7b,32 in the following discussion the various types of emulsions are merely enumerated. Ilford, Kodak (England), and Eastman Kodak (Rochester) manufacture various types of emulsions of different sensitivity. The most sensitive emulsions, which are selected for work with fast electrons and in the field of high energy are: G6 by Ilford, NTd by Kodak and NTB-, by Eastman-Kodak. The emulsion in widest use is the G-5 emulsion which is available in various sizes and thicknesses either with glass backing or as free sheets called pellicles. The emulsions next in sensitivity are the CZ, NTZa, and NTB emulsions from Ilford, Kodak, and Eastman Kodak respectively in which protons up to 5G70 Mev and electrons up to 30-100 kev can be detected. In El, NTA emulsions, where protons up to l(f20 Mev can he detected, the discrimination between proton and (Y particle tracks is very good. Ilford’s D and Eastman Kodak’s NTC emulsions detect only slow particles and no protons. They are designed for the detection of fission products which can easily be distinguished from particles in these emulsions. Ilford manufactures another type of emulsion, the Go, with a sensitivity between El and CZ emulsions; it can be developed in the same way as G6 emulsions and is very useful if sandwiched between GSpellicles for the detection of multiple charged particles of high energy (e.g., heavy primaries in cosmic radiation). Ilford manufactures G5 emulsions in gel-form, with which one can prepare fresh emulsion layers and avoid background tracks, due to cosmic radiations. This technique is used whenever the intensity of the radiation under investigation is very low as in cosmic-ray experiments at great depths below the earth’s surface or in measurements of the radiation of pure isotopes. It is furthermore useful in geological and biological problems where the emulsion can be directly poured over the substances whose radioactivity is to be measured. Within the last few years, Ilford Ltd., had introduced new types of small grain emulsions, sensitive to particles of minimum ionization, with essentially the same constitution as G-5, which has a mean crystal diameter of 0.27 j~ (microns). The new emulsions, K-5 and L-4, with mean crystal sizes of 0.20 p and 0.15 p diameter respectively, thus exhibit developed grains of smaller diameter than does G-5. The fine grain emulsions prove especially useful in the identification of dense tracks, in the analysis of large stars, and in general provide sharper resolution in the measurement of small distances. However, these emulsions must be 32 C. Waller, J . Phot. S&. 1, 41 (1953).

1.7.

PHOTOGRAPHIC EMULSIONS

215

processed very soon after exposure, because the fading of the latent image is more rapid in emulsions of smaller crystal size. Among other types are the so-called diluted emulsions which have a smaller silver halide content than other nuclear emulsions. These emulsions are used usually together with ordinary emulsions in experiments in which interactions of particles with light and heavy elements are compared; the former being more frequent in diluted emulsions. A similar purpose is served by plates with alternating emulsion and thin gelatin layers where the trajectories of particles emitted in the interactions with the light elements of the gelatin can be followed into both adjacent emulsion layers. Furthermore, all companies supply emulsions loaded with boron, lithium, and bismuth for experiments in which certain reactions with these elements are studied. Unfortunately, the amount of foreign element which can be introduced is small. Experiments connected with loaded and sandwiched emulsions are described in references 1-7b, mentioned above. Most nuclear emulsions have very similar chemical composition, with the exception of diluted emulsions and Eastman-Kodak N T C emulsions which contain smaller amounts of silver halides. TABLE I. Composition of Dry Ilford Gg Emulsions. Element

Weight in gm/cm3

Silver Bromine Iodine Carbon Hydrogen Oxygen Sulfur Nitrogen

2.025 1.496 0.026 0.30 0.049 0.20 0.011

0.073

a Ilford Nuclear Research Emulsions (Ilford Research Lab., Ilford, London, England, 1949).

The composition of the Gsemulsion is given in Table I. The density of the dry emulsion is 4.18. However, emulsion density changes if the emulsion is brought into surroundings of higher relative humidity; because the changes take place very slowly, an appreciable length of time will elapse before equilibrium is reached. These problems were recently investigated with great care by Barkas and co-workers and are discussed below. An exact knowledge of the emulsion composition, which of course includes the water content, is very important in investigations of the range-energy

216

1.

PARTICLE DETECTION

relation in emulsions and for cross section experiments. Since the increased water content changes the volume of the emulsion layer, the spatial relationship between particle trajectories can only be evaluated if the actual volume a t exposure is known. During processing, the emulsion layers experience several large density changes, the most important occurring during fixing, when most of the original silver halide is dissolved. After drying, the thickness of the emulsion layer is considerably decreased with respect to the original value. The ratio of emulsion thickness before and after processing (provided that the emulsion was mounted on glass during the processing), the socalled shrinkage factor, determines the relationship between geometrical conditions a t exposure and in the processed emulsions. However, inasmuch as the processed emulsion is also hygroscopic, emulsion work should be done in humidity controlled laboratories and the emulsion thickness should be checked frequently through repeated emulsion thickness measurements. The magnitude of the shrinkage factor depends slightly on the processing conditions and is greatly influenced by the concentration of the glycerin, or other plastisizer solutions, which is used in the last bath to which the emulsions are subjected. Details on shrinkage factor measurements may be found in references 1-7b. The large thickness of emulsion layers required the development of new processing methods in order to achieve uniform development throughout the emulsion and to avoid distortion. The latter is very important not only for the true reproduction of the geometrical condition prevailing a t exposure, but even more so on account of multiple scattering measurements which represent one of the most important measuring techniques in nuclear emulsions. Emulsions, if not developed immediately, should be kept in deep and narrow wells both before and after exposure in order to minimize exposure to cosmic radiation. The latent image tends to fade under the influence of humidity and high temperature, necessitating special care in the storage of exposed emulsions. The fading effect is discussed in references 1-7b and is treated exhaustively in a more recent paper b y Demers et 1.7.3. Processing of Nuclear Emulsions

1.7.3.1. Processing Techniques. The development of various types of emulsion of thicknesses u p to 2 0 0 p has been discussed in detail in references, 3, 4, and 5. The larger the emulsion thickness, the more difficult it is to obtain uniform development because of the time needed for thorough penetration of the developer. This difficulty was removed by the so-called temperature 33

P. Demers, T. Lapalme, and T. Thonvenin, Can. J . Phys. 31, 295 (1953).

1.7.

PHOTOGRAPHIC EMULSIONS

217

development method.34-36The principle of this technique is based on the fact that a developer is chemically inactive at very low temperatures (about 4%). Amidol has been generally adopted as developing agent. The plates are soaked in cold developer until its diffusion in the emulsions is completed. Then the plates are removed from the solution and warmed to the required temperature in stainless steel containers. After the development at the higher temperature is concluded, th e developing action is stopped by a weak solution of acetic acid. The emulsion is then washed and finally readied for the fixing solution. The temperature of the dry-development stage determines the degree of development, i.e., the size of the developed grains and the number of minimum ionizing tracks. The fixing procedures for thick emulsion layers have also been completely changed just as in the development stage. While fixing a t room temperature would be rapid, it has been found to lead to grave distortion in the emulsion; hence the fixing solution is also kept a t very low temperatures. I n order t o minimize distortion, the fixing solution is never changed abruptly during the fixing process which lasts several days, but is slowly and carefully replenished by fresh solution. When the fixing process is concluded, the solution is slowly removed and replenished b y cold water. The plates are then soaked in a plasticizing solution, placed between guard rings (old emulsion sheets), and dried under controlled temperature and humidity conditions. The plasticizer is introduced in order to avoid stripping of the emulsion from the gIass and to restore part of the original thickness; the latter has been considerably decreased by the dissolution of all the silver halide which was not activated during the exposure. The dry emulsions are often covered with a thin plastic coat in order to protect the emulsion and t o avoid excessive fluctuations in the water content of the processed pellicle, even if the ambient relative humidity should change abruptly. Some authors introduce a clearing solution after fixing, especially if plates thicker than 600 p are used. The purpose of all these involved procedures is t o ensure the most homogeneous development possible and to minimize distortion. The latter is accomplished by avoiding abrupt changes in the pH of all solutions brought into contact with the emulsion, and by avoiding sharp temperature changes. To this end, it has been proposed that the temperature of the hot stage be lowered, or that only cold developer be used, consequently increasing the development time. However, i t has been found that such 34 C. C. DiIworth, C. P. S. Occhialini, and L. Vermassen, Bull. centre phys. nucltaire, univ. libre Bruselles No. 13a (1950). 35 A. Bonetti, C. C. Dilworth and C. P. S. Occhialini, R 711L. centye phys. nucldaire, univ. libre Bruxelles No. 13c (1951).

218

1.

PARTICLE DETECTION

emulsions show a smaller ratio of grain density to background density.3e In the processing t e ~ h n i q ~ e sfor ~ ~ thick , ~ emulsions ~ ~ ~ ~ - (Ilford, ~ ~ G-5), Amidol is used as the basic developing agent. However, the Brussels group use boric acid Amidol while the Bristol group uses a combination of Amidol and bisulfite. TABLE 11. Temperature Development for 600 p Plates Used in Belgian Laboratories. Boric Acid Amidol Developera Amidol Sodium sulfite (anhydrous) Potassium bromide (10% solution) Boric acid Distilled water PH a

4 . 5 gm 18 gm 8 cma 35 gm 1000 ml 6.4

From Dilworth, Occhialini, and V e r m a ~ s e n . ~ ~

TABLE 111. Temperature Development for GOO p Plates Used in Belgian Laboratories" Operation

Bath

Preliminary soaking Distilled water Cold stage Boric acid Amidol Warm stage slow Dry (after wiping the heating plate surface with a soft tissue) Development Dry Slow cooling Dry Stop bath Acetic acid, 0.2% Silver deposit cleaning Washing Running water Fixing Hypo 40% (sodium sulfate u p to 10% is added if swelling is excessive) Slow dilution Water (adding sodium sulfate) Glycerin bath Glycerin, 2% Slow drping with guard rings

Temperature

Time

Cooling down to 5°C 5°C 5°C to 28°C

120 min 120 min 5 min

28°C 28°C to 5°C 5°C to 14°C

60 rnin 5 min 120 min

14°C 14°C cooling to 5°C

5°C

5°C to room temperature 20°C

120 min Until clear

100 h r 120 min 7 days

a From Y. Goldschmidt-Clermont, Photographic emulsions. Ann Rev. Nuclear Sci. 3, 149 (1953).

*OA. J. Her2 and M. Edgar, Proc. Phys. SOC.(London) A66, 115 (1953). 87 A. D. Dainton, A. R. Gattiker, and W. 0. Lock. Phil. Mag. 171 42, 396 (1951). 38A.J. Herz, J . Sci. Znstr. 29, 60 (1952). 39 B. Stiller, M. M. Shapiro, and F. W. O'Dell, Rev. Sci. Instr. 26, 340 (1954).

1.7.

219

PHOTOGRAPHIC EMULSIONS

Tables 11, 111, IV, and V describe processing solutions and processing procedures used in Brussels and in the Naval Research Laboratory in Washington. TABLE IVa. Developer for 600 p Emulsions in Development Procedures by Stiller, Sha.piro, and O'Dell of the Naval Research Laboratory88 Amidol Sodium suIfite (anhydrous) Potassium bromide (10% solution) Boric water Distilled water

4 . 5 gm 18 gm 8 cm 35 gm 1000 ml 6.6

PH TABLE IVb. Fixing Solution (pH 5.3) ~~

Distilled water Sodium thiosulfate cp Sodium bisulfite Ammonium chloride

~

1000 400 7 7

om3 gm gm gm

500 15 5 5

cm2 gm gm gm

TABLE IVc. Clearing Solution (pH 5.2) Distilled water Ammonium acetate Citric acid Thiurea

TABLE V. Processing Procedure

Presoaking in distilled water Penetration of cold developer Warm dry development Dry cooling Acid stop (0.5%) Fixing Clearing solution Washing Plasticizing solution (Flexoglass) Coating and drying

Temperature

Time

Room temp. to 5°C 5°C 18°C 18"C-5"C 5°C 5°C 5°C 5°C 5°C Room temp.

150 rnin 150 rnin 180 min 5 min 150 rnin Clearing time 50% 24 hr 36 hr 1h r 5 days

+

After each exposure a sample emulsion should be carefully examined for the degree of development by grain density measurements on the tracks of fast electrons. The uniformity of development may be investigated by determining the variation of grain density of fast tracks which

220

1.

PARTICLE DETECTION

traverse the thickness of the emulsion, or by measuring the grain density in electron tracks near the top, center, and bottom of the pellicle. Finally, one has t o determine the degree of distortion in the processed emulsion; this may be accomplished through a technique described by Cosyns and Vanderhaeghe.40 First, several steeply dipping tracks are chosen at various positions in the emulsion; the distortion is then determined by measuring the angle, fi - a, between the tangent and the chord of the curved line representing the projection of the distorted track. The quantity

Distorted trock

FIG.1. Perspective drawing of a distorted track inclined to the emulsion surface.

d sin(@- a) as shown in Fig. 1, where d is the chord length of the track, is then the component of the distortion vector perpendicular to the projection of the track on the emulsion surface. Here AC would be the path of the particle if no distortion were present; AB is the chord length of the track; and ARB is the actual distortion path of the particle. The problem of distortion measurements and the various types of distortion configurations are treated in greater detail in various papers on multiple scattering, where methods for distortion diminution in scattering measurements are also described (see Section 1.7.5). I n some laboratories, emulsions are subjected to erradication processes 4 0 M. G. E. Cosyns and C . Vanderhaeghe, Bull. centre phys. nuclbaire, univ. libre Bruxelles No. 16 (1951).

1.7.

PHOTOGRAPHIC EMULSIONS

221

before expoeure. The latter is based on the destruction of the latent image by water v a p ~ r . ~ ~ , ~ ~ The wide use of large stacks of emulsions necessitates a description of the assembly and processing techniques peculiar to stacks. Pellicles were first used by D e m e r ~ and , ~ ~ the first commercial pellicles were manufactured by Eastman-Kodak-NTB emulsion sheets, 250 p thick. The pellicles most commonly used a t present are Ilford G-5 emulsions which are available in thicknesses ranging from 200 to 1000 p. Inasmuch as the pellicles suffer considerable lateral expansion and contraction during processing, it is necessary to mount them on glass plates prior to development. 43,44 The mounting techniques in various laboratories differ slightly from each other, although the essential point is the uniform wetting of both the emulsion surface and the glass plate in order to ensure freedom from air bubbles. The pellicle and glass are generally soaked in cold water and are then pressed to each other either with a rubber roller or by passing the pellicle and glass plate through a mangle in which the pressure can be adjusted. Again, care must be taken that the pressure is uniform over the entire emulsion sheet. After mounting, the pellicle is developed and fixed in exactly the same way as are ordinary plates. For exposure the emulsion sheets are tightly pressed together by blocks of Bakelite to ensure close contact of emulsion surfaces. Holes are sometimes punched in the pellicles to aid in the packing and subsequent alignment of the stack. Prior to development the plates are sometimes exposed to narrow beams of X-rays passing near the edges of the stack in order to facilitate, after development, the aligning of emulsion sheets relative to each other and thereby aid in tracing tracks from one plate to another. Other more elaborate have also been developed for successful alignment. One method which has gained wide acceptance is the printing of labeled grids on the surface of each pellicle. Before printing each plate is carefully adjusted in a jig with respect to small holes which were previously punched in the stack. Another method is the placement of brass tabs at the corners of the emulsion which then allow each emulsion to be mounted a t the same relative position of the microscope stage. The alignment of the emulsions by this method is generally within 50 microns. M. Wiener and H. Yagoda, Rev. Sci. Znstr. 21, 39 (1950). P. Demers, Can. J. Research A24 628 (1950). 43 B. Stiller, M. M. Shapiro, and F. W. O’Dell, Phys. Rev. 86, 712 (A) (1952). 4 4 C. W. F. Powell, Phil. Mag. [i]44, 219 (1953). 46 R. W. Burge, L. T. Kerth, C. Richman, and D. H. Stork, U. C. R. L. 2690 (1954). 46 G. Goldhaber, S. J. Goldsack, J. E. Lannuti, and H. L. Whetstone, U. C. R. L. 4l

42

2928 (1955). 47

E. Silverstein and W. Slater, J . Sci. Instr. 33, 381 (1953).

1.

222

PARTICLE DETECTION

I n a novel a p p l i c a t i ~ nof~ ~the stripped emulsion technique the two outside emulsion sheets move relative to a fixed stack, providing a time record for particles entering or leaving the stack. This method is especially suited for investigation of the heavy primary component in cosmic radiation. 1.7.3.2. Water Content of Emulsions. The emulsion is composed of a mixture of silver halides, gelatin, and glycerin; the last two constituents are hygroscopic and will normally contain a certain amount of adsorbed wat,er. A precise knowledge of the water content of emulsion is of great importance in cross section and range measurements and for a n exact determination of angular relations in scattering, disintegration, and decay events. The influences of the water content of emulsion on shrinkage factor and range measurements have been treated e ~ t e n s i v e l y . ~ ~ ~ ~ ~ - ~ ~ It is assumed that the absorption of water by gelatin and the consequent swelling of the emulsion occur without strong chemical interaction. On the strength of this assumption, one obtains that the absorption of w gram of water, of density one, by 1 cm3 of emulsion, density p , should result in a final volume, (1 - w) cm3, of material of density d. The latter is then related t o the emulsion densit,y p through the expression d = [ ( p to)/(l w)] gm cm-3. However, detailed experiment^^^-^^ have shown that Av/Aw, the volume change in cm3per mass change in grams, due to absorption or evaporation of water, is a quantity smaller than unity. The deviation of this ratio from unity is particularly evident if the time interval between the accrual or removal of water from the emulsion and the actual measurement is short. After long time intervals-several days -Av/Aw reaches an equilibrium value, which is 0.875, 0.84, and 0.94 for G5 emulsions, according to determinations by Batty, Ilford Lab., and Barkas respectively. This effect, which Barkas attributes to the porosity of the emulsion, requires th at both the mass and volume of the emulsion be obtained when precision measurements of particle ranges are needed. The slow diffusion of water vapor into and out of emulsions was measured carefully by Oliver and B a r k a ~ .Table ~ ~ , ~VI~ gives the loss of

+

+

J. J. Lord and M. Schein, Phys. Rev. 80, 304 (1950). F.K.Goward and T. T. Wilkins, Proc. Phys. Soc. (London) A63,662, 1171 (1950). 6o H. Bradner and A. S. Bishop, Phys. Rev. 77, 462 (1950). 61 J. Rotblat, Nature 166, 387 (1950). 62 J. J. Wilkins, A. E. R. E., Harwell c/r 664 (1951). 6 8 A. J. Oliver, Rev. Sci. In&. 25, 326 (1954). 6 4 W. H.Barkas, Rev. Sci. Instr. 25, 329, (1954). 66 C.J. 'Batty, Nuclear Znstr. 1, 138 (1951). 6 e W. H. Barkas, P. H. Barrett, P. Cuer, H. H. Heckman, F. M. Smith, and H. K . Ticho, Nuovo cimento [lo]8, 185 (1958). 48 49

1.7.

PHOTOGRAPHIC EMULSIONS

223

weight due to the ambient humidity of a 1000 fi G5 emulsion kept a t 50% r. h. (relative humidity) after unpacking. Only after 30 days in a vacuum do 1 0 0 0 ~plates lose their water content entirely, exhibiting then a density of 4.03 gm/cm3. TABLE VIm Days at 50% r. h.

Weight loss in grn per gm of emulsion 5.02 X 10-3 8.07 8.36 9.07 9.'16 9.22

1 4

5 11

15 18

"From A. J. Oliver, Rev. Sci. Instr. 26, 327 (1954).

Table VII gives the ratio of thicknesses of plates, kept a t various values of relative humidity, to the thickness at 50% r. h.; the thicknesses represent equilibrium values. Thickness rather than volume determinations are permissible if changes in length and width of the emulsion are negligible. TABLEVIIa % r. h.

T/Taor.h.

10 20 50 60 70 81

0.9657-0.0035 0.9720-0.0011 1.ooo 1.0202-0.0016 1.0466-0.004 1.1090-0.0035

5From A. J. Oliver, Rev. Sci. Instr. 26, 326 (1954).

If the physical conditions a t the time of exposure are known precisely, the shrinkage factor of the processed emulsion may be determined by measuring the thicknesses of the processed and unprocessed emulsions. Because this factor depends strongly on the humidity of the surroundings, measurements must be made repeatedly if the work is not done in humidity controlled rooms. Only if the shrinkage factor is known exactly can one relate the measurements in the processed plate to the situation prevailing at the time of exposure. The shrinkage factor also depends strongly on the concentration of plasticizer used in the last step of processing, on the concentration of hardener used in the fixing solution, and according t o Oliver, on the time of fixing and washing.

224

1.

PARTICLE DETECTION

One of the most accurate measurements of the shrinkage factor employs a narrow beam of particles (e.g., a particles from a radioactive source) inclined at a small angle to the emulsion surface. The ratio of the tangent of this angle to the tangent of the angle observed in the processed emulsion gives the shrinkage factor directly. The density of emulsion may be made by comparing residual ranges of particles from accelerators or in decay processes (e.g., p mesons from r - p decay) with the ranges found in standard emulsion.

1.7.4. Optical Equipment and Microscopes The contents of this section do not purport to be a complete description of microscope procedures for nuclear emulsions, but rather to highlight certain aspects which differ from ordinary microscopic work. Because many hours are spent in searching the emulsions and in measuring events, the use of a binocular microscope is necessary for the viewer’s comfort. The total magnification depends on the specific problem at hand, and may vary between 100 and 2000 times. The eyepiece lenses should be of the best possible quality; the magnifications which are commonly used may go from 6 to 20X. Dry objectives, lox, 20X, or 25X can be used when low magnification is desired, while oil immersion objectives, 45-70X and 90-100X, are commonly used for higher magnifications. The aperture of these objectives should be as high as is possible on account of depth measurements and in order to ensure optimum working conditions. Emulsions which are thicker than 400 p (or emulsions whose shrinkage factors are reduced by special processing) require objectives with long working distances, such as those now manufactured by Cooke, Throughton, and Sims, Leitz, and Koristka. The relatively low numerical aperture (n. a.) is a drawback which must be tolerated in order to allow observation of the entire emulsion thickness. Table VIII describes the characteristics of various objectives with long working distance. TABLE VIII

Manufacturer Cooke, Throughton, and Sims Leitz KS Objective Leitr KS Objective Leitr KS Objective Koristka Koristka Koristka

Magnification 45 x 22 x 53 x 100 x 30 X 55 x 100 x

n. a.

0.95 0.65 0.95 0.95

Working distance

1.50mm

1.05

2.30 m m 1.00m m 0.370m m 3.00m m

0.95 1.25

1.35 m m 0.530 mm

1.7.

PHOTOGRAPHIC EMULSIONS

225

The low magnification oil immersion objectives are useful because they permit better visibility and because their use avoids changing from dry to immersion lenses when switching magnifications. Dry objectives with large working distances such as the Newton and reflecting objectives which are manufactured by Beck (England) have the advantage of enabling one to place one emulsion atop another, and then to trace directly a track which passes from one pellicle to the other; this procedure is often convenient when one is examining a stack of emulsions. However, the magnification is so small that it is difficult to follow minimum tracks; furthermore, the visibility is impaired by the emulsion-glassemulsion sandwich, and the degree of optical alignment which is necessary to permit easy tracing of tracks is quite critical. Measurements of length are performed with an eyepiece micrometer which has been calibrated against a stage micrometer. The other eyepiece may contain a reticle in which a line, or two parallel lines, are engraved; this line is used as a fiducial line in making angular measurements in the plane of the emulsion. The rotational movement needed to superimpose this line on the track under consideration may be determined by a protractor device connected to the eyepiece. Angular measurements to within fractions of a minute may be made with precise eyepiece goniometers. For very accurate length measurements, so-called filar eyepiece micrometers are available; these devices contain one or two hairlines which are moved normal to a calibrated scale by a micrometer screw. Depth measurements are performed with the fine vertical adjustment screw, the emulsion being viewed with an oil immersion objective of highest magnification and n. a., in order to minimize the depth of focus. Another method of depth determination utilizes depth gages for measuring the vertical motion of the objective. Microscopy with nuclear emulsions differs markedly from ordinary applications in its requirements for precise and extended stage movements. The microscopes which are most widely used in emulsion work are the Cooke, Throughton, and Simms, type M4000, and the Leitz Ortholux; both instruments possess sufficiently smooth movements along the two stage axes. The former microscope has the advantage of a micrometer movement, thereby enabling one to read the stage position to within 2 p , while the rigid structure of the Ortholux is a desirable feature. A serious drawback in the use of both microscopes is the fact that neither can accommodate plates which are larger than 3 in. X 4 in. These disadvantages are especially evident in high-energy work, where large emulsion sizes are now widely used. In order to overcome these difficulties, nuclear emulsion workers have themselves designed and built completely new stages, or modified the

226

1.

PARTICLE DETECTION

original stages, to allow them to accommodate the large plate sizes. The simplest kind of modification consists of merely extending the linear dimensions of the stage itself; the chief disadvantages of this method are the fact that the very largest plates cannot be used and that the relatively short movements must be tolerated. A further step towards versatility is the construction of a superstage which is mounted atop the original stage on runners, allowing movement of the plates in addition to that of the stage. Many workers have retained only the frames and the optics of their microscopes, and have constructed entirely new stages, according to their specifications. Others have gone even further and have utilized only the optical components from commercial microscopes and have, in effect, manufactured their own microscopes. Such an instrument has been built by ZornS7from a universal table and drill press stand; it can accommodate the largest emulsionfi used and its movement is adequate for use as a scattering microscope. Another complete microscope has been constructed by Schein at the University of Chicago. In all of these endeavors the requirements of precision and care are understandably high. The above-mentioned microscopes are not always satisfactory for scattering measurements, where the movement must be accurately linear over a range of several inches. One must check each microscope for the linearity of its movement, which is found to vary with the individual instrument. 1.7.5. Range of Particles in Nuclear Emulsions

1.7.5.1. Measurement of the Residual Range of Particles in Nuclear Emulsions. The length of a track in the emulsion is determined by first ) then measuring the projection of the track in the focal plane ( 2 , ~and finding the angle of inclination (dip angle) to this plane. The former measurement is executed with a carefully calibrated eyepiece micrometer, while the latter is found by measuring the z coordinates, of two more grains in a track, on the fine adjustment depth screw of the microscope. If the direction of the trajectory changes, separate determinations of these quantities must be made for each segment of track with a different dip. The actual length of a track is sometimes given by

R

=

(P

+ S2z2)1/2 Z(l + S2tan2 =

+

a)lI2

(1.7.1)

where 2 = AX)^ ( A Y ) ~ .The shrinkage factor S is defined as the ratio of the original emulsion thickness to t o the thickness after development t d , while a is the angle of inclination as measured in the processed emulsion. 0’

G, Zorn, Rev. Sci. Instr. 27, 628 (1955).

1.7.

227

PHOTOGRAPHIC EMULSIONS

Inasmuch as the track length depends on the magnitude of the shrinkage factor, the latter must be determined with great accuracy. The shrinkage is a function of the water content of both the undeveloped and shrunken emulsions. Furthermore, the slow diffusion of water into and out of the emulsion63requires that before exposure the plates be kept in surroundings of constant humidity for an extended period of time (several days). The humidity content of the developed emulsion should also be kept constant during search and measurement. Methods of shrinkage factor measurements are described by many a~thors.4~~8-6~ Dip angle measurements must be made under the highest magnification (smallest depth of hcus) and with oil immersion objectives. The index of refraction varies slightly with the water content of emulsions; for G5 emulsions, it has the values 1.539, 1.533, and 1.521 for 31, 51, and 75% r. h., respectively. The projected length, dip angle, and shrinkage factor must be determined separately for each emulsion sheet when a track passes through several sheets in a stack. These measurements presuppose a knowledge of the thickness of each plate a t exposure, the value of which may vary as much as 5% throughout the stack. As a consequence of this fluctuation in thickness, when accurate range determinations are required, it is not adequate merely to measure the total thickness of the stack and then to divide by the number of sheets to obtain a mean value. describe the measuring techniques and all the necesBarkas et uZ.66,62a sary precautions in obtaining accurate values of the density and of water content of emulsions. A precise knowledge of the emulsion composition is a prerequisite for range measurements, the latter being meaningful only in a well-defined medium. The residual range Ad, of a particle in a dry emulsion (it may be dried in a vacuum or over Hzso4) of density do is related to its range in an emulsion of density d and water content w through the Eq. (1.7.2) :

-Ad

rd - 1 r(da - d ) Ado rdo - 1 -l- rdo - 1

L‘

’

(1.7.2)

Here, Ado and Ad are the ranges in emulsions of densities do and d, respectively. A, is the range of the particle in water, and T [ = ( A v l A w ) 5 I] is the ratio of the increase of volume to that of weight of an emulsion after L. Vigneron, J . phys. radium 10, 309 (1949). J. RotbIat, Nature 167, 550 (1951). E o V. L. Telegdi and W. Zunti, Helv. Phys. Acta 23, 754 (1950). 61 F. A. Roads, in “Fundamental Mechanism of Photographic Sensitivity” (J. W. Mitchell, ed.), pp. 327-330. Academic Press, New York, 1951. 82 M. Gailloud, C. H. Heanny, and R. Weill, Helv. Phys. Acta 27, 337 (1954). 8% W. H. Barkas, F. M. Smith, and W. Birnbaum, Phys. Rev. 98, 605 (1955). 6*

68

228

1.

PARTICLE DETECTION

the absorption of a certain amount of moisture. In the case of dry emulsions, the authors find r = 0.94 and 4.004.03 gm/cma for the density, while Ilford Laboratories obtain 0.84 and 4.033 gm/cm3 for the same quantities. Another factor which, if not properly taken into account, may lead to inaccuracies in range determination is emulsion distortion. Cosyns and Vanderhaeghe40 were the first workers to treat the distortion problem mathematically. Distortion calculations are based on simple geometrical considerations, which describe the coordinate displacement of an originally straight track, due t o the action of stresses present in the emulsion before solidification or induced d\ring the processing and drying of the emulsion. The maximum displacement A O will occur a t the air surface of the emulsion, since the other surface is firmly attached to the glass and will remain fixed. The displacement A of any other depth h in the emulsion will be a function of the ratio h :t, where t is the thickness of the emulsion sheet. The authors introduced a unit distortion vector, called the “Covan” and defined by C = Ao/t2, where the surface displacement A0 is measured in microns and the emulsion thickness t in mm. The relation of distortion to range measurements is discussed by Barkas et aE.6sand its connection with angular and scattering measurements by La1 et u Z . ~ ~ The apparent range of a particle in distorted portions of the emulsion differs from the value in the undisturbed part. If A0 = Ct2is the maximum displacement occurring on the emulsion surface and if the second derivative of the distortion vector with respect to the depth coordinate in the emulsion is constant (C-shaped curvature, which is the most widely observed), then the change p in the position of a point in the xy plane is given by p = Ao(1 - h2/t2) where h is the z coordinate of the point, measured from the emulsion surface. The range variance arising from small local distortions is often called “microscopic distortion straggling,” which is related to cavities and irregularities in the emulsion that are caused by the fixing process. The variance due to this type of straggling depends on the grain diameter, and lies between 0.02fi and 0.03E where R is the mean particle range. The finite grain size and separation of the grains introduces another uncertainty into range measurements in that the actual range may be larger than the measured range. However, the effect is generally small, except in the case of trajectories of very small residual range. Finally, the observer error must also be taken into account. While this contribution is small for experienced workers, it may be considerable for steeply dipping or strongly scattered tracks. The resultant of all errors and uncertainties responsible for range 68

D. Lal, Y. Pal, and B. Peters, Proc. Indian Acad. Sn’. A38, 398 (1953).

1.7.

PHOTOGRAPHIC EMULSIONS

229

straggling, which have been mentioned thus far, are smaller than the effect of “Bohr straggling.” The latter is inherent to the process of energy loss and will be discussed below in the section on range-energy theory. 1.7.5.2. Range-Energy Relation in Nuclear Emulsions.* The residual range of an ionizing particle is a function of its velocity, charge and mass.

R

=

F(v,Z,mi).

(1.7.3)

The range-energy relation in emulsions is derived partly from theoretical considerations and partly from actual range measurements on particles whose energies have been accurately determined. Because the stopping power of nuclear emulsions is relatively high (it has more than 1000 times the stopping power of air) even fast particles, available from high-energy machines, can be brought to rest in the emulsion, provided th a t thick layers or stacks of emulsions are employed. This feature is one of the greatest assets of the photographic method in th a t it permits the observation of the entire trajectory of a particle and its investigation b y a variety of experimental methods. Although the relation is derived from protons, the ranges of other singly charged particles with velocities equal to th a t of protons of range R , can be obtained immediately from th e following equation:

where mi and m p are the masses of the particle and proton respectively. This relationship follows from the energy loss equation (1.7.5), which asserts t ha t the energy loss is independent of particle mass. The range of a particle, R(miz), with a velocity equal to that of a proton, but with different mass mi and charge z is given by the expression

(1.7.5) The quantity f(z) represents a range correction due to electron capture, and will be discussed later in connection with the ranges of multiply charged particles. The range-energy relation for e l e c t r ~ n s ~differs ~ - ~ ~somewhat from the case of heavier particles; but will not be discussed here, inasmuch as range

* Refer to Section 1.7.6. H. Ross and B. Zajac, Nature 162, 923 (1946). R. H. Hertz, Phys. Rev. 76, 478 (1949). 6 6 J. Blum, Compt. rend. 228, 918 (1949). B. Gauthe and J. Blum, Compt. rend. 234, 2189 (1952). 68 J. P. Lonchamp and C. GBgauff,J . phys. radium 17, 132 (1956).

64

66

230

1.

PARTICLE DETECTION

measurements of electrons are rarely performed in emulsions. The trajectories of slow electrons are not straight, but, due to scattering phenomena, curved in a complicated way. The rectified length of electron tracks or the number of grains rendered developable in electron tracks of certain energy is of importance in problems of &ray emission and will be discussed in the section on particle charge (2.1.1.3). The residual range of particles of given mass, charge, and energy can be calculated if the emulsion composition and the differential stopping power of the constit,uent emulsion elements are known. This follows from the fact that the stopping power of a homogeneous mixture is equal to the sum of the contributions from each element. I n earlier emulsion experiments the differential stopping power relative to air was used because the a particles from radioactive elements were residual ranges-in air-of well-known quantities. The range-energy relation in emulsion was first calculated by Webbz3 for Eastman-Kodak emulsions. Similar calculations were later performed by Wilkins,62based on the experimental datasg-72 for the computation of the irlntegral stopping power of emulsions. These data were adjusted to provide a smooth curve of stopping power versus particle velocity. Wilkins calculated range-energy curves for protons and a particles with velocities up t o p = 0.31; the proton values, with the exception of the very low energy region, are in excellent agreement with the latest and most carefully determined data of Barkas. 7a Wilkins’ calculations are extremely useful in that they allow evaluation of the residual range of particles in emulsions of various compositions-in th e so-called “loaded emulsions,” and in emulsions which contain a higher percentage of water as a result of humidity conditions. A more direct approach to the calculation of the residual range is the evahation of the integral of the reciprocal of the energy loss per unit length which a n article suffers along its trajectory through matter.

(1.7.6) The energy loss, * d E / d x is a function of the mean excitation potential I, of the elements of the stopping material and of the charge z, and velocity B = v / c of the ionizing particle. The following equation shows th a t the

* See Eq. (1.7.13) Section 1.7.6. M. S. Livingston and H. A. Bethe, Revs. Modem Phys. 9, 245 (1937). J. D. Hirschfelder and J. L. Magee, Phys. Rev. 73, 207 (1948). ’1 R. Warshaw, Phys. Rev. 76, 1759 (1949). 7 1 E. L. Kelly, Phye. Rev. 76, 1006 (1949). 73 W. H. Barkas, Nuovo cimento [lo] 8, 201 (1958). 88

To

1.7.

PHOTOGRAPHIC

EMULSIONS

231

energy loss does not depend on the mass mi of the particle, provided that P / m c < mi/m, where p is its monentum and m the electron mass.

Here N is the number of stopping atoms per unit volume of atomic number Z and Ci is a correction term for non-participating electrons in the i t h shell of the atoms. The Bethe-Bloch equation (1.7.7) is valid for heavy particles with velocities greater than approximately 1.5 X lo9 cm/sec. The factor ZCi becomes negligible if the particle velocity is well above the orbital velocity of K electrons. At still higher velocities a different correction term must be introduced into the Bethe-Bloch equation which represents the reduction of energy loss arising from the polarization of the medium. This polarization effect or the “density effect,”* so called because of its evidence in dense media, was first treated mathematically by Fermi.74 The density correction term (6), which must be added to the energy loss equation, depends both on the particle velocity and the mean excitation potential of the medium. This effect will be discussed in greater detail in the section on grain density and energy loss. The energy loss, and hence the residual rangeof the particle [Ey. (1.7.6)], can be determined in the region of validity of Eq. (1.7.7) if I and Ci are known. However, until recently, because these quantities were not known directly from experiments, appreciable uncertainties existed in the rangeenergy relation. It is generally quite difficult t o perform very accurate ionization potential measurements. The magnitude of ionization potentials I, may be obtained from energy loss and range measurements on particles with known momentum; however, in both cases In I , and not I, enters into the equation. Therefore, extremely accurate measurements are required in order to determine the value of I and to decide about the velocity dependence of the ionization potential. Another difficulty arises from the dependence of dE/dx on the electron density of the medium, and therefore on the density and composition of the emulsion. A number of a ~ t h o r s 7 ~ - have 7 ~ attempted to represent the rangeenergy relation empirically by power law equations which, however, are valid only in a restricted energy region. A semiempirical relation, * Refer to Section 1.7.6. E. Fermi, Phys. Rev. 67, 485 (1940). U. Camerini and C. M. G. Lattes, Ilford technical data (Ilford Research Lab., London, 1948). 76H. Bradner, F. M. Smith, W. H. Barkas, and A. S. Bishop, Phys. Rev. 77, 462 (1950). 77 W. M. Gibson, D. J. Prowse, and J. Rotblat, Nature 173, 1180 (1954). 78 H.Fay, K. Gottstein, and K. Hain, Nuovo cimento [91 11,Suppl. No.2,234 (1954). 74 7b

232

1.

PARTICLE DETECTION

R E 2 = K . Ral * RPb, for high-energy particles has been proposed.79 This equation compares the range in emulsions, RE, with ranges in aluminum and lead, as determined from absorption experiments where K is a slowly varying constant. The first theoretical range-energy curve was calculated by VignerongQ for particles with velocities p 5 0.3 in dry emulsions of density 3.815 gm/cm3, employing a constant value of I = 332 ev. The curve is based on the most reliable experimental data known a t that time, and extends down to proton energies of 0.1 MeV. The low-energy portion of the curve is based on measurements b y Mano.81 Baroni et aLsz have calculated the range-energy relation for energies up to several Bev, taking into account the density effect by utilizing Sternheimer’se3 equation. The emulsion density used in deriving this curve is taken t o be 3.92 gm/cm3. Barkas and Youngs4 have also extended Vigneron’s curve t o higher energies by calculating the ranges from the Bethe-Bloch equation, in which they used an ionization potential of 331 ev and a n electron density corresponding to an emulsion density of 3.815 gm/cm3, the same value used by Vigneron. The calculated values for high energy protons are based on Sternheimer’s work. I n a more recent paper, B a r k a refines ~ ~ ~ the approach to this problem by performing much more detailed calculations. The correction factor C, in the Bethe-Bloch equation is evaluated for the K and L shells of all emulsion nuclei, except hydrogen, by using Walske’sss calculations. Sternheimer’s expression for the density effect correction is used at very high energies; the mean excitation potential, which enters this correction, is determined experimentally. The value of the mean ionization potential can, in principle, be obtained from a single accurate measurement of the range in a n emulsion of known composition (humidity content) and shrinkage factor on a particle of welldefined momentum. The energy of the particle must lie below the value where the density effect becomes noticeable, in order to utilize the simplest form of the Bethe-Bloch equation. Once I has been determined, however, the exact ranges or higher energy particles may be employed for the direct evaluation of the density effect correction.

R. R. Daniel, G. G. George, and B. Peters, Proc. Indian Acud. Sci. A41,45 (1055). 8 o L .Vigneron, J. phys. radium 14, 145 (1953); Compt. rend. 232, 1199 (1951). M.G. Mano, Compt. rend. 197, 1759 (1933); Ann. phys. [ l l ] 1, 407 (1934). 8 2 G. Baroni, C. Castagnoli, G. Cortini, C. Franrinetti, and A. Manfredini, Bureau of Standards, Bull. No. 9, CERN, Geneva (1956). 81 R. M. Sternheimer, Phgs. Rev. 103, 511 (1956). 81 W. H. Barkas and D. M. Young, U. C . R. L. 2579 (1954). 8 b M. C. Walske, Phys. Rev. 88, 1283 (1952), 101, 940 (1956). 70

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PHOTOGRAPHIC EMULSIONS

233

A mean value of the excitation potential of 331 zk 6 ev has been established from several very exact range determinations. If one uses the Bloch relations6 I = kz, and substitutes for Z the mean atomic number of the emulsion elements, one obtains the value 12.25 -t 0.22 ev for k. The latt,er value is in satisfactory agreement with the measurement of Bichsel et aZ.,s7 who find that k varies between 12.5 and 13.1 in absorption measurements of low-energy protons in various elements. The above value of k is definitely greater than that of 9.1 which was found from the experiments of Mather and Segr&8s The excellent agreement between calculated and measured values presents a strong argument in favor of adopting a mean excitation potential near 331 ev; furthermore, it seems to strengthen Caldwell’s*9 assumption concerning the constaiicy of I . The calculated values of I have ~ ~ with ~ experimental not only been compared with the work of B a r k a but data from other laboratories as well. The calculated range, (602 2.2)p, of the p meson in the T-P decay is in excellent agreement with the Berkeley results and with measurements on the G stack.g0The ranges of mesons from the decay of K,, mesons agree with Barkas-Birge tables91 to within an experimental error in the determination of the K meson mass. The measurements of Heinzg2on 342 Mev protons are about 1.5% lower, those of Friedlander el aL93on protons of 87, 118, and 146 Mev are lower by about 1%,and of De Carvalho and Friedman94 on 208 Mev protons are in good agreement with Barkas’ values. Figure 2 gives the residual range of protons as a function of kinetic energy in the emulsion; the curves are drawn according to the tabulated range data in Barkas’ paper.73 For energies below 1MeV, the ranges were calculated from the E3’2 relation of Geiger and Bohr. The range-energy relation due to Baroni et aZ.S2deviates from Barkas’ curve only at proton energies greater than 1 Rev, while the curve from Fay et aZ.,78which is based on a power law, exhibits considerable deviation a t proton energies as low as 150 MeV. Figure 3, taken from Barkas’ paper,73gives the percentage increase in F. Bloch, 2. Physik 81, 363 (1933). H. Bichsel, R. F. Morley, and W. A. Aron, Phys. Rev. 106, 1788 (1957). 88 R. L. Mather and E . SegrB, Phys. Rev. 84, 191 (1951). *9 D. 0. Caldwell, Phys. Rev. 100, 291 (1955). G. Stack Collaboration, Nuovo cimento 1101 2, 1063 (1955). $1 R. W. Birge, D. H. Perkins, J. R. Peterson, D. H. Stork, and M. N. Whitehead, Nuovo cimento [lo] 4, 834 (1956). *z 0. Heinr, Phys. Rev. 94, 1726 (1954). *3 M. W. Friedlander, D. Keefe, and M. G. X. Menon,Nuovocimento [lo] 6,461 (1957). *4 H. G. De Carvalho and J. I. Friedman, Rev. Sci. Instr. 26, 261(1955). 86

234

1. PARTICLE

DETECTION

FIG.2. Range-energy relation for protons in emulsions. The energy of protons in Mev is plotted versus the range in cm. (According to tabulated data by Barka~'~.)

-

2 -

I

I

I

I

I

I

I

1

I

FIG.3. The percentage increase of ionization potential causing a 1% increase in emulsion range is plotted versus particle velocity (after Barkas73).

1.7.

PHOTOGRAPHIC

235

EMULSIONS

the mean excitation potential which would cause a one per cent increase in emulsion range, as a function of particle velocity. Similarly, Fig. 4, also from Barkas’ work,66gives the relative decrease in range resulting from a one per cent increase in emulsion density, as a function of P ; these curves are drawn assuming that the ratio of water volume decrease to water I .o

I

0.9

-

0.8

-

0.7

-

0.6

-

0.5

-

0.4

-

0.3

I

I

I

I

I

I

I

I

I I l l

I I I l l

0.01

I

I

I

I

1

I

0.1

I

I

I

I

l

l

I I l l

I

B FIG.4. The percentage range decrease for one per cent increase in emulsion density is plotted versus particle velocity. The curves are calculated for 3 different assumptions about the ratio of the water volume to water weight decrease in emulsions (after Barkas et al.66).

-

weight decrease is A * * equal to unity, B * * equal t o 0.94, and C * equal to 0.84. It may be noted how critically the residual range is affected by density variations, especially a t high particle velocities. Fowler and Scharffg6propose a simple range-energy formula which is believed to be accurate to within 5% for ranges lying between 0.1 and 3000 gm/cm2:

--

E 96

=

+(R)[l . 1R

+ 25 z / R - 21.

(1.7.8)

P. H. Fowler and M. Scharff, cited by Friedlander et al., see reference 93.

236

1.

PARTICLE DETECTION

Here R is measured in gm/cm2 and the factor 9 is close to unity but varies somewhat with emulsion density. Another range-energy relation which is valid within a rather wide energy interval and which was used extensively in the early fifties, may be written in the form:

E

=

aZzn&fl--nRn.

(1.7.9)

1.7.5.3. Range Straggling.* The effect of range straggling on the range-energy relation, due to fluctuations in the rate of energy loss, must be considered if experimental range data is*to be interpreted correctly. Range straggling in homogeneous matter was first studied by B0hr.~6The problem was treated relativistically by Lindhard and Scharff,97 who showed that the exact straggling may be obtained from the equation:

where n is the electron density; E the kinetic energy, dE/dR, the mean the mean value of R2. rate of energy loss, R the mean range, and Lewisgs has applied several corrections to the nonrelativistic form of Bohr’s equation without appreciably altering the magnitude of the Bohr straggling effect in emulsion. Furthermore, the difference between the mean range and the most probable range found by Lewis to be caused by a slight skewness of the range distribution, is of negligible magnitude. Barkas et aLg9 have tabulated the percentage range straggling for protons in emulsions. Inasmuch as both quantities u = Z2ZB/mi1/2and r = Z2E/mi,where u is a measure of the range straggling and r is a measure of the residual range, depend only on the velocity of the particle, the percentage straggling 100a/r is also a function, albeit slowly varying, of the velocity alone. The quantity (100a/r) = (100mi1’2Z~/R),and hence the percentage straggling, does not depend on the charge of the particle, but varies inversely with the square root of its mass, since varies directly with the mass. According to Barkas, the percentage straggling is greater than 2% for very slow protons and about 1% for very fast protons. However, the actual range straggling in emulsion is greater than the Bohr effect alone, as a result of straggling due to distortion and inhomogeneity

0,

* Refer to Section 1.7.5.1. N. Bohr, Phil. Mag. [6] SO, 581 (1915). J. Lindhard and M. Scharff, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 27, No. 15 (1953). 96 91

98 99

H. W. Lewis, Phgs. Rev. 86,20 (1952). W.H.Barkas, F. M. Smith, and W. Birnbaurn, Phys. Rev. 98,605 (1955).

1.7. PHOTOGRAPHIC

EMULSIONS

237

of emulsion, reading errors, and end effects, as has been noted in previous sections. 1.7.5.4 Range-Energy Relation for Multiply Charged Particles.* Residual ranges of a particles of various energies were known even before those of protons, because the former were available a t well-defined energies, emitted from radioactive sources. These experiments are described in references 1-7b. The problem of a-particle ranges was also extensively studied by Wilkins,62 Cuer and Lonchamp,'O' and Neuendorffer et c z L 1 0 2 Wilkins determined the ratio of ranges of a particle and proton of equal velocities by means of a comparison with the like ratio in air (Livingston and Bethesg), and by baking into account the stopping powers of both media. He found t h a t the relation obeyed in emulsion by a particles is given by (1.7.11) where the range extension C = 1.5 N for CZ emulsions of density 3.92 gm/cm3. The excess of the range of multiply charged particles over the range of protons of the same velocity is caused by the occasional capture of orbital electrons from atoms of the traversed medium. This effect becomes important, when the ion velocity is equal or smaller than the orbital velocity of electrons. Consequently during a portion of its trajectory through the emulsion the effective charge of the a particle ZHe< 2. Thus the energy loss is not proportional to Z2,but is given by -dE/dR = Zf(P), wheref(P) depends only on the particle velocity. The effect of range extension becomes very noticeable for ions of still higher charge, since the ion velocity a t which electron pickup sets in increases with higher ion charge. It is usually assumed that the cross section for electron capture and electron loss become comparable when the ion velocity approaches the velocity of the most loosely bound electron, and that the capture cross section increases rapidly and hence the effective ion charge decreases when the ion velocity drops below this value. Knipp and TellerlO3 have calculated the ratio ionic t o nuclear charge for slow ions in gases, using range data, available at this time. The authors describe in detail the process of orbital electron capture, basing their calculations on the Thomas-Fermi charge distribution, and calculate the effective charge as a function of the ion velocity. I n these calculations

* Refer to Sections 1.1.3.2and

1.1.3.3.

P. Cuer and J. P. Lonchemp, Compt. rend. 232, 1824 (1951). 102 T. A. Neuendorffer, D. R. Inglis, and S. S. Hanna, Phys. Rev. 82, 79 (1951). 103 J. Knipp and E. Teller, Phys. Rev. 69, 659 (1941). 101

238

1.

PARTICLE DETECTION

enters a parameter, y, assumed to be constant, which has to be determined by comparison with experimentally determined range energy curves. A semitheoretical study of the range-energy relation of heavy ions in emulsions, based on the calculations of Knipp and Teller was made by WilkimS2The range-energy relation for heavy ions is given by

(1.7.1l a ) where the range extension C Z ( @is a function of velocity: C Z @ )is different for ions of different charge. Wilkins determined C Z @ )for various heavy ions in emulsions of density 3.92 gm/cm3 and compared the results with the experimental data of various authors. Since then a great number of range measurements of heavy ions, Li, Be, B, C, and N in emulsions have been made. The heavy ions used for the purpose of analysis originated from disintegrations of light emulsion nuclei or from boron atoms (boron-loaded emulsions), caused by irradiation with particles of well-defined energy. In other experiments carbon or nitrogen ions, recoiling in elastic collisions with monoenergetic beam protons, and finally magnetically analyzed ions, produced in the target of accelerators were used. The latter method was used by Barkaslo4who has measured the ranges of H and He isotopes and of Li8 and Be8-the last two being easily identified by their decay schemes (“hammer” track). The particles were emitted in the bombardment of a thin target by a! particles from the 184-inch cyclotron. The difference in range extension can be measured for particles of equal velocities. Barkas derives from Knipp and Teller’s work a general relation for the quantity BZ which B Z is thus a function is defined by the equation r = (Z2R/mi)- B z ; r of velocity only. He assumes the validity of the relation Bz = aZ3 and determines a from experimental data to a = 1.2 X cm. The range extension CZ is then given by CZ = Bz(mi/Z2). The value of C Z for hydrogen is just equal to BZ = 1.2 X cm and thus evidently a negligible quantity; CZ was determined for a particles Li7, Be9, and C12 ions to Cz = 0.8 p, Cs = 2.4 p, Cq = 4.2 p, and Ca = 8.5 p, where the Cz values of the heavier ions were found with reference to the range-energy relation of a! particles. A very similar procedure was previously used by Lonchamp,105 while in his later paper106range-energy relations of heavier ions are calculated with reference t o protons. There is furthermore an essential difference in the

+

W. H. Barkas, Phys. Rev. 89, 1019 (1953). J. P. Lonchamp, J. phys. radium 14, 89 (1953). lo8J. P. Lonchamp, J. phvs. radium 18,239 (1957). 104

106

1.7.

239

PHOTOGRAPHIC EMULSIONS

treatment of the relationship of effective charge versus ion velocity. The ratio between effective charge and ion velocity, y, which in the work of Knipp and Teller,’Wilkinsand Barkas was assumed to be constant, is now treated as a variable. This assumption is justified, since according to measurements of Reynolds et aZ.lo7in gases and Reynolds and Zuckerlo8 in emulsions, y decreases with increasing velocity. The experiments were performed with nitrogen ions in the energy interval between 4-28 MeV. The range-energy relation for various heavy ions in CZemulsions were investigated by Lonchamplo6 and the corresponding values for range extension were derived; Lonchamp’s values are in general somewhat lower than Barkas’ and Wilkins’ values, with exception of Cz, which was found to be Cz = 1.6 p instead of 0.8 p , given by Barkas. The later paper of Lonchamplo6is dedicated to the investigation of the range-energy relation of Li7 ions. The ions were emitted from the target of the 184-in. Berkeley cyclotron, and after being deviated by the cyclotron magnet, strike photographic emulsions, situated a t various distances from the target ; each position corresponds to a different radius of curvature and therefore different particle velocity. In a painstaking way the author compares the energy loss in trajectories of protons and Li7 ions, starting in each case from a point of known energy; the values of d E / d X were measured within very small energy intervals. Since the range energy relation for protons is known, one finds the energy loss of Li7 ions according to the relation (1.7.12)

provided that the velocity of protons and Li7 ions is the same. 2, in the above equation is equal to 1, a t least down to energies larger than 0.32 Mev.lo9In this way Lonchamp has determined the effective charge of Li7* ions for various energy values, from which in turn the range extension and thus the complete range-energy relation can be derived. Figure 5 gives the range-energy relation for Li7 ions, drawn according to the tabulated values in Lonchamp’s paper. From the data of Li7 the curves for Li8 and Lie ions can be obtained easily, considering the proportionality between mass and residual range for particles of equal charge and velocity. The author compares the calculated data with experimental values of various authors and especially with the more recent range measurements of Livesy,l10 which are in excellent agreement. Livesy discusses in detail the difficulties confronting any theoretical approach to the problem of H. L. Reynolds, J. W. Scott, and A. Zucker, Phys. Rev. 96, 671 (1954). H. L. Reynolds and A. Zucker, Phys. Rev. 96, 393 (1954). 109 T. Hall, Phys. Rev. 79, 504 (1950). 110 D. L. Livesy, Can. J . Phys. 34, 219 (1956). 107

108

240

1.

PARTICLE DETECTION

effective charge versus particle velocity. He proposes a semiempirical method in which the range-energy relation is approximated by a power law with coefficients determined from experimental data. However up to now the range energy relation in emulsions is exactly known only for He and Li ions, while the question of range extension in heavy ions is still unsettled. It is evident that the knowledge of CZvalues is especially important for slow ions, if energy determination from range measurements is needed. Problems of this kind arise in investigations of I00

I

0.1

FIG.5. Range-energy relation for Li7, Lie, and Lie particles. The energy E is given in Mev and the range in microns.

binding energies and studies of energy levels of disintegrating atoms and calculations of the total energy release in stars caused by the capture and subsequent decay of unstable particles. Fortunately several heavy ion accelerators were recently built and it is to hope that in the near future a great number of experimental data will become available, so that the problem of velocity dependence of effective ion charges can be solved in a general way.

1.7.6. Ionization Measurements in Emulsions The problem of energy loss in emulsions as a function of particle charge and velocity has been discussed in a previous section in connection with the range-energy relation in emulsions. The theory is baaed on the BetheBloch relation and modified for high-energy particles to include the

1.7.

241

PHOTOGRAPHIC EMULSIONS

relativistic deiisit y effect . Sternheimer 83, l s l l2 performed detailed calculations of the density effect in various materials, including nuclear emulsions. Inasmuch as the theory is fully treated in Section 1.1, here we will discuss only Sternheimer’s equation for fast particles in connection with the corresponding ionization parameters in the emulsion. This relation is :

( ) $, [In 2me2W0 Iz + In

g,=- 1 dE P

r i b wo

=

PZ 1 - 02

~

- 61.

(1.7.13)

In this equation, which is valid for singly charged particles heavier than electrons, the energy loss per gram cm-2 is represented as a function of particle velocity, p, and depends on the value of the mean excitation . latter represents the potential I , the polarization effect 6 , and W OThe maximum energy transfer, which contributes to the local ionization, limited to the silver halide crystals within the path of the particles; g, does not include the energy spent in the production of fast knock-on electrons (6 rays), whose range in silver bromide exceeds the mean grain diameter. The choice of W Ois somewhat arbitrary and estimated values between 2 and 30 kev can be found in various publications; m is the mass of the electron, and A is a constant defined by A = 4?me4/mc2p,where n is the number of electrons per cm3in the substance; thus A = 0.0698 MeV/ gm cm-2 for Ilford emulsions, and A = 0.0671 Mev/gm cm-2 for AgBr. The importance of the mean ionization potential I for energy loss calculations was already mentioned in the section dealing with the rangeenergy relations in emulsions. Sternheimer has chosen C a l d ~ e l l ’ svalue, ~~ I Ei‘ 132. The value of 6 in AgBr is given by 6 = 4.606 loglo(py)

- 5.95 + 0.0235[4 - logl~(pr)]4.03 (1.7.14a)

for all values of loglo(py) 0.30

< logio(P7) < 4

and 6 = 4.606 loglo(py) - 5.95

*

*

10glo(Pr) < 4.

(1.7.14b)

The constants in these equations were calculated by Sternheimer, taking into consideration mean values of excitation potentials of Ag and Br. According to Eqs. (1.7.13), (1.7.14a), and (1.7.14b) the ionization loss decreases slowly for high-energy particles until a minimum is reached a t 0 = 0.95. For still higher velocities, the second term in Eq. (1.7.13) rapidly increases so that a continuous increase in energy loss would be expected if it were not for the 6 term, which a t very high energies increases proportionally to loglo(0y). 111 111

R. M. Sternheimer, Phys. Rev. 88, 851 (1952). R. M. Sternheimer, Phys. Rev. 91, 256 (1953).

242

1.

PARTICLE DETECTION

It has been found that the grain density closely follows, within a wide energy range, the energy loss versus velocity curve. The grain density curve goes through a broad minimum, reaching the lowest point a t y = 4 and then increases slowly for still higher energies. The relativistic increase was first observed by Pickup and V o y ~ o d i c ; ~these ~ 3 authors found that the plateau value, which is reached for y = 50, is about 14% higher than the minimum grain density. Other authors110-118have more recently confirmed the realtivistic rise in grain density. However, there is still disagreement among the various authors about the ratio gPl,t,,u/g,i, as well as the rate of increase of grain density with y beyond the value y = 4, qplateaubeing the maximum value the grain density reaches for values of y > 4. The experimental results of Stiller and Shapiro"' and Fleming and LordlZ1seem to be in good agreement with Sternheimer's calculations. Alexander and Johnston122have determined the rate of plateau to minimum grain density to be 1.133; the error in these measurements is estimated to be less than 1 %. Actually this investigation was not based on grain densities but on blob densities; the relation between these two parameters will be discussed in the next paragraph. The authors, furthermore, determine from the ratio of blob densities Bpiateau/Bmio the parameters I and Wowhere Bplatellu is the maximum blob density beyond y = 4 and I and Wo have the same meaning as in Eq. (1.7.13). This can be done because the energy loss at the plateau value does not depend on Eo/I but only on Wo =

A[ln 2mc2W0 - 21n(hvp)]

(1.7.15)

where v p is the plasma frequency of the mean emulsion nucleus,l13m the mass of the electron, and A the constant defined in Eq. (1.7.13). On the other hand In(2mc2Wo/12)can be calculated from (1.7.16) (1.7.16) E. Pickup and L. Voyvodic, Phys. Rev. 80, 89 (1950). A. H. Morrish, Phil. Mag. 171 43, 555 (1952). us A. H. Morrish, Phys. Rev. 91, 425 (1953). llE M. M. Shapiro and B. Stiller, Phys. Rev. 87, 682 (1952). 117 B. Stiller and M. M. Shapiro, Phys. Rev. 171 92, 735 (1953). 11* R. R. Daniel, J. H. Davies, J. H. Mulvey, and D. H. Perkins, Phil. Mag. [7] 43, 753 (1952). ' 1 9 M. Danysz, W. 0. Lock, and G. Yekutieli, Nature 169, 364 (1952). 120 R. P. Michaelis and C. E . Violet, Phys. Rev. 90, 723 (1953). 121 J. R. Fleming and J. J. Lord, Phys. Rev. 92, 511 (1953). 122 G. Alexander and R. H. W. Johnston, Nuovo n'mento [lo] 6, 363 (1957). 118 114

1.7.

PHOTOGRAPHIC EMULSIONS

243

where j3 is the particle velocity (determined from the residual range) for which minimum ionization is observed. In these investigations ?r and p mesons from K,, and K,, decays were used. The authors find for W othe value (2.9 & 0.5) X lo4 ev and for the mean excitation potential I = 12.92; the latter value is in good agreement with Caldwell’s value but differs somewhat from the value found by B a r k a ~ ’for ~ particles of lower energy. The above value of W Oseems high, considering that an electron ejected with this energy produces an easily visible &ray track in the emulsion. Before analyzing the theoretical basis of various ionization parameters, it is well to discuss experimental det,ails and methods. The measurement of grain density requires oil immersion objectives of 90-1OOx magnification and eyepieces with carefully calibrated micrometer scales. One counts the number of grains lying within given scale intervals, being careful to choose track segments which lie near the center of the field of view. When the trajectory is inclined to the horizontal plane, the true grain density is found by multiplying the measured value by cos a,where cr is the dip angle of the track in the emulsion before development. It is generally assumed that there exists a simple relationship between the probability of activating an emulsion crystal and the ionization loss of the particle traversing the emulsion. This relationship can be represented, assuming a (Poissonian) distribution law, P = 1 - exp( - p ~=) 1 - exp( - y) (1.7.17) where P = n,/nt is the probability of rendering developable nu crystals out of a total number nt crystals. In Eq. (1.7.17) y is given by y = qv2 where q is a parameter which depends only on development conditions, 2 is the mean path length of the ionizing particle through the crystal, and v is the number of ionization acts per crystal; the latter must be identified with the restricted ionization loss, since, otherwise (including 6 rays), would depend on the velocity of the particle activating the grain. Equation (1.7.17) assumes that all crystals have equal size and that their centers are aligned. If one assumes with Demers2 that the crystals are spheres of equal size, distributed at random about the trajectory of the particle then

] exp (:)- - .

(1.7.18)

Fowler and PerkinsIzapropose a more general approach accounting for fluctuations in crystal sizes; they assume random distribution of crystals 1zSP.

H. Fowler and D. H. Perkins, Phil. Mag. [7] 46, 587 (1955).

244

1.

PARTICLE DETECTION

and a distribution of crystal sizes in which all diameters between 0 and 22 are equally probable. P is then given by 1

+ (I + y)exp(-zy)

].

(1.7.19)

Because of the wide distribution chosen (the distribution of crystal sizes is sharper) Eq. (1.7.18) can accommodate variations in the parameter q, by expressing y = f[(QZ)v].

FIQ.6. A plot of the relations between mean development probability and restricted ionization loss. The curves refer to Eqs. (1.7.17), (1.7.18), and (1.7.19) in the text. The experimental points are: o experiment A; experiment B; 0experimental values which have been corrected for the apparent loss of grains in clusters; X track with dip angle 40", A tracks with dip angle of 30", calculated according to Eq. (1.7.43a).

+

In Fig. 6, the values of defined by Eqs. (1.7.17), (1.7.18), and (1.7.19) are plotted as functions of y. (The experimental points in this figure will be discussed later.) The three curves are nearly identical €or small values of y, but differ considerably for larger values. From grain density measurements in tracks of lightly ionizing particles it is known that values of normalized grain density (g* = g/go) as a function of ( S / g o ) ? can be easily fitted to the initial part of the 3 curves. However, thus far, it has not been possible to decide which one of the 3 curves, or if indeed any one of them, does truly represent g* = f[(g/go),] for g* 2 6. The reason for this failure is due to difficulties of performing grain density measurements in dense tracks.

1.7.

PHOTOGRAPHIC EMULSIONS

245

Grain counting is a simple procedure in tracks of very fast particles, especially if the development is light. I n this case the grains are well separated from each other and clogging of grains occurs only occasionally. However, in denser tracks cluster formation is quite frequent and it becomes difficult t o resolve the clusters into single grains. The number of grains per cluster is either estimated b y the observer, or determined by the application of length criteria based on the mean grain diameter of developed grains, or finally by investigating the diffraction patterns of grain clusters. However, all these methods are subjective and tedious, leading t o the belief that the grain counting method should be replaced by other more objective procedures.

1.7.7. Ionization Parameters: Blob Density, G a p Density, Mean G a p length, and Total G a p Length

At present the general practice is to measure “blob” density instead of grain density, a blob being a single grain as well as a cluster of grains. An advantage of the blob counting method is that it leads directly to another ionization parameter, the gap ” density, where a gap is the blank space between blobs. However, it is necessary to emphasize that the grain density, or the number of silver crystals per unit length, made developable by the traversing particle, and not the blob density, is directly related t o the ionization loss. The blob density depends upon the spacing of crystals in the emulsion and upon the size and configuration of the developed grains, these being influenced by the processing conditions. As in the case of grain density, one introduces, instead of blob density 3,the normalized value 3*,which refers t o the ratio of actual blob densities t o the respective value at minimum ionization, or more often to blob densities a t the plateau value. It has been found th a t for low ionization densities-singly charged particles of high energy-the normalized blob densities are reasonably independent of development. Therefore, by exposing emulsions to particles in this energy region one can obtain calibration values of B”, which are valid also in emulsions of a different batch or which were processed under slightly different conditions. However, for higher ionization densities the independence of B values of development conditions ceases to be valid, and blob density, therefore, loses its value as an ionization parameter. There is still another reason why blob density measurements are not meaningful in the region of higher ionization density. The blob density increases with increasing ionization loss up to a certain maximum value and then decreases slowly for still higher ionization values; thus in a certain energy interval, blob density is a bivalent function of energy loss.

246

1.

PARTICLE DETECTION

H o d g s ~ nrecommended ~~~ gap length, the total blank space between blobs per unit length, as a useful parameter in ionization measurements in tracks with blob densities larger than 2 times plateau value. Shortly afterward Renardier and AvignonlZ6modified this technique by counting, instead of measuring, the total number of gaps per unit length. Since that time gap measurements in a variety of forms have played a n important role in nuclear emulsion problems. Gap measurements are now generally used instead of, or in combination with, blob density measurements if the ionization exceeds 2 times the minimum value or when light but steeply dipping tracks have to be measured. The following parameters are used in ionization measurements : (a) blob density B , the number of blobs per unit length; (b) H , the number of gaps per unit length; H is, of course, equal to B, the number of blobs per unit length; (c) L H , gap length or the total width of gaps per unit length; (d) A, mean gap length, defined as the mean distance between the inside edges of neighbor grains. The last parameter was proposed by O'Ceallaigh. l Z 6 Ritson127 has devised a very convenient method for gap measurements : The track is aligned in the microscope with one of the two stage movements which in turn is driven by a motor a t a low constant speed. The observer is provided with two counters, driven by a single pulser. One of the counters runs continuously with the stage movement, while the other one is activated only when the observer presses a button which is done whenever the hairline in the eyepiece lies over a region of track unoccupied by grains. The ratio of the two counter readings gives directly the total gap length in the track under investigation or the gap length LHper unit length if counter readings are made at certain predetermined time intervals. Baroni and CastagnolilZ8describe a similar arrangement, using a motor-driven stage moving with constant velocity. The authors add several improvements, the most important being a n RC circuit connected with a counting device which enumerates only those gaps which are longer than a certain predetermined minimum value. Therefore, one can separately record, with this apparatus, the total number of gaps, the number of gaps longer than a minimum value I, the total width of gaps P. E. Hodgson, Phil. Mag. [7] 41, 725 (1950). M. Renardier and Y. Avignon, Compt. rend. 233, 393 (1951). IaSC. O'Ceallaigh, Proc. Intern. Union Pure and Appl. Phys. Conf. on Cosmic Radiation, BagnBres, France, 1953 (unpublished). l27 D. M. Ritson, Phys. Rev. 91, 1572 (1953). l28 G. Baroni and C . Castagnoli, Nuovo cimento [9] 12, Suppl. No. 2, p. 364 (1954). Iz4

'26

1.7.

PHOTOGRAPHIC EMULSIONS

247

per unit length and the total width of gaps surpassing a minimum length. The introduction of a minimum gap length is connected with the findings of various authors th at the inclusion of gaps below a certain minimum length in gap measurements makes the measurements unnecessarily tedious and dependent on observational errors and optical conditions (resolution). Therefore, only those gaps are included whose widths exceed a certain minimum value 1 which should be large enough for accurate and rapid measurements, but small enough to prevent loss of information, especially in tracks of high ionization density. The choice of 1 depends upon the problem, the length, density, and dip angle of the track. The corresponding parameters referring to data above a minimum value 1 will be denoted in the following expressions: H(1), X(I), and A H @ ) For . tracks inclined t o the emulsion surface the optimum value of gap length 1 is given by 1 = I’ sec a,where (Y is the dip angle and 1‘ the minimum value corresponding t o a track with equal gap density, but negligible dip angle. I n the following paragraphs we will discuss the relationships which exist among blob density, total gap length, mean gap length, and grain density. Grain density, as pointed out earlier, is a direct measure of the energy loss of the traversing particle; it depends, of course, also upon the processing conditions; however, the dependence is of such a nature th a t the normalized value g* becomes independent of development conditions and so represents, for a given emulsion type, the true ionization parameter. The normalized blob density B* is, as stated before, also reasonably independent of development in tracks of low ionization density; this is due t o the fact th at only a small percentage of grains will coalesce to greater complexes. However, for higher ionization densities, the process of cluster formation becomes more and more important and the degree of coalescence depends on the strength of development which determines the final size t o which the crystals grow during processing. The relationships between ionization loss and blob density or any other parameter connected with gap measurement, are much more complicated than in the case of grain density. An analytic expression for the former relationship must be based on a theory of track formation in the emulsion. Various a~thors123~12~-1a7 have worked on this problem and several models describC. O’Ceallaigh, Bureau of Standards Document No. 11, CERN, Geneva (1954). M. G. K. Menon and C. O’Ceallaigh, Proc. Roy. Soc. A221,292 (1954). 131 R. H. W. Johnston and C. O’Ceallaigh, Phil. Mag. [7] 46, 424 (1954). 132 M. Della Corte, M. Ramat, and L. Ronchi, Nuovo cimento [9] 10, 509, 958 (1954). 133 M. Della Corte, Nuovo cimento [9] 12, 28 (1954). 134 M. W. Happ, T. E. Hull, and A. H. Morrish, Can. J . Phys. SO, 669 (1952). 1 3 5 A. J. Hem and G. Davis, Australian J . Phys. 8, 129 (1955). 136 J. M. Blatt, Australian J . Phys. 8, 248 (1955). 137 C. Castagnoli, G. Cortini, and A. Manfredini, Nucwo cimento [lo] 2, 301 (1955). 139 130

248

1.

PARTICLE DETECTION

ing this process were proposed. The essential difference between these models lies in the treatment of spatial distribution of silver crystals. O’Ceallaigh’s work126~12p-1s1is based on the assumption, shared by Happ et al.,la4 that there exists no correlation between the position of silver crystals in the emulsion, while Herz and Davis136visualize the emulsion as a lattice with silver crystals spaced a t regular intervals; other authors use somewhat modified models supporting their assumptions by comparison with experimental data. BlattL3‘jand Fowler and PerkinslZ3discuss in great detail the theory of track formation and the importance of the theoretical assumptions for the practical use of emulsions in ionization measurements. In comparing their theories with experimental data the authors arrive at different conclusions, Blatt supporting a modified constant spacing model, while according to Fowler and Perkins’ findings, agreement between experiment and theory can be obtained with a variable spacing model; the authors introduce certain refinements in the earlier theory which are connected with fluctuations occurring in the distribution of crystal positions, size, and developability. For low ionization densities all theories give satisfactory agreement with experiments, due to the fact that the blob separation in lightly ionized tracks is not greatly influenced by any type of fluctuation, and exceeds greatly such distance8 as required in the constant spacing model. We will discuss here first O’Ceallaigh’s work based on the simplest model. O’Cealla.igh126~12v and Menon and O’Ceallaigh130have shown that the distribution in the length of gaps along tracks is exponential and given by (1.7.20)

where h is the mean gap length. The connection between mean gap length A, grain density g, and total gap length LH,is given by the equation 1

-

9

- a!= x and LH

=

1 - ag

(1.7.21)

where (Y is the mean diameter of the silver halide crystal.’30 During development the grain increases in size and reaches finally a diameter which is about twice the size of the original value. Therefore, the free space between grains becomes smaller, and some free spaces may disappear completely while the grains coalesce and form larger complexes. However, because of the exponential distribution, the homogeneous growth of all grains does not change the mean gap length A; the latter is, therefore, a parameter which is largely independent of the diameter of the developed grains and hence of developing conditions. The model is based

249

1.7. PHOTOGRAPHIC EMULSIONS

on the assumption of a random distribution of crystal centers along the track. The blob density B which is equal to the number of gaps H ( 0 ) can then be written B

=

gexp(--)

(1.7.22)

where K is a constant, depending strongly on developing conditions, must be determined for each stack and it is related to the crystal diameter, a, by the relation K = d - a e, where d is the diameter of the developed grain and e is the smallest distance between grains which can be resolved microscopically. K can be found by the simultaneous measurement of B and X in track segments of constant ionization. The expression exp( - K/X) represents the conversion factor for grains into blobs, which for constant energy loss depends only upon developing conditions. Alexander and JohnstonlZ2develop a relationship which allows the conversion of normalized blob densities from one stack to another (different development); however, this relation is limited to tracks of low densities for reasons previously mentioned. In this case one can replace B* = (B/BO) by

+

B*

= g*

exp[-K(g - go)]

= g*

exp[-K(B

- Bo)]

(1.7.23)

where go and Bo refer to the values of grain and bIob density in the plateau or minimum ionization region. If now K1 and K z are the developing constants for two different stacks, it can be easily seen that the two blob densities B1* and Bz* are related by B1* = Bz* exp([Kz(Bz - B231

- [K1(B1 - BIO)]). (1.7.24)

The procedure of finding the mean gap length can be greatly simplified by using the method proposed by Johnston and O’Cealleaigh.’31 The authors propose to count the number of gaps H(I1) and H(E2),exceeding two predetermined lengths l1 and l z ; these data are related to each other and the mean gap length by (1.7.25) With a set of 4 t o 5 different values of I , the mean gap length X can be quite accurately determined. Such measurements, performed with conventional methods (microscope and eyepiece scale), ar’e lengthy and time consuming. This difficulty can be removed by the use of “gap analyzers” ;

250

1.

PARTICLE DETECTION

recently a number of such devices have been c o n ~ t r u c t e d . ~ ~ *The -'~~ principle of such gap analyzers is a system of counter circuits, which are actuated by the closing of a microswitch; the microswitch is closed by a number of equidistant contacts on a motor-driven disk; if the system is actuated by photoelectric means, then the contacts are replaced by holes in the disk. I n this way the gap lengths are chopped into increments which are determined by the constants of the quantizer and the gaps are measured in subunits of time, in a time dependent circuit, or in subunits of length if the stage and disk are driven by synchro motors. Another method for the measurement of X was proposed by Fowler and perk in^,'^^ in which blob measurements are combined with the measurements of gaps exceeding a predetermined length I

H ( I ) = B exp

(-

$a

(1.7.26)

This method has the advantage, that i t can be performed with conventional methods, fixed eyepiece scale and manually driven stage, especially if a conveniently large value of I is chosen; however, one is likely t o lose information if E is too large since, especially in denser tracks, the number of small gaps is large and thus important for the measurement. The authors have calculated the optimum gap length A, which, of course, depends on the track density. They found a relatively broad optimum region between 1.5X < 1 < 2.5X. A great advantage of mean gap length measurements by (1.7.25) or (1.7.26) is, th at these methods can be easily adapted for dipping tracks. Mean gap length measurements, instead of blob measurements alone, should be made, even in very light tracks, if the ~ - ' ~measured ~ distance I dip angle exceeds 15" (developed e m ~ l s i o n ) . * ~The in (1.7.25) or (1.7.26) is the projected length and has t o be replaced by It which is related t o I and dip angle a b y It =

I sec a

+ d(sec a - 1)

(1.7.27)

where d is grain diameter and d(sec a - 1) accounts for the obscuration due to the apparent increase of blob sizes. The "smooth model" by O'Cealleigh neglects any type of correlation, which may exist among the positions of silver crystals in the emulsion. I n la* J. E. Hopper and M. Scharff, Bureau of Standards Document No. 12, CERN, Geneva (1954). la8 K. Enstein, Electronic Eng. 29, 277 (1957). 140 A. DeMarco, R. Sanna, and G . Tomasini, Nuovo cimento [8]9, 524 (1928). 141 S. C. Bloch, Rev. Sci. Instr. 29, 789 (1958). 144 M. Della Corte, Nuovo cimento [lo] 12, 28 (1959). 148 H. Winzeler, Nuovo cimento [lo] 4, Suppl., p. 259 (1956). 14( R. C. Kumar, Nuovo cimento [lo] 6, 757 (1957).

1.7.

251

PHOTOGRAPHIC EMULSIONS

fact, such a correlation may be of no consequence in light tracks with widely separated grains; however, it will become important in dense tracks. Fowler and PerkinsIz3 assume that the gaps, found in very dense (saturated) tracks, can be identified with the gelatine gaps existing between neighboring crystals. Furthermore, they assume (based on experiments) that the gap length distribution is exponential, but cuts off sharply a t a distance-lco-equal to the mean crystal diameter. The distribution function of gaps between developed grains is found in the following way. If u is the distance between the grain centers of two grains, bounding a gap of n undeveloped crystals, then the frequency with which a gap can ) p2(1 p ) n is appear will depend on the product a(n)F,(u)du. ~ ( n = the probability that one grain is developed and followed by an undeveloped and finally another developed grain. The function Fn(u) is defined by

-

Wn n!

Fn(u) = - exp(-W)

(1.7.28)

+

where W = (u/ko) - (u 1)2 is the path length through gelatine, expressed in units of lco. The total differential distribution, or the number of gaps per unit length between u and u du, is obtained by summation over all values of n, and is given by

+

The calculation of the integral distribution is quite complicated, because the sum appearing in (1.7.29) cannot be simplified. Numerical evaluation shows for small grain separations a certain degree of roughness in the distribution curve; which strongly depends on the choice of the numerical values for LO and 2 . However, for grain separations (center to center) greater than about 0.8 p and reasonable assumption for the values of Lo and 2, the curves appear smooth. Fowler and Perkins assumed LO = 2 = 0.2 p. I n general it will be possible to describe the total integral gap length by (1.7.30)

which is a n asymptotic approximation of the distribution function for u -+ a.I n this equation X is defined by (1

- $)exp(-

:)

=

1 - F.

(1.7.31)

252

1.

PARTICLE DETECTION

In one assumes that the asymptotic distribution form is also valid for the calculation of blob densities and if u’ is the distance between neighboring crystal centers, then B is given by

B

=

+(u’P) = -exp X

5k03 (- ): [ + 4l l k8o +~m3 -

1

*

*

*

1

(1.7.32)

where the series in the squared parenthesis converges rapidly and reaches a maximum value for ( k o / X ) = 1. Thus far we have neglected to take into account fluctuations in the size of the developed grains in the distribution function. The existence of such fluctuations can be easily observed; Fowler and Perkins showed that the distribution function is Gaussian and determined the standard deviation t o be 0.14 p or 20%, with a mean grain diameter of 0.7 p. They then calculated a correction factor C = [ l f (p2/4X2)] by which both blob and gap densities have t o be multiplied in order to account for fluctuations in the size of the developed grain. It is easily seen that this correction factor is negligible for light tracks. Fowler and perk in^'^^ have plotted 1 / X as a function of residual range for proton and pion tracks in two sets of emulsions which were developed in different ways. The curves for both emulsions coincide over a wide range of energy, down to about 100 Mev (protons) ; but from there on the curves diverge considerably and hence A * in this region cannot be considered as a parameter which is independent of development conditions; A * = X/Xu, where Xo is the mean gap length found in tracks of high-energy electrons; X* was determined from blob density in the region of small energy loss and in the denser region from blob and gap measurements [Eq. (7.1.26)]. The constant spacing model of Herz and Davis (H-D) is based on the following assumptions: the crystals in the emulsion are arranged in a definite lattice, so that each silver crystal can be assumed to lie within a cell of certain small dimensions. Therefore, there exists a strong correlation between crystal positions, which will affect the results in the dense region of the track; however, in light tracks where grain spacing is usually many times the length of the small unit cells, the correlation will cease to be effective and the (H-D) model leads to the same results as the two previously discussed models. For very high densities the gap length will not reduce to zero, but will assume a constant value which is determined by the original lattice spacing and the dimensions of the developed grain. The grain size is assumed to have a constant value (grain size fluctuations are neglected) and the model has been calculated in “linear approximation” (the crystal centers are assumed to be aligned along the path of the particle). The intrinsic difference in the calculations of the

1.7.

PHOTOQRAPHIC EMULSIONS

253

various parameters of the (H-D) model in comparison with other models stems from the assumption of a discrete gap length distribution. I n the following we call b the length of the unit cell containing one crystal and d the diameter of the developed grain; the zero gap length is 1) - d, where r is the largest integer less than then given by Go = b(I’ d/b, and it is the integral part of a factor determining the growth of the crystal during development. The size of the next gap (first-order gap) will 2) - d and the probability of finding a n n t h order gap is be G = b(I’ One can now calculate the integral given by ~ ( n=) P z ( l - P)(r+n-l). gap length distribution and the various track parameters exactly as in the case of other models. Although the problem was calculated rigorously by Blatt,136we indicate here only the results in the simplified form used ’ following equations (1.7.33-1.7.35) refer to by Castagnoli et ~ 1 . l ~The blob density B = H(O), the number of gaps larger than 1 per unit length H(Z), and, the mean gap length X :

+

+

(1.7.33) (1.7.34) where

rl is the integral part of

(d

+ Z)/b and (1.7.35)

The expression for X in (32) becomes identical with O’Cealleigh’s value for P << 1 and X >> b. The total gap length LH per unit length is equal t o the product B X X. Blatt has compared the O’Ceallaigh and (H-D) models and various other slightly modified models with experimental data, and has found good model. Castagnoli et al. and agreement with the predictions of the (H-D) model more recently O’Brien, 146 based their investigation of the (H-D) on a large number of experimental data. Castagnoli et al. measured about 100 particle tracks in an energy region from 600 Mev (protons) down to very small energies using the gap counting machine, previously mentioned. Equations (1.7.33)-(1.7.35) show that the various parameters are related to each other through the probability P . I n order t o calculate P one has to know the mean grain diameter d, which can be found by direct measurements and b and r. The unit cell is not strictly defined; it is at least the size of the mean crystal diameter and can probably be identified X where 8 is the projected mean distance between centers with 8 = ko of successive crystals as defined by Fowler and Perkins. 12* Castagnoli 146 B. T.O’Brien, Nuovo cimento [lo] 7, 147 (1958).

+

254

1. PARTICLE

DETECTION

a

et aZ. define b by b = 1/0.455($a), where is the mean crystal diameter and 0.455 the fraction of emulsion volume occupied by silver bromide. Finally r, which by definition is an integer number, can for experimental reasons assume only values between 1 and 3. Although neither b nor I’ are exactly known, one can determine these magnitudes from the following considerations. Only certain pairs of values of b and r are compatible with the observed grain diameter and the crystal diameter known to be between 0.2-0.3 p. The best combination of b and I? can be found by comparing calculated and observed values at the maximum of the B versus P curve, since B,, = ( l / b ) [ l / ( l r)][l- 1/(1 I?)]’ depends only on b and y . The value of r, of course, is development dependent, and must be determined for each set of emulsions. With the values of r and b found at the blob density maximum, B versus X curves were calculated and compared with experimental data. The authors found good agreement between experimental and calculated curves within a wide region of ionization, starting with near minimum densities down to values of about 8 times minimum value. O’Brien145tested the (H-D) model for relativistic heavy primaries in G-5 emulsions which were very lightly developed; he found good agreement for light and medium tracks, but disagreement for very dense tracks (relativistic Fe nuclei). ~ 7 a) plot of the measuring parameters Figure 7 (Castagnoli et ~ 1 . ~ is B, A, LH,and H(Z) per 200 track length versus R / m = r, where m is the electron mass and R the residual range measured in microns; the range of a particle with a mass mi times the electron mass is then given by r X mi. It can be seen that for dense tracks (small values of r ) the slope of the LH curve is steep (contrary to B and X curves) and therefore, in this region, L H is sensitive to changes in ionization loss. Furthermore, it has been found, that LH in the dense region is less subject to personal errors than other parameters, since the occasional omission of small gaps in the determination of the total gap length does not seriously affect the results. In the light region the slope of the Lcr curve decreases and LH ceases to be sensitive to changes in ionization loss; in this region the best parameters are R and A, the latter having the additional advantage of being independent of development conditions. The role of the parameter H(1) (the curve refers to 1 = 0.8 p ) has been discussed before. It must be emphasized that in the (H-D) model the parameter X is not, as in the (F-P) model, derived from the slope of the gap distribution curve. but is found from the ratio L H / B . The experimental values should be identical in both cases if the gap length distribution were truly exponential However, experiments show that this is only approximately true for light tracks, but certainly not for medium and heavy tracks. The discrepancy

+

+

1.7.

PHOTOGRAPHIC EMULSIONS

255

between the two X values is quite serious, because Fowler and Perkins (and subsequently many other authors) express the ionization loss in tracks by the parameter l/X*. In general, the value 1 / X derived from the slope of the gap distribution curve is larger than the value found by L H / B ,and consequently the values of the probability derived from Eqs. (1.7.31) and (1.7.35) will be quite different when tracks of medium or large densities are considered.

200 100

50

1

1 -

-

I

0/200p

-

20 10

5.0 H(l)/200p

0.5 2.0 1.0

_/_-------

-

k(microns)

0.2

-

The situation is rather complicated for the following reasons. The ionization parameters proposed thus far are applicable only within certain density intervals, and furthermore, their mutual interdependence as well as their relationship to are based on assumptions (models) and emulsion parameters [Eqs. (1.7.31) and (1.7.35)] which cannot be verified in a simple experimental way. While the exact values of these emulsion parameters are of no consequence in light tracks, they are of great importance for tracks of medium and large density. Thus the problem of

256

1.

PARTICLE DETECTION

ionization measurements in the latter regions has not yet been solved satisfactorily, even though this is an important region for the identification of slow singly charged, and multiply charged particles of all velocities. 100

90 00 70 60 50 40

30

-8

20

---

u

I

10

9 8

7 6

5 4

3 2

\ ,

I

0

1

2

3

4

5

6

GAP LENGTH (IN MICRONS)

FIQ.8. Gap length distribution curves (normalized). Experiment A: pion, 1.3Bev/c. Experiment B: proton, R = 9300 p.

The discrepancy between the X values, measured by different methods can be partly understood from observations recently published by Cortini et ~ 1 . The ' ~ ~authors found that the gap length distribution deviates from an exponential curve for small gap lengths, an observation which was confirmed by other authors. Figure 8 gives the semilogarithmic plot for a pion with momentum I46 G.

Cortini, G. Luzzatto, G . Tomasini, and A. Manfredini, Nuovo cimento [lo] 9,

706 (1958).

1.7.

PHOTOGRAPHIC EMULSIONS

257

1.3 Bev/c and a proton of 9 mm residual range and it can be easily seen that the curves deviate from a straight line for gap lengths 1 5 0.5 p. If h is calculated from the straight line (F-P), the result will be smaller than the value found from h = LH/B and consequently the parameter 1/X in the (F-P) model becomes too large. This effect will be more pronounced, the denser the track, since the fraction of small gaps (which are neglected in the slope measurements) will increase with ionization loss. Cortini et ul.l4*explain their observations as the result of a development effect and propose a modification of Eqs. (1.7.26) to (1.7.36)

[

H ( Z )= B exp -

1 - F(Z)

(1.7.36)

where 6 is a function, which is zero at 1 = 0 and increases to a constant value a t the gap length, where (1.7.36) becomes exponential in 1. If h is defined by Eq. (1.7.36), then evidently Eq. (1.7.31) which relates h and P is no longer valid, but must be replaced by another relation which now contains in addition to ko and x still another parameter 6. The exact value of 6 is difficult to determine, since it involves distances, which are close to the limit of optical resolution. Because of the ambiguities contained in the use of the parameter A, another more empirical approach to the ionization problem has been attempted, based on the use of a parameter which is more directly related to grain density.ld7 The method is based on the blob length distribution; the importance of the latter has already been pointed out by Della Corte132-133 who investigated the differential blob length distribution in tracks of various density. The differential distribution curves consist of a distinct Gaussian part and a tail, which later becomes more important with increasing ionization. The integral blob length distribution curves for tracks of various densities are plotted in Fig. 9. (The curves refer to measurements in two different experiments A and B.) The number of blobs B(Z)as a function of blob length decreases exponentially with increasing blob length, except for the initial part of the curve. The curved part in the integral distribution reflects the Gaussian part of the differential distribution. It is reasonable to assume that all blobs falling within the Gaussian part are single grains, while all blobs outside the Gaussian part are clusters, composed of at least two grains. In the following the density of single grains will be denoted by N , and the cluster density by N,. The dividing point between grains and clusters can be determined from the shape of differential and 147 M. Blau, S. C. Bloch, C. F. Carter, and A. Perlmutter, Rev. Sci. Instr. 31, 289 (1960).

258

1.

PARTICLE DETECTION

integral distribution curve. If S is the abscissa of the dividing point, then all blobs with 1 5 S are considered to be single grains while all blobs with 1 > S are clusters with a mean cluster length of w = A S , if A is the reciprocal slope of the exponential curve. The percentage a of blobs, being single grains ( S 5 I) decreases with increasing ionization loss, while A and therefore, w increases (see Figs. 10a, 10b and 1Oc). Since the

+

FIG.9. Integrated blob length distribution curve; the blob length is given in microns. The curves represent measurements on trajectories of the following particles. Experiment A: A 1 - proton, Reff = 1890 p ; A B - proton, Reff = 1.5 cm; A 3 proton, R,rr = 3.66 cm; A , pion with equivalent proton range, R = 17.7 cm. Experiment B: I?, - antiproton, Ref{= 1.01 cm; Bf - antiproton, R,ff = 1.45 cm; B3 - 680 Mev/c pion. The curves were fitted t o the experimental points by the method of least squares. The errors shown are statistical.

-

-

exponential behavior of the curves is probably related to the randomness of the agglomeration process, one can assume that w is a function of the number of grains contained in a cluster. In order to evaluate the blob length distribution curves certain assumptions have to be made: The first assumption is connected with the number of grains per cluster in minimum tracks and it was assumed that the mean cluster length w o corresponds to just two grains per cluster; w o is smaller than 2d, where d

1.7.

PHOTOGRAPHIC EMULSIONS

259

is the mean grain diameter found from the Gaussian distribution. I n the minimum case the size of the second grain is defined by d, = wo - d, which is less than d, due to geometrical conditions or reasons connected with development conditions. It is furthermore assumed that the number of grains per cluster, g,, in denser tracks with w > W O is given by ge

= 1

+ wwo --d d

(1.7.37)

*-

This assumption seems to be reasonable, as a first approximation, at least in tracks of medium density. Linearity of the value of g, - 1 as a function of w - d u p to P = 0.6 follows for instance from the (H-D) model, using parameters /3 = 0.35, 'I = 2, and d = 0.7 p . However, it is clear that the linearity relation will break down for large densities for a variety of reasons connected with the spatial distribution of silver crystals as well as with phenomena appearing in the saturation region. Therefore, relation (1.7.37) is no longer valid a t high ionization densities and a certain value of w might correspond to a larger number of grains than expected from the proportionality relation; the number of apparent grains per cluster is too small, because a certain number of grains are not efficiently observed, while others, due to saturation processes, are not efficiently produced. This statement is equivalent to the assumption that the grain diameter in clusters d, decreases with increasing ionization and this effect was considered tentatively by writing d,

= (w, -

(

3

d) exp - -

(1.7.38)

The ratio in the exponent is the ratio of the mean blob length to the free space in a measuring cell. The procedure of evaluating the data is as follows. First one determines in minimum tracks the number of blobs B and the total blob length LB per unit cell, while the mean blob length A in minimum tracks is obtained from the integral blob length distribution, and the mean grain diameter from the differential distribution. (The use of a blob length analyzer increases considerably the measuring speed.) From the measurements one obtains readily the following equations :

+ (1 - ao)wo] no, - Bo[ao + (1 - ad21

LB, = Bo[aod and

( 1.7.39) ( 1.7.40)

based on the assumption that clusters in minimum tracks contain just two grains.

1.

260

PARTICLE DETECTION

RESTRICTED IONIZATION

LOSS

FIG.10(a). Experimental values of function a, determining the percentage of blobs which are considered t o be grains.

.I

4

I

2

4

RESTRICTED

6

0 10

20

I

IONIZATION LOSS

Fro. lO(b). Experimental values of the reciprocal slope A of the exponential part of the integral blob length distribution. The dimensions of A are microns.

For tracks heavier than minimum ionization one uses the equation LB = B[ad

+ (1 - a).]

(1.7.41)

and because of (1.7.42)

1.7.

PHOTOGRAPHIC

261

EMULSIONS

one finds the total number of grains as given b y (1.7.43) Finally, in very dense tracks, the number of grains per cluster gc has to be corrected in accordance with Eq. (1.7.38). Equations (1.7.41) and (1.7.43) have to be modified, if tracks with dip angle >20° are analyzed, because all the lengths measured b y the analyzer are projected lengths, which have to be converted in true lengths.

ioJ 1

2

4

RESTRICTED

6

8 10

20

IONIZATION

I

LOSS

FIG. 10(c). Experimental values of the total blob length L g per unit length in microns per 100 microns. The abcissas for all three curves are restricted ionization loss. The curves were drawn as best fits t o the experimental points. The symbol 0 refers t o experiment A, t o experiment B.

+

This can be accomplished by multiplying the number of blobs by cos 0 and the blob lengths I' by sec 8. However if the integrated blob length distribution is plotted versus 1' sec 8, one would measure each blob too long by a n amount of d(sec e - 1); (see Della Corte133). Therefore the abscissa of the integrated distribution, from which the inverse slope is determined should be chosen to be 1' sec e - d(sec e - 1).Applying this correction Eq. (1.7.43) becomes w-d

The triangles and crosses in Fig. 6 represent dippingItracks of 40" and

262

1.

PARTICLE DETECTION

30" dip angle calculated in this way. The formula probably gives a value which is somewhat too high because of the possibility of including some single grains within the clusters. There are several reasons why the parameter n, is preferable to others, mentioned before. no is a sensitive parameter in the entire range of ionization. This can be seen from Figs. 10a, b, and c, representing the plot of the parameters: -a-, -A-, and L g as functions of the restricted ionization loss (cut off value 5 kev). For near minimum tracks the parameter -B- (not plotted here) increases rapidly with ionization loss, and, since -a- is near unity, while A and therefore w is small, the value of n, is essentially blob density which here is nearly equal to grain density. With increasing ionization loss -a- decreases rapidly while -Aincreases, so that the second term becomes more important. Even in the dense region some of the parameters, used in (1.7.41) and (1.7.43) are still sensitive to changes in ionization, so that n, is a sensitive parameter in the entire range. In the dense region subjective errors in blob density measurements might be considerable, but this fact does not affect the results greatly because the mean blob length, measured simultaneously by blob length analyzer compensat.es the error, as can be seen from Eqs. (1.7.41) and (1.7.43); since in dense tracks the value of -a- is small, the compensation will not suffer through fluctuations in the value of -a-. If the parameter no,defined by Eq. (1.7.43), is really identical with the true grain density, then no must be equal to n, = Pn,; where n1 is the total number of developable crystals per unit length and P is the probability of development defined by Eqs. (1.7.17) to (1.7.19). Therefore, n,, calculated from Eq. (1.7.43), must be a function of y = QZV,where u, the number of ionization acts per unit length, can be identified with the restricted ionization loss 4,. The experimental points in Fig. 6 represent experimental values found in two different experiments. Since the absolute value of y = q f u is not known, the value of y for the minimum track was found by normalization to the theoretical curve, while all other points were plotted by using the experimental values P = n,/nr and for y the value of the minimum track multiplied by g / g o , (ratio of restricted ionization values; cut off at 5 kev). The points ( 0 . . . experiment A, * * experiment B) lie fairly close to the curve within the error limits, with exception of the points referring to dipping tracks (A * . 40" dip angle, X . * 30" dip angle) which have been calculated according to Eq. (1.7.43a) and the very dense region ( 0 )which were corrected for the apparent loss of grains according to Eq. (1.7.38). The fact that measurements from two different experiments show agreement with a single theoretical curve indicates that n, is a parameter which is independent of development.

+

+

-

1.7.

PHOTOGRAPHIC EMULSIONS

263

The problem of finding a suitable model is b y no means trivial; there is a n urgent need for the standardization of ionization parameters in the medium and dense region. The present situation is quite involved, because the results of various authors do depend not only on the method of measurements, but also on the model used for the evaluation of experimental data. This dependence extends still farther to the problem of error evaluation. For instance, in O’Ceallaigh’s model, which assumes random distribution of crystals along the track, the standard deviation is simply given by the square root of the numbers counted, provided th a t systematic errors and irregularities in the emulsion constants can be neglected. For other models, however, the problem is more involved, because then, one has t o account not only for the statistical fluctuations of the parameters but also for the correlation between the fluctuations of each of the parameters involved. The error problem in emulsion work was studied in great detail by Blatt introduces fluctuation parameters for various authors. the evaluation of errors; the parameters are defined as the ratio of standard deviation t o the square root of the mean value of the respective parameters measured. Blatt’s fluctuation theory was experimentally verified by O’Brien for the case of the (H-D) model. I n connection with the problem of heavily ionizing particles, which ~ 4 ~recently includes particles near the end of their range, Alvial et ~ 1 . have proposed another method. This method is based on the measurement of track profiles; the tracks are projected on a screen and the distances of both borders of the track from a fiducial line are measured. The authors found in a preliminary investigation that the thickness of tracks for slow particles depends on the velocity of the particle and reaches a maximum near p = 0.1. The increase in thickness and the maximum is explained as being caused by short 6 rays, ionizing crystals in the immediate neighborhood of the tracks; for p 7 0.1 the track width diminishes, because the range of the emitted 6 rays is too short to reach crystals of the main track. If further measurements should confirm that the position of the maximum can be easily established, the method would be very valuable for mass determination of slow particles, since the position of the maximum varies with particle mass. Unfortunately it seems that the observability of the maximum depends on developing conditions and on the dip of the track; therefore, the measurement of each individual case affords a great amount of precalibration work. 123.136v146,148

G. Lovera, Nuovo cimento [lo]8, 1476 (1956). G. Alvial, A. Bonetti, C. C. Dilworth, M. Ladu, J. Morgan, and G. Occhialini, Nuovo cimento [lo] Suppl. No. 4,244 (1956). 148

149

264

1.

PARTICLE DETECTION

1.7.8. Photoelectric Method I n photoelectric methods the observer’s eye is replaced by photoelectric cells. In all measurements photomultiplier cells are used, and the image of the track is projected through a slit, in the image plane of the eyepiece of the microscope, onto the sensitive screen of the photocell. In most photoelectric devices the stage is motor driven and the output of the photocell is recorded, so that the function of the observer is limited to the task of keeping the track image in focus and in the center of the slit. The width of the slit is usually 2-3 p ; it should not be wider, because otherwise the photocell sees and records in addition t o the track segment a considerable amount of the background. However, the slit should not be too narrow, because then the adjustment of the track within the slit becomes difficult; furthermore the slit must be wide enough to accommodate also tracks of slow particles, which frequently suffer deviations from the initial direction. The length of the slit is determined by the purpose of the measurements. With short slits and high magnification the photocell acts as a blob and gap counter; in this case the measurements are relatively independent from background conditions and the depth position of the track. For long slits the photocell acts more as a densitometer, covering, but not resolving larger segments of the track; it is clear that this type of measurement depends greatly on the background and position of the track and it becomes necessary to perform accurate background measurements in order to evaluate the net photoelectric effect arising from the track alone. The length of the slit is somewhat limited by the dip of the track, because during the measurement the whole exposed track segment must be kept in focus. There is still another type of photoelectric measurement in which the profile of a track is measured by sweeping out the cross section of a track with oscillating prism or mirrors. A great number of authors have performed photoelectric measurements and details of measurements and devices used are given in the following papers.lS0-l68We will come back to this section in the chapter on mass measurements and will describe in greater detail the work of the Lund school, where the method has been applied with great success and has been constantly improved. The author warmly thanks Dr. A. Perlmutter for reading the manuscript and suggesting many style improvements. llr0

M. Blau, R. Rudin, and M. Lindenbaum, Rev. Sci. Znstr. 21, 978 (1950).

S. yon Friesen and K. Kristiansson, Arkiv Fysik 4,505 (1951). 152 M. Ceccarelli and G. T. Zorn, Phil. Mag. [7]43, 356 (1952). 161

153

C. Kayas and 0. Morellet, Comp. rend. 234, 1359 (1952).

P.Demers and R. Mathieu, Can. J . Phys. 31, 97 (1953). 15.5 L.Van Rossum, Comp. rend. 236, 2234 (1953). 154

M. Della Corte and M. Ramat. Nuovo cirnento [9]9,605 (1952). M. Della Corte, Nuovo cimento [lo] 4, 1565 (1956). 168 P.C. Bizzeti and M. Della Corte, Nuouo cimento [lo]7 , 231 (1958). 156

157

1.8. Special Detectors 1 A.1. The Semiconductor Detector* 1.8.1.1. Introduction. The junction region of a reversed-biased p-n diode is essentially a solid state version of the conventional gaseous ionization chamber. For the purpose of analogy, it is well to recall the essential features of a gas-filled ion chamber which are shown schematically in Fig. 1. The region between the plates of a charged parallel-plate capacitor is filled with a gas such as argon a t a pressure near one atmosphere. The externally applied voltage V establishes an electric field E = V / d ,where d is the interelectrode spacing. The field E is of sufficient magnitude to prevent recombination of the positive ions and electrons, but not large enough to permit gas multiplication. If a single a-particle,

f

i

PULSE OUT

FIG.1. Essential features of a gas-filled ionization chamber.

say, passes through the chamber it will lose energy by elastic and inelastic collisions with the argon atoms. The net effect of these interactions is the formation of a number of positive ion-electron pairs which are swept apart by the electric field. Frequent collisions with gas molecules preventj both the ions and electrons from obtaining enough energy from the field between collisions to produce secondary ion-electron pairs. As the electrons and ions drift apart, the collector electrode potential rises from zero to ne/C, where n is the number of ion pairs formed, e is the electronic charge, and C is the chamber capacitance. The rise-time and shape of the leading edge of the output pulse is dependent upon the interelectrode distance d, the mobilities p e and pi of the electrons and ions in the chamber gas, and the applied potential V . The decay time is determined by the RC time constant where R is an external resistor (see Fig. 1) and C is the chamber capacitance. A discussion of the principles of operation of a semiconductor detector

* Section 1.8.1 is by S. S.

Friedland and 265

F. P.

Ziemba.

266

1.

PARTICLE DETECTION

is aided by referring to Fig. 2. A p-n junction’ is formed close to one surface of a slab of high-resistivity p-type (boron-doped) silicon by a shallow diffusion of phosphorous. A reverse bias applied to the junction establishes a depletion region (space-charge region) on both sides of the junction. The thin phosphorous doped (n-type) region near the junction has a positive space-charge due to ionized donors, whereas the boron doped (p-type) region has a negative space charge due to ionized acceptors. This distributed dipole layer resembles the charged parallelplate capacitor of the conventional gaseous ionization chamber. If an a particle passes through the space-charge region, electron-hole pairs are produced by inelastic collisions with the silicon atoms. These carriers are swept apart by the electric field set up by the dipole layer, giving rise to an electrical pulse similar to that obtained in a gas chamber. Another variation of the semiconductor detector, often referred to as a “surface barrier counter,” has also been developed. These counters are made by evaporating a thin layer of gold (100-2OOOw) onto highresistivity n-type silicon (or germanium). A distributed p-type layer is formed by surface states at the interface between the metal and the semiconductor. 11,12 Positively charged ionized donors in the n-type material along with the p-type states form a dipole layer. The region in the semiconductor which is nearly stripped of conduction electrons is called a surface barrier. The charge distribution, potential gradient, barrier capacitance, and barrier depth can be calculated from Poisson’s equation and the Fermi-Dirac distribution of charge carriers. The results are 1 I. Van der Ziel, “Solid State Physical Electronics.” Prentice-Hall, Englewood Cliffs, New Jersey, 1957. J. W. Mayer and B. R. Gossik, Rev. Sci. Instr. 27, 407 (1956). J. W. Mayer, J. Appl. Phys. 30, 1937 (1959). a F. J. Walter, J. W. T. Dabbs, L. D. Roberts, and H. W. Wright, Oak Ridge National Laboratory, CF 58-11-99 (1958). F. J. Walter, J. W. T. Dabbs, and L. D. Roberts, Oak Ridge National Laboratory, ORNL-2877 (1960); Bull. A m . Phys. Soc. [11]3,181 (1958); 3,304 (1958); 6,38 (1960); 6, 22 (1960). F. J. Walter, J. W. T. Dabbs, and L. D. Roberts, Rev. Sci. In&. S1, 756 (1960). J. M. McKenzie and D. A. Bromley, Phys. Rev. Letters 2, 303 (1959); Bull. A m . Phys. Soe. [11] 4, 457 (1959). 7 E. Nordberg, Bull. Am. Phys. Soc. [11]4, 457 (1959). M. L. Halbert, J. L. Blankenship, and C. J. Borkowski, Bull. Am. Phys. Soc. [II] 6, 38 (1960). J. L. Blankenship and C. J. Borkowski, Bull. Am. Phys. Soc. [11] 6, 38 (1960). l o M. L. Halbert and J. L. Blankenship, Nuclear Instr. and Methods 8, 106 (1960). 11 W. Schottky, Z . Physik 118, 539 (1942). l2 R. H. Kingston, ed., “Semiconductor Surface Physics.” Univ. of Pennsylvania Press, Philadelphia, 1957.

1.8.

SPECIAL DETECTORS

267

similar to those to be described for the p-n junction. The nuclear characteristics of both the p--n diffused junction detector and the surface barrier counter are the same, since they are dependent upon the nature of the interaction of radiation with the semiconductor material. On the average, 3.5 ev of incident particle energy is required to produce one electron-hole pair in silicon (as opposed to 32 ev for a typical gas), To dat,e, all of the experimental evidence indicates that this value is independent of the particle type. Thus any particle losing energy E (electron volts) in the space-charge region produces n = &/3.5 electronhole pairs (as compared to &/32 ion-electron pairs for the gas chamber).

T-

SURFACE

CONTACT . r P TYPE SILICON BASE CONTACT D = DIFFUSION DEPTH Xp= WIDTH OF DEPLETION REGION I N P MATERIAL X,,= WIDTH O F DEPLETION REGION I N N MATERIAL d = X, + X, TOTAL WIDTH O F DEPLETION REGION

=

x= L

0

N TYPE S I L I C O N

FIG.2. Scheniatic diagram of a p-n junction under reverse bias.

Statistical arguments show that the fundamental device resolution limit is set by the characteristic fluctuation. One, therefore, theoretically expects and experimentally obtains a significant improvement in the energy resolution of the solid state chamber over that obtained with a gaseous chamber. The high carrier mobilities and drift velocities in silicon combined with the small width of the depletion region result in pulses with millimicrosecond rise times. Since the range of energetic particles in silicon is measured in microns as compared to centimeters in a gas, the physical size of the detector is several orders of magnitude smaller. For the p-n junctions under consideration the thickness of the depletion layer in microns is given approximately by d = (pV)l'2/3 where p is the resistivity of the p region in ohm centimeters and V is the applied reverse bias voltage. It is necessary that the incident particle lose all of its energy within the depletion region in order that there be a linear relationship between the pulse height and particle energy. It is obvious from the above result that the product pV be made as large as possible when the detectors are to be used as spectrometers for penetrating particles such as protons and electrons. The results of several workers using

268

1.

PARTICLE DETECTION

detectors with p = 10,000 ohm-cm and V = 400 volts are shown qualitatively in Fig. 3. The relative pulse height is linearly related to the electron energy to about 1 MeV, for protons to 10 MeV, a's to 40 MeV, heavy ions and fission fragments > 100 MeV. These results are consistent with the known range-energy relationships of these various particles in silicon. The typical noise level of amplifiers is shown to be near 20 kev. Resolutions of 0.3% for 5-Mev a particles have been obtained. It should be pointed out that if one wishes merely to count individual events (not I I I 1 1 1 1 1 ~ I l l l l l l l ~ I 1111111~ I 1 1 1 1 1 1 1 ~ I 1 1 1 1 ~

HEAVY

IONS

/

ELECTRONS

a .01

I lfllll'

I I 111111'

I I 1 1111J

I I I 111d

0.1 1.0 10 PARTICLE ENERGY IN

100 MEV

I ''I

1000

FIG.3. Relative pulse height versus energy of a semiconductor detector for various nuclear particles.

measuring the particle energy) then even minimum ionizing particles may be detected inasmuch as they will expend energy at approximately 0.35 kev/micron in silicon. Early attempts to use solid state devices as particle detectors date back to the work of Jaffe13and Schiller14who observed small changes in the conduction current of crystals irradiated with cr particles. Measurable pulses produced by individual & particles penetrating a AgCl crystal were first obtained by Van Heerden.16 Extending that work, McKayl8 studied conductivity changes in diamond under electron bombardment, and Ahearn17tested a large number of crystals for conduction pulses proG. Jaffe, 2.Physik 33, 393 (1932). H. Schiller, Ann. Physik [4] 81, 32 (1926). 16P. J. Van Heerden, "The Crystal Counter, A New Instrument in Nuclear Physics." North Holland Publ., Amsterdam, 1945. l6K. G. McKay, Phys. Rev. 74, 1606 (1948). 1 7 A. J. Ahearn, Phys. Rev. 76, 1966 (1949).

i

13

l4

1.8.

269

SPECIAL DETECTORS

duced by a particles. It was generally found that the crystal counter was not useful as a spectrometer because of poor resolution and polarization effects. Review articles by Chynowethls and Hofstadterlg summarize much of the work with crystal counters. McKay20 reported the response of point contact germanium rectifiers to a particles and suggested the use of p-n junctions. Orman et McKay,22and Airapetiants et measured the voltage pulses produced by germanium p n junctions struck by a particles. In 1955 Mayer and GossikL8used gold-germanium surface barrier counters as a-particle spectrometers. They found that the pulse height from such counters was proportional to the a-particle energy up to about 8 Mev and obtained good resolution from units with 0.8 to 16.0 mm2 active surface area. The Ge-Au surface barrier counter was developed ~,~ and Bromley16and Dearnaley and further by Walter et C L Z . , ~ ~ McKenzie Whitehead.25For good resolution, Ge-Au counters have had to be operated a t low temperatures. Room temperature operation was obtained with the introduction of silicon which has a larger energy gap, and, therefore, a much lower (reverse) saturation current. Si-Au surface barrier counters were developed by McKenzie and Bromley16Nordberg17 Blankenship and Borkowski,Yand Halbert and BlankenHalbert et d.,* ship.l o Such detectors have excellent properties a t room temperature, but until recently the fabrication has been most difficult. Use of a p-n junction diffused in silicon which resulted in an operational room temperature spectrometer was developed by Friedland et aLZ6 Further properties of the p-n diffused junction detector have been reported by a number of ~ o r k e r s . ~ ~ - ~ l ~

1

.

~

~

~

9

~

~

A. G. Chynoweth, Am. J . Phys. 20, 213 (1952). R. Hofstadter, Proc. IRE (Znst. Radio Engrs.) 38, 726 (1950). ,OK. G. McKay, Phys. Rev. 70, 1537 (1949). 2 1 C. Orman, H. Y. Fan, G . T. Goldsmith, and K. Lark-Horowitc, Phys. Rev. 78, 646 (1950). z* K. G. McKay, Phys. Rev. 84, 829 (1951). 28 A. V. Airapetiants and S. M. Ryvkin, Zhur. Tekh,Fiz. 2 7 , l l (1955); Soviet Phys., Tech. Phys. (Eng. Trans.) 2, 79 (1958). z 4 A. V. Airapetiants, A. V. Logan, N. M. Reinov, S. M. Ryvkin, and I. A. Sokolov, Zhur. Tekh. Fiz. 27, 1599 (1957); Soviet Phys., Tech. Phys. (Eng. Trans.) 2, 1482 (1957). zs G. Dearnaley and A. B. Whitehead, Atomic Energy Research Establishment, Harwell, Berkshire, United Kingdom, AERE-R 3278 (1960). 26 8.S. Friedland, J. W. Mayer, J. M. Denney, and F. Keywell, Rev. sci. Instr. 31, 74 (1960). 27 Report on the Seventh Scintillation Counter Symposium, Washington, D.C. February 25-26, 1960. Nucleonics 18 (5), 98 (1960). 28 Complete Proceedings of the Seventh Scintillation Counter Symposium, Washl9

270

1.

PARTICLE DETECTION

1.8.1.2. Junction Capacitance and Junction Width. The potential distribution due to the distributed dipole layer in the space-charge region of the plc junction may be calculated by solving Poisson’s equation with the appropriate boundary conditions. For sufficiently large-area diodes, edge effects may be ignored and the resulting one-dimensional problem is readily handled. The result so obtained allows for the determination of the capacitance and width of the space-charge region. The details of the calculation are omitted and may be found elsewhere.’ For a “Schottky-type potential barrier” where there is a sudden step from p type to n type, the thickness of the transition is d = x p x,; xp is the distance from the junction into the p-type material and xn is the distance from the junction into the n-type material. The thickness d is readily found to be given by

+

d = [2eeo(Vd

+ V)(N,, $- Nd)/eNoNd]112.

(1 A.1. I)

The acceptors and donors per unit volume are N , and Nd; ED

=

8.85 X 10-lafarad/m

and e is the relative dielectric constant; Va is the diffusion potential (built in voltage) and V is the applied bias. For the devices under consideration V >> Vd, Nd >> N,, and we obtain d

[2ee0V/eNd]’/~.

(1.8.1.2)

Since the total charge must be zero for over-all charge neutrality in ington, D.C., February 25-26, 1960. I R E Trans. on Nuclear Sci. NS-7,2-3 (1960). *9 G. L. Miller, W. L. Brown, P. F. Donovan, and I. M. Mackintosh, Bell Laboratory-Brookhaven National Laboratory Report BNL 4662 (1960). 3o G. F. Gordon, Univ. of California Radiation Lab. Rept. 9083 (1960). J. Beneveniste, Univ. of California Radiation Lab., private communication. a* H. Mann, Argonne National Laboratory, private communication. E. L. Zimmerman, “Comments on the Use of Solid State Detectors for Neutron Detection.” Solid State Radiations, Inc., Culver City, California, unpublished. 34T. A. Love and R. B. Murray, Oak Ridge National Laboratory, CF 60-5-121 (1960). 3 6 C. T. Raymo, J. W. Mayer, J. S. Wiggins, and 5. S. Friedland, Bdl. Am. Phys. SOC.[I11 6, 354 (1960). ae S. 6. Friedland, J. W. Mayer, and J. S. Wiggins, Nucleonics 18, 54 (1960). J. D. Van Putten and J. C. Vander Velde, Bull. Am. Phys. SOC.[11] 6, 197 (1960). 38 R. L. Williams, Bull. Am. Phys. SOC. [11]6, 354 (1960).

J. W. Mayer, R. J. Grainger, J. W. Oliver, J. 5. Wiggins, and S. S. Friedland, BuU. Am. Phys. SOC.[11] 6, 355 (1960). ‘O

P. F. Donovan, G. L. Miller, and B. M. Foreman, Bull. Am. Phys. SOC.[I11 6,355

(1960). 41

J. M. McKenaie, Bull. Am. Phys. SOC.[11] 6, 355 (1960).

1.8.

27 1

SPECIAL DETECTORS

the junction we must have NdXn

=

( 1.8.1.3)

NoXp.

Typical values for zn and r p are 0.1 micron and 0.5 mm respectively. The transition region capacitance per unit area is given by

c = [ccoeN,N,&(V,j

+ V)(Na+

Nd)]'/'

= tco/d.

(1.8.1.4)

The space-charge layer thus acts as a parallel plate capacitor with plate separation d. For a sudden step junction, one obtains the approximate expression C = [ee0eN~/2V]l/~. ( 1.8.1.5) In terms of resistivity and mobilities, Eqs. (1.8.1.2) and (1.8.1.4) take on the more convenient forms d = [2eeoVpspo]1'2 = ( + ) ( P V ) microns ~/' ZZ [Eeo/2VphP,p,]"'C (+)(pV)-"' 10' ppf/Cm2

c

(1.8.1.6)

where p. and ph are the mobilities of electrons and holes respectively with V expressed in volts and p in ohm-em. The above relationships have been combined by Blankenship27into a very useful nomograph. 1.8.1.3. Properties of the Semiconductor Detector. The response of a semiconductor detector is linearly related to the energy of the incident particle energy provided the range of the particle (in silicon) is less than the width of the depletion region. Equation (1.8.1.6) demonstrates that the depletion width is proportional to the product (pV)l/'. Detectors with resistivities up to 13,000 ohm-cm have been described and e ~ a l u a t e d . ' ~ Operating such a device with the not unreasonable bias of 750 v would lead to a depletion depth of approximately 1 mm which corresponds to the range of about a 15-Mev proton in silicon. Equation (1.8.1.6) shows that the capacitance is inversely proportional '~. the quantity pV therefore reduces the to the product ( P V ) ~Increasing capacitance, increases the output voltage V o= ne/C, and increases the signal energy E. = (ne)'/2C. The signal-to-noise ratio, however, does not in practice increase with bias voltage. The detector reverse leakage current increases monotonically with the applied voltage. At large voltages this current gives rise to a diode noise energy which increases more rapidly with voltage than the signal energy. The frequency distribution of the noise, the effects of surface leakage, and other matters relating to signal-to-noise ratios have not been studied extensively a t this time. At relatively low bias voltages, the detector noise level is usually below typical amplifier noise levels. This state of affairs has initiated a considerable effort to develop low noise preamplifiers for use

272

1.

PARTICLE DETECTION

with solid state detectors.27 Preamplifiers with noise levels as low as 7.5 kev have been reported. As discussed above, the capacity of an abrupt junction should vary with the voltage according to the relation W 2 C = constant. In a “graded” junction such as that obtained in a grown junction, the acceptor and donor densities N , and Na are slowly varying functions of position in the region of the junction. For a “graded” junction one expects the relation

I

10

100

REVERSE BIAS IN VOLTS

FIG.4. Approximate high-frequencyequivalent circuit of a semiconductor detector. The junction capacitance Cj and resistance Rj determine the pulse rise-time. The resistance Rb is due to the bulk silicon outside of the space charge region.

to be V I W = constant.2 Experimental measurements indicate that VfC = constant with 4 < f < $; typical data for capacitance versus reverse bias for typical detectors are shown in Fig. 4. The temperature dependence of the capacitance is negligible. An approximate equivalent circuit of a solid state detector is shown in Fig. 5. The radiation source is replaced by a charge generator charging the junction capacitance Cj through a resistance Rj which may be estimated by setting the time constant RjCj equal to the transit time of the carriers through the space charge region. Rise times in the millimicrosecond region are commonplace in contrast to the microsecond rise

1.8.

273

SPECIAL DETECTORS

times obtained with gaseous ion chambers. The development of lownoise wide-band amplifiers appears to be quite desirable. The resistor Ra is due to the bulk silicon outside of the space charge region and the ohmic contact with the p-type material. It may be reduced by: (i) having the wafer thickness comparable to the depletion depth; and (ii) a proper doping at the ohmic contact. The latter should also have the effect of producing a more uniform field within the space charge region, whence a more uniform collection efficiency and resolution in large-area detectors. Surface leakage current, space-charge-generated current, and diffusion current all contribute to the reverse current of a semiconductor deis ~ difficult t e ~ t o rIt. ~ ~ ~ ~ to determine the relative contributions in a par-

CJ

I

CHARGE GENERATOR

I

FIQ.5. The dependence of the junction capacity upon applied bias voltage. ticular device; however, the absence of a well-defined breakdown in many units indicates that surface leakage is usually the most important of the three sources. The temperature and voltage dependence of the reverse current are at present under considerable investigation. Surface effects will have to be minimized before any definite conclusions are drawn. 1.8.1.4. Experimental Results. The response of a semiconductor detector to various types of nuclear radiations is shown in Figs. 6 to 11. The energy range over which the device responds linearly to the different types of radiation and the resolution for each of the particle types are discussed and illustrated with experimental data. 1.8.1.4.1. HEAVYIONS AND FISSION FRAGMENTS. Fission fragments have relatively short ranges in silicon and there is no difficulty in obtaining depletion depths which are wide enough to ensure linearity with energy. The same remark applies to heavy ions such as C12. Figure 6 demonstrates that the device is linear for CI2 ions with energies to 120 M ~ v . ~The O kinetic energy spectrum of fragments from the spon42 J. H. Shive, “Semiconductor Devices.” Van Nostrand, Princeton, New Jersey, 1959.

274

1.

PARTICLE DETECTION

taneous fission of Cf262observed29with a p-n detector is in agreement with the accepted time-of-flight measurement^.^^ The detector appears to be “windowless” for fission fragments provided the phosphorous surface layer (n-type) is less than -0.1 p . The semiconductor detector does not exhibit the “ionization defect ’m7 characteristic of gaseous chambers and negligible columnar recombination appears to exist

ENERGY IN

MEV

FIG-.6. The relative pulse-height versus energy of a semiconductor detector for Cl* ions with energies from 30 to 120 Mev.

along the tracks of fission fragments even though carrier densities are -lozo ~ m - ~ . 1.8.1.4.2. PROTONS AND a PARTICLES. The proton response31 of a 5 mm X 5 mm area detector made from 10,000 ohm-cm silicon, and operating a t a reverse bias of 400 v, is shown in Fig. 7. The device is linear for protons to about 10 MeV. Energy-range relationships for protons in silicon show that the range of a 10-Mev proton is about 700 1.1. This agrees well with the calculated width of the depletion region. The resolution versus bias of a typical detector for 8.78-Mev a particles from Pb2I2is shown in Fig. 8. The poor resolution a t low bias voltages 4a

J. C. D. Milton and J. S. Fraser, Phys. Rev. 111, 877 (1958).

1.8.

SPECIAL DETECTORS

275

FIQ.7. The relative pulse-height versus energy of a semiconductor detector, 6,000 ohm cm, 400-v bias, for protons with energies from 2 to 12 MeV.

I

0

0

I I I 0 100 200 250 150 REVERSE BIAS (VOLTS)

50

FIQ.8. The energy-resolution for 8.78-Mev a particles and reverse current versus bias voltage of a semiconductordetector, 6000 ohm cm, 6 mm X 5 mm.

276

1.

PARTICLE DETECTION

is probably due to a combination of a poorer signal-to-noise ratio and a nonuniformity in collection efficiency over the sensitive area. The absence of a well-defined avalanche breakdown in the current-voltage characteristic would indicate considerable surface leakage. A maximum resolution of 0.3% for 5-Mev a! particles which has been obtaineds2 is currently limited by the noise level of the amplifiers. 1.8.1.4.3. ELECTRONS. Figure 9 shows that the detector response is linear for electrons to nearly 1 MeV. Using a calibrated charge-sensitive amplifier, Mann32shows that about 3.7 ev of incident electron energy

ENERGY IN

KEV

FIG.9. The relative pulse-height versus energy of a lo4ohm cm semiconductordetector for electrons with energies from 50 to 800 kev. Bias, 360 v; X, Pml47; 0, Agllom.

is required to produce one electron-hole pair in silicon. The result is in reasonable agreement with the results obtained for protons and a’s in silicon. The data are shown in Fig. 10. Internal conversion lines in CsI3’ are shown in Fig. 11. The K and L lines are distinctively resolved. The data were kindly supplied by C. S. Wu. 1.8.1.4.4. N E u T R o N S . ~ ~Since the semiconductor detector is an excellent device for observing heavy charged particles, it is obvious that its usefulness may be extended to include neutron detection by applying coatings which react with neutrons to produce heavy charged particles. Efficient thermal neutron detectors can be realized by BIO, Lis, and UZ35 coatings. Such devices are not directly useful as neutron energy spec-

1.8.

SPECIAL DETECTORS

277

trometers since the reaction energies are large compared to the incident neutron energy. A combination of bare and cadmium-covered detectors will, however, give some indication of the thermal neutron distribution. Threshold detectors based upon the Np239(n,f) and U238(n,f) reactions will be useful for high-energy neutrons. Neutron energy spectrometers based upon “proton recoil techniques,”* Lie (n,a) HS, and SiZ8(n,p) AlZ8

FIQ.10. The number of charges collected versus energy of a semiconductor detector for electrons with energies from 50 to 800 kev.

reactions hold considerable promise. I t is interesting to note that no coating is needed for the SiZ8(n,p) A128detector. Preliminary results of Love and on a Lie ( n , ~H3 ) neutron spectrometer are most encouraging. The promising ranges of usefulness of the variously coated semiconductor neutron detectors are illustrated in Fig. 12. The height of each curve is a rough indication of the degree of utility of each of the possible arrangements. 1.8.1.4.5. PHOTONS. The only experimental data available on the

* Refer to Section 2.2.2.1.

278

1.

PARTICLE DETBCTION

photon response of semiconductor detectors is some preliminary work with high-energy CosOgammas.36 The device is relatively insensitive to gamma radiation in this energy range (Compton effect) due t o the low-absorption cross section of silicon. The p-n detector is quite sensitive

K - LINE

2000

624 KEV

1500 w

i

0

z

3 1000

I-

0

u

500

BIASED SO THAT ZERO CHANNEL AT - 5 6

100

150 CHANNEL

200 NO

FIQ.11. The internal conversion lines in Cs1*7 aa measured with a semiconductor detector.

to photons with energies comparable to the gap energy in silicon (1.1 ev) and should not be exposed to light when used as a nuclear particle detector. 1A.1.4.6. HIGH-ENERGY PARTICLES. Recently, several laboratories have made investigations t o determine whether high-energy particles in

1.8.

279

SPECIAL DETECTORS

the minimum ionizing region can be measured with semiconductor detectors. The results obtained so far are quite e n ~ o u r a g i n g . ~ ~ Figure 13 shows the energy spectrum of a positive pion and proton beam with a momentum of 750 Mev/c from the Brookhaven Cosmotron obtained in a silicon junction detector. The resistivity of the silicon junction detector is 10,000Q-cm and it was operated a t a reverse bias

si 28cn.p) ( n, Li’

In.

AL~~SPECTROMETER

!/---

cx

INTEGRAL ~)H31NTEGRAL

i \

/I Li6 (h.oC)H3 N$”(n,f)

THRESHOLD

Cn.p) P

.dl d.i

i!o

Ib Ib2

1’0. I3 I2 I’O’

NEUTRON ENERGY-EV

!&.I

I107

SPECTROMETER

THRESHOLD

- RECOIL

Ibe

ilop

;do

4.5 MEV

0.26 MEV

FIQ.12. The relative utility of variously neutron sensitized semiconductor detectors versus the neutron energy.

of 100 v. At this momentum the pions are close to minimum ionization whereas the protons are twice minimum. The ionization losses for the pions and protons are found to be 110 kev and 200 kev, respectively, indicating a linear response, and the pion and protons are very clearly separated. 1.8.1.5. Conclusion. The small size of the semiconductor detector makes it possible to arrange a linear array of detectors in the focal plane of a spectrometer or a t several angles inside of a reaction chamber. Two 44 L. C. L. Yuan, Application of solid state devices for high energy particle detection. Intern. Conf. on Instrumentation for High Energy Physics, Berkeley, California, September, 1960.

280

1.

PARTICLE DETECTION

dimensional arrays can be assembled to obtain large-area a-survey instruments, for example, with low power requirements. A three-dimensional array along with an appropriate data handling system would obtain the equivalent of a “solid state cloud chamber.”46 The effects of radiation on semiconductor devices have been summarized in many report^.^^.^^ Radiation damage studies in semiconductor

300

302

>-

550

w 7T+ AND PROTON BEAM, 750

y,

IOK COUNTER, 1.4 CM DIAMETER,

2oc

100 VOLT BIAS.

Y

z

g5

?oo

a-

I50

LL

sJ! 0

> U

w-

IOC

z

100

W

50

-

)

1

I

I

I

I

20

30

40

50

60

I

m

I 80

90

I 100

CHANNEL NUMBER

FIG.13. 750 Mev/c positive pion and proton spectrum obtained in a Brookhaven Cosmotron beam. Number of counts per pulse-height analyzer channel is plotted versus the channel number which is a measure of the particle energy.

detectors is not available at the present. The data available a t present indicate that no changes in detector operation are observed after exposure to 10l214-Mev protons. The physical properties, low power requirement, wide range of linear relationship of pulse height versus particle energy, and high speed suggests that the semiconductor detector will soon become one of the basic operational devices in the field of radiation detection. S. S. Friedland, “The Solid State Cloud Chamber.” To be published. G. J. Dienes, Radiation effects in solids. Ann. Rev. Nuclear Sci. 2, 187 (1953). 41 F. J. Reid, The effect of nuclear radiation on semiconductor devices. REIC Report No. 10, Battelle Memorial Institute, Columbus, Ohio (1960). 46 *6

1.8.

SPECIAL DETECTORS

281

1.8.2. Spark Chambers* Frequently it is desirable to have good spatial resolution of high-energy particle interactions because so many modes of interaction are possible. Generally, the interesting processes also have a small cross section so that good time resolution is desirable too. Bubble chambers have been very effective in experiments where it is reasonable to expand the chamber and then have approximately ten charged particles incident on the chamber. If the chamber is large enough and if there are enough interesting events, then this is a good technique. The electromagnetic spectrometers which have been used to select the desired mass of the incident charged particles have helped to extend the use of bubble chambers. However, these will be considerably less effective at higher energies and new techniques will be desirable. Cloud chambers have been used in conjunction with scintillation counters to select interesting events and then the chamber is expanded to get tracks. Also diffusion chambers have been used in this manner, the lights being flashed to detect an interaction. The resolving time of such chambers is greater than 10 psec and background radiation is a rather severe limitation of these chambers. The scintillation chambers are being developed which have both good spatial and time resolution. For some experiments these have many desirable features. The chief disadvantages are the still rather small size and the limited flexibility for a variety of experiments. In some experiments' where only moderate spatial resolution is required, such as differential cross section scattering of antiprotons and K mesons, it has been quite practical to use arrays of scintillation counters and Cerenkov counters. I n these experiments the constraints due to twoparticle interactions have simplified the analysis. I n many cases it is now necessary to investigate specific details with good statistics. This frequently requires that rare events be selected from a background of many other interactions. The spark chamber used in conjunction with scintillation counter and Cerenkov counter telescopes is a device that permits both good spatial and time resolution of highenergy charged particle interactions. An early type of discharge chamber2 consisted of bundles of slightly "2. A. Coombes, B. Cork, W. Galbraith, G. R. Lambertson, and W. A. Wenzel, Phys. Rev. 112, 1303 (1958). M. Conversi and A. Gorrini, Nuovo cimento [lo] 2, 189 (1955);'M. Conversi, S. Focardi, C. Franzinetti, A. Gozzini, and P. Murtas, Nuovo cimento Suppl. [lo] 4, 234 (1956). __

* Section 1.8.2 is by

Bruce Cork.

282

1.

PARTICLE DETECTION

conducting glass tubes filled with one-half atmosphere of neon and placed between parallel metal plates. When a high-energy charged particle was transmitted through the tubes and plates a high electric field pulse was applied between adjacent plates. The tubes that transmitted the particle would then give a glow discharge and the light from the ends of the tubes could be photographed. These devices had a long recovery time, approximately one second, due to the electrostatic charges on the glass. 1 I

ji

i

4

FIQ.1. A parallel plate spark chamber. The plates are made of +-in. thick aluminum separated by a gap of in.

4

A similar device has been described by Cranshaw and DeBeer.3 However, they omitted the glass tubes, immersed the metal plates in air at one atmosphere, and applied a 20-kv pulse to the plates when a charged particle was transmitted. The efficiency for minimum ionizing particles was 99%, for a 3-mm gap. Then Fukui and Miyamotoe immersed the plates in an atmosphere of neon. They observed that minimum ionizing particles could be detected with nearly 100% efficiency and by applying a +psec pulse to the plates, the sensitive time was observed to be approximately 10 Fsec. Several other groups have built similar spark chambers. The details 8

T. E. Crawhaw and J. F. DeBeer, Nuovo cimento [lo] 6, 1107 (1957). S. Fukui and S. Miyamoto, Nuovo eimento [lo] 11, 113 (1959).

1.8.

283

SPECIAL DETECTORS

of a chamber6 built by Beall, Cork, Murphy, and Wenzel are given below. The chamber consisted of seven parallel plates of g i n . thick aluminum, separated by gaps of s i n . thickness, Fig. 1. The chamber has been filled with one atmosphere of argon or neon. This chamber has been tested in a beam of high-energy pions and protons a t the Bevatron.

Coincidence circuit

Discriminator 8 gote circuits

1--D

2i----Jy .

T h'.>'"."",. urntvnm

II

Thyratron 2

FIG.2. Block diagram of electronics, scintillators C , Cz, Cs and spark chamber.

The diagram, Fig. 2, shows a charged particle being detected by a scintillator coincidence telescope. The output of the coincidence circuit is used to operate a hydrogen thyratron which applies a 20-kv pulse, approximately 0.2psec, to alternate plates of the spark chamber. A battery supplies a dc clearing field between the parallel plates so that electrons produced in the gap between the plates can be swept away, thus reducing the sensitive time of the chamber. The efficiency of the chamber when filled with argon or neon is given by Fig. 3 for a clearing field of 0 v/cm and 270 mFsec delay after the traversal by a minimum ionizing charged particle. To determine the 6 E .Beall, B. Cork, P. G . Murphy, and W. A. Wenzel, UCRL-9313 (1960).

284

1.

PARTICLE DETECTION

sensitive time of the chamber for various values of clearing field, varying time delays were inserted in the trigger line to the thyratron. The efficiency as a function of delay time is given by Fig. 4. It is noted that the sensitive time can be made less than 0.5 psec. The recovery time of the chamber should be of the order of the deionization time of an inert gas at one atmosphere. The observed recovery time (Fig. 5) was long compared to the deionization time. This was measured by selecting a charged particle that was transmitted by the chamber, firing the chamber, and then selecting a second charged particle a t a predetermined time and again firing the chamber. One reason for the appar-

I

1

I

a

5

10

15

20

Pulse voltoge

1

25

(kv)

FIG.3. Efficiency of a single +in. gap in 1 atmos of argon or neon as a function of pulsed voltage across the t i n . gap.

ent long recovery time may be due to impurities in the gas. This is not a practical limitation for proton synchrotrons where the beam time is of the order of 100 msec. Typical photographs of the 6-gap chamber are given in Fig. 6. Minimum ionizing pions enter from the left. In Fig. 6(a) one pion interacted with the plate and a second pion entered during the sensitive time of the chamber. A second interaction is shown by Fig. 6(b) where one reaction product was scattered at an angle of 25 degrees, the second a t 36 degrees. When a magnetic field of B equal to 13 kg was applied parallel to the plates, the efficiency was still nearly 100% per gap. If a clearing field E of 80 v/cm was applied and the time of applying the hv pulsed electric field was delayed for 1psec, the electrons from the ion pairs were displaced an amount proportional to E X B, and the delay time. The photograph (Fig. 7) shows a displacement of the tracks for the above conditions of

FIQ.4. Efficiency of a single *in. gap in 1 atmos of argon as a function of delay in application of the high-voltage pulse. The zero of the delay axis is the time at which the particle passed through the chamber. I

I

3 100 E c 'p

c

80

0 V

$

60

8

Argon

A Neon

5

10

15

20

Time between partieter (maec)

FIG.5. Efficiency for a single gap to a spark on a second particle as a function of the time between particles. The clearing field was -40 v/cm. 285

286

1.

PARTICLE DETECTION

(b)

FIG.6. Typical photographs of the 6-gap chamber. In Fig. 6(a) one pion interacted with the aluminum plate and a second pion entered during the sensitive time of the chamber. Figure 6(b) shows two large angle scatters.

1.8.

SPECIAL DETECTORS

287

FIG.7. A magnetic field of 13 kg parallel to the plates and a clearing field of 80 v/cm cause a displacement of the sparks of ki in. if the high-voltage pulse is delayed for 1 usec.

288

1.

PARTICLE DETECTION

approximately 1 cm. Particles arriving “off time” could be detected by this means. Besides the good spatial and time resolution of the spark chamber, the chamber can be arranged in a manner that is appropriate to the particular experiment. For example, the quantity of light from the spark is so great that stereographic photography is easy from an extensive assembly of chambers. The plates can be made of metal or graphite, for example, to preferentially scatter polarized protons. It should be possible to make the plates of scintillator or cerenkov material so that the spark chamber and counter telescope are integral. Large solid angles for detecting short-lived particles can be obtained by this method.

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 .l. Rutherford Scattering.* Rutherford, in 1906, first noticed that the angular deflections experienced by a rays while passing through thin layers of air, mica, or gold were occasionally very large. He recognized that the large electrostatic field strengths required to cause such deflections could be produced only at very small distances from an electric charge. In 1911 Rutherford introduced his nuclear model of the atom, in which all the charge of one sign (now known to be positive) was concentrated into a central region, or nucleus, smaller than 10-l2 cm in radius, with an equal amount of charge of the opposite sign (now known to be the atomic electrons) distributed throughout the entire atom in a region whose effective radius is of the order of lo-* cm. It can be shown quite generally' that, when any unbound incident nonrelativistic particle (such as an CY ray) interacts with a target particle (such as an atomic nucleus) according to an inverse-square law of force (either attractive or repulsive), both particles must, in order to conserve angular momentum, traverse hyperbolic orbits in a coordinate system whose origin is at the center of mass of the interacting particles. When conservation of the sum of kinetic and potential energg is imposed as a third restriction on the orbits, it is found that the angle of scattering 6 in the center-of-mass coordinates is given by 5 =

b 6 -cot -* 2 2

(2.1.1.1.1)

Here the impact parameter x is the distance a t which the two particles would pass each other if there were no interaction between them, and b is the collision diameter defined by

["gb?;] ~

= lZzl

(2.818 X low1* cm)

(2.1.1.1.2)

1 R. D. Evans, "The Atomic Nucleus," pp. 12-19, 838-851. McGraw-Hill, New York, 1955.

*Section 2.1.1.1 is by Robley D. Evans. 289

290

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

where ze = charge on incident particle, Ze = charge on target particle, V = mutual velocity of approach, and Mo = reduced mass M 1 M 2 / ( M 1 M 2 ) of colliding particles. The absolute value of Zz is to be taken without regard to sign. The collision radius b/2 is the value of the impact parameter for which the scattering angle 0 is just 90" in center-of-mass coordinates, both for attractive and repulsive forces. For the special case of repulsive forces, as in the nuclear scattering of a rays, the collision diameter b is also equal t o the closest possible distance of approach, that is, to the minimum separation between the particles during a head-on collision. At the instant of minimum separation the particles are stationary with respect to one another, and therefore their initial kinetic energy (+)MoV2in center-of-mass coordinates is just equal to their mutual electrostatic potential energy. For this head-on collision x: = 0 and 8 = 180°. For other scattering angles 0, the minimum distance of approach pminfor repulsive forces is larger than b and is given by

+

pmin =

[

I ' )+: (

2 1 -k 4 1

=

5 [ + -11

1 sin(0/2)

(2.1.1.1.3)

I n center-of-mass coordinates there is no transfer of kinetic energy between the colliding particles. But in the laboratory coordinates there is an energy transfer and the target particle M , which was initially a t rest emerges from the collision with the kinetic energy

where the bombarding particle has mass M I and a n incident kinetic energy (+))n4',V2in the laboratory coordinates. Equation (2.1.1.1.4) is entirely a consequence of conservation of energy and linear momentum, and i t applies to all nonrelativistic elastic collisions regardless of the nabure of the forces between the colliding particles. Collisions between two nuclei will involve only the inverse-square Coulomb force whenever b is significantly larger than the combined effective radii of the interacting nuclei. If b is too small then specifically nuclear forces may become effective, some nuclear barrier penetration may occur, and Eq. (2.1.1.1.1) will become invalid. Such a deviation from the Rutherford scattering law, which is based on inverse-square Coulomb forces, is called anomalous scattering. It can be shown quite

2.1.

DETERMINATION OF CHARGE AND SIZE

29 1

generally that the classical Rutherford scattering relationships will be valid whenever (2.1.1.1.5)

in which 0 = V / c is the mutual velocity of approach in units of the velocity of light c, and X = h / M o V is the rationalized de Broglie wavelength of relative motion of the collidiiig nuclei. The cross sections for Rutherford scattering, that is, for scattering by inverse-square Coulomb forces, are the same in classical and in wavemechanical theory because the cross section is independent of Planck's constant h. Classically, the differential cross section du for Rutherford scattering between angles 6 and 6 d 8 is the area of a ring of radius x and width dx, or da = 27rxdx. This leads to the Rutherford scattering differential cross section 1 cm2 (2.1.1.1.6)

+

where dQ is the solid angle a t mean scattering angle 8 in center-of-mass coordinates, and 2, is the collision diameter as defined by Eq. (2.1.1.1.2). This marked preponderance of forward scattering is characteristic of long-range forces, such as the inverse-square interaction. I n the language of wave mechanics, these interactions involve interference between many partial waves whose angular momentum quantum numbers I extend from zero up to a t least 1 'V MoVx/h = x/X. Therefore the foreand-aft symmetry which characterizes interactions involving only one value of 1 is not seen in Rutherford scattering. Geiger and Marsden completed in 1913 a beautiful series of experiments which verified the dependence of Eq. (2.1.1.1.6)on the scattering angle 0, the incident a-ray velocity V , and t,he nuclear charge Ze. Figure 1 shows Geiger and Marsden's results for a-ray scattering from a thin foil of gold. For such a heavy target nucleus ill2>> M I , hence M ON M I , and the scattering angle 8 in center-of-mass coordinates is substantially equal to the scattering angle 8 in laboratory coordinates. The curve in Fig. 1 is proportional to l/sin4(8/2), as predicted by Eq. (2.1.1.1.6), and is fitted t o the arbitrary vertical scale a t 8 = 135". The agreement a t all angles shows that, under the conditions of these experiments, the only force acting between the incident CY rays and the gold nuclei is the inverse-square Coulomb repulsion. The Coulomb parameter, Eq. (2.1.1.1.5), has the value 22z/1378 N 36 for the collisions between Au and the 7.68-Mev a rays from RaC', and the closest distance of cm for 150" scatapproach in these experiments was p,,,,,, = 30 X tering. Therefore the positive charge in the gold atom is confined to s

292

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

small central region which is definitely smaller than this, or about lo-' of the atomic radius. Using the 7.68-Mev RaC' a rays from a source containing RaB RaC, and thin scattering foils of Au, Ag, Cu, and All Geiger and Marsden showed experimentally for the first time that the nuclear charge Ze is approximately proportional to the atomic weight, and that 2 is about

+

MEAN ANGLE OF SGATTERlNG,8

FIQ. 1. Geiger and Marsden's relative differential cross section measurements for the single scattering of RaC' 01 rays by a thin foil of gold [from Evans'].

one-half of the atomic weight. Their experimental uncertainty in the absolute value of 2 was about 20%. For an accurate determination of 2, many experimental precautions must be observed. Especially the scattering foils must be so thin that the a rays lose only a small fraction of their energy by ionization and therefore have an accurately known mean velocity in the foil. Chadwick introduced in 1920 a n ingenious experimental arrangement which greatly increases the observable scattered intensity for any given angle, source, and thickness of scattering foil. The foil is arranged, as shown in Fig. 2, as an annular ring around an axis between the source and the detector. This annular geometry for the scattering body has subsequently been widely adapted to a variety of

2.1.

293

DETERMINATION O F CHARGE AND SIZE

*

other scattering problems, especially with neutrons. Chadwick’s precision a-ray scattering experiments with this arrangement gave the absolute value of the nuclear charge of Cu, Ag, and Pt as 29.3e, 46.3e, and 77.4e, with an estimated uncertainty of 1 to 2 %. This direct measurement was a welcome confirmation of the atomic numbers 29,47, and 78 which had in the meantime been assigned to these elements in 1914 by Moseley on a basis of their characteristic X-ray spectra. /

SOURCE

\

BAFFLE ORAY TO STOP DIRECT BEAM OF U RAYS

SCINTILLATION DETECTOR FOR 0 RAYS SCATTERED BETWEEN ANGLES AND 8 2

+

/#

ANNULAR RING OF SCATTERING FOIL

FIQ. 2. Chadwick’s arrangement of source, annular scatterer, and detector for increasing the intensity of (Y rays scattered between 61and 62, as used for his direct measurement of the nuclear charge on Cu, Ag, and Pt [from Evrtnsl].

Rutherford scattering is of course applicable to the scattering of any charged projectile (proton, deuteron, a ray, or fission fragment) under the restrictions imposed by Eq. (2.1.1.1.5). Modern scattering-experiment techniques with artificially accelerated projectiles usually provide for momentum or energy measurements on the scattered particles. t Then the energy of the Scattered particle is given by Eq. (2.1.1.1.4) and can be used to determine the mass number Mz of the scattering nucleus, while the charge Ze of the scattering nucleus determines the intensity of scattering according to Eq. (2.1.1.1.6).

2.1.1.2. Characteristic X-ray Spectra.* $ Following the proof of the existence of atomic nuclei by the a-ray scattering experiments, Niels Bohr in 1913 assigned the principal part of the atomic mass t o nuclei and introduced his quantum theory of the origin of atomic spectra. This step completed the basic concepts of the Rutherford-Bohr model of the

t See also Vol. 4, B, Part 9.

4 See also Vol. 1, Chapter 7.10. -

* Section 2.1.1.2 is by

Robley D. Evans.

294

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DETERMINATION OF FUNDAMENTAL QUANTITIES

nuclear atom. To the extent that the simple theory for hydrogen-like atoms is valid, the energy hv of characteristic X-ray quanta is given approximately by

where the principal quantum numbers are nl for the initial electron vacancy and n2 for the final electron vacancy, a = e2/hc ‘V & is the fine-structure constant, and mgc2 = 0.511 MeV. For characteristic X-ray lines of the K , series nl = 1, 722 = 2; for the Laseries n l = 2, n2 = 3. Equation (2.1.1.2.1) gives a reasonably good representation of the gross behavior of characteristic X-ray spectra, but of course does not give accurate values nor any information on the fine structure because it ignores the orbital ( E ) , spin (s),and total ( j ) quantum numbers, because it is nonrelativistic, and because it ignores screening of the nuclear charge by atomic electrons. Moseley in 1913 applied the then new principles of Bragg reflection to the study of X-ray lines and introduced a new era of X-ray spectroscopy by measuring the mean wavelength of the unresolved K , doublets for 21 elements from I d 1 to 4,Ag, and the mean wavelength of the . showed the unresolved La doublets for 24 elements from 40Zr to 7 9 A ~He existence of a linear relationship between the atomic numbers of the light elements, as previously assigned from chemical data, and v 1 / 2 for the characteristic K , and L, X-ray lines. The overlap between 4oZrand 47Ag oriented the La series with respect to the K , series, and thus permitted the use of the La series for bridging over and going above the rare earth group of elements. Moseley’s work was the first to show that a total of 15 places (Z = 57 to 71) had to be reserved for the rare earths, and the first to assign atomic numbers to the heavier elements, for example to gold (Z = 79). Screening of nuclear charge by atomic electrons. Moseley’s original data / the ~ K, are shown in Fig. 1. The plot of atomic number against v ~ for series does not pass through the origin but has an intercept of about unity on the scale of atomic number. If the nuclear charge Z is assumed to be the same as the atomic numbers which had been assigned to the light elements from chemical data, then Moseley’s data on the K , series have the form ( ~ V ) I /= ~ const X (Z - 1). (2.1.1.2.2) Moseley correctly interpreted this result as indicating that for the K , series the effective nuclear charge Ze is reduced to about, (2 - l ) e because

2.1.

DETERMINATION O F CHARGE AND SIZE

295

of the screening of the nuclear potential by the potential due t o the other K , L, . . . electrons present in the ionized atom both before and after the X-ray transition. Similarly, Moseley’s data on the Laseries exhibit a substantially linear relationship given by (hv)’I2 = const X (2 - 7.4).

(2.1.1.2.3)

Under the same physical interpretation, the effective screening constant is about 7.4e for the La series. These screening constants are physically reasonable.

ATOMIC NUMBER

FIG. 1. Moseley’s original data (1914) showing the frequency Y of the K , and La X-ray lines. There is a uniform variation of ~ 1 ’ 2with integers 2 assignable as atomic numbers to the 38 elements tested, if aluminum is assumed to be 2 = 13 in accord with the chemical evidence.

It is concluded th at the atomic number is equal tBot,he number of elementary positive charges 2 in the atomic nucleus and hence also to the number of atomic electrons in the neutral atom, as had been proposed first by van den Broek in 1913. The approximately linear relationship between (hv) for the lines of any particular X-ray series and the atomic number 2, as illustrated by Eqs. (2.1.1.2.2) and (2.1.1.2.3),is commonly referred to as Moseley’s law. Identification of new elements. The new elements which have been produced by transmutation processes in recent years have no stable isotopes. But each of these new elements (2 = 43, 61, 85, 87, 93, 94,

2.

296

. . .)

DETERMINATION O F FUNDAMENTAL QUANTITIES

does have a t least one isotope whose radioactive half-period is sufficiently long t o permit the accumulation of milligram quantities of the isotope. I n every case, the atomic number has been assigned first by combining chemical evidence and transmutation data, a t a time when the total available amount of the isotope was perhaps 10-lo gm or less. Confirmation of most of these assignments of atomic number has been made by measurement of the K and L series X rays, excited in the conventional way by electron bombardment of milligram amounts of the isotope.1'2 Measurements of the characteristic X-ray emission lines are regarded as conclusive evidence in the identification of any new element. X rays from radioactive substances. Whenever any process results in the production of a vacancy in the K or L shell of atomic electrons, the subsequent rearrangement of the remaining electrons is accompanied by the emission of one or more X-ray quanta of the K or L series, or by Auger electrons, or both. Such vacancies are always produced spontaneously in two types of radioactive transformations, electron capture and internal conversion. In electron-capture transitions it is generally more probable that a K electron will be in the vicinity of the nucleus and will be captured than that an L, M , . . . electron will be captured. The majority of the vacancies therefore are produced in the K shell. If 2 is the atomic number of the parent radioactive substance, then (2 - 1) is the atomic number of the daughter substance in which the electron vacancy exists and from which the X rays are emitted. For example, several isotopes of technetium (2 = 43) decay predominantly by electron capture, and the early identification of element 43 was aided by the observation of molybdenum (2 = 42) X rays which are emitted in the decay of these technetium isotopes. A number of transuranium isotopes decay by electron capture, in competition with a-ray emission, and can be identified unambiguously by the characteristic X rays which accompany the electron-capture transitions. Internal conversion is an alternative mode of deexcitation which always competes with y-ray emission. The nuclear excitation energy hv is transferred directly to a penetrating atomic electron, which is expelled from the atom with a net kinetic energy hv - B,where B is the initial binding energy of the electron. In the most common cases, internal conversion is more likely to expel a K electron than an L, M , . . . electron from the atom. Thus the majority of the vacancies are produced in the K shell of atomic electrons or in the L shell if the available energy hv is insufficient, to eject a K electron. Internal conversion transitions are always accom1L.E.Burkhart, W. F. Peed, and B. G. Saunders, Phye. Rev. 78, 347 (1948). W.F. Peed, E. J. Spiteer, and L. E.Burkhart, Phys. Rev. 76, 143L (1949). 95,

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DETERMINATION O F CHARGE AND SIZE

297

panied by X-ray emission spectra. Because internal conversion involves no change in nuclear charge, the X-ray spectra are characteristic of the element in which the internal conversion and the competing y r a y transitions took place. The chemical identification of a number of radioactive nuclides among the transuranium elements ( Z > 92) has been made or confirmed by observations of the L series X rays which accompany the internal conversion of excited nuclear levels produced in the decay products, following a-ray or @-rayemission by the parent nuclide. For the measurement of these X-ray spectra it is frequently convenient to use a bent-crystal X-ray spectrometer.a In this way the fine structure of the L series X rays, which occur in the energy range of 10 to 20 kev for Z 2 80, can be resolved and studied in detail. The K series X rays of the heavy elements occur a t much higher energies, and are in the domain of 70 to 120 kev for Z 2 80. Measurements of the energy spectrum of the internal conversion electrons, with a &ray spectrometer, provides an additional method for determining the atomic number of the element in which the internal conversion transition takes place. If hv is the energy available, then a conversion electron from the K shell will have a kinetic energy hv - BK, where BK is the binding energy of a K electron, that is, the energy of the K edge. Similarly, a conversion electron ejected from the L shell will have a kinetic energy hv - BL, where BL is the energy of the L edge. Under sufficiently high resolution the fine structure of the L conversion electrons can be resolved into discrete groups from the L,LII, and LIIIelectronic levels. If hv is known, the absolute values of the energies of the conversion electrons serve to determine B K , BL, . . . and hence Z of the element. Even if hv is unknown, the energy difference between the conversion electron groups from a single nuclear transition can be used for a determination of 2. For example, the energy difference between the L-conversion electrons and the K-conversion electrons from the same nuclear transition is (hv - BL) - (hv - BK),which is equal to (BR - BL)and therefore to the energy of a K , X ray from the same element. cE Tables of the binding energies in kev of the various groups of K, L,M , N , and 0 electrons have been compiled by Hill et a1.7 and ~ t h e r s . ~ . ~ ]

G. W. Barton, Jr., H. P. Robinson, and I. Perlman, Phys. Rev. 81, 208 (1951). C. D.Ellis, Proc. Roy. SOC.AQQ,261 (1921);A101, 1 (1922). 6L.Meitner, 2. Phgaik Q, 131, 145 (1922). G.A. Graves, L. M. Langer, end R. J. D. Moffat, Phys. Rev. 88, 344 (1952). ’R.D.Hill, E. L. Church, and J. W. Mihelich, Rev. Sci. Znstr. 23, 523 (1952). a Y. Cauchois, J . phys. radium 13, 113 (1952). S. Fine and C. F. Hendee, Nucleonics 13(3),36 (1955). a

298

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DETERMINATION OF FUNDAMENTAL QUANTITIES

Atomic Number, 2

FIG.2. Binding energy of the x electrons in the K , L, and M shells of the elements [from Evans”J1.

Figure 2 shows the variation wit.h Z of the binding energies of the most tightly bound groupsl the s electrons, whose binding energies correspond to the K, L I ,and MI edges.

2.1.1.3. Charge Determination of Particles in Photographic Emulsions.* The application of the most obvious method for charge deter-

mination-deflection in a magnetic field-to nuclear emulsion technique is fraught with several serious difficulties; these include the large amount of multiple scattering a particle suffers in the emulsion and the relatively short path in this dense medium. Two magnetic methods, namely the lo R. D. Evans, in “Handbuch der Physik-Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 34. Springer, Berlin, 1958.

*Section 2.1.1.3 is by M. Bfau.

2.1.

DETERMINATION O F CHARGE AND SIZE

299

sandwich method’’ and the magnetic deflection in the emulsion itself, have have treated by various authors. l-’% The “sandwich method ” consists in the measurement of deflections of particles traversing the air gap between two parallel emulsion sheets. The curvature of the trajectory in the magnetic field, existing in this gap, is determined by the angle between the particle’s exit and entrance directions in the two adjacent emulsions. Although the accuracy of this method could be increased by using wider gaps, the maximum separation is limited by the fact that it becomes increasingly difficult to follow a t,rack, which is not visible, over a considerable segment. Distortion of the emulsion may also impede the usefulness of this technique by causing large errors, especially for tracks with dip angles exceeding 10”. However, even if the results are inadequate for accurate charge determination, the method can be used to obtain the sign of the charge. The other method of magnetic deflection also requires distortion-free emulsions. It will yield accurate value of the charge only if the magnetic field is large, the path in the field long, and the mean scattering angle &, small compared to the magnetic deviation angle am.If the latter is defined by a, = ( t / p ) , where p is the radius of curvature of a trajectory which describes a path of circular arc t in a region where the magnetic field is H , then one obtains am = ( t H z / p ) ,where z is the charge of the particle and p is its momentum. If p can be determined independently, and if the particle trajectory is nearly perpendicular to the magnetic field, then z can be measured with considerable accuracy. However, the sign of the charge may be obtained from the direction of deflection, even if the momentum cannot be determined. It should be emphasized again th a t these measurements can be made only if multiple scattering does not, obscure the direction of the magnetic deviation a r n / a ~>c> 1. Since it can be shown that am/aAC H p dj, the magnetic deviation method is dependable only if H , t, or 0 are large. On the other hand, when 4 1, t a n d H must be very large to give a measurable value for am.The authors working in this field have adopted, as a general rule, that the sign of the charge of particles in the energy range 200-2000 Mev can be determined with 80%

-

I. Barbour, Phys. Rev. 76, 820 (1949). C. Franzinetti, Phil. Mag. 171 41, 86 (1950). a Y. Goldschmidt-Clermont and M. Merlin, Nuovo cimento 191 7 , 220 (1950). 4 C. C . Dilworth, S. J. Goldsack, Y. Goldschmidt-Clermont, and F. Levy, Phil. Mag. [7] 41, 1032 (1950).

C. C . Dilworth and S. J. Goldsack, Nuovo cimento [9] 10, 926 (1953). M. Merlin, Nuovo cimento Suppl. [lo] 2, 218 (1954). 1 M. Merlin and G. Someda, Nuovo n’mento [9]11, 73 (1954). 7s H. P. Furth, Rev. Sci. Znstr. 26, 1097 (1955). 6

0

300

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DETERMINATION OF FUNDAMENTAL QUANTITIES

probability if the path length is at least 2 cm and the applied magnetic field is 30,000 gauss. Somewhat better results can be achieved, if diluted emulsion with a smaller content of silver bromide are used. This and the recent availability of large transient magnetic fields may well improve the conditions for magnetic methods in nuclear emulsions. There are other methods which can be used for the measurement of particle charge, but not the sign, of the charge. In the previous section, emphasis was placed on singly charged particles. It should be noted that the ionization of two particles of equal velocity is in the ratio of the squares of their charges. Therefore the grain density may be expressed by the relation g = f(z2j3). Hence, the grain density alone gives no information on z or j3 separately. Even if a large segment of track is available, the situation is not improved, since the rate of change of the grain density depends on j3, M , and z. Thus, one must supplement such observations with an independent method, such as scattering, which is a measure of momentum and charge (see Section 2.2.1.1).

Another important parameter for charge determination, closely related to ionization, is the production of 6 rays. The latter arise from collisions with atomic electrons when the energy transfer is greater than the average energy given to grains, forming the particle trajectory. Thus, the atomic electrons acquire considerable velocities, and hence are able to render several grains developable, thereby forming short trajectories which protrude from the original track. &ray measurements were first used to discriminate among charges of heavy primaries which were discovered in cosmic radiation experiments by Bradt and Peterss and Freier et aL9 The frequency of 6 rays increases with decreasing velocity, although at very low velocities the number of visible 6 rays decreases again. The latter decrease is mostly due to the smaller energy and therefore shorter residual range of the emitted 6 rays, a t very low velocities-near the end of the range of the multiply charged particle-there are no distinctly visible 6 rays, but the tracks seem merely somewhat thicker than the trajectories of singly charged particles. This phenomenon is known under the name of “thin down length”; the extension of the thin down length is a function of charge and mass of the particle, since both parameters determine the rate of velocity loss of the particle, The maximum energy which particles heavier than electrons can transfer to electrons is given by Em, = 2mJ2(1 - a2), where m. is the electron mass and B the particle

* H. L. Bradt and B. Peters, Phye. Rev. 74, 1828 (1948). OP. F‘reier, E. J. Lofgren, E. P. Ney, and F. Oppenheimer, Phys. Rev. 74, (1948).

1818

2.1.

DETERMINATION OF CHARGE AND SIZE

301

velocity. If is small, the range of the ejected electron is not sufficient to produce a clearly visible trajectory. The &ray density depends not only upon velocity but also on particle charge, and it is necessary to know the exact relationship in order to evaluate the charge of particles traversing the emulsion. Bradt and Peters8*10-11 treated the problem of &ray frequency versus velocity and charge by applying a modified Mott12 equation, based on Rutherford's law for elastic-coulomb scattering. The number of 6 rays of energies between Emin and Em, is given by (2.1.1.3.1)

Here me is the electron mass and A is a constant, which depends on the composition of the emulsion used, and on the efficiency in observing 6 rays. Emin represents the smallest &ray energy which can be detected or is being considered in the count, and Em,, the greatest energy which the electron can obtain in the collision with a particle of velocity P, or the maximum energy which can be observed in an emulsion of given sensitivity. Bradt three to four grains protrudand Peters have chosen as criterion for Emin ing from the track while Freier et aL9 use a range criterion of 1.5 p for = the minimum length of the track, corresponding in both cases t80Emin 10 kev. Bradt and Peters calculated a family of &ray density versus residual range curves for particles of various charges and masses. The calculations are based on the assumption that the &-raydensity Na(zM) is proportional to the square of the charge, and inversely proportional to the particle velocity. Furthermore, Na(zM) is calculated with reference to the &ray densities in a-particle tracks, which, because of their relative abundance in cosmic radiation, are very well investigated. The &ray density is given by

where R is the residual range. Delta-ray counting is a quite difficult technique, as one can see, when looking at the microphotograph of multiply charged particles. Therefore it is easy to understand that the agreement is not too good if the results H. L. Bradt and B. Peters, Phys. Rev. 77, 54 (1950). H. L. Bradt and B. Peters, Phys. Rev. 80, 934 (1950). 12 N. F. Mott, Proc. Roy. SOC.(London) Al24, 425 (1929).

10

11

302

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DETERMINATION OF FUNDAMENTAL QUANTITIES

of various author^^^-'^ are compared. The work of these authors does not clearly verify or disprove Bradt and Peters results. Dainton et al.,lb however, found in their experiments a distinct deviation of the &ray distribution from the Rutherford law. The &ray problem became increasingly more important in connection with investigations about the charge distribution of heavy primaries in cosmic radiation. The earlier experiments, performed especially by the Rochester and Bristol group, gave

-

G 14 g r o h (0)

-

1.5

-

x Dainton eta/ Rutherford W,,:

l5keV

.

FIG. 1. Comparison between experimental and theoretical results. (a) Number of 6 rays with four or more grains as a function of the particle velocity. (b) Number of d rays with residual range equal to or larger than 1.58 p . (The figure is taken from the paper by Tidman et aZ.9 KEY:X, Dainton et al.l6;0, Tidman et aZ.17

quite different results, which a t least in part were caused by differences in the measuring technique and interpretation of the &ray distribution. The &ray problem has been reinvestigated by Tidman et a1.l’ Their treatment of the &ray distribution function is similar as in the work of Bradt and Peters with the exception that the atomic electrons were assumed to be neither free nor a t rest. The authors treat the problem rigorously, considering separately close and distant collisions; they find a l / E z dependance for the former interactions and a l/E4 for the latter. They establish minimum criteria (for the slowest 6 rays accepted in the count) on a more realistic basis, by using experimental range-energy and S. 0. Sorensen, Phil. Mag. 171 40, 947 (1949). L. Voyvodic, Can. J. Research 28, 315 (1950). l6 P. Demers and L. Wasinkynska, Can. f. Research 31, 480 (1953). l 6 A. D. Dainton, P. H. Fowler, and D. H. Kent, Phil. Mag. [7] 43, 729 (1952). l7 D. A. Tidman, E. P. George, and A. J. Herz, Proc. Phys. Soc. (London) A66,1019 (1953). l3

l4

2.1. DETERMINATION

OF CHARGE AND SIZE

303

number of grains versus energy-relations, 18.19 and taking into consideration straggling and scattering effects. The results, which are based on partly theoretical and partly empirical considerations, can be expressed most conveniently by the curves given in Figs. 1(a) and 1(b). The curves compare the calculated results of these authors with the experimental data of Dainton et a1.16 for the grain criterion g 2 4 and the range criterion R 2 1.58 p , both corresponding to energies 2lFj kev. The curves refer to &-ray densities of singly charged particles which are plotted versus the particle velocity 0,. The agreement between experimental and theoretical values, especially in the range criterion case [Fig. l(b)] is truly remarkable, in view of the fact, that the &ray density is given in absolute units. Curve B in Fig, l(a) represents the &ray distribution according to Rutherford’s law and it seems evident that the experimental data are in much better agreement with the rigorous calculations of Tidman et al. (curve A ) . Also later investigations by Dainton and Fowlerz0on the &ray distribution in singly charged particle tracks are in good agreement. The results of measurements on more than 200 tracks of protons (with ranges between zero and 2800 p ) are presented in Table I; the data refer to the minimum criterion of at least 4 grains. The problem of charge identification in the case of heavy primaries is relatively easy, if the exposures were made in such conditions (altitude and geomagnetic latitude) that all primaries are of relativistic velocities. In this case the number of 6 rays, exceeding a certain energy Emi.[see Eq. (2.1.1.3.1)l depends only on the particle charge, and is given by N , = az2 b ; z is the charge of the particle and the constants a and b have to be determined by calibration experiments, which are best performed with the multiple scattering method.21J2Since the &ray densit,y in very heavy primaries is large, it is useful to abandon the grain criterion and to use a range criterion which includes only 6 rays of more than 8 p that protrude distinctly from the core of the The charge of relativistic primaries also can be determined by photometric opacity measurements alone, or in combination with scattering or &ray m e a ~ u r e m e n t s . * ~The - ~ ~photometric method essentially represents a measurement of the ionization, produced along the particle tra-

+

lsR. H. Hem, Phys. Rev. 76, 478 (1949). M. A. S. Ross and B. Zajac, Nature 164, 311 (1949). 20 A. D. Dainton and P. H. Fowler, Proc. Roy. SOC.(London) 221, 414 (1954). 2 1 0. B. Young and F. E. Harvey, Phys. Rev. 109, 529 (1958). 2 2 0. B. Young and F. W. Zurheide, Nuovo cimento [lo] 14, 90 (1959). 23 M. Koshiba, G. Shultz, and M. Schein, Nuovo cimento [lo] 9, 1 (1958). 24 B. Waldeskog and 0. Mathiesen, Arkiv Fysik 17, 427 (1960). 26 0. Mathiesen, Arkiv Fysik 17,441 (1960). 28 K. Kristiansson, 0. Mathiesen, and B. Waldeskog, Arkiv Fysik 17, 455 (1960). 19

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DETERMINATION O F FUNDAMENTAL QUANTITIES

jectory; because of the high ionization density, this method is preferable to any other ionization method, based on blob or gap measurements. The method which has been developed to highest perfection by the Lund g r o ~ p , has ~ ~ the - ~ great ~ advantage of freedom from personal errors, but TBLE I. &Ray Density as a Function of Residual Range (Proton Track) ~

Residual range in microns

No. of 6 rays

Standard deviation (%)

0-100 100-200 200-300 300-400 400-500 500-600 600-700 700-800 800-900 900-1000 1000-1100 1100-1200 1200-1300 1300- 1400 1400-1 500 1500-1600 1600-1700 1700- 1800 1800-1900 1900-2000 2000-2100 2100-2200 2200-2300 2300-2400 2400-2500 2500-2600 2600-2700 2700-2800

0.01 0.02 0.03 0.06 0.12 0.22 0.28 0.43 0.61 0.71 0.66 0.87 0.81 0.92 0.96 0.84 0.84 0.97 1.01 0.94 1.03 1.06 0.87 0.90 0.97 0.83 0.99 0.96

60 40 40 30 20 15 14 13 11 10 10 9 9 9

8 8 8 8 8 8

8 8 8 8 8

8 8

the disadvantage to depend strongly on normalization and calibration measurements; also depth and dip corrections are more critical than in any other method. I n these e x p e r i m e n t ~ ~ 7the - ~ ~photometric instrument consisted of a Leitz microscope with a K.S. 53 objective; the place of the eyepiece is taken by an eyepiece of special construction. There is a slit in the image plane and above this slit a photomultiplier which registers the *'S. von Friesen and K. Kristiansson, Arkiv Fysik 4, 505 (1952). ** S. von Friesen and L. Stigmark, Arkiv Fysik 8, 127 (1954). 2 9 K. Kristiansson, Arkiu Fysik 8, 311 (1954). *O B. Waldeskog, Arkiu Fysik 10, 447 (1956).

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DETERMINATION OF CHARGE AND SIZE

305

light passing through it; the dimensions of the slit, referred to the objective plane, were 54 p X 4.3 p. The slit is wide enough to cover not only the core of the track but also a great part of the 6 rays; some of the 6 rays are cut off by the slit, but the amount will be the same for all tracks of equal charge. The opacity or MTW (mean track width), as called by the Lund group, is measured along the track and is compared with background measurements, which are taken alternately after each measurement, at distances 10 p below and above the track. In these experiments only tracks lying in the middle of the emulsion and with dip angles less than 25' (unprocessed emulsion) were accepted for measurements. The dip correction is somewhat more complicated here than in the case of singly charged particles, since one has to consider separately the change in the light transmission through the nearly compact core and through the loose structure of the protruding 6 rays. Therefore special correction factors have to be introduced, which can be determined by calibration experiments. Charge calibration was performed by comparing the experimental results with those obtained by the &ray method (Na = uz2 b). Breakup events and interactions of heavy primaries with emulsion nuclei were used to obtain another independent correlation between mean track width and particle charge. The resolving power of the photometric method, concerning the discrimination between consecutive charges, depends of course on the error made in the mean track width determination. The error, as a function of track length, is given by

+

Here, n is the number of individual measurements per track, uo the standard deviation, and eo an error which is independent of track length and mostly due to irregularities in the emulsion. The error for a track passing through N pellicles is given by e ( N ) = e(L)N-''Z. The error can be made small, if the acceptance criteria are rigorous. Only tracks from the center part of the emulsion with a minimum length of 3 mm per pellicle are accepted, and the authors find that the method gives a better charge discrimination than the &ray method. The problem of charge determination of fast, but not relativistic tracks is much more difficult to solve. In this case not only the mean track width, but also the slope of the mean track width versus residual range curve has to be determined. Kristiansson et aL3I investigated this problem for heavy primaries with charges 6 5 z 2 26. They came to the conclusion that correct charge identification is possible if the following conditions 31

K. Kristiansson, 0. Mathiesen, and B. Waldeskog, Arkiv Fysik 17, 485 (1960).

306

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DETERMINATION O F FUNDAMENTAL QUANTITIES

are satisfied: (1) the track length must be longer than 25 mm; and (2) the momentum of the particle with charge z must be smaller than a certain minimum value p , ; e.g., for magnesium p12 1.2 (Bev/c) per nucleon, 1.5 (Bev/c) per nucleon. This method can be used and for iron p26 together with the multiple scattering method, or in place of the latter, if the emulsion is distorted. For tracks of multiply charged particles ending in the emulsion, a combination of the range-energy relation, multiple scattering method (constant sagitta), 6 ray and ionization measurements may lead to correct charge determination if the conditions are favorable. Alvial et recommend for slow particles (e.g., hyperfragments and stable fragments, emitted from stars) a method, based on the count of slow 6 rays, which are observable only as blobs attached to the edges of the track. The fluctuations in the track width, caused by these attachments, are measured with a n eyepiece micrometer (Clausen micrometer). The authors claim that the integral number of these short 6 rays, measured from the end of the track up to a certain residual range R1is a charge-sensitive parameter, giving a distinct discrimination between consecutive charges. The method presumes the accurate measurement of the mean grain diameter and accepts as 6 rays only blob attachments, which cause a broadening of the track, equal or larger than 1.8 times the grain diameter. A somewhat similar method equally based on micrometric width measurements of short ending tracks was proposed by Nakagawa et The authors use for calibration purposes 01 particles of radioactive elements, and Be8, Lis and BEtracks emitted from stars. Using only flat tracks (dip angle 5 12.5') and very short cell lengths, they find that the track width increases proportional with
-

-

3* G. Alvial, L. Grimaldi, J. Riquelme, E. Silvia, and S. Stantic, Nuovo cimento [lo] 16, 25 (1960). a3 S. Nakagawa, E. Tamai, H. Huzits, and K. Okudaira, J. Phys. SOC.Japan 12, 747 (1957). 34 E. Tsmai, Nuovo cimento [lo] 14, 1 (1959). 36 G. Ammar, Nuovo cimento Suppl. [lo] 16, 181 (1960).

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DETERMINATION O F CHARGE AND SIZE

307

small cell sizes (0.57 p ) and omitting the last 1.5 1.1 of the ending tracks. Studying the track width distribution curve, the dispersion of the distribution

u = v“ ( n - 1 ) and the skewness n

a3 =

u3

the author arrives to the conclusion that the skewness or third momentum a3is an even better, more sensitive charge parameter than the mean track width, and that this parameter should be used in charge determination. An additional advantage is that in the case of the third momentum no normalization is needed. A disadvantage is the rather large amount of measurement errors, which is an inherent difficulty of all investigations when dealing with low-energy particles.

2.1.2. Principal Methods of Measuring Nuclear Size* 2.1.2.1. Introduction. The principal methods of measuring nuclear size may be divided into two main classes as follows: (A) Electric or Magnetic Methods; and (B) Nuclear Force Methods. Certain other determinations fit partially into each class and we may therefore consider a third group: (C) Combinations of Methods (A) and (B). By nuclear size we shall mean something more than just a single measure of nuclear dimensions for we must recognize that nuclei consist of masses, electric charges, and magnetic moments. I n fact the charges, to take one example, will be spread out continuously from a nuclear center to a nuclear “periphery,” defined as that region of space where the charge density approaches zero. We must therefore speak of nuclear densities and their variation throughout the nuclear volume and not simply of size or dimensions such as we might, e.g., ascribe to idealized metal blocks or spheres. Furthermore, we must also note that nuclei need not necessarily be spherically symmetric objects although it is clear ~

* Section 2.1.2 is by Robert Hofstadter.

308

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

that many nuclei possess such symmetry. This implies that an ultimate description of nuclei will have to take into account the angular variation of the various densities of a nucleus, as well as the radial variation of its densities. We shall discuss this subject briefly in Section 2.1.2.2.4. From these remarks we can see that a determination of nuclear “size” presents a formidable problem. In a certain sense it is fortunate that our information is at present quite limited for this fact allows us to keep many of the considerations simpler than they would otherwise be. We are only in the very early stages of distinguishing electric charge density from mass density or from magnetic moment density. But we shall come upon these distinctions as we describe the various methods now available for studying nuclear size. Since we shall discuss a large number of methods in the subsequent text it proves to be quite impossible to present the actual physical details of the experimental arrangements. However, a reading of the principal references, such as the survey articles, will disclose sufficient detail concerning experimental arrangements. Therefore, in the material to follow, we shall present the essential ideas of the various methods of measuring nuclear sizes and a very brief accounting of some of the results in order to indicate the type of accuracy that can be obtained. 2.1.2.2. Discussion of i h e Principal Methods of Measuring Nuclear Size. In this section we shall list and discuss most of the principal methods of determining nuclear size. Note that, unless otherwise specified, in the following material all distances will be quoted in units of cm. A. Nuclear Force Methods 1. Alpha-Particle Scattering 1. Alpha-Particle Radioactivity 3. Proton Scattering 4. Neutron Scattering 6. Pion Scattering 6. Antiproton Studies 7‘. Scattering of Nuclei by Other Nuclei 8. Diffraction Phenomena at High Energies B. Electric or Magnetic Methods 1. Electron Scattering 1. Mu-Mesic Atoms 8. Mirror Nuclei 4. Isotope Shift 6. X-ray Fine Structure 6. Magnetic Hyperfine Structure Anomalies C. Combination Methods I . Weissiicker Semiempirical Formula 1. Neutral Pion Photoproduction D. Angular Shapes of Nuclei

2.1.

DETERMINATION OF CHARGE AND SIZE

309

It is not possible in the space allotted to describe and illustrate each of the methods listed above. However, it is felt that if a few of the more important methods are dealt with in some detail the general features of the entire field can be well represented. 2.1.2.2.1. NUCLEAR FORCE METHODS. 2.1.2.2.1.1.Alpha-Particle Scattering. Historically this is the original method of observing the effects of nuclear size and the method depends on the observation that deviations from Coulomb scattering from a point charge are observed when the alpha particle approaches or penetrates the periphery of the nucleus. Actually, it is only necessary that the alpha particle and the nucleus approach within the range of nuclear forces. However, as far as concerns the concept of size we are employing in the nuclear force methods, this is equivalent to determining a nuclear “radius.” In consequence, the range of nuclear forces is automatically included in the distance separating the centers of the two geometrical bodies involved in the collision, e.g., the alpha particle and the nucleus. It is understandable that a simple deviation from Coulomb scattering will not be able to give more than a rough indication of nuclear size. cm Early observations of this kind indicated radii of the order of for the heavier nuclei. In recent years the method has advanced strikingly so that quite sophisticated techniques are now required to analyze the experimental data. Typical examples of the experimental elastic scattering are shown in Fig. 1 and are taken from the work of Igo and Thaler.‘ It is to be noted that the lighter elements indicate pronounced diffraction maxima and minima but the angular behavior for heavier nuclei shows a much smoother decrease a t larger angles. Such data have been evaluated in two ways. In the first method,2 the so-called “sharp cutoff model,” the Coulomb behavior of the potential is assumed to hold rigorously beyond the “cutoff” radius R, and within the spherical nuclear volume having this radius, the nucleus is assumed to be completely absorbing or “black.” Such an analysis fits the data better at small angles than a t large angles3 and certainly reproduces the main trend of the data. When analyzed in this way a general formula for the nuclear radius may be obtained as follows: R = 1.41A’’3 2.19 (2.1.2.1) in units of 10-13 cm. In the second method the nucleus is represented by an optical model having a diffuse boundary, and a potential energy term with real and imaginary (absorption) parts is used. The same kind of model can be

+

G. Igo and R. M. Thaler, Phys. Rev. 106, 126 (1957). J. S. Blair, Phys. Rev. 96, 1218 (1954). 3 D. D. Kerlee, J. S. Blair, and G. W . Farwell, Phys. Rev. 107, 1343 (1957). 1

2

310

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

-

0.001 20 30 40 x)

e

80 70 80 SO

(c.M.)

FIG.1. This figure shows the experimental values of the ratios of elastic-scattering cross sections to pure Rutherford scattering for 40.2-Mev CY particles. This figure is due to Igo and Thaler.1

used for both real and imaginary parts, although this is not strictly necessary. A typical form of this potential energy is given in Eq. (2.1.2.2). potential energy

=

(V,

+ iW,)(1 +,exp[(r - R)/a]

1-l

+ 2e$(r).

(2.1.2.2)

Typical parameters fitting the data are given by Igo and Thaler.'

R

+ 1.3; a = 0.5, V , = -45

= 1.35A1'3

MeV, W , = -10 MeV. (2.1.2.3)

I n Eqs. (2.1.2.2) and (2.1.2.3), A is the mass number of the nucleus in question and 2e4(r) is the electrostatic interaction energy. Unfortunately,

2.1.

DETERMINATION O F CHARGE AND SIZE

311

these parameters are not unique, as has been demonstrated by different Igo6*6has shown that a common property of all a ~ t h o r s . ~However, -~ the potentials which give good fits to the data is the shape of the nuclear surface. This is shown in Fig. 2 where the real parts of the optical model

r (FERMIS)

FIG.2. Igo's representation of the best real potentials for the elastic scattering of 40-Mev (Y particles from Cu (V and W are in MeV; r,, and d i n fermis). (a) (V,W,ro,d) = (-110, -20, 6.30, 0.5); (b) (-49.3, -11, 6.78, 0.5); (c) (-19, -13, 7.22, 0.6).

potentials for copper are plotted that fit the 40-Mev alpha-particle scattering data on that element,. Note that the central parts of the potential, lying within radii smaller than 7 fermis," may differ by nearly a factor of 10 while the surface features are all very nearly the same. This result is not accidental and has been shown to apply t o the results in several elements.6 A similar result had been found to apply in electron scattering studies. Figure 3 shows how the optical model potential can

* 1 fermi

= 10-13

em.

W. B. Cheston and A. E. Glassgold, Phys. Rev. 106, 1215 (1957). G. Igo, Phys. Rev. Letters 1, 72 (1958). G. Igo, Phys. Rev. 116, 1665 (1959).

312

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

be fitted to the data in the case of Cu a t 40 Mev for a particular choice of parameters shown in the figure. Note that these parameters differ somewhat from those given in Eq. (2.1.2.3) demonstrating the kind of latitude allowed by the analysis. It is not clear whether the same parameters will suffice at all alpha energies and there is some evidence that at

@(DEGREE)

FIG.3. The points represent the experimental work of Igo, Wegner, and Eisberg for 40-Mev 01 particles scattering elastically from Cu. This is typical of the type of fit one obtains by using the optical-model potential.

lower energies the real part of the potential is less deep. There seems to be little doubt that through such analyses the alpha-particle scattering data lead to information about the nuclear surface parameters and to a smaller amount of information, if any, concerning the center of the nucleus. Blair' and Brussard* appear to have come to this conclusion independently on theoretical grounds. As a result of the work of Igo, the portion of the nuclear potential a t the surface of the nucleus has the J. S. Blair, Phys. Rev. 108, 827 (1957).

* P. J. Brussard, Thesis on the theory of alpha decay. University of Leyden, Holland, 1958.

2.1.

313

DETERMINATION OF CHARGE AND SIZE

following appearance for a particles in the neighborhood of 50 Mev: potential = - 1100 exp -

1

1.7~1’3 0.574

-

- i 45.7exp -

[

T

) MeV.

- 1.40A1la 0.578

(2.1.2.4)

It can be seen that while Eq. (2.1.2.3) allows for something like an alpha-particle radius (1.3 X 10-13 cm) this is absent in Eq. (2.1.2.4). Again, this illustrates the kind of latitude allowed in present nuclear force determinations with a particles. The imaginary part of the potential is less well known than the real part. Critical discussions of many of the points related to the above type of analysis are given in the comments following an article by Glas~gold.~ While the alpha-particle nucleus potential can be determined by these methods, the exact relationship of this potential to the nuclear charge or mass distributions is a t present theoretically indeterminate. 2.1.2.2.1.2. Alpha-Particle Radioactivity. This is a large subject and one with a long history. Standard textbooks containlo,” the main part of the story, and while there is no doubt that the basic explanation of the decay times is understood in terms of the penetration of the nuclear Coulomb barrier by the alpha particle, the quantitative details have resisted complete analysis since 1937. At that time Bethel2 showed that the nuclear radius depended on the lifetime of the alpha particle within the nucleus in the absence of a barrier. In other words, the time of formation of an alpha particle is important in computing the probability of radioactive decay. More recent attempts to study the relationship between nuclear radii13,14and lifetimes of the radioactive nuclei also introduce another element of arbitrariness since there is no doubt that the diffuseness of the nuclear surface plays a role in the phenomenon. Shell model features are also important,. For general reviews see references 14 and 15. A. E. Glassgold, “Comptes rendus d u congres international de physique nucleaire,” pp. 23-37. Dunod, Paris, 1959. l o G. Gamow and C. L. Critchfield, ‘ l Theory of Atomic Nucleus and Nuclear Energy Sources,’’ 3rd ed. Oxford Univ. Press, London and New York, 1949. I I R. D. Evans, “The Atomic Nucleus.” McGraw-Hill, New York, 1955. l2 H. A. Bethe, Revs. Modern Phys. 9, 69 (1937). H. A. Tolhoek and P. J. Brussard, Physica 21, 449 (1955). 1 4 I. Perlman and J. 0. Rasmussen, Alpha radioactivity. I n “Handbuch der Physik -Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 42, pp. 109-202. Springer, Berlin, 1957. l6 J. 0. Rasmussen, Revs. Modern Phys. 30, 424 (1958).

3 14

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Nuclear radii from studies of a-particle radioactivity generally fall in the range R = 1.50 X A1/3 (2.1.2.5) but allowance must be made for the size of an a particle and for the range of nuclear forces, in seeking the true geometrical sizes of the nuclei. 2.1.2.2.3. Proton Scattering. Ever since the pioneering experiments of Cohen and NeidighlBit has become apparent that the prominent diff raction features observed in the elastic scattering of protons on nuclei could be interpreted in some way in terms of nuclear size. Unfortunately, the scattering phenomena are fairly complicated and the analysis of the experiments must be carried out with nuclear potentials having a diffuse boundary, just as in the case of alpha-particle scattering. In fact, the potentials are usually taken t o be of the same form as that given by Eq. (2.1.2.2).17Whereas a n alpha particle has no spin, a proton does, and the additional possibility of a spin-orbit type of interaction must be allowed for. I n spite of the enormous complexity introduced in the analysis by this large number of parameters, important progress has been made by choosing the parameters, as nearly as possible, similar to those found by other methods. In this connection the parameters found from neutron scattering can be very valuable. (See Section 2.1.2.2.4.) Another reason why the proton-scattering data yield to analysis is that the experimental data are very rich indeed and, at the same time, quite accurate. The advantages in investigating proton scattering arise from the experimental ease of detecting protons and in the copious production of beams of these particles. Van de Graaff machines, cyclotrons, linear accelerators, synchrocyclotrons have all been very successful in producing intense sources of well-collimated protons in narrow energy intervals. Scattering experiments can then be carried out on thin targets and detection and analysis can be made with scintillation counters and magnetic spectrometers. The accuracy of the experimental data is demonstrated in Fig. 4 by an example taken from Dayton and Schrankl8 a t 17 MeV. This figure shows the elastic scattering curves for C, Al, Cu, Ag, and Au. The dots in this figure refer to experimental points. Again, one sees the gradual reduction in the sharpness of the diffraction features when the nucleus becomes heavier. Note the regular spacing of the diffraction maxima and minima, a phenomenon which first appeared in the work of Cohen and Neidigh.16 The solid lines in Fig. 4 represent the theoretical analysis made by the B. L. Cohen and R. V. Neidigh, Phys. Rev. 93, 282 (1954). R. D. Woods and D . S. Saxon, Phys. Rev. 96, 577 (1954). 18 I. E. Dayton and G . Schrank, Phys. Rev. 101, 1358 (1956).

16 17

2.1.

DETERMINATION O F CHARGE AND S I Z E

315

FIQ.4. Elastic scattering of 17-Mev protons on various nuclei. The data are given in terms of the ratio of the elastic scattering t o the pure Rutherford scattering. The data are taken from Dayton and Schrank.18

316

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

Minnesota groupl9 using the optical model and a complex potential with diffuse walls of the basic types given in Eqs. (2.1.2.2) and (2.1.2.4), but, of course, with different values of the parameters. The agreement is quite good except at large angles in light elements. These discrepancies have been shown by Bjorklund and FernbachZO to be caused by neglecting spinorbit effects in the medium-weight elements. Glassgold19 states that the earlier Minnesota results on the central potential are unaffected by inclusion of the spin-orbit potential. The successful optical potential representing the interaction between a proton and a nucleus can be put in the form

where (2.1.2.7) I n this equation V and W are functions of the proton’s kinetic energy (13) in conformity with our earlier remarks on the behavior of these quantities at different alpha-particle energies. e4(r) represents the Coulomb interaction energy and the last term provides the effect of the spin-orbit potential. This potential is also taken t o be of complex form and the strengths are energy dependent. L and d/2 are the orbital and spin angular momenta (in units of h). In Eq. (2.1.2.6) V ( 0 )is the shell model potential for protons. From many analyses of the data1$ the value of R in Eq. (2.1.2.7), which is interpreted as a radius of the proton-nucleus scattering potential, is given by

R

=

T ~ A= * /(1.25 ~ f 0.05)A1/3

(2.1.2.8)

and a choice of T O in this range will lead to suitable values of the other parameters giving agreement with the experiments. After r0 has been chosen it is possible to find more specific values for “ a ” and the other quantities in Eqs. (2.1.2.6) and (2.1.2.7). The value of the diffuseness parameter “ a ” has been found to be a = 0.65

f 0.05

(2.1.2.9)

and was derived from experiments above 40 Mev while a =

0.50 f 0.05

(2.1.2.10)

A. E. Glassgold, Revs. Modern Phys. SO, 419 (1958); see also the items listed under reference 17 of that article. 2o F. Bjorklund and S. Fernbach, see reference to these authors in our reference 19.

2.1.

DETERMINATION O F CHARGE AND SIZE

317

a t lower energies. However as Glassgold has remarked, it is not certain that the energy variation of this quantity represents a real effect since the low-energy analysis did not include the effect of the spin-orbit interaction. The e v i d e n ~ e , ’however, ~ - ~ ~ seems to favor the larger value of “a” when most interactions are taken into account. A complete discussion of the effects of the spin-orbit interaction must be left t o the originaI sources. 19,20 Polarization measurements on scattered protons will help to resolve many of the present ambiguities.21J2Proton scattering experiments yielding both differential and integral cross sections have been carried out in the region of 1000 Mev and higher and have also been analyzed with optical model potentials. The original references should be consulted for this w o r k . 2 3 ~Most ~ ~ of the work has been done on nucleon-nucleon scattering a t the high energies.26.26 2.1.2.2.1.4. Neutron Scattering. Descriptions of the method of analyzing neutron-scattering data a t low and high energies are given in review articles by FernbachZ7and by Glassgold,22where references t o the earlier literature will also be found. I n studies of neutron scattering the theoretical analysis is simplified somewhat by the absence of a Coulomb potential between projectile and target nucleus. However, the experimental data are more difficult to obtain because of the relative difficulty in obtaining well-collimated monoenergetic beams of neutrons. Nevertheless, in recent years surprising progress has been made in both the experimental scattering work and in its theoretical analysis. There is no opportunity to present a critical summary of this work in our article but the reader may refer to references 22 and 27, for details. I n a general way the results appear t o be similar t o those observed in proton scattering but more emphasis has been given t o reaction, or inelastic cross sections in the case of neutron studies. As in the analysis of proton scattering, it is possible to define a phenomenological optical-model potential of the type introduced originally by Fern~ 9 the addition of a spin-orbital bach et aL28 and by Feshbach et ~ l . , with potential used in proton-scattering work [Eqs. (2.1.2.6) and (2.1.2.7)]. However, the most successful work27 which has been reported has H. A. Bethe, Ann. Phys. 3, 190 (1958). A. E. Glassgold, Progr. in NucEeur Phys. 7, 124 (1959). *a B. Cork, W. A. Wenzel, and C. W. Causey, Phys. Rev. 107, 859 (1957). 2 4 W. N. Hess, Revs. Modern Phys. 30,368 (1958); and see other references contained in this article. 2s C. J. Batty and S. J. Goldsack, Proc. Phys. Soe. (London)A70, 165 (1957). 2 6 G. E. Brown, Proc. Phys. Soe. (London)A70, 361 (1957). 21 S. Fernbach, Revs. Modern Phys. 30, 414 (1958). 28 S. Fernbach, R. Serber, and T. B. Taylor, Phys. Rev. 76, 1352 (1949). 19 H. Feshbach, C. E. Porter, and V. F. Weisskopf, Phys. Rev. 90, 166 (1953). 21

*a

318

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

u

lo -8 0

60

LAB.

ANGLE

I20

I

I60

DEGREES)

FIQ.5. This figure shows the experimental and theoretical differential cross sections for 7-Mev neutrons scattered from various nuclei. The figure is taken from F e r n b a ~ h . * ~

employed a slightly different form of the imaginary potential from the type used in proton-scattering work. The expression found to be most successful for the potential has the form2’ given in Eq. (2.1.2.11) :

+

V N = V ( E ) f ( r ) iW0 (2.1.2.1 1)

where f is given, as before, by Eq. (2.1.2.7). Bjorklund and F e r n b a ~ h ~ O ~ ~ ~ so F.

Bjorklund and S. Fernbach, Phye. Rev. 109, 1295 (1958). F. Bjorklund and S. Fernbach, University of California Radiation Laboratory Rept. No. 5028 (1958). *I

2.1.

DETERMfNATfON OF CHARGE AND SIZE

319

also use a formverysimilar to Eq. (2.1.2.6) with a real and imaginary part of the spin-orbit term, but, of course, without the Coulomb term. In either case the agreement observed with experiment has been very good and this is demonstrated by the results in Fig. 5. For the model given in Eq. (2.1.2.11) the parameters of Table I provide the best fit:

Energy (MeV)

-V (MeV)

(MeV)

x

a (fermi) [Eq. 2.1.2.7)J

4.1 7.0 14.0

50.0 45.5 44.0

7.0 9.5 11 .o

35 35 35

0.65 0.65 0.65

0

WO

b (fermi) 0.98 0.98 0.98

From Fernbach.27

and

R

= (1.25

k 0.05)A1’3X

cm.

(2.1.2.12)

Fernbach remarks that the parameters presented in Table I are independent of the atomic mass number A . There is no doubt that there will be further tests and refinements of the Fernbach results when experimental polarization studies of neutron scattering become available. It is remarkable that the proton and neutron results agree in so many of the parameters obtained from analysis of the experiments. For a study of high-energy neutron work we must refer the reader to the original ~ a p e r P -where ~ ~ reaction cross sections are obtained. For critical remarks on such work see reference 22. 2.1.2.2.1.5. Pion Scattering. As in the case of alpha-particle, proton, and neutron scattering the theoretical analysis of the experimental data is beset with the difficulty of not knowing the projectile-nucleon interaction in adequate detail. This difficulty has been taken account of in an approximate fashion by Kisslinger3’who notes that the fundamental pionnucleon interaction is preeminently of a p-wave type. The optical model, when coupled with this assumption, then leads to appropriate analyses 32 T.Coor, D. A. Hill, W. Hornyak, L. W. Smith, and G. Snow, Phys. Rev. 98,1369 (1955). 33 R. W. Williams, Phys. Rev. 98, 1387 (1955). 3 4 V. P. Dzhelepov, V. I. Satarov, and B. M. Golovin, Doklady Akad. Nauk S.S.S.R. 104,717 (1955). 36 V. I. Moskalev and B. V. GavrilorskiI, Doklady Akad. Nauk S.S.S.R. 110, 972 (1956). 38 N. E.Booth, G. W. Hutchinson, and B. Ledley, Proc. Phys. SOC.(London) A71, 293 (1958). 37 L. S. Kisslinger, Phys. Rev. 98, 761 (1955).

320

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

of the pion-complex nuclei scattering data. A slight modification of the Kisslinger theory has been used by Baker et ~ 1 . 3 8 t o find the nuclear density functions for various light nuclei. The experimental data were obtained by some of the same inve~tigators3~**0 for nuclei of lithium, carbon, aluminum, and copper. I n this case negative pions of energies 69.5,80.0, and 87.5 Mev were scattered from various nuclei in such a way

-

0.1

1

1

I

I

I

I

'

I

'

I

I

1

'

that essentially pure elastic scattering could be studied. Such a procedure involves an elaborate method of counting the incident and scattered pions with a telescopic array of scintillation counters arranged in a coincident and anticoincident scheme. Details will be found in reference 39. Figure 6 shows the results obtained in carbon with the modified Kisslinger model. The three theoretical curves refer to slightly different choices of the W. Baker, H. Byfield, and J. Rainwater, Phys. Rev. 112, 1773 (1958). E. Williams, Phys. Rev. 112, 1763 (1958). 4oR.Edelstein, W. Baker, and J. Rainwater, Bull. Am. Phys. SOC.6, No. 4, EAlO (1960). aD W. Baker, J. Rainwater, and R.

2.1.

DETERMINATION OF CHARQE AND SIZE

321

amount of p-wave interaction introduced into the theory. The results of such work show that a nuclear density model of the same type as Eq. (2.1.2.7) will fit the data very well if the ro parameter in Eq. (2.1.2.8) is chosen to be 1.08 and the thickness parameter ‘(a” is chosen to be 0.25. Thus the radial parameter ro is smaller than the value obtained from proton or neutron-scattering data, but as we shall see later, is in excellent agreement with the value for the charge density obtained by electronscattering methods. The value for “a” found in the pion measurements is only half of the value obtained by electron-scattering and nucleonscattering experiments. Abashian and associates41 attempted to detect a difference in the density distributions of protons and neutrons in the peripheries of heavy nuclei by studying comparative interaction cross sections for positive and negative pions a t 700 Mev in lead. Their data could be interpreted as showing no evidence for the neutrons having a larger extent than the protons. 2.1.2.2.1.6. Antiproton Studies. While antiproton work is still in its initial phases, because of the recent discovery of the antiproton, it seems likely that this subject will open still another avenue of approach to the study of nuclear sizes. The relatively large scattering cross sections of antiprotons in nuclear matter (about twice geometrical cross section) seem to be due to the magnitude of the nucleon-antinucleon cross section and to the diffuseness of the nuclear surface. For brief introductions to this subject we refer the reader to an article by Seg1-8~~ and to a section on antiproton scattering in reference 22 by Glassgold. 2.1.2.2.1.7. Scattering of NucEei by Other Nuclei. Recent work with heavy ion accelerators has provided an additional new source of information on nuclear sizes. The scattering of nitrogen nuclei by stationary nitrogen target nuclei43has been interpreted in terms of classical, quasiclassical12and optical model a n a l y s e ~All . ~ methods ~ ~ ~ ~ give radii in the same general range and correspond to T O values of 1.45 X 10-13 cm or 1.6 X 10-13 cm. Other new work by McIntyre and associates46 on the scattering of Nr4from Aulg7by a neutron-transfer reaction: Au’~~(N’*, Nl3)Aulg8furnishes a value of T O S 1.5 X cm. Such a radius corresponds to surface regions of the two nuclei where tunneling of a neutron from one to the other takes place. This radius seems to be rather sharply A. Abashian, R. Cool, and J. W. Cronin, Phys. Rev. 104,855 (1956). 1L SegrB, Revs. Modern Phys. 30, 550 (1958). 4 5 R. Reynolds and A. Zucker, Phgs. Rev. 102, 1378 (1956). 44 C. E. Porter, Phys. Rev. 112, 1722 (1958). 46 J. S. McIntosh, S. C. Park, and J. E. Turner, Phys. Rev. 117, 1284 (1960). 48 J. A. McIntyre, T. Watts, and F. Jobes, Phys. Rev. 119, 1331 (1960). 41

41

322

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

defined. The angular distributions and the total cross sections appear to be in good agreement with a theory of Breit and Ebe14’ for the t,unneling process, Deuteron-stripping processes or other pickup processes belong in the same general category of scattering of nuclei by other nuclei but we do not have an opportunity to discuss specific details of these methods in this article. The connection of these methods with nuclear size determinations has so far indicated large radii which may be ascribed to the weak binding energy of the deuteron. 2.1.2.2.1.8. Diffraction Phenomena at High Energies. It is likely that the diffraction peaks observed in the forward direction in the scattering of high-energy incident particles may yield rough information on nuclear size. A beginning in this analysis has been made by Williams48 who has considered the case of 1.5-Bev negative pions. Using an optical-model approach he showed that a nucleus with a tapered or diffuse edge gives better agreement with the data than a nucleus with a square edge. 2.1.2.2.2. ELECTRIC OR MAGNETIC METHODS. In this section we wish to discuss the second main group of methods used t o study nuclear sizes, namely the Electric or Magnetic Methods. It will be clear that such methods have an inherent advantage over the nuclear force methods described in Section 2.1.2.2.1 since the interaction between the probing particle and the probed nucleus is the well-known and well-understood electromagnetic interaction. The theoretical calculations can be made quite directly without the introduction of important unknown parameters or doubtful assumptions. Moreover the “radii” so obtained will not involve the range of nuclear forces and may, therefore, perhaps be expected to be smaller in value. 2.1.2.2.2.1. Electron Scattering. One of the principal electromagnetic methods that has been used is the elastic scattering of high-energy electrons on nuclei. In this case the electrons have reduced de Broglie wavelengths smaller than the size of nuclei and consequently it may be expected that some of the finer features of nuclear density variations may be discovered. This method has been exploited at Stanford University by Hofstadter and his collaborators who use a linear electron accelerator as the source of electrons. The angular scattering patterns are investigated with the use of large magnetic spectrometers. The latter are employed in order to resolve the elastically scattered electrons from those undergoing inelastic processes. Elastic scattering is required in order to study the stable ground state of nuclei. The inelastic scattering provides useful information on the transition matrix elements between the excited and 47 48

G.Breit and M. Ebel, Phya. Rev. 103, 679 (1956); 104, 1030 (1956). R. W. Williams, CERN Symposium, Geneva p. 324 (1956).

2.1.

DETERMINATION OF CHARGE AND SIZE

323

ground states. Much of the information on electron scattering appears in several review articles or b0oks.4~-~4 Figure 7 shows a typical experimental curve for 187-Mev incident electrons which demonstrates how the elastic peak, at the right, is separated by the magnetic spectrometer from the inelastic scattering peaks in C12 a t 80” in the laboratory system. By determining the angular

ENERGY IN MEV

FIG. 7. The electron-scattering peak from C for 187-Mev incident electrons and the inelastic scattering peaks from excited states of the C nucleus. The peak near 180.7 Mev is associated, e.g., with the 4.43-Mev level.49

distribution of the intensity of this peak one can obtain sufficient experimental information to find a satisfactory nuclear model fitting the data at all angles and at all energies. We show two examples below which illustrate the angular dependence of the data in CI2 and both the angular dependence and energy dependence of the data in 0 ’ 6 . In Figure 8 the angular distribution of the elastic R. R. 6 1 R. 62 D. 63 R.

Hofstadter, Revs. Modern Phys.28, 214 (1956). Hofstadter, Ann. Rev. Nuclear Sci. 7 , 231 (1957). Hofstadter, F. Bumiller, and M. Yearian, Revs. Modern Phys. 30, 482 (1958). G. Ravenhall, Revs. Modern Phys. 30, 430 (1958). Herman and R. Hofstadter, “High Energy Electron Scattering Tables.” Stanford Univ. Press, Stanford, California, 1960. 5 4 R. Huby, Repts. P r o p . i n Phys. 21, 59 (1958).

324

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

peak in CI2is shown a t 420 Mev close to the solid line of the figure.66One notes the deep minimum near 50". The position of this minimum is related quite closely to the nuclear size, The figure also shows data (long dashes) giving the angular distribution at this energy of the first inelastic peak at

SCATTERING

ANGLE IN DEGREES

FIG.8. Elastic and inelastic scattering of 420-Mev electrons for

0 2 (Ehrenberg et ~ 1 . 6 6 ) . Note the deep minimum near 50" in the elastic-scattering peak. The uppermost dashed curve is a theoretical representation of the inelastic scattering corresponding to the 4.43-Mev level. Letters (a), (b) and (c) refer to slight changes of the parameters used in fitting the charge distributions.

4.43 MeV. No diffraction structure is apparent in the inelastic scattering. Three theoretical curves enclose the experimental elastic points and nuclear models corresponding to these curves are readily calculated to yield a typical close-fitting model for carbon shown in Fig. 9. In light 6sH. F. Ehrenberg, R. Hofstadter, U. Meyer-Berkhout, D. G. Ravenhall, and S. Sobottka, Phys. Rev. 113, 666 (1959).

2.1.

DETERMINATION O F CHARGE AND SIZE

325

nuclei the Born approximation which is used in such cases is adequate for the cal~ulations.*9~~3 Figure 10 shows three elastic angular distributions for oxygen taken at 240,360, and 420 One may see how the minimum is displaced as a function of energy. The behavior of this minimum, as well as the remainder of the curve, must, of course, be consistent with the nuclear model A model of Ole providing good agreement finally chosen to represent 0l6. with these features of the scattering curves is shown in Fig. 9. 7

I

I

I

FIG.9. This figure shows the charge distributions for C l a and 0 1 8 which provide a very good fit to the experimental elastic-scattering data (Ehrenberg et ~ 1 . ~ 6 ) .

Heavier nuclei, such as, e.g., Ca, In, and Au, have also been studied by this method as well as light nuclei such as H, D, He, Li, et cetera. Thus the electron-scattering method provides a universal technique for finding nuclear charge density curves as a function of radial distance from the center of the nucleus. Figure 11 gives a summary of many of the nuclei investigated by this method. For very light nuclei, such as the proton, the deuteron, and He8 the scattering from the magnetic moment can be important and the experimental data may be analyzed to find the density distribution of magnetic moment in such nuclei. This has been done in the case of the proton with relatively high accuracy.s1 For the charge distributions of heavier nuclei the nuclear model P =

P I P + exp{(r - c ) / ~ I l - ~

(2.1.2.13)

has been found to be very satisfactory. However, the present techniques

Tn-rrTl

16'

i

OXYGEN 360 MEV HARMONIC WELL SHAPES

40

50

SCATTERING ANGLE IN DEGREES

--\+

OXYGEN 420 MEV HARMONIC WELL SHAPES

-

__

40

50

60

I

FIG.10. These figures represent three elastic angular distributions for 0 taken a t 240, 360, and 420 Mev.66 The slightly different theoretical curves are obtained in the fitting procedure and the letters (a), (b), (c) refer to different choices of fitting parameters. 326

2.1.

DETERMINATION O F CHARGE AND SIZE

327

are still unable to distinguish between such a model and a trapezoidal Nevertheless analysis shows that model or a modified Gaussian m0de1.~”,5* the distance to the half-density point “c” and the skin thickness “t,” are two parameters which the data determine with good accuracy. Figure 12

FIG. 11. This figure gives a summary of the charge distributions found in various nuclei by the method of electron scattering (HofstadterfiO).

provides a definition of the two parameters. Experiment showss6 that these two parameters are given approximately by the relations c = 1.07A1’3X 10-13 cm

t = 2.4 X 10-13 cm

=

constant

I

(2.1.2.14)

These results are analogous to the findings of the nuclear force methods since the ‘(a”corresponding to formula (2.1.2.7) is given approximately by

t 66

=

4.4a

(2.1.2.15)

B. Nahn, D. G. Ravenhall, and R. Hofstadter, Phys. Rev. 101, 1131 (1956).

328

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

and thus a E 0.55 X lo-'% cm for the model given by Eqs. (2.1.2.13) and (2.1.2.14). This compares well with the value given in Eqs. (2.1.2.9) and (2.1.2.10). However, the R values obtained in the nuclear force work from Eq. (2.1.2.7) are always larger than the c values obtained from Eq. (2.1.2.13). For example, whereas

R c

=

1.25A1/% X 10-13cm 1.07A1/% X lo-'% cm

One should note that these two radii do not refer to the same quantities, since R is representative of the radial extent of an optical-model potential

r-FERMIS

FIQ. 12. This figure gives the shape and the parameters used in a specific model (Fermi model) which has been successful in providing a fit t o the charge distributions of many nuclei (Hahn et aLba).

while c is representative of the actual charge distribution. It is not at all unreasonable that c is therefore smaller than R. A convincing theoretical explanation of the exact value of this difference between R and c has not yet been developed. Though the electron-scattering method is still in its infancy it has already surveyed the region from the proton to uranium. For the proton a root mean square radius of 0.8 X lO-l3 cm has been found for the charge radius and also 0.8 X cm has been obtained for the rms radius of the magnetic moment distribution. Further information on the proton may be obtained from reference 57. A value of 0.8 X 10-13 cm has been found for the magnetic moment distribution in the neutron.61 For the heavier 67

R. Hofstadter, Nuovo cimento

[lo] 12, 63 (1959).

2.1.

DETERMINATION OF CHARGE AND SIZE

329

nuclei, an equivalent T O [see Eq. (2.1.8)] for a nucleus of uniform density, lies in the range 1.18 X 10-13 cm to 1.25 X 10-13 cm,62and thus verifies the theoretical expectation that an “electromagnetic radius” should be smaller than a “nuclear force radius” 1.5 X 10-13 cm. It is quite clear, from the foregoing, however, that nuclear charge densities or sizes cannot be adequately described in terms of a single parameter for a uniform density model. 2.1.2.2.2.2. Mu-Mesic Atoms. Although this interesting method was introduced in a practical way in 1953 it has been applied to only a few nuclei for the purpose of determining nuclear radii. The method is based on the formation of transient mesic atoms when a muon approaches a nucleus of an atom. The muon cascades down the series of levels corresponding to a series of Bohr orbit terms. On a crude model, because of the small radii of its Bohr orbits, the muon falls quickly inside the K-shell electronic orbits of heavy atoms. There it undergoes the cascade of transitions mentioned above and ends up in the lowest-lying excited states of the mesic atom such as 2P and 1S states. In S states the muon will penetrate the nucleus a very large number of times before it is finally captured so that in lead, for example, a muon spends 50% of its life in an S state within the nucleus. I n many cases the muon makes, e.g., a 2P-1s transition, and the transition energy lies in the gamma ray or X-ray region for heavy or light elements respectively. Since the muon is shielded for a considerable fraction of the time by its presence in the nucleus, the energy of the gamma ray is reduced below the value it would have for a point nucleus. In the case of heavy elements the reduction factor may be as large as three. For light elements the reduction in quantum energy is very small. By measurement of the energies of the resulting gamma rays or X-rays the nuclear size may be determined approximately. The method appears to be capable, at present, of determining essentially only a single parameter for the nuclear charge distribution, and in this sense, gives less detailed information than the electron-scattering studies. The existence of mesic atoms and the above-described method were pointed out in 1949 by Wheelerb8who attempted to explain the observations of Chang.69 The method was first turned to practical application by Fitch and Rainwater.6O The detailed theory of the method has subsequently been given by many a ~ t h o r s . ~ ~ ~ ~ J. A. Wheeler, Revs. Modern Phys. 21, 133 (1949). W. Y.Chang, Revs. Modern Phys. 21, 166 (1949). 6o V. L. Fitch and J. Rainwater, Phys. Rev. 92, 789 (1953). L. N. Cooper and E. M. Henley, Phys. Rev. 92, 801 (1953). 6 2 D. L. Hill and K. W. Ford, Phys. Rev. 94, 1617 (1954). 63 E. M.Henley, Revs. Modern Phys. SO, 438 (1958). 68

s8

330

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

I n the method outlined above the gamma rays are detected by a NaI(T1) crystalad-66 which can be calibrated with known sources of gamma rays. * Figure 13 shows one of the original spectra of the transition gamma rays from lead. It is in principle possible to obtain an experimental accuracy in the determination of the energy of the line of about 1% although in heavy elements the interpretation of the observed peaks is not entirely clear-cut. It would be very desirable to repeat the early experiments with improved techniques that are now available using the large total-absorption crystals of NaI(T1). 300

v)

240

I-

z

3

8

180

-1

a

120

c W

z

60

0 3

4

6

ENERGY

- UEV

7

I

FIG.13. The spectrum of y rays obtained from the p-mesic atom of lead.60 The principal peak near 5 Mev is interpreted as a pair-production peak in NaI(T1) from the X-rays of the 2-PaI2to 1-Stransition.

The type of results that can be obtained with this method are given . ~ ~results found thus far by the method of in Table 11, due to H e n l e ~The mesic atoms are essentially in agreement with those of electron-scattering methods, although, unfortunately, there is a paucity of mesic-atom data to compare with the results of other methods. 2.1.2.2.2.3. Mirror Nuclei. The binding-energy difference of nuclear isobars provides one of the early and now well-known methods of estimating nuclear radii. 67 The binding-energy difference depends on the neutronproton mass difference, the electrostatic or Coulomb repulsive energy of the protons, and various contributions from nuclear forces. When pairs of isobars are selected the binding energy will be the same for the members of

* In this volume, see Section 2.2.3.3. R. Hofstadter, Phys. Rev. 76, 796 (1948). J. A. McIntyre and R. Hofstadter, Phys. Rev. 78, 617 (1950). 66 R. Hofstadter and J. A. McIntyre, Phys. Rev. 80, 631 (1950). 67 H. A. Bethe, Phys. Rev. 64, 436 (1938). 64

65

TABLETI

Experimental 2PU2 - 1s1/2 Element 2 energy (Mev)

Calculated T o (R = ,,A”’) to fit experiment ( X l O W 3 cm) p =

210m,

p =

Energies for R = 1.3A113X ern (Mev)

Energies for a point charge Wev)

207m,

r

A1 Ti Cu Sb Pb a

13 22 29 51 82

0.35 0.955 1.55 3.50 6.02

From Henley.63

1.17 1.21 1.22 1.17

1.09 1.17 1.17

0.484 1.393 2.432 7.707 21.33

0.363 1.046 1.829 5.833 16.41

0.363 1.044 1.823 5.763 15.86

.

1.782 2.12

5.22 10.11

0.935 1.52 3.41 5.48

0.933 1.51 3.37 5.30

2.

332

DETERMINATION OF FUNDAMENTAL QUANTITIES

the so-called "mirror-nuclei" pair if 2 2 = A & 1 or 2 2 = A f 2, etc., for example, (O1'-F1'). This is true if nuclei can be described by shellmodel rules with charge-independent nuclear forces. Present experimental data are essentially in agreement with this view. For such pairs, then, the Coulomb energy difference can be found simply after allowing for the neutron-proton mass difference. Now the Coulomb repulsive energy depends on the distribution of charge in the nucleus and will be related to its size and shape. Since precise binding-energy differences can be found experimentally from measurements of reaction energies and from the end points of j3-ray spectra this method provides an independent approach to the problem of electromagnetic sizes of nuclei. Unfortunately, nuclear theory is not far enough advanced so that the determinations of nuclear radii can be made without ambiguity. This& true because the Coulomb energies depend on the overlap and exchange integrals of nuclear-wave functions and these, in turn, depend on the nuclear model employed. Consequently mirror-nucleus estimates of nuclear radii differ from one investigator to another. I n recent years there has been a tendency for the radius values to converge to common values. A crude idea of the basis of the method can be obtained from the classical expression for the Coulomb energy of a homogeneously charged sphere : E, = Q(Z2e2/R). (2.1.2.16) The Coulomb energy difference between mirror nuclei possessing values 2, 2 1, is then

+

Ec(Z

+ 1, 2)

(2.1.2.17)

=

Because this difference includes a contribution from the self-energy of the proton Kofoed-Hansen68(see also Cherrysg) subtracts the quant,ity Ec(l,O) and thus the radius R is determined from a formula

Ec(Z

+ 1, 2)

=

O.Ge*

("",' ') ~

where the Coulomb energies are given by

+ +

E, = Emax 1.804 Mev for j3+ decay E, = Emax 0.782 Mev for K capture E c -- Emax - 0.782 Mev for 0- decay where Emax are the upper end points of the j3-ray spectra. o8 0.Kofoed-Hamen, Revs. Modern Phys. 30, 449 (1958). R. D. Cherry, Phys. Rev. 116, 1243 (1959).

(2.1.2.19)

2.1.

DETERMINATION OF CHARGE AND SIZE

333

The quantum-mechanical improvements on this formula have been developed by several but we do not have the space to discuss such calculations in this article. One must, of course, recognize that it is the outer proton which is involved most significantly in evaluating the Coulomb energy. Furthermore, several small corrections are usually ignored in obtaining radii from mirror nuclei, as pointed out in reference 68.

1.70

-

t 2.01 1.98 1.83 1.89 ELECTRON SCATTERING

s

MODEL

1.60

3

8

1.50

8

0

OTHER DATA

W

THIS EXPERIMENT

I

ELECTRON SCATTERING

E

‘2

1.40

5 Lo 1.30

1.20

A

FIG.14. This figure, taken from Wallace and Welch,” shows the rms radius const versus A and orbital of the odd nucleon. The electron-scattering results are also shown in the figure.

As a result of such considerations, a summary of the results can be 41 by R = T o A ~ / ~ givenas for mirror nuclei in the range A Gi 13 to A where

r0 = (1.28 k 0.05) X

cm.

(2.1.2.20)

A recent redetermination of many beta-ray spectra end points was made by Wallace and Welch.74A diagram summarizing their results is reproduced in Fig. 14. It may be seen that these results are in reasonably good agreement with those of Kofoed-Hansen,6sand also with the results of the electron-scattering method. B. G. Jancovici, Phys. Rev. 96, 717 (1954). B. C. Carlson and I. Talmi, Phys. Rev. 96, 436 (1954). 78 0. Kofoed-Hnnsen, Nuclear Phys. 2, 441 (1956). 79 P. C. Sood and A. E. S. Green, Nuclear Phys. 6, 274 (1958). 7 4 R. Wallace and J. A. Welch, Jr., Phys. Rev. 117, 1297 (1960).

70

71

334

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

2.1.2.2.2.4. Isotope Shift. This method is concerned with the optical shifts in the hyperfine structure observed among isotopes of a given element, that is, among nuclei differing in their number of neutrons but possessing the same number of protons. Usually the nuclei involved are among the heavy elements. An excellent summary of the basic ideas involved in the method is given by Breit7swho points out the weaknesses of the present theory of the isotope shift. It is to be noted that the isotopeshift method will not give results which depend directly on nuclear size but only on the slight changes of nuclear size between isotopes. Nevertheless, important information can be obtained from a knowledge of such differences. The quantity measured by the isotope shift can be shown to be given approximately by a quantity similar to a root mean square radius for S states but modified by a factor containing (2/137)2, viz. r2p where

p =

1 / 1 - (2/137)’.

(2.1.2.21)

Unfortunately, conclusions cannot be drawn uniquely from the information presently available and other nuclear properties are apparently strongly involved, such as the symmetry energy76and the compressibility of nuclear matter.’? Until further theoretical results are obtained and new experimental information added, it is quite probable that nuclear size effects will be inserted into the theory of this subject in order to obtain results on nuclear matter properties such as those mentioned above. Furthermore, angular shapes of nuclei enter into isotope-shift eff ects78 and offer additional material to be evaluated. Other reviews of isotopeshift studies are presented in several reference~,?~-~l but we think it is fair to say that the conclusions to be drawn from these studies should be considered to be tentative. 2.1.2.2.2.5. X - R a y Fine Structure Measurements. This method is related to the precise measurement of the X-ray fine structure splitting of the atomic 2p1/2 level of heavy elements. While the method involves the effect of the finite extension of the nuclear size on the LII(2pl12)- LIII(21)3,2) X-ray level splitting, unfortunately, other electrodynamic effects have been shown to contribute to the splitting.s2 Such effects include the anomalous magnetic moment of the electron, the polarization of the G. Breit, Revs. Modern Phys. 30, 507 (1958). A. R. Bodmer, Nuclear Phys. 9, 371 (1958/59). 77L. Wilets, I).L. Hill, and K. W. Ford, Phys. Rev. 91, 1488 (1953). 78 P. Brix and H. Kopfermann, Z . Physik 128, 344 (1949). i9 D. L. Hill, an “Handbuch der Physik-Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 89. Springer, Berlin, 1957. 80 1’. Brix and H. Kopferman, Revs. Modern Phys. 30, 517 (1958). 8 1 J. E. Mack and H. Arroe, Ann. Rev. Nuclear Sci. 8, 117 (1956). 82 A. L. Schawlow and C. H. Townes, Phys. Rev. 110, 1273 (1955). 75

76

2.1.

335

DETERMINATION O F CHARGE AND SIZE

vacuum, and radiative or Lamb-shift influences. The subject has been reviewed recently b y S h a ~ k l e t t who , ~ ~ with DuMondS4had previously made extremely precise measurements of the X-ray splitting for several heavy elements. These measurements, together with the eleetrodynamic corrections mentioned above, now give agreement between the nuclear radii measured b y electron scattering and those obtained b y the X-ray fine structure method. 2.1.2.2.2.6. Magnetic Hyperjine Structure Anomalies. I n time the distribution of nuclear magnetization may become known with sufficiently high accuracy that the nuclear size parameters associated with that distribution will be determined. The largest effect of the distribution of nuclear magnetization appears to be on its interaction with the S electron which is responsible for a hyperfine structure. In place of assuming a simple point dipole magnetic moment a distribution of dipole moment density can be considered in the hyp,erfine interaction e n e r g ~ . ~ ~Again, -~7 by comparison of the modifications caused by the magnetization distributions in neighboring isotopes the quantitative features of the distribution may be determined. Such a method is now only a t its very earliest stages of development and we must refer the reader to a review article b y Eisinger and Jaccarino88 and t o the references appearing therein. It is t o be noticed that some progress has been made in this field already in the case of the proton, where i t was found that the assumption of a point dipole is in disagreement with the experimental facts.s1 2.1.2.2.3. COMBINATION METHODS. 2.1.2.2.3.1. WeiszackerSemiempiricat Formula. A method based on the semiempirical mass equation has been used for many yearsag-91 and is still currently being studied and revised.92-94 The input data for this method are provided b y the accurately known values of atomic masses. The nuclear binding-energy curve derived from these masses may be considered to be a phenomenological function of several terms involving N , 2, and A : AE

=

-aA

+ bA2l3+ C Z ~ A -+~d/(~N - Z)2A-1 + 6.

(2.1.2.22)

R. L. Shacklett, Revs. Modern Phys. 30, 521 (1958). R. L. Shacklett and J. W. M. DuMond, Phys. Rev. 101,843 (1956). H. Kopfermann, “Kernmomente.” Akademische Verlagsges., Leipzig, 1940. 85 F. Bitter, Phys. Rev. 76, 150 (1949). 87 A. Bohr and V. F. Weisskopf, Phys. Rev. 77, 94 (1950). J. Eisinger and V . Jaccarino, Revs. Modern Phys. 30, 528 (1958). 89 C. F. von Weiszilcker, 2. Physik 96, 431 (1935). H. A. Bethe and R. F. Bacher, Revs. Modern Phys. 8, 165 (1936). 9 1 E. Feenberg, Revs. Modern Phys. 19, 239 (1947). 92 A. E. S. Green, Revs. Modern Phys. 30,569 (1958). 9 s W. D. Gunter and R. A. Hubbs, Phys. Rev. 113,252 (1959). g 4 A. G. W. Cameron, Can. J . Phys. 36, 1021 (1957). 88

84

336

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Here N , 2, and A are the neutron number, charge number, and mass number of a nucleus and a, b, c, d, are constant coefficients which can be determined. 6 is a quantity which depends on the local even-odd properties of the nucleus. 6 does not need to be taken into precise account in obtaining the general trends of nuclear binding energies. In Eq. (2.1.2.22) the first term is a volume term and represents the energy per nucleon in an infinite extent of nuclear matter at normal “nuclear” density. The second term takes into account the fact that the nucleus is actually finite and is, therefore, proportional to the nuclear surface The third term represents the repulsive Coulomb energy of the protons in the nucleus. The fourth term describes the empirical fact that the stable nuclei tend to cluster around N S 2, in the absence of other effects. It is the third term, namely the repulsive Coulomb term, varying approximately as $ e 2 / R[see Eq. (2.1.2.16)], where R is the nuclear radius of a uniform charge distribution, which contains the greatest sensitivity to nuclear size effects. The constant c in Eq. (2.1.2.22) can be determined empirically, or equivalently if R = rOA1I3, a value of ro can be determined by fitting with least error Eq. (2.1.2.22) to the known values of the binding energies of nuclei. Clearly many small corrections have been omitted in a formula of the type Eq. (2.1.2.22).I n particular, shell-model effects have been ignored, and these will certainly influence the value of 6, if not other terms in Eq. (2.1.2.22). Shell-model calculations of such effects have been made by Talmi and Thiebergera6and by Thieberger and de Shalitgs with considerable success. When all the above matters are properly taken into account it appears as if the radii from the semiempirical formula are in good agreement with the Stanford electron-scattering studies. I n other words, they yield a value of ro S 1.2 X cm. However, the Stanford results give two parameters rather than an equivalent uniform-density radius and it cannot be expected that the semiempirical method will be accurate enough or sufficiently unique to determine these parameters. However the introduction of two parameters for the nuclear density distributions in the semiempirical method is consistent with the electron-scattering studies.92g93 Much more remains to be done in this field as nuclear mass determinations improve still further, and as additional electron-scattering data are obtained. 2.1.2.2.3.2. Coherent Neutral Pion Photoproduction. For want of a more suitable place this method is included in Section 2.1.2.2 because on the p6 98

I. Talmi and R. Thieberger, Phye. Rev. 105, 718 (1956). R. Thieberger and A. de-Shalit, Phys. Rev. 108, 378 (1957).

2.1.

DETERMINATION O F CHARGE AND SIZE

337

one hand the production of pions occurs through the medium of an indirect electromagnetic interaction of a photon with nucleons and on the other hand nuclear forces surely play an important role. This method proves to be sensitive to the distribution of protons and neutrons in the nucleus, in other words, to the mass distribution. The method has been worked out by Leiss and S ~ h r a c k , ~ Schrack ’ et aLlg8and by Davidson99 and has developed out of the earlier observations of Goldwasser et ~ 2 . ’ ~ ~ on the “elastic” photoproduction of neutral pions from He4. The term “elastic ” has the connotation in this experiment that the target nucleus recoils as a whole, hopefully in its ground state. Under such conditions an incident photon strikes the nucleus, a T O meson is produced and the nucleus recoils (‘coherently” to conserve energy and momentum. The threshold for a carbon nucleus for such an elastic reaction is 135.6 MeV. If the production takes place from a single neutron or proton in the nucleus, which would correspond to the so-called “incoherent production,” the threshold would be 152 MeV. By using relatively low y-ray energies, near threshold, it would be expected that the production process is coherent. Now the ?yo meson cannot be detected directly since it decays very quickly. Instead its presence, and, in fact, its direction also, are deduced from the pair of gamma rays into which it disintegrates. By using a system of two telescopes including an angle of about 120-150”, the nomesons may be successfully detected. The mid-line of the counter telescope, corresponding to an average T O direction, is rotated about an axis perpendicular to the incident gamma-ray beam and gives the approximate angular distribution of the produced neutral pions. Born-approximation predictions of the production cross sections can be made by utilizing the known neutral pion production photo-production cross section on a single proton. In this case the coherent production would then be expected to be proportional to A*. Experimentally a 1.85 power low has been observed in place of 2.00. The assumptions about the process, therefore, seem to be borne out, at least approximately. Both angular distributions and yield curves as a function of peak bremsstrahlung energies are observed. Figure 15 shows angular distributions observed by Schrack et aLlgBwhile Fig. 16 gives the results of these authors for mass-density distribution corresponding to those found previously by electron-scattering methods. The agreement is impressive, 87

9*

J. E. Leks and R. A. Schrack, Revs. Modem Phys. 30, 456 (1958). R. A. Schrack, S. Penner, and J. E. Leiss, Nuovo cimento (to be published).

G. Davidson, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, September, 1959. looE. L. Goldwasser, L. J. Koester, Jr., and F. E. Mills, Phys. Rev. 96, 1692 (1’954).

338

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

cos e

FIG.15. The angular distributions of coherent neutral-meson production for various elements. The solid curves are Born-approximation calculations using the electronscattering parameters (Schrack et aZ.99.

except possibly for aluminum. This method would seem to hold considerable promise in obtaining further interesting results. 2.1.2.2.4. ANGULAR SHAPES OF NUCLEI.This most interesting subject, has been reviewed recently by Temmerlol and Hill.79 I n the static ground state of a nucleus only the even electric-quadrupole (octupole, etc.) G. M. Temmer, Revs. Modern Phys. 30, 498 (1958).

2.1.

DETERMINATION O F CHARGE AND SIZE

339

distortions can exist. The simplest of these types is a n ellipsoid which corresponds to the quadrupole type with axial symmetry. Most nuclei which show distortion have this shape, a t least approximately. This fact has been discovered through the careful spectroscopic observations of Schiiler and Schmidt102on the europium nuclei. In such cases the hyperfine splitting is affected in a specific way demanding this conclusion. 1000

I

I111111~

I

1

I I I I I ' J

AI.85

+5%

Pb

d

FIG. 16. The observed cross section for neutral-pion production divided by a theoretical form factor. The half-density radius corresponding t o T O = 1.07 is used in all cases. Except for A1 the agreement between this method and the electron-scattering method is highly satisfactory (Schrack et d9*).

More recently microwave spectroscopy, atomic-beam spectroscopy, and paramagnetic resonance techniques confirm such conclusions by determining the values of the associated quadrupole moments. As we have pointed out earlier, the measurements of isotope shiftTs also demonstrated the existence of quadrupole distortion of nuclei. Just a few years ago good observations of low-lying nuclear rotation levels by Coulomb excitation methods led t o new determinations of quadrupole moments of various nuclei and also to the determinations of lo*

H. Schtiler and T. Schmidt, Z. Physik 94, 457 (1935).

340

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

the nuclear moments of inertia. As is well known, these observations have led to an understanding of many properties of nuclei related to the socalled "collective" motions. Quite specific nuclear models have been suggested by these results and all such models show large departures from sphericity for nuclei lying far from magic-number nuclei. The magicnumber nuclei are known to be quite spherical. Electron-scattering studies of nuclei such as Ta, W, and U show that the characteristic diffraction features observed for spherical nuclei are absent

AT-

180 MEV

I 03

IOP

10'

-

loo

I lo'

20°

I I I Y

40'

60'

00'

100"

120'

SCATTERING ANGLE

FIG. 17. This figure shows the effect of the quadrupole deformation on the electron scattering from heavy nuclei a t 180 Mev.66 The cross section was divided by the Matt scattering in order t o emphasize the differences in diffraction features between the curves for Pb, Bi, and Au and the other nuclei. Pb, Bi, and Au are known to be spherical nuclei while Ta, W, and U are known to be strongly deformed nuclei.

for the above distorted n ~ c l e i . ~This 6 is shown clearly in Fig. 17. From such observations it can be concluded that the above nuclei show angular departures from sphericity.56~101 Temmer'O' has also reported the spectacular results of Weiss, Petree, and Fuller (NBS) who have found a splitting of the giant dipole resonance in Ta and not in Au. According to Danos108 these results are to be interpreted in terms of giant dipole lines associated with the different axes of a distorted nucleus. In Au, which is nearly spherical, this splitting does not occur, Splittings of the muon capture gamma-ray spectra should also lead to loa

M. Danos, Nuclear Phys. 6, 23 (1958).

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

341

information about quadrupole moments and, therefore, angular shapes of nuclei, but such experimental splittings have not yet been ACKNOWLEDGMENTS

I wish to thank Drs. R. A. Schrack, S. Penner, J. E. Leiss, J. A. McIntyre, T. L. Watts, and F. C. Jobes, Jr., for informing me of their experimental work before publication. I also wish t o thank Mr. F. W. Bunker and Mrs. M. Lochner for helping with the manuscript. This work was supported in part by the Office of Naval Research and t h e U.S. Atomic Energy Commission, and by the U.S. Air Force, through the Office of Scientific Research of the Air Research and Development Command.

2.2. Determination of Momentum and Energy 2.2.1. Charged Particles

2.2.1 .l.Measurement of Momentum. Electric and Magnetic Analysis.*? 2.2.1.1.1. INTRODUCTION. There exist a variety of possibilities to measure the momenta of charged particles. The most common and precise methods are based on the interaction between the charge of the particle and external magnetic or electric fields. The magnetic or electric force will curve the trajectories. The radius of curvature is determined by the momentum or the energy and of the strength of the ap'plied field. By making use of a suitable system of shutters an instrument can be designed that will transmit to the detector only particles of a given momentum (or energy). Such momentum or energy selective instruments are known as spectrometers or analyzers. For relativistic particles the magnetic field is the most suitable choice. Electrons approach relativistic velocities already at relatively moderate energies. A magnetic field of only 1000 gauss will give the same radius of curvature as does an electrostatic field of 300,000 volts em-'. These high electrostatic fields are difficult to stabilize and keep under perfect control. Furthermore the particle trajectories in the magnetic spectrometers are relativistically invariant. Assume a particle of charge e moving with the velocity v in a plane perpendicular to the uniform magnetic field B. The equation of motion L. Wilets, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 29, No. 3 (1954). B. A. Jacobsohn, Phys. Rev. 96, 1637 (1954). See also Vol. 4, A, Chapters 3.3 and 3.4.

104

Io6

t * Section 2.2.1.1 is by T. R.

Gerholrn.

342

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

then becomes mv2/p = evB

(2.2.1.1.1)

and Bp

1

= -mu =

e

1 -p. e

(2.2.1.1.2)

This means that for a given radius of curvature p the momentum of the particle is proportional to the magnetic field. The momenta of charged particles can therefore conveniently be expressed in terms of their “ B p values.” From the Bp value one can easily calculate the energy according to the relation

where Z e is the charge of the particle and mo is the rest mass. Extensive tabulations of this formula have been published. l-za In the electrostatic case we have to replace the Lorentz force ev X B with the force eE. Assume for simplicity an electric field of spherical symmetry, obtained by means of two conducting spheres of radii R , and Rz with a potential difference V . Let p be the mean radius of curvature p =

+ Rz).

+(RI

The equation of motion then becomes

“Rl

.-

RlR2 1 mu2 -- R Z p2 P

(2.2.1.1.4)

and (2.2.1.1.5) As seen in (2.2.1.1.5) the electrostatic instrument is energy rather than momentum selective. However, contrary to the magnetic case (2.2.1.1.1) and (2.2.1.1.2) the electrostatic equations are not relativistically invariant. For velocities comparable to c (2.2.1.1.4)and (2.2.1.1.5) have to be replaced with more complicated-and energy-dependent-expressions. Physically this means that the particle trajectories in an electrostatic spectrometer varies with the energy of the focussed particles. At higher energies it therefore becomes necessary to readjust the baffles if optimum conditions of operation should be maintained. 1 T. R. Gerholm, in “Handbuch der Physik-Encyclopedia of Physics” (S. Flugge, ed.), Vol. 33, Appendix, pp. 678-684. Springer, Berlin, 1956. * A. H. Wapstra, G. J. Nijgh, and R. van Lieshout, “Nuclear Spectroscopy Tables,” pp. 31-36. North Holland Publishing, Amsterdam, 1959. g*L. Marton, C. Marton, and W. G. Hall, Electron Physics Tables. Natl. Bur. Standards Circ. No. 671 (1956).

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

343

For these reasons electrostatic instruments have not found much use hitherto. However, for heavy particles of moderate energies and for lowenergy electrons electrostatic spectrometers may have some advantages. A recent analysis3 shows that under these conditions electrostatic spectrometers compete favourably with the best magnetic instruments. A few electrostatic spectrometers are described in the An extensive analysis of the focusing properties has been published by Rogers6 in a series of papers. So far little use has been made of combined electrostatic and magnetic fields. However, it has been shown recently by Bergkvistg that the figures of merit of the double focusing instrument can be substantially improved by means of suitable electric fields. 2.2.1.1.2. PARAMETERS DESCRIBING THE PERFORMANCE OF A MAGNETIC SPECTROMETER. I n the following we will consider mainly beta-ray spectrometers. The general arguments, however, apply also to the various magnetic spectrometers designed to determine the momenta of charged particles from nuclear reactions. The resolving power determines the capability of the instrument to separate particles of different momenta, i.e., to resolve closely spaced “lines” in the spectrum. Obviously the resolving power also determines the degree of accuracy that can be achieved in the determination of the Bp values and corresponding energies. For precise absolute measurements the instrument has to be calibrated with a corresponding degree of accuracy. For beta-ray spectrometers a suitable set of reference lines has ) . charged been established (cf. tables in Gerholm’ and Nijgh et ~ 1 . ~ For particle spectrometers there exist a number of nuclear reactions with wellknown Q values. These may be used for calibration purposes. By analogy to the nomenclature in optical spectroscopy the resolving power R is defined as (2.2.1.1.6)

3 R. H. Ritchie, J. S. Cheka, and R. D. Birkhoff, Nuclear Znstr. 6, 157 (1960); 8, 313 (1960). 4 S. K. AlIison and H. Casson, Phys. Rev. SO, 880 (1953). 6B. V. Thosar, Thesis, University of Birmingham, England, 1949; M. C. Joshi and B. V. Thosar, Proc. Indian Acad. Sci. A38, 367 (1953). 6 C. P. Browne, D. S. Craig, and R. M. Williamson, Rev. Sci. Tnstr. 22, 952 (1951). 7 Y. Kobayashi, J . Phys. Soc. Japan 8, 135, 440, 648 (1953). 8 F. T. Rogers, Jr., Rev. Sci. Znstr. 8, 22 (1937); 11, 19 (1940); F. T. Rogers, Jr., and C. W. Horton, ibid. 14, 216 (1943); F. T. Rogers, Jr., Phys. Rev. 69, 537 (1946); Rev. Sci. Znstr. 22, 723 (1951). 9 K. E. Bergkvist, to be published.

344

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

where (Bp)‘ is the momentum and A(Bp)’ is the width at half-maximum of the line profile when plotted in an intensity versus Bp diagram. It is assumed that the particles are monoenergetic. In nuclear spectroscopy, however, it has become customary to refer to the quantity resolution q rather than to the resolving power R. The resolution is defined as the inverse of R (2.2.1.1.7)

It is usually expressed as a percentage. Closely related to q is the quantity base spread qo. The concept base spread is frequently used in theoretical considerations of the focusing properties of magnetic fields. The base spread is defined as (2.2.1.1.8)

where Ao(Bp)’ is the base width of the line in the intensity versus momentum diagram. q o is also expressed as a percentage. An exact relation between q oand q can be derived only after taking into account the theoretical line profile. Such an analysis has been performed only in a few special cases. As a general rule, however, one may take q o to be about two times q if the instrument is adjusted to optimum conditions of operation. It is important to notice that q and qo are constants independent of the absolute value of (Bp),i.e., the momentum of the focused particles. However, this statement holds only as long as the trajectories of the particles transmitted through the instrument remain constant and independent of the actual magnetic field. These conditions are generally not exactly fulfilled in particular if iron is used to shape the magnetic field. At lower fields remanence effects distort the field distribution and a t higher fields saturation effects appear. Iron-free spectrometers are more reliable but residual fields such as the earth magnetic field may interfere and at higher field settings temperature effects in the magnetic coils may give rise to deviations from the desired field distribution. However, q and qo and in fact the whole line profile will be constant as long as the magnetic field in each point is strictly proportional to the current. Therefore, according to (2.2.1.1.7) the momentum band pass is proportional to (Bp)’. If we wish to determine the true intensity distribution, i.e., intensity per unit momentum interval I, we must plot (2.2.1.1.9)

versus (Bp)‘.N(Bp)’ is the counting rate corresponding to the momentum

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

345

(Bp)'. This should be borne in mind if one wants to study a continous spectrum such as the beta spectrum or if one determines relative intensities from the areas under the line profiles, rather than from the peak counting rates. The latter are of course directly comparable as long as the finite widths of the lines can be neglected, which is generally not the case due for instance, to absorption in the source. 0.9

/I

a5 -LA. '2

'2 i. t. ,'15. . .p. ,' , ?. . .j, . . . . ,10xi03 , . , . 'Bp

I . .

2

1

0

M cV

E-

FIQ. 1.

>/Yr AB BP

momentum resolution to energy resolution versus Bp and E .

I n this connection it may be worth while to call the attention to the relation between energy resolution and m o m e n t u m resolution. Diff erentiating (2.2.1.1.3)we obtain

-dE_ - E + 2Eo d(Bp) -. E

+

+

E

+ Eo

Bp

(2.2.1.1.10)

In Fig. 1 E' E o / E 2Eo is plotted versus Bp. For nonrelativistic particles the energy resolving power is one-half of the momentum resolving power. The relation approaches unity with increasing energy. Another important parameter describing the performance of a spectrometer is the dispersion y defined as (2.2.1.1.11)

where x is the appropriate coordinate specifying the position of the image of the focused particles. Obviously closely spaced spectral lines can only be resolved if the geometrical distance is greater than the widths of the two adjacent lines. The efficiency of the instrument is determined by the spectrometer transmission T which is the effectivesolid angle expressed as a percentage of 4n.

346

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

The effective solid angle T should not be confused with the entrance solid angle or gathering power w . If we assume that the particles from the source have no preference direction of emission (isotropic emission), the gathering power can be interpreted as the probability that a particle emitted from the source will be accepted by the spectrometer entrance baffles. Correspondingly the transmission is the probability that a particle emitted from the source will be transmitted through the exit baffles onto the detector. In order to get the probability that the particle is actually detected we have to multiply T with the detector efficiency. The latter is generally 100 % for high-energy particles, but for low-energy particles the counter efficiency will be much less in particular if the counter window is thick or if a scintillation detector is used. Obviously

T 2

w.

(2.2.1.1.12)

The exact relation between T and w depends on the conditions of operation. For optimum conditions of operation w is roughly two times T. Just as the resolution 9 is constant the transmission is also a constant as long as the magnetic field in each point is proportional to the current. When the instrument is to be used for accurate relative intensity measurements the validity of this basic assumption should be checked. The transmission is the appropriate figure of merit as long as the finite extension of the source can be neglected (point source). However, in practice this is generally not the case. For example in beta-ray spectroscopy one is often limited not by the total activity of the source but by its speciJic activity, i.e., the total activity divided by the weight. Sources have necessarily to be made very thin in order to avoid distortion of the lines due to absorption in the source itself. If the specific activity is low the source has to be spread out over a comparatively large surface. Under these conditions the luminosity L rather than the transmission is the appropriate figure of merit. L is defined as

L

=

uT

(2.2.1.1.13)

where u is the area of the source generally expressed in cm2. Obviously L determines the capability of the instrument to deal with sources of finite dimensions. By analogy to (2.2.1.1.12) one can also introduce an aperture luminosity defined as A = uw. (2.2.1.1.14)

It should be noticed that contrary to the transmission T the luminosity L depends on the geometrical dimensions of the spectrometer. The source area u is to be compared with the dimensions of the instrument itself. For

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

347

a fair comparison between different focusing principles all instrument should be reduced to unit dimensions. Such a comparison, however, necessarily involves a considerable degree of arbitrariness. The volume of the magnetic field, the source t o detector distance, or the mean radius of curvature may all be considered appropriate figures for the comparison of the relative sizes of different instruments. It is generally not realized that whenever the luminosity is decisive a small transmission can be more than compensated for if the focusing principle allows for large source areas. 2.2.1.1.3. DIFFERENT TYPES OF MAGNETIC SPECTROMETERS. The magnetic spectrometers can be divided into two different groups: flat spectrometers and lens spectrometers. In the flat spectrometers the central trajectory is confined to a plane perpendicular to the magnetic field, while in the lens spectrometers the trajectory spirals along the field lines. The prototype of the flat instruments is the semicircular spectrometer (and spectrograph). To this family belongs also the third-order focusing spectrometer, the double focusing spectrometer, and various sector field spectrometers, among these the orange spectrometer, and the spiral orbit spectrometer. The uniform field solenoidal spectrometer is the prototype of the second group. Other members of this family are the short lens and the long lens spectrometers, the intermediate image spectrometer and the triangular field spectrometer. For all magnetic spectrometers there exists a certain relationship between the resolving power on one hand and the transmission and the luminosity on the other. An increased resolving power can only be obtained at the expense of reduced transmission and luminosity. Obviously the aim is to obtain the highest possible transmission and/or luminosity for a given resolving power. A direct comparison between the various types of instruments may be based on such “figures of merit.” The “figures of merit,” however, should be taken only as a rough guide. As already pointed out for a comparison of the luminosity of two different instruments adjusted to the same resolving power the two spectrometers should be reduced to “unit size” which involves a certain degree of arbitrariness. Furthermore there are several instrumental properties that are not taken into account in these “figures of merit.” For instance it is in practice not always possible to prepare the source in the most suitable size and form. If one wants to study very weak lines in the presence of much stronger transitions or of an intense gamma-ray background, the “signal-to-noise” ratio is an important measure of the performance of the instrument. The “signal-to-noise’’ ratio depends among other

348

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

factors on the size of the counter, the possibilities to arrange an effective shielding against gamma rays from the source and of the discrimination against scattered radiation. There is a wide range of problems studied in nuclear spectroscopy and the required experimental techniques vary considerably from one case to another. It is not possible to single out one particular type of instrument which quite generally can be said to have superior qualities. Different problems have to be met with different techniques. Perhaps one may say that the flat spectrometers such as the semicircular and double focusing spectrometers are to be preferred for high P

P'

S

FIQ.2. Direct deflection method.

resohition and low transmission experiments, such as precise energy measurements. Lens spectrometers are most suitable in the high transmission and moderate resolution range. They are preferrable for instance for coincidence experiments. The orange spectrometer, however, is an example of a flat spectrometer, with moderate resolution but very high transmission. And lens spectrometers have been used also for very accurate energy measurements. lo 2.2.1.1.3.1. T h e Semicircular Focusing Principle. As shown in Fig. 2 and Fig. 3 the semicircular focusing principle is a straightforward extension of the direct deflection method. The direct deflection method was used already in 1910 by von Baeyer and Hahn" to study beta rays from radioactive samples. In 1914 Danysz12realized that for simple geometrical reasons there will be a focusing of the electrons after 180".Electrons of 10 H. Craig and C. F. Dietrich, Proc. Phys. SOC. (London)B66, 201 (1953); H. Craig, Phys. Rev. 86, 688 (1952). l1 0. von Baeyer and 0. Hahn, Phys. Z. 11, 488 (1910); 0. von Baeyer, 0. Hahn, and L. Meitner, Phys. 2. 12,273, 378 (1911); 13,264 (1912). 12 J. Danysz, Compt. rend. 163, 339, 1066 (1911); Radium 9, 1 (1912); 10, 4 (1913).

2.2.

DETERMINATION O F MOMENTUM A N D E N E R G Y

349

different momenta will be focussed a t different points, determined b y their Bp values. B is the strength of the field and 2 p the source to image distance. If a photographic plate is used for detector the instrument will record a wide range of the spectrum simultaneously. Such a n instrument is called a spectrograph. When carefully designed semicircular spectrographs are capable of the highest degree of resolution. See Fig. 4. T h e inherent low transmission (and luminosity) is to a certain extent compensated for by the “multichannel” capacity. The main limitation is due to the inaccuracy in intensity measurements since the relative intensities can only be roughly estimated from the blackening of the photographic plate. The semicircular spectrograph is very useful, however, as a survey instrument in particular for the study of very complex spectra.lZa

FIG.3. Semicircular focusing principle.

If the photographic plate is replaced by a counter placed at a certain distance 2 p o from the source the instrument will record only particles with a momentum corresponding to Bpo. B y varying B, particles of different momenta will be recorded. The spectrum is obtained b y measuring the counting rate as a function of B. Such a “single channel” instrument is called a spectrometer. The first beta-ray spectrometer was built already in 1914 by Chadwick.13 With the aid of this spectrometer Chadwick was able to record the focused electrons individually. This led him t o the discovery of the continuous beta spectrum. With minor modifications the semicircular spectrometer is still frequently used in nuclear spectroscopy. As shown in Fig. 5(a) and Fig. 5(b) the focusing properties are not perfect. There are several contributions to the width of the image, i.e., to the finite width of the line in the intenH. Slatis, Nuclear Znstr. 2, 332 (1958). 13

J. Chadwick, Verhandl.deut. physik. Ges, 16, 383 (1914),

2.

350

DETERMINATION OF FUNDAMENTAL QUANTITIES

N

I

M -.

f’

c-C’

FIG. 4. Semicircular spectrograph. Vertical section through the spectrograph chamber. T4e source rings are placed in two holes in a slide D, which is operated from outside by means of a precision screw E. The wire is seen i n the microscope F against a hair cross G which is fked i n the vacuum chamber. The light from the lamp H passes the source position and t h e slit I and is then deflected by the mirror K into the microscope. The film is placed i n a casset which is put in from the end L. In front of it is a shutter system consisting of two “doors” M also operated from outside by knobs N. [E. Karlsson and K. Siegbahn, Nuclear Instr. 7, 113 (1960).]

2.2. DETERMINATION

OF MOMENTUM AND ENERGY

351

sity versus Bp spectrum. Obviously these focusing errors determine the resolving power of the instrument. Let 2p be the angle between the two extreme rays in the (zy) plane (cf. Fig. 5), and let $ be the angle between the two extreme rays in the

X

ZP

FIG.5. Semicircular focusing.

(yz) plane. The angle cp is determined by the width and position of the entrance slit and $ is given by the relation

+ = -h

(2.2.1.1.15)

=P

where p is the mean radius of curvature. For symmetry reasons the height of the source h should be equal to the height of the entrance and exit slit openings. If we assume that the particles emitted from the source are monoenergetic we obtain for the base width of the image Aop = 2p(l -

C O S ~ COS+) ~. = p(p2

+')

(2.2.1.1.16)

+

since cp and are small angles. Taking into account also the width of the source s we have to add a third term AOP =

4- p(9' 4- $').

(2.2.1.1.17)

352

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

This expression represents the actual width of the line when it is recorded on a photographic plate. However, there will be a further broadening of the line when the position of the line on the photographic plate is determined by means of a microphotometer. This additional broadening is due to the finite width d of the defining slit. I n a spectrometer, d corresponds to the width of the exit slit. Thus we obtain for the effective base width (Aop)eff= d s p(p2 1L.'). (2.2.1.1.18)

+ +

+

From this expression we can determine the base spread use of the dispersion dx - d x dp = 2 Y=d ( B p ) - & ' d(Bp) B

00

by making (2.2.1.1.19)

and the following relations

From (2.2.1.1.18) and (2.2.1.1.20) we get d + s

+-v 22+ P

= -

00

2P

(2.2.1.1.21)

and finally we get for the aperture luminosity A

=

shw

= &~pp#~.

(2.2.1.1.22)

The mean radius of curvature p is limited by the geometrical dimensions of the magnetic field. The problem is then to adjust the four instrumental parameters s, d, 'p, and 1L. t o optimum conditions of operation, i.e., t o maximize the luminosity for a given resolution. We can certainly not be far away from optimum conditions if all four contributions to the base width in (2.2.1.1.22) are made equal, i.e.,

d

= s = pp2 = plL2.

(2.2.1.1.23)

For an instrument adjusted in this way we obtain qo =

(2.2.1.1.24)

2v2

w = - 00

(2.2.1.1.25)

4*

and A=---

&-

P2q06'2.

(2.2.1.1.26)

I n practice, however, we are generally more interested in resolution, transmission, and luminosity rather than base spread, gathering power,

2.2.

353

DETERMINATION OF MOMENTUM AND ENERGY

and aperture luminosity. I n order to derive these expressions from the formulas above we have to take into account the theoretical line profile. This will not be done here. However, the following simple assumptions are at least approximately correct: q = +qo, T = +w, and L = +A. We get from these expressions T 0.08q (2.2.1.1.27) L 0.25~~q~1~. (2.2.1.1.28)

--

Geoffriont4has made a more careful analysis of the problem and found that there exist more favorable conditions of operation than those of (2.2.1.1.23), namely d = S = 2pcp2 (2.2.1.1.29) and b! = cp2/2. (2.2.1.1.30) This choice of parameters gives a somewhat higher luminosity a t a given resolution. According to Geoffrion14one has under these conditions

L

=

0.456~~q~1~.

(2.2.1.1.31)

The increased luminosity is due t o an increase in the height of the source with respect to d, s, and cp. G e o f f r i ~ n lhas ~ ~ built an instrument of this “high source” type. The experimental results are in good agreement with the theoretical predictions. 2.2.1.1.3.2. T h e Third-order Focusing Spectrometer. One of the four contributions to the finite width of the image obtained in the semicircular instrument can be eliminated by means of a suitably shaped inhomogeneous field, namely, the spherical aberration p(p2. This was first realized by Korsunsky and co-workers16 who arrived a t the appropriate field form by a stepwise ray-tracing technique. The principle is illustrated in Fig. 6. I n the shaped field the extreme rays R1 and R Zwill intercept a t the focus P because R1 and Rz pass through weaker magnetic fields. A detailed theory has been published by Beiduk and Konopinski. l 6 They showed that for a field of the form

+ & (y)6 + . . .).

(2.2.1.1.32)

C. Geoffrion, Rev. Sci.Instr. 20, 638 (1949). C. Geoffrion, private communication (1954). 16 M. Korsunsky, V. M. Kelman, and B. Petrov, J . Phys. USSR 9, 7 (1945). 16 F. M. Beiduk and E. J. Konopinski, Rev. Sci. In&. 19, 594 (1948). 14

148

354

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

The spherical aberration will be of the order pq4. One therefore obtains a third-order focusing of the trajectories in the median plane. In (2.2.1.1.32) B,(p) is the z component of the field in the median plane at a distance p from the axis of symmetry and po corresponds to the radius of curvature of the central, semicircular trajectory. A couple of beta-ray spectrometers utilizing this focusing principle have been designed. 17*18 However, the Beiduk-Konopinski field has found its by far most important application in large scale electromagnetic isotope separators (“calutrons”).

a a A

P‘ P

FIG.6. Third-order focusing principle.

2.2.1.1.3.3. The Double Focusing Spectrometer. Let us consider a magnetic field which is of rotational symmetry and also symmetric with respect to a median plan perpendicular to the symmetry axis. Such a field can be expressed as follows N P ) = &o)

(1

+

PO

+p

( y y

+6

(

!

3

3

+

.

,

,]

(2.2.1.1.33)

where B(p) is the strength of the field in the median plane a t a distance p from the axis of symmetry and po corresponds to the mean radius of curvature. By this definition the properties of the field can be described 1’

L. M. Langer and C. S. Cook, Rev. Sci. In&. 19, 257 (1948).

** J. A. Bruner and F. R. Scott, Rev. Sci. Instr. 21, 545 (1950).

2.2.

355

DETERMINATION O F MOMENTUM A N D E N E R G Y

in terms of the parameters a, p, and 6. As discussed above the particular choice a = 0, p = 2, and 6 = +$ gives a perfect third-order focusing. I n the semicircular spectrometer (a = p = 6 = 0) there is no restoring force along the x axis [cf. Fig. 5(b)]. A particle emitted a t a n angle with respect t o the plane perpendicular to the lines of force will spiral away from this plane. The projection of the trajectory in the (zy) plane corresponds t o a smaller radius of curvature. For this reason the aperture angle has t o be kept very small in order to avoid a broadening of the line and a corresponding reduction in the resolving power. However, i t is easily seen that if the field decreases with increasing radius there will be a force which tends to bring the particles back to the median plane. This is precisely the same effect that is used t o obtain axial focusing in for instance betatrons or synchrocyclotrons. This axial focusing implies that there will be an axial focus-at a n angle &and a radial focus-at an angle #+. I n general these two focii will not coincide. However, it was shown by Svartholm and Siegbahn,Ig that for a particular field form, namely

+

+

the following relation holds (2.2.1.1.35) i.e., the two focii will coincide after a deflection angle of 255.6'. The theory shows that this double focusing effect will be obtained only if a = -+, while j3 may be chosen within certain limits. The most favorable since only in choice for the parameter p, however, is definitely /3 = this case the double focusing conditions will be realized for all values of p. It may be noticed that the field distribution (2.2.1.1.34) corresponds to field which decreases as p-*l2. From the general theory of the double focusing spectrometer one obtains in analogy with the semicircular case

+;

Formula (2.2.1.1.36) gives the base spread as a function of the instrumental parameters s, d, 'p, +, and h, and of the field parameter p. It should be observed that for p = $jthe third term vanishes, i.e., we will have a higher order focusing effect in the axial direction. Correspondingly, the 19 N. Svartholm and K. Siegbahn, Arkiv Mat. Astrcrn. Fys. A33, No. 21 (1946); K. Siegbahn and N. Svartholm, Nature 167, 872 (1946).

356

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

+

choice 0 = gives a higher order focusing in the radial direction. This is of importance for the design of the shutters. In order to improve the resolution-at the expense of transmission-we should reduce the cp aperture angle only (but not the $ aperture angle) for /3 = $ and vice versa for fi = $.

A

FIG.7. Double focusing principle.

Comparing formulas (2.2.1.1.21) and (2.2.1.1.36) we notice another important advantage of the double focusing spectrometer: the dispersion is twice as high as in a corresponding semicircular instrument. Due t o the double focusing action and the doubled dispersion one might expect a, considerable gain in luminosity. In order t o derive a n explicit formula for the luminosity one has to find the optimum conditions of operation. This problem has been studied by Lee-Whiting and Taylor.'g8 If one makes all four contributions to the base spread equal one obtains (for /3 = $) (2.2.1.1.37) lga G. E. Lee-Whiting and E. A. Taylor, Cun. J . Phys. 85, 1 (1957); see also, Chalk River Report, CRT-668 (October, 1956).

2.2.

357

DETERMINATION OF MOMENTUM A N D E N E R G Y

Under these conditions 770

=

bv2

(2.2.1.1.38)

and (2.2.1.1.39) Assuming, arbitrarily, a special case where the aperture is square (2.2.1.1.40) For the luminosity we obtain (2.2.1.1.41) and (2.2.1.1.42) Finally, making use of the same approximate relation as before ( q T = +w, and L = +A) we get T,, L.,

--

0.24q 1.9p0~7~’~.

=

h-7,

(2.2.1.1.43) (2.2.1.1.44)

Comparing (2.2.1.1.28) and (2.2.1.1.44) we see that the luminosity has been increased by almost an order of magnitude. This substantial gain in luminosity can be used either to increase the intensity at a given resolution or to increase the resolution for a given luminosity simply by using a larger instrument. For this reason the double focusing spectrometers are frequently used for precision measurements. In fact the figures of merit for the double focusing spectrometer will be even higher than formulas (2.2.1.1.43) and (2.2.1.1.44) indicate since, as already pointed out, the focusing in the axial direction (for p = 9) is of higher order. It is therefore not necessary to reduce the # aperture in the same proportion as the p aperture is reduced in order to obtain higher resolution, i.e., it is preferable to operate the instrument with a rectangular rather than, as assumed in the derivation of (2.2.1.1.43) and (2.2.1,1.44), a quadratic aperture. Empirically one has found that under optimum conditions of operation the following relation holds: T = q . The required field distribution can be obtained in different ways. An iron yoke-with parabolic polepieces for @ = +can be used to shape the field (cf. Fig. 8). Several iron yoke spectrometers have been constructed

358

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

by Hedgran et by Bartlett and Bainbridge,21 and others.22t23For highest precision, however, the use of iron has certain disadvantages. Due to hysteresis effects the desired field distribution will not be perfectly realized over the entire volume for all field strengths. This gives rise to small deviations from perfect focusing and will ultimately limit the resolution. Furthermore it becomes necessary to measure the fieldgenerally by means of a rotating coil system for each magnetic field

!

FIG.8. Alternative designs for iron yokes for the double focusing spectrometer.

setting. Alternatively one can calibrate the instrument with suitable reference lines in which case the accuracy is limited by the degree of accuracy available for the calibration lines. This limitation is perhaps not too serious since there exist nowadays a considerable number of precisely determined reference lines. However, the double focusing field can also be obtained in an iron-free spectrometer. Such an instrument was first designed by Siegbahn and EdvarsonZ4(cf. Fig. 9). Recently several iron-free spectrometers have been c o n s t r ~ c t e d . ~ ~ - ~ ~ 2o A. Hedgran, K. Siegbahn, and N. Svartholm, Proc. Phys. Soc. (London) A63, 960 (1950). A. A. Bartlett and K. T. Bainbridge, Rev. Sci. Znstr. 22, 517 (1951). 22 F. Shull and D. Dennison, Phys. Rev. 74, 917 (1948). 23 F. N. D. Kurie, J. S. Osoba, and L. Slack, Rev. Sci. Instr. 19, 771 (1948). 2 4 K. Siegbahn and K. Edvarson, Nuclear Phys. 1, 137 (1956). 2 6 R. L. Graham, G. T. Ewan, and J. S. Geiger, Nuclear Znstr. 9, 245 (1960). M. Mladjenovic, in “Proceedings of the Rehovoth Conference on Nuclear Structure” (H. J. Lipkin, ed.), p. 537. North Holland Publishing, Amsterdam, 1958. 27 C. de Vries and A. H. Wapstra, Nuclear Znstr. 8, 121 (1960).

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

359

The iron-free double focusing spectrometer is capable of a very high resolving power. Energies of conversion electrons have been measured which is quite comwith an accuracy of the order of a few parts in 105,24,*8 parable to the degree of precision obtainable in X-ray spectroscopy. An

FIG. 9. Iron-free double focusing spectrometer due to Siegbahn and Edvarson (cf. reference 24).

interesting and important application of precision beta-ray spectroscopy is the measurement of the absolute binding energies of atomic electrons by the photoelectric m e t h ~ d . ~ ~ - ~ l 2.2.1.1.3.4. Prism and Sector Field Spectrometers. The focusing prin28 G. E. Lee-Whiting and E. A. Taylor, Can. J . Phys. 36, 1 (1957); see also, Chalk River Report, CRT-668 (October, 1956). 29 E. Sokolowski, C. Nordling, and K. Siegbahn, Arkiv Fysik 12, 301 (1957). 30 E. Sokolowski, Arkiv Fysik 16, 1 (1959). 31 C . Nordling, Arkiv Fysik 16, 397 (1959).

360

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

ciple is schematically illustrated in Fig. 10. The source and the detector are both located outside the magnetic field which is perpendicular to the plane of the paper. There exist a variety of different designs. The magnetic field can be homogeneous32~~3 or inhomogeneous.34-37 The profile of the pole pieces can be shaped in order to reduce the effect of the fringing field or contrary one can utilize the focusing properties of the fringing field to obtain twodirectional focusing. Several sector field spectrometers (and spectro-

SOURCE

FIQ.10. Sector field focusing principle.

graphs) have been described.38-42 Due to the large distance between the source and the detector the dispersion is particularly high. This implies that the instrument is capable of high r e s o l ~ t i o nThe . ~ ~ transmission and the luminosity however, is considerably lower than in a semicircular or double focusing spectrometer adjusted to the same resolution. This drawback can be compensated for by making use of short lens spectrometers introduced between the source and the magnet and between the magnet and the detector. The lens spectrometer acts as collimators. Kelman M. Camac, Rev. Sci. Instr. 22, 197 (1951). W. G. Cross, Rev. Sci. Instr. 22, 717 (1951). 8 4 N. Svartholm, Arkiv Fysik 2, 115 (1950). 3 6 D. L.’Judd, Rev. Sci.Instr. 21, 213 (1950). 8 E E. S.[Rosenblum, Rev. Sci. Instr. 21, 586 (1950). 37 H. Richardson, Proc. Phys. Soc. (London) ASS, 791 (1947). 3s R. E. Siday, P ~ o cPhys. . Soc. (London)A59, 905 (1947). as M. Korsunsky, J . Phys. USSR 2, 7 (1945). 40 R. E. Siday and D. A. Silverstone, Proc. Phys. Soc. (London) A66, 328 (1952). 41 K. T. Bainbridge and R. S. Bender, private communication to M. Deutsch, L. G. Elliott and R. D. Evans, Rev. Sci. Instr. 16, 178 (1944); R. S. Bender, Thesis, Harvard, Cambridge, Mass., 1947; L. Lavatelli, Thesis, Harvard, Cambridge, Mass., 1950. 4 2 C. Mileikowsky, Arkiv Fysik 4, 337 (1952); 7, 33 (1953). 32 33

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

361

et a1.43*44 have described a somewhat complicated system of this kind which seems to have figures of merit comparable to those of the double focusing spectrometer. The so-called “orange spectrometer’’ is an ingenious extension of the sector field focusing principle developed by Kofoed-Hansen et al.46 As shown in Fig. l l b the orange spectrometer consists of a number, in this

10-

cm

I

FIQ. 11fa). Sector field spectrometer due t o Kofoed-Hansen, Lindhard, and Nielsen (cf. reference 45).

case of 6 , sector field spectrometers operated in parallel with the common source and detector located on the symmetry axis of the magnetic field which decreases as l / r when measured from the axis of symmetry. In this way the transmission and the luminosity is substantially increased without sacrifice in the resolving power. Kofoed-Hansen, Lindhard, and Nielsen obtained a transmission as high as 12% a t a resolution of 2%. In this instrument the mean angle of emission is about 90’ with respect to the symmetry axis. This implies that some 50% of the electrons emitted from the source have to pass through the backing support. At 43 V. M. Kelman, V. A. Romanov, R. Ya. Metskhvarishvili, and V. A. Kolyunov, Nuclear Phys. 2, 395 (1956). 44 V. M. Kelman, Vestnik Akad. Nauk SSSR 7 , 75 (1958). 46 0. Kofoed-Hansen, J. Lindhard, and 0. B . Nielsen, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 26, No. 16 (1950).

362

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

lower energies, say below 100 kev, the absorption in the backing will broaden the line and for this reason only a fraction of the instrument can be utilized if the full resolution should be maintained. This is, however, not a necessary limitation. It has been shown b y Mallmann46t ha t boundary conditions for the magnetic field can be found such that the orange field focusing can be realized also for a n aperture opening where the maximum emission angle is less than 90". This also opens up the possibility of arranging two orange spectrometers end t o

FIG. 1 l(b). Orange spectrometer due to Kofoed-Hansen, Lindhard, and Nielsen (cf. reference 45).

end with a common source as an electron-electron coincidence spectrometer.47 Mallmann and his co-workers have under construction a large double orange coincidence spectrometer. Each orange consists of 8 gaps. As shown in Fig. 11(b) pole pieces of iron are used t o obtain the proper boundary conditions of the magnetic field. It is, however, also possible t o obtain the required field distribution b y means of an iron-free magnetic coil system. A highly engineered large double orange coincidence spec~~ trometer is under construction by Freedman and his c o - w ~ r k e r sat Argonne. A spatially confined field of the required form ( H a l / r ) can be generated inside a n iron-free toroidal coil with specially shaped cross section. C. A. Mallmann, Publ. Corn. Nacl. Energia Atdmica Argentina, Ser. Fis. 1, No. 1 Spanish. C. A. Mallmann, Physica 18, 1139 (1952); see also reference 46. M. S. Freedman, F. Wagner, Jr., F. T. Porter, J. Terandy, and P. P. Day, Nuclear Inslr. 8, 255 (1960). I6

(1953)-h

2.2.

DETERMINATION O F MOMENTUM AND EN ERG Y

363

The double beta coincidence spectrometer (Fig. 12a) constructed a t A r g ~ n n eemploys ~~ two 100-turn, IOO-gap, l-meter diameter “orange ” toroids, in a nonferrous construction. The individual tubular conductors are accurately preformed and located to define the boundary surfaces through which electrons enter and exit from the field. These surfaces are symmetrical and correspond to the focusing parameter b = p / r = 0.59, to which value the performance of the spectrometer is critically sensitive.4s

FIG.12(a). Schematic cross section of double-beta coincidence spectrometer.

Representative transmissions ( T )for source diameters (D) and resolutions ( q ) achieved by each independent unit are: T = 19%, D = 3 mm, q = 0.9%; T = IS%, D = 10 mm, q = 0.9%; T = IS%, D = 3 mm, q = 0.4%, and T = 1.6%, D = 1 mm, q = 0.1%. Thus very high luminosity and coincidence efficiency are obtained a t fairly good resolution. The leakage field reduces to less than of the internal toroidal field within a few centimeters from the coils, so mutual interference between the two units is negligible. Angular correlations (electronelectron and electron-gamma) can also be measured using adjustable baffles which select small ( T = 1%) pencils of radiation into each spectrometer or to a gamma scintillation detector near the source. A word of caution should be said here. When considering the focusing properties of the orange spectrometer it is necessary to take into account

364

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

the defocusing effect of the fringing field. The fringing field acts in two ways. It gives rise to a defocusing in the axial direction due to the curvation of the trajectories in the fringing field. Secondly it will cause a deflection out of the plane of the ideal trajectory. Both effects give rise to a focusing error and will therefore reduce the figures of merit. These fringing field effects introduce great complicatioiis in the theory. The trajectories can only be computed by numerical ray tracing technique. These problems have been discussed in some detail in a recent paper by Jaffey and his c o - w o r k e r ~The . ~ ~ effects of the fringing field are less critical, however, if a great number of gaps is used, such that the width of each gap is small. The sector field instruments have been much used as heavy particle spectrometers. The reason is that they require smaller magnets in order to focus particles of a given momentum. The inherent low transmission is less serious in cases where the particles have a fairly well-defined direction of emission. I n charged particle reactions with light nucleis the center-of-mass velocity gives rise to an energy spread correlated to the direction of emission with respect to the direction of the bombarding particle. A small aperture angle is necessary to reduce the effect of this energy spread on the resolution of the instrument. Mileikowsky60 has shown, however, that it is also possible t o eliminate this effect by making use of an astigmatic double focusing spectrometer. Finally, for angular distribution studies the small aperture of the sector field spectrometers constitutes no serious limitation. 2.2.1.1.3.5. Spiral Orbit Spectrometer. Among the family of flat instrument there is finally one interesting spectrometer which we will briefly mention. This is the so-called spiral orbit spectrometer, first suggested by MiyamotoS1and by ShipneLs2In this instrument (cf. Fig. 12b) the magnetic field is of rotational symmetry and decreases radially. Charged particles of a given momentum emitted from a source located on the axis of symmetry and in the median plane will describe spiral trajectories and asymptotically approach an equilibrium orbit. The radius of this stable equilibrium orbit is determined by the Bp value of the particles. Due to the radial decrease of the magnetic field there will be an axial focusing effect and in fact for a proper field distribution the theory indi49A. H. Jaffey, C. A. Mallmann, J. Suarez-Etchepare, and T. Suter, Argonne National Laboratory Report ANL-6222 (1961). 60 C. Mileikowsky, Arkiv Fysik 4, 337 (1952); 7, 33 (1953); L. Bianchi, E. Cotton, and C. Mileikowsky, Nuclear Instr. 3, 69 (1958). 61 G. Miyamoto, Proc. Phys.-Math. SOC.Japan 17, 587 (1943)-in Japanese; G. Iwata, G . Miyamoto, and M. Kotani, J . Phys. SOC.Japan 2 , l (1947)-in Japanese. 62 V. S . Shipnel, Compt. rend. acad. sea'. U.R.S.S. IS, 793 (1946)-in English.

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

365

cates a n acceptance solid angle of nearly 411..According to Sakai63 a transmission as high as 7801, has been experimentally obtained at a resolution of 1.25%. However, in practice one has t o introduce defining slits in order t o single out the equilibrium orbit and one also has to provide some sort of shielding against gamma radiation coming from the source. This will substantially reduce the figures of merit.54For these reasons the spiral orbit fipectrometer has so far found very little use in practice. There exist, however, certain experimental situations where the remarkable

BEAM

TARGET 0 THE AXIS OF MAGNETC F

RADIATION SHIELD

FIG.12(b). Spiral orbit spectrometer used for focusing of ?r mesons (cf. reference 54).

properties of this instrument can be used to good advantage. For instance Sagane and Dudziak66have used the spiral orbit spectrometer for the focusing of ?r mesons [cf. Fig. 12(b)]. 2.2.1.1.3.6. Lens Spectrometers. 2.2.1.1.3.6.1. The solenoidal spectrometer with uniform magnetic Jield. The prototype of the lens spectrometers is the solenoidal spectrometer with uniform magnetic field. I n this instrument the source and the detectors are both located on a n axis parallel to the field lines. Charged particles emitted from the source with an angle to the axis will describe solenoidal trajectories [cf. Fig. 13(a)] and will return t o the axis after a certain distance which is proportional to the momentum of the particles. 6 s M . Sakai, J . Phys. SOC. Japan 6, 178, 184 (1950)-in English; 6, 529 (1951)-in French. 6 4 L. Marquez, Nuovo cimento [lo] 1, 785 (1955). 6 6 R. Sagane and W. Dudziak, Phys. Z2ev. 92, 212 (1953).

2.

366

DETERMINATION O F FUNDAMENTAL QUANTITIES K

z

r

SOURCE

FIG. 13. (a) Electron trajectories in the solenoidal uniform field spectrometer. (b) Electron trajectories in the meridional plane.

By means of suitable baffles the aperture angles are limited within a certain cone. A lead shield along the axis is used to prevent gamma rays from the source to reach the detector. The trajectories are most conveniently expressed in cylinder coordinates (2, T , 9).Assume that the direction of emission corresponds to 6 = 0. Figure 13(b) shows the traces of the trajectories in a meridional

2.2.

DETERMINATION OF MOMENTUM A N D ENERGY

367

plane revolving with the electrons around the z axis, with the angular velocity* (2.2.1.1.45)

Thus 9=-

z

D cos a

(2.2.1.1.46)

D sin a cos tp

(2.2.1.1.47)

and

r

=

where (2.2.1.1.48)

Obviously D is equal to the diameter of the circle that the particles describe if they are emitted perpendicular to the lines of force. According to (2.2.1.1.47)we will get a first focus at 9 = T . This means that the focal length f becomes

f

=

TD cos a.

(2.2.1.1.49)

+

As shown in Fig. 13 the traces of particles emitted at angles LY Sa and a - 8a will intercept at a certain point defined by (2.2.1.1.46) and (2.2.1.1.47)and the condition

-0. 6r 6a

(2.2.1.1.50)

At this point the bundle of particles accepted within the cone defined by LY 6a and a - 8a will have a minimum width and in fact form a ring-shaped image, the “ring focus.’’ Obviously the trajectories will be most precisely defined if the exit slit is located a t the ring fo~us.~6-69 There exist a variety of different designs of ring focus baffles. The most favorable, however, is the one first suggested by Hubert.6oThis consists of a number of baffles defining the outer part of the envelope of the trajectories and a single inner baffle. The edge of this inner baffle should coincide with the point of interception of the extreme rays (cf. Fig. 14). With this arrangement the ring focus baffle will serve as a combined entrance and exit baffle. If the mean emission angle is chosen in a suitable way the point of interception between the extreme rays will always be

+

* Note that

2$ is equal to the “cyclotron frequency.”

s a c . M. Witcher, Phys. Rev. 60, 32 (1941).

S. Frankel, Phys. Rev. 73, 804 (1948). E. Persico, Phya. Rev. 73, 1475 (1948). 6 9 E. Persico, Rev. Sci. Znstr. 20, 191 (1949). 60 R. Hubert, Compt. rend. 230, 1464 (1950); Physica 18, 1129 (1952). 67

68

368

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

located at a fixed distance from the z axis. This means that the relation between the transmission and the resolution can be varied at will by varying only the axial setting, but not the diameter of the defining (inner) ring focus slit, which is an obvious advantage from the constructional point of view. In a uniform field solenoidal spectrometer this may be realized by choosing the emission angle Q! = 40" 25'. However, it remains to be shown that this choice will correspond to optimum conditions of operation. According to Persico'jl one has f.02

-

= f(a).

(2.2.1.1.51)

9

Persico has computed f(a) for a point source and has shown that f(a) has a fairly broad maximum around 45" 37'. However, a somewhat

FIG.14. Ring focus baffle design due to Hubert (cf. reference 60).

smaller emission angle is justified from the economical point of view since it results in a considerable reduction in the size of the magnetic coils. No detailed theory exists for the case of a source with finite dimensions. The problem has been studied by PersicoG1and by DuMond.e2,GaAccording to Persico the following conditions should not be far from optimum (cf. Fig. 15): angle of emission source radius entrance slit (vertical) exit slit (horizontal) minimum radius of counter window focal length gathering power luminosity resolution

a = 40" 25' s =2.840~~ Z A = 1.020; T A = 0.630; h = 0.880~ Z E = 1.630; rh = 0.540; 5 = 1 2 . 3 0 0 ~ ~ W = 1.74Dea f = 2.390 o = 0.658 A = 16.5D2d 7 = 3.569

E. Persico, see reference 59; also E. Persico and C. Geoffrion, Rev. Sci. Znslr. 21, 945 (1950). J. W. M. DuMond, Rev. Sci. Inslr. 20, 160, 616 (1949). 83 J. W. M. DuMond, Ann. Phys. 2,283 (1957).

2.2.

369

DETERMINATION O F MOMENTUM AND ENERGY

From DuMond’s analysis of the theoretical line profile one gets the following relations T 0.17q1’2 (2.2.1.1.52) L 0.41D2q6’2. (2.2.1.1.53)

--

While in the flat spectroscopes T is proportional to q it is here proportional to q1I2. For this reason the solenoidal spectrometer will give a good resolution (-0.5%) also a t a fairly high transmission (“3%).

FIQ.15. Solenoidal spectrometer under optimum conditions of operation. [Quoted from Persico and Geoffrion (cf. reference Sl).] The solenoidal spectrometer and in fact all lens spectrometers will not distinguish between positively and negatively charged particles of the same momenta, since the traces in the meridional plane are independent of the sign of the charge. However, as is easily realized, positive and negative particles will revolve in opposite directions since dd d2

eB - f l Zp cos D cos a

(2.2.1.1.54)

It is sometimes desirable to separate positive and negative particles. This can be done, as first suggested by Deutsch et al.64by means of a twisted baffle system. See Fig. 16. If this is carefully matched to the trajectories the transmission of the instrument for particle of the proper sign will be reduced by only some 10% while the particles of opposite sign will be completely screened off. On the other hand the fact that lens spectroscopes normally transmit positive as well as negative particles with the same momenta can be utilized to convert the magnetic spectrometer into a pair spectrometer for high-energy gamma ra~s.6~5-67 B4

e6 O7

M. Deutach, L. Elliott, and R. Evans, Rev. Sci. Znstr. 16, 178 (1944) K. Siegbahn and S. Johansson, Rev. Sci. Znstr. 21, 442 (1950). D. E. Alburger, Rev. Sci. Znstr. 23, 671 (1952); 26, 1025 (1954). S. J. Bame and L. M. Bagget, Phys. Rev. 79, 415 (1950).

370

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Several solenoidal spectrometers have been described in the literature. Careful designs have been published by Feldman and Wu16*by SchrnidtlBg and by DuMond.’O 2.2.1.1.3.6.2. O n the focusing properties of inhomogeneous, axially symmetric magnetic fields. The uniform field has the particular advantage that the trajectories can be described by simple equations. This means that the focusing properties can be derived in a straightforward way.

FIG. 16. Twisted baffle system used to separate positron and negatrons.

However, there is no reason why the uniform field should be particularly favorable and it turns out that inhomogeneous fields shaped in different ways have, in fact, focusing properties superior to those of the uniform field. A theoretical study of the focusing properties of such fields, however, is hampered by the formidable mathematical complications. For axially symmetric magnetic fields the radial component B,(r,z) as well as the axial component Bz(r,z)are both known if we know the field along the L. Feldman and C. S. Wu, Phys. Rev. 87, 1091 (1952). 6QF. H. Schmidt, Rev. Sci. Znstr. 28, 361 (1952). J. W. M. DuMond, Ann. Phys. 2, 283 (1957); J. W. M. DuMond, L. Bogart, J. L. Kohl, D. E. Muller, and J. R. Wilts, California Institute of Technology, Pasadena, Spec. Tech. Rept. No. 16 (1952). 6*

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

371

axis of symmetry: r2

B,(r,z) = B(O,z) - 4 B"(O,z)

r4 +a B""(0,z) -

. . (2.2.1.1.55)

and

B,(r,z) =

- r B'(0,z) + r 3 B"'(0,z)

-

.

The expressions (2.2.1.1.55) and (2.2.1.1.56) correspond t o (2.2.1.1.33) in the case of flat spectroscopes. While in (2.2.1.1.33) only the first few terms are of importance we have in (2.2.1.1.55) and (2.2.1.1.56) to take into account the complete function B(r,z) with all its derivatives. For this reason practically all investigations have been made experimentally by trial and error techniques. However, a few qualitative conclusions may be drawn. As seen in Fig. 13(b) the lens spectrometer will be characterized by a spherical aberration. A way to compensate for this effect, somewhat similar to the third-order focusing principle discussed above, would be to let the outer trajectories pass through a weaker magnetic field. As seen in (2.2.1.1.55) this implies B"(0,z) > 0, i.e., the field when measured along the axis, should be concave upwards [cf. Fig. 17(a)]. A single short lens gives a field of just the opposite shape [cf. Fig. 17(b)] and should therefore be expected to be characterized by a large spherical aberration. This is also the case. Short lens spectrometers have focusing properties much inferior to those of the uniform field. Some short lens spectrometers have been desgribed in the Deutsch et aZ.7' have given a theory of the focusing properties (in the paraxial ray approximation, i.e., valid for small emission angles only). It is possible to design a long lens spectrometer, with or without iron, which will give a field distribution resembling the ideal field form corresponding to complete compensation of the spherical aberration. Sieg71

7*

M. Deutsch, L. Elliott, and R. Evans, Rev. Sci. Instr. 16, 178 (1944). J. M. Keller, E. Koenigsberg, and A. Paskin, Rev. Sci. Znstr. 21, 713 (1950). W. W. Pratt, F. I. Boley, and R . T. Nichols, Rev. Sci. Instr. 22, 92 (1951).

372

2.

DETERMINATION OF FUNDAMENTAL QUANTITXES

bahn74has shown that the field shown in Fig. 17(c) gives a considerably reduced spherical aberration. A properly designed long lens spectrometer should therefore give figures of merit superior to those given in (2.2.1.1.52) and (2.2.1.1.53). Several designs have been published.74-Ts

nn

FIQ.17. Field shapes in different lens spectrometers.

The intermediate image spectrometer77 is a special type of long lens instrument. In this spectrometer the field distribution is chosen in such a way that the ring focus image is formed in the middle of the instrument. For symmetry reasons the trajectories form a second, nearly point-shaped image at the axis. The intermediate image spectrometer developed by SlBtis and Siegbahn??is characterized by a high transmission (8%) a t a 74

K. Siegbahn, Phil. Mag. [7] 37, 162 (1946).

n W. Zunti, Helv.Phys. Acta 21, 179A (1948). w E. A. Quade and D. Halliday, Rev. Sci. Znstr. 7’

19, 234 (1948).

H. Slatis and K. Siegbahn, Arkiv Fysik 1, 339 (1949).

W

-I

W

Y

FIG.18. Intermediate image spectrometer due to Slatis and Siegbahn (cf. reference 77). coils used to give the magnetic field the appropriate shape iron yoke and flanges I and Y N1 and N2 iron pole pieces source S Geiger-MUer counter G-M

R1 and Rz L D TI and Tf W P

baffles Lead shield against gamma radiation lock device for the source inlet valves for the filing of the GM counter cooling water inlet and outlet vacuum-pump connection.

374

2.

DETERMINATION O F FUNDAMENTAL Q U A N T I T I E S

moderate resolution (4%).77aA great advantage is the formation of the point image which enables one to use a very small counter. The desired field form can be realized both with77 and w i t h o ~ t ~the ~ ~use 7 ~of iron. An alternative way to reduce or eliminate the spherical aberration has recently been described by Dolmatova and Kelman.80 They used a uniform field with a correcting magnetic field superimposed. The correcting field is generated by means of a special coil system placed inside the spectrometer. According t o Dolmatova and Kelman they have obtained a resolution of 1.4% at a transmission of 5.2% using a 1 X 1 mm2 source and a resolution of 1.9% a t a transmission of 6.5% with a source of 5 mm in diameter. However, for a source of finite dimensions the spherical aberration is not the only contribution to the width of the ring focus. I n fact the most important contribution to the finite width of the ring focus is due t o the width of the source. It is therefore more profitable t o try to reduce this contribution t o the line width rather than concentrate on reduction of the spherical aberration. Gerholm81 has studied a "triangular" field distribution, i.e., a field which increases proportionally to the distance from the source when measured along the axis of symmetry. It was found that such a field gives a magnification, which is less than 1. See Fig. 19. This means t ha t the effect of the width of the source is reduced. A more careful theoretical study of focusing properties of the triangular field has recently been published by Lindgrexg2 According to Lindgren the minimum width of the beam bundle is for all emission angle of 20" less than one-fifth of the width of the source. This minimum does not exactly coincide with the ring focus-where the spherical aberration has its minimum-but in spite of this the triangular field gives a higher luminosity than a uniform field does. The theory also indicates th a t one might find a field where the two minima coincide. This would give rise to a large increase in the luminosity. The triangular field has the additional advantage that the source is located in a field-free region. This makes it possible to arrange two ' T 8 These figures refer to a source diameter of 5 mm. With smaller sources the resolution is improved, for instance, one obtains T = 4.0%, 9 = 2.0% for 4 2 mm and 5" = 2.0%, 7 = 1.0% for 4 1 mm. It should also be observed that the intermediate image focusing can be obtained with sources placed in a field-free region. This makes it possible to arrange two spectrometers end-to-end for coincidence studies [cf. C. J. Herrlander and H. Slatis, Arkiv Fysik 20, 71, (1981)l. 78 W. Bothe, Naturwissenschaften 37, 41 (1950). 7 9 D. E. Alburger, Rev. Sci. Znstr. 23, 871 (1952); 26, 1025 (1954). 8 o K. A. Dolmatova and V. M. Kelman, Nuclear Znstr. 6, 269 (1959). T. R. Gerholm, Rev. Sci. Znstr. 26, 1069 (1955). 82 I. Lindgren, Nuclear Znstr. 3, 104 (1958).

2.2.

375

D E T E R M I N A T I O N O F MOMENTUM A N D E N E R G Y

r/D

IMAGE MINIMUM

SOURCE DIAMETER

z /D

FIG. 19. Electron trajectories in the “triangular field.” The width of the image corresponds to the contribution due t o the width of the source only. It is shown that the image width is about one-fifth of the source width (cf. reference 82).

spectrometers “end to end ” for coincidence studies and similar experiments.81 Lens spectrometers have been used almost exclusively for beta-ray spectroscopy. The short lens seems to have no part.icular advantage compared t o the sector field instruments and the long lens instruments require heavy magnets in order t o focus high-energy particles.

2.2.1.1.4. MEASUREMENT OF MOMENTUM WITH CLOUDCHAMBERS OR

BUBBLE CHAMBERS.* The momentum of a charged particle can be determined by observation of the deflection of the particle in a magnetic field.? Various methods are available for determination of t,he deflection; here we discuss the use of cloud chambers and bubble chambers, which make the track of the particle visible. If a particle of charge 2 in units of the electron charge and momentum p travels at right angles to a magnetic induction B gauss, it will move in a circle of radius p cm, such that p c = 300Z(Bp)ev.

t See also Vol. 4, A,

Chapter 3.4.

* Section 2.2.1.1.4

is by W. B. Fretter.

(2.2.1.1.4.1)

376

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

The experimental procedure is to measure the radius of curvature of the track in a known magnetic field, and calculate the momentum from (2.2.1.1.4.1).There are, of course, many complications which enter in actual practice, and various techniques of measurement and reduction of observat,ions which may be used. We discuss first the determination of the radius of curvature. Measurement of p on jilm. The track of a particle in a cloud or bubble chamber is presented to the observer as a series of photographic images of droplets or bubbles more or less in focus on a piece of film. There are several ways of determining the curvature on the film, which are presented approximately in order of increasing accuracy. Comparison curves. Circles of known radius may be drawn on paper, reduced in size photographically, and compared directly with the photographic image by placing the circles over the tracks until a match of curvature is obtained. More commonly the photograph is projected to some standard magnification, usually the original size of the chamber, and circles of known radius are compared with the tracks. This requires a set of curves differing in radii by 5% to lo%, accurately drawn, and estimates of the radii accurate to 5 % can be made on good quality tracks in a fairly uniform magnetic field. Such measurements can be made rapidly; a minute or so is required for each track. The accuracy of the method is not well-defined and may vary with the observer, and no permanent record of the measurement is made which can be checked independently for mistakes. If the track is not accurately circular, as it rarely is, because of inhomogeneities in the magnetic field, optical effects, or energy loss in the medium, the method loses accuracy. For a quick survey of the tracks in a photograph the method is invaluable and widely used. Sagitta method. Measurement of the position of three points on a circle is sufficient to determine the radius of curvature of the circle. A typical procedure is to place the track in a measuring microscope, aligning it approximately with one of the directions of motion. Readings are then made at points ABC, and the sagitta 6 is determined. The length of chord AB and the sagitta 6 are related to the radius by the equation (see Fig. 1)

This equation is exact, but if 6 << p, the simpler form p = (AB)2/86may be used. Although the measurement of the chord and sagitta in a measuring microscope does not require more than a few minutes, the principal time being taken in lining up the track along the direction of motion, other procedures have been developed to speed the process and make it more

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

377

accurate. Leightonl has made an optical device to magnify the sagitta while keeping the chord constant, and Jones and Hughes2used a mechanical device for measuring the sagitta of a reprojected track. The sagitta method is probably about comparable in accuracy to the comparison curve method. It suffers from the same troubles when the curve is not a circle, and does not use a very large amount of the total information available in the track photograph, since only three points on the track are measured. It has the advantage that the calculations of the error due to scattering are done on the assumption that the sagitta method is used to measure the apparent curvature of the track. Repeated measurements may also be used for estimate or error due to uncertainty in setting. c

FIG.1. Three-point measurement of curvature on a circular track.

Curvature compensation. Another fairly rapid method for measurement of the radius of curvature has been developed by Blackett3 and his collaborators and is described in detail by Wilson.3BThe photograph of the track is projected through an optical system that transforms the circle into a straight line. A prism which may be rotated about an axis introduces a curvature that is easily variable, and the observer sights along the image of the track, turning the prism until the track appears straight. This method makes use of the entire track image, and is based on the ability of an observer to detect very small deviations from a straight line. It has the advantage that parts of the track such as delta rays, which are prominent but not necessarily on the path of the particle, show clearly their displacement. The procedure requires that the track be aligned properly relative to the zero position of the prism, and two readings are taken with the track position differing 180”. The entire procedure takes only a few minutes, and is comparable in time required to the sagitta method. The presence of noncircular components of curvature make the method R. B. Leighton, Rev. Sci. Znstr. 27, 79 (1957). H. Jones and D. J. Hughes, Rev. Sci. Znstr. 11, 79 (1940). 3P. M.S. Blackett, Proc. Roy. Soc. A169, 1 (1937). 38 J. G. Wilson, “The Principles of Cloud-Chamber Technique.” Cambridge Univ. Press, London and New York, 1951. 1

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DETERMINATION O F FUNDAMENTAL QUANTITIES

less satisfactory since these cannot be compensated, leaving a residual curve. Errors can be estimated by repeated independent readings. The optical components of the compensator must be very carefully made, and the completed device represents a considerable expenditure of money compared to the devices previously described. Three prisms are r e q u i r ~ dto~ ~cover a range of curvatures up to C = 12.5m-', which corresponds to a momentum of 2.4 X 108 ev/c in a field of lo4gauss. Tracks that are curved more than this must be measured by other methods. Thus the compensator is used mostly for measurement of high-momentum tracks. I

3 w

zJ

$

s

1.5

0

a

Iu)

I 1.0 0 a LL

I-

z

yw 0.5 0

a J

a

v, 0

5 10 15 20 DISTANCE ALONG TRACK ON FILM ( M M )

FIG.2. Coordinate plot measurement of curvature on a cloud-chamber track.

Coordinate method. The track is placed in a measuring microscope, lined up with one direction of motion, and readings are made at regular intervals along the track of the coordinates of the center of the track. These readings are then used to give the value of the radius of curvature in the following ways. 1. The points are plotted on graph paper, with a magnified scale on the deflection coordinate. A curve, in this case a parabola, is fitted to the points by eye, and the sagitta and chord are measured (see Fig. 2). The radius of curvature is then calculated from the sagitta formula. This method takes longer than the three-point method, but makes use of all the tracks. Setting errors can be estimated from the scatter of the points, and repeated measurements may be made. If the original track is noncircular or if various distortions occur, they are immediately apparent on

2.2.

DETERMINATION OF MOMENTUM A N D E N E R G Y

379

the coordinate plot, which forms a permanent record of the measurement. 2. The radius of curvature may be calculated from a least-squares fit of a circle or a parabola to the measured track coordinates. It was pointed out by Thomson4 that the least-squares reduction is particularly simple if the points along the track are equally spaced, and odd in number. Measurement and calculation together take about 10 minutes for a track where 31 points are measured. The m*ethod is most useful for tracks of rather small curvature when accurate determinations are required. Its principal advantage, aside from the obvious ones of accuracy and use of the entire track, is that it permits an objective evaluation of the error in the momentum determination. Such errors will be discussed in a later section. 3. The coordinate measurement method is also used when the observations are to be reduced in a computing machine. The coordinates may be read in a measuring microscope, punched into a card, or otherwise fed into a computing program. Automatic machines for measuring the coordinates and following the track may also be used. A machine built a t the University of California Radiation Laboratory is designed to follow the track automatically under the supervision of the operator, who punches a button periodically to permit the machine to read the coordinates of the track and transfer them to a punched card. Clearly such a machine is so complicated and expensive th at its use is only justified if many thousands of tracks must be measured. But for photographs of bubble chambers operating in the beams of high-energy particle accelerators, where thousands of photographs are taken per day, automatic analysis machines are probably necessary to handle the data. Conversion t o p i n Eq. (2.2.1.1.4.1).The procedure described above determines a kind of average value of radius of curvature on the film. It is then necessary t o transform this value into a number which can be substituted as p in Eq. (2.2.1.1.4.1).The most obvious correction needed is the magnification factor, which depends on the position of the track in the chamber and is thus variable if the track is not in a plane perpendicular to the axis of the lens. I n general the image on the film represents a “point” or “conical” projection of the object in the chamber. There may be other optical corrections to be made, such as displacement of the image due to refraction in the medium of the chamber, the glass, or the medium surrounding the chamber. All of these corrections invoIvc the distance of the track from the lens, and thus stereoscopic photographs must be taken to determine this distance. The distance between the lenses of the stereoscopic camera, or cameras, must be carefully considered. Although 90” photographs are in principle

R. W. Thompson, Nuovo eimento [lo] 1, 735 (1955).

380

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

the most accurate, and are discussed in detail by Wilsonlaathey are rarely obtainable because of geometrical or illumination difficulties. Instead, it is usually convenient to use lens separations of from a few inches up to the full width of the chamber. Stereoscopic photographs taken at separations of four t o six inches can readily be seen visually in a stereoscopic viewer, although the depth dimension is exaggerated. Such viewing is exceedingly valuable in certain scanning operations, and for drop counting. Regardless of such possible uses, stereoscopic photographs are necessary for the reconstruction of the track in space. Reconstruction of tracks in space. It is convenient to provide fixed fiducial marks or lines in a cloud chamber or bubble chamber to serve as frame of reference for the reconstruction of the track. Measurements of the position of a point on track on the film relative to the fiducial lines registered on the film in each picture give the depth of the particular point on the track. The procedures available for reconstruction include (1) actual measurement point by point and calculation of the depth for many points along the track, and (2) optical reprojection of the track images either side by side, superposed, or one after the other. The first of these is used when computing machines do the actual numerical work. When fewer tracks are to be analyzed optical reprojection is sufficiently accurate and reasonably fast. Procedures have been discussed by Barker6 and by Fretter and Friesen.6 For cloud-chamber work, the principal correction to be made to the curvature of the track as measured on the film, after the gross correction due to the magnification factor is made, is due to the conical projection. This correction may be large if the camera is close to the chamber or if the track is tilted far out of the plane perpendicular to the camera axis. Another correction is due to the variable dilation of the track during the expansion, if the track is not perpendicular to the direction of motion of the gas. In general for thin chambers photographed from a large distance, e.g., a chamber of 2-in. illuminated depth photographed a t a distance of 40 in., these corrections will be small, but for thick chambers photographed from short distances, the corrections are large and rather complicated to make. Determination of the momentum of a single track in case all corrections must be made takess from two to four hours, if done with a desk calculator. I t is thus desirable to program these calculations on a computer if many must be done. Corresponding calculations for bubble chamber tracks may be considerably more complicated. In a cloud chamber only refraction in the front K. H. Barker, Nuouo cimento Suppl. [9] 11, 309 (1954). W.B. Fretter and E. W. Friesen, Rev. Sci. Instr. 26, 703 (1955).

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

38 1

glass must be considered, but in a bubble chamber the refraction in the medium is not negligible and for large bubble chambers the corrections for displacement are nonlinear. It is difficult to arrange for reprojection through an optically identical arrangement, particularly if the medium is liquid hydrogen. Even if the corrections are programed for a computer, the computations are quite long. A solution may be to reproject through an optically equivalent system. Curvature errors. Errors in the determination of the radius of curvature of the track may be divided into three classes: (1) track noise; (2) distortion of the track; (3) scattering of the particle. The track noise has two origins. First, there is an error of setting on the track due to the finite width of the collection of photographic images that define the track. In a cloud chamber the track may be widened by diffusion during the time between the passage of the particle and the fixing of the ions. Blackett' has discussed the error in momentum due to diffusion and finds

relative error in momentum due to diffusion

L = length of track in cm B = magnetic induction

D = diffusion coefficient T, = expansion time n = number of drops per cm. This error is important when the track is short, for large values of the momentum, and in low-pressure chambers where the diffusion is large and n small. Barker6 shows that the setting error, as determined by repeated settings on a given point, is smaller than the uncertainty due to the diffusion. Another factor contributing to the track noise is the occurrence of more energetic secondary electrons which may form ions at a distance from the main track. These ions often form prominent groups of drops which are easy to set on, but the error that arises from setting on these is generally smaller than that due to diffusion. For setting on bubble chamber tracks, the effect of diffusion is negligible. The tracks are formed by bubbles which represent formation centers having much higher energy than is required to form an ion in a cloud 7

P. M. S. Blackett, Nuovo cimento Suppl. 191 11, 264 (1954).

382

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

chamber. The photographic images of the bubbles reported'" are about 40 p in diameter, and Bradner* reports that on a row of 10 bubbles the setting error may be as low as one micron. The bubbles may occasionally be displaced from the main line of the track by distances as great as 10-20 ji, since they are probably formed on delta rays, which tend to deposit energy at a small distance from the path of the particle. Thus the ultimate limits of track noise in bubble chamber and cloud chamber are probably not very different. Improvement in bubble chamber photographic t,echnique may alter this situation somewhat. Track noise enters into all methods of computation of curvature. Blackett's calculations7 were made for the case of the sagitta method, but Barker has extended these to the coordinate method. It is important to note that since the error in setting on the track is presumed to be symmetrical and Gaussian, the error in the curvature C is symmetrical, rather than the error in the radius of curvatures. Here

Thus the error on the momentum is an asymmetrical error. It is convenient therefore to discuss errors in the curvature rather than errors in the radius of curvature. Wilson38and Barker6 discuss the errors due to track noise in measurement of the curvature by the compensation method. The precision of measurement appears to be somewhere between AC = 0.04 m-l and AC = 0.10 m-l measured on the film, for a counter-controlled track 1 cm long on the film. Under usual conditions, the spurious curvature due to the thermal motion of the gas is greater than this. Track noise also affects the measurement of curvature by comparison curves. No systematic study of this method has been made, but in the hands of experienced observers it is probably nearly as accurate as the compensation method. Regardless of which method is adopted, it is advisable to make a thorough study of setting errors and track noise since these are likely to be different for various experimental conditions. Distortion of the track. The track of a charged particle in a cloud chamber or a bubble chamber may be distorted on the photographic film in several ways. These may be divided into three classes: optical, expansion motion of the medium, and pre-existing motion of the medium. Optical distortions coming from point projection effects are systematic and have been treated elsewhere. Other similar optical distortions, such as displacements due to the index of refraction of the glass on medium, '8

D. A. Glaser, D. C. Rahm, and C. Dodd, Phys. Rev. 102, 1653 (1956).

H. Bradner, Private communications.

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

383

or lens distortions are also systematic and in principle can be corrected. Wilsonsadiscusses some features of lens distortion. Lenses should be carefully selected for low distortion and flatness of field for high-precision work, and maps of lens and other optical distortion are usually made by photographing straight lines in the position of tracks and measuring the apparent curvature. Generally it is possible to make optical distortion errors negligible in a cloud-chamber experiment. I n bubble chambers, aside from the necessity of correcting for known optical distortions, a more serious problem may arise. The index of refraction of the medium may vary with temperature and if temperature gradients exist, local distortions may occur. These may or may not be systematic. Such effects may seriously affect the precision of curvature measurements, but no quantitative measure of their importance has been made. Expansion motion of the medium. I n cloud chambers and in bubble chambers the track of the particle is photographed soon after the medium has undergone a rapid expansion. If turbulent motions are set up during the process of expansion that continue up to the time of photography, the track may be distorted by these motions. Various precautions must be taken t o avoid expansion turbulence. In a piston-type cloud chamber, the motion of the piston should be closely controlled, the motion of all parts of the piston parallel. At the end of the stroke the motion must be critically damped t o avoid shock waves in the medium. I n a diaphragmtype cloud chamber baffles are usually installed to smooth the motion of the gas in the sensitive region. Diffusion chambers do not suffer from this problem, but in bubble chambers expansion turbulence may present design problems. Expansion turbulence in a carefully designed system does not, however, limit the accuracy obtainable in momentum measurements. Pre-existing motion of the medium. For high-momentum tracks, the limit of momentum measurement is usually set by the motion of the medium existing before the passage of the track. This motion is due t o convection in the medium produced by various temperature gradients, or to turbulence residual from the previous expansion. Convection currents may be quite slow and still cause serious spurious curvature. Consider the convection velocity required to produce a spurious curvature of 0.02 m-l in a track 40 cm long. This requires a displacement of the sagitta by 0.04 cm which takes place in the time of photography, say 0.1 sec. The gas must then have a velocity of only 0.4 cm/sec to cause this distortion, which would be intolerable in a n accurate experiment. Temperature gradients in the chamber which tend to cause convection

384

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

currents must be minimized in accurate work. Side-to-side temperature difference of more than 0.01"C usually cause trouble in cloud-chamber work. It is customary to work with a top-to-bottom temperature gradient of about O.Ol"C/cm which stabilizes the gas. I n bubble chambers the topto-bottom temperature gradient is usually somewhat larger. Thermostatic control with accurate determination of temperature in various parts of the system is essential for all accurate work. Some sources of heat are inevitable. The lights used for photography or for viewing supply heat to the walls and to the medium. In the most accurate cloudchamber work, the heat supplied to the windows is sufficient to cause distortion if the waiting time between pictures is too short. The higher viscosity of the medium in a bubble chamber reduces the convection velocities, but other thermal effects after the expansion are likely to cause trouble. Spurious curvature due t o pre-existing motion of the medium may be checked by measuring tracks of fast particles in the absence of a magnetic field. If the momentum of the particle is large enough so that scattering effects are unimportant, the spurious curvature of no-field tracks gives a measure of the accuracy of the chamber. No-field tracks should be taken under conditions as closely as possible similar to the field-on conditions, and a sufficient number observed so that possible systematic effects as well as random curvatures can be determined. Study of these tracks will reveal the type of curvature, which may not be circular, and which may be dependent on length and location of the track, characteristic of a given chamber. Wilson3aand Barker6 give discussions of gaseous convection, but the distortions in bubble chambers have not yet been systematically studied and reported. The most accurate curvature measurements have been made in a cloud chamber designed by Thompsong where the probable error due to spurious curvature is c8 = rt4.3 km-' for tracks 56 cm long. Maximum detectable momentum. Wilson3ahas defined the maximum detectable momentum PO as the particle momentum for which true field curvature is equal to the probable uncertainty of curvature measurement. For momenta higher than this, even the charge of the particle is uncertain; for lower momenta, the probable uncertainty is given by p 2 / p 0 . I n the case of Thompson's chamber referred to above, the maximum detectable momentum is 50 Bev/c. I n principle, if the cloud chamber is made larger so that longer tracks may be measured or if the field is higher, such a limit can be exceeded. The technical problems are so complicated and interrelated that it is difficult to predict the magnitude of improvement possible. R.W. Thompson, J. R. Burwell, and R. W. Huggett, Nuovo cimento Suppl. [lo] 4, 286 (1956).

2.2.

DETERMINATION O F MOMENTUM A N D E N E R G Y

385

Scattering. When a charged particle passes through the medium in the chamber it will be deflected by interaction with the charge centers in the medium. The error in the determination of curvature of a track due to scattering has been discussed by Williamslo and by Bethel' and results summarized by Wilson and Blackett.' Blackett gives the relation

=

($)80att

relative rms momentum error due to scattering

e = 4.8 X 10-lo esu 2 = atomic number of the medium N = number of nuclei per cm3 p = v/c B = magnetic induction in gauss, and

A

=

where

em,,

(log, -

the maximum angle of single scattering that would not be recognized as such, and mcZ1I3 6m3n . =137p mc = 0.51 X 10" ev/c p = momentum of particle in ev/c. =

The term A varies rather slowly with velocity, and being logarithmic, is not very sensitive to the value of em, chosen. Suppose, for example, em,, = 0.1 radian, and p = 108 ev/c, 2 = 18. Then A = 2.6, and if we take 2 = 1, A = 2.8. In the following, we have taken A = 3 for order of magnitude calculations. Putting in numerical values we find

where p = density of the medium in g/em2, and M = molecular weight of the medium. For fast tracks of length 25 cm, 3! E 1, and for B = lo4, the percentage error is of the order of magnitude 10% in a propane bubble chamber, 2% in a hydrogen bubble chamber, and 1% in a cloud chamber filled with argon a t atmospheric pressure. The error increases as p decreases, but the c a l c ~ l a t i o nis~more ~ complicated if /3 is small or changes appre10 11

E. J. Williams, P h p . Rev. 68, 292 (1940). H. A. Bethe, P h p . Rev. 70, 821 (1946).

386

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

ciably along the track because of energy loss. However there is generally a value of p below which momentum measurements have little meaning, which is defined by the particular experimental situation. The magnetic Jield. It is not the purpose here to discuss in detail the design of magnets for cloud chamber or bubble chamber work. There are some problems, however, which should be mentioned. Most of these are concerned with the uniformity of the magnetic field. Ideally, the chamber would be placed in a uniform magnetic field such as that available in Helmholtz coils. Since the power consumption would then be large if large fields are to be obtained, most experimenters use iron to make the magnet more efficient. The field is then less uniform and measurements of the value of the magnetic induction over the volume of the chamber are required. Since the measurement of momentum involves the magnetic field linearly, if the magnetic field varies by more than a per cent or so along the track, its variation must be taken into account when the momentum is computed. In cloud chambers the mass motion of the gas may be sufficient to move the track appreciably from its original position in the magnetic field, and account must be taken of this. If the field varies appreciably along the length of the track, the track will not be circular (or helical) and some average value of the magnetic field must be taken to correspond to the average value of the curvature measured. The averaging process has been discussed by T h o m p ~ o n , ~ by Fretter and Frieseq6and by Barker.6 If the measurement of curvature is made by the sagitta, comparison curve, or compensation method, it is usually sufficiently accurate to use the value of the field a t the center of the track unless the field is very inhomogeneous. For a five-point measurement of the field along the track, Fretter and Friesen used Beff

=

B,

+ 32B2 + 60B3 + 32B4 + €Is 126

I n their case the gradient of the field was sufficiently large to produce substantial nonaxial components of the magnetic induction. Thus it was necessary to measure the direction of B as well as its magnitude. The transverse field components give rise to a curvature in a plane perpendicular to the main axis of the magnetic field, and some of this curvature appears in the image due t o the conical projection. The correction for this effect may be as much as several per cent in extreme situations. Even if the magnetic field is fairly nonuniform, with appropriate corrections the error due t o magnetic field uncertainties may be about 2%. With uniform fields the error can be made several times smaller than this, and generally is negligible compared to the error in curvature.

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

387

Measurement of momentum with magnetic Jield external to the chamber. The previous discussion has been concerned with the measurement of momentum when the chamber detecting the particle is immersed in the magnetic field. It is also possible to use the cloud chamber external to the field, as a precision indicating device showing the deflection of the particle after it has passed through a magnetic field. Various system^'^^'^ which have been used are shown in Fig. 3(A) and (B). CLOUD CHAMBER I I

I

CLOUD CHAMBE

,

\ \

1 \I

CLOUD CHAMBER

\

\\

*

\

I

A

CLOUD CHAMBER 3

(A)

(B)

FIG.3. Measurement of momentum of high-energy particles using cloud chambers and magnetic fields external to the cloud chambers.

In the apparatus (A) described by Brode12the momentum was determined by measuring the direction of the track in the upper and lower chambers and obtaining the deflection of the particle in the magnetic field. The field was produced in the air gap of a permanent magnet. This arrangement has the advantage that there is no source of heat near the cloud chambers, and needs no direct current power supply. The maximum detectable momentum was about 20 Bev/c. The Manchester apparatusI3 (B) uses three flat cloud chambers to determine the coordinates of the particle before and after passage through the magnetic field. The maximum detectable momentum in this appaR. B. Brode, Revs. Modern Phys. 21, 39 (1949). J. C. Lloyd, E. Rossle, and A. W. Wolfendale, Proc. Phys. SOC.(London) A70, 421 (1957). 12

13

388

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

ratus as it was used by Lloyd el al. was 290 Bev/c, which is several times larger than that in any other apparatus in existence. When the multiple scatteringt of a particle passing through a medium can be measured, the quantity p p , the product of momentum and velocity can be determined. Although this is done mostly with photographic emulsions, it may also be done with high-pressure cloud chambers, cloud chambers with plates, or bubble chambers. A cloud chamber containing argon at 75 atmos pressure was used by Baxter and Stannard14to determine p/3 of particles up to 150.Mev/c. In such a chamber the effects of distortion of tracks due to motion of the gas may be important. The method of measurement of scattering in unit cells was used, and the projected angle was measured. In a multiplate cloud chamber, the procedure is to measure the angle of deflection, or more often the projected angle of deflection, in each plate, and take the mean square deflection. The techniques are discussed by Olbert,16 Annis and associates16 and summarized by Peyrou.” The cell of the photographic emulsion technique becomes the plate itself, and the total number of measurements cannot exceed the number of plates. Thus the statistical errors are likely to be large, and are in general about 20 to 40% on an individual measurement. The distribution in scattering is usually assumed to be Gaussian, even though there is 8 tail due to single scattering. I n a bubble chamber containing a material heavier than hydrogen or helium, multiple scattering measurements may also be made on the tracks. In such a case, measurement by unit cells and other techniques similar to those described for photographic emulsion work will prove useful.

2.2.1.1.5. MOMENTUM MEASUREMENT IN NUCLEAR EMULSIONS.* In the section on charge determination, $ the feasibility of using magnetic deviation methods in emulsions to obtain either charge or momentum was discussed. It was pointed out that this method, which requires extremely large magnetic fields, is not only very costly, but also entails the

t Refer to Section 2.2.1.1.5.1. P. Baxter and F. R. Stannard, Proc. Phys. SOC.(London)A70, 19 (1957). S. Olbert, Phys. Rev. 87, 319 (1952). 1e M. Annia, H. S. Bridge, and S. Olbert, Phys. Rev. 89, 1216 (1953). l7 C. Peyrou, Nuovo cimento [9] 11, Suppl. No. 2, 322 (1954). $ See Section 2.1.1.3. 1*

l6

* Section 2.2.1.1.5 is by M. Blau.

2.2.

DETERMINATION OT MOMENTUM A N D E N E R G Y

389

use of regulated power supplies and cooling mechanisms to protect the temperature sensitive emulsions. The other parameters which govern the use of this mode of measurement-the length of the trajectory and the mean scattering angle-may be varied to render the magnetic method practicable by using the large emulsion volumes now available and by decreasing the silver content in the emulsion in order to lower the number of high Z scattering centers. The most useful method of momentum determination in emulsion, however, is that of multiple scattering. A brief outline of the theoretical basis of the multiple scattering method is given below in Section 2.2.1.1.5.1. Then the following section describes the experimental methods used for scattering measurements in emulsions. 2.2.1.1.5.1. Multiple Scattering Method. 2.2.1.1.5.1.1. Theoretical outline. Charged particles, in their passage through matter, are observed to deviate from their original direction of motion. It is believed that this deviation is the resultant of many small elastic deflections, which the particle suffers in passing the Coulomb field of atomic nuclei. The probability for elastic Coulomb scattering, as first calculated by Rutherford, is inversely proportional to the fourth power of the sine of the deflection angle, thereby explaining the high frequency of small-angle deviations. The probability for small-angle deviation is given, to a first approximation, by Rutherford’s law and depends upon the nature of the scattering nuclei, as well as the mass, charge and velocity of the scattered particle, although not upon its spin. For more accurate calculations, the screening effect of the nuclear field by outer electrons must be considered and quantum-mechanical calculations have to be introduced. I n these calculations both limiting cases-scattering as a pure refraction effect for zZ/137p >> 1 (classical approximation), and scattering as a pure diffraction effect, zZ/137p << 1 (Born approximation)-have been considered; in both expressions z is the particle charge, Z the charge of the nuclei of the scattering material, and the ratio of particle to light velocity. The probability function for single scattering events serves as a point of departure for the treatment of the “multiple scattering” problem, which deals with the resultant deviation of particles, having traversed a finite thickness of matter. It is the resultant deviations which are observed in experimental investigations, e.g., the broadening of a narrow particle beam, after passing through a metal foil, or the deflections of the trajectories of particles in cloud chambers or photographic emulsions. The aim of the multiple scattering theory is to correlate the resultant deviation per unit path length with the characteristics of the scattered particles. Before discussing the theory we must consider the fact that,

390

2.

DETERMINATION O F FUNDAMENTAL Q U A N n T I E S

in addition t o many small deviations, large single scattering events occur occasionally which will contribute to the resultant deviation. On the other hand “plural scattering” (resultant of only a few single scattering events) can be neglected, unless extremely thin foils or very short track segments are considered. have contributed t o the solution of the multiple Many scattering problem. There are certain differences in the treatment of the problem, which are due t o differences in the choice of the scattering potential, in the mathematical methods used, and the rigor of the calculations. They differ also in the manner in which the limits of small and large scatt,ering angles are approximated and in the solution of the screening problem. I n some papers the calculations refer to the total spatial angle 8, while in others to 9,the projected angle in the plane of the initial particle trajectory. I n the latter case the mean square value of 4 (which, a t least for small angles, is one-half of the mean square value of 8 ) or the lateral displacement is evaluated. A critical survey of the various theories has been presented by Bethe.lo I n the following we will discuss the multiple scattering problem as it is treated in Williams’ theory, this being the first one t o include multiple as well as single scattering events. He has derived the following expression for the mean of the absolute value of the resultant deflection:

<4>

=

(1

;:)1’2 2e2xZ(Nt)1/2 n PP

= 6L

(2.2.1.1.5.1)*

where N is the number of atoms per unit volume, Z is the atomic number of the scattering medium and x , p (in units of Mev/c) and /3 (in units of c ) are charge, momentum, and velocity, respectively, of the particle; t is the path length in the scattering medium; L, on the right side of (2.2.1.1.5.1), is the logarithmic part of the equation, while 6 represents

* One can account [Z(Z

+ 1)]1’2.

for the scattering on atomic electrons by replacing Z with

1 E. J. Williams, Phys. Rev. 47, 568 (1939); Proc. Roy. SOC. A169,:531 (1940); Revs. Modern Phys. 17, 217 (1945). 2 5 . A. Goudsmit and J. L. Saunderson, Phys. Rev. 67, 24 (1940).1 3 B. Rossi and K. Greisen, Revs. Modern Phys. 13, 240 (1941). L. A. Kulchitsky and G. D. Latyshev, Phys. Rev. 61, 254 (1942). G. Moli&re,2. Naturforseh. 2a, 133 (1947); 3a, 78 (1948). 6 H. S. Snyder and W. T. Scott, Phys. Rev. 76, 220 (1949). 7 Y. Goldschmidt-Clermont, Nuovo cimenlo [9] 7, 331 (1950). 8H. W. Lewis, Phys. Rev. 78, 526 (1950). 9 W. T. Scott, Phys. Rev. 86, 245 (1952). 10 H. A. Bethe, Phys. Rev. 89, 1256 (1953).

2.2.

DETERMINATION OF MOMENTUM A N D ENERGY

391

an angular unit, which is basic t o all theories; the physical meaning of 6 may be derived from the statement that the total scattering probability through an angle greater than 6 is exactly one. The calculations refer to deviations projected in the plane of the trajectory. Equation (2.2.1.1.5.1) was derived by using Rutherford's law for the single scattering probability in the approximation of small angular deflections (sin 41 = $z), and integrating between the limits $1, and $z. Williams approximates the upper limit by choosing an angle of such magnitude, that there is, on the average, only one deviation $ > $2 throughout the whole scattering thickness t . This angle is then $2

=

(7r/2)'/26

where 6 is given in Eq. (2.2.1.1.5.1).The lower limit $1 of the deviation is determined by the screening of the nuclear charge for particles passing the nucleus a t distances equal or smaller than the K-shell radius. Williams calculates the minimum deflection, assuming the validity of the FermiThomas model; he then calculates the screening effect for slow particles [y = (zZ/137p) >> I] according to classical methods, and for fast particles [y = (zZ/137p) << 11 by applying wave mechanics in which he utilizes the Born approximation. Introducing the calculated values for $2 and in Eq. (2.2.1.1.5.1) one obtains2equations

(s)']'''

In, 0.20n~t2-2/3

=

6(ln, M , l ) 1 / 2 (2.2.1.1.5.2a)

and

~ l E ~ Bare the mean angular deviations calculated (a) for where ~ 1 and y >> 1 and ( b ) for y << 1. In these equations h is Planck's constant divided by 27r and m the electronic mass. The quantities M are a measure of the mean number of collisions, a particle suffers in passing through a medium of thickness t. Thus far the calculations of the mean angular deflection refer only to the contribution of the multiple scattering, which is assumed to follow a normal distribution law. However, the total distribution function is determined by both, the multiple scattering contribution and the single scattering tail. Williams approximates the total distribution of all pro), jected angles by the probability function P ( a ) = G(a) s ( ( ~where G ( a ) is the multiple scattering or Gaussian portion and X(a) the single

+

392

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

scattering contribution. The lower limit of the single scattering disthereby modifying the Gaussian distribution occurs a t the angle c$~, tribution. The transition from multiple t o approximately single scattering occurs a t a n angle as,which is the intersection of the functions G ( a ) and X ( a ) and is given by as = 5 . 1 -~ 4.0, ~ ~ambeing the arithmetic mean of the Gaussian function G(a). The final result for the arithmetic mean projected deflection (due t o multiple and single scattering) is given by 5 =

(1.456

+ 0 . 8 0 ~ 1 )= 6[0.80(lne M)1/2+ 1.451.

(2.2.1.1.5.3)

Because of the approximations made in the derivation, Eq. (2.2.1.1.5.3) is only approximately valid, and then only within certain limits. However, comparison of Williams’ theory with the results of other authors show that the error is only a few per cent, if M e is larger than 10. For large values of M B , or correspondingly large path lengths, the contribution of single scattering events becomes negligible and the distribution is normal, being determined only by the multiple scattering probability. MoliBre’s theory, while also limited to small angular deviations, treats the problem more rigorously. Molihre calculates the single scattering cross section by using exact quantum-mechanical methods; he then derives an expression from this cross section for the screening angle 41 in the form of a n interpolation formula. The latter is valid not only for y >> 1 and y << 1, but also in the intermediate region with sufficient accuracy. The screening angle for a Fermi-Thomas potential is given by =

0.’”’’ 468513 [1.13 + 3.76y2]1/2.

(2.2.1.1.5.4)

is the only parameter in MoliBre’s theory inasmuch as the angular distribution depends only on 6/41, the ratio of the angular unit to the which screening angle $ 1 . Molihre introduces the parameter Qb = is a measure of the number of collisions suffered by a particle traversing the distance t in the scattering medium. The transition from the multiple scattering region t o the single scattering tail is found analytically without the introduction of the rather arbitrary upper limit, +2. Molihre has shown that this transition occurs much more slowly than assumed by Williams. The resultant deviation of multiply scattered particles is calculated from successive single scatterings to which Wentzel’s statistical method is applied. The results are expressed in the form of a series of decreasing powers of the parameter B, which is determined by the equation:

B - In, B

=

In,

a b

- 0.115 or %

=

1.116 e B / B .

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

393

The distribution function for projected deviations is given by

where cp = 2/@] and f1 and f z are functions whose values for various angular variables q are given in Table I. fl(cp) and f2(cp) can be TABLE I

0 0.2 0.4 0.6 0.8 1.o 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.5 4.0

0.0206 0.0246 0.1336 0.2440 0.2953 0.2630 0.1622 0.0423 0.0609 0.1274 0.147 0.042 0.1225 0.100 0.078 0.059 0.045 0.0316 0.0194

0.416 0.299 0.019 0.229 0.292 0.174 0.010 0.138 0.146 0.094 0.045 0.049 0.071 0.064 0.043 0.029 0.010 0.001 0.006

approximated by the following equations : (2.2.1.1.5.Ga) and

where C is the Euler constant. Equation (2.2.1.1.5.5) is only an approximation, inasmuch as the series expansion does not include powers of 3 larger than 2. However, it can be shown that the error is only about 1% if B is a t least = 4.5 or Qe = 20. The first term in the expansion is the well-known Gauss function, while the second and third terms represent the single scattering tail and the transition from one region to the other.

394

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

Finally, the projected mean deviation,

< +>, is given by

+

1

* * = 6Lm. (2.2.1.1.5.7) B2 This expression is similar to Williams’ with the exception that the equation is valid for all values of y and the value of L, differs from the corresponding term in Eq. (2.2.1.1.5.1). The theory of Molibre is today the most generally accepted because it provides relatively precise results without an inordinate amount of mathematical The theory is analytical (with the exception of the functions f 1 and f 2 , which, however, can be approximated by simple equations), while most other exact theories present the data only in numerical form (tables). Molibre’s theory which is calculated for total spat,ial angle, projected angle, and lateral displacement has the additional advantage that it can be easily applied to various special problems. The case for emulsions will be treated in the following section. Bethelo has reinvestigated the transition from multiple scattering to the single scattering t,ail and represents this region by an asymptotic formula. This theory simplifies the calculations and seems to be in excellent agreement with experimental data. Of all the other theories we mention only the work of Scott and Snyder,6 Scott,s and Goudsmit and Saunderson.2The mathematical treatment by the first two authors is identical with Molibre’s except for the fact th a t a different single scattering potential is used. Their theory is limited to small angles and high-energy particles and has the disadvant,age that the results are given in numerical form. The latter shortcoming is also characteristic of Goudsmit and Saunderson’s theory, which, however has the great advantage of being valid for all scattering angles. For comparison with experimental values, we can write the projected mean absolute scattering angle in a form which is common to all theories:

(2.2.1.1.5.8)

K is called the scattering constant and is given by K = 2e2ZN1/2L, where L has the same meaning as in Eq. (2.2.1.1.5.1). Since L varies slightly with particle velocity and length of the scattering cell, K is not actually l 1 B. P. Nigam, M. K. Sundaresan, and Ta-You Wu [Phys. Rev. 116, 491 (1959)] point out that approximations made in the derivation of Moli&re’sdistribution function neglect terms in zZ/137 which would become important for large scattering angles. Therefore, the scattering of electrons traversing large thicknesses of material must be treated differently. l2 Calculations solving this problem were recently carried out by G. Molibre, Z . Physik 166, 318 (1959).

2.2. DETERMINATION

OF MOMENTUM AND E N E R G Y

395

a constant. However, to a first approximation it may be considered as such, and then the mean scattering angle depends upon the nature of the scattering medium, the length of the scattering cell, a n d the particle charge and the product of the particle momentum and velocity. Therefore, scattering measurements give information on particle charge, mass and velocity. The product p p has twice the value of the kinetic energy for slow particles and is equal to the momentum for relationistic particles ( b + 1). The scattering constant can be calculated for various scattering materials from Eq. (2.2.1.1.5.8),taking into account the slight dependence K can also be found experiof K upon p and t [see e.g., Eq. (2.2.1.1.5.2)l. mentally, if particle trajectories of carefully determined characteristics are available. I n mixed media the computation of K is complicated, since all corrections due to screening vary with the atomic number of the scattering medium. 2.2.1.1.5.1.2.Experimental method. The occurrence of successive angular deviations in low-energy particle trajectories is quite evident; it is often possible, with the aid of experience, to estimate qualitatively the momentum of a particle simply from the geometric appearance of its track. However, in order to obtain a precise value of the momentum, or rather the product of momentum and velocity, it is necessary to construct techniques of measurement which will yield the mean scattering angle. Between the years 1948 and 1950, various methods of accomplishing this task were proposed and attempted. Goudsmit and ScottI3 proposed the difference between the actual track length and the chord length (a straight line between two points in the track) as a measurable parameter. Lattimore14 used the angle between adjacent chords of segments of given lengths of track, or cells, as they are called, while GoldschmidtClermont et aZ.16 and Davies et a1.16 measured the angle between the tangents to successive cells. While most of the earlier researchers employed a projection method of measurement, Goldschmidt-Clermont and Levi-Setti" abandoned the latter in favor of direct microscopic observation. The angular measurements were then performed with a highprecision protractor which could measure to within a fraction of a minute. The relation between mean scattering angle and pfi for singly charged S. A. Goudsmit and W. T. Scott, Phys. Rev. 74, 1537 (1948). S. Lattimore, Nature 161, 267, 518 (1948). 16Y. Goldschmidt-Clermont, D. T. King, H. Muirhead, and D. M. Ritson, Proc. Phys. SOC.(London) 61, 183 (1948). 16T. H. Davies, W. 0. Lock, and H. Muirhead, Phil. Mag. [7] 40, 1250 (1949). R. Levi-Setti, Nuovo cimento [9]8, 96 (1951). 13

l4

396

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

particles was calculated from equation a = Kt1/2/p/3.*In the expression for the scattering constant, K = 2e2ZN1/2L1,Z W 2 was replaced by ZiZ&P, where Ni is number of atoms of type i per unit volume of the emulsion, and Zi is the corresponding atomic number. The evaluation of L is more troublesome, inasmuch as it depends not only on properties of the emulsion, but also on particle velocity and length of the cell used in measuring c, Once the mean scattering angle has been measured, the momentum of a singly charged particle may be obtained from the relation, p . p = K [ L ] 112. (2.2.1.1.5.9) n,(t) 100 Here, t is the cell length employed in the measurement of e,(t), p is the momentum in Mev/c, and p is the velocity in units of c. The mean scattering angle in the above expression is of course the through the true angle, and is related to the experimental value, cerp expression cexp = +Y2, Et2 (2.2.1.1.5.10)

+

where E~ is the total error. The latter quantity may be expressed as the resultant of several contributing factors, Ef =

z/n,z

+ + Er2

Eo2

(2.2.1.1.5.10a)

according to the work of Goldschmidt-Clermont et U Z . ’ ~ The error c8arises from imperfections in the microscope stage movement (see Section 1.7.4), which may indicate angular deviations which do not really exist or exaggerate the magnitude of true deviations. The stage error varies with the cell size as tz, where x lies between 0.2 and 0.5. This quantity must be determined experimentally through careful interferometric measurements. The Bannister stage (Bristol) is accurate to within 0.1 p per cm of movement, while the Koristka and Chicago movements’8 are supposed to deviate less than 0.05 p per ern and 0.03 p per cm, respectively. The observer error, E,, may be made very small by increasing the number of measurements made on a single track segment. Finally, E ~ which , is generally named grain error or “spurious scattering,’’ is attributed to the irregular distribution of grain centers about a straight line or to other irregularities and dislocations in emulsion. The spurious error becomes very important for the trajectories of particles with high momentum, although it seems to have been underestimated in earlier experiments. The total noise level, at, may be obtained from observations of particles of very high energy, whose scattering angle is known to be negligible. * S e e Eq. 2.2.1.1.5.8forz = 1. l 8 M. J. Berger, J. J. Lord, and M. Schein, Phys. Rev. 89, 850 (1951).

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

397

A statistical deviation must be added to the noise level. If the distribution of scattering angles were purely Gaussian, then the root mean square of measured angles would be the best representation of standard deviation. However, as has been pointed out earlier, the admixture of single and plural scattering events gives rise to a non-Gaussian tail in the distribution. The effect of the non-Gaussian contribution on the measurements is minimized by the introduction of a cutoff angle which is commonly chosen as four times the mean value of the distribution. Although the method of the Brussels group is sufficiently accurate even for measuring very small angular d e v i a t i ~ n , ’ this ~ ? ~technique ~ has been largely replaced by the “coordinate” or “sagitta” method,20-22which provides a measurement of angles between successive chords. There are, however, other methods in use, like that of M ~ D i a r r n i dwho , ~ ~ effectively measures angles between tangents by considering the resultant deviation of coordinates in distant cells (not neighboring cells). M a b b ~ u utilizes x~~ a further variation in technique in which he measures deviations in the vertical coordinates; the author claims that such measurements are less dependent on stage movement and emulsion distortion. 2.2.1.1.5.1.3. Experimental procedure in the coordinate method. Fowler’s method is more convenient to use, faster, and gives results which are at least as accurate as the direct angular measurements. The track is first alligned parallel to one of the microscope stage movements-the X axis, for example-thereby permitting the y coordinates, which are measured a t the endpoints of consecutive cells, to record the multiple scattering of the particle trajectory. The measurement is performed under high magnification (90 X or 100X immersion objective) and the ordinary eyepiece scale is replaced by a filar micrometer, which is read to within a few hundredths of a micron. The scale of the filar micrometer is adjusted normal to the track, while the moving hairline is placed along an imaginary line passing through the centers of several grains. This procedure has the effect of smoothing the data and must be considered in the calculation which relates coordinate differences to angles between chords. The difference between coordinates of adjacent cells, (AYlt) = [(Yi -

2/1+1)/4

R. Levi-Setti and G. Tomasini, NUOVO cimento 8, 994, 1951. H. Fowler, Phil. Mag. [7]41, 169 (1950). 2 1 K. Gottstein, M. G. K. Menon, J. H. Mulvey, C. O’Ceallaigh, and 0. Rochat, Phil. Mag. [7] 42, 708 (1951). 22 M. G. K. Menon, C. O’Ceallaigh, and 0. Rochat, Phil. Mag. [7] 42, 932 (1951). 93 I. B. McDiarmid, Phys. Rev. 84, 851 (1951). 24 C. Mabboux, Compt. rend. acad. sci. 232, 1091 (1951). 19

28P.

2.

398

DETERMINATION O F FUNDAMENTAL QUANTITIES

divided by the cell length t, is a measure of the slope of the track with respect to the X movement. For accurate measurements, the slope angle should be very small inasmuch as the small angle approximation is used. Second differences between the y coordinates of successive cells, again divided by t, approximate the angles between successive chords, in radians.

The mean scattering angle, in degrees, is then given by (?I = ( I ) / t )(18O/?r), where D = A2y = XlAi2yl/n is the arithmetic mean of second differences in a given segment of track; the energy loss in this segment must be negligible if E is to be a meaningful quantity. It is necessary, in addition to the above computation, to investigate separately the sums of the positive and negative deviations from the mean scattering angle. If the deviations are the result of random processes, then the positive and negative deviations should be equal, whereas, if they differ considerably, then distortion is suspected. The problem must then be treated in a manner which will be discussed later. If no distortion is present, one eliminates all second differences which exceed largely the mean value considering the fact that some of these large deviations might be due to single and not multiple scattering (see Section 2.2.1.1.5.1). The criterion commonly accepted is the elimination of second differences, forwhich 1A2yil 2 4 after this elimination a new value is found which is smaller than the originally calculated value. Finally, it must be recalled that the observed mean scattering angle, iti,,,,is related to the true value, &, by Eq. (2.2.1.1.5.10). The total noise level in the angular measurement, ( ? I ~is derived from three major sources of error, according to the relation (2.2.1.1.5.10a). These quantities are assumed to be statistically independent of each other and to have a Gaussian distribution. It then follows that the mean value of the second differences (or the mean scattering deviation) is given by

n,

a

a;

q0= D& -

€2

(2.2.1.1.5.11)

where 6 represents the noise level as it appears in the coordinate method. are the true and measured mean scattering deviations respectively. Unless otherwise noted, they refer to the standard 100 fi cell length. It is clear that the results are reliable only if the true scattering deviation exceeds the deviation due to the noise level. When the true scattering angle is small [Eq. (2.2.1.1.5.1)], it is measurable only if relatively large cell sizes are chosen. The mean scattering deviation is proportional to Pi2,while the stage error, which varies with t

Dm,and D,,,

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

399

to a power which is less than +, is the only source of error which is a function of the cell length. It is therefore evident that the signal-tonoise ratio will increase as the cell length increases. On the other hand, a larger cell size implies fewer cells and consequently a greater statistical error. Therefore, an optimum cell length must be chosen with considerable care. Although the choice depends primarily on the energy of the particle, it will also be affected by emulsion variables, e.g., distortion and degree of development. Most workers endeavor to obtain a mean scattering deviation not smaller than 0.1 p and a signal-to-noise ratio of at least 4 to 1. Levi-Settilg systematically investigated the problem of noise level in several modes of scattering measurement, including the angular and ordinary coordinate methods, as well as various modifications of the coordinate method. He found a slight but definite superiority of the angular method over the others. The contribution of the noise level to errors in scattering measurements has also been treated extensively by other authors. 248-29 Biswas et aL30 describe the existence of another type of error, called spurious scattering. It is believed that this effect is related to dislocations in the emulsion which sometimes affect a macroscopic section of the plate. That this is true was proved by comparing the contours (mean scattering deviation versus the length of track in emulsion) of neighboring parallel tracks. The contours of these tracks exhibit peaks and troughs in the deviation which are different in magnitude, but at the same position in the emulsion; similar results were found in many different sets of plates. This effect is negligible for particles of low energy, but is significant for high-energy trajectories, where the spurious scattering is often larger than the true scattering. I n their observations on the tracks of heavy primaries, the authors determine the error which would be introduced if the spurious scattering were not taken into account. They find that spurious scattering is proportional to t 3 I 2 . The existence of spurious scattering was later confirmed by Fay31 and Lohrmann and T e ~ c h e r ; ~ ~ these workers, however, find that the effect is directly proportional to 24a

26

26 *?

G. Moli&re,2. Naturforsch. !la, 78 (1948).

G.Molihre, 2.Naturforsch. 10a, 177 (1955).

J. E.Moyal, Phil. Mag. [7]41, 1058 (1950). W.T.Scott, Phys. Rev. 86, 245 (1952). M. J. Berger, Phys. Rev. 88, 59 (1952).

B. D’Espagnat, J . phys. radium 13, 74 (1952). 80 S. Biswas, B. Peter, and Rama, Proc. Indian Acad. Sci. A41, 154 (1955). 31 H . Fay, Z . Naturforsch. 10a, 572 (1955). 32 F.Lohrmann and M. Teucher, Numo cimenfo [lo]6, 59 (1956).

2Q

400

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

the cell length t. Brisbout et ~ 1 find . that ~ ~the magnitude of this type of error is smaller and not as widespread as reported by the other authors. While the solution of the complex problem of noise level is important for all energies, it acquires a special significance in the study of highenergy particles. If not analyzed carefully the noise level may vitiate the usefulness of the scattering measurement; the latter is often the only means of obtaining the momentum of the particle if the grain density has a minimum or plateau value. One must, therefore, either devise methods for the measurements of all the noise components, or find procedures which will lead to the elimination of noise. The latter has been generally accomplished by the comparison of scattering angles or scattering deviations which were obtained from measurements on the same track with two or more different cell lengths. Such a procedure was first applied to Fowler’s coordinate method by Menon et aLZ2and Gottstein and M ~ l v e ybefore , ~ ~ the spurious scattering effect was known. 2.2.1.1.5.1.4. Noise elimination. This method is based on the combination of two scattering results which are performed on the same track element with two different cell lengths. As a first approximation the scattering constant is assumed to be independent of cell length. Recalling that the mean scattering angle varies with tl’z [Eq. (2.2.1.1.5.9)], then it is apparent that the mean second difference due to true scattering will increase as iY2.The square of the mean scattering deviation, D,.“,,, is then given by: QXP = D33 €2 (2.2.1.1.5.12)

+

where Z is the total contribution of the noise to the deviation, D.. is the true scattering deviation for a cell of length t, while t in this expression is measured in units of 100 p . If were independent of cell length, it could be eliminated by comparing the mean scattering deviations obtained by measurements on the same track using two different cell lengths. If small cell sizes are employed, as is generally done in the case of low momentum, the error due to stage noise, as well as its percentage increase with t, is negligible compared to grain and reading error. Then, the mean scattering angle for 100 p cell lengths (the standard t o which all measurements are referred) is (2.2.1.1.5.13)

where B is a constant which depends on the microscope magnification 88 F. Briebout, C. Dahanayake, A. Engler, P. H. Fowler, and P. Jones, Nuovo eimento [lo] 3, 1400 (1956). a4 K. Gottstein and J. H. Mulvey, Phil. Mag. [7] 42, 1089 (1951).

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

40 1

and on the conversion of radians to degrees, and the subscripts refer to the measurements with the two different cell lengths; DIz and DZzare the second differences measured with cell sizes tl and tP. Brisbout et aE.33and G ~ t t s t e i nrepresent ~~ the dependence of stage noise on cell length by e:t t2z, where x is determined empirically for each microscopic stage. The mean scattering angle is then given by

-

where & O O refers as in (2.2.1.1.5.13) to the mean scattering angle calculated for a 100 p cell, A is a factor, converting the readings in scale divisions or microns to degree/100 p and tl and t z are the two cell lengths, used for the measurement. However, &OO is a measure of the true scattering angle only if certain additional conditions are satisfied. The authors found that the inequality (DI/&)~ > ( t ~ / t represents ~ ) ~ ~ a necessary, but not sufficient condition. Equation (2.2.1.1.5.14) does not include the dependence of spurious scattering on cell size. Biswas et ~ 1investigated . ~ ~ carefully the noise level problem in the more complicated case of large spurious scattering. Because these authors find that deviations due to spurious scattering obey a t 3 I 2 law, it appears that measurements with two different cell sizes will not yield the true scattering, which exhibits a similar behavior. It is indeed unfortunate that the issue of spurious scattering is not completely resolved, as it may seriously limit the reliability of scattering measurements in the Bev energy range. If, however, the tracks of highenergy particles are parallel and lie near each other (within the same field of view at high magnification), then the problem may be solved expeditiously with the aid of the method of relative ~ c a t t e r i n g . ~Here, ~-~~ the mean deviation between the readings on two parallel tracks contains only grain and reading errors, but is independent of stage error and spurious scattering. If the mean relative scattering is compared with either or both of the parallel tracks, then it is possible to evaluate the magnitude and distribution of the spurious scattering. If eBp is then known, the true scattering deviation of other tracks situated in the same part of the emulsion may be obtained through application of appropriate correction terms. Nearly parallel tracks may be found easily in plates exposed to accelerator beams, in the showers of high energy interactions, or the fragmentation of heavy primaries (Biswas et aLao lists many references on this subject). 36

K. Gottstein, Nuovo cimento [9] 7, 619 (1954).

402

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

2.2.1.1.5.1.5. Dip and distortion in scattering measurements. The above discussion has been limited to scattering measurements on flat tracks. When measurements are to be made on inclined tracks, appropriate corrections must be applied in order to convert projected lengths into true lengths. Because of the difficulty of measuring the scattering in steeply inclined tracks, such observations are often unreliable. Dipping tracks are also more frequently subject to large distortion errors, as has been pointed out earlier. The presence of emulsion distortion may be recognized, as mentioned above, by the difference between positive and negative second differences. This effect is, of course, very important and must be considered. Fowlerz0corrects for distortion by introducing a parameter 4, which added to the true scattering deviation D, represents the value for the apparent or observed deviation: D, = Dm (6. The distortion parameter is approximately equal to 4 = (ZD,/n), where ZD, is the algebraic sum (not the sum of the absolute values) of the observed deviations. I n this way one takes into account that distortion deviations are all of a single sign, while in true scattering phenomena positive and negative deviations are approximately equally distributed. However the method of third differences, and in severe cases even fourth differences, is more commonly used for distortion elimination. The mean sagitta D 3is in this case given by

+

D 3

=

J'D.3)" 3(n3

(D13)2

- 1)

where Dn3 and DI3 are the third differences taken with cell lengths in the ratio n : 1; the factor Q takes into consideration correlations between third and second difference^.^^ Later investigation^,^^^^^ however, have shown that the mean sagitta, derived from this expression is too high and should be replaced by the empirical equation:

Rosendorf and Eisenberg3* have carefully evaluated the numerical factor using MoliBre's theory and found that the factor depends somewhat on the number of collisions the particle suffers per unit path length. 36 S.Biswas, E. G. George, B. Peters, and M. S.Swami, Nuovo cimento (Suppl.) [9] 12, 369 (1954). 37 M. DiCorato, D. Hirschberg, and B. Locatelli, Nuovo cimenlo (Suppl.) [9]12, 381 (1954). S. Rosendorf and Y. Eisenberg, Nuovo cimento [lo] 7, 23 (1958).

2.2.

DETERMINATION O F MOMENTUM A N D ENERGY

403

It is apparent that only simple types of distortion-those which can be approximated analytically-can be considered, while complex distortions often render scattering measurements useless. 2.2.1.1.5.1.G. Scattering measurements on trajectories of slow particles. The technique of measuring multiple scattering in slow particle trajectories differs from that for high-energy particles where the rate at which energy is lost is negligible. On the other hand slow particles lose energy more rapidly in traversing the emulsion. This necessitates the subdivision of the available track length into segments of such lengths as are compatible with the rate of energy loss. The mean scattering angle of each segment is then computed. However, because of the small number of cells, the statistical accuracy of the measurements may be low. Another difficulty which arises is the problem of choosing an appropriate cell length for long tracks which stop in the emulsion. The optimum length at the high-energy portion of the track will certainly be different from that near the end, where the scattering is large and the energy loss rapid. Menon and R ~ c h a approach t~~ the problem by making scattering measurements only on the higher energy part of the track; the particle momentum may then be determined from these measurements and from the range-energy relation. The method of “constant sagittal’ is now used almost exclusively to measure the multiple scattering of slow particles, stopping in the emulsion. This idea was proposed independently by Biswas et aL40 and Dilworth et aL41 The principle of the method differs from the technique described above in that the cell length is varied in such a way as to maintain the measured scattering deviation at a constant value. It is clear that the cell sizes must be determined in advance in order to perform this type of measurement. Both groups of a u t h o r ~have ~ ~ ,con~~ structed cell schemes for singly charged particles of given mass on the basis of the range-energy relation. The “constant sagittal’ method has proved successful in the mass determination of unknown particles and will be treated in greater detail in Vol. 5, B, Chapter 2.3. 2.2.1.1.5.1.7. The scattering constant. K = 2e2ZN1’2L,can be determined experimentally as a function of particle velocity and cell size by scattering measurements on trajectories of particles with known characteristics. The Bristol g r o ~ p ~has~ performed , ~ ~ - scattering ~ ~ ~ ~ ~ ~ ~ measurements on fast electrons and positrons and on 336-Mev protons, comparing the results with the predictions of Molihre’s theory. They M. G. K. Menon and 0. Rochat, Phil. Mug. [7] 42, 1232 (1951). S. Biswas, E. G. George, and B. Peters, Proc. Indian Acud. Sci AS8, 418 (1953). 4 1 C. C. Dilworth, S. J. Goldsack, and L. Hirschberg, Nuovo cimento [9] 11,113 (1954). 4 2 C. O’Ceallaigh and 0. Rochat, Phil Mag. [7] 42, 1050, 1951. 39

40

404

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

find that the dependence of L on cell length for p + 1 is given by

L(t) = 1.45

+ a (log

t)l12

the experimental value of a being 1.19. I n order to obtain the dependence of L on particle velocity the authors have evaluated Qb/t MoliBre’s parameter, characterizing the number of collisions per 100 p, as a function of 0. From f i b the parameter B [fh,= 1.167eB/B, (see Section 2.2.1.1.5.1)lcan be found and therefore the scattering constant K expressed as function of B [compare Eq. (2.2.1.1.3.7)l.

K

=

8.203B112(1

+ 0.982 7 - 0.117 B2 __

degrees

(2.2.1.1.5.15)

can be calculated and compared with experimental values. The scattering refers to angles between chords and to constant given in (2.2.1.1.5.15) a sampling method in which all second (or third) differences, regardless of magnitude, are included. More frequently used is the scattering conreferring to measurements from which all differences exceeding stant KO,, four times the mean value have been omitted. The relation between the two scattering constants can be calculated by integrating the probability function from zero up to four times the minimum value of the observed deviation. The experimental values of the scattering constant Kc, with values of various a ~ t h o r are s ~given ~ ~ in ~ ~Table 11. The table is taken from a paper by Voyvodic and P i c k ~ pIn. ~a par~ ~ ~ ~ ticularly detailed analysis, these authors have compared their experimental data-electron pairs of y rays from Be8-with calculations based essentially on Williams’ theory. * The authors compute the parameter M [ M = (%/a)]* as a function of p and t and combine it with MoliBre’s ~ )providing the transition from the classical expression (1.13 3 . 7 6 ~ ,t to the Born approximation case (y >> 1 + y << 1). The scattering constant is then given by

+

K O= 11.9(1

+ 0.837[10g10 0.94t/(P2 + 0.30)”2])

(2.2.1.1.5.16)

where KOrefers to angles between chords and to a sampling without cut off;in the derivation of K Oinelastic electron scattering has been considered. The scattering constant with cutoff a t 4n is given by

KO.= 8.21ac0

* See Section 2.2.1.1.5.1.

t This procedure has been used first by Goldschmidt-Clermont (see reference 15). D. R. Corson, Phys. Rev. 83, 850 (1951). L. Voyvodic and E. Pickup, Phys. Rev. 86, 91 (1952). 4J Compare also L. Voyvodic, Progr. in Cosmic Ray Phys. 2, 217 (1954).

48

44

405

2.2. DETERMINATION OF MOMENTUM AND ENERGY

where ao0= [n - ( r / 4 a ) ] / [ l- (r/32n2)] and 8 is the mean scattering angle without cutoff. The values of KO. calculated by Voyvodic and Pickup are given in Ta,ble 11. These are about 1% higher than the values derived from Molihre’s theory, if in both cases corrections for inelastic TABLE I1 Scattering constant

KO, Cell length microns

Experiment

--

Gottstein et aZ.21Js Positrons 105 Mev Positrons 185 Mev Protons 336 Mev Protons and mesons 5-50 Mev Corson4a Electrons 40 280 Mev Berger et ~ 1 . 1 8 Protons 337 Mev Voyvodic and Pickup44 Electron pairs Be8 gamma rays (Mean energy 16.7 Mev)

(expt.)

-

(theor.)

-_-

1 1 0.46 0.14

200 400 600 80

26.2 24.0 29.2 26.1

f 0.6

1

Not given

26

k1

0.46

250 500 750

24.4 f 0 . 8 24.5 f 0 . 8 24.6 f 0 . 9

26.5 27.4 28.0

1

45 (20-70)

21.2 f 0 . 7

22.1

k 0.8 k 1.0 f 0.7

25.3 26.4 27.7 25.6

electron scattering are applied.44The experimental values of Kooin most calibration experiments (compare also Brisbout et u Z . ~ ~ )seem to be in fair agreement with the scattering constant cdculated by Voyvodic and Pickup. In the comparison of experimental and theoretical data it is very important to consider carefully the actual composition of the emulsion. Fichtel and Friedlar~der~~ have evaluated the scattering constant for “standard emulsion” (Barkas) and the curve in Fig. 1 gives the correction factor with which the scattering constant has to be multiplied in order to account for changing humidity conditions; the “standard emulsion” has a water content corresponding to 61 % relative humidity. The curves in Fig. 2 are plots of K Oand KO,for singly charged particles (calculated for standard emulsion) as a function of cell length; the curves are drawn for three different velocities; p = 1, 0.75, and 0.5. The authors have also computed the scattering constant for a particles and particles of higher charge. They find for Q particles KO,= 25.9 ( p + 1, t = 100). The scattering constant increases slightly with increasing particle charge. 48

C. Fichtel and M. W. Friedlander, Nuovo cimento [lo] 10, 1032 (1958).

406

2.

DETERMINATION O F FUNDAMENTAL Q U A N T I T I E S

Relative humidity ('4

FIG. 1. Correction factor for the scattering constant plotted versus relative humidity (per cent). (This figure is taken from the paper of Fichtel and fried land^.'^)

and KO,(Kcut0fr)as a function of cell FIG.2. Scattering constants K O (Kno size for singly charged particles with velocities p = 1.0, 0.75, and 0.5. (This figure is taken from the paper of Fichtel and F ~ i e d l a n d e r . ~ ~ )

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

407

For particles of charge 4, K,, = 26.2 (p+ 1, t = loo), and does not further increase with increasing charge. 2.2.1.1.5.1.8. Error determination in scattering measurements. Because of uncertainties in the determination of the scattering constant, it has been proposed4' that the statistical error be increased by 8%; if, however, the scattering constant has been found for a particular set of emulsions through calibration measurements, then the calibration error must be used instead. The relative statistical error in scattering measurements is taken /as 0 . 7 5 / d q , where N is the number of distinct, nonoverlapping cells; the above formula assumes that normal distribution has been reached. The noise level error is given by (Z/D) (0.75/.\/z), where N , is the number of cells used in the noise level determination, r ) is the measured total scattering deviation, and i the mean sagitta of noise. If the noise is eliminated by employing two different cells, then the relative root mean square error in D., is: 0.75

(2.2.1.1.5.17)

Dw

where N 1 refers to the smaller cell, and Z/D is the noise-to-signal ratio. The latter expression includes neither systematic errors which may depend on cell length nor any error resulting from distortion or strongly dipping tracks. DiCorato et aL3' have calculated the relationship between D/Z and the ratio of mean deviations for large and small cells; the calculations are based on the theories of Molihre and d'Espagnat, the results of which are given in the form of curves for second and third differences. More detailed experimental and theoretical investigations concerning various sampling statistics have been carried out by Johnston48and the above-mentioned authors. 37 According to these investigations the numerical factor in Eq. (2.2.1.1.5.17) depends on the cutoff procedure used and is generally ~ 1 . times 3 larger than the given value. This increase is supposed to take into account the fact that the distribution of second or third differences deviates from a normal distribution. The maximum value of p p which can be measured successfully depends on the length of available track, the physical condition of the emulsion, and the precision of the stage movement. Because (pp),., varies as t3/2, it is desirable to choose cell lengths which are as large as possible; on the other hand the track must be sufficiently long to provide enough cells to ensure a small statistical error. Fortunately, the use of large emulsion 4 7 Recommendation for the Standardization of Scattering Measurements, Varema Conference, 1954. Published in Nuovo cimento 191 Suppl. Nos. 11 and 12 (1954). 48 R. H. W. Johnston, Nuovo cirnento (Suppl.) [lo] 4,456 (1956).

408

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

stacks in recent years has afforded researchers the opportunity of finding long trajectories. However, the presence of distortion and dislocations (spurious scattering) in the emulsion and imperfect stage movements introduce uncertainties which might increase the error, and may even render the scattering measurement useless. Scattering measurements on tracks of singly charged particles can determine the product p p provided /3 is already known. The reason for this lies in the dependence of the scattering constant on P. Only when p -+ 1, do scattering measurements lead directly t o a determination of the particle momentum. It is therefore necessary that scattering data be combined with ionization measurements (grain-blob or gap measurements), in order to establish the momentum of the particle. If the particle is multiply charged, then another parameter which is a function of p and z as, e.g., &ray count or the gradient of the grain density, must also be measured. In order to facilitate the identification of a particle it is convenient to employ calibration curves which relate Q (or an equivalent measure of the ionization) to p p for particles of known mass. It is evident that the combination of scattering and ionization measurements, providing the separate knowledge of particle momentum and velocity, leads to the determination of particle mass and energy. Finally we should mention a new approach to the scattering problem, introduced by Lipkin et al.49and Rosendorf and E i ~ e n b e r gThe . ~ ~authors recommend the use of the mean value of the cosine function instead of the mean absolute angle for the evaluation of p p . The factor q is a constant chosen to minimize the dispersion of p p , and the best value of 9 must be determined by trial and error. The cosine expression lends itself to analytic treatment and the errors for first, second, third, and even higher differences can be exactly determined. Therefore here, it is possible to obtain a theoretical dispersion of the measured values of pp, while in the mean absolute angle case, dispersion calculations can be performed only if cutoff procedures are introduced. 40

H. J. Lipkin, S. Rosendorf, and G . Yekutieli, Nuovo eimento [lo] 2, 1015 (1955).

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

409

2.2.1.2. Determination of Energy. 2.2.1.2.1. ENERGY MEASUREMENT IONIZATION CHAMBERS.*The energy loss of a charged particle passing through matter has been considered in Chapter 1.1. The processes involved are ionizing collisions, atomic excitations, and (for relativistic particles) cerenkov radiation. For a pure noble gas, the total amount of ionization liberated for a given energy loss is constant to about 1 %, over a wide range of type and energy of particle. Collection of ionization in a gas-filled counter therefore affords measurement of energy loss, or total energy if the particle stops in the counter. This technique, using an ionization chamber or proportional counter to measure the amount of ionization, was the most rapid and convenient method of energy measurement before the development of the scintillation counter, and it is still superior in some applications. The basic facts of the energy-ionization relationship are discussed in Section 1.2.4, which includes a table of W ,the average number of electronvolts energy loss corresponding to one ion-pair, in various gases; W for argon is 26.4 ev per ion pair. The lower limit to the amount of ionization which can be directly measured is set by amplifier noise; this problem is discussed in Section 1.2.5. The rms voltage due to noise corresponds to a few hundred ion pairs, so that the smallest energy loss which can be measured even approximately is a few thousand ion pairs or a t least a hundred kev. Where the signal is too small to use an ionization chamber, the proportional counter (Section 1.3.1, and below) offers a similar instrument, augmenting electrical amplification with gas amplification. When the energy available in the chamber is greater than 1 Mev-as is generally the case for fission and alpha particles, and often for protonsthe accuracy of the ion chamber is usually greater than that of the proportional counter, since the additional uncertainties due to gas amplification are avoided. However, the larger signal of the proportional counter is often a great convenience. If a particle loses all its energy in an ionization chamber, the statistical fluctuation in number of ion pairs is small (Section 1.2.4). If it loses only a fraction of its energy, the fluctuation in number of primary collisions’ will also have to be considered. [For a proportional counter one has also the statistics of gas amplification, Eq. (1.3.2).] Various difficulties which may decrease the accuracy and resolution WITH

t For some aspects of the phenomena here described see also Vol. 2, Chapter 4.1 and Vol. 4, A, Section 2.1.5. 1 See B. B. Rossi, ‘ I High-Energy Particles,” p. 29. Prentice-Hall, Englewood Cliffs, New Jersey, 1952.

* Section 2.2.1.2.1 is by R. W.

Williams.

410

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

of the ionization chamber as a quantitative instrument are considered in Sections 1.2.3, 1.2.4, and 1.2.6.1, and will not be discussed here. It must be pointed out, however, that for all electron-collection chambers, that is for the commonly used type of chamber which has a fast-time response, the variation of pulse height with point of liberation of the electrons constitutes a built-in limit to resolution (Section 1.2.2), even for relative energy measurements. For absolute measurements one must also consider the fact that the electron-collection time is not arbitrarily short compared to the clipping time (Section 1.2.2) and is not very well known. Wilkinson2 considers the resolution of a number of geometrical arrangements in detail. The highest resolution, when neither time-resolution nor pileup of background pulses is a problem, can be achieved with ion-collection chambers, in which one measures the total charge collected by the chamber, and avoids the uncertainties of electron-collection. I n the work of Jesse and collaborators, for example, on alpha particles, absolute energy measurements to about one per cent and relative measurements to a few tenths of a per cent were achieved. The energy measurement by ionization ultimately takes the form E = WQ/e,where Q is usually calculated from the measured output voltage V Oof an amplifier of gain G, Q = VoC,/Gf; the “electron-collection” fraction f (Section 1.2.2, 1.2.3) is, of course, unity for the ioncollection chamber. The capacity of the collector, C,, can be measured with the aid of a standard ~ a p a c i t a t o r when ,~ absolute measurement is involved. Usually, however, CJG is determined by calibration with alpha particles from Po21oor some other well-known source. Details of such a calibration procedure are given, for example, in reference 6 of Section 1.2.3.2. For extreme-relativistic particles, whose energy U is much greater than their rest energy Mc2, an energy-loss measurement with an ionization chamber affords a measure of U/Mc2. The relativistic rise in specific energy loss (Chapter 1.1) is not inhibited by the density effect in gases at moderate pressure (as it is in condensed matter), so that the specific ionization of an extreme-relativistic particle increases with energy. Energy measurements of this type, by using cloud chambers, are discussed in Section 2.2.1.2.3. It is noteworthy that an extreme-relativistic particle passing through a pure high-pressure heavy noble gas gives three distinct signals: cerenkov light, scintillation light, and ionization. 2 D. H. Wilkinson, “Ionization Chambers and Counters,” Chapter IV. Cambridge Univ. Press, London and New York, 1950. 3 W. P. Jesse and J. Sadaukis, Phys. Rev. 97, 1668 (1955).

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

41 1

2.2.1.2.2. SCINTILLATION SPECTROMETRY OF CHARGED PARTICLES.* t The technique of scintillation spectrometry represents a major advance in nuclear instrumentation. As a charged-particle spectrometer it possesses the advantages of simplicity, generally small dimensions, high efficiency for moderate resolution, and high speed. A disadvantage of the method is that its resolution, which is poor a t low energies, may be insufficient for some applications. I n this discussion, scintillators for charged-particle measurements will be grouped according to their composition, viz., inorganic, organic, or gas scintillators. 2.2.1.2.2.1. Inorganic Scintillators. The inorganic scintillators in current use will detect any nuclear radiation to some extent, so the choice of a scintillator for a particular application will depend not only on the sensitivity to the desired radiation, but also on the sensitivity to extraneous radiations which may be present. For example, the inorganic scintillators are very gamma-sensitive by virtue of their high density and high atomic number; therefore, to reduce the background interference from gamma rays, a minimum thickness should be used for measurements of charged particles. While the response of NaI(T1) and CsI(T1) to electrons produced by gamma-ray interactions within the crystal has been useful in gamma-ray spectroscopy (cf. Section 2.2.3.3), application of these scintillators to detection of electrons from external sources is not recommended,’ because their high effective atomic number leads to excessive electron scattering. This difficulty has been overcome to some extent by growing a NaI(T1) crystal with a trace of a beta emitter added to the melt.2 All of the beta rays are absorbed by the crystal except at the surfaces. If coincident gamma rays follow the beta transitions, this technique gives rise to summing effects which may be very difficult to interpret. A somewhat similar technique which achieves 47r geometry by sandwiching a thin source between two NaI(T1) crystals has been used by Bannerman, Lewis, and Curran3 in a study of HgZo3,and b y Ketelle, Thomas, and Brosi4 for the measurement of internal conversion coefficients and L / K capture ratios. Sodium iodide, activated by 0.1 mole per cent thallium iodide, is the scintillator generally used for spectrometry of heavy charged particles. Its

t See also Vol. 2, Section 11.1.3.3 and Vol. 4,A, Section 2.1.3. P. R.Bell, in “Bets- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), Chapter 5. Interscience, New York, 1955. 2 E. der Mateosian and A. Smith, Phys. Rev. 88, 1186 (1952). 3 R. C. Bannerman, G. M. Lewis, and S. C. Curran, Phil. Mag. [7]42, 1097 (1951). 4 B. H.Ketelle, H. Thomas, and A. R. Brosi, Phys. Rev. 103, 190 (1956).

*Section 2.2.1.2.2 is by G. D. O’Kelley.

412

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

fluorescence decay time (0.25 psec) is the shortest of the inorganic scintillators, and is fast enough to be employed in millimicrosecond timing applications, since its high light output makes it possible to obtain a signal from the first photoelectron of a high-speed, high-gain photomultiplier tube. Although the pulse height-energy relationship is important to spectrometry, a complete study of the response of NaI(T1) to energetic ions has not been made; however, several investigators have studied the linearity of this scintillator over limited ranges, from which it is possible to establish some important features.* The response of NaI(T1) to protons was reported by Allison and Casson6 to be linear in the energy range of 6&400 kev, with an intercept at the origin. Other workers confirmed the linearity for proton energies of 1-8 Mev6-8 and for molecular hydrogen ions of 1-11 Mev,6 but found that the pulse-height energy curve extrapolated to an intercept on the energy axis. These results imply a nonlinearity in the low-energy response between 1 Mev and the region investigated by Allison and Casson. The energy intercept was measured by Brolley and Ribe7 as 0.276 f 0.005 MeV, which disagrees with a recent result by Whetstone et aL8 of 0.46 i-0.10 MeV, probably because of differences in preparation of the NaI(T1) surfaces. The response of NaI(T1) to deuterons is similar to the proton situation: a linear pulseheight energy relationship was obtained from 60-630 kev,6 but the linear curve for the region of 1-11 Mev6.7 extrapolated to an intercept of 0.78 f 0.11 MeV. For alpha particles6v9it has been shown that the pulseheight response curve is quite nonlinear below about 10 MeV; above this energy the response is almost a straight line, although the final slope is not attained until well above 18 MeV. Scintillations in NaI(T1) induced by fission fragments are about twice as large as those produced by 5-Mev alpha particles,1° and the light and heavy fragments may be readily separated. The larger scintillation pulses are associated with the more energetic light-fragment group. Some data on the fluorescent response of NaI(T1) to heavy particles are shown in Fig. 1. Although the nonlinear response of inorganic scintillators to heavy charged particles is not as pronounced as for organic scintillators, both

* For pulse measurements, see also Vol. 2, Chapter 9.6. S. K. Allison and H. Casson, Phys. Rev. 90, 880 (1953).

C. J. Taylor, W. K. Jentschke, M. E. Remley, F. S. Eby, and P. G . Kruger, Phys. Rev. 84, 1034 (1951). 7 J. E. Brolley, Jr. and F. L. Ribe, Phys. Rev. 98, 1112 (1955). * A. Whetstone, B. Allison, E. G. Muirhead, and J. Halpern, Rev. Sci. Instr. 29, 415 (1958). F. S. Eby and W. K. Jentschke, Phys. Rev. 96, 911 (1954). lo J. C. D, Milton and J. S. Fraaer, Phys. Rev. 96, 1508 (1954). 6

2.2.

413

DETERMINATION O F MOMENTUM AND ENERGY

exhibit their deviations from linearity a t low energies, where the specific energy loss dE/dx is high. This behavior is believed due to a “saturation” effect in which the specific fluorescence dL/dx fails to increase linearly with dE/dx. I n the case of NaI(Tl), Eby and Jentschkeg find that dL/dx increases linearly with dE/dx to dE/dx 11 0.11 Mev/mg-cm-2, after which dL/dx increases more .slowly, attaining a peak value a t about 0.4 Mev/mg-cm-2; beyond this point, dL/dx decreases with increasing dE/dx. It may be noted that a nonscintillating layer, in which energy is lost as particles enter the scintillator, would cause the sort of low-energy behavior under consideration here ; however, the investigations of der Mateosian and Yuan’l into the surface effect of NaI(T1) for Po2loalpha particles showed any such effect to be less than 5%, and so it appears that saturation of dL/dx is the important mechanism. 2000

I

I

I

I

0

2

4

6

8

I

I

I

10 (2 44 ENERGY [MeV)

I

I

I

46

18

20

22

FIG.1. Pulse heights in NaI(T1) as a function of heavy particle energy and type (Taylor et aZ.8).

Information on the relative scintillation efficiencies of NaI(T1) for excitation by various charged particles is also fragmentary, but a few data may be noted for comparison purposes. Allison and Cassons reported relative scintillation efficiencies for positive ions in the energy range 60-600 kev of 1.00, 0.96 k 0.03; 0.75 i-0.05, and 0.54 k 0.03 for H, D, He, and Ne, respectively, and also noted that the response for electrons appeared to be about the same as for protons. In the intermediate energy region, Whetstone et aL8found the ratio of alpha-particle energy to proton energy for equal pulse heights to be 1.93 k 0.04 for 4.48-Mev alpha particles. At 20-Mev alpha-particle energy this ratio has been reduced to about 1.3, using as a guide the curves of either Taylor el aZ.6 or Eby and Jentschke.9 Cesium iodide, available commercially with 0.1 mole per cent thallium iodide as activator, has seen ever-increasing application in chargedLL

E. der Mateosian and L. C. L. Yuan, Phys. Rev. 90, 868 (1953).

4 14

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

particle scintillation spectrometry, Because it is not deliquescent, it is very easy to use, especially where thin wafers of scintillator must be prepared from a large piece of stock material. Some additional advantages of CsI(T1) will be discussed below, and for general properties of this scintillator see Section 2.2.3.3. Fluorescence decay times for a number of inorganic and organic scintillators have been measured, and have been found to depend upon factors such as temperature and activator concentration; however, CsI (Tl) exhibits what appears to be a unique dependence of fluorescence decay time 7 on ionization density, i.e., particle type. Some measurements by Storey, Jack, and Ward,12using various particles, showed r to be 0.425 k 0.025 psec for 4.8-Mev alpha particles, 0.52 k 0.01 psec for 2.2-Mev protons, 0.595 +_ 0.02 psec for 8.6-Mev protons, and 0.700 k 0.025 psec for electrons produced by 0.66-Mev gamma rays. The variation of T with average ionization density is similar to that observed by Knoepfel, Loepfe, and Stoll13 for the variation of 7 with temperature for alpha particles, which suggests that the dependence on ionization density may be due to variations in the effective local temperature. Such differences in fluorescence decay time will, of course, be manifested as differences in the rise times of pulses from the photomultiplier. A plot of the pulse height attained after 0.5 psec versus the height after 4 psec for alpha particles, protons, and electrons yielded three distinct loci for the experimental points, one for each type of particle. This prompted Storey, Jack, and Ward12 to suggest that a single CsI(T1) spectrometer might be employed to determine simultaneously the particle type and the energy. Becker and Biggerstaff l4 have developed an electronic circuit for distinguishing between different types of charged particles based on the variation of 7 in CsI(T1) with particle type. Protons, tritons, and alpha particles are readily separated by their decay time differences. The decay time separation of the alpha particles and protons is 6 times the instrumental resolution of the alpha particles, and the triton-proton separation 4 MeV, is twice the instrumental resolution of the protons for E , E, 3 to 5 MeV, and Et 3 MeV. This approach is particularly valuable for characterizing particles of energy less than about 5 MeV, at which energies the particle ranges are very short and more conventional methods, such as a combination of detectors to measure dE/dx and E , are difficult to use.

-

-

-

R. S. Storey, W. Jack, and A. Ward, Proc. Phys. SOC.(London) 73, Part 1, 1 (1958). H. Knoepfel, E. Loepfe, and P. Stoll, 2. Naturforsch. laa, 348 (1957). 1 4 R. L. Becker and J. A. Biggerstaff, Bull. Am. Phys. SOC.[2] 4, 326 (1959); J. A. Biggerstaff, R. F. Hood, and M. T. McEllistrem, ibid. 121 6, 441 (1960); J. A. Biggerstaff, R. L. Becker, and M. T. McEllistrem, Nuclear Instr. and Methods 10,327 (1961). la

l3

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

415

According to Halbert,lS the saturation of light output from CsI(T1) begins at a specific energy loss of about dE/dx = 0.35 Mev/mg-cm-2, while for NaI(T1) Eby and Jentschkeg find this value to be a little larger than 0.1 Mev/mg-cm-2. Thus, the response of CsI(T1) should be more nearly linear than NaI(T1) a t low energies. I n particular, the proton response must be linear a t all energies, since d E / d x for protons on CsI never exceeds the saturation “threshold” of 0.35 Mev/mg-cm-2. The proton response of CsI(T1) was studied by Galonsky, Johnson, and Moak, l 6 who found the pulse-height energy relationship to be linear from 0.88 to 4.33 MeV, with an energy intercept a t zero pulse height of 0.07 f 0.02 MeV. Whetstone and co-workerss recently determined a zero intercept of -0.17 L- 0.15 Mev from data acquired at 1.33,5.46,and 8.29 MeV. The small difference in the intercepts obtained may be ascribed to the preparation of the crystal surfaces; at least, these results show that saturation effects are small. Although the CsI(T1) fluorescent response to heavy particles is more linear at moderate energies than that for NaI(Tl), saturation effects are still significant. Using alpha particles from Pozl0, whose energy was varied by absorbers, Halbert l 6 measured the response to alpha particles of energy 1.95 to 5.3 MeV, and obtained a linear pulse-height energy curve above 3.5 MeV, with an intercept a t about 1 MeV. Similar measurements, to an energy of about 5 MeV, were made by Bashkin et al.” using alpha particles from nuclear reactions, and by Souch and Sweetman,18 whose results a t 6.1 and 8.8 Mev are in good agreement with the lower energy data. A summary of the fluorescent response to various particles has been included in Fig. 2. The light output from C12 ions (0.32 to 1.83 MeV) is linear with energy, but exhibits a negative energy intercept;17 N14 ions (2.9 to 23.8 MeV) yield a linear fluorescent response to about 16 MeV, above which the light output increases.16By using a magnetic analyzer to select median-masses of either light or heavy fission fragments and by varying the gas pressure within the analyzer, Fulmerlg was able to determine the fluorescent response of CsI(T1) to fission fragments over an energy range of about 20 to 100 MeV. The response was nearly linear within experimental error. Because of saturation effects the slopes of all heavy-particle response curves are less than the proton case a t low energies. For example, at an alpha-particle energy of 5.3 MeV, the M. L. Halbert, Phys. Rev. 107, 647 (1957). A. Galonsky, C. H. Johnson, and C. D. Moak, Rev. Sci. Instr. 27, 58 (1956). l7 S. Bashkin, R. R. Carlson, R. A. Douglas, and J. A. Jacobs, Phys. Rev. 109, 434 l6

18

( 1958). l9

A. E. Souch and D. R. Sweetman, Rev. Sci. Instr. 29, 794 (1958). C. R. Fulmer, Phys. Rev. 108, 1113 (1957).

416

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

0

5

10 15 PARTICLE ENERGY (Mev)

20

5

FIG.2. Fluorescent response of CsI(T1) to heavy ions. The pulse-height scale of Halbert16 has been normalized to that of Bashkin el d.17 by the factor 1.74.

energies of 4" ions to alpha particles to protons for the same light output in CsI(T1) are in the proportion 5.2: 1.7: 1.0, r e s p e c t i ~ e l y . * J ~ ~ ~ ~ I n an attempt to account for the decreasing scintillation efficiency of activated alkali halide scintillators with increasing particle mass and the nonlinearity of the pulse height-energy curve for heavier particles, Meyer and Murrayz0 used a model in which the incoming particle produces a certain number of energy carriers per cm. These carriers may be excitons or electrons holes successively captured at T1 sites. The carrier density thus is related to d E / d x by a recombination function. When a diffusion equation for the density of energy carriers is coupled with a n expression for the time dependence of N,, the density of unoccupied activator sites, the solution indicates that the depletion of available activator sites by particles with a high d E / d x accounts for the experimental results. The data of Bashkin et aZ,'7 can be fitted by this calculation with only one

+

90

A. Meyer and R. B. Murray, IRE Trans. on Nuclear Sci. NS-7 (2-3), 22 (1960).

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

417

parameter to be determined. This model contains the activator concentration as a parameter, and predicts relative scintillation efficiency and saturation effects as a function of concentration. Other inorganic scintillators have been employed occasionally, but none have proved as useful as NaI(T1) and CsI(T1). Historically, activated ZnS is the most interesting scintillator because it was used both in early visual scintillation counting and in the first scintillation counter using a photomultiplier. Although ZnS (Ag) is still used extensively in scintillation counting, it is not used for spectrometry because it is available only as a multicrystalline powder whose light transmission is poor. Calcium and cadmium tungstates are not available as large crystals and have long decay times. While KI(T1) can be grown in large, nonhygroscopic single crystals and yields a usable amount of light when bombarded by charged particles, it possesses the disadvantages of a long fluorescence decay time (>1 psec), a long-lived phosphorescence (>1 msec), and the natural radioactivity of K40. I n spite of these difficulties, KI(T1) has been used for spectrometry of fission fragments,lOalpha particles to about 38 and Clz ions to 113 The fluorescent response to Cl2ions was linear within the rather large experimental errors, indicating a surprisingly small saturation effect; on the other hand, the response to fission fragments and alpha particles shows a large saturation effect. Further information on the properties, application, and theory of inorganic scintillators may be found in severaI review articles on the subject.1~z2-26 Lithium iodide occupies a special place in charged-particle detection because of its use as a detector of the alpha particle and triton produced in a LiI crystal by the Lia(n,a)H3reaction (Q = 4 78 Mev). Crystals of LiI, enriched in Lis and preferably activated by EuC13,26make sensitive neutron detectors, since the (n,a) cross section for Lie is favorable over a wide energy range. Murrayz7 has investigated the properties of thin LiI(Eu) scintillators in the neutron energy range 1-15 MeV. As a spectrometer, the magnitude of the scintillation pulse from a neutron capture event in LiI(Eu) depends upon the sum of the neutron energy and the Q value, and in principle the pulse height should increase linearly with the energy. It was found by Murray that the width and irregular shape of the W. E. Burcham, Proc. Phys. Soc. (London) A70, 309 (1957). J. B. Birks, ‘lScintillation Counters.” McGraw-Hill, New York, 1953. 28 S. C. Curran, “Luminescence and the Scintillation Counter.” Academic Press,

I1

I*

New York, 1953. 24 G. F. J. Garlick, Progr. in Nuclear Phys. 2, 51 (1952). 26 R. K.Swank, Ann. Revs. Nuclear Sci. 4, 111 (1954). 2 6 J. Schenck, Nature 171, 518 (1953). 21 R. B. Murray, Nuclear Instr. 2, 237 (1958).

418

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

fast-neutron peak could be attributed to a significant difference in the scintillation efficiency for the alpha particle and triton and to the distribution of energies available to these particles. For thermal-neutron excitation, however, the energy division between the alpha particle and the triton is uniquely determined, and the pulse-height distribution for this case is a sharp Gaussian. At low temperatures the resolution is greatly improved; for example, the pulse-height resolution for 5.3-Mev neutrons was reduced from 18%at room temperature to 10% a t - 142°C. This improvement in the fast-neutron response suggests that the differ-

1 - 0 . 4

Mylar 0.00025" thick + Al Foil mg/cmzthick

s"O" Ring -No1 (T1) Scintillation Crystal (0.040"thick x ih"diameter) -Aluminurn Frame -Glass --R343

( i / 2 " thick x (+"diameter) Binding Glass to Aluminum

-406

Centistoke Silicone Oil at Each Interface

DuMont No. 6292 Photomultiplier T u b e

0"

i/2"

1"

r u FIG.3. A charged-particle detector for use with inorganic scintillators (Whetstone et ~ 2 . ~ ) .

ence between the alpha and triton scintillation efficiency is minimized a t low temperatures. It must be remembered that the pulse-height resolution was obtained in the usual way, i.e., by dividing the full width at halfmaximum counting rate by the pulse height of the maximum. Because the zero of the energy scale falls at the slow-neutron peak, the neutron energy resolution is greater than the pulse-height resolution; e.g., the neutron resolution a t -142OC is 18%. At the temperature of liquid nitrogen the pulse height as a function of neutron energy is almost linear in the range of 1 to 14 MeV, although it is slightly convex to the energy axis. The preparation of NaI(T1) or CsI(T1) crystals for use in charged-

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

419

particle spectrometry is similar to that used for gamma-ray spectrometry (cf. Section 2.2.3.3), although the thin window sometimes required for charged particles and the smaller scintillator thickness necessitate some important changes in the design of the crystal mounting. A scintillation detector for charged particles developed by Whetstone et al.8 is shown in Fig. 3, and will be described briefly. The crystals used were ground from commercially available NaI(T1) or CsI(T1) disks l k i n . in diameter and +in. thick, using progressively finer abrasive papers ending with number 400. Final polishing of NaI(T1) was accomplished by wiping the crystal surfaces on a Kleenex tissue soaked with a solution of 75% acetone and 25% chloroform. Water served a s the polishing agent for CsI(T1). The crystals were dried on tissues immediately after polishing. A glass light pipe, &in. long with rough-ground sides and polished ends, was found to improve the resolution, presumably because the light was spread over the photomultiplier tube face and hence reduced the effect of nonuniform photocathode sensitivity. Glass is recommended for the light pipe because plastics react with the substances commonly used as optical seals, making the interfaces cloudy. It was also found that the use of a n aperture smaller than the scintillator gave improved resolution; in Fig. 1 the aperture is 1 in. and the scintillator is 1; in. The resolution of such detectors using NaI(T1) crystals varied between 3.2 and 3.8% for 4.48-Mev alpha particles. If CsI(T1) is used it is unnecessary for the crystal to be hermetically sealed, and the window of Fig. 3 may be omitted. However, the light collection is improved by covering the crystal with a thin A1 foil. An arrangement due to Souch and Sweetman18 eliminates the need for covering a CsI(T1) crystal. The wide end of a 60° truncated cone was attached to the photomultiplier face as shown in Fig. 4; the narrow end of the cone (-+-in. diameter) was covered by a +-in. aperture. The white, inner surface of the cone served as a good light reflector. The crystal was polished with small amounts of cerium oxide in ethylene glycol. A resolution of less than 3% full width at half-maximum counting rate was obtained for 8.8-Mev alpha particles, which implies a resolution of about 4.1% at a n energy of 4.48 MeV, since Bashkin et a l l 7 found that the resolution of CsI(T1) spectrometers varies inversely as the square root of the energy. Comparing this value with the 3.2-3.8% obtained for 4.48-Mev alpha particles on NaI(Tl), it is seen that the resolution attained using CsI(T1) can be nearly as good a s that from NaI(Tl), even though the pulse height per Mev absorbed by CsI(T1) is smaller than for NaI(Tl), a t least when using photomultiplier tubes with a n S-11 response. When a scintillation spectrometer is used for energetic ions, corrections must be made for a small “tail” below the main peak of the pulse-height

420

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

0” 60’

L A T t u r e , 3/8 in. diameter

?zZZm Metal Cone White Surface

Wire Springs

GsI Crystal, 2 in. diameter Photomultiplier Tube

4/2 to 4 in. diometer, 4/2 to 3 mm thick

FIQ.4. A mounting arrangement which does not require covering the scintillator with a light reflector foil (Souch and Sweetman18).

distribution arising from nuclear reactions within the scintillator. These corrections have been evaluated by Johnston, Service, and Swenson28for NaI(T1) and a plastic scintillator excited by 10- to 68-Mev protons. At 10, 28, and 40 Mev the contribution of the low pulse-height “tail” from NaI(T11 amounted to 0.41, 1.2, and 2.3% of the total counting rate, respectively. When the plastic scintillator was irradiated by protons of 10, 40, and 68 Mev the “tail” accounted for 0.82, 2.8, and 6.4% of the counting rate, respectively. These results suggest that, a t very high energies, the usual scintillation techniques are rendered useless, because most of the incident protons will make nuclear collisions before they can be stopped by the electronic interactions which produce useful light. 2.2.1.2.2.2. Organic Scintillators. Spectrometry of heavy charged particles by the scintillation method is largely the domain of the inorganic crystals; however, a number of organic compounds are also very useful, either as pure crystals or in various combinations as liquids or solid solutions. These organic fluors generally cost less than inorganic ones, and scintillator assemblies of large size and irregular shape can be made easily and rather inexpensively using either organic liquid or plastic solutions. A further advantage is the low effective atomic number of all organic Z S L . H. Johnston, D. H. Service, and D. A. Swenson, I R E Trans. on Nuclear Sci. NS-6 (3), 95 (1958).

2.2.

DETERMINATION OF MOMENTUM A N D E N E R G Y

42 1

fluors, which is desirable for low gamma-ray sensitivity and small electron backscattering probability. Their fast fluorescence decay time (a few mpsec) is very useful for fast timing applications, although this feature is somewhat weakened by the low light output compared to that of inorganic fluors. A further general disadvantage for spectrometry is that the fluorescent light output depends on the ionization density, and so the light output is not proportional to energy for heavy ionized particles. A number of recent re vie^^^^-^^^^^^^^ of this subject may be consulted for information beyond the survey included here. The scintillation properties of anthracene have been studied more extensively than the other pure organic scintillators and, because it is the standard against which other fluors are usually compared, it is a n interesting example to consider in detail. Recent work (Chapter 1.4 and References 29, 30, 31) on the absolute scintillation efficiency (defined a s the fraction of the energy dissipated in a fluor which reappears as useful light) of anthracene favors a value of 0.03 to 0.05, although efficiencies ranging from 0.01 t o 0.10 have been reported. The variation in measured absolute efficiencies arises from experimental difficulties associated with preparing and retaining very pure anthracene, photomultiplier response, crystal size, optical efficiency, and the like. Table I shows a comparison between the characteristics of anthracene and a number of other typical organic crystal and plastic scintillators. It is seen that while anthracene has a higher scintillation efficiency than the other organic scintillators, its fluorescence decay time is the slowest listed; therefore, experiments requiring very fast pulse timing may require a scintillator with faster response (e.g., trans-stilbene or a plastic), in spite of the lower pulse height. An important difference in the characteristics of inorganic and organic scintillators is the poor response of the organic materials to heavily ionizing radiations. It will be seen from Fig. 5 that the electron response for anthracene is essentially linear, but for heavier particles considerably more nonlinearity in the pulse-height energy relationship is observed. The pulse heights from anthracene, trans-stilbene, and a polystyrene plastic similar to that of Table I are compared32 in Fig. 6 for excitation by Ba137 internal conversion electrons, protons, and deuterons. The impaired linearity of organic scintillators has been ascribed t o a quenching F. D. Brooks, Progr. in Nuclear Phys. 6,252 (1956). W. E. Mott and R. B. Sutton, in “Handbuch der Physik-Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 45, pp. 86-173. Springer, Berlin, 1958. R. C. Sangster and J. W. Irvine, J . Chem. Phys. 24, 670 (1956). 32 M. M. Hoffmann, R. W. Peterson, and M. Janco, Some characteristics-of plastic fluor. U.S. Atomic Energy Commission Rept. LA-2069 (1956). Unpublished. eQ

30

422

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

TABLE I. Organic Scintillatorsa Wavelength of maximum emisForm sion, .&

Material*

Rela- Time for tive decay to pulse l/e of height initial for p light excita- intensity, tion mpsec

Remarks Large crystals difficult to grow

Anthracene

Crystal 4400

100

trans-Stilbene

Crystal 4100

60

6.4

Diphenyl acetylene

Crystal 3900

45

5.4 Good crystals readily obtained

p-Terphenyl

Crystal 4000

40

5.0 Fair crystals readily obtained

p,p'-Quaterphenyl

Crystal 4200

85

7

4450

39

4

Plastic

4450

45

.4.0

Liquid

3700

49

<2.8

4150

42

53 2

+

Polystyrene 36 gm/liter Plastic TP 0.2gm/liter T P B

+

Good crystals readily obtained; very fragile

Good crystals difficult to grow

+

Polyvinyl toluene 36 gm/liter TP 0.2 gm/liter T P B

+

32

+ 8 gm/liter Toluene + 5 gm/liter

Toluene PBD

TP Liquid

+ 0.2 gm/lit.er a-NPO

These data were selected from a review article by Swank.26 Solute abbreviations: TP = p-terphenyl; T P B = lJ1,4,4-tetrapheny1-1,3-butadiene; a-NPO = 2-(l-naphthyl)-5-phenyloxazole;P B D = 2-phenyl-5-(4-biphenylyl)1,3,4-oxadiaeole. a

of the luminescence, due t o damage suffered by the organic molecules when the specific energy loss is very high. The dependence of specific luminescence dL/dx upon specific energy loss d E / d x has been studied by BirksZ2J3and by Wright,34 whose results both fit the available data within the experiment'al uncertainties. A summary of current theories of organic scintillator response will be found in Chapter 1.4 and in the review article by Brooks.29 33

34

J. B. Birks, Proc. Phys. Soc. (London) A84, 874 (1951). G.T. Wright, Phys. Rev. 91, 1282 (1953).

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

423

PARTICLE ENERGY (MeV)

FIG,5. Fluorescent response of anthracene as a function of particle energy and type. Solid lines are theoretical; points are experimental (Wright$*).

The pulse-height energy relationship for electrons on anthracene has been the subject of exhaustive study and may be regarded as typical of organic scintillators in general. hop kin^^^ and Taylor et aLs observed a definitely linear dependence of pulse height on electron energy from about 125 to 3200 kev, but below 125 kev the pulse height appeared to vary less rapidly with energy. The linear part of the curve extrapolated to an intercept of about 25 kev a t zero pulse height. The behavior of anthracene for low-energy electrons is difficult to interpret, since surface effects arising from the treatment of the crystal become very important. Differences between the response to electrons incident on the crystal surface and to electrons liberated inside the crystal have been attributed by Wright3'Jto escape of fluorescence from the surface and to trapping of excitation energy by non-fluorescent impurities in the surface. I n an experiment designed to study the anthracene response at low energies, Johnston et al.31 measured the pulse height for accelerated electrons of 3e

J. I. Hopkins, Rev. Sci. Znstr. 22, 29 (1951). G. T. Wright, Phys. Rev. 100, 588 (1955); Proc. Phys. Soc. (London) B68, 701

(1955). 37L. W. Johnston, R. D. Birkhoff, J. S. Cheka, H. H. Hubbell, Jr., and B. G. Saunders, Rev. Sci. Instr. 28, 765 (1957).

424

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES 45

I

f4 - -

ANTHRAGENE

t3 -

I

-- STILBENE

42

I4 I-

$ 40

i1z

9

;e

2 7 W

z

6

6 5 a 4

3 2

I

0 ENERGY (Mev)

FIG.6. Relative fluorescent response of anthracene, stilbene, and a polystyrene plastic scintillator. The points A, S, and P denote the pulse heights obtained when anthracene, stilbene, and plastic scintillator, respectively, are excited by the internal conversion electrons from a Cs187 source (Hoffmann et aLs2).

energy 10 to 120 kev incident on cleaved anthracene crystals 0.011- and 0.060-in. thick. A linear response curve was obtained, with an intercept of about 4 kev. The scintillation eaciency of anthracene at low energies may depend upon the surface treatment used, and so it is not clear how close an agreement should be expected between the results of Johnston et al. using thin, cleaved crystals, and those of Hopkins using crystals 2j- and 1-in. thick with polished surfaces. Birkhoff3* recently derived an equation for the slope of the pulseheight energy curve as a function of the electron energy, using Birks’ theory. The response of anthracene thus derived was linear a t high energies, but showed some curvature a t low energies. The intercept of a straight line on the energy axis was calculated from the slope equation for various energies. At 60 kev (approximately the middle of the energy range used by Johnston et al.), the intercept was about 4 kev; however, 88 R. D. Birkhoff, private communication, 1960; U.S. Atomic Energy Commission Rept. ORNL-2806, p. 153 (1959). Unpublished.

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

425

a t 1.6 Mev (approximately the middle of the energy range of Hopkins' measurements), the intercept was about 20 kev. Therefore, the two experiments are consistent with each other and with the theory of Birks. Useful scintillators can be made from liquid or solid (plastic) solutions of two or three organic compounds, and typical examples will be found in Table I. For information on the preparation of liquid and plastic scintillators, the reader may consult Chapter 1.4 and the review articles by SwanklZ6Mott and S ~ t t o n , ~and O Brooks.2gBy their very nature, liquid scintillators have the advantage of easy fabrication in almost unlimited volumes, and large scintillator tanks make good detectors for high-energy particles and electromagnetic radiation because their stopping power can be made very high. Notable advantages of the plastic scintillators are their ruggedness and ease of machining to desired shapes. Both liquid and plastic solutions have fluorescence lifetimes at least as short as the best crystal scintillators. Following a report by Wrightsg that scintillations produced by alpha particles and electrons in anthracene showed decay times of 53 mpsec and 31 mpsec respectively, it was subsequently reported29 that other organic scintillators, notably stilbene, quaterphenyl, and some liquid scintillators, showed decay time properties similar to those observed in anthracene. In all of these cases, the decay times were longer for heavy ionizing particles, such as alpha particles or protons than for electrons. This effect has been very u s e f ~ 1 ~ ~in- 4the ~ detection of recoil protons from fast neutrons in the presence of Compton electrons from gamma rays. Brooks42 found that, using a l-in. thick stilbene crystal, 2-Mev neutrons could be detected with 9.5% efficiency while the detection efficiency for 2-Mev gamma rays was reduced by decay-time discrimination to less than 0.007%. Organic scintillators are to be preferred for spectrometry of electrons and beta distributions, because their low effective atomic number is responsible for a considerable improvement in backscattering over that of inorganic materials. Nevertheless, backscattering of electrons incident on a flat organic scintillator may be as high as S%,l and spectra will be distorted at low energies by beta particles which scatter out of the scintillator, leaving only a fraction of their energy behind. This effect is worst at low incident beta energies. Above several Mev the backscattering probability is low enough to make practical beta-ray energy determinations G . T.Wright, Proc. Phys. SOC.(London) B69, 358 (1956). M. Forte, Proc. 9nd Intern. Conf. on Peaceful Uses of Atomic Energy, Geneva 14, P/1514 (1958). 4 1 R. B. Owen, IRE Trans. on Nuclear Sci. NS-6 (3), 198 (1958). 4 * F. D. Brooks, Nuclear Znstr. and Methods 4, 151 (1959). 39 40

426

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

using only a cylindrical organic scintillator optically coupled to a photomultiplier tube, if the shape of the low-energy portion of the spectrum is not important. Some improvement in the flat-crystal spectrometer performance may be made by collimating the incident beam to insure that the particles will enter the surface near normal incidence, and so will, on the average, penetrate deeply into the crystal before scattering. All scintillators are gamma-sensitive to some degree, so whenever the beta source is known to emit gamma rays a correction must be made. Generally this spectrum is obtained by interposing a beta absorber between source and detector, and calling the observed gamma-induced spectrum the ((gammabackground.” It will be realized that the gammaray contribution is determined only approximately by this measurement, as the beta absorber itself contributes bremsstrahlen and scattered photons whose influence cannot be precisely evaluated. At low electron energies, more elaborate experimental methods are required to circumvent the electron scattering effect. One such method is the ((split-crystall’spectrometer first employed by Ketelle,43in which two scintillators are mounted close together on a photomultiplier tube. The source is positioned between the two scintillators such that electrons scattered from one crystal will be detected by the other; therefore, all of the electron energy is given up to the scintillator, and none is lost by scattering. This 47r geometry makes it difficult to measure the gamma-ray induced background by addition of a beta-ray absorber, and enhances the probability that electrons produced by coincident gamma rays in the sample shall be detected simultaneously with the beta particles. Such summing contributions will distort the pulse-height distribution in a manner which is very difficult to interpret. I n some cases it is possible to gate the split-crystal beta spectrometer by an external gamma-ray detector, such that the beta pulse-height analyzer records an event only if all of the coincident gamma energy escapes the beta scintillator. Although this method is limited to gating by single gamma rays or by gamma-ray ((sum” peaks if the beta particles are in coincidence with a gamma cascade, excellent spectra of some “inner” beta groups have been obtained. 44 When beta-particle energies are very low, the response is somewhat sensitive to the treatment and history of the scintillator surfaces, and is greatly influenced by the preparation of the beta-ray source. These obstacles may be overcome to some extent by adding the radioactivity B. H. Ketelle, Phys. Rev. 80, 758 (1950). R. L. Robinson, Inner beta spectra of Trn171, RulOJ,Aglll, and Rb*6. Ph.D. Thesis, Indiana’University, Bloomington, 1958; R. L. Robinson and L. M. Langer, Phys. Rev. 44

109, 1255 (1958).

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

427

directly to an organic liquid ~ c i n t i l l a t o rAs . ~ ~the source is an integral part of the scintillator, the problems arising from external sources are eliminated, although it may be difficult to find a chemical form in which to introduce the radioactivity which will not also quench the fluorescence of the solution. Beta-gamma summing effects inherent in the 4n geometry systems just described led Bell’ to suggest a “ hollow-crystal ’’ spectrometer, which reduced summing through the use of a low solid angle. Backscattering SOURCE ON ALUMINIZED PLASTIC FILM HOLLOW PLASTI SClN TI LLATOR

STEEL PHOTOTUBE HOUSING

BLAC FELT PHOTOMULTIPLIE

FIQ. 7. A hollow-crystal scintillation spectrometer for beta particles, using two pieces of plastic scintillator optically coupled (Gardner4e).

from the crystal was reduced by collimating the incident electrons into a conical hole in the scintillator, from which the probability of escape was low (cf. Fig. 7). Air scattering becomes important at low energies, and may be reduced by placing the equipment either in a vacuum or an atmosphere of hydrogen or helium. Another advantage of this arrangement is that the gamma background may be determined using a beta absorber just as conveniently as for a flat scintillator. Gardner46examined the responses of three hollow-crystal spectrometers, which were made from scintillating plastic and designed for use with maximum beta energies of 1.5, 2.2, and 3.6 MeV. These scintillation spectrometers proved very convenient for studies of beta-ray spectra, and gave improved perform46 E. Steinberg and L. E. Glendenin, Argonne National Laboratory, personal communication, November, 1958. 46 D. G. Gardner, “Nuclear Decay Scheme Analysis and Characterization Studies of (d,m) Reaction Products,” U.S. Atomic Energy Commission Rept. AECU-3514 (1957). Unpublished; D. G. Gardner and W. W. Meinke, Beta-ray spectroscopy using a hollow plastic scintillator. Intern. J . A p p l . Radiation and Isotopes 3, 232 (1958).

428

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

ance over a flat scintillator, both as regards resolution and backscattering. The response of an organic scintillator irradiated by monoenergetic electrons is mainly a Gaussian peak whose width depends inversely on the square root of the e n e r g ~ . ~ This ~*~T implies that the statistics associated with the number of photoelectrons at the photocathode and the electron multiplication processes within the photomultiplier tube primarily determine the energy resolution, and any intrinsic scintillator resolution is small. Because the light output from the organic scintillators is so much lower than NaI(Tl), the resolution is correspondingly poor. For an anthracene hollow-crystal spectrometer, a resolution (full width a t halfmaximum counting rate) of about 10% can be obtained at 624 kev, and if a plastic hollow-crystal spectrometer is used the resolution is about 14%, because of the smaller scintillation efficiency of the plastic (cf. Table I). Pulse-height distributions of beta-ray continua must be corrected for finite instrumental resolution and scintillator backscattering before careful analysis of the spectral shape can be made. Corrections for finite resolution have been applied by Ketelle43and by Palmer and L a ~ l e t t , ~ ~ using the method of Owen and P r i m a k ~ f f .The ~ ~ scintillator response assumed in these cases was a Gaussian whose width varied as E-l12, but did not include a backscattering contribution. Such a procedure eliminates a curvature of the Fermi-Kurie plot near the maximum energy, but does not prevent a n upturn at low energies. The shape of the beta spectrum can be corrected more accurately using the technique of Freedman, Novey, Porter, and Wagner,49who developed an iterative numerical integration to correct for finite resolution, including the effect of backscattering from the scintillator. Four internal conversion electron peaks were measured with a flat anthracene crystal spectrometer in the energy range of 0.18-1.0 MeV, and it was shown that all peaks were accompanied by a low-energy tail, whose height relative to the peak counting rate stayed constant at all energies. The typical response of a plastic hollow-crystal spectrometer to the internal conversion electrons of Bal3Trnis shown in Fig. 8; the height of the tail is about 5 % of the peak height. Beta particles from Cs13’ in the source were excluded in this measurement by requiring coincidence with the X-ray following internal 47 J. P. Palmer and L. J. Laslett, “Beta-Ray Spectrometry with an Anthracene Scintillation Spectrometer,” U.S.Atomic Energy Commission Rept. ISC-174 (1950). Unpublished. (8 G. E. Owen and H. Primakoff, Phye. Rev. 74, 1406 (1948); Rev. Sci. Instr. 81, 447 (1950). 49 M. S. Freedman, T. B. Novey, F. T. Porter, and F. Wagner, Jr., Rev. Sci. Instr. 27, 716 (1956).

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

429

conversion. Two analyses of a spectrum of the low-cnergy beta group of Cs13’ are shown in Fig. 9. The upper curve has been corrected for the unique spectral shape by use of the usual factor “a,” and for instrumental line width by the method of Owen and Primakoff. The lower curve shows the improvement in the Fermi-Kurie analysis when the correction for scintillator backscattering is applied. 500 I

I

I

I

I

I

I

PULSE HEIGHT

FIG.8. Organic scintillator response to monoenergetic electrons from Bala’n, using a Cs187 source. The beta particles of Cs137 were excluded by requiring coincidences between X-rays and internal conversion electrons (Gardner‘n).

The standard correction procedures mentioned here break down below 100 t o 200 kev, depending on the detector resolution. This is because the low-energy response cannot be well defined and, when the resolution is very poor, most of the counts recorded in the high-energy half of the spectrum arise from low-energy electrons which fall within the large resolution width. Freedman, Novey, Porter, and Wagner proposed the practical criterion that their correction procedure should not be used when the slope of the upper energy side of the monoenergetic lines becomes comparable to the slope of the upper half of the true spectrum, if they are normalized a t the peaks. This situation led Steinberg and Glendenin46to use empirical correction curves for analyzing low-energy beta spectra recorded with a liquid scintillator which contained the beta activity. The empirical corrections took into account both the low-energy nonlinearities of the scintillator and the instrumental resolution, and were derived from measurements of several low-energy beta activities whose energies and spectral shapes were determined by other methods.

430

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

1

FIG.9. Analysis of the lorn-energy beta ray group of C S ' using ~ ~ a hollow-crystal scintillation spectrometer. The upper curve was obtained after correcting for the unique spectral shape and the finit,e instrumental resolution, while the lower curve shows the improvement obtained in the Fermi-Kurie plot when the correction for scintillator backscattering is applied (Gardner'B).

When intense gamma rays or internal conversion electrons are in coincidence with a beta-ray group under investigation, the measured beta-ray spectrum may be distorted by summing of pulses from the coincident radiations. Although conversion electrons are detected with high efficiency by an organic scintillator, gamma rays are detected very inefficiently, and thus the gamma-ray summing problem can usually be solved by maintaining a low geometry. If such a procedure does not prove adequate it may be possible to compute the sum spectrum, using an approach similar to that described in Section 2.2.3.3 for gamma-ray summing. In most practical cases, for which the decay schemes are relatively complex, such computations are impractical, and reduction of the solid angle is the main avenue left open to the experimenter. A special case of coincident summing may occur when measuring positron spectra, since energy from either or both of the two 0.511-Mev positron annihilation photons can be transferred to the crystal in addition

2.2.

DETERMINATION O F MOMENTUM AND EN ERG Y

43 1

to the kinetic energy. These events are in prompt coincidence, and the summing contribution is primarily due to pulses from the Compton effect of 0.511-Mev photons in the scintillator which sum with the positron pulses. The source of photons is the scintillator proper, so the effect cannot be reduced by lowering the solid angle. Fortunately, the gamma-ray stopping power of organic scintillators is small, and hence the probability of escape for annihilation photons is very high; if scintillator dimensions are kept t o the minimum required by the positron range and other experimental parameters, then the annihiiation summing effect will be small. For example, the positron spectrum of 14.6-hour NbgO was determined by Lazar et aE.,60 using a n anthracene hollow-crystal spectrometer for which the diameter was l+ in. and the thickness of crystal seen by the direct beam was 0.70 gm/cm2; a n independent measurement of the same nuclide was made by Sheline,61 using a 4n geometry, split-crystal detector. Both determinations yielded the same positron endpoint of 1.50 Mev within an experimental error of about 2%; data obtained using the hollow-crystal spectrometer were free of summing effects, while the spectrum from the split-crystal spectrometer showed a small annihilation contribution, probably because of the larger scintillator dimensions. When annihilation summing must be eliminated, a coincidence method can be used similar to th a t described above for suppressing gamma-ray summing in split-crystal experiments. I n this case, two gamma detectors, set to respond to 0.511 MeV, are placed in line with the positron detector between them. The pulse-height analyzer is set to record a positron pulse only if the two gamma detectors signal the escape of both annihilation photons. A further summing effect which may distort beta-ray spectra is random summing. This arises from the addition of pulses which happen to be detected simultaneously within the resolving time of the amplifier. The pulses which add t o the pulse of interest may be due to high gamma-ray activity, another intense beta group, or internal conversion electrons present in the source. Such an effect is difficult to compute exactly, but as it is dependent on counting rate, the effect can usually be practically eliminated by reducing the source strength. I n very serious cases it may be necessary to derive an empirical correction for the data from measurements taken a t several source strengths. 2.2.1.2.2.3. Noble Gas Scintillators. The first successful gas scintillation counter using a photomultiplier tube was reported by Eggler and HudEON. H. Lazar, G. D. O’Kelley, J. H. Hamilton, L. M. Langer, and W. G. Smith, Phys. Rev. 110, 513 (1958); N. H. Lazar and G. D. O’Kelley, Bull. Am. Phys. SOC.1 (4),163 (1956). 61 R. K. Sheline, Phwsieu 23, 923 (1957).

432

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

dleston. 52 Since that time gas scintillation counters have been developed and used in various laboratories because of their short decay times (-10-9 sec), large light output, constancy of light output per Mev independent of ionization density, and their availability in a wide range of Z and density. An extensive literature on experimental technique of gas scintillation counters has been published in the last few years. However, because most of the primary scintillation light lies in the far ultraviolet, direct observations of the fundamental processes are very difficult to make, and no adequate description of the scintillation mechanism exists. The rather large number of workers actively building and using these counters are in qualitative agreement on the determination of some properties, but the data are frequently so difficult to correlate as to yield very little net progress toward an understanding of these systems. The light output of a gas scintillator is very sensitive to impurities. Northrop and Nobles63have assessed the effects of various contaminants by measuring the decrease in pulse height from a xenon gas counter upon the addition of various partial pressures of hydrogen, nitrogen, oxygen, and methane. A 5 to 10% admixture of hydrogen or nitrogen reduced the pulse height about a factor of 2; the same pulse-height reduction was obtained with only 1 to 2% of oxygen or methane. These quenching effects are small compared to those from natural impurities, such as hydrocarbons from gaskets or counter walls, which may reduce the pulse height to an unusable value. Several methods have been used to insure purity of the scintillating and by gas. A very simple technique, used by Boicourt and Br01ley~~ Villaire and Wouters,66is that of continually flowing fresh gas through the counter. Nobles66 obtained improved stability from a counter using a wavelength shifter when the vapor pressure of wavelength shifter was reduced by cooling. To obtain good resolution it is necessary to purify the gas very carefully, and when using the more expensive gases such as Kr or Xe, it is also advisable to recirculate them. The usual techniques for purification of gases may be used, taking care to avoid rubber gaskets and other materials which contribute noxious vapors. Sayres and Wu67 recirculated their Xe or Kr counter gas through a stainless steel trap filled with calcium metal at 4OO0C, and obtained very good results. It was found by Northrop and Nobles63 that a furnace containing uranium 62

C. Eggler arid C. M. Huddleston, Phys. Rev. 96, 600 (1954).

J. A. Northrop and R. A. Nobles, I R E Trans. on Nuclear Sci. NS-3 (4), 59 (1956). 5 4 (2. P. Boicourt and J. E. Brolley, Jr., Rev. Sci. Instr. 26, 1218 (1951). 66 A. E. Villaire and L. F. Wouters, P h p . Rev. 98, 280 (1955). G 6 R. A. Nobles, Rev. Sci. Instr. 27, 280 (1956). 67 A. Sayres and C. S. Wu, Rev. Sci. Instr. 28, 758 (1957). 63

2.2.

DETERMINATION OF MOMENTUM A N D E N E R G Y

433

turnings a t 8OO0C,followed by a trap of cold uranium to absorb hydrogen, provided adequate purification of noble gases. Barium getters have been used by Bennett5* to prepare noble gases of extremely low nitrogen content. The short wavelength of the emitted light makes it necessary to view the scintillations of noble gases with either a n untraviolet-sensitive photomultiplier tube or a wavelength shifter capable of absorbing the short wavelengfh light and re-emitting light in a wavelength band which more nearly matches the response of conventional photomultipliers. For measurements of high-energy charged particles, e.g. fission fragments, in which the pulse heights are large, the scintillations may be viewed with an ultraviolet-sensitive photomultiplier such as the DuMont K-1306 or ~' about the RCA 6903.* Optimum pulse height is 0 b ta in e d ~ ~ 0when 20-75 pgm/cm2 of wavelength shifter is deposited on the walls and photomultiplier tube window by vacuum evaporation. It is important to cover the optical reflector with wavelength shifter because the reflector TABLE 11. Relative Scintillation Efficiencies for Various Noble Gas-Wavelength Shifter CombinationsWaveshif ter /gas Q.P.* D.P.S.O T.P.B.d PO POP^ POPOP-D.P.S.1 a-N.P.O.0

Ar

Kr

Xe

0.202 0.144 0.071 0.066 0.037

0.247 0.567 0 356 0.172 0.170 0.090

0.438 1.000 0.699 0.366 0.337 0.176

From an article b y Northrop.69 p-Quaterphenyl; Pilot Chemicals, Inc., 36 Pleasant Street, Watertown 72, Massachusetts. Diphenyl stilbene; Pilot Chemicals. 1,1,4,4Tetraphenylbutadiene;Pilot Chemicals. p-Bis-2-(5-phenyloxazolyl)-benxene;Pilot Chemicals. First the POPOP and then the D.P.S. were evaporated on the phototube face. 0 2-(l-Naphthyl)-5-phenyl oxaxole; Arapahoe Chemicals, Inc., Boulder, Colorado. a

efficiency is rather poor a t short wavelengths. N o r t h r ~ phas ~ ~measured the relative efficiencies of several organic wavelength shifters for alphaparticle scintillations in xenon, krypt,on, and argon. His results are given in Table 11, and have been normalized to the xenon-diphenyl stilbene combination, which was found to give about the same pulse height as

* Further 68

6*

information on photomultipliers can be found in Vol. 2, Section 11.1.3. W. R. Bennett, Jr., Rev. Sci. Znstr. 28, 1092 (1957). J. A. Northrop, Rev. Sci. Znstr. 29, 437 (1958).

434

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

obtained from alpha particles on NaI(T1). Some difficulty was experienced with a mechanical instability of the evaporated tetraphenylbutadiene, but diphenyl stilbene and quaterphenyl were mechanically and optically stable over long periods. Sayres and Wu57 and Northrop, Gursky, and Johnsrud60 have measured relative pulse heights obtained for several noble gases in combination with glass-window (S-11 response) and quartz-window (5-13 response) photomultiplier tubes, both with and without quaterphenyl wavelength shifter. Their results show th a t a quartz-window photomultiplier tube without wavelength shifter produces almost the pulse height of a photomultiplier with a glass window in combination with wavelength shifter. The greatest pulse height was achieved by using both the quartz-window phototube and wavelength shifter. The relative amounts of light from alpha excitation of several noble gases, determined with a 6292 photomultiplier tube and quaterphenyl wavelength shifter, were compared by Northrop and Nobless3 with the light output of NaI(T1) under the same conditions, except that no wavelength shifter was used with NaI(T1). The relative light outputs were NaI(T1) : X e :Kr :A :Ne :He = 72 :32 : 16 :5 : 1 : 10. Similar measurements on xenon, krypton, argon, and helium by Sayres and Wu are in good agreement with these ratios. If the more efficient diphenyl stilbene had been used in these measurements instead of quaterphenyl, a scintillation efficiency approaching that of NaI(T1) would probably have been possible (cf. Table 11) for xenon. The light output from binary mixtures of xenon, krypton, argon, neon, and helium was also investigated by Northrop, Gursky, and Johnsrud, who found a characteristic, large drop in the light output for mixtures containing a small proportion of the heavier gas in a major fraction of the lighter. Unlike the alkali halide and organic scintillators, the noble gas scintillators do not exhibit saturation effects for charged particles of high energy loss. The linearity of the pulse height-energy relationship for a pure xenon spectrometer was investigated by Nobles66 in the energy range 2-5 MeV. Data for protons, deuterons, and alpha particles of various energies fell on the same straight line, which intercepted the energy axis a t about 0.5 MeV. Using a krypton scintillation spectrometer, Boicourt and Rr01ley~~ obtained the proper ratio between the energies of the light and heavy fission fragment mass groups, but each was too large in comparison with 4.76-Mev alpha particles. Such behavior is actually the inverse of the saturation effects noted for conventional scintillators. Clearly, a systematic investigation of the fluorescent response of noble gas scintillators 6o J. A. Northrop, J. M. Gursky, and A. E. Johnsrud, IRE Trans. on Nuclear Sci. NS-6 (3), 81 (1958).

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

435

to different charged particles is much needed, as this information should be valuable not only from a practical point of view, but also as a contribution toward the understanding of the scintillation mechanism. The resolution of xenon, krypton, and argon spectrometers was studied by Sayres and Wu5' as a function of experimental conditions. For all gases investigated, the resolution (and pulse height) improved with increasing gas pressure and finally reached a limiting value. Using a quaterphenyl wavelength shifter and a 6292 photomultiplier tube, the best resolution (4.8% full width at half-maximum counting rate) for 5.3-Mev alpha particles was obtained with xenon a t 45 psi absolute; for krypton the best resolution was 5.8% a t about 50 psi, and for argon about 9% at 60 psi. Optimum resolution occurred a t the pressure which yielded maximum pulse height, and the system which yielded the greatest pulse height of those tried, a quartz-window phototube in combination with xenon and a wavelength shifter, was also found to give the best resolution, less than 3%.67r61 The combined properties of speed, linearity, and large light output make these counters particularly suitable in certain applications. Fission counting in the presence of a large alpha-particle background is simplified using the gas scintillation counter, especially if the gas pressure is kept low compared to the range of the alpha particles. Because of the low stopping power and small pulse height for nuclear gamma rays, energetic charged particles may be analyzed in the presence of a n intense gammaradiation field. It has been found that large amounts of helium in xenon will not impair the good performance of the xenon a10ne,67~6~ which suggests that it is possible to use such a counter containing He3 as a highspeed neutron detector, by counting the proton and triton from the He3(n,p)H3reaction. A gas scintillator has been used by Sayres and Wu to detect slow neutrons by the reaction Bl0(n,a)Li7.Both the Li7 and alpha peaks in the pulse-height distribution were well resolved, not only from each other but also from the low-energy background caused by gamma rays from the Ra-Be neutron source. Further remarks on scintillation counters using noble elements in the form of gases, liquids, and solids will be found in Chapter 1.4. C. S. Wu, private communication, 1958.

436

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

2.2.1.2.3. MEASUREMENT OF RANGE AND ENERGY WITH CLOUD CHAMBERS BUBBLE CHAMBERS.* The measurement of the kinetic energy of any particle in mechanics can only be done indirectly. If the particle can be stopped, the energy will convert into work or heat or both, which can be measured. Fast-moving charged nuclear particles can sometimes be stopped in material media since they lose energy by ionization and excitation of the medium. Thus if the range, or distance traveled by the particle is measured, and a relation between the range and energy particle can be found, the energy can be inferred from the range measurement. The energy of a particle can also be found if the mass and velocity are known or if the mass and momentum or momentum and velocity are known. We will be concerned here, however, with the determination of energy from range measurements in cloud chambers and bubble chambers. The measurement of range of a particle in the gas of a cloud chamber or the liquid of a bubble chamber is a relatively simple matter. The track may become quite curved a t the end of the range due to multiple scattering, but length measurements are normally possible. It is important to know accurately the composition, pressure, and temperature of the medium so as to be able to apply the range-energy relation. At the end of the range there is a statistical straggling’ which makes an error in the range measurement, usually of the order of 2%. The upper limit of energies which can be determined in this way depend on the size of the chamber and the stopping power of the material. Table I gives examples of limiting energies of particles which can be stopped in a chamber where tracks of length 30 em are observable. AND

TABLE I. ENERQIES OF PARTICLES WHICHWILL STOPIN 30 CM RANGEIN CHAMBERS CONTAINING VARIOUSMATERIALS Energies in Mev Argon 1 atmos

Argon 75 atmos

Hz liq. ~~

2 5 18

7

9

16

21 80

60

Propane liq. ~~

70 150 550

Range in Heavy Material. When the range of high-energy particle is greater than the size of the chamber, the particle may be stopped by E. Rutherford, J. Chadwick, arid C. 1 ) . h X s , “R.adiations from Radioactive Substances.” hlanmillan, New York, 1930.

* Section 2.2.1.2.3is by W.

B. Fretter.

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

437

placing plates of a heavy material such as graphite, copper, or lead, in the chamber, leaving spaces between for viewing of the tracks. Another procedure is to place the heavy material between separate chambers. The first method has been widely used with expansion cloud chambers, so-called multiplate chambers. The problems involved in the operation of such chambers have been discussed by Peyrou. la The range of a particle is determined by measuring its length of path through the plates, making sure of the angle and depth corrections. The density and composition of the plates must be known. The actual range cannot be determined because the particle stops in the interior of the plate. If an estimate of measurement of the ionization of the particle can be made before the particle enters the plate in which it stops, the accuracy of the measurement can be improved. Otherwise the total uncertainty is essentially the thickness of the plate. The errors involved in range measurements have been treated2 by Armenteros et al. They define three quantities: RI-minimum range-is the amount of matter traversed before the particle disappears. R3-available range-is the amount of matter it would have traversed before leaving the illuminated volume of the chamber. R2-maximum range-is the amount of matter it would have traversed if it had stopped at the bottom of the last plate entered. R2 is corrected for the additional path due to scattering in the last plate. This is of the order of 4% for 1.1 mesons in copper. Ra is important in case there is a spread of ranges, and geometrical biases can be introduced, or in case the particles enter in different directions with different potential ranges. These writers then compare the predicted Gaussian distributions of error with the rectangular distribution with limits at R1 and Rz and show that there is an appreciable tail t o the distribution. Various attempts have been made to use plates in conjunction with diffusion cloud chambers and bubble chambers, but these have not been very successful. Range-Energy Relations.* In order to determine the energy from the range, the nature of the particle (mass and charge) must be known, as well as the relationship between range and energy. Range-energy curves are largely empirical and the many problems involved in their use are beyond the scope of this article. The basic assumption involved is that the particle loses energy in many small collisions, so that a statistical

* Refer to Section 1.1.3. C. Peyrou, Nuovo cimento [9] 11, Suppl. No. 2, 322 (1954). R. Armenteros, B. Gregory, A. Hendel, A. Lagarrigue, L. Leprince-Ringuet, F. Muller, and C. Peyrou, Nuovo cimento [lo] 1, 915 (1955). 1s

2

438

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

treatment is possible. Under these conditions, a “universal ” range-energy curve may be constructed. Usually the proton-range energy is given in graphical or tabular form3 and the equivalent relation for other particles is obtained as follows: “The range of a particle of charge Ze, mass M , and kinetic energy E may be obtained from the range of a proton or energy ( M , / M ) E where M p is the proton mass by the relation:

2.2.1.3. Determination of Velocity. 2.2.1.3.1. TIME-OF-FLIGHT METHt The velocity of charged particles can be measured quite accurately using a time-of-flight method over a wide range of particle velocities. The time-of-flight method is generally accomplished by means of a scintillation counter telescope consisting of, say, two scintillators spaced a t a distance OD.*

+I= -1 S C I N T I L L A T I O N DETECTORS

(I

CHARGED, PARTICLE

\(

2)

to+At

D E L A Y LINE

I

I

1

FIQ.1. Schematic of the time-of-flight coincidence method.

d apart (Fig. 1). The particle to be measured passes through the first scintillator (1) at a time t o and then it passes through the second scintillator (2) at a later time t o A t ; where At = (d/v),v being the velocity of the particle. The essence of this method is to measure accurately the time difference At. This can be achieved in a number of ways, such as the coincidence method, measurement of pulse separation on an oscilloscope display, a chronotron, etc. The most commonly used method is the coincidence

+

t Refer to Section 2.6.2,

Vol. 5 B. “American Institute of Physics Handbook,” Section 8-23. McGraw-Hill, New York, 1957. 8

* Section 2.2.1.3.1is by Luke C. L.

Yuan and S. J. Lindenbaum.

2.2.

DETERMINATION OF MOMENTUM A N D E N E R G Y

439

method. A coincidence signal can be obtained in a fast coincidence circuit if a time delay of magnitude At is introduced in the signal path from the first scintillator (1) (Fig. 1).By measuring t,he delay time At, and knowing the distance d , one can easily obtain the particle velocity v. The time-of-flight method has been commonly used because of its simplicity and its wide velocity range. There are limitations, however, in how precisely At can be measured. Some of the most common limitations are as follows: The resolution time of the coincidence circuit or chronotron circuit. Rise time and time dispersion of signal output pulses from the detectors and photomultiplier tubes. * Transit time dispersion in photomultiplier tubes. Rise time of amplifier circuits when used. Referring t o (I), a typical fast coincidence circuit employing a 6BN6 gated beam tube is shown in Fig. 2, where the input signals are applied

FIQ.2. A typical fast coincidence circuit (simplified version).

to the two grids of the 6BN6 (connections 2 and 6). A resolution of the order of sec can usually be obtained with this circuit without difficulty. One can push the resolution higher if extreme care is exercised in the design and construction of the circuit and in the careful selection of the 6BN6 tubes.

* See also Vol. 2, Section

11.1.3 and Vol. 4, A, Part 2.

440

2,

DETERMINATION OF FUNDAMENTAL QUANTITIES

Using a time-to-pulse-height converter’ type of chronotron circuit a resolution of -5 X 10-10 sec has been obtained. If one wishes to measure the mean velocity of a group of particles instead of the velocity of an individual particle, one can measure the position of the peak of the resolu-

FIG.3. A circuit diagram of the time-to-pulse height converter.

tion curve for such a group of particles and a much higher resolution of -lo-” sec can be achieved. A diagram of the time-to-pulse-height1 converter circuit is shown in Fig. 3, and a typical resolution curve is shown in Fig. 11, Section 1.8.1. sec The factors (2) and (3) usually limit the time resolution to for measurements of velocities of individual particles depending on Dhe R. E. Green and R. E. Bell, Nuclear Znstr. and Methods 3, 127-132 (1958) ; G . C. Neihon and D. B. James, Rev. Sci. fnstr. 26, 1018-1024 (1955).

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

441

characteristics* and size of the scintillators and the types of photomultipliers used. If semiconductor detectors are used instead of scintillation detectors, the limitations due to photomultipliers are eliminated and the transit time of the electron-hole pairs created in the detector alone becomes the limiting factor. I n the factor (4), for amplifiers which follow immediately the signal output from the photomultipliers, the rise time in the commonly used wide band amplifiers is usually fast enough that they do not limit the time resolution. Furthermore, when high gain photomultiplier tubes such as Amperex type 56AVP or RCA type 7264 are used, the output signal from these tubes is large enough that no amplification is needed before feeding into the coincidence or chronotron circuits. However, in the case where semiconductor detectors are used, the output signal from these detectors is so small (approximately a fraction of 1 mv for minimum ionizing particles and up to several mv for LY particles) that low noise characteristics as well as fast response in the amplifier is required. At the present, the best rise time of such amplifiers obtainable2is -2 X 10V see. Current developments in parametric amplifiers* indicate promising sec. application with a rise time of The velocity of protons of 800 Mev/c momentum (p = 0.64) was measured4 using the time-of-flight coincidence method to an accuracy of +1%. Using a time-to-amplitude converter type of chronotron circuit, At for pions, protons, antiprotons, K mesons, etc. of momentum up to -5 Bev/e has been m e a ~ u r e dThe . ~ time-of-flight path d is 27 meters, which corresponds to a time of flight of 90 mpsec for light, and the time resolution is about 0.9 X see. Figure 4 shows the time of flight versus momentum for the various particles. The curves are the delays relative t o light for a flight path of 27 meters, calculated for 7r mesons, K mesons, protons, and deuterons. The experimental points are the average values of the measured delays. A typical time delay spectrum for 3.30 Bev/c negative pions and antiprotons is shown in Fig. 5, which also shows the clear separation of the two types of particles. The time-to-amplitude converter type of circuits were widely used * Refer to Chapter 1.4. W. Higginbothem, Proc. Solid State Detector Conf., Gatlinburg, Tennessee, October, 1960. 3 Proc. Intern. Conf. on Instrumentation for High Energy Nuclear Physics, Berkeley, California, September, 1960. 4 J. C . Brisson, J. Detoef, P. Falk-Vaivant, L.Van Rossum, G. Valladas, and Luke c. L. Yuan, Phys. Rev. Letfers 3, 561 (1959). 6V. T. Cocconi, T. Fazzini, G. Fidecero, M. Legros, N. H. Lipman, and A. W. Merrison, Phys. Rev. Letters 6, 19 (1960). 2

442

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

in the neutron time of flight measurements.* As an example, in one such application,6 a neutron beam was produced in a Van de Graaff target by sweeping the Van de Graaff proton beam past the target 7.4 X lo6 times per second. This is accomplished by applying a n alternate voltage

20 18 16

\

14

0

Positive particles

o

Negative particles

2 12 al m

-2 10 zl 8 6

4

2 1

2

3

4

5

6

FIG.4. Time of flight versus momentum for t h e various particles. The curves are the delays relative t o light for a flight path of 27 meters, calculated for pion, K mesons, protons, and deuterons. The experimental points are the average values of the measured delays. The statistical errors are not indicated, as they are smaller than t h e size of the circles.

from 3.7 Mc power oscillator to a pair of deflection plates across the proton beam. The time of production of the neutrons was inferred from the phase of the oscillating electric field. I n order to measure the time interval between a given phase of the rf voltage and the neutron detector

* Refer t o Section 2.2.2.2. W. Weber, C. W. Johnstone, and L. Cranberg, Rev. Sci. IrLstr. 27, 166 (1956).

2.2.

DETERMINATION OF MOMENTUM A N D E N E R G Y

443

signal, the rf voltage was first applied to a circuit which generates a very short pulse or “pip” for each rf cycle. A measurement was then made of the time interval between a neutron signal pulse and an adjacent rf pip. The resolution in At obtainable in this case is about sec. Using this technique, the neutron spectra of the decay K40 and the mean life of the first excited state of K40 were obtained.’~~

ij-

,600/ Countslchonnel

p = 3.30 Gevlc

-

1400 -

I mpsec

I,

‘

1200 -

1000 -

800

--

600

-

400

-

/0.9

6:

mpsec

b

40 -

1

”

-40

p

Time sorter channel

FIG.5. A typical time delay spectrum.

Both the above results obtained are the mean values of the velocity or the time of flight, At. To measure accurately the velocity of individual particles, the time-of-flight method is generally employed for particles in the energy region of from a fraction of a Mev to about 1 Bev. For particles of higher energies, the time-of-flight path required for accurate velocity measurements would be extremely long and becomes rather impractical in most instances. For such cases, cerenkov counters are much better suited (see Section 2.2.1.3.3). ’R. E. Holland and F. J. Lynch, Phys. Rev. 119, 903 (1959). F.J. Lynch and R. E. Holland, Phys. Rev. 114, 825 (1959).

444

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

2.2.1.3.2. MEASUREMENT OF VELOCITY.* Cloud chambers or bubble chambers can be used to determine the velocity of a particle if measurements of energy loss or ionization can be made in the chamber. The energy loss of a particle, under the assumption that many small individual energy losses make up the total, is given by

- p2 - 6 where

1

(2.2.1.3.2.1)

q = upper limit to energy loss considered in an individual event

N

=

number of electrons per cm3

m = mass of the electron

e

=

charge of the electron

v = velocity of the ionizing particle

p Ze

I 6

= = = =

v/c charge of the ionizing particle average ionization potential (determined experimentally) density effect correction.

Thus we can write ( d E / d x ) , , = f(p) and if q and I are not functions of p, but are constants and are known, the energy loss versus velocity curves can be calculated. Energy loss occurs by ionization and excitation of the atoms in a cloud chamber. The ions produced form droplets in a cloud chamber and can be counted, or their number estimated in other ways. If we make the further assumption that the ratio of energy loss to ionization and excitation is independent of 0, we can find the velocity by determinifig the ionization. In a bubble chamber, as in nuclear emulsions, the relationship is not so obvious. The number of bubbles per unit length may be counted and related to the velocity, and techniques for determination of velocity in bubble chambers are developing rapidly. Some general remarks can be made about the regions in which velocity measurements can be made. A typical ionization-velocity curve is plotted in Fig. 1. For low velocities the ionization is approximately proportional to l/P2 and if no other effects occurred would fall to a constant value as P 3 c. However the relativistic contraction of the electric field of the charged particle, which occurs a t velocities near c causes atoms farther from the track to be ionized and produces the “relativistic rise” in ionization. Eventually the density effect term 6 [Eq. (2.2.1.3.2.1)]cancels this relatJivistic effect and the curve flattens out. There is, however, a

* Section 2.2.1.3.2is by W. B. Fretter.

2.2.

DETEEMINATION O F MOMENTUM AND ENERGY

445

minimum of ionization, which occurs at approximately 0 . 9 7 ~ .For velocities less than this the ionization is approximately proportional to l/p2 but for higher velocities a logarithmic increase occurs. The most accurate measurements are made in the l/p2 region. In the relativistic region measurements of velocity can be made if accurate ionization measurements are possible, up to the region where the density

34

1.6 32

"34

f;.

30

I.

4

28

e6

.e 24

0

x

ELECTRONS p MESONS

22

.O 20

I

I

10

100

PY

1000

I(

DO

is for tracks in a cloud chamber containing argon and FIQ.1. Ionization versus helium, each a t 0.2 atmosphere pressure. The vapor was a mixture of alcohol (70%) and water (30%).

effect sets in. Since in dense materials this effect is large, measurements in the relativistic region are only possible in gaseous media, not in bubble chambers. Cloud-Chamber Ionization Measurements. An experienced observer of cloud-chamber tracks can estimate the ionization of such tracks relative to the minimum ionization, and such estimates often prove valuable in qualitative interpretation of cloud-chamber photographs, Many factors influence the appearance of cloud-chamber tracks however, such as intensity of illumination, expansion ratio, position in the chamber, tilt of the track, film development, etc., and such estimates are not very

446

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

quantitative. It is possible, however, to measure ionization either b y drop counting or densitometer measurements on the track. Drop Counting. It has been shown that the number of droplets formed on ions and the total number of ions depends on the nature of the vapor and the expansion ratio. I n general the expansion ratio for 100% condensation is different on the positive and on the negative ions. I n alcohol and in alcohol-water mixtures the drops condense preferentially on the positive ions (Fig. 1, Section 1.6). Substantially all of the positive ions form drops' when only 20% of the negatives do. It is thus common procedure to split the track with a clearing field so that the two columns of ions can be counted separately in accurate absolute measurements of ionization. The expansion of the chamber is delayed while the ion columns separate. Such a procedure insures accurate ionization measurement but tends to introduce errors into the momentum measurement if it is made on the same track because of distortion in the ion columns due to inhomogeneous electric fields. If several tracks occur i n the same picture, as is often the case, or if the operating conditions of the chamber are stable for long periods, this splitting process may be avoided2and calibration tracks may be used, just as is done in photographic emulsion work. The minimum level is determined by counting droplets in tracks of known velocity, the unknown then being compared t o the known. This procedure has been successful recently3in extending the range of velocity measurements far into the relativistic region. 1. The cloud chamber should be temperature-controlled for short- and long-time variations so th at its fluctuation is not more t,han 0.1OC. 2. Accurate and stable control of the expansion ratio must be provided. 3. The light should be adequate so th at individual drop images can be clearly seen. 4. The photographic apparatus must have sharp lenses and sufficient depth of focus. 5. The film must be fine grained with high resolution; otherwise the drop images will be poor. 6. The development of the film must be carefully controlled. It must give botJhfine grain and high contrast, two properties which are normally opposed in photographic practice. 7. The chamber must be kept clean. The drop background must not be allowed to develop and the front glass must be clean. C. E. Nielsen, Ph. D. Thesis, University of California, Berkeley, California, 1941. W. B. Fretter and E. W. Friesen, Rev. Sei. Znstr. 26, 703 (1955). 3 R. G. Kepler, C. A. d'Andlau, W. 3. Fretter, and L. F. Hansen, Nuovo cimento [lo] 7, 711(1958).

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

447

Many of these problems are discussed by Fretter and Frieseq2 who used drop counting techniques to measure ionization of cosmic-ray tracks i n helium. Wilson4 describes the photographic techniques necessary for drop-counting work. It is not always possible to provide photographic conditions adequate for drop-counting work, and in these cases other methods must be used, such as photometric methods. Drop counting is usually not possible in a multiple-plate chamber, for example, because of lighting conditions. Let us consider here some of the fundamental limitations. Image Diameters. The photographic image of a droplet (r E cm in the chamber) is a diffraction pattern produced by the aperture of the lens. The size of the diffraction pattern is usually determined by the aperture of the lens, not by the size of the droplet. The half-angle of the diffraction pattern is given by el,, = 1.221,’~. Then D , the diameter of the diffraction pattern is D = 2.44Xv/a where v = image distance, a = aperture of the lens. We may take X =4X

cm

and rounding off, find D = 10-4v/a cm. I n a typical arrangement12 16, giving D = 16 p. Clearly the wings of the diffraction pattern are not registering on the film, since the drop image diameter is reported as 10 p. High-contrast film and development is responsible for this reduction in size. On the other hand, if the film is overdeveloped or has low resolution, the droplet image may be much larger than this. Magnification. Although the droplet image size on the film does not depend on the demagnification factor M of the optical system, the track width on the film does. If it is necessary to measure curvature and momentum simultaneously, the track width in the chamber w c h cannot be much more than 2 mm. Accurate drop counting requires th a t the drop images on the film be somewhat separated, otherwise corrections for drop overlapping become large and uncertain. Let R = Wj/D, the ratio of track width on the film to drop image diameter. Now M = W&/Wj or M = (WOh/Rd)= [Woh/(2.44XRv/a)].I n a typical case, W o h = 1.5 mm, R = 10, giving M ( v / a ) = 150. Depth of Focus. The Rayleigh limit of focus, corresponding to a path difference of X/4 between central and extreme rays is4 6u = f2~%.1~(v/a)~X in the object space, where v is the image distance, normally not much greater thanf. If we take the practical limit of depth of focus as twice this, and put in our previous values, we get cm = k3.6 cm. 6u Es 4(150)2 4 X v/a

-

4

J. G . Wilson, “The Principles of Cloud-Chamber Technique.” Cambridge Univ.

Press, London and New York, 1951.

448

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

Note that the only way this can be improved is by increasing the product M(v/a). We recall that M ( v / a ) = (Wch/2.44XR).The value of R can be as low as 10 for tracks of fairly low ionization, and if stereoscopic counting is employed, but generally cannot be much less than 10. Thus W , is the only disposable quantity, and it is limited either by the necessity to determine the momentum of the track, or a t least to see that the track exists. In any case, it is useless to attempt drop counting in a chamber more than about 15 cm in depth, if only a $ngle camera is employed. With more than one stereoscopic camera, focused at different depths, the range may be extended. Film Resolution. Another factor, the resolution of the film, circumscribes the design of drop-counting experiments also. The practical limit of resolution. ,of commercially available fast .filmsl\is about 90 lines/mm. Thus a droplet diameter of about 10 pis the smallest that can be obtained. It seems clear that the best use of the scattered light from the droplet is to concentrate it into the smallest possible disk. If we Jiz a value of D , M is immediately determined from M = W,,/W, = W,h/RD. For Wah= 1.5 mm, R = 10, d = 10 microns, M = 15, and f/a = 10. If more light is available, and D can be larger, M can be correspondingly smaller, otherwise the depth of focus will suffer because of the necessity to decreasef / a . If an excess of light is available, and development procedures are carefully controlled, the depth of focus can be increased by increasing f l u . The tendency of drop images t o increase in size can be reduced by high contrast development. In summary, the properties of film and the limitations of physical optics place severe limitations on the depth of focus and the optical system to be used in drop-counting experiments. The actual value of the focal length can be adjusted to fit the experimental conditions, within practical limits of available lenses. The size of the film compared to the size of the chamber determines the lower limit of the value of M . Drop Counting. The procedures for drop counting are relatively simple. It is desirable to obtain stereoscopic photographs and count the droplets while viewing them stereoscopically. Magnification of about 40X produces images readily seen. If drop counting by cell is desired, the observer places a scale reproduced on a clear film base over one view and counts drops between lines of the scale. Usually the separation is adjusted so that about 10 drops per cell are counted on the average. If the track is counted full length, which is a quicker procedure, some indication of starting and stopping such as crossing fiducial lines must be made. The value of 7,the maximum energy loss in a single event, is determined by setting an upper limit on the number of droplets that can readily be counted in a blob. It is generally desirable to set the maximum blob size

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

449

such that fairly accurate counts may be made to determine whether or not a given blob passes the limit set. Otherwise errors in rejecting or accepting a blob may be appreciable. To obtain the number of drops/cm in the chamber, the magnification factor must be applied to the length as measured on the film. This involves a knowledge of the location in depth of the track in the chamber, and for this, of course, stereoscopic photos are necessary. Use of a relatively small stereoscopic angle, so that the track may be viewed in space makes possible both depth measurement and stereoscopic counting. The latter is desirable to eliminate background droplets. Once the procedure is established, the process of drop counting is simple, quick, and accurate, and where relatively low ionizations are involved (up to 4X minimum) preferable to photometric measurements. The errors in drop countingZare readily established. They are: statistical errors, counting errors, errors in the overlap correction, and errors in the correction for the effect of temperature gradient. I n order of magnitude, the standard error in ionization for an individual track 40 cm long containing 1000 drops may be somewhere between 5 and 10%. Photometric Measurements. The photographic density of a track on film may also be used in determinations of ionization. It was shown above t,hat drop-counting procedures are possible only over a rather limited depth of focus, and in many cases ionization measurements are required for deep chambers or where photographic techniques are not good enough to permit drop counting. In such cases it is possible to make a photometric determination of the density of the track. Even when drop counting is possible for minimum ionization tracks, heavily ionizing tracks cannot be counted, and photometric measurements may be required. The techniques of photometric measurements of ionization of cloudchamber tracks have been discussed6 by Bjornerud. His apparatus consisted of a precision microphotometer, a voltage-regulated light source together with projection apparatus, a d i t upon which the image was projected, a phototube which measured the light coming through the slit, and an amplifier and recorder arrangement. The method is based on the assumption that the amount of light transmitted through the photographic image of the track is related to the density of ionization in the track. In order to use this method, it is necessary to decide first on the slit width. If the slit is too wide, the presence of the track has little relative effect, on the transmission through the given area of the film, if it is too narrow, small fluctuations in background and in the track become important. A slit which is approximately as wide as the track appears 6

E. K. Bjornerud, Rev. Sci. Instr. 26, 836 (1955).

450

2.

DETERMINATION OF FUNDAMENTAL QUANTlTIES

to be suitable. The procedure is then t o scan across the track, recording the density of the developed image, and compare the density at the track with the background. It is necessary to calibrate the photometric readings obtained against readings obtained on tracks for which the ionization can be determined by other means. For example, if electrons or mesons are present they may

PHOTOELECTRIC MEASURE, D

FIG.2. Photographic density as a function of ionization in cloud chamber tracks.

be identified, and their ionization predicted from this momentum. A calibration curve is established (Fig. 2) between the photoelectric measure which can then be used on unknown particles. Corrections must be made if the track is tilted in the chamber. Various difficulties are inherent in this procedure. 1. The transmission of a n out-of-focus track may vary with the degree of out-of-focus. The size of the optical image of a droplet will clearly be different and the overlapping of images will be different. Bjornerud estimates that in a 5 X minimum track the total transmission could change by about 60% if the image size doubled in area. To overcome this

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

451

difficulty, comparison tracks in the same region of the chamber, equally out-of-focus, must be measured. 2. The illumination may not be uniform over the entire chamber. Coupled with this is the variation of scattered light as a function of scattering angle. Again, comparison tracks nearby must be used for accurate results. 3. The method is limited in its range of applicability. The calibration curve saturates a t high value of ionization, and the method cannot be used beyond 8 X minimum ionization. For lightly ionizing tracks, 2 X minimum seems a lower limit, since variability in background density enters in an important way when the density of the track is small. 4. The method is relatively slow. For accurate measurements, Bjornerud reports that on the average about an hour is required to determine the ionization of a particle, although this could possibly be shortened. This implies several measurements along the track, and the use of nearby calibration tracks. For rough estimates on short tracks not so much time would be required. The errors involved in ionization are estimated by Bjornerud to be 10 to 15% in the most favorable situation. This assumes that calibration tracks are conveniently available, and under these circumstances the error in measurement compares favorably with the drop-counting procedure. On the other hand, if calibration tracks in the same region of the chamber are not to be seen, the standard error in an individual track may be 30 to 40%. This accuracy is considerably poorer than that obtainable by drop-counting techniques, If the track is of high quality the error should be lower, however, and it may be possible to extend the range of ionization measurements by drop counting t o higher values by the use of photometric measurements. Measurement of Velocity i n a Bubble Chamber. The possibility of counting bubbles along the track of a particle in a bubble chamber to determine the velocity of the particle was first recognized by Glaser, and a systematic study of procedure was published by Glaser, Rahm, and Dodd in 1956. The technique has rapidly become refined,617and it now appears to be an accurate and useful method for measurement of velocity. Just as in a photographic emulsion, where the grain-density versus velocity must be determined experimentally, so in a bubble chamber, the bubble density versus velocity must be determined experimentally. Methods developed in connection with emulsion methods have in fact proved to be very useful in bubble density measurements. It has been shown6.7 that the bubble density g (number of bubbles per D. A. Glaser, D. C. Rahm, and C. Dodd, Phys. Rev. 102, 1653 (1956). G. A. Blinov, I. S. Krestnikov, and M. F. Lomanov, Soviet Phys. JETP 4, 661 (1957). 6

7

a 0

a

a e

2.2.

DETEXMINATION OF MOMENTUM AND ENERGY

453

unit length along the track) can be written as g = goZ2//32where Z is the charge of the particle, p the velocity, and go is a coefficient depending on the medium. (See Fig. 3.) Instead of the number of bubbles per unit length, as in earlier measurements, the mean gap length between bubbles or the number of gaps per unit length is determined. Only for low track densities are actual bubble counts used. Straightforward bubble counts at high track densities are inaccurate for two reasons: the bubbles in the chamber may be close enough together to coalesce while growing, or the images on the film may overlap. However, for low track densities these problems are not serious and bubble counting is used because it is a simple and quick procedure. In this case7 g =

1

1

D ln-. 1 - k D

Here g is the track density, D is the bubble image diameter, and k = N / L , N being the observed number of bubbles and L the length of the track, and the logarithmic form takes into account the overlapping of the images. The statistical error in this method is Ag/g = N-1/2. If the track is more dense, the mean gap length method is used. It takes longer to make the measurements, but the results do not depend on the bubble image diameter. A minimum measurable gap length E is first determined. Then g = l/(Z, - B) where I , is the mean gap length, and the statistical error is Ag/g = N,-lI2, N , being the number of gaps counted. For the densest tracks, the number of gaps larger than B is counted. Then8

N,

= gLe-O(D+Q.

The statistical treatment in this case is more complicated, but the method is good for ionizations up to 80 times the minimum ionization. The fact that the bubble density varies as l/p2 supports the idea that bubbles are formed on small 6 rays along the track of the particle, since the frequency of 6 rays also varies as l/p2. The quantity go must be determined experimentally. It depends on the temperature and expansion conditions and generally must be obtained from measurements on known tracks. Once go is determined, the velocity can be determined from the measured value of g and Eq. (2.2.1.3.2.1). Blinov et al.? have shown that the relativistic rise of ionization beyond the minimum is appreciable in propane. From measurements on electron tracks, they find that the ionization rises by 12.5% over the value at minimum, for tracks of p-y N 20-200.

* M. G. K.

Menon and C. O’Ceallaigh, Proc. Ray. SOC.A221, 292 (1954).

454

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

2.2.1.3.3. MEASUREMENT OF VELOCITY USINGCERENKOV COUNTERS.* Referring to Chapter 1.5,we note that accurate measurements of velocities of charged particles can be obtained using a properly designed Cerenkov counter provided that the radiator of the counter possesses a n index of refraction above the threshold value for the particles to be measured (see Section 1.5.1). We note also that the intensity of the Cerenkov radiation is considerably lower than that of the light output from a corresponding scintillation detector (generally speaking, the ratio of photoelectrons generated by charged particles in plastic scintillators per centimeter to that generated by Cerenkov radiation in optical media per centimeter is greater than 30 to 50). Therefore Cerenkov counters are much more difficult to use in practice than scintillation counters, and they are usually employed when conventional detectors do not provide sufficiently accurate information for the velocity measurements required.? Of course, a long enough radiator, say several inches to a few feet long, would give sufficient signal pulse t o overcome the above-mentioned difficulties. I n general, there are several advantages of the Cerenkov counter, which are listed below. (1) The Cerenkov pulses are extremely fast, since there is no inherent decay time constant involved in the Cerenkov light-emitting processes, whereas the scintillation processes involve a decay time constant of 2 sec. There is a limitation in the time resolution of the Cerenkov pulse due t o the difference in the transit time of the particle in transversing the length of the radiator and the propagation time of the wavefront in the forward direction. This time difference is a second-order effect, however, and is generally negligible when compared with other limitations in time resolution imposed by photomultiplier tubes and electronic circuitry, etc. Thus, when measurements are performed in a large flux of desired particles or in a large background of undesired particles, a pileup or jamming will result in a scintillation type of detector if the size of the detector is such that the rate of the total number of particles detected approaches the resolution limit of the scintillator or the electronics. The Cerenkov detector on the other hand, is in many cases not subjected t o this difficulty. Especially in the case of focusing-type Cerenkov counters, the effects due to background are much smaller owing to the narrow range of velocities acceptable in such counters. (2) Scintillators respond to charged particles of all energies, whereas Cerenkov counters respond only to those particles whose velocities are

t Refer to Section 2.2.1.3.1. -

* Section 2.2.1.3.3is by Luke C. 1. Yuan and S. J. Lindenbaum.

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

455

above the velocity threshold of the radiator (see Section 1.5.1.1)) and in the focusing type of counter only those particles within the accepted velocity interval are not counted. Thus, when a counter telescope is used in coincidence, the chance coincidences are much smaller in the case when cerenkov counters are employed than when scintillators are used. This is evident, since the chance coincidences are directly proportional to the product of the total number of counts obtained in each individual detector. (3) I n the detection of short-lived particles, the detection system should be placed as near the source of the particles as possible, so as to avoid the loss of intensity due to decay. Cerenkov counters can be placed very close to the source for a n accurate velocity measurement, even for particles in the relativistic region ; whereas other methods of velocity measurement, such as the time-of-flight method (see Section 1.2.1.3.3)) would require an extremely long time-of-flight path in order t o obtain a reasonably accurate measurement in the velocity in the relativistic region. (4) When using gas as radiator, the application of such cerenkov counters plays a specially important role in the high-energy nuclear physics experiments. One can vary the index of refraction n by varying the pressure of the gas, and, hence, the selection of the desired particles can be obtained easily over a wide range of velocities. ( 5 ) Since cerenkov radiation has a directional property, one can take advantage of this property to ascertain the direction of the particle measured as well as its velocity. Of course, there is a disadvantage in this too, because cerenkov counters can only be used to detect particles within a limited direction. 2.2.1.3.3.1. Nonfocusing 6erenkov Counters. The nonfocusing type of Cerenkov counter does not give information on the angle of emission of the eerenkov radiation and, hence, is not, suitable for the highest accuracy of velocity measurements. I t is used generally to bracket those particles whose velocities lie above the threshold velocity of the radiator. An example of such an application is the detection of n’s in the Bev region (p from 0.980 to 0.999) in the n-p scattering experiment by Devlin et aZ.,l where a 6-ft long, 4-in. diam, nonfocusing gas cerenkov counter was used essentially to separate nf from protons. The gas used as radiator was sulfur hexafluoride (SFe) or Freon 12 (CC12F2)a t a pressure of up to 150 psi. Two or more of such nonfocusing counters, with slightly different 1 T. J. Devlin, B. C. Barish, W. N. Hess, V. Peres-Mendez, and J. Solomon, Phys. Rev. Letters 4, 242 (1960);J. H.Atkinson, W. N. Hess, V. Peres-Mendes, and R. W. Wallace, ibid. 2, 168 (1959).

456

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

t,hreshold velocities, are often used in coincidence-anticoincidence pairs to bracket those particles whose velocities fall within a comparatively narrow range defined by these threshold velocities. An example of such method is the K+ production experiment in p - p collisions at 3.0 Bev by Lindenbaum and Yuan,2 where the counter system included cerenkov counter elements in coincidence and anticoincidence to select a velocity interval, and a magnet was used to select a momentum interval such that the combination required a positive particle of rest mass equivalent to 495 5 100 Mev in order to count. This excluded all known particles except K+. The background, due to the accidentals in the telescope, was less than 5%. A second example of this method is the experiment on the ~ one of the measurement of K+ lifetimes by Fitch and M ~ t l e y .In Cerenkov counters used, the light collection is such that only light that gets out of the radiator (in this case CSz was employed, which has an index refraction, n = 1.62, /3 min = 0.62) is detected. Total internal reflection results from the cerenkov angles corresponding to velocities greater than 6 = 0.78, and the counter responds only to particles with velocities 0.62 < /3 < 0.78. The velocity range defined by a telescope of nonfocusing type counters as those described above is generally not sufficiently narrow to give a highly accurate velocity determination, since the threshold velocities are usually not sharply definable in practice. 2.2.1.3.3.2. Focusing 6erenkov Counters. Using the focusing-type Cerenkov counters, as described in Section 1.5.2, we note that t,he velocity resolution is expressed by [Eq. (1.5.7)]:

Thus, by choosing n to obtain and the smallest @ permissible to give minimum usable light intensity, which is proportional to sin2 B, accurate velocity measurements (A0 < l O V ) can be obtained. Liquid Cerenkov Counters. An example of such a n application is the antiproton experiment by Chamberlain et u Z . , ~ where particles of momentum of 1.19 Bev/c from a target in the 6-Bev Bevatron were selected by a deflection and focusing magnet system. The antiprotons were carefully selected from the'presence of an overwhelming number of ?r mesons in the beam by means of a focusing cerenkov counter (see Section 1.5.2.1, Fig. 10) in conjunction with a time-of-flight scintillation telescope. The a

a

S. J. Lmdenbaum and Luke C. L. Yuan, Phys. Rev. 106, 1931 (1957). V. Fitch and R. Motley, Phys. Rev. 101, 496 (1956). 0. Chamberlain, E. SegrB, C. Wiegand, and T. Ypsilantis, Phys. Rev. 100, 947

(1955).

2.2.

DETERMINATIOX O F MOMENTUM AND E N E R G Y

457

radiator used for the focusing counter was fused quartz (no = 1.458). Only particles with 0 in the range between 0.75 and 0.78, i.e., AD = 0.03, (corresponding to a A0 of -5') were counted. A nonfocusing counter employing C8FlB(no= 1.276) was also used in anti-coincidence in this experiment to further eliminate the d s (0 = 0.99). A second example is the experiment by Gilly et aL6 to search for unusual particles produced by the CE RN (European Organization for Nuclear Research) ProtonSynchrotron operated a t 24 Bev. The mass of the particle was identified by measuring its momentum and velocity (Cerenkov angle). The basic element in the detection system is a differential, isochronous, self-collimating Cerenkov counter employing liquid as radiator (see Fig. 1). The

FIG.1. Basic design of a differential isochronous, self-collimating Cerenkov counter.

range of 0 covered by this counter is 0.85 2 0 5 0.96, which corresponds t o the maximum yield of particles with masses from a few hundred to a few thousand &lev produced in the forward direction. Figure 2 shows typical spectra a t 0 = 0.95. The mass of the particles agree to within 1% of the accepted values. Gas Cerenkov Counters. There are a number of gases which are suitable for use a s radiator. However, when selecting the gas radiator, one should bear in mind certain conditions which are listed below. (1) The gas chosen should have a low dispersive power. (2) The desirable range of the index of refraction should be obtainable niostly in the gaseous state at various not excessively high pressures under 6L. Gilly, B. Leontic, A. Lundley, R. Meunier, J. P. Stroot, and M. Szeptycka, Proe. Ann. Rochester Conf. on High Energy Nuclear Phys. Interscience, New York, 1960

458

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

the operating temperature range. Once the liquid phase is reached, very little increase in the index of refraction can be expected by further increasing the pressure. (3) Gases having low 2 and low mass numbers are preferred, to reduce the effects due to Coulomb scattering and nuclear interactions. (4) Gases having high index of refraction a t comparatively low pressures are desirable for mechanical convenience in the counter construction. (5) One should make sure that the gas chosen as radiator does not scintillate. I

I

I

I

I

I

I

I

I

P

3 Gev C

FIG.2. Particle spectrum having p = 0.95 from 24Bev protons striking a 50 target in the CERN Proton-Synchrotron.

fi

Al

A gas focusing counter using gaseous fluorochemical chemical 0.75 a t elevated temperatures and high pressures was employed to detect K+ particles in the K+-p scattering experiment by Burrowes et aL6 (see

-

Section 1.5.2, Fig. 12). The velocity resolution is such that the counting rate dropped by a factor of 20 when AD 0.006. An appreciably different, focusing-type gas counter such as the one shown in Section 1.5.2, Fig. 14, possesses a much better velocity resolution (AD < than the one 6 H. C. Burrowes, D. 0. Caldwell, D. H. Frisch, D. A. Hill, D. M. Ritson, and R. A. Schluter, Phys. Rev. Letters 2, 117 (1959).

2.2.

DETERMINATION OF MOMENTUM A N D ENERGY

459

mentioned above. Such counters have been used to analyze the highenergy particle spectra (from 3 to 10 Bev/c momentum) emitted from thin targets placed in the 28-Bev proton beam of the CERN ProtonSynchrotron, as well as from a similar target in the 32-Bev Brookhaven alternate gradient proton synchrotron. CERN used ethylene gas as radiator (COZ was used at Brookhaven), and with the optical system so adjusted such that cerenkov radiations emitted at 0 = 6" are accepted. One of the essential advantages of this counter is that 6 is small and, hence, good resolution results. Furthermore, the main optical system of the counter, including the resolution defining annular slits, is completely

.020 .018 W V

z a 0

.016 ,014

z I3

.012

a

.OlO

m

.008 .006 .004

.002 .990

992

.994

.996

.998

1.000

V/C

FIQ.3. Velocity spectrum of the negative beam of 8 Bev/c momentum, defined in 5" lab. CERN beam.

situated inside the radiator, thus avoiding any possible aberration in the cerenkov light resulting from its passage through a thick lens or window. The optical system of this counter consists of a spherical concave mirror having a focal length of 130 cm, a plane mirror placed at 45" for bringing the light out of the beam, and a specially shaped conical mirror for concentrating the light transmitted by the annular slit onto the photocathode of a 2-in. 56 AVP photomultiplier. The angular resolution, as defined by the annular slit, is f O . O 1 radian. The width of the slit is the main limitation in resolution for particles of @ > 0.99. The velocity Figure 3 shows a typical velocity resolution of this counter is A/3 2 spectrum of the negative secondaries of 8 Bev/c momentum emitted a t

460

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

5" from the internal beam direction (the internal beam energy was 28 Bev). The antiproton, I<- and T-, peaks are clearly separated in the spectrum. A method for improving the rejection ratio of the desired to undesired particles can be achieved by collecting simultaneously in the same counter the Cerenkov light emitted by the undesired particles as well as by the desired particles. This light is collected in separate channels, and I

I

Pressure of CO,

FIG. 4. Analysis of a negative beam of 8.5 Bev/c momentum at Golab. production angle at Brookhaven AGS operated at 20 Bev machine energy.

the Cerenkov light from the desired particles is used in coincidence, whereas the light from the undesired particles are used in anticoincidence with signals from the other elements of a counter telescope. Such a counter (see Fig. 14 in Chapter 1.5) with both a signal and anti-coincidence channel has been used by the authors and their collaborators to analyze beams at the Brookhaven 32 Bev Alternating Gradient Proton Synchrotron. A typical analysis of a negative beam of 8.5 Bev/c momentum a t -4$" lab. production angle from a beryllium target is shown in Fig. 4.An improvement in the 7r rejection ratio by a factor of -10-100 was obtained by Cork' in the detection of 2.5-Bev K particles a t the Bevatron by using an anti-coincidence channel. 7

B. Cork, private communication.

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

46 1

2.2.2, Neutrons 2.2.2.1. Recoil Techniques for the Measurement of Neutron Flux, Energy, Linear and Spin Angular Momentum.* t 2.2.2.1.1. INTRODUC-

By virtue of being electrically neutral in charge, the neutron is rather less amenable to easy observation as compared with a charged particle. Rapidly moving charged particles locally ionize the matter they traverse and hence their trajectories are identified by observing the trail of ionization. Moreover, since the moving charged particle constitutes a current it may be deflected in a magnetic field as well as an electric field. The neutron perforce lends itself to none of these modes of observation or manipulation. However, it does possess a magnetic moment which can be used to rotate its intrinsic spin orientation. This property is also shared by the proton though the two moments are of differing sign and magnitude. Contemporary methodology detects the neutron only by its interaction with another nucleon or aggregate of nucleons. (The interaction of the neutron with electrons is thought to be very small in comparison to the very strong interaction of charged particles with electrons.) The nuclear interaction may be either elastic (the neutron and target fly apart after collision with no internal excitations) or inelastic (absorption of the neutron in the target, etc.). This note will concern itself with elastic processes only. 2.2.2.1.2. PRINCIPLES OF FLUX,$ENERGY, AND LINEARMOMENTUM MEASUREMENTS. From such a collision we desire to calculate the neutron energy, linear momentum, and spin angular momentum from a n observation of the recoiling target nucleus. A theoretical lower limit of applicability is of course immediately set on such a technique since the neutron must impart sufficient velocity to the recoil nucleus that it sheds some or all of its electrons and hence registers as a charged particle. For the purposes of this discussion the theoretical lower limit may be ignored. From measurements on ionization trails of the recoiling nuclei we seek to infer three characteristics of the neutron beam, namely: the flux of neutrons in the beam, the energy of the neutron, and the momentum. The evaluation of these quantities from an observation of the ionization trail of the recoil particle cannot be broken down into three independent TION.

t A virtually nionumental body of literature exists on much of the content of this chapter which fact has led to the following point of view. The general principles of t h e measurements will be enunciated and a nonexhaustive set of specific examples of various techniques will be discussed and briefly commented on. A brief Appendix will consolidate some design data. Literature cited will provide adequate guidance to other publications. 3 Refer t o Vol. 5, B, Chapter 2.8.

* Section 2.2.2.1 is by John E. Brolley, Jr.

462

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

procedures but rather is a somewhat interacting process. To evaluate the beam flux1.2 a knowledge of the probability of collision of the neutron with the target nucleus is required. This probability or cross section will depend on the angle of recoil with respect to the original direction of the neutron. Many standard texts discuss the concept of cross ~ e c t i o n . ~ . ~ Thus the method begins with a limitation that the direction of the linear momentum vector of the neutron is known. Such a limitation is however DETECTOR OF RECOIL NUCLEI

/

TARGET OF

NEUTRON SOURCE

’dw,

Fro. 1. Schematic diagram of an idealized scattering process.

only a practical one as will be presently indicated. I n general, having the source of neutrons sufficiently far from the recoil detection system defines the neutron direction with the desired accuracy, assuming the source and detector are of finite dimensions. The direction may also be defined by collimators.s*6 Hence if we know the angle of recoil and the probability for such a collision we may compute the flux of neutrons if we know the density of target nuclei and the rate of production of recoils at a given angle. The counting rate of the detector, as schematically depicted in Fig. 1, 1 H. H. Barschall, L. Rosen, R. F. Taschek, and J. H. Williams, Revs. Modern Phys. 24, 1 (1952). * L . Cranberg, R. B. Day, L. Rosen, R. F. Taschek, and M. Walt, i n “Progress in Nuclear Energy” (R. A. Charpie, J. Horowitz, D. J. Hughes, and D. J. Littler, eds.) Ser. I, Vol. I, p. 107. Pergamon, New York, 1956. 8 See, for example, L. I. Schiff, “Quantum Mechanics,” 2nd ed. McGraw-Hill, New York, 1955. 4 N. F. Mott and H. S. W. Massey, “The Theory of Atomic Collisions,” 2nd ed., p. 19. Oxford Univ. Press, London and New York, 1949. 6 H. E. Adelson, H. A. Bostick, B. J. Moyer, and C. N. Waddell, Rev. Sci. Instr. 31, 1 (1960). 6A. Langsdorf, Jr., in ‘(Fast Neutron Physics” (J. B. Marion and J. L. Fowler, eds.), Vol. IV, Part I, p. 721. Interscience, New York, 1960.

2.2.

463

DETERMINATION OF MOMENTUM A N D E N E R G Y

will then be related to the flux from the neutron source by the relation

C

=f

dwi dNg(E,,$)

do2

(2.2.2.1.1)

where

C

number of recoils counted, per second, by the detector) f = number of neutrons per steradian per second in the direction of the target,* dN = number of target nuclei, dwl = solid angle subtended at the source by the aggregate d N , dwz = solid angle subtended by the detector at dN, a(E,,J.) = cross section (for a neutron of incident energy E n ) for the recoil to proceed in the direction $; cm2 per steradian, =

and we have assumed that every recoil entering the detector is counted. Relation (2.2.2.1.1) is of course idealized. Frequently the design of a particular experiment will be such that relatively large solid angles must be used in order to accumulate information at a satisfactory rate. Then it is necessary to integrate relation (2.2.2.1.1)over the finite dimensions of the source, target) and detector. There exist several types of generalized calculations,7-10which may be adapted to some particular experiments. Thus a flux measurement in principle requires a geometrical calculation, a knowledge of the number of scattering centers) and the cross section for scattering. There are other small details as we shall see in a n illustrative example. The second characteristic of the neutron beam we should like to consider is the energy. I n some cases the energy of the beam will be sharply defined; in others it may consist of bands of energy or a continuum or an admixture of, both. The collision of the neutron with the target nucleus will deposit a definite amount of kinetic energy with the recoil particle. In the nonrelativistic caset (2.2.2.1.2)

* The neutron emission from the source will be anisotropic in many cases.

t Appropriate kinematics of the scattering, both relativistic and nonrelativistic, are given in Appendix A. G. Breit, H. M. Thaxton, and L. Eisenbud, Phys. Rev. 66, 1018 (1939). 8 E. A. Silverstein, Nuclear Instr. and Methods 4, 53 (1959). S. J. Bame, Jr., E. Haddad, J. E. Perry, Jr., and R. K. Smith, Rev. Sci. Instr. 29, 652 (1958). lo

E. Haddad and J. L. Warren, Rev. Sci. Instr. 30, 664 (1959).

2.

464

DETERMINATION OF FUNDAMENTAL QUANTITIES

where

E , = incident neutron energy, E R = recoil energy, .M, = neutron mass, A I R = recoil mass, )I = laboratory angle of the recoil with respect to the incident neutron direction. We have then to measure the recoil energy a t angle Ic/. There are many methods of measuring the energy. Those that we shall consider utilize nonnuclear interactions. In general if one attempts to utilize the nuclear interaction, one greatly sacrifices speed of data accumulation. Some methods currently in vogue are the following: range measurements, deflection in electromagnetic fields, light output of scintillations, cerenkov devices, time-of-flight equipment, or combinations of these. 2.2.2.1.3. PRINCIPLE OF SPINANGULAR MOMENTUM MEASUREMENT. Lastly there is the question of the momentum of the neutron. There are two parts to this query: namely, what is the linear momentum and what is the spin angular momentum? The first part has in principle been considered. The direction of the linear momentum is usually specified by the design of the experiment. However, there may be occasions when one does wish to establish the direction of a neutron beam. Collimators6-6 could be used in conjunction with a suitable recoil detector. Such an arrangement is relatively bulky, for example, for installation in extraterrestial observational vehicles. In the extreme relativistic case the problem is simple because the recoil will proceed in essentially the same direction as the incoming neutron. However as the energy of the neutron decreases the recoil particles will be found inside a cone centered about the incident neutron axis whose solid angle opens to 2ir at a low neutron velocity. Thus in the low-velocity limit the direction of the neutron is still given since there will be no recoils in the backward hemisphere. The spin angular momentum of the neutron remains to be ascertained. In contradistinction to the linear part of the problem the magnitude of the neutron's spin angular momentum is known and we have only to determine the direction of the angular momentum vector or polarization."*l2 This question is actually relevant only if we have a number of neutrons in the beam and seek to ascertain the average spin orientation in the beam. In all of the preceding remarks it, was assumed that there was no L. Wolfenstein, R n r ~ Rev. . Nuclear Sei. 6, 43 (1956). Proceediugs of the Internationd Symposium on Polarization Phenomena, Basel, Switzerland, 1960, Helv. Phys. A d a Suppl. VI (1961). l1

2.2.

DETERMINATION O F MOMENTUM AND EN ERG Y

465

preferred spin orientation in t,he neut,ron beam. If now we consider a neutron beam having some preference for spin orientation, we must have some quantitative description. The projection of the neutron spin on any axis is known to be either +h/2 or -h/2. For t,he purposes of this discussion one may choose the axis of reference as t,hat defined by the perpendicular to the plane of scattering containing the incoming and outgoing neutron directions as indicated in Fig. 2. Thus we choose the axis to be in the direction k,,,inX k,,,eut.k, is the wave vector of the neutron and hk, is the linear momentum of the neutron. Now the relevant question

-n,in k

’kn,out

FIG.2. Vector diagram defining positive polarization.

is: for a given current of neutrons passing point 0 in unit time, what fraction have their spin projection up and what fraction down? We define the polarization P by (2.2.2.1.3) Thus if the fraction of the beam having spin parallel to k,,i,, X k,.,,, is 1, then P = +1. If all spins are antiparallel, P = -l.* In the event that we are dealing with a neutron beam having P # 0, there will be a difference in intensity of the left and right scattering of some recoil nuclei. The following discussion will be restricted to spin-zero recoil nuclei. The force which scatters the neutron from the target nucleus is spin dependent. If the force experienced by the neutron when the spin and orbital angular momentum vectors are parallel is different from the force experienced when the spin and orbital angular momentum are antiparallel then there will be preferential scattering for one side. The practical expression” of this fact is that the differential scattering

* As defined here the signs of P are i n accord with t h e “Base1 Convention.”’* The opposite definition occurs in the literature occasionally but may be less frequent in the future.

466

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

cross section for the neutron is modified to be

u(En,O) = ap=.o(En,t9)[1

+ PrPnR(En,B)cosCp]

(2.2.2.1.4)

where Cp is the azimuthal scattering angle measured from the plane normal to P,, and we have introduced the quantity, P n R , which is the polarization that would be induced in a n initially unpolarized beam of neutrons scattered by the target nucleus a t angle 8. One may then infer that the left :right scattering ratio is

(2.2.2.1.5)

if cos d = 1. Relation (2.2.2.1.5)then provides us with information on the average orientation of the angular momentum vectors in the incoming beam if we know P n ~ . In summary we have seen in a schematic way how the recoil method can provide information on flux, energy, linear and spin angular momentum of the neutron beam. I n succeeding sections we shall consider specific examples to illustrate various approaches to the problem. 2.2.2.1.4. INSTRUMENTATION FOR FLUXAND ENERGY MEASUREMENTS. In the following sections we shall consider, for the most part, instrumentation that has already contributed useful data or is incipient. The specific examples have been used in certain neutron energy intervals. However, the ideas involved will often lend themselves to an extension to other energy intervals. We shall seek mainly to understand some of the principles and problems involved in the design of such equipment. Some auxiliary data for the design are presented in the Appendix. The data are not all-inclusive but are useful for the early stages of design. 2.2.2.1.4.1.Flux Measurement with a Tandem Counter. Many devices have been developed for the measurement of fast-neutron flux. Of this family the tandem recoil counter, often termed a “ counter telescope,” has proved to be one of the most quantitative and best understood. In its modern form it provides information on energy distribution as well as flux. Earlier models did not give precise energy information. There are four components in this technique: the source of neutrons, the radiator or lamina containing the recoil nuclei, thin AE recoil counter(s), thick E recoil counter (Fig. 3) which function in the following manner. Neutrons from the source impinge on a thin radiator. Some of the neutrons will elastically collide with the target nucleus in the radiator and the forward recoils from the collision will traverse the AE and E counters. If the radiator is relatively thin the recoils originating near the

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

467

source side of the radiator will differ little in energyI3 from those proceeding from the counter side. The recoils deposit a small fraction of their energy in the AE counter(s), sufficient to generate a usable signal, and then stop in the E counter where most of their energy is deposited. The E counter then provides a signal that is approximately proportional to the energy deposited. The AE and E counters may assume various forms. We shall consider the following specialized form. The radiator in this case is thin polyethylene. Recoil protons from the polyethylene pass through two proportional counters and stop in a NaI scintillator. If the crystal were used alone to observe the recoil protons, in many cases accurate identification of the particles would be difficult because of background effects. The crystal will respond to neutrons and gamma rays in the ambient radiation field. However, if a coincidence is required between the two proportional counters and the scintillator it is most likely due to RADIATOR

E COUNTER

FIG.3. Block diagram of a tandem counter.

a particle coming from the radiator: the coincidence imposes a directional condition as well as time restriction. If the background radiation is sufficiently intense there will be high random singles counting rates in each of the counters which will lead to appreciable accidental coincidences. These spurious counts will arise from charged-particle reactions induced by neutrons, as well as spurious recoils in the counter gas. I n some cases a strong gamma-ray background may produce electrons copiously in the counters and give pileup signals. There are several ways in which one may hope to mitigate the problem of accidental counts in a flux measurement with this device. One may simply increase the thickness or area of the radiator, thereby increasing the number of true proton counts. This procedure entails a loss of energy resolution but is tolerable in some circumstances. A second alternative is to decrease the resolving time, but there are practical electronic limits to this procedure. Also if the neutron flux is being produced by a pulsed machine of the cyclotron type for example there is a n absolute lower 13 There are numerous compendia of energy-loss tables. See, for example, W. Whaling, in " Handbuch der Physik-Encyclopedia of Physics" (S. Flugge, ed.), Vol. 34, p. 193. Springer, Berlin, 1958.

468

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

limit beyond which no gain is to be achieved. This will occur when the resolving time becomes comparable to the period of radio-frequency excitation of the dees. In such situations it is also desirable to assure that the beam of the cyclotron has minimum modulation other than that of the radio-frequency. A third consideration is the use of high-2 materials in counter areas which are visible to the actual counter components. I n general at moderate neutron energies, neutron-induced charged-particle reactions have lower probabilities in h i g h 4 than in low-2 materials.

FIG.4. Side view of the tandem counter of Johnson and Trail.

The effect of registering spurious recoiling gas nuclei can be minimized by operating the proportional counters at low pressures. However as the gas pressure is reduced the true recoil from the radiator will deposit less energy. Eventually a lower limit of gas pressure is reached when the true signal decreases to a value approximating the electronic noise. The passage of the true recoil through the proportional counter will give a signal of definite mean value with a characteristic spread.l4vl6 Additional rejection of spurious counts can be achieved by requiring the proportional counters to register only if the counter signal falls within this spread. Some measures can also be taken with the scint.illntor. Evidently the D. Landau, J . IJhys. U.S.S.R. 8, 201 (1944). K. R. Symon, referred to by B. Rossi, (‘HighEnergy Particles,” p. 32. PreriticeHall, 1952. l4L. 16

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

469

smaller the volume the less the number of spurious events that will be registered. Thus the thickness of t8hecrystal should be no greater than the residual range of the recoil. Moreover, if the gamma-ray background is especially severe one may wish to choose a scintillator of minimum gamma-ray detection efficiency. At this point we may consider the instrument of Johnson and Trail.16 This device is schematically indicated in Figs. 4 and 5 .

FIG.5. Detail of the wall and field-tube structure of the Johnson and Trail counter.

Since the device was intended to cover the neutron spectrum 2-20 Mev, provision was incorporated for using several radiator thicknesses. This is clearly desirable since the n-p cross section drops with increasing neutron energy. Energy resolution can be approximately maintained despite the increasing thickness of the radiator since the rate of energy loss of the recoil proton diminishes with increasing energy. Johnson and Trail find that if optimum resolution is maintained the net result is a linear rise in efficiency from 2 to 15 MeV. The recoil protons traverse two proportional counters of the field-tube type. 17s1* Certain advantages accrue over the usual tandem arrangement 15

C. H. Johnson and C. C. Trail, Rev. Sci. Insir. 27, 468 (1956).

A. L. Cockroft and S. C. Curran, Rev. Sci. Instr. 22, 37 (1951). 18 S. C.Curran, in “Handbuch der Physik-Encyclopedia of Physics” (S. Flugge, 17

ed.), Vol. 45,p. 174. Springer, Berlin, 1958.

470

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

when the recoil particles pass through counters perpendicular to the counter wires. With the present field-tube arrangement the actual counting volume is quite well defined. Moreover loss of recoils arising from multiple scattering in the gas19 is partially compensated in contradistinction to the more conventional designs. Wall-scattering of the recoils in the

500

400

300 t

z 6

I

0

; ; 200 Iz

3 0 0

4 00

0

0

10

20

30

40

50

60

70

CHANNEL NUMBER

FIG.6. Coincidence pulse-height spectrum of the scintillator obtained with 13.7-Mev neutrons.

proportional counter system has been greatly minimized by using a wire wall that allows most particles to go through without scattering or possible loss of energy while still maintaining appropriate electric fields in 3% COZ.* the counter. The filling gas was A After generating signals in the two proportional counters, the recoil stops in a N a I scintillator where most of its energy is deposited. The light pulses from the crystal are detected b y a photomultiplier via a simple

+

* When using gas counters it is often convenient to dispense with static systems and purifiers. One may let a steady stream of gas go through the counter from a large reservoir and control the pressure very accurately with a Cartesian manostat. l9 W. C. Dickinson and D. C. Dodder, Rev. Sci. In&. 24, 428 (1953).

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

471

light-pipe. The photomultiplier pulses are analyzed whenever a threefold coincidence occurs between the two proportional counters and the crystal. I n Fig. 6 we note a crystal pulse-height spectrum of 13.7-Mev neutrons. Integration of the peak area gives a measure of the neutron flux. Summary. We have considered one of the examples of the tandem counter flux-and-energy-measuring instruments. The general principles extend to other energy intervals though the exact form of the components may change. Thus a t several hundred Mev all components may be crystals or the E counter may even be a cerenkov counter.20At very low This device can accuenergies there may be no distinct E counter a t mulate data quickly and provide immediate answers. It can be made relatively small and light if required, but is not well suited for handling a large number of events coming from a very short burst. 2.2.2.1.4.2. Flux and Energy Measurements with Bulk Counters. I n the preceding section we considered counters wherein the source of recoil particles was confined to essentially a point or plane. Somewhat inferior devices utilizing recoil sources spread out in three dimensions can also be used. They may provide information a t a faster rate but possibly of a lesser quality than the tandem counter. Basically the problem is to obtain a n electrical signal from anywhere in a given volume and to interpret it in terms of flux and energy. The device may take on a variety of physical forms though we will consider only two for the purposes of illustration. Firstly we shall consider the proportional counter filled wholly or partially with hydrogen and partially with some other gas. Thus the recoil source is distributed throughout the active volume. It is immediately apparent that several undesirable effects may occur; namely, recoils originating near the boundaries may escape the active volume before depositing much energy and the directionality of the recoils has poor geometric definition, (We shall consider an exception later.) The boundary leakage effect can be minimized by keeping the range of the recoil small compared with the dimensions of the active volume.* I n the present example this can be done either by increasing the geometrical size of the counter or by increasing the pressure of the gas. There will be practical limitations to both procedures. Estimates of the boundary effects have been obtained for a monochromatic beam of neutronszz by Skyrme et al. They have applied their

* Leakage effects can also be minimized by using a blanket of anticoincidence counters around a primary counter which has transparent wire walls. 20 V. A. Nedzel, Phys. Rev. 94, 174 (1954). 2 1 G. A. Perlow, Rev. Sci. Insir. 27, 460 (1956). 22 T. H. R. Skyrme, P. R. Tunnicliffe, and A. G. Ward, Rev. Sci. Znstr. 23, 204 (1952).

472

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

calculations to a field-tube proportional counter using either methane or hydrogen. Their counter is illustrated in Fig. 7. Characteristically a counter of this type is constructed with thin walls and a minimum of mass to reduce neutron absorption and scattering. Such a device has some slight gamma-ray sensitivity which can be taken into account by standard background methods.23 A typical pulse-height distribution obtained with this counter with nearly monochromatic neutrons is shown in Fig. 8. The solid curve is a

PRE-

LYPLlFlER

0

II

1

2

3

4

1

6L

INCHES

FIQ.7 . Field-tube counter of Skyrme, Tunnicliffe, and Ward used for flux measurements of monochromatic neutrons. KEY:(1) steel case (0.015 in.); (2) steel end window (0.020 in.); (3) the base plate; (4) and (5) Iiovar glass seals; (6) metal vacuum tap; ( 7 )the counter cathode, diameter = 2 in., thickness = 0.010 in.; (8) two mica spacers; (9) and (10) two field tubes, 2 in. into the cathode; (11) the center wire, 0.004 in. Pt wire; (12) lead glass insulator; (13) four +'gin. diameter pins; (14) and (15)thinwalled brass tubes.

normalized calculation. Agreement is quite good and indicates that such a device is indeed a quantitative tool. The authors expect that flux measurements to 5% accuracy or better can be obtained in the neutron energy region between 0.1 and 1 MeV. The departure from the theoretical expectation near the maximum pulse height is not completely understood. It may be due to slight neutron energy spread and circuit noise of the amplifier. Evidently if the edge of the distribution can be located accurately one has a measure of the incoming neutron energy. If several energies are present there will be several steps in the curve. If there is a continuum of energies there will, in general, be a very complicated pulse-height curve. Manifestly it will be very difficult to analyze 23 J. L. Fowler and J. E. Brolley, Jr., Revs. Modern Phys. 28, 103 (1956).

2.2.

DETERMINATION OF MOMENTUM AND E N E R G Y

473

such a distribution in order to get the nriginal spect,rum. We shall reiurn to this point in the next section. A considerably higher density of recoil atoms in the bulk counter can be achieved by replacing the gas by a solid. One such technique is to use the proton recoils in solid (or liquid) organic scintillators. The recoil proton will then produce a scintillation of magnitude nearly proportional to its energy,24 which is converted to an electrical signal by a viewing photomultiplier. Boundary effects will be present as with the gas counter.

f 2000

A

W

t

z d 2

0 \

v) I-

z

1000

3 0 0

0 PULSE HEIGHT

FIG.8. Pulse distribution produced by the counter of Skyrme, Tunnicliffe, and Ward for 640-kev neutrons.

The general shapes of the scintillator curves will differ somewhat from those of gas counters because of the nonlinear response of the scintillator as compared with the gas counter. In the crystalline types of organic scintillators still another factor will affect the pulse-height distribution. Coon and FelthauseF have found that shape of the pulse-height distribution will depend on t’he orientation of the neutron beam with respect to the crystal axes. Their observations are shown in Fig. 9. The plastic scintillator, which has no ordered structure as does stilbene, was observed to be isotropic in directional response. s t C. D. Swartz and G. E. Owen, in “Fast Neutron Physics,” Vol. IV, Part I, p. 211. Interscience, New York, 1960. 15 J. H. Coon and H. E. Felthauser, Los Alamos Scientific Laboratory, private communication, 1960.

474

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

96

tI [3 W

1 W -I v)

108 -

3

STI LBE N E

W

2

.!'

5 W

a

,.q 1

Ib Ic AXES ; L 8 0 "

a

1

\J

.1 - . --.-.- .--- *-.-.-.-.-

Io2 100

c

L8

8O

C.-.

-.-.-

98

FIG.9. Observations of Coon and Felthauser on the response of oriented stilbene and a plastic scintillator to a monochromatic 14Me v neutron beam.

As an example of energy resolution for a stilbene crystal we note the in a stilbene crystal as indicated two-group observation of Ames el dZE in Fig. 10. Part of the low-energy tail is attributed to gamma rays. While it is possible to obtain a reasonable theoretical understanding of this curvez4on the basis of the known properties of the scintillator, i t does 280. Ames, G. E. Owen, and C. D. Swartz, Phys. Rev. 106, 775 (1957).

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

475

not seem likely that complex neutron spectra can be studied as precisely as with the tandem counter. Moreover problems of electronic gain stability may become quite acute. Summary. Bulk counters are useful for flux measurement of monochromatic neutrons. Spectral analysis is more difficult and the technique is still evolving. The organic scintillators are capable of extremely high counting rates but are not suited for observation of high intensity short bursts.

PULSE HEIGHT IN VOLTS

Fro. 10. Pulse-height spectrum observed in stilbene by Ames, Owens, and Swartr for the ground and first excited states produced by Bl1(d,n)Cl2.

2.2.2.1.4.3. Measurement of Flux and Energy with Nuclear Emulsions.* The previously described electronic methods are characterized by semicontinuous sensitivity and rapid translation of a n observed event into useful information; such a time interval may be of the order of millimicroseconds. I n contrast, nuclear emulsions are continuously sensitive and indeed may usefully register many events simultaneously. However, there is a very considerable time lapse between the recording of the event in the emulsion and its translation into useful information. There are several other important distinctions between the two methods: emulsions

* Refer to Section 2.2.1.1.3.

476

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

provide no facility for the usual coincidence techniques, and they may intercept only a fixed amount of background radiation before detection of the desired recoils is vitiated. As in the previous electronic considerations, nuclear emulsions may be used to detect recoils originating either from a separate radiator or those arising from collisions of neutrons with hydrogen in the emulsion.27 I n the former case the arrangement is usually such that the radiator only is illuminated by a collimated neutron beam, The recoil protons then proceed in vacuum to the detector where they register as tracks st.arting

U

+-

/

RECOIL

NUCLEAR EMULSION

RADIATOR

A

U“

NEUTRON SOURCE

COLLIMATOR

FIG.11. Schematic arrangement of the separate-radiatornuclear-emulsionapparatus for the measurement of neutron flux and energy.

on the emulsion surface. Scanning of the developed emulsion is greatly facilitated by the requirement that all desired tracks start from the surf ace. Equipment for this type of measurement has appealing simplicity and is essentially that of Fig. 11. Simple geometrical relationship~27~~~ apply and no angles need be measured precisely with the analyzing microscope nor is emulsion shrinkage of importance. If the emulsion is used for both radiator and detector, i.e., a bulk detector, considerable economy of space and weight may be achieved. However, a concomitant increase in microscope analysis time is consequential as well as more stringent development procedures. 27-28 Bulk emulsions are well suited to rocket missions provided ambient temperatures do not rise too high. L. Roseq Nucleonics 11, No. 7, 32 (1Y53), and No. 8, 38 (1953). L. Cranberg and L. Rosen, i 7 ~“Nuclear Spectroscopy” (Fay Ajeenberg-Selove, ed.), Part A, p. 3Y5. Academic Press, New York, 1960. 28

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

477

As an example of the application of nuclear emulsion as a bulk detector one may consider the work of Stewartz9on the leakage neutron spectrum of Jezebel, a pure plutonium critical assembly at the Los Alamos Scientific Laboratory. As in many other problems of neutron measurement great

300

250

-'>

g

200

cv

'S 0

v)

a c

I50

EARTH

3

w

2

100

50

0

2

4

6

0

NEUTRON ENERGY, MEV

FIO.12. Leakage neutron spectrum of a bare critical aeeembly measured by Stewart with the bulk emulsion technique.

care must be taken to insure that the spectrum observed Gas a minimum of contamination from neutrons riot proceeding directly from the source to the detector. I n this case Jezebel was operated out of doors at thirteen feet from the ground. Emulsions (200 Ilford C2 and El) were placed 2 9 L. Stewart, NucleaT Sci. and Eng. 8, 595 (1960).

478

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

125 cm from the center of Jezebel; their small mass and volume contribute

little perturbation to the measurement. Some of the criteria for track analysis were that the range be greater than 3 p (maximum track lengths observed were of the order of 500p) and that the horizontal and dip angles of each track be less than 16". With the aid of computer techniques (IBM 704) the neutron spectrum shown in Fig. 12 was obtained from track analysis. The lower energy limit for this method is about 0.5 MeV. It is interesting to note the whole spectrum required only two nuclear plates exposed at the same time, thus minimizing errors th a t would occur if many separate irradiations were needed to measure various portions of the curve. Summary. Nuclear emulsions are continuously sensitive and can record many simultaneous events, but cannot establish simultaneity of two events in general. The time-integrated tolerance to background radiation is limited. A considerable time lag exists between the registering of a n event and its translation into physical knowledge. Small weight and volume and freedom from electronics problems are useful features. 2.2.2.1.4.4. Measurement of Neutron Energy by the Bubble-Chamber Method.* The hydrogen bubble chamber is another example of the bulk counter. Very high efficiency of detection may be anticipated because of the large mass of hydrogen that can be used. As in the case of nuclear emulsions, the bubble chamber may accommodate registry of a number of simultaneous events, the number being limited by increasing complication of analysis of the chamber photographs. Problems of recoil range measurements impose a lower limit on the neutron energy that may be studied which is rather higher than previously mentioned methods. There is of course no firm line of demarcation but neutron energies below 5 Mev do not lend themselves to easy measurement in the bubble chamber with the present state of the art. Conceivably the bubble chamber could be used with lower-energy neutrons by studying exothermic neutron-induced reactions in liquids other than hydrogen. Since bubble chambers are not continuously (or nearly so) sensitive, the apparent high efficiency of detect,ion may not materialize as more rapid accumulation of data compared with the other bulk counters. Cycling time may be of the order of 5-10 sec and longer so that relatively high efficiency will be realized only in conjunction with slowly pulsing sources of neutrons of comparable periods. It seems likely that automated data-processing equipment may be able to analyze proton-recoil spectra at a rate comparable to the rate of production, in which case the hydrogen bubble chamber should be able to produce neutron energy spectra at quite a significant rate.

* Refer to Section 2.2.1.2.3.

2.2.

DETERMINATION OF MOMENTUM AND E N E R G Y

479

It is instructive now to consider an actual application of the bubble chamber t o neutron spectra. The 4-in. Berkeley bubble ~ h a m b e r ~ Ois8 ~ ~ indicated in Fig. 13. In this bubble chamber the expansion system is external to the chamber. Dark-field illumination is used with stereoChamber recompression piston Expansion piston

Safety valve

Gaseous hydrogen Regulated

4 TO vacuum pump Fro. 13. Schematic diagram of the four-inch liquid hydrogen bubble chamber of Adelson, Bostick, Moyer, and Waddell.

photography a t 90" to the neutron beam. The illumination was provided by a xenon flash tube. Expansion and photography are suitably phased and synchronized with the accelerator supplying the neutrons. The expansion operation precedes the beam pulse slightly and requires about 4 msec. The limit of sensitive time is approximately 50 msec. Strobe so

H. E. Adelson, H. A. Bostick, B. J. Moyer and C. N. Waddell, Rev. Sci. Instr. 31,

1 (1960). *l

D. Parmentier, Jr., and A. J. Schwemin, Rev. Sci. Instr. 26, 954 (1955).

480

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

flashing is done between 2 and 5 msec after the beam pulse. It, cannot, be delayed much longer because the bubbles grow too large and turbulence effects from the expansion become manifest. To evaluate the performance as a neutron spectrometer 14-Mev neutrons from bhe D-T reaction were used. The source was somewhat

‘’1

D -T

5/8” Collimator

I

En (MeV)

Fro. 14. 14-Mev neutron spectrum observed with the four-inch bubble chamber. The neutron spectrum has been somewhat distorted by thick-source and collimator effects.

thick and was estimated to furnish a neutron group centered a t 14.1 with a full width of 0.6 Mev at half-maximum. Using a collimator and suitable track criteria, the spectrum of Fig. 14 was obtained. Spectra obtained without a collimator had similar widths but more neutrons in the lowenergy tail. From this study it was estimated that resolution of the over-all system was about 10% for 14-Mev neutrons. It is estimated that

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

48 1

track lengths and averages could be measured to about 1% and that the major distortion of the spectrum was introduced by the collimator. Summary. Hydrogen bubble chambers can provide accurate knowledge of neutron energy and are well suited to efficiently record data from pulsed machines operating a t rates comparable to the bubble chamber. The over-all amount of apparatus and personnel required is rather extensive compared with other methods. 2.2.2.1.5. DIFFERENTIAL SCATTERING: A LINEARMOMENTUM MEASUREMENT. In the first section we noted that the problem of momentum measurement was partly one of establishing the direction of the momentum vector be it linear or angular. I n the remainder of this chapter we shall discuss some specific examples of instrumentation used for momentum observations. The measurement of the differential scattering of neutrons is essentially the study of the probability of finding the neutron linear momentum vector pointing in some given angular interval after scattering. Techniques can be applied in several ways to measure the angular distributions of scattered neutrons. In a straightforward way one could use any of the recoil detectors discussed previously to measure the number of neutrons scattered at some angle by a scatterer with suitable shielding and collimation. If the detector is electronic, perhaps time-of-flight could be incorporated. We shall, however, consider another variant which will also lead us into the problem of measuring the spin direction. 2.2.2.1.5.1. Angular Distribution from Recoil Scatterer Measurements. If the recoil nucleus is light enough, the colliding neutron will impart significant kinetic energy. If the recoil nucleus is a constituent of the gas filling of a proportional counter, a signal will be produced in the counter proportional to the energy of the recoil. For all gas fillings, and for a monochromatic neutron beam, the amplitude of the signal will then be a measure of the scattering angle of the neutron. * The pulse-height spectrum gives the center-of-mass angular distribution on a cosine scale. Angular distributions of scattered neutrons from the hydrogen isotopes up to nuclei as heavy as neona2have been measured by this method. As a n illustrative example we shall consider the case of neutron-helium scattering as this will lead us naturally to the problem of spin analysis. One could of course observe the helium recoil by a variety of methods: liquid-helium bubble chamber, liquid-helium scintillator, cloud chamber, etc. * Strictly this is not true for the case of the hydrogen scatterer since the neutron is slightly heavier and this leads to double-valuedness in the scattering. Practically the mass difference between neutron and proton is often ignored. 32 H. 0.Cohn and J. L. Fowler, Phys. Rev. 114, 194 (1959).

482

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

This method has been practiced by S e a g r a ~ eusing ~ ~ a proportional counter to observe the helium recoils. Helium is not a n especially tractable gas for counter fillings since it does not permit very much stable gas multiplication and has low stopping power. To some extent this situation is alleviated by using a mixture of helium and krypton. Signals from the krypton recoils are of course of much lower amplitude than those from helium. Boundary effects were diminished by collimating the neutron beam as shown in Fig. 15. Corrections need only be made for the back wall since all other recoils will stay within the active volume of the counter.

NEUTRON SOURCE

COLLIMATOR

PROPORTIONAL COUNTER

FIG.15. Schematic diagram of Seagrave’s proportional-counter arrangement for the study of neutron-He* scattering.

It is necessary to ensure that pulse distributions are reasonably independent of the physical position of the recoil trajectory in the counter. Seagrave obtained angular distributions of neutrons at energies of 2.6, 4.5, 5.5, 6.5, and 14 MeV. With the exception of the 14-Mev point these data, in the center-of-mass system, are exhibited in Fig. 16. The theoretical calculation is th at of Dodder and Gamme1.34It will be noted that this method does not lend itself to study of the forward scattering since the recoil signals grow smaller and the resolution will deteriorate. If a cloud chamber is the same method can be used a t considerably smaller forward scattering angles. J. D. Seagrave, Phys. Rev. 92, 1222 (1953). D. C. Dodder and J. L. Gammel, Phys. Rev. 88, 520 (1952). 36 D. F. Shaw, Proc. Phys. Soc. (London) A67, 43 (1955). 33

54

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

483

Summary. Observations of the recoil signal in a proportional counter provide a simple and rapid technique for obtaining neutron-scattering angular distributions. Measurement of forward scattering is somewhat restricted. Quality of counter filling is quite important. 680 640

600 560

520

FIG.16. Examples of differential scattering data obtained by Seagrave with a proportional counter.

2.2.2.1.6. POLARIZATION. Our final consideration of the neutron beam is to ascertain its average polarization by recoil techniques. The approach to this question is rather less direct than in the previous remarks and we will briefly examine some of the relations involved from a pragmatic viewpoint. If one measures the probability or cross section for the scattering of a beam of initially unpolarized neutrons by a target having no intrinsic angular momentum (i.e., spin zero), the data may be represented by a

484

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

general relation. This relation, of quantum-mechanical origin, is an expansion of the cross section in terms of angular momentum and spin of the neutron. It, has the form (center-of-mass system)36 u(0) = IAIz

+ IB12

(summed over spins)

(2.2.2.1.6)

where

A

1 2ik

{(Z

= -

+ l)[e2isr+- 11 +

Z[e2"l-

- l]}P~(cose) (2.2.2.1.7)

2

and (2.2.2.1.8)

The P's are Legendre polynomials. The unknown quantities are the 6's. They measure the strength of participation of a state of orbital angular momentum 1 and spin up (+), J = 1 &, or down (-), J = I - +, in the scattering process. For our purposes the 6's may be regarded as obtained by fitting the formulas to experimentally determined scattering cross sections. In practice, only a few values are required to reasonably fit the corpus of experimental data. Thus for neutrons of 10 Mev or less good fits are obtained for the set I = 0, 1, 2, on He4. If the 61% phases are determined, the polarization produced in the scattering can be computed from the relation3'

+

p(e) = -

AB* AA*

+ BA* + BB*

(2.2.2.1.9)

where n is a unit vector having the direction X kn.out. The phases for n-He4 scattering are becoming moderately well known. Seagrave has obtained a set from his data cited in the previous sections. Additionally, Dodder and Gamme13*have inferred n-He4 phases from p-He4 phases; their results are consonant with Seagrave's phases.? The latter set have been used by Levintov et ~ 2 1 to . ~ compute ~ a polarization map for n-He4 scattering. 2.2.2.1.6.1. Helium Polarimeter. There are many possible experimental techniques for the observation of the recoil He4's in polarization studies. We shall consider the method of Levintov et U Z . , ~ ~which is essentially a variant of the bulk-counter technique with additional geometrical constraints. Bundles of long, thin paraxial proportional counters filled with

t See the Appendix. 86

F. Bloch, Phys. Rev. 68, 829 (1940).

wJ. V. Lepore, Phys. Rev. 79, 137 (1950). We use the negative of Lepore's expression to conform to the Base1 Convention. *a I. I. Levintov, A. V. Miller, and V. N. Shamshev, Nuclear Phys. 3, 221 (1957).

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

485

helium were employed to make a left :right measurement. Since there is a known and unique energy for the helium recoil for each angle of scattering, it is possible by pulse-height analysis of the proportional-counter signals to define the helium scattering angle. Geometry, in this case, improves the accuracy of angular knowledge. The counter pressure is set at that value where the helium recoils of desired angle just spend their range in the counter. Referring to Fig. 17 we see then that if signals of a certain maximum amplitude only are counted, then only recoils originating at one end of the counter, A , and having the appropriate energy corresponding to stopping a t B will register. Thus a target area of recoils is defined withDno window barrier between the target and counter, a desirable arrangement to save recoil energy.

f =

HELIUM RECOIL

FIG. 17. Helium-recoil counter-polarimeter of Levintov, Miller, and Shamshev. Detection of maximum-range particles is accomplished by high bias setting.

These refinements over the simple bulk counter, e.g., Seagrave’s, are necessary since we require not only the value of the angle of scattering, but a knowledge of whether the scattering was left or right. Complete 1eft:right symmetry of the neutron field and counters must be achieved or satisfactorily accounted for to avoid spurious data. It is possible to compute the counting rate of this device as a function of discriminator setting and obtain good experimental agreement. The preceding polarimeter, by virtue of using helium-filled counters, will require systems and problems tolerant of quite long pulse lengths (-10 psec). With scintillation techniques it is possible to construct highefficiency polarimeters wit8hcharacteristic pulse-times nearly a thousand times shorter. We briefly note a scintillation polarimeter under development by Perkins and Simmoris.”YThey have utilized the fact that a recoil He4 in 39 R. U. Perkins and J. E. Simmons, Lot; Alamov Scientific Laboratory (private communications).

2.

486

DETERMINATION O F FUNDAMENTAL QUANTITIES

liquid helium generates a scintillation of rather short duration.40 I n principle pulse-height analysis of the scintillation value gives the angle of recoil but not the sign. In practice i t is found convenient to establish the angle and sign of scattering by detecting the neutron after its helium scattering with a high-efficiency organic scintillator. Thus the helium scintillation furnishes a fiducial signal which is then in coincidence with the organic scintillator. Evidently time-of-flight can be imposed so that neutrons of only the desired energy are measured. Such a fast system should be well adapted to “slavingff to a pulsed machine such as the cyclotron to further reduce background. Summary. Gaseous and liquid helium polarimeters can cover a broad spectrum of neutron energies: the former being relatively slow and the latter fast. Knowledge of the analyzing power of helium or other material is not very extensive or accurate yet. 2.2.2.1.7. APPENDICES. 2.2.2.1.7.1. Appendix A . Kinematics.* A few relations and some data useful for preliminary design considerations are given in this Appendix. The nonrelativistic kinematical relations41,42 between the incident particle and the elastic recoil are listed below. We use the following notation:

MI = rest mass of the incident, particle,

M z= rest mass of Eo

= El= Ez = 6 = $= 6’ =

$’

=

the target or recoil particle, laboratory energy of the incident particle, laboratory energy of the incident particle after collision, laboratory energy of the recoil particle after collision, laboratory scattering angle of the incident particle, laboratory scattering angle of the recoil particle, c.m. scattering angle of the incident particle, c.m. scattering angle of the recoil particle.

In terms of the abbreviations

* Refer to Appendix B. H. Fleishman, H. Einbinder, and C. 8. Wu, Rev. Sci. in st^. SO, 1130 (1959). B. Carlson, M. Goldstein, L. Rosen, and D. Sweeney, Los Alamos Scientific Laboratory Rept. LA-723 (1949). Unpublished. 42N. Jarmie and J. D. Seagrave, eds., “Charged Particle Cross Sections.” Los Alamos Scientific Laboratory Rept. LA-2014 (1956). ‘0

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

487

the energies can be simply expressed as

z -- 2 c(1 - cos e’) E2

= 4c cos2 $

and

I n the last expression, use both signs if M I > M 2 ( D < l ) , in which case sin Omax = D. Use only the plus sign if M1 I M 2 ( D 2 1). The angles are related by the equations

D sin 2$ 1 - D cos 2$ 0’ = ?r - 2$, D sin(0’ - 0)

tan 0

=

and =

?r

-

el

=

=

sin 0

2+.

The cross sections transform from the laboratory system to the c.m. system in the ratios u(0’) --

u(0)

- sin 0 d 0 - C ( D z - sin2 0 ) I I 2 sin 0’ de’ Ei/Eo

u(8’) --

- sin $ d$

and u($)

sin e’de’

-

~. 1

4 cos $

Although the instruments described in this chapter have not operated in the extreme relativistic range some are capable of much extension. Moreover, if one wishes to do very precise work with neutrons even as low as 25 or 30 Mev it is necessary to use relativistic transformations. Blumberg and Schlesinger43 have compiled the relativistic kinematical relations. From this compilation we quote the following results (using the previously defined symbols for angles but the notation T O ,T O ,and T+ for the relativistic kinetic energies corresponding to 3 0 , El, and E 2 in the nonrelativistic case, respectively). The energies are given by the expressions

and

4 3 L. Blumberg and L. Schlesinger, “Kinematics of the Relativistic Two-Body Problem.” Los Alamos Scientific Laboratory Rept, LAMS-1718 (1955). Unpublished.

488

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

where YI

and

= I

+ (TOlMlC2)

The angles are related by the equations cos

- yk2 tan2 + *’ = 11 + Y : ~ tan2 $

and tan 0 = Y2’(cY

sin $‘ - cos *’)

where

Finally, the recoil cross sections transform in the ratio

’ [(a)’ Mzc2 + 2 (“)]’” M

Y2

$2

-

[ + g2]1

(Y;2

1)1’2COS

*

The behavior of some of these relations is indicated in Fig. 18, Fig. 19, and Fig. 20, for the particular case of n-p scattering, 2.2.2.1.7.2. Appendix B. Neutron-Proton Cross Sections. Gamme144has considered the problem of generating a useful set of n-p cross sections for E , up to 40 Mev from the fragmentary experimental data now extant. He has found that the total cross section may be represented by uT(Eo) =

+

3~[1.206E (- 1.860

+ 0.09415E + 0.0001306E2)2]-’ + ~[1.206E+ (0.4223 + 0.1300B)2]-’

and the differential cross section in the center-of-mass system by

where EOis the laboratory neutron energy in Mev and b

=

2(E0/90)~.

4 4 J. L. Gammel, to be published in “Fast Neutron Physics” (J. B. Marion and J. L. Fowler, eds.), Vol. IV, Part 11. Interscience, New York, 1960. (I am indebted to the author for this advance communication.)

2.2.

10.000

DETERMINATION OF MOMENTUM AND ENERQY

489

F

1000

-> W

2

Y

I-

150

ic

10

I

I

10

00

1000

10,000

l00,000

To (MEV) FIG.18. Relation of recoil proton energy T#to incoming neutron energy T oin the laboratory system, for n-p scattering.

490

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

1000

I

10

1000

100

l0,OOO

100,000

To (MEW FIG. 19. Cross-section transformation for t h e recoil proton from t h e laboratory system into the center-of-mass system, in n-p scattering.

2.2.

DETERMINATION OF MOMENTUM AND E N E R G Y

491

J8C I70 160

160

140

I30 120 I10

100

JI'

90 80

70 60

50 40

30 20 10

0

,

1

1

1

1

1

1

,

I

10

,

I

1 , 1 1 1 ,

I

1

I

,

,

,

,

100

,

L]

1000 TO

,,,,,,,,, l0,OOO

I l00.000

(MEV)

FIG. 20

+

FIQ.20. ReIation between the laboratory angle and center-of-mass system angle $' of the recoil proton, in n-p scattering.

2.

492

DETERMINATION OF FUNDAMENTAL QUANTITIES

33

-

31

-

27

-

P

156 MEV

Z 7 MEV

4 I

W Iu)

a

- 4

- 2 20

-

I

I

- -

4 0 MEV

I

I

I

I

I72 MEV

- 12

I8 .-

I

I

I

4--I

I

0

- 15

215 MEV

QOMEV

0 >'IS

I

3OL 0

v I

t

30

60

I

I20

6

-

I

1800 30 6 C 90 120 C.M NEUTRON SCATTERING ANGLE 8' 90

12

150

Is0

180

FIG.21. Family of n-p scattering data collated with the calculations of Clementel and Villi.

FIG.22. Theoretical n-He4 phase shifts of Dodder and Gammel compared with the experimental values of Seagrave.

2.2.

493

DETERMINATION OF MOMENTUM AND ENERGY

Lo

I

I

I

I

I

1

0.8 0.0 0.6 0.4

0.2

P

o -0.2 -0.4 -0.6 -0.8

-I

.o 0

I

I

I

I

I

I

I

I

I

2

4

6

8

10

12

14

16

18

20

En (LAB) FIG.23. n-He4 polarization map inferred by Levintov, Miller, and Shamshev from the phases of Dodder and Gammel and of Seagrave.

Gammel and Thaler46and Clementel and Villi4Ehave also assembled n-p scattering data at higher energies and have analyzed them. For orientation some results of Clementel and Villi are displayed in Fig. 21. 2.2.2.1.7.3. A p p e n d i x C. Neutron-He4 Scattering. Design cross sections at any energy may be computed from the phase shifts according to Eqs. (2.2.6), (2.2.7), and (2.2.8). I n Fig. 22 are plotted the theoretical phase shifts of Dodder and Gamme134 together with experimental points of Seagra~e.~~ Polarization can be calculated from the phase-shift equation (2.2.2.1.9). Such calculations have been done by Levintov, Miller, and Shamshev and are shown in Fig. 23. For polarization at higher energies one may use a proton-alpha polarization map4’ shown in Fig. 24, and equate neutron and 46 J. L. Gammel and R. M. Thaler, in “Progress in Elementary Particle and Cosmic Ray Physics” (J. G. Wilson and S. A. Wouthuysen, eds.), Vol. 5, p. 155 North Holland Publ., Amsterdam, 1960. 48E. Clementel and C . Villi, Nuovo cimenlo [lo] 6, 1167 (1957). 47 J. L. Gammel and R. M. Thaler, Phys. Rev. 109,2041 (1958).

494

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

E L A B( M e V )

FIG.24. Quasi-theoretical p-He4 polarization map calculated by Gammel and Thaler.

proton energies. This is not a strictly correct procedure but will provide a reasonable number. ACKNOWLEDGMENTS

I should like to thank N. Jarmie, L. Rodberg, L. Rosen, J. D. Seagrave, and L. Stewart for criticism.

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

495

2.2.2.2. Time-of-Flight Method.* 2.2.2.2.1. INTRODUCTION. One of the most successful methods for determining the energy dependence of the interaction of neutrons with matter is the time-of-flight technique. t With this technique bursts of neutrons which have a wide distribution in energy are produced for a short interval of time a t a specific place, a neutron detector is placed a measured distance from the effective source of the neutrons, and the time between the production of the burst of neutrons and the detection of a neutron is measured. Knowing the time required for the neutron to travel over the measured path enables one t o calculate the velocity of the neutron and its energy. B y interposing samples between the source and the detector to absorb or scatter the neutrons, one can determine the effect of the sample on a neutron of a particular energy. It is this variation in energy of the interaction of the neutrons which gives information of value both in the study of the structure of the nucleus and in the study of the structure of solids and liquids. A summary of the important properties of the neutron and its interactions, is given in Table I, pages 496 and 497. Neutrons can only be produced in nuclear reactions. The reactions usually used for the production of neutrons are U236(n,f,2.43n); H2(d,n)He3; U(r,n). H3(d,n)He4;Li7(p,n)Be7; Beg(d,n)B'O; B e 9 ( ~ , n ) N 1Beg(y,n)Be*; 2; The energy of the neutron emitted in the nuclear reaction depends on the initial energy of the incident particle or photon, the particular nuclear reaction, the angle between the direction of the incident particle or photon, and the direction in which the neutron is emitted. The time-of-flight technique has been used t o study neutrons of enerev and as high as 5 X lo9ev. Because this energy gies as low as 2 X range is so wide, no single apparatus can be used to study the entire spectrum. The details of the neutron source and the experimental apparatus used t o measure the time of flight vary markedly with the energy of the neutron. It is therefore convenient to divide the entire energy range into three different energy intervals because of the differences in techniques used. These are shown in the tabulation. (1) Slow neutrons (a) Thermal (b) Epithermal (2) Fast neutrons (3) High-energy neutrons

E < 0.5 ev 0.5 ev < E lO4ev < E lOsev < E

t Refer to Section 2.2.1.3.1.

* Section 2.2.2.2 is by W. W.

Havens, Jr.

< 106 ev < lo8 ev

496

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

The boundaries of these categories are somewhat arbitrary, and experiments which are designed primarily to measure the effects of epithermal neutrons can also be used to measure the effects of fast neutrons. The techniques used for time-of-flight measurements on high-energy neutrons are substantially the same as those used for fast neutrons, so category 3 will not be discussed in detail. Neutrons produced in nuclear reactions are usually fast, and it is therefore necessary to slow them down before their interactions can be studied. The slow neutron category is divided into two parts, thermal and epithermal, because the information obtained from studying neutrons with energies below 0.5 ev is primarily-although by no means exclusively-of interest in the study of the structure of liquids and solids, whereas the information obtained from studying neuTABLE I. Summary of Properties and Interactions of Neutrons Data

Property General

Mass, 1.00898; charge, 0; spin +; half-life 10-12 min., statistics, Fermi-Dirac

Wave length, energy, and velocity conversions

X = h/mv = h / . \ / m

~~

X(A.)

=

~~

3.96 X 108/v(meters/sec)

=

0.286/=

t(psec/meter) = l/u(meters/sec) X

lo8 = 253X(A)

3.96 X 10-at

= 72.3/.\/E(ev) E(ev) = 5.226 X 10-Dva(meters/sec) = 5226/t2 = 0.0818/X2(A) = 3.96 X 103/X(A) v(meters/sec) = 1O6/t = 1.383 X lo4 X = wavelength, cm; A(& = wavelength in 1 units; h = Planck’s constant, 6.624 X 10-81 erg sec; rn = mass of neutron in grams; v = velocity, cm/sec; v(meters/sec) = velocity in meters per E(ev) = kinetic second; E = kinetic energy in ergs = energy in electron vo1ts;a t = time of flight in microseconds per meter.

m)

~~

= 1 ev, = 0.286 d; a t E = 0.026 ev, X = 1.8 A; wavelength in this energy range is of the order of distance between atomic planes in crystals. Bragg diffraction occurs: nX = 2d sin e where n = order of reflection, d = distance between atomic planes, and 0 = angle of incidence with plane of atoms. Interaction with Pass through matter much more readily than charged particles. matter Practically no ionization produced. Fast neutrons knock protons from hydrogen-containing material. Prolonged irradiation may change color, thermal conductivity, or electrical conductivity. Bonds may be broken with decomposition of molecules. Owing to their magnetic moment, slow neutrons interact with electron magnetic moment of paramagnetic and ferromagnetic atoms.

Diffraction by crystals

At E

2.2.

DETERMINATION OF MOMENTUM AND ENERQY

497

TAEILH I. (Continued) Property Detection

Data Fast neutrons : recoil protons and nuclear reactions. Example: IP SP(n,p)A12* b Sias 2.3 min. 3.0 Mev

Slow neutrons: radioactivation of foils of In, Mn, Au, Ag, Rh, etc. Example: 8MnK6 (n,y)Mn66 Fe66 1.69 hr, 2.8 Mev

Counters lined with B or Li or proportional counters filled with B'OFa. Example: B10 n -+ Li' (Y L i ' + n + H * +(Y Fission Example: U2as n -+high energy fission fragments Photographic plates containing elements that become radioactive by interaction with neutrons. Scintillation counters arranged to detect y-rays emitted when neutrons are absorbed or to detect other nuclear reactions caused by neutrons.

+

+

+

Scattering Elastic nuclear scattering; inelastic nuclear scattering; resonant nuclear scattering; coherent crystal scattering (diffraction) ; procem, (n,n) reactions ferromagnetic scattering; paramagnetic Scattering; inelastic molecular scattering; neutron-electron scattering. Absorption process

The neutron is retained by nucleus and a photon or other particle is emitted: (n,y), (n,p) (%,a).Also (n,2n) and (n, fission).

One electron volt is the energy acquired by an electron when it is accelerated by a potential difference of 1 volt. The electron acquires an energy of E = Ve = 4.802 X 10-1°/299.8 = 1.602 X erg and a velocity given by E = #mu2or v = m m = 1/(2)(1.602 X 10-12)/9.107 X 10-28 = 5.931 X lo7 cm/sec Per mole of electrons, the energy would be (1.602 X 10-12)(6.023 X loz3) = 9.648 X 10" ergs/mole = 96,480 joules/mole = 23,055 cal/mole A neutron with an energy equivalent to one electron volt would have an energy of 1.602 X 10-12 ergs and a velocity of u = d(2)(1.602 X 10-12)(6.023 X 10z3)/1.00898 = 1.383 X lo6 cm/sec or 1.383 X lo4 meters/sec. ( I

trons of energies between 0.5 and lo6 ev is primarily of interest in the study of the structure of the nucleus. 2.2.2.2.2. THERMAL NEUTRONS. 2.2.2.2.2.1. Sources.* Among the important factors determining the experiments which can be performed

* Refer to Vol 5B, Section 3.2.1.4.

498

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

with slow neutrons are the availability, intensity, and total number of neutrons emitted by the neutron source. The increase in the number of research reactors, as well as the increase in the neutron flux available with new reactors, will certainly increase the number and types of neutron experiments performed. Although some of the early work in the study of thermal neutrons was done with neutrons produced with radioactive sources and charged particle accelerators, practically all of the research in this energy range is now performed using nuclear reactors. No other source can give anywhere near the average intensity or total number of thermal neutrons that can be obtained from a nuclear reactor. Of course, this does not mean that it is impossible to perform ingenious experiments with thermal neutrons using natural radioactive sources or particle accelerators, and some outstanding experiments using these sources have been done recently when a reactor was not available. I n most cases, however, it would have been much simpler to perform the experiments if a reactor had been available. It has been suggested on many occasions that particle accelerators can be used to compete with reactors in all types of neutron studies. Although particle accelerators, especially those which naturally produce neutrons in bursts, have definite advantages for neutron studies with epithermal and fast neutrons, a simple order-of-magnitude calculation shows that they cannot compete with a reactor for studies with thermal neutrons. Although neutron production rates as high as 1019n/sec in the pulse are possible with a pulsed synchrocyclotron,l the more usual continuous production rate with a Van de Graaff accelerator or a Cockcroft-Walton machine is 10%/sec. This number of neutrons would be produced per second with a 300-pa beam of 2-Mev deuterons incident on a heavy ice target or a l-ma beam of deuterons of 250-kev energy incident on a zirconium-tritium target. The neutrons from the H2(d,n)He3reaction using 2-Mev deuterons would have an energy about 5 MeV, which would require a sphere of paraffin of about 10 cm radius to moderate the neutrons. Thus the neutron flux emitted from the sphere if all the neutrons were moderated and none were captured would be

Q

=

[1011/4s(10)2]= 8 X 107n/cm2/sec.

If the H3(d,n)He4reaction were used, the neutrons would have an energy of about 14 Mev and a sphere of approximately 20-em radius would be required to moderate the neutrons to thermal energies, reducing the flux 1.6 x 1O7n/crn2/sec. Since even a small a t the edge of the sphere to Q reactor of enriched uranium operating at 2 kw has a central neutron flux of about 10"n/cm2/sec and an edge flux greater than 101°n/cm2/sec, it is

-

J. Rainwater, W. W. Havens, Jr., J. S. Desjardins, and J. L. Rosen, Rev. Sci. 31, 481 (1960).

znstr.

2.2.

DETERMINATION OF MOMENTUM A N D ENERGY

499

obvious that the flux and the total number of neutrons available from a reactor is a t least two orders of magnitude greater than the flux available with a particle accelerator. Since the reactor flux of thermal neutrons is so much higher than can be obtained with particle accelerators, our discussion in this section will deal primarily with experiments which use reactors as a thermal neutron source. 2.2.2.2.2.2. Mechanical Chopper. The energy and time distribution of the neutrons emitted by a reactor are both continuous. Since a pulsed source is required for time-of-flight studies, a device called a chopper is usually used to produce the burst.* FIXED MASS O F FISSIONABLE MATERIAL SLIGHTLY LESS THAN C R I T I C A L \ ~

MASS OF FISSIONABLE MATERIAL SLIGHTLY LESS THAN CRITICAL MOUNTED ON ROTATING DISK

FIG.1. Schematic diagram of pulsed neutron source using two subcritical masses.

A collimating system is placed in the reactor shield to form a beam of neutrons as shown schematically in Fig. 2. A mechanical shutter (called a chopper because it chops the beam off in a short time interval) intercepts the continuous beam emitted by the reactor and thus produces the short bursts required. The detector for neutrons in the thermal region is usually a proportional counter or ionization chamber filled with boron trifluoride.

* Several methods of obtaining pulsed reactors for neutron spectroscopy have been suggested, both seriously and in jest. One device called the “dragon” would use a fixed mass of fissionable material just smaller than a critical mass and several subcritical masses on the rim of a wheel which rotated as shown schematically in Fig. 1. When the two subcritical masses were closest, the assembly would be above criticality and a large pulse of neutrons would be produced. The bursts could be very intense and also could be very short if the wheel were rotated at a sufficiently high speed. Although it might be practical, no one has as yet published any description of the operation of such a device. An atomic bomb gives a short burst of neutrons of very high intensity and bombs have been used for neutron spectroscopy.e Although the pulse of neutrons is extremely large, only one burst is available from each bomb and therefore few results have been obtained. * G. A. Cowan and A. Turkevich, Bull. Am. Phys. SOC.[2] 4, No. 1, p. 31 (1959).

500

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

These devices make use of the boron disintegration reaction

B’O

+ n<

+a Li7 + a 4- ~(480 kev).

Li7

The ionization caused by the a-particle and the lithium ion in the gas of the proportional counter or ionization chamber is detected. For thermal neutrons the efficiency of these devices can be made very high by using high gas pressures. The speed of collection of the ions does not limit the time-of-flight resolution for thermal neutron studies. Counters have been

- - -- - - - RANSMlSSlON

NEUTRON BEAM

Y

FIG.2. Schematic diagram of “neutron chopper ” used with a nuclear reactor.

developed which have excellent plateaus, long time stability, and reproducibility, making them simple and excellent detectors for thermal neutrons. Critical assemblies have been pulseda to give extremely large numbers of neutrons in the puke, but the burst time and repetition rate were usually much longer than would be useful in neutron spectroscopy. Keepin and co-workers, using a pulsed critical assembly, have obtained some very interesting results on delayed neutron and short half-life fission product isotopes which are of interest for the understanding of the fission process. Some idea of the speed of rotation required for a neutron chopper can be obtained from the relationships given in Table I. For a 0.1-ev neutron, the time required for the neutron to travel 1 meter is 229 psec. If the time resolution is to be 1%, the mechanical shutter must be open less than aT. F. Wimett, R. H. White, W. R. Stratton, and P. P. Wood, Nuclear Sci. and Eng. 8, 691 (1960);G. R. Keepin and T. F. Wimett, Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1966 4, 162 (1956);R. E. Peterson and G. A. Newby, Nuclear Sci. and Eng. 1,112 (1956). Ct. R. Keepin, T. F. Wimett, and R. K. Zeigler, Phya. Rev. 106, 1044 (1957).

2.2.

DETERMINATION O F MOMENTUM A N D E N E R G Y

501

2.29 psec, if a 1-meter flight path is to be used. A slit 0.1 cm wide must be traveling at a speed of 4.38 X lo4cm/sec past another fixed slit of the same width in order to give a burst length of 2.29 psec. If this slit is a t the edge of a rotating wheel of radius 10 cm, the speed of rotation must be 697 rev/sec, which is close to the limiting speed at which ordinary materials can be rotated without disintegrating. Of course, it is not necessary to have a 1% resolution for most experiments and flight paths longer than 1 meter can be used, so that the speed of rotation can be lower than that calculated above. However, choppers used at reactors must be carefully designed and dynamically balanced to run smoothly at the high speeds required for neutron time-of-flight studies. NEUTRON SOURCE

ION CHAMBER

I

FIQ.3. Schematic diagram of the mechanical velocity selector of Dunning, Pegram, Fink, Mitchell, and Segrb.

For thermal neutrons, the design of a neutron chopper is not nearly as difficult as it is for neutrons of higher energy, because the time of flight is longer and cadmium absorbs practically all neutrons of energy less than 0.5 ev in a thickness of a few tenths of a millimeter. Thus it is possible to construct a slit for slow neutrons in the same sense as a slit is used in optical instruments. For higher energy neutrons there is no substance which will absorb neutrons in a short distance and the collimating and chopping systems necessarily become more complex. The first slow neutron velocity selector was the completely mechanical device of Dunning et aL6 which is shown schematically in Fig. 3. This device used a fixed disc of aluminum on which were pasted sections of cadmium. Aluminum is practically transparent to neutrons, whereas cadmium is opaque. A similar rotating disc was placed close to the fixed disc to pass bursts of neutrons from a radium-beryllium source. On the same shaft as the rotating disc 54 cm away was a third aluminum disc 6

J. R. Dunning, G. B. Pegram, G. Fink, D. P. Mitchell, and E. SegrB, Phys. Rev.

48, 704 (1935).

502

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

with cadmium sectors, and a fourth disc was fixed just in front of the third disc. The first and fourth discs acted as source and detector collimating slits, and the second and third discs chopped the neutrons. An ionization chamber behind the fourth disc detected the neutrons. The cadmium sectors subtended an angle of 3.7" with 3.5" spacing between the sectors. Using this device Dunning et al. were able to demonstrate that neutrons moderated in paraffin had a Maxwellian velocity distribution. The energy of the neutrons passed b y this velocity selector system depended on the separation of the discs, their speed of rotation, and the relative phase of the rotating discs. Mechanical considerations limited the speed of rotation and confined the operation of the selector to well below 0.1 ev. Because of the relatively large fast neutron background inherent in this system, LIGHT SOURCE%-

SH'ELD7

yPHOTOELECTRIC

TUBE

.TECTOR

LNEUTRON BEAM FROM THERMAL COLUMN

FIG.4. Schematic diagram of a Fermi-type mechanical velocity selector or chopper.

the resolution was poor. Although a great deal of significant information was obtained with the velocity selector, it was limited in its application because of the very weak neutron sources then available and the energy range over which it was useful. The first mechanical velocity selector to operate with a reactor was designed by Fermi and his associates.6 A schematic diagram of this mechanical velocity spectrometer is shown in Fig. 4. A rotating cylinder of alternate thin laminations of aluminum and cadmium was placed next to the reactor shielding. Neutrons can pass through the cylinder only when the direction of the layers is parallel to the beam. A mirror rotates with the cylinder, and a reflected light beam activates a photoelectric tube, which in turn activates a BF3 neutron counter through a n electronic delay circuit. By adjusting the position of the mirror, the photoelectric tube and time delay circuits, the counter can be activated a t time t after the neutrons have passed through the cylinder. Thus only those neutrons are counted whose velocity is equal to E. Fermi, J. Marshall, and L. Marshall, Phys. Rev. 72, 193 (1947).

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

503

d / t . An absorber can then be placed in the beam, and the transmission of neutrons through the absorber for a particular velocity can be measured. The time t can then be changed, and the transmission can be measured a t other velocities. A chopper which extends for some distance in the direction of motion of the neutrons in the beam, such as the Fermi-type chopper shown in Fig. 4, also partially monochromatizes the beam. For example, if the diameter of a Fermi-type chopper with straight slits is 10 cm, the slits 0.1 cm wide, and the chopper rotating a t 9000 revolutions per minute, the burst width will be 21.1 psec. Therefore, a neutron which has a speed of less than 211 psec/meter will not traverse the 10 cm in 21.1 psec and will not pass through the chopper. This means that a neutron with a n energy less than 0.116 ev will not pass through the chopper. Thus a chopper running a t a particular speed has a low-energy neutron cutoff. It is possible to design a chopper with curved slits which will pass neutrons of one energy band for a particular rotational speed. Without this monochromatization, the duty cycle of a chopper must be low. Sufficient time must be allowed between bursts for the slowest neutrons from a burst to pass the detector before the fastest neutrons from the next burst arrive. A low duty cycle decreases the average neutron counting rate and is therefore undesirable. With monochromatization, i.e., curved slits, the counter can in principle be used continuously. If the counter is to be used continuously, the curvature of the slits must change with the speed and a new rotor will be required for each velocity. A device which requires a new rotor for each new speed is not practical, so in practice a compromise is usually made. The rotor is designed to give partial monochromatization, and the duty cycle is thereby increased over what could be obtained with straight slits for the same resolution. Several curved-slit Fermi-type choppers are in operation and have proved very satisfactory.’ A chopper similar in principle to that originally built b y Dunning et al., but, using electronic techniques for measuring the time of flight instead of a second shutter, has recently been constructed by Jacrot8for use with the reactor a t Saclay. The rotors are 1 meter in diameter and operate at speeds up to 6000 rpm. The slits are 1.5 cm wide and 5 cm long. There are 8 slits on the rotor, b u t the number can be reduced to 4 or 2 where it is necessary to reduce the repetition rate of the neutron burst. The rotors 7F. G. P. Seidl, H. Palevsky, R. F. Randall, and W. Thorne, Phys. Rev. 82, 345 (1951); K. E. Larsson, R. Stedman, and H. Palevsky, J . Nuclear Energy 6 , 2 2 2 (1958); V. V. Vladimirsky and V. V. Sokolovsky, PTOC. 8nd Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1958 14,283 (1958). * B . Jacrot, Proc. Meeting on the Use of Slow Neutrons to Investigate the Solid State, Stockholm, October, 1957, p. 115, Swedish Atomic Energy Commission.

504

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

operate in a vacuum to reduce power dissipation. An example of the type of results obtained with slow choppers is given in Fig. 5 where a reactor spectrum is shown and in Fig. 6 where the spectrum of neutrons passing through a liquid nitrogen cooled Be filter is shown. 2.2.2.2.2.3. Application of the Mechanical Choppers. Slow neutron choppers have been used both to determine total and absorption cross sections at very low energies and to study inelastic (neutron loses energy) and hyperelastic (neutron gains energy) neutron scattering. The absorption cross section of any substance increases as l / d E . Therefore, a t very low energies the absorption cross section is large compared to its value a t higher energies. Since the absorption cross section is so large a t very low energies, the corrections which must be applied to the measured cross section to obtain the absorption cross section, such as the scattering cross section and the cross sections of impurities and diluents in the sample, become relatively small. A more accurate value of the absorption cross section can therefore be obtained. The absorption cross sections of boron and gold, which are secondary standards, and of the fissionable materials have been most accurately determined by measuring the total cross section a t 0.000818 e ~ . ~ The inelastic scattering of neutrons by liquids and solids has been studied by measuring the variation of total cross section with neutron wavelength and temperature on materials which have low-absorption cross sections. The theory of this process has been worked out by Weinstocklo and Placzek,J' and an example of the results obtained by measuring the total cross section of lead using neutrons of 10 %, wavelength is shown in Fig. 7. The open circles are the experimental data and the solid lines are the two curves for the two different approximations used in the theory by Weinstock and by Placzek. When the measured inelastic scattering cross sections are compared to the theoretically calculated values, the agreement is only fair. Exact theoretical calculations are impossible because the frequency spectrum of the lattice vibrations is not known. The incoherent approximation assumes that each atom scatters the neutron wave incoherently. The expression for the cross section using this incoherent approximation involves an integration like the specific heat integral. In analogy with specific heat measurement, one might expect the neutron cross section to be fairly insensitive to the detailed shape of the frequency spectrum. However, Placzek and Van Hovel2 have shown G. J. Safford and W. W. Havens, Jr., Nucleonics 17, 134 (1959);J. E.Evans and R. G. Fluharty, Nuclear Sci. and Eng. 8, 66 (1960); H.Palevsky, PTOC. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1966 4, 311 (1956). l o R. Weinstock, Phys. Rev. 66, 1 (1944). G.Plaozek, Phys. Rev. 86, 377 (1952). l2 G.Placzek and L. Van Hove, Nuovo cimenlo [lo]1, 283 (1955).

4

Wowlength. .Xi

FIG..5. The distribution of neutrons emitted by a natural uranium heavy water moderated reactor. From K. E. Larsson, R. Stedman, and H. Palevsky, J . Nuclear Energy 6(3), 222 (1958).]

FIG. 6. Distribution of neutrons filtered through a liquid nitrogen cooled Be filter. [From K. E. Larsson, R. Stedman, and H. Palevsky, J . Nuclear Energy 6(3), 222 (1958).]

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

507

that interference effects in cold neutron scattering are quite sensitive to the shape of the spectrum. Therefore the study of inelastic nucleon scattering furnishes a good method for determining the frequency spectra of lattices. One of the difficulties always encountered in using slow choppers a t reactors is the fast neutron background which passes through the chopper. This background can be greatly reduced by inserting in the beam filters which have a high stopping power for neutrons in and above the Max-

T IN

O K

FIG.7. Total cross section of lead as a function of temperature for 10 A neutrons. The line labeled “coherent” represents the theoretical prediction calculated by t h e theory of Weinstock and the “incoherent approximation” is t h a t calculated using Placzek’s theory.

wellian distribution but which are practically t,ransparent to neutrons of long wavelength. The most useful filters for this purpose are beryllium, BeO, graphite, bismuth, lead, and quartz. A curve of the transmission of 15 cm of Be as a function of wavelength a t 100’ and 300°K is shown in Fig. 8. At wavelengths greater than 5 the transmission is very much higher than for shorter wavelength neutrons, and consequently the shorter wavelength neutrons are preferentially filtered out. Gould13 has made a careful study of the properties of filters useful for reducing the higher energy neutron component in a low-energy neutron beam. One of Gould’s curves for the transmission of the Brookhaven reactor spectrum F. T. Gould, The neutron spectrometer for subthermal neutrons and the cross sections of gold and metallic hydrides in the 4-11.5 range. Columbia Uoiversity Report CU-179 (October, 1958).

a

508

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

as a function of thickness and temperature is shown in Fig. 9. This shows that for any filter of thickness greater than 15 cm almost all of the neutrons of higher energy have been eliminated. The properties of the filter are improved as the temperature of the filter is reduced because of the reduction in thermal inelastic scattering. Therefore, reducing the tem-

7 T = lOO*K

-T=

300' K

VJ

!!

w

I

I

1

perature of the filter improves its filtering properties. Gould was interested in applying filters to remove the higher order neutrons in a crystal spectrometer, so his results are not directly applicable t o a slow chopper. However, the general properties of filters can be determined from his data and demonstrate their usefulness in experiments with subthermal neutrons.

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

509

Neutron filters which pass very low-energy neutrons but stop neutrons of higher energies are useful for two purposes. The first, mentioned above, is to reduce the background in transmission experiments. The second, which is by far the more interesting application, makes use of the fact that a reactor spectrum filtered through cold beryllium or graphite comes out with a relatively small energy spread. The small energy spread occurs because the intensity of the spectrum emitted by a reactor decreases very

Io30

2

4 6 8 10 12 THICKNESS OF B e , INCHES

14

FIG.9. Transmission of Be a8 a function of thickness for 5 A neutrons. [From F. T. Gould, T. I. Taylor, W. W. Havens, Jr., B. M. Rustad, and E. Melkonian, Nuclear Sn'. and Eng. 8, 453 (1960).]

rapidly (about he4) as the wavelength of the neutron increases, and the filter will transmit neutrons of low energy very much more readily than it will transmit neutrons of high energy. The neutron spectrum emerging from the filter has a sharp cutoff at the short wavelength end of the spectrum and has a maximum intensity at a wavelength slightly larger than the Bragg cutoff wavelength for the filter used. The spectrum obtained by Ghose et aZ.l4 for the Brookhaven reactor spectra filtered through a thick polycrystalline refrigerated beryllium filter corrected for a fast neutron background is shown in Fig. 10. A. Ghose, H. Palevsky, D. J. Hughes, I. Pelah, and C. M. Eisenhauer, Phys. Rev. 113, 49 (1959).

510

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

2.2.2.2.2.4. Combination of Filter and Chopper. The beam of low-energy neutrons obtained by filtering reactor neutrons through the beryllium is then incident on a solid or liquid. The neutrons scattered by the sample making a specific angle with the initial beam of neutrons are then passed through a slow chopper, and the time of flight of the neutrons to a detector, placed some distance from the chopper, is measured. Since the energy of the neutron incident on the sample is much smaller than the characteristic vibrational or rotational energies of the structure of the liquid or the

FIG.10. Distribution of neutrons filtered through thick polycrystalline beryllium at liquid nitrogen temperature. [From A. Chose, H. Palevsky, D. J. Hughes, I. Pelah, and C. M. Eisenhauer, Phys. Rev. 113, 49 (1959).]

solid, the neutron will gain energy from the sample and therefore the energy of the neutron emitted by the sample will be characteristic of the sample. It is these measurements which enable the frequency distribution of elastic waves in solids to be determined. ' ~ study A schematic diagram of the apparatus used by Carter et ~ 1 . to the inelastic and hyperelastic scattering of cold neutrons is shown in Fig. 11. An example of the data taken on a single crystal of aluminum at several orientations to the incident beam is shown in Fig. 12. These data demonstrate the reality of the phonon scattering. The resolution is rather poor and the counting rate low, but comparison of neutron and X-ray 1 6 R . 5. Carter, H. Palevsky, and D. J. Hughes, Phys. Rev. 108, 1168 (1957).

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

511

data shows fair agreement. Where more differences do appear, the neutron data are not sufficiently accurate or detailed to be significant. Several methods have been used to improve the intensity of cold neutrons. At Brookhaven a very large area beam emitted from a graphite thermal column was used.I6 Egelstaff" placed a large Dewar vessel of liquid hydrogen in a reactor and obtained an improvement in the intensity

Reactor Shield

-

FIG. 11. Schematic diagram of experimental arrangement for the cold neutron studies a t the Brookhaven reactor. [From R. S. Carter, H. Palevsky, and D. J. Hughes, Phys. Rev. 106, 1168 (1957).]

by a factor of about 25 for neutrons of 10 8. However, very few experiments have been reported using liquid hydrogen moderation because of the danger of introducing hydrogen into a reactor. A good example of the kind of results which can be obtained with this type of apparatus is the data on the inelastic scattering of neutrons by vanadium given in Fig. 13. The light line represents the scattered intensity corrected for background, while the heavy line includes further corrections for detector efficiency and chopper transmission. The scattering l6 C. M. Eisenhauer, I. Pelah, D. J. Hughes, and H. Palevsky, Phys. Rev. 109, 1046 (1958). l7 P. Egelstaff, Proc. Meeting on the Use of Slow Neutrons to Investigate the Solid State, Stockholm, October, 1957, p. 40. Swedish Atomic Energy Commission.

Fro. 12. Energy distribution of inelastically scattered neutrons from aluminum for 4 different orientations of the crystal. [From R. S. Carter, H. Palevsky, and D. J. Hughes, Phys. Rev. 106, 1168 (1957).]

.5ooc

NEUTRON WAVELENGTH IAI

FIG 13. Spectrum of neutrons inelastically scattered from vanadium corrected for chopper efficiency and background. The dashed line is the calculated contribution from two phonon interactions and multiple scattering in the vanadium sample. [From C. M. Eisenhauer, I. Pelah, D. J. Hughes, and H. Palevsky, Phys. Reu. 109, 1046 (1958).] 512

2.2.

DETERMINATION OF MOMENTUM AND ENERQY

513

of vanadium is almost completely incoherent, so the equation for conservation of energy completely describes the scattering process. For the case of an incoherent scatterer in a cubic lattice, the energy distribution of neutrons scattered by one phonon isla

where nu is the flux of scattered neutrons, e-zw is the Debye-Waller factor, [exp(hw/kT) - 11-l is the Boltzmann factor, and g(w) is the frequency distribution of the normal modes. The peaks observed in the spectrum are probably peaks in the frequency spectrum of vanadium. The measured energies of the peaks correspond to frequencies of approximately 5 X 10l2 and 6.5 X 1OI2eps. If the sharp drop in intensity at 0.03 ev is interpreted as the limiting Debye frequency, a Debye temperature of 350" is obtained. This is to be compared with a 00 = 338" obtained from specific heat measurements.19 Further experiments of this type should increase our knowledge of the properties of solids. Since the neutron time-of-flight methods are so useful in furnishing information on the structure of solids, more complex devices are being constructed to improve the resolution. Egelstaff and Cockingz0 have recently reported on a four-rotor thermal neutron analyzer. The apparatus employs a double neutron spectrometer in which four choppers operate in a predetermined phase relationship to one another and produce bursts of monochromatic neutrons. The bursts fall upon a sample and the energy of the scattered neutrons is measured by the standard time-of-flight methods. Up to 30 counters can be arranged around the sample covering the angle from 10" to 160". The data from these counters are recorded on magnetic tape with a system described by Rae.21The intensity and resolution attained with this apparatus is better than any previously reported. Very interesting results have already been obtained with this apparatus. Liquid water has been shown to act more like a solid than a gas, indicating that the classical ideas about atomic motion in liquid may be in error. The phonon frequency distribution in graphite has been shown to be temperature dependent. This field promises to be very fertile. 2.2.2.2.2.5. Other Devices for Xtudying Neutron Time of Flight. A novel method of producing short bursts of cold neutrons has been reported by G. Placzek and L. Van Hove, Phys. Rev. 93, 1027 (1954).

W. S. Corak, B. B. Goodman, C. B. Satterthwaithe, and A. Wexler, Phys. Rev. 102, 656 (1956). zo P. Egelstaff and S. J. Cocking, Proc. Symposium on Inelastic Scattering of Neutrons in Solids and Liquids. International Atomic Energy Agency, Vienna, October, 1960. 11 E. R. Rae and F. W. K. Firk, Nuclear Eng. 1,227 (1957). l8

514

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES NEUTRON BEAM BERYLLIUM F I L L E R 7

7

MAGNETIC PICKUP COLLIMATING SLIPS

;d i\ LBEAM

MOTOR

CATCH TANK

FIG.14. Experimental arrangement for the spinning sample method of measuring inelastic scattering events. [From R. M. Brugger and J. E. Evans, Proc. Symposium on Inelastic Scattering of Neutrons in Solids and Liquids. International Atomic Energy Agency, Vienna, October, 1960.1

Beom from reactor

--Quartz filter

~

,d5 @ \Pulsed

/

”

I

monochromatic beom

5 Meters

I

4

I 1

Time of flight spectrometer

FIG.15. Schematic diagram of a rotating crystal time-of-fight spectrometer. [From A. B. D. Woods, W. Cochran, and B. N. Brockhouse, Phys. Rev. 119, 980 (1960).]

2.2.

DETERMINATION O F IdOIdENTdM AND ENERGY

515

Brugger and Evans.22The sample is passed rapidly through a beam of Be-filtered neutrons by spinning the sample on the end of a rod as shown schematically in Fig. 14. Bursts of scattered neutrons are produced and the energies are measured by time-of-flight analysis. An apparatus for producing monochromatic neutrons with a crystal and then causing the neutrons to strike the sample in bursts, so that the time of flight of the neutron scattered by the sample can be measured, has been reported by Woods et aLZ3I n this device, which is shown schematically in Fig. 15, the monochromatizing crystal is rotated around its axis. Bursts of monochromatic neutrons strike the sample and are scattered. 400 KEV-DEUTRONS FROM CASCADE ACCELERAE ANALYSER

AIR- COOLED

-

rn

SHUTTER Cd

@?J B 2 0 z 94c

FIG. 16. Schematic diagram of the apparatus used to measure neutron spectra emitted by a moderator.

The time of flight is measured for the neutrons scattered through an angle 8 which are recorded by the detector after traveling a distance d. The time of flight can be measured in the usual manner with the start pulse coming from a mirror on the rotating crystal. Each of the various devices described above for studying the inelastic and hyperelastic scattering of thermal and subthermal neutrons has certain advantages and disadvantages. The proper device t o select for a particular problem depends on the resolution and counting rate desired for the experiment to be performed with the source available. 22 R. M. Brugger, L. W. McClellan, G. B. Streetman, and J. E. Evans, Nuclear Sci. and Eng. 6, 99 (1959). 23 A. B. D. Woods, W. Cochran, and B. N. Brockhouse, Phys. Rev. 119,980 (1960).

516

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

The time-of-flight technique has been used very successfully for studying the moderation of thermal neutrons. A schematic diagram of the apparatus used for such studies is shown in Fig. 16. Fast neutrons produced by a pulsed accelerator strike the moderator blocks, where they are slowed down and thermalized. After a period of time which is a t the discretion of the investigator, a fast acting mechanical shutter opens and allows neutrons to pass into a flight path, at the end of which they are

v (rn/s)

FIG.17. Spectrum of the neutrons emitted from the center of a 15 cm cube. The solid curve is a theoretical Maxwellian distribution for the temperature of the moderator, [From K. H. Beckurts, EAES Meeting on In Pile Spectra and Pulsed Neutrons. Ris6, Denmark, July, 1960.1

detected by beam counters. This system measures the energy spectrum of the neutron emitted by the moderators. Some of the data obtained by Beckhurts for the spectra emitted by a 15 em cube, after the neutrons had come to complete equilibrium with the moderator, are shown in Fig. 17. The results are fitted excellently by a Maxwellian distribution for the temperature of the moderator. The neutron source for this experiment was a 400-kw cascade accelerator using a pulsed rf source and the H2(d,n)He4reaction to produce the neutrons. The neutron pulses were 40 psec in duration with approximately 108 neutrons per burst. The flight path was 3.35 meters long. The neutron shutter was a piece of cadmium on the surface of an aluminum disc spinning at an angular

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

517

velocity of 14,000 rpm. A multichannel (100 and 256) analyzer was used for the time analysis. Data taken with apparatus of this type will be extremely useful in leading to a better understanding of the operation of reactors. 2.2.2.2.3. EPITHERMAL NEUTRONTIME-OF-FLIGHT SPECTROMETERS. Most of the data which exist on slow neutron resonances in the epithermal region have been obtained by time-of-fight spectroscopy. There are two types of device which have been used to accumulate these data: (1) fast choppers at nuclear reactors; and (2) pulsed particle accelerators. Most of the early data on which the original nuclear energy program was foundedz4were obtained with pulsed accelerator time-of-fight spectrometers. However, with the introduction of high-intensity reactors and the elaborate development of fast choppers, these latter have been used t o accumulate a large fraction of the data in the epithermal region which are now available. However, pulsed accelerators can produce shorter pulses than fast choppers simply because it is easier to modulate rapidly a beam with an electromagnetic shutter than one with a mechanical shutter. Recently most of the extremely high resolution work has been done with pulsed accelerators. 2.2.2.2.3.1. General Design Factors. The resolution of a time-of-flight system depends on the duration of the neutron pulse, the duration of the detector modulation pulse, and the distance from the effective source or shutter t o the detector. The resolution that can be obtained with any time-of-flight apparatus is theoretically infinite since, in principle, the detector can be placed very far from the source. In practice, of course, the intensity available limits the resolution, since as the source detector distance increases, the intensity per burst at the detector decreases rapidly because of the increased distance and because a slower repetition rate is needed to eliminate the overlap of neutrons from the previous bursts. When the area of the detector is small compared with the area of the neutron beam, the counting rate is given by the formula

N

=

( k .I T , T d A R ) / d a

(a.2.2.2.2)

where the constant k includes the collimator and detector efficiencies, and all numerical factors, I is the burst intensity, rr is the time the source is active, T d is the time.the detector is active, A is the area of the detector, R is the repetition rate, and d is the source detector distance. 24 See, for instance, Neutron cross section compilation. Report AECU 2090. Technical Information Division of the U.S. Atomic Energy Commission, Oak Ridge, Tennessee, 1952. Compiled by the Nuclear Cross Sections Advisory Group of the U.S. Atomic Energy Commission.

51 8

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

For measurements above 0.5-ev neutron energy, the pulse repetition rate is determined by the time of flight at which cadmium reduces the intensity to almost zero by resonance absorption so that overlap of successive cycles may be avoided. Cadmium is practically opaque to neutrons of energies Iess than 0.5 ev. This means that the time between bursts must be greater than about 150d psec or R = 6700/d per sec where d is in meters, when cadmium filtering is used. When the pulse repetition rate is fixed, the path length that can be used must be equal to or less than that given by the above relationship. Cadmium has its own characteristic resonance structure which must be taken into account when cadmium filtering is used. However, a very thin sheet of cadmium will reduce the thermal intensity by a very large factor, so some cadmium filtering is often advantageous. If the characteristic resonance structure for cadmium cannot be tolerated in a particular experiment, then boron filtering must be used. The boron cross section is proportional to l / d E and will greatly reduce the thermal neutron background, but it will also reduce the intensity in the epithermal region. The boron filter to be used with a particular apparatus depends so much on the exact design of the time-of-flight system and the characteristics of the neutron source with which the time-of-flight system is to be used that no general statements can be made on the thickness of the filter to be used. For extremely long flight paths where it is necessary to have very low repetition rates, the overlap problem can be eliminated by placing a crude slow chopper in the neutron beam. A chopper for this purpose need only have poor resolution and need not be critically phased with the neutron burst. To obtain a relationship between the counting rate and the resolution, we define the resolution At of an ideal velocity spectrometer as At =

+ 7d)/2d.

(T~

(2.2.2.2.3)

For many velocity spectrometers it is most efficient to use T* = ~ d in , which case we have N = const [(AQ2/d] (2.2.2.2.4)

where the constant now also includes the pertinent factors for a particular apparatus. Equation (2.2.2.2.4) shows that there is a definite advantage to working a t small distances. Of course, the other problems encountered in the design of the time-of-flight apparatus become more difficult. As the distance decreases, the burst time and the electronic circuits must be made faster and the background also increases. Further, for a pulsed accelerator velocity spectrometer the resolution width is increased be-

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

519

cause of the uncertainty in the time a t which the neutron is emitted from the moderator. The detector must also be made smaller, for the shorter the flight path the greater the uncertainty that is introduced due to the lack of knowledge of the position in the detector at which the neutron is detected. For some velocity spectrometers, the burst time is negligible compared with the detector gate time. I n this case At = rd/2d

(2.2.2.2.5)

and the counting rate dependence on the resolution width and source detector distance is N = const (A.r/d2). (2.2.2.2.6) Here a much greater advantage is gained by using shorter distances. If the beam is well-collimated so that the detector is always larger than the area of the beam, then the counting rate does not fall off as the inverse square of the distance. For the case in which the burst width is equal to the detector width, there is a definite advantage to using longer distances since N a AT)^^. I n the case where the burst time is negligible, the counting rate is independent of distance. This means the distance to be used should be as large as is practicable to minimize other problems. 2.2.2.2.3.2. Pulsed Accelerator Velocity Spectrometers. The design of a pulsed accelerator velocity spectrometer is affected by many factors, some of which depend on the experiments to be performed, while others depend on the type and characteristics of the device with which the spectrometer is used. The advantages of pulsed accelerators for use as velocity spectrometers are: (a) the accelerator is not producing neutrons while the neutrons are being counted so that the background is low; (b) the pulse of neutrons produced by the accelerator can be made extremely short because electromagnetic, rather than mechanical, changes are involved; (c) the production is restricted to a well-defined time interval. A schematic diagram of the Columbia University synchrocyclotron neutron velocity spectrometer is shown in Fig. 18 to illustrate the size and complexity of a modern time-of-flight spectrometer. This velocity spectrometer probably has the highest resolution of its kind in the world. 2.2.2.2.3.2.1. Moderation time. I n computing the timing resolution of a pulsed accelerator system for slow neutron work, account must be taken of the moderation time for fast neutrons of initial energy Eo to be emitted from the moderator slab a t a final energy E. Groenewold and GroendijkZ5 have treated the problem of a delta function burst of fast neutrons, a t t = 0, uniformly illuminating a thick slab of hydrogenous material. By *6

H. J. Groenewold and H. Groendijk, Phusica 13, 141 (1947).

I N S I D E OF CYCLOTRON BUILDING

OUTSIDE OF CYCLOTRON BUILDING DETECTORS

CYCLOTRON

DETECTOR

VACUU CHAMB

C

N THYRATRON ON CIRCUIT

BACKSTOP FLIGHT PATH 35.37 METERS NEUTRONS IN

POLYETHYLENE

HODERA

( L A R G E O I M E N S I O N S OF ALL APERTURES I N P L A N E O F THE P A P E R )

FIQ. 18. (Geometry of the Columbia University synchrocyclotron slow neutron velocity spectrometer (top view). [From J. Rainwater, W. W. Havens, Jr., J. S. Desjardins, and J. L. Rosen, Rev. Sei. Instr. 31, 481 (1960).]

2.2.

DETERMINATION O F MOMENTUM AND ENERQY

521

making reasonable simplifying assumptions, they obtain the distribution in time of the flux N ( E , t )dE dt of moderated neutrons of energy E leaving the face at time t . For a given E, the time dependence has the functional formf(s) = +x2e-2 which has been normalized t o unit area. Here x = t / r m and 7, is the mean collision time of a neutron of energy E with a proton in the moderator. If a! = mean free path for such a collision, then 7, = a/v, where v = neutron velocity a t final energy E and a! = 0.62 cm in polyethylene for E less than a few kev. Figure 19 gives a plot off(z) which has the following properties, expressed in terms of t. The maximum comes a t t = 27, while t,, = 37, and the rms spread At = -\/3rm. The full width

"EQUIVALENT"

0

I

I

I

I

I

1

2

3

4

5

6

1

-

0

X

FIQ.19. A plot of the unit area functionf(z) = (&)r2e-2, where z = t / T m and T,,, is the mean time between n-p collisions a t final energy E. For a delta function fast neutron burst in a semi-infinite hydrogenous slab a t t = 0, this gives the time distribution of neutrons of final energy E from a face.

at half-maximum is 3.47, which is almost identical to 2At = 3.467,. For polyethylene r m = 0.45 Ell2 psec, where E is in ev. The moderation time contribution to the over-all timing smearing in terms of the full width at half-maximum is 3.47,. For comparison with the rectangular weighting function of a detection channel (0.1, 0.2, etc.) it is perhaps better, although more conservative, to use 67, = 0.086 psec (1000 ev), 0.27 psec (100 ev), 0.86 psec (10 ev). The mean moderation time for energy E varies inversely as the velocity and can be regarded conveniently as an increase and smearing in path length. The effective path is increased by 3a = 1.86 cm and there is an rms smearing -\/S a! = 1.07 cm. The equivalent rectangular smearing function has a full width 6, = 3.72 cm. 2.2.2.2.3.2.2.Spectrum shape. For a very thick moderator the energy

522

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

dependence of the source spectrum would be of the form dE/E over the region E 2 a few kev, where the n-p cross section has essentially its low energy value. For a relatively thin source where the escape probability p after a given moderating collision is not zero, the energy spectrum is relatively richer in higher energy neutrons. A simple formula may be derived as follows. If the n-p cross section and the escape probability p are assumed constants, and N Oneutrons of energy EOhave a first moderating collision, then Nop dE/Eo escape in an energy interval dE after one collision. Similarly,

after the second 'collision and

N o p ( l - p)n(dE/Eo)[ln(E~/E)n]/n! escape after the (n

+ 1 ) collision. Summing the series to n =

00

gives

Nop (dE/ E0) (Eo/E) (I-*). The efficiency of the moderator is usually governed by the geometry of the accelerator. For maximum efficiency, the moderator should be as close to the neutron source as possible. 2.2.2.2.3.2.3.Neutron source intensity. The neutron intensity in the burst is determined from the energy and current of the charged-particle beam and the particular type of nuclear reaction used for the production of the neutrons. Since the neutron intensity is directly proportional to the current in the burst, this factor need not be considered further. Very few data on the production of neutrons in thick targets as a function of the energy of the incident charged particle are available in the literature, since yield measurements are difficult to make and cannot be readily interpreted in terms of fundamental nuclear constants. In the energy region below a few MeV, the total neutron yield of a thick target can be determined from the cross section integrated over the energy spectrum of the particle as it is slowed down in the target. The dominant factors determining the yield are the ionization loss and the Coulomb barrier. Since both the ionization loss and the Coulomb barrier rise wit,h increasing 2, light nuclei are favored. I n practice, the only targets which need be considered for the production of neutrons by protons and deuterons below 50 Mev are deuterium and tritium, their compounds, and beryllium. The cross section for the production of neutrons in the D(d,n)He3 reactionz6 as a function of energy is given in Fig. 20. For deuterons of 2 6 N. Jarmie and J. Seagrave, eds., Charged particle cross sections. Report LA-2014, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, March, 1956.

100

80 60

40

20

10 8

6 4

2

I' I Kev

2

4

6

0

10 Kev

20

40

60 00

100 Kev

Deuteron

200

400 600800

I Mev

2

4

6 0

10 Mev

20

40

energy

FIG.20. Cross section for t h e D(d,n)He3 reaction as a function of energy. [From Los Alamos Charged Particle Cross Section Compilation, LA-2014, unpublished.] Q = 3.267 MeV. KEY:0 , W. A. Arnold, J. A. Phillips, G. A. Sawyer, E. J. Stovall, Jr., and J. L. Tuck, Phys. REV.93,483 (1954). P.E. = +4%. [Note: Values revised upward 3 t o 12 %, using t h e newer angular-distribution of Preston et al. as suggested by K. G. McNeill, Phil. Mag. [7] 46, 800 (1955).] +, R. K. Smith and J. E. Perry, LASL Preliminary unpublished data. P.E. = +6%. v, G. T. Hunter and H. T. Richards, Phys. Rev. 76, 1445 (1949). P.E. = +15%. A, J. M. Blair, G. Freier, E. Lampi, W. Slater, Jr., and J. H. Williams, Phys. Rev. 74, 1599 (1948). P.E. = +4%.O, G. Preston, P. F. D. Shaw, and S. A. Young, Proc. Roy. SOC.A226, 206 (1954) P.E. = f8%. x, K. W. Erickson, J. L. Fowler, and E. J. Stovall, Jr., Phys. Rev. 76, 1141 (1949). P.E. = + l o % .

2.

524

DETERMINATION OF FUNDAMENTAL QUANTITIEB

energy less than 0.5 Mev it is impractical to use a gas target because the thickness of the foil required to contain the gas degrades the energy of the charged particle beam by an amount which greatly decreases the neutron production. Most of the neutron work using this reaction has been done with heavy ice targets or deuterium adsorbed on a metal such as zirconium or tantalum. The production of neutrons as a function of the energy of the incident deuteron in a thick, heavy ice target as given by Hanson, Taschek, and Williams, is shown in Fig. 21 up to an energy of 1 MeV.

"0

0.2

I

I

I

I

0.4

0.6

0.8

I.o

Deuteron energy

- Mev

FIG 21. The total yield of neutrons from a thick ice target of D20as a function of incident deuteron energy. [From A. 0. Hansen, R. F. Taschek, and J. H. Williams, Revs. Modern Phys. 21, 635 (1949).]

Above about 1.5 Mev the cross section does not change appreciably with energy, as can be seen from Fig. 20. If we assume that the cross section is constant, the calculation of the neutron production reduces to that of a target penetration problem. Since the energy loss is about proportional to 1/E, the yield should be approximately proportional to E2. Actually, since the cross section decreases as the energy increases, and since the energy loss is a more complicated function than 1/E,the yield should probably increase with energy somewhere between and E2. For very low-energy particles, H 3 gas or H3 adsorbed on zirconium serve as excellent targets for neutron production when bombarded with deuterons. The existence of the resonance in the H3(d,n)He3reaction at 124.3 kev with peak cross sections as high as 4 barns makes it possible to

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

525

obtain extremely large numbers of neutrons a t low bombarding energies. The cross section curve for this reaction is shown in Fig. 22. The thick gas target yield for 600-kev deuterons is reportedz7as 5 X los neutrons per microcoulomb. The production of neutrons by 250-kev deuterons on

FIQ.22. The cross section for the T(d,n)He3reaction as a function of energy. Open circles, A. 0. Hanson, R. F. Taschek, and J. H. Williams, Revs. Modern Phys. 21, 639 (1949); closed circles, T(d,n)He4, W. R. Arnold, J. A. Phillips, G . A. Sawyer, E. J. Stovall, and J. L. Tuck, Phys. Rev. 93, 483 (1954).

deuterium and tritium-impregnated zirconium targets has been measured by Coon2*as a function of the number of coulombs of charge bombarding the target, with the results shown in Fig. 23. For energies of heavy charged particles above a few MeV, beryllium is probably the best target to use because of its excellent mechanical properties. A thin sheet of beryllium is soldered to a large block of copper, which is water-cooled so that the target can withstand bombardment with several kilowatts of beam power without adverse effects. I n the Columbia ‘7 J. H. Coon, Targets for the production of neutrons. I n “Fast Neutron Physics,” (J. B. Marion and J. L. Fowler, eds.), Part I, p. 677. Interscience, New York, 1960. 1 8 J . H. Coon, ibid.

526

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

36-in. cyclotron, a beryllium target prepared in such a manner has sustained up to 5 kw of beam power concentrated on an area less than 5 sq mm. Below about 1.8 Mev the neutron production in beryllium is fairly low because of the Coulomb barrier. Above this energy, however, the cross section should be approximately constant and the yield should follow the to E 2 dependence. Smith and KrugerZ9 report the yield of neutrons from IO-Mev deuterons on a thick beryllium target as 3.7 X 1Olo neutrons per microcoulomb. Allen and co-workers30 have measured the fast neutron

Zr TARGET, D - D YIELD

Zr T TARGET

I

1.0

2.0

D-OYlEtO

3

COULOMBS

FIG.23. Neutron yield vs coulombs for 250 kev deuterons on various gas-containing targets. [From “Fast Neutron Physics” (J. B. Marion and J. L. Fowler, eds.), Interscience, New York, 1960.1

yield a t 15 Mev and report 1.9 X 1Olo neutrons per microcoulomb. The discrepancy between these two numbers probably arises from the fact that in the 10-Mev measurements neutrons of all energies were counted, whereas for the 16Mev deuterons the results apply only to those neutrons 2 which was used as which are capable of exciting the S 3 2 ( n , p ) P 3reaction detector. For protons and deuterons well above 50 MeV, the ionization loss gradually ceases to be the predominant factor in the yield, since the energy lost by a nuclear collision approaches the energy lost by ionization. If we assume that about half of the energy of a proton goes into ionization and, on the average, half of the particles produced in a spallation reaction 29

30

L. W. Smith and P. G. Kruger, Phys. Rev. 83, 1137 (1951). A. J. Allen, J. F. Nechaj, K.-H. Sun, and B. Jennings, Phys. Rev. 81, 536 (1951).

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

527

are neutrons, then from energy considerations alone about 40 to 50 Mev would be required per neutron produced. For 350 to 400 Mev protons several neutrons are produced per incident proton. Although no absolute measurements of the number of neutrons produced by high-energy protons have been reported, rough measurements of the strength of the neutron source using the 400-Mev synchrocyclotron of Columbia University indicate that about 10 neutrons are produced per proton incident on the target. This number is almost independent of the target material. The feasibility of using a high-energy electron accelerator as a neutron velocity spectrometer was first pointed out by Cockcroft el aL31These accelerators are naturally pulsed and have exceedingly high currents in the pulse. Using a 3.2-Mev electron linear accelerator which had a current pulse of 100 ma and a pulse width of 2 psec, a lead target was bombarded to produce y-rays. The y-rays were used to produce the neutrons using the (y,n) reaction in heavy water and beryllium. This arrangement gave a neutron intensity of 3 X 10l2neutrons per second in the burst. Using the old Harwell linear accelerator, Wiblin32 reported that, with an output energy of 13 Mev and a current pulse of 35 ma, the production during the neutrons per second when the electrons strike a thick uranium burst is loL4 target. The existence of the giant (7,n) resonances above 10 Mev in a large number of nuclei33means that almost any target can be used for the copious production of neutrons above this energy. Uranium seems to be , ~ ~ is lower best, since the (y,n) resonance occurs at about 11 M ~ vwhich than most of the other (y,n) resonances; in addition, a (7,fission) resonance exists, which increases the neutron production. The problem of determining the thick target yield from the total (7,n) cross section as a function of the energy of the incident electron is difficult because of the uncertainty in the bremsstrahlung spectrum. Feld3Shas given a simple theory for this process which, when applied to the data of Duffield and Huizenga, gives the curve shown in Fig. 24 for the energy dependence of the yield per electron in a thick target. The production of neutrons as a function of the energy of the incident electron in a thick uranium target has been measured by Baldwin et J. D. Cockcroft, J. C. Duckworth, and E. R. Merrison, Nature 163,869 (1949). E. R. Wiblin, Atomic Energy Research Establishment, Harwell, England, private communication. 33 R. Montalbetti, L. Kate, and J. Goldenberg, Phys. Rev. 91, 659 (1953). 34 R. B. Duffield and J. R. Huizenga, Phys. Rev. 89, 1042 (1953). 36 B. T. Feld, Nucleonics 9, 5 1 (1951). G. C. Baldwin, E. R. Gaerttner, and M. L. Yeater, Knolls Atomic Power Laboratory of the General Electric Company, Schenectady, New York, private communication. 31

a2

528

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

using the General Electric betatron with the results shown in Fig. 25. T o determine the number of neutrons produced, they measured the activity induced in a rhodium foil surrounded by paraffin. The activity of the rhodium foil is plotted as a function of the energy of the incident electron. Above about 25 MeV, where shower theory should hold for uranium, the neutron production is expected to increase linearly with energy, as indicated by the dashed curve in Fig. 25. Above 80 Mev the deflector system in the GE betatron is not capable of deflecting all of the beam into the target and consequently the neutron production falls off. This falloff is not a characteristic of the production process but is due to the character of the GE betatron. The positive curvature of the results obtained by the General Electric Company group should not be considered to be in disagreement with the

ENERGY OF ELECTRONS

FIG.24. Calculated neutron yield from a uranium target bombarded by electrons. [Taken from B. T. Feld, Nucleonics 9, 51 (1951).1

results obtained from the calculations using Feld’s theory and the results of Duffield and Huizenga, since there is very little overlap between the two curves. Recently the production of neutrons by electrons as a function of energy has been measured for several thick targets by Barber and George3’using the Stanford electron linear accelerator. The neutron yield per radiation length of target as a function of 2 for 34.3-Mev electrons is shown in Fig. 26. The points calculated from theory are in good agreement with the experimental determinations. Several pulsed accelerator velocity spectrometers have been described in the literature. a* The Columbia University synchrocyclotron velocity W. C. Barber and W. D. George, Phys. Rev. 116, 1551 (1959). W. W. Havens, Jr., Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1966 4, 74 (1956); E. Wiblin, ibid. p. 35; V. V. Vladimirsky and V. V. Sokolovsky, Proc. Znd Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1968 14, 283 (1958); M. J. Poole and E. R. Wiblin, ibid. p. 266. 37

a8

2.2.

DETERMINATION OF MOMENTUM AND ENERQY

529

~pectrometer~~ probably has the highest intensity in the burst and the highest resolution known at present. The new Harwell linear a c ~ e l e r a t o r , ~ ~ which has operated at 30 Mev with an electron current of 250 ma in the burst, is used almost entirely for neutron spectroscopy. 2.2.2.2.3.3. Fast Choppers. A fast chopper produces sharp bursts of neutrons by using a heavy shutter to open and close quickly on a beam of neutrons. The large size needed to stop neutrons of all energies, and

ELECTRON BEAM ENERGY, MEV

FIQ.25. The activity of a rhodium foil in a standard paraffin geometry is plotted as a function of the energy of the electrons that strike a thick uranium target. This gives the neutron yield as a function of energy for a constant current in the accelerator. [From W. W. Havens, Jr., Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1966 4, 74 (1956).]

the high rotational speed necessary to produce a sharp burst, make the fast chopper a fairly complicated device. A schematic diagram of the fast chopper velocity spectrometer would be very similar to the schematic diagram of the slow chopper shown in Fig. 4. 8*J. Rainwater, W. W. Havens, Jr., J. S. Desjardins, and J. L. Rosen, Rev. Sci. Znstr. 31, 481 (1960). 4oM. J. Poole and E. R. Wiblin, PTOC.2nd Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1968 14,266 (1968).

530

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

ONE RADIATION LENGTH TARGETS ~

E,= 3 4 . 3 M E V

0

EXPERIMENTAL POINTS CALCULATED POINTS

ATOMIC NUMBER

2

FIG.26. Experimental and calculated yields of neutrons per incident electron for 1-radiation-length targets a t 34.3 Mev as a function of atomic number 2.[From W. C. Barber and W. D. George, Phys. Rev. 116, 1551 (1959).]

The requirement of short burst length for the fast chopper leads to neutron beams of small area, hence low intensity; this latter tendency must be counteracted as much as possible to obtain a finite counting rate at the distant detector. These conflicting requirements result in a rather heavy, rapidIy rotating shutter, containing shaped slits for the passage of neutrons, slits whose cross-sectional area is of the order of 0.025 cm by 2.5 cm. As the slits in the moving shutter pass similar slits in a stationary collimator, neutron bursts of duration about 0.5 psec to 5 psec, depending on the particular chopper, are produced. The requirement of small area slits to obtain a short burst leads to one of the greatest advantages of a fast chopper. The beam must be collimated down to the small area and a very small sample can be placed near the position of the slit. Thus fast choppers have made measurements with samples of only a few milligrams, and most of the results on separated isotopes have been obtained with fast choppers. Another advantage of the fast chopper is the fact that the problems associated with the opera-

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

53 1

tion of the chopper can be separated from the operation of the reactor. The reactor just serves as a continuous source of neutrons. This does not mean that the design of a chopper can be independent of the reactor with which i t is to be used. In fact, the design of a particular chopper is strongly affected by the size and characteristics of the reactor with which it is to be used. The first step in the design of a fast chopper is to decide on the kind of experiments for which the chopper is to be used. For the measurement of total cross sections b y the transmission method, it is usually of prime importance to employ the narrowest resolution width that can be obtained. This objective is met by forming a narrow burst of neutrons and by using a long flight path, for which a low repetition rate is desirable. The low neutron intensity which results from both of these requirements on the design can be tolerated in a transmission experiment because of the relatively high efficiency of the neutron detector. For the measurement of most of the partial cross sections, on the other hand, the efficiency of the detector is low and the background may be high. Thus the neutron intensity, which can only be increased by sacrificing resolution, is of primary importance. One solution to this conflict of requirements is to have a system using two interchangeable rotors of differing designs. Excellent experimental results have been produced a t Argonne National Laboratory on both total and partial cross sections by using a chopper which has two rotors of different design. Having determined the general nature of the experiments for which the chopper is intended, one must next consider in more detail the interrelationship between the three major components of the system, namely, the mechanical chopper itself, the neutron detector, and the multichannel time analyzer. If sufficient effort is put into the development of the time analyzer, it can always be made compatible with the other two components. However, the known techniques of neutron detection have not reached such a high degree of refinement. Therefore, for optimum performance of the system, the chopper and the detector should be considered as a unit. For partial cross sections, it is desirable to use a high efficiency detector having a relatively low response time such as a boron-loaded liquid scintillator. In this case the burst width need be no shorter than the response time of the detector. The detailed design considerations of fast choppers have recently been reviewed by Bollinger el aL4I and by Vladimirsky and S o k o l ~ v s k yA. ~ ~ schematic drawing of the Argonne fast chopper taken from Bollinger’s 4 1 L. M. Bollinger, R. E. CotC, and G. E. Thomas, Proc. 9nnd Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1958 14, 239 (1958). 42V. V. Vladimirsky and V. V. Sokolovsky, Proc. 2nd Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1958 14, 283 (1958).

I

-

0

2

4

I

6

INCHES

FIQ.27. The Argonne fast chopper. The essential components designated by the numbers on the figure are the following: (1)magnetic pick-up unit, the signal generator for speed control; (2)load supporting ball bearing, a single row, prelubricated, type 207 radial bearing; (3)oil-cooled vacuum seal; (4) oil slinger to protect mirror surfaces; (5)emergency bumper ring, bronze; (6)mirror to reflect light beam for zero-time signal; (7) monel metal outer shell of rotor; (8) uranium core of rotor; (9) damper, approximately # in. lateral movement is permitted; (10)ball bearing in contact with bottom pin of rotor; (11) light source and photomultiplier of zero-timesignal generator; (12) 5-horsepower dc motor supplied by Electric Specialty Co., Stamford, Conn.; top speed 15,000rpm. The weights of Rotors I and I1 are about 190 and 150 lb, respectively. [From L. M. Bollinger, R. E. Cot6, and G. E. Thomas, Proc. dnd Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1968 14,239 (1958).] 632

2.2.

DETERMINATION OF MOMENTUM A N D E N E R G Y

533

article is shown in Fig. 27 to give the reader some idea of the complexity of a fast chopper. For more detail on the design considerations the reader should study the original papers. 2.2.2.2.3.4. Neutron Detectors for Epithermal Neutrons. The rapid growth of the time-of-flight technique for measuring neutron cross sections at epithermal energies has required the development of slow neutron detectors having special characteristics. An ideal counter for this application should have a fast response time, a high counting efficiency, an insensitivity to y-radiation and fast neutrons, and a large active surface area. None of the systems that are now in use has all of these desirable characteristics; in each case it has been necessary to sacrifice one or more property in order to emphasize another which is considered more important. In most of the early work with neutron velocity spectrometers, BF, proportional counters and BF8 ionization chambers were used as detectors. The advantages of the proportional counters over the ionization chamber are that the pulses are shorter and the electronic amplification required is less. The counter is, therefore, less sensitive to pickup. The ionization chamber, on the other hand, gives more uniform pulses and less drift, and can be run a t higher pressure without the necessity of excessively high voltages for the size of the detector required. For the detection of neutrons of energies below a few electron volts, the boron cross section is very large and it is quite easy to make excellent, efficient, high-pressure BF3 counters designed for a particular apparatus. However, as the energy of the neutron increases, the detection efficiency decreases and the time gates also decrease. Eventually the time uncertainty of the response of a BF3 counter becomes the limiting factor in the resolution of a time-of-flight apparatus. For a BF3 counter, 5 cm in diameter and filled to a pressure of slightly less than one atmosphere, this uncertainty is about 2 psec. In order to decrease the time uncertainty in the response of BF3 counters, a large number of counters of small diameter have been used to obtain a large area detector. One such system43makes use of an array of 128 proportional counters within a single pressure vessel. Individual counters within the array are 2.2 cm in diameter and 23 cm long; the wire cathode is 0.15 mm in diameter and the anode is made of steel tubing having a wall that is 0.25 mm thick. The assembled system has a thickness, as seen by incident neutrons, of 9 cm and a sensitive area of 1600 cm2. The pressure vessel containing the counter array is filled with enriched (96% B10) BF, gas to a pressure of 115 cm of Hg. The system is operated with cathodes at ground potential and 7000 volts on the anodes. For this 48

H. Palevsky, N. G. Sjostrand, and D. J. Hughes, Phys. Rev. 91, 451 (1953).

534

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DETERMINATION OF FUNDAMENTAL QUANTITIES

voltage the gas amplification factor is of the order of 1000. A single amplifier is used for the entire array of counters. Careful selection of individual counters for the system caused all to have a similar pulse-height response, making the discrimination against counting of y-rays effective. The counter “jitter time” (the maximum pulse collection time) is small, being only 0.4psec. The drawbacks of the system for time-of-flight measurements are those that are inherent in BF3 gas counters. These defects are, first, an uncertainty as to the time of arrival of a slow neutron because of the counter thickness. If, on the other hand, the time uncertainty is reduced by making a thin detector, the counting efficiency becomes small; it is only 0.5% at 1 kev for this gas counter. I n an effort to improve the efficiency and eliminate the time uncertainties of gas counters without sacrificing their insensitivity to y-radiation, solid scintillation counters have also been developed a t B r ~ o k h a v e n ~ ~ for use in time-of-flight measurements. Intimate mixtures of 2 gm enriched B z 0 3and 1 gm of ZnS were fused together and formed into thin translucent scintillating discs having surface areas of 20 em2, Each disc was mounted on an RCS 5819 photomultiplier. The tube and scintillator were then placed in light-tight containers to form independent counting units. An array of 64 units of this kind operate in parallel to form a detection system of large surface area. The effective gain of each unit is controlled by adjusting the voltage difference between the cathode and first dynode. All units deliver their signals to a common amplifier. The characteristics of this scintillation detector are significantly better than those of the BF3 counters for time-of-flight measurements. In particular, time uncertainties are negligibly small; the pulse rise time is less than 0.1 psec and the neutron capture time is even less for the energies involved. The over-all detection efficiency is also somewhat improved, being about 1% a t 1 kev. At the present time, the efficiency cannot be significantly raised by increasing the thickness of the scintillating discs because of their low optical transmission. Again, because of lack of transparency, the counting rate versus bias curve obtained with these scintillators has no definite plateau. In spite of this drawback, y-radiation is effectively avoided by using a relatively high bias, and stability in neutron detection efficiency is achieved by employing an especially steady high voltage supply. Several glass and glass-like scintillators have been developed recently44 which will have special applications, but no single material seems to be the answer to the epithermal neutron velocity spectrometer detector problem. r4R.J. Ginther, I R E Trans. on Nuclear Sci. NS-7, 28 (1960); L. M. Bollinger, G . E. Thomas, and R. J. Ginther, Rev. Sci. Znsti. 30,1135 (1959); P. A. Egelstaff, Nuclear Znetr. 1, 197 (1957).

2.2. DETERMINATION

OF MOMENTUM AND E N E R G Y

535

A detector which realizes an exceptionally high counting efficiency a t the expense of a sensitivity to background radiations is the boron-loaded liquid scintillation counter.46The scintillation from the disintegration of BIO after capture of a neutron is detected by a photomultiplier. The principal technical problems encountered in using a liquid detector of this kind are caused by the small magnitude of the light signal produced; the pulse given is equal to that from electrons having an energy of only

FIG.28. A boron-loaded liquid scintillation counter for neutron transmission measurements by the time-of-flight method. [From L. M. Bollinger, Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1956 4, 47 (1956).]

40 kev. The most serious difficulty attributable to the small signal is the broad pulse-height distribution obtained for slow neutrons, a distribution which makes discrimination against y-rays and fast neutrons somewhat ineffective. A lesser problem is that the signal produced by slow neutrons is in the same pulse-height range as that caused by photomultiplier dark current noise. A counter design which eliminates the latter of the above difficulties and minimizes the former is illustrated in Fig. 28. The counting solution used is 4 gm/liter of phenylbiphenylyloxadiazole and 20 mg/liter of POPOP (1,4-bis-[2-(5-phenyloxazolyl)]-benzene)in equal parts of toluene and enriched methyl borate (70% B'O). This solution is contained in a sheet steel vessel having the dimensions 14 by 18 by 2.5 cm. The interior surface of the vessel is coated with vitreous enamel, thus forming a good 46L.

M. Bollinger and G. E. Thomas, Rev. SFi. Instr. 28, 489 (1957).

536

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DETERMINATION OF FUNDAMENTAL QUANTITIES

reflecting surface. The counting liquid is viewed by four Dumont 6292 photomultipliers which are selected for their good resolution and low noise. These tubes are paired by joining the anodes of those diagonally opposite, the signal from each pair being independently amplified. The gains of the photomultipliers are equalized by an adjustment of the voltage across each tube. The electronic system used requires a coincidence between the two amplifier outputs, thus eliminating noise counts, whereas counting of y-ray background is minimized by requiring that the sum of amplifier outputs be in a given pulse-height range. Electronic difficulties caused by a high noise counting rate are reduced by cooling the counter to 10°C. The operating characteristics of the counter shown in Fig. 28 and others of a similar design have been found to be very satisfactory for use in transmission measurements with the Argonne fast chopper. The pulseheight distribution obtained with slow neutrons is approximately triangular in shape, the full width a t half maximum being about 65%. The over-all counting efficiency E of the system is given approximately by t = 1.OE-O.lafor neutrons in the energy range 10 < E < 10,000 ev; thus, it has an efficiency of about 40% a t 1 kev. The mean neutron capture time of the solution is about 0.6 psec for a medium of infinite extent, and for the geometry used it is reduced to about 0.4 psec a t 1 kev. The background rate of the counter, when heavily shielded on all sides, is about 8 counts per second; an effort was made to minimize the counting of y-radiation by making the counter thickness small. It should be emphasized that, for reasons of background, the boron-loaded liquid counter may be used in the straightforward way described above only when the source of neutrons is essentially free of y-radiation; this condition can be achieved in some time-of-flight experiments because of the separation in time of y-rays and neutrons. Another disadvantage of this liquid scintillation counter is its slow response time, making it useless for the highest resolution work. For high resolution work using short time gates, it is absolutely necessary to use a detector with an extremely short response time. The only two detectors known which have a response time shorter than sec are scintillation detectors4@ and the recently developed solid state detectors.47 * The development of the solid state detector is so new that it has not been used as a regular detector in time-of-flight studies, although it *Refer to Section 1.8.1. 46 See, for example, K. Siegbahn, ed., “Beta- and Gamma-Ray Spectroscopy.” North-Holland Publ., Amsterdam, 1955. ‘7 S. S.Friedand, J. W. Mayer, J. M. Denney, and F. Keywell, Rev. Sci. Instr. 31, 74 (1960).

2.2.

DETERMINATION OF MOMENTUM A N D E N E R G Y

537

has been used to measure the fission cross section of U233.48 It can, however, be used for time-of-flight studies by coating a piece of Si with a substance which absorbs neutrons and then detecting the charged particle which is emitted. Unfortunately, such detectors will be extremely inefficient because the coating which absorbs the neutrons must be extremely thin to enable the charged particles to emerge. However, for particular applications, such as studies of the fission process, the solid state detectors promise to be very valuable. For several years scintillation detectors have been used in time-of-flight systems. The scintillators detect y-rays or recoil charged particles. The system used with the Columbia synchrocyclotron velocity spectrometer employs 16 NaI crystals to detect the y-rays emitted by a sample when the neutron is captured. If a nonresonant detector is desired, then a slab of boron can be inserted in the sample position, and the 0.48-Mev y-ray emitted when the neutron is captured by BIO will be detected by the crystal. The window of a single channel pulse-height analyzer is centered near 0.480 Mev to eliminate the y-rays due to capture in other substances. Such a system has been described in detail by Rae and B o ~ e y and * ~ is used regularly with several time-of-flight spectrometers. A property of organic scintillators has recently been discovered which has greatly improved the signal-to-background ratio. The rise time for the pulse resulting from the passage of a heavy charged particle through the scintillator is from 50 to 400 mpsec, whereas the rise time of the pulse resulting from an electron passing through the scintillator is from 6 to 12 mpsec. Thus events caused by heavy charged particles can be distinguished from events caused by light charged particles by making use of this difference in rise time. This method of separation of the events is called “pulse shape discrimination.” Fast neutrons are observed by detecting the recoil proton from a neutron proton collision in a scintillator. Most of the background in fast neutron work is caused by y-rays which give rise to electrons in the scintillator. Thus fast neutrons can be distinguished from y-rays. Brooks has recently published articles60 describing a simple system for separating the neutron from a y-ray. For neutrons with energies above 2 Mev, the discriminator can’be made almost perfect. For neutron energies below 2 Mev the discrimination is not as good. A choice must be made between the signal-to-background ratio desired and the loss of intensity that can be tolerated. If the ratio Melkonian, Bull. Am. Phys. Soc. [ 2 ] 6, No. 1 (1961). E. R. Rae and E. M. Bowey, Proc. Phys. SOC.(London) A66, 1073 (1953). 6oF. D. Brooks, Nuclear Instr. and Methods 4, 151 (1959); F. D. Brooks, R. W. Pringle, and B. L. Funt, IRE Trans. on Nuclear Sci. NS-7, 35 (1960). 48E. 49

538

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

of the signal from the neutron to the signal from the y-ray is made large, then many of the fast neutrons will not be detected. Obviously the intensity will therefore be reduced. Order of magnitude improvements in the peak height to background value have already been obtained in the study of inelastic neutron scattering using fast time of flight. For fast time-of-flight studies, the scintillation detectors are no different from those used for scintillation spectroscopy except that special care must be taken with the response time of the associated electronics. Such scintillators have been adequately described in the literature and will not be discussed here.61 2.2.2.2.3.5. Applications of Epithermal Neutron Spectrometers. Epithermal neutrons are used primarily to study resonances in the compound nucleus for nuclei of intermediate and heavy mass. The variation of the slow neutron cross section with energy gives a picture of the levels of the compound nucleus near the binding energy of the neutron which cannot be obtained in any other way. Time-of-flight studies for neutrons with energies less than 10 kev provide the only method which gives high enough resolution for the detailed investigation of these levels. The level spacing of the compound nucleus for A > 100 near the binding energy of the neutron is usually less than 1 kev and sometimes less than 1 ev, so experiments which use particles of several kev energy cannot hope to resolve these levels. The measured neutron cross sections in the epitherma1 region are not only of great value for the study of nuclear structure but are also very important for their application to reactor physics. We will not deal with reactor physics applications, which are treated elsewhere,62but with the application to the study of nuclear structure. However, several millions of dollars have been saved in the design of production and experimental reactors because our knowledge of neutron cross sections at thermal energies is so extensive. One of the first contributions of neutron time-of-flight spectroscopy to the theory of nuclear structure was the experimental verification of the Breit-Wigner theory for isolated resonances. A plot of the experimental total cross section of cadmium,53which varies in magnitude from 5.7 to 7800 barns, is shown in Fig. 29 in the energy interval from 0.008 to 10 ev. The solid curve is the theoretical curve for EO = 0.178 ev, I’ = 0.114 ev, uo = 7800 barns, and u p = 4.9 barns, where EOis the resonance energy, 61 See, for example, K. Siegbahn, ed., “Beta- and Gamma-Ray Spectroscopy.” North-Holland Publ., Amsterdam, 1955. E. P. Wigner and A. Weinberg, “The Physical Theory of Neutron Chain Reactors.” Univ. of Chicago Press, Chicago, 1958. azL. J. Rainwater, W. W. Havens, Jr., C. S. Wu, and J. R. Dunning, Phys. Rev. 71, 65 (1947).

2.2.

DETERMINATION OF MOMENTUM AND E N E R G Y

539

ANL-fc7

I 3.01

0.I

I.o

FIG.29. Experimental and theoretical cross sections of the neutron resonance in the compound nucleus Cd1I4at a neutron kinetic energy of E = 0.178 ev.

r is the total level width, uois the cross section at the resonance, and up is the potential scattering cross section. The theoretical curve fits the experimental curve to within the experimental accuracy, even though the cross section varies by a factor of about 1500 and the energy varies by a factor of 1250. There are few processes in physics in which the theoretical curve fits the experimental data over so wide a range for both the ordinate and the abscissa. Several more experimental of the validity of O 4 R. E. Wood, Phys. Rev. 104, 1425 (1956); R. K. Adair, C. K. Bockelman, and J. M. Peterson, Phys. Rev. 76, 308 (1949); J. M. Peterson, H. H. Barschall, and C. K. Bockelman, Phys. Rev. 79, 593 (1950).

I

I

I

'

1.2

"23@

.:

J

...... I

I000

FIG.30. Backmound mlhtmntd noiintsl ncr ohnnnd fnr at the detector position waa viewed by large plastic scintillation detectors to detect (n,?) capture ?-rays. For the D T curves a sample having c1 = 475 bams/atom was also in a transmission setting. The plots show counts per 0.1 wec detection interval width for a 35.37-meter neutron path. [From J. L. Rosen, J. S. Desjardins, J. Rainwater, and W. W. Havens, Jr., Phys. Rev. 118,687 (1960).] ~

+

~~

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

54 1

the application of resonance theory to the analysis of experimental data have been made and this theory is now regularly assumed to hold in determining the parameters of a particular resonance level. Energy level spacing and distribution of energy level spacings. A recent set of data showing the slow neutron resonance levels in the compound nucleus U2a9 in the energy interval from 100 to 1300 ev is shown in Fig. 30. The most striking feature of these data is the sharp peaks which occur. Each of these peaks represents a resonance level in the compound nucleus.

60/

FIG. 31. The sum E = 0 to E of the number of observed resonances in Uz88 to energy E. The slope of the curve yields the average level spacing d. The number of resonances contained in the indicated energy subintervals are shown in brackets. [From J. L. Rosen, J. S. Desjardins, J. Rainwater, and W. W. Havens, Jr., Phys. Rev. 118, 687 (1960).]

The actual positions of the levels are not of particular theoretical interest at the present time because the complexity of the nuclear many-body problem does not allow the observed positions of the levels to be compared with theory. However, the average energy level spacing and the distribution of energy level spacings are of theoretical interest. The average energy level spacing is obtained by plotting a histogram of the integral distribution of energy levels vs energy as shown in Fig. 31 for the UZa8 data given above. The slope of the curve provides the average level density or the inverse quantity, the average level spacing D. The dis-

542

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

tribution of level spacings is obtained by determining the number of level spacings which are within a convenient energy interval AE, and plotting a histogram of the number of energy levels per energy interval AE versus the neutron energy. The convenient energy interval for the U238data shown in Fig. 30 is 5 ev and the dist,ribution of the 54 energy levels observed is shown in Fig. 32.

S

P

A

C

I

N

G

( c v )

FIG.32. The distribution of leveI spacings per 5 ev interval for the first 54 level spacings in U’38 from 0 to 1000 ev. The smooth curves represent the “repulsion” formula proposed by Wigner and an exponential function corresponding t o a random distribution of spacings. [From J. L. Rosen, J. S. Desjardins, J. Rainwater, and W. W. Havens, Jr., Phys. Rev. 118, 687 (1960).]

The energy level spacing and the distribution of energy level spacing were investigated theoretically by Bethe some time agob5and interest has recently revived.56The experimental results on the average energy level spacing and the distribution of energy level spacings is in good agreement with the trends predicted by theory, but the theory is not sufficiently well developed to make detailed comparisons. 5 7 Neutron spectroscopy gives information on the details of the decay H. Bethe, Revs. Modern Phys. 8, 82 (1957). C. Block, Phys. Rev. 93, 1094 (1954); T. D. Newton, Can. J . Phys. 34,804 (1956); N. Rosenzweig, Phys. Rev. 108, 817 (1957); J. M. B. Lang and K. J. LeCouteur, Proc. Phys. SOC.(London) A67, 486 (1954); S. Blumberg and C. E. Porter, Phys. Rev. 110, 786 (1958); N. Rosenzweig, Phys. Rev. Letters 1, 24 (1958); C. E. Porter and N. Rosenzweig, Ann. Acad. Sci. Fennicae, Ser. A , V I . Physcia No. 44 (1960). 6’See J. A. Harvey, Proo. Intern. Conf. on Nuclear Structure, Kingston, Canada, 1960, p. 659. University of Toronto Press, Toronto, 1960. 66

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

543

of the compound nucleus. The neutron width rnis related to the phase of the nuclear wave function a t the nuclear boundary. Early theories of nuclear structure predicted that the neutron width of a resonance would be directly proportional to the square root of the energy of the resonance. However, wide variations from resonance to resonance are observed in the neutron widths and the fidependence does not hold in detail. A new concept had to be introduced to correlate the experimental data with

ev)llZ FIG.33. The distribution of reduced width amplitudes (F%a)i'zper 0.5 interval for 54 resonances in U'-'38from 0 to 1000 ev. The smooth solid curve represents the Porter-Thomas distribution ( V = 1) while the dashed curve corresponds to a random distribution of reduced widths ( v = 2 ) . [From J. L. Rosen, J. S. Desjardins, J. Rainwater, and W. W. Havens, Jr., Phys. Rev. 118, 687 (1960).]

the theory. This new concept assumed that the phases of the nuclear wave function a t the nuclear boundary for different resonances were statistically independent and random. After this new theoretical assumption was introduced, it was possible to compare the observed distribution of rnwith the distribution to be expected on this random phase basis. A comparison is shown in Fig. 33 of the distribution of neutron widths for U238with a Porter-Thomas distribution, which is the name given to the random phase distribution,

544

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

The parameter Fn0is the neutron width reduced to a standard energy, usually 1 ev, to remove the l / d E dependence expected theoretically, i.e., rn0 = I',/dE. The agreement between the theoretical curve and the experimental data is very satisfactory. The total y-ray width I??, which is the probability for a compound nucleus to decay by emission of a y-ray, has been measured for many nuclei for transitions from a highly excited state of the compound nucleus t o the ground state of the compound nucleus. The total y-ray width is expected to be constant for a particular spin state in a specific isotope. Sufficient y-ray widths have recently been obtained6s so that detailed comparisons can be made between the experimental data and some of the predictions of nuclear theory. The agreement is only fair. Partial y-ray widths to a particular level in the compound nucleus have been observed recently and found to differ markedly from level to This variation has stimulated interest in measurements of partial y-ray widths.6DInvestigation of partial y widths will undoubtedly give us a better understanding of the y-decay process after neutron capture. The strength function, defined as S = F / D , where 2 is the average reduced neutron width and D is the average level spacing, is of theoretical interest since it represents the penetrability of the nuclear surface. This strength function is not a property of the nuclear energy level system but of the nuclear surface itself and is therefore of great importance in connection with the cloudy crystal ball model of the nucleus. The experimental values of the strength function for many isotopes have been compared with those predicted by various cloudy crystal ball models of the nucleus, and modifications of theory have resulted which give us a better picture of the nucleus. 2.2.2.2.4. FASTNEUTRON TIME-OF-FLIGHT TECHNIQUES. Slow neutron time-of-flight techniques have been used for many years, but the extension of these techniques to higher energies is just developing. The delay in this development is primarily technical. The flight times to be measured for fast neutrons are in the range of 10-7 to 10-9 sec, a range which has only recently become accessible because of developments in photomultipliers, scintillators, and amplifiers. Also, time-of-flight techniques have been slow to develop in the fast neutron region because monoenergetic sources of neutrons are readily available from Van de Graaff, Cockcroft-Walton, J. S. Desjardins, J. L. Rosen, W. W. Havens, Jr., and L. J. Rainwater, Phys. Rev. iao, 2214 (1960). s9 L. M. Bollinger, R. E. CotB, and T. J. Kennett, Phys. Rev. Letters 3, 376 (1959). Eo J. R.Bird, M. C. Moxon, and F. W. K. Firk, Nuclear Phys. 13,525 (1959);L. M. Bollinger and R. E. CotB, Bull. Am. Phys. SOC.[2]0, 294 (1960);D.J. Hughes, H. Palevsky, R. E. Chrien, and H. H. Bolotin, ibid. p. 295.

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

545

and cyclotron accelerators, and therefore it is seldom necessary to use a time-of-flight analysis. Recently, however, interest has focused on the study of fast neutron spectra, particularly those resulting from inelastic neutron scattering, and it is primarily because of this problem that timeof-flight techniques have been developed. Interest in these spectra comes from the ever-increasing demand for good data on nuclear energy levels to check models of the nucleus and from the fact that inelastic scattering cross sections are needed for the design of intermediate and fast reactors. A study of the inelastic scattering of neutrons is particularly interesting in the study of the energy levels of heavy nuclei. To study heavy nuclei with charged particles, the energy of the particle must be sufficiently high to have a high probability of penetrating the barrier. When the energy of the particle is this high, the competition of various reaction processes may render the determination of the level structure extremely difficult. Neutron scattering experiments may be carried out at energies where inelastic scattering dominates other processes, and therefore the interpretation of the results should be very much simpler. Time-of-flight techniques developed for neutron spectroscopy have also found valuable application in charged particle spectroscopy. For instance, time-of-flight spectroscopy in conjunction with magnetic spectroscopy makes possible distinctions between particles of equal momentum but different mass. It has also been found profitable to use time-of-flight spectroscopy in order to eliminate background which comes from unknown sources. This is equivalent to applying a modulated signal to a particular device and using a detector which is tuned to the modulation frequency to eliminate noise. The relationship between the flight time and energy of the neutron for the flight path d measured in meters is t = 72.3 d / l / E , where t is measured in millimicroseconds, and E is in MeV. Table I1 gives the energy TABLE 11. Energy Resolution for Different Time Spreads

At(mpsec)

0.1

0.3

1.0

3.0

10.0

1 2

0.00087

0.0045

0.0017

0.009

0.0035

0.018

0.14 0.29 0.58

0.87

4

0.028 0.056 0.11

~

1.74 3.5

spread for the flight path of 1 meter corresponding to the given energy in a given uncertainty in time At. From this table, it is clear that, to obtain resolution at high energies, time resolutions approaching the limitations

546

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

of presently available techniques must be used. At lower energies the time resolution is not important. In fact, for low energies the beam homogeneity and consequently the energy spread of the primary neutron beam is the important factor in determining the resolution. Thus if the spread in energy of the primary neutron beam is AE, then the spread in the energy of any neutron group excited by inelastic scattering will also be AE if the yield curve for exciting a particular level does not vary appreciably over this energy interval. Thus the conditions for optimum resolution for high-energy groups (corresponding to the excitation of levels just above the ground state of the residual nucleus) are different from those for low-energy groups (high-lying levels). For the lower end of the fast neutron spectrum a thin target is required and a short neutron flight path may be used, whereas for the higher energy end of the spectrum the target need not be thinner than the energy resolution expected from the fast time-of-flight measurements and the flight path should be as long as possible. Van de Graaff, Cockcroft-Walton, and cyclotron accelerators have been used for fast neutron time-of-flight studies. In all of these accelerators great care must be exercised to assure that the charged particle beam hits the neutron producing target for a very short period of time. In the Van de Graaff and Cockcroft-Walton machines, which usually accelerate a beam continuously, the beam must be modulated by an external device. The charged particle beam can be deflected past an aperture in the beam pipe of the accelerator by a pair of parallel plates to which an rf voltage has been applied. The beam deflection system used at Los Alamos is of this type and is shown schematically in Fig. 34. A typical value for the rf frequency is 4 mc/secel which gives a beam pulse of about 2 mpsec duration and 250 mpsec between pulses if the beam is forced to traverse the slit once in an rf cycle. In a cyclotron the phase bunching which results from the normal operation of the cyclotron is usually of a few millimicroseconds duration. This self-bunching has been used to form the pulse for fast timeof-flight studies. 6 2 When dealing with times as short as a few millimicroseconds, it becomes necessary to determine the exact time a t which the burst occurs. I n principle this can be determined from the rf deflection pattern, but in practice L. Cranberg and J. S. Levin, Phys. Rev. 105, 343 (1956). H. H. Landon, A. J. Elwyn, G. N. Glasoe, and S. Oleksa, Phys. Rev. 112, 1192 (1958); C. 0. Muehlhause, S. D. Bloom, H. E. Wegner, and G. N. Glasoe, ibid. 105, 720 (1956); R. Grismure and W. C. Parkinson, Rev. Sci. Znstr. 28, 245 (1957); G. F. Bogdanov, N . A. Vlasov, S. P . Kalinin, B. V. Rybakov, and V. A. Siderov, Intern. Conf. on Neutron Interactions with the Nucleus, New York, 1957. Columbia University Report CU-175 [U.S. Atomic Energy Commission Report TID-7547 (1957)l. 6*

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DETERMINATION O F MOMENTUM AND EN ERG Y

this is done by calibrating the time scale with a known structure which is readily identifiable, such as the y-rays from inelastic scattering, the elastically scattered neutrons, or the inelastically scattered neutrons from prominent levels which are well known from other studies. Another technique for determining the start time, which is particularly useful for fast neutron time-of-flight studies, is the associated particle method. This method is best described by using the reaction H3(d,n)He4for the production of neutrons. I n this reaction a neutron and an alpha particle are 390 Crn

8.3 Crn

3 C r n target

Scalterer

Defleclor plales

Plastic scinlillalor

Mu metal shield RCA 5519 photomultiplier

\

FIQ.34. Schematic diagram of physical layout of apparatus for measuring the inelastic scattering of neutrons by time of flight. [From L. Cranberg, Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1965 4, 43 (1956).]

produced a t exactly the same time. The pulse from the alpha particle detector, which is close to the source, gives the start pulse for the timeof-flight measurement. Although the methods of establishing the start time differ markedly, all of them seem to be limited in accuracy to a few millimicroseconds. 2.2.2.2.4.1. Measurement of the Flight Time. Numerous devices are in use for the measurement of transit times of the millimicrosecond range. These include delayed coincidence arrangements with one or more channels and a special version of this principle embodied in a device called a

548

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

chronotron. Lefevre and RussellG3have developed a chronotron which has gate widths of the order of 0.5 mGsec, which are stable to 10-4 mpsec. The systems usually used are not as complicated as the chronotron. A time-tovoltage conversion circuit usually suffices, and the pulse distribution is then fed into an ordinary commercial multichannel analyzer. In practice it is not the electronic timing system that limits the resolution which can be obtained but the size and jitter of the detector. 2.2.2.2.4.2. Detectors for Fast Time of Flight. Most of the detectors used for fast time-of-flight studies have been liquid or plastic scintillators, although the recent development of the solid state detector (see Section 1.8.1) may furnish another device which can be used as a detector for fast time-of-flight studies. At present the detector is the component which limits the time resolution in fast time-of-flight spectroscopy. In order to detect a large number of neutrons, it is desirable to have the detector as large as possible. However, if the detector is large, the difference in time between a neutron arriving a t the point in the detector nearest to the source and one arriving a t the furthest extremity of the detector can be large compared with the resolution time desired. For example, if a plastic scintillator which has a 2-cm length in the direction of the beam is used as detector, a 1-Mev neutron takes 1.4 mpsec to traverse the detector. This means that a neutron detected a t the front of the detector will be detected 1.4 mpsec before a neutron detected a t the rear of the detector, thus introducing a 1.4-mpsec time spread. In order to obtain better time resolution, it is necessary to use smaller detectors. Since the intensity decreases in direct proportion to the volume of the detector, very small detectors cannot be used with source intensities presently available. In order to improve fast time-of-flight spectroscopy, it is clearly necessary to increase the yield of pulsed neutron sources. by several orders of magnitude. 2.2.2.2.4.3. Recent Improvements in Fast Time-of-Flight Equipment. The effective yield of pulsed neutrons has recently been increased in Van de Graaff accelerators by pulsing the source of charged particles in the high voltage terminal of the machine, which allows the pulsed beam intensity to be increased because the average current accelerated remains low. The background is also decreased because ions are not accelerated except a t the times they are wanted. Another recently developed method for increasing the pulsed intensity from accelerators is the magnetic bunching system, which is used in conjunction with a regular fast time-of-flight system like that described by Cranberg.64 H. W. Lefevre and J. T. Russell, Rev. Sn'. Instr. 30, 159 (1959). s4L. Cranberg, Bull. Am. Phys. SOC.[ Z ] 6, No. 1 (1961). tj3

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DETERMINATION OF MOMENTUM AND ENERGY

549

The method used for the magnetic bunching originally suggested by Mobleya6is shown schematically in Fig. 35. The group of particles labeled (1) in Fig. 35 emitted by the pulsed accelerator is traveling with a velocity v , extends laterally over a distance d, and has a width w. If a target is placed in the path of this group of particles, the time At the group will take to strike the target will be At = d/v. For a 1-Mev beam which has a time spread of 2 X 10-9 sec in a regular time-of-flight arrangement, the lateral extension of the beam is 13.8 em. The group of particles is injected into a magnetic field which is shaped to cause the particles a t a larger disnEFLECTING PLATES

CHOPPING SLITS

TARGET

FIG.35. Schematic diagram of magnetic bunching system to be used with standard fast time-of-flight beam pulsing system.

tance from the center of curvature of the path of the particles in the magnet to travel on the circle of radius Rzwhile those particles closer to the center of curvature travel along a path of radius R1.If the path difference is exactly d in a deflection of 180°, the particles will leave the magnet a t exactly the same time. The geometrical width of the beam will be exactly the same as when it entered the magnet. Thus, in principle it is possible to shorten the pulse in time without loss of intensity, thereby increasing the burst intensity. The magnetic bunching system gives an order of magnitude improvement in the intensity for the same time-of-flight resolution, or conversely gives an increase in the time-of-flight resolution of a factor of 10 for the same intensity. The use of a time-of-flight system to eliminate extraneous background 6sR. C. Mobley, Phys. Rev. 88,260 (1952).

550

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DETERMINATION OF FUNDAMENTAL QUANTITIES

is best illustrated by its recent use in the study of neutron polarization.66 The time-of-flight system is used to separate those neutron events of interest from a variety of background events. The background can be separated into three categories: (1) time-independent events, e.g., cosmicray counts, electronic noise pulses, and y-rays from neutrons which are moderated and radiatively captured in the general vicinity of the counter; (2) events which occur a t a given time within the rf cycle but at a different time from a neutron event of interest, e.g., neutrons or y-rays produced

I

r=o

CHA "€1

NUMBER

FIQ.36. Time-of-flight spectrum at 20" from the reaction Beg(d,n)B'O with a bombarding energy of 1.85 MeV, a fight path of 150 cm, and channel width of 0.594 X sec. Detector bias for recoil protons in t h e plastic scintillator was about 800 kev, and the deuteron burst width was 1.1 mpsec. The time scale proceeds linearly from left t o right,. The time scale can be made t o run in either direction in this as well as in any vernier instrument. [From H. W. Lefevre and J. T. Russell, Rev. Sci. Znstr. 30, 159 (1959).]

by the beam on the target or surrounding equipment; (3) events which occur at such a time within the rf cycle as to arrive a t the detector a t the same time as the event of interest. The time-of-flight system will decrease the background from time-independent events by a factor which is the ratio of the time duration of the pulse to the time between pulses. It will eliminate the background from events which occur a t a given time in the rf cycle but at a time different from the neutron events of interest. It will not, of course, eliminate the background from those events which occur at the same time as the event of interest. Time-of-flight as a background J. A. Baicker and K. W. Jones, Nuclear Phys. 17, 424 (1960); L. Cranberg, Phys. Rev. 114, 174 (1959).

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DETERMINATION OF MOMENTUM AND ENERGY

551

eliminator has been used in many studies.67Neutron time-of-flight data are improved considerably by using the pulse shape discrimination method developed by Brooks for the separation of pulses due to y-rays from those due to heavy particles in a scintillation detector.6s This system is described in the section on neutron detectors. 2.2.2.2.4.4. Applications of Fast Time-of-Flight Spectroscopy. Fast neutron time of flight has been used to study individual levels in nuclei and

6000

I‘

I

I

I

I

E LAST I C A L LY SCATTERED 7

‘-1

-

CHANNEL NUMBER INCREASING T I M E OF FLIGHT

FIG.37. Time spectrum obtained a t 90 degrees for neutron scattering at 2.45 Mev from gold. The “sample out” background has been subtracted. [From L. Cranberg and J. S. Levin, Phys. Rev. 103, 343 (1956).]

nuclear temperatures. An example of its use for the study of individual levels is shown in Fig. 36, taken from the work of Lefevre and Russell, where data obtained on the Beg(d,n)B’O reaction illustrate the resolution that can be achieved. When a heavy nucleus is studied by inelastic scattering, the level density is so large that it is impossible to resolve individual levels. Figure 37 shows the spectrum obtained by Cranberg and Levinegfor gold. As expected, the curve shows no peak due to nuclear energy levels and the results can be interpreted in terms of a nuclear temperature. However, 67 See, for example, Conf. on Neutron Physics by Time-of-Flight, Gatlinburg, Tennessee, November, 1956. Oak Ridge National Laboratory Report ORNL-2309

(1956). 88 F. D. Brooks, Nuclear Znstr. and Methods 4 , 151 (1959). e9L. Cranberg and J. S. Levin, Phys. Rev. 103, 343 (1956).

552

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

new phenomena arise when new techniques are applied to old problems. Figure 38 shows the time spectrum of 2-Mev neutrons scattered from UZa8. Unexpectedly, the spectrum of U238 shows resonance structure. The large peak comes from the y-rays which are emitted when the neutron is captured; the small peak comes from structure in the UZ38 nucleus. It

d:l

Om

0.5

0.86

1.98

En(MEV)

FIG.38. Time spectrum of neutrons from the interaction of 2 Mev neutrons with Ups*.[From L. Cranberg, Proc. Intern. Conf. on the Neutron Interactions with the Nucleus, Columbia University Report CU-175 (1957).]

seems unlikely, both from general considerations of level density and the width of the observed peak, that a single level has been resolved. It is preferable to say that the levels centering about 1.98 Mev having a spread of 150 kev or less are excited with substantially higher cross section than adjacent levels. Fast neutron time-of-flight spectroscopy is capable of supplying more unique data on nuclear energy levels and nuclear temperatures. It is necessary to improve the resolution by an order of magnitude to obtain

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

553

the resolution necessary to separate low-lying nuclear energy levels. Because of the physical limitations of the system, this improvement can only be obtained by increasing the intensity by an order of magnitude. New developments in accelerator techniques seem capable of providing the necessary increase in intensity, so fast neutron time-of-flight studies should prove a fruitful field of research. 2.2.2.2.4.5.Cross-Section Measurements in the kev Region. An apparatus has been developed to measure neutron cross sections in the kev region70 which competes with fast neutron spectroscopic equipment at its lower energy range and with epithermal neutron spectroscopic equipment a t its upper energy range. The apparatus is described here because the techniques used are the same as those used in fast neutron time-of-flight measurements. Epithermal neutron spectroscopic equipment uses a continuous energy distribution of moderated neutrons and determines the energy by time of flight, whereas fast neutron spectroscopy usually makes use of monoenergetic neutrons produced by a Van de Graaff machine or a cyclotron. The apparatus developed for studies in the kev region is a hybrid of both of these techniques. The method employs a pulsed Van de Graaff machine to produce bursts of neutrons of about lo-* sec duration from the Li7(p,n) reaction. By suitable choice of target thickness and proton bombardment energy, a spectrum of neutrons is produced a t 0' to the proton beam for covering the kilovolt range. These unmoderated neutrons constitute the source from which neutrons of a specific energy are chosen by time-offlight measurements. Thus the Van de Graaff supplies only a limited spectrum of neutrons whose precise energy is determined by time of flight. The resolution does not depend on the energy spread of the neutron beam, as it does when the Li7(p,n) reaction is regularly used to produce monochromatic neutrons for cross-section measurements. The resolution is obtained by timing the neutrons over a measured path using ultra fast timing techniques. The intensity of the neutron source is fairly low compared with the intensity of the neutron sources used for epithermal neutron spectroscopy. However, because no very low-energy neutrons are produced, a high repetition rate can be used. Average counting rates are comparable with those obtained with other neutron spectrometers in the energy range from 1 to 40 kev. The stray neutron background for this apparatus is considerably less than for other neutron spectrometers, because no neutrons are emitted in the backward direction from the initial direction of the proton when the bombarding energy is not more than 40 kev above the threshold of the Li7(p,n) reaction. 70

W. M. Good, J. H. Neiler, and J. H. Gibbons, Phye. Rev. 109, 926 (1958).

554

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

The neutron bursts are produced by using a device which deflects the proton beam across a slit in the high-voltage terminal of an electrostatic generator. The pulsed proton beam is then accelerated and hits the target assembly. The voltage applied to the deflector has a frequency of 0.5 Mc. The resultant beam pulse has a time duration of less than see. For intermediate neutron energies, the detector that has been used is a 10-cm slab of Bl0 closely backed by a NaI(T1) crystal. A single-channel window is set over the full peak of the 480-kev y-ray from the Bl0(n,a,y)Li7reaction. The signal from the detector starts a time-to-pulse-height conversion circuit which is stopped by a signal from the target. The resultant voltage

I5

5

0 I

2

5

10

20

E, (kevl.

FIG.39. Total cross section of Ye9 as a function of neutron energy. [From W. M. Good, J. H. Neiler, and J. H. Gibbons, Phys. Rev. 109,926 (1958).]

is fed into a pulse-height analyzer to give the time-of-flight spectrum. The over-all resolution of the system described by Good et aZ.70 is 6 to 8 mpsec full width at half-maximum. Five different flight paths varying from 0.55 to 2.0 meters have been used to give the resolution desired for the particular energy neutron studied. An example of the results obtained in the measurement of the total cross section of Yes is given in Fig. 39. 2.2.2.2.5. TIMEANALYZERS FOR THERMAL AND EPITHERMAL NEUTRON SPECTROSCOPY. Time analyzers used in neutron spectroscopy range from a simple gate a t a variable delay time after the neutron burst to a complex electronic system equivalent to a modern computing machine. The complexity of the device depends on the speed required for the recording of the data, the counting rate, the repetition rate of the pulse, and the

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

555

number of discrete intervals into which one would like to divide the time interval over which data can be taken. The phrase ‘(time gate” is used constantly in neutron time-of-flight instrumentation and means an electronically controlled switching circuit which permits signals from a neutron detector to pass through when the time gate is on or open and which prevents such signals from being transferred when the time gate is closed. This “time gate” is also referred t o as a (‘channel.” The complexity of the analyzer increases rapidly as the number of channels in the analyzer increases. One of the simplest time analyzers was that used by Baker a n d ’ B a ~ h e r , ~ ~ who pulse-modulated the ion source of a cyclotron. In this analyzer, a delay circuit w&s triggered at time t = 0 when the neutron puke was formed. The delay circuit was adjustable from 0 to 10,000 psec after the start of the time gate which pulse-modulated the cyclotron. The width of the time gate was adjustable from 10 to 1OOOpsec. The time delay circuits and the time gates used in a simple analyzer of this type can be unsymmetrical univibrators or phantastrons. Such circuits are fairly standard and are described in textbooks on electronic^.^^ * A single-channel analyzer of the type used by Baker and Bacher recorded only a small fraction of the available data, so multichannel analyzers for this purpose were quickly developed by adding channels in a straightforward manner. However, this simple method of adding channels caused the electronic circuitry to increase linearly with the number of channels, so neutron spectroscopists then resorted to modern digital computer techniques for recording the data. Block diagrams for simple analyzers are described in the early papers on neutron s p e c t r o ~ c o p yMuch .~~ better circuits than those described in these early papers are now available due to the remarkable advances in electronic techniques. 2.2.2.2.5.1. Multichannel Analyzers for Low Counting Rates. If the counting rate is low enough so that there is a very small probability of detecting two neutrons from one burst, then the time-of-flight analyzer can be very simple. If a multichannel pulse-height analyzer is available, the simpIest system to use is one in which the time a t which the neutron is detected is converted into a pulse height and the pulse height is then recorded by the pulse-height analyzer.

* See also Vol. 2, Part 7. C. P. Baker and R. F. Bacher, Phgs. Rev. 69, 332 (1941). J. Millman and H. Taub, “Pulse and Digital Circuits.” McGraw-Hill, New York 1956. 7 s J . H. Manley, J. Haworth, and E. A. Luebke, Phys. Rev. 61, 316 (1940); C. P. Baker and R. F. Bacher, ibid. 69, 332 (1941); J. Rainwater and W. W. Havens, Jr., ibid. 70, 136 (1946); J. Rainwater, W. W. Havens, Jr., C.S. Wu, and J. R. Dunning, ibid. 71, 65 (1947). 71 72

556

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Almost all of the time-of-flight analyzers for millimicrosecond time-offlight studies operate in this manner. Several time-to-amplitude converters for the millimicrosecond and microsecond range have been described in the literat~re.'~ If it is desirable to gather data for a time interval which is large compared to the length of the individual channels, then it is sometimes difficult to construct a time-to-voltage converter which is linear over the whole range. However, it is always possible to break the total time interval

ClRCUlT

1'5 RING COUNTER

7 1 41

%IDELAY

LOO'S RING COUNTER 10

1st

srwc

I

RESET

FIG.40. Brookhaven analyzer. [From W. A. Higinbotham, Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1966 4, 56 (1956).]

up into several smaller intervals and construct time-to-voltage converters which are linear over a more limited range. If a multichannel pulse-height analyzer is not available, or if it is undesirable to tie up a multichannel pulse-height analyzer for the time-offlight work, then it may be desirable to construct a time analyzer using matrix circuits. The time-of-flight analyzer used at the Brookhaven National Laboratory, designed by Graham, Higinbotham, and Rankowitz and shown schematically in Fig. 40,is of this type. A start pulse from the photocell on a chopper or from the target of a pulsed accelerator a t time TOturns on the bistable trigger circuit and starts the "gated" 2-Mc oscillator. The oscillator frequency may be divided as shown to give F. Lepri, L. Mazzetti, and G . Stoppini, Rev. Sn'. fnstr. 26,936 (1955); W. Weber, C. W. Johnstone, and L. Cranberg, ibid. 26, 166 (1956); P. R. Orman, Nuclew Instr. 2, 95 (1958); P. R. Orman and F. H. Web, Proc. 6th Tripartite Instrumentation Conf., Chalk River, Canada. Report AECL-804 of Atomic Energy of Canada, Ltd., 1959.

2.2.

DETERMINATION O F MOMENTTJM .4ND ENERGY

557

channel widt,hs of 0.5 to 4 psec. Three ring c:ount>crsare employed. Thc first two ring counters are used to define trhe 100 channels. The t,hird ring is used t o delay the start of the sensitive period hy requiring that, a detector pulse be in coincidence with one of the stages of t,he slow ring. The delay may also be varied in steps of 10 channels by starting the “tens” ring counter in different configurations. The tens ring counter is made up of 11 bistable elements, ten of which are connected to the matrix. The eleventh stage eliminates any ambiguitJywhich might, arise due to cumulative delay in the several counting circuits. The sum of the numbers adjacent to the switches on the “tens” and the “hundreds” ring counters is the delay in unit#sof a channel width to the start of the 100 channel recording interval. If a pulse is received from the detector within the selected interval, the bistable trigger circuit is turned off and the ring counters store the channel number. Delay 1 is manually set to follow the last channel of interest and to precede the next To pulse. If a detector pulse has been received, the number stored in the ring counters is recorded. All circuits are reset by a pulse from delay 2. The frequency divider circuit and the units ring counter circuits are derived from the Chalk River scaler The slower ring counter circuits can be similar to those of Gatti.76 If an analyzer of the Brookhaven type were to be constructed a t the present time, it would be desirable to use recently developed beam switching tubes, such as the Haydu M-l0R rather than the ring counters. 2.2.2.2.5.2. Multichannel Analyzers f o r Higher Counting Rates. If the probability of detecting more than one neutron from one burst of the source is not small, then the time analyzer becomes much more complex. The starting system and time delay circuits can be the same as in the simple system, but the detection channels must have a much faster riseand-fall time and must be much more precisely timed than the detection channels used in analyzers which are not required to record more than one neutron per burst,. The detected neutron pulses occur randomly in time. If the edges of the successive time gates are not infinitely steep, some counts may be lost between channels or counted in two adjacent channels. Poor timing might, cause the channels to be unequal. The analyzers described in this section perform in a manner similar to the analyzers described briefly earlier. However, these circuits make use of B matrix in order to provide a large number of channels with a relatively small number of tubes. The first analyzer of this type was designed by deBoisblanc and McCol76N. F. Moody, W. 1). Howell, W. J. Battell, and R. H. Taplin, Rev. Sci. Znstr. 22, 439 (1951). 76 E, Gatti, Rev. Sn’. Znstr. 24, 345 (1053).

558

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DETERMINATION OF FUNDAMENTAL QUANTITIES

lum.” A block diagram of the analyzer is shown schematically in Fig. 41. The shaded portion a t the top left represents a face of the reactor. The chopper, which has four slits, is shown next to the reactor. The detectors are located 16 meters distant from the chopper. A light source and photomultiplier tube generate a pulse at To which starts the variable delay. The variable delay is a commercially manufactured unit which uses a temperature regulated phantastron circuit and is stable to 1 part in 1000. The 100 adjacent counting channels start at the end of the delay period.

FIG.41. 100 channel analyzer, Phillips Petroleum Co. [From W. A. Higinbotham, Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1955 4, 57 (1956).]

The crystal oscillator has a frequency of 8 Mc. The channel width selection unit contains binary frequency dividers to give channel widths of 0.5, 1.0, etc., psec. The electronic switch opens the gate circuit which permits pulses from the channel width unit to enter the units ring counter. The tenth stage of the units ring counter opens a gate permitting one pulse from the channel width unit to advance the tens ring counter one step. This insures that both ring counter circuits advance in phase. After one complete cycle of 100 steps, the gate is shut off and all circuits are reset. The two sets of ten leads form a raster or matrix with 100 intersections. A triple coincidence circuit is located at each junction. The third 77 D. R. deBoisblanc and K. A. McCollom, “ M T R Time of Flight Instrumentation,” Report No. IDO-16159 (1954). Obtainable from the U.6. Atomic Energy Commission, Office of Technical Services, Oak Ridge, Tennessee.

2.2.

DETERMINATION O F MOMENTUM A N D E N E R G Y

559

input is from the detector pulse shifter. Each coincidence circuit in succession switches the pulses from the detector into a channel counting circuit. Each coincidence circuit consists of two germanium diodes and three resistors. Each counting circuit consists of a glow discharge decade counter tube, a hot cathode gas tetrode, and a message register. A detector pulse shifter circuit is used in order to ease the stringent timing requirements mentioned a t the start of this section. Each pulse from the neutron detector-amplifier is delayed by an amount (not more than the duration of one channel) such that the corresponding pulse delivered to the coincidence circuits occurs in the middle of a channel interval. The circuit consists of a bistable trigger circuit. Pulses from the channel width unit, delayed by one-half channel interval, are connected to drive the trigger circuit in one direction. The amplifier lead is connected to drive it in the other direction. The normal state of the trigger circuit is determined by the pulses from the channel width unit. When a detector pulse is received, it throws the trigger circuit into the other state. The next pulse from the channel width unit throws the trigger circuit back to its normal state and a pulse is passed to the coincidence circuit. The figure shows circuits for recording the background counting rate and for recording all pulses received during the selected 100 channel interval. 2.2.2.2.5.3. Time-of-Flight Analyzers Using Digital Storage Techniques. The first analyzer to use digital storage techniques was described by Schultz et al.7aA fused quartz acoustic delay line was used as the storage or “memory” element. The analyzer has been used with a microwave cavity linear electron accelerator which produces neutrons by the (7,n) reaction. This accelerator can be synchronized from the time-of-flight circuits. This analyzer is shown in Fig. 42. The operation and circuits master pulse are very similar to those used by Hutchinson and S c a r r ~ tA. ~ ~ is generated automatically to start operation of the circuits. The master pulse modulates the 40-Mc oscillator and causes an acoustic pulse to travel through the 1000-hsec quartz delay line. The pulse is amplified and passed through a detector (or rectifier) circuit. It continues to recycle through the pass master pulse and reshaper units. The master pulse triggers a delay circuit which causes “channel marker” pulses and “clock” pulses to be generated for a time just less than 1000 psec. The master pulses pass through a frequency divider to trigger the liner accelerator. A detector pulse is time shifted so as to arrive in synchronism with a 500-kc channel pulse. The adder circuit causes a binary digit to be inserted in the corresponding channel. The binary digit continues to circu78 79

H. L. Schultz, G. F. Peeper, and L. Rossler, Rev. Sci. In&. 27, 437 (1956). G. W. Hutchinson and G. G. Scarrott, Phil.Mag. [7] 42, 792 (1951).

560

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DETERMINATION OF FUNDAMENTAL QUANTITIES

late through the quartz line, the amplifiers, the reshaper, and adder circuits until another detector pulse is received in the same channel. Then the adder circuit adds one to the binary number stored in the channel interval; 21° counts may be stored in each channel interval, since the binary digits have 0.2-psec spacing. The information may be displayed on a cathode-ray tube oscilloscope : the horizontal sweep circuit is triggered by the master pulse. A vertical sawtooth waveform, triggered by the channel pulses, is applied to the vertical deflection plates. And the “information out ” terminal is coiinect,ed to the control grid of the cathode-ray tube. Schultz has designed

,

CIRCUIT

INFWATIW

OUT W K C

PULSES

OLZECTOR PULSE

I

OUANnZEA

FIG. 42. Five-hundred channel analyzer with acoustic delay line storage. [Froin

w. A. Higinbotham, Proc. 1st Intern. conf. Peaceful Uses Atornic Energrj, Geneva, 1,956 4, 59 (1956).]

an analog printing circuit, which also uses about 40 tubes: a delay circuit, triggered from the master pulse, is slowly varied from 0 to 1000 wsec. It selects the first channel pulse for a number of cycles, then the secoxid, and so -on. While the delay selects one channel in this manner, the digits stored in the selected channel operate gas tetrodes which control a set of relays. The relay contacts switch a set of resistors in such a manner as to generate a voltage proportional to the number in the channel. The successive channel readings are plotted 011 a moving (.hart by a pen recorder. I n most cases it is not possible to trigger the iieutroii source from the analyzer circuits, so the quartz line method cannot! be used. The remainder

2.2.

D E T E R M I N A T I O N OF MOMENTUM ANI) E N E R G Y

56 1

of the instruments to be described here may lw triggered hy a Y’,,signal. Schumannsohas designed and constructed an analyzer with 1024 Channels which uses magnetic core storage. The time is measured by counting pulses from a n oscillator. The number stored in the selected position is transferred to an arithmetic circuit. One is added to the number and it is transferred back into the magnetic core memory. The analyzer takes about 16 psec to record each pulse. I t stores up t o 216 counts per channel. It includes a cathode-ray tube display and has circuits for printing out the contents of the memory at the end of a run. Control

-

I

I

- Memory

I

I

I I

i I

I I I I

Detector pulses

I

switch

j

Add 32

I I

I I

Monostab!: trigger ckt duration data period

I

Initiate storage cycle Signal from end of I delay reset address Dscater after last I channel of interest I

I I

FIG. 43. 1024 channel analyzer with magnetic core storage. [From W. A. Higinhotham, Proc. Inlprn. Con,f. F’pacpfifitl 1 T . w ~.!lornir EriPrqij, Geveva, 19.56 4, 59 (1956).1

Schumann’s analyzer is shown schematically in Fig. 43. The 7‘0 pulse turns on the trigger circuit, which is adjusted t o turn itself off after the last channel of interest and before the next T Opulse arrives. The oscillator is a free-running crystal oscillator. The trigger circuit permits oscillator pulses t o enter the divider circuit ( + 2 ) . The output of the divider is 2 Mc, which defines time channels of 0.5 psec duration (the divider may also be set to divide by 3 , 4, 6, 8, etc., for wider channels). The channel pulses from the divider circuit pass through a second electronic switch, which is normally closed, t o the “address scaler.” The address scaler is a 10 stage binary scaler which is normally permitted to “count” 1024 channel pulses and is then shut off. When a detector pulse occurs, it passes through an electronic switch, controlled by the trigger circuit, and R. W. Schumann, Rev. Sci. Instr. 27, 686 (3056).

562

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

stops the address scaler by opening the electronic switch which follows the divider. The address scaler remains in a state determined by the time when the detector pulse was received for 16 p e c or 32-channel periods. During this time the binary number stored a t the corresponding address or position in the magnetic core matrix is transferred to a set of trigger circuits; one is added to the binary number and the new number is transferred back into the memory. Thirty-two counts are added to the address scaler to compensate for the time it was off, and channel pulses from the divider are switched on again. The magnetic cores are arranged in arrays 32 X 32. There are 16 such arrays, giving capacity for 216 counts in each channel. The operation of magnetic core memories has been described in a number of places.” A combination of diode coincidence circuits and transformers is connected to the address scaler to direct the “read” and “write” signals with a small number of tubes. The analyzer is arranged so that the channels may be separated into four sets of 256. There are circuits for recording the background counting rate in an interval 32 channels wide. The accumulated data may be presented in analog form on a cathode ray tube display. At the end of an experiment the data will be automatically punched on teletype tape and plotted on a moving chart with a pen recorder. Fifty-eight tubes are used in the control and timing circuits; 104 tubes in the magnetic memory and arithmetic circuits; 40 tubes for the cathode ray tube display, the pen recorder, and the teletype punch operations. The time-of-flight analyzer with the shortest time gates, the smallest dead time and largest number of channels has recently been described by Hahn and Havens.82The analyzer consists of four parts: (1) an electrostatic temporary storage system; (2) a magnetic drum intermediate storage system; (3) a visual monitor using an analog oscilloscope display; and (4) an output system for transferring the drum contents to paper tape or punched cards. A simplified block diagram of the system is shown in Fig. 44(a), and a sequence diagram indicating the positions of the start pulse, the write cycle, and the read cycle with respect to magnetic drum position in Fig. 44(b). After initial synchronization, the system awaits the start pulse. This pulse initiates a train of 10-Mc pulses which cause the address generator and storage tube deflection circuitry to step sequentially through the 2000 addresses. After the writing cycle is completed, the system waits J. Rajchman, Proc. I.R.E. (Inst. Radio Engrs.) 41, 1407 (1953); RCA Rev. 13, 183 (1952); W. N. Papian, Proc. I.R.E. (Inst. Radio Engrs.) 40, 475 (1952). 82 J. Hahn and W. W. Havens, Jr., Rev. Sci. Instr. 31, 490 (1960).

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

r---i I

563

I ELECTROSTATIC ADDRESS

GENERATOR

PULSES FROM

Y

-i

ELECTIMSTATIC STORAGE

MAGNETIC STORAGE

I

\

START OF R E A D CYCLE

Fro. 44. (a) Simplified block diagram of 2000 channel analyzer; ( b ) sequence diagram indicating cyclotron start pulse, write cycle, and read cycle with respect to magnetic drum position. [From J. Hahn and W. W. Havens, Jr., Rev. Sci. Znstr. 31,490 (1960).]

until the drum is in the proper position, a t which time the drum sends the first of 2000 address advance pulses to the address and deflection circuitry and the 2000 addresses are sequentially interrogated. During the period in which data are taken, the number of counts in each channel is displayed on a cathode-ray tube. After the data-taking period is completed, the data are transferred from the magnetic drum to punched cards or punched paper tape. The general principles of electrostatic storage have been covered in the l i t e r a t ~ r e .The ~ ~ system utilizes focus-defocus storage. A slightly defocused beam is used for writing and a focused beam is used for reading. In order to store 2000 channels, it is necessary t o create a raster of 2000 dots, or addresses, on the surface of the tube face. Actually, the system 83 R. C. Williams and T. Kilburn, Pioc. Zns2. Elec. Engrs., Pt. ZI 96, 183 (1949); J. P. Eckert, H. Lukoff, and G. Smoliar, Proc. I.R.E. (Znst. Radio Engrs.) 38, 498

(1950).

564

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

generates a rectangular array of 64 by 32 dots, or a total of 2048 dots, of which only 2000 are used for the storage of information. The deflection pattern required is that of a stairstep. During the write cycle the entire raster is traced out in 200 psec (assuming 0.1-psec channels). Each step is triggered by a pulse from a 10-Mc gated oscillator. I n this manner a process of time to position conversion takes place in the storage tube. Whenever a neutron is detected, the corresponding spot on the tube is intensified and a count is stored a t th a t position on the phosphor. During the read cycle the same 2000 spots are traversed, with each step corresponding to an address advance pulse which comes from timing tracks on the magnetic drum. The rate during “read” is 7 or 8 psec per channel. Pulses from the timing track are fed to the dividers and stepping circuits so that the cathode-ray tube beam is deflected through 2000 positions exactly as it was previously deflected by the oscillator. At each position the beam of the cathode-ray tube is turned on (intensified). I n this way, the stored counts cause pulses to appear at the pickup plate and all storage positions on the cathode-ray tube are reset to the zero position. Whenever a pulse occurs a t the pickup plate, the number stored in the corresponding position on the drum is “read out.” One is added to the number and the new number is ‘(written”back onto the drum surface. Separate recording heads are used for the “read” and “write” operations. The storage is 213 counts in each of the 2000 time channels. Channel widths are of 0.05, 0.1, 0.2, 0.4, and 0.8 psec duration. At the end of the experiment, the data which have been stored on the drum are punched out on teletype tape a t the rate of ten five-digit octal numbers per second, or 200 sec read-out time for the 2000 channels. A recording oscillograph has also been connected to the analog output voltage to plot a spectrum of the results on two 48-in. lengths of paper simultaneously with the teletype print-out to permit immediate visual inspection of the results. A time-of-flight analyzer which can be used to record simultaneously the time of flight of the neutron and the spectra of the y-rays emitted after the capture of the neutron has been developed by Rae and Firk The analyzer uses a twin track magnetic and improved by Bird et dS4 tape to record both the time of flight of the neutron and the pulse height of the y-ray. I n this way the spectra due to the capture of a neutron into a number of different resonances are recorded simultaneously and can be reproduced individually by selecting the y-ray pulses corresponding to the appropriate group of timing pulses and playing them back into :i E. R. Rac and F. W. K. Firk, Nucleur f n s t . 1,227 (1957); J. R. Bird, J. R. Witters’ and F. €I. Wells, I R E Trans. on & * d e a r Sci. NS-7, 89 (1960).

2.2.

585

D E T E R M I N A T I O N O F MOMENTTJM A N D E N E R G Y

niult~ichannc~l pulse-hcight analyzer. The speed of the tape for recording is 1.5 cm/sec and the playback speed is 250 cm/sec. Therefore the analyzing time is a small fraction of the recording time; one analyzer may thus be used for the several recording systems. This type of system can obviously be extended by the use of multichannel tape to record several different events simultaneously. The device as presently designed is limited to analyzing those events in which there is a very small probability of obtaining more than one SAMPLE

/

PULSED NEUTRON BEAM /

AMPLIFIER AND PRE-AMPLIFIER

I

START' PULSE TO MAONETIC TAPE RECORDING HEAD 1 PULSE AMPLIT~DE a T ~ M E OF FLIQHT

a

TO MAQNETIC TAPE RECORDINQ HEAD 2 PULSE AMPLITUDE

a y-RAY

ENEROY

b

FROM RECORDINQ HEAD -2 (Y-RAY PULSES)

I

1..pl

MULTI-CHANNEL ANALY SER

FIG.45. (a) Block diagram for simultaneously recording the time of flight of neutrons and the energy of the corresponding -prays; ( b ) Block diagram for analyzing -pray pulses associated with a particular neutron resonance level.

count per burst. This limitation is not a severe one, since the experiments with which a multidimensional analyzer of this type is used usually have very low counting rates. If the pleasant situation is ever achieved where a high counting rate is a problem, then some intermediatje storage can be introduced to eliminate the difficulty. A block diagram of the recording section of the magnetic tape analyzer of Rae and Firk is shown in Fig. 45(a). The time of flight of the neutron is determined by producing a pulse whose amplitude is proportional to the neut,ron time of flight using a time to pulse-height converter. The time t,o pulse-height conversion circuit is arranged to cover a period of about

566

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

100 psec following a preset delay. This method is used so as not to place too stringent requirements on the linearity of the time to pulse-height conversion circuit. The time to pulse-height converter circuit is triggered by the signal pulse from the sodium iodide y-ray detector. The signal pulse is stretched and recorded on one channel of the tape and the timing pulse stretched and recorded on the other. The signal pulses are gated by the timing pulses before recording on the tape so th a t only pulses which occur during the 100 psec when the timing circuit is sensitive are recorded. A schematic diagram of the playback arrangement is shown in Fig. 45(b). On playback, the neutron time of flight of the resonance is selected by means of a single channel analyzer and the output of this analyzer is used to gate the y-ray pulses which are fed into the multichannel analyzer. In a n article of this nature, it is impossible to describe any system completely and the reader is referred for more detail to the two reviews of time-of-flight analyzers b y Higinbotham and by T s i t o v i ~ h . ~ ~

2.2.2.3. Crystal Diffraction.*? I n the use of crystal diffraction for measurement of neutron momentum, a particular neutron wavelength is measured or selected from a momentum distribution, by Bragg reflection a t a crystal surface. The principle of the measurement is expressed by the Bragg equation, which gives the neutron wavelength that will be diffracted when incident on a set of lattice planes (h, k , 1) a t a glancing angle 8, nX = 2d sin 8 (2.2.2.3.1) where n is the order of the reflection. I n this equation, d is the spacing of the planes denoted by the Miller indices h, k, 1, and for a cubic unit cell B 6 W. A. Higinbotham, Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1955 4,53 (1956);A. P. Tsitovich, Proc. 2nd Intern. Conf. Peaceful Uses Aloniic Energy, Geneva, 1958 14, 258 (1958).

t Editors’ note: The rough draft of this manuscript was received shortly before the untimely death of Dr. Hughes. The editors wish to express their deep appreciation to Dr. Harry Palevsky who has kindly checked the manuscript and supplied the list of general references.’-4 E. 0. Wollan and C. G. Shull, Phys. Rev. 73, 830 (1948). D. J. Hughes, “Neutron Optics.” Interscience, New York, 1954. 3 G. E. Bacon, “Neutron Diffraction.” Oxford Univ. Press, London and New York, 1955. C. G. Shull and E. 0. Wollan, Solid State Phys. 2, 137 (1956).

* Section 2.2.2.3 is by

D. J. Hughes.

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

567

of side ao, d is given in terms of the indices as (2.2.2.3.2)

This simple equation relates the glancing angle to the neutron wavelength, but it is also necessary t.o estimate the intensity in a consideration of actual instruments. The intensity of a particular reflect'ion, arising as it does from the coherent addition of amplitudes, is proportional to the square of the net amplitude of the wave function scattered from nuclei in a particular set of lattice planes. The net amplitude will be a definite function of the number and positions of the atoms in the planes because the waves scattered by the nuclei will have definite phase differences at the neutron detector. The amplitude per unit cell of the crystal for a particular reflection is given in terms of the coherent amplitudes of the individual nuclei by the structure factor of the unit cell,

where the summation is over the j atoms of the unit cell, located a t the coordinates xj, yi, z j , in units of the unit cell dimensions. The coherent amplitudes aj in the structure factor are less than the coherent bound amplitudes that would apply if the atoms of the crystal were held a t rest. As the atoms in a crystal are in constant motion, which is approximated by the Debye spectrum of lattice vibrations, the changing positions of the scattering nuclei introduce variabions in phase. The phase variations decrease the intensity of the Bragg reflections and produce diffuse scattering, the decrease being given by the Debye-Waller factor, which must be applied t o the coherent bound amplitude to obtain aj :

a 3. --

abounde-D(8in

UW*.

(2.2.2.3.4)

The constant D is a function of the temperature of the scatterer and the frequency spectrum of its lattice vibrations. The structure factor of Eq. (2.2.2.3.3) differs from the x-ray structure factor in one important respect-a respect th a t simplifies the neutron structure factor greatly. Because the scattering of slow neutrons has an isotropic angular distribution, the amplitude in Eq. (2.2.2.3.3)for each atom is a constant, independent of scattering angle. In the case of x-ray scattering, the atomic form factor, which appears in place of the constant nuclear amplitude, is a function of angle because the electrons th a t give rise to x-ray scattering are distributed over distances of the same order of magnitude as the x-ray wavelength, hence appreciable phase differences

568

2.

DETEHMINATION OF FUNDAMENTAL QUANTITIES

are introduced by changes in scattering angle. I n addition, the x-ray scattering from a n individual electron is not isotropic and an angular variation arises from this cause as well. Another difference between x-ray and neutron diffraction results from the fact that the absorption of neutrons is usually much smaller than it is for x-rays. As a result, measurement of wavelength by diffraction of neutrons can be performed either by observation of the scattered neutrons, as is done for x-rays, or of the neutronb removed from a beam by Bragg scattering, that is, by “transmission” measurements. I n considering the intensity scattered from a crystal, the depth of penetration of the neutrons into the single crystal grains must be taken into account. I n this respect, the results of the dynamical theory of x-rays are applicable, which show that the neutrons incident a t the Bragg angle penetrate to the order of lo-* cm before a 1/e decrease occurs. Concerning the reflectivity of the crystal, the same x-ray theory can also be applied with the result that the reflectivity is found to be complete for an angular range of about one second of arc a t the Bragg angle for a single neutron wavelength. This figure applies to an ideal crystal and is illcreased to several minutes for a real crystal because of the angular variation ot the crystal planes introduced by mosaic structure. The effect of mosaic structure on the reflectivity has an important bearing on the debign of moiiochromating crystals used to select narrow wavelength bands of neutrons, or for accurate measurement of neutron wavelength. The simplest method for selection of neutron wavelength by diffravtion is by transmission of a thermal neutron beam through a filter. The type of filter that concerns us here is based on the fact th a t in a polycrystalline material no Bragg reflection occurs for any neutron which the wavelength is longer than twice the largest lattice spacing. This result is obtained directly from Eq. (2.2.2.3.1)with n = 1 and d taken as its maximum value, d,, given by the smallest sum in the denominator of Eq. (2.2.23.2). In principle, the latter sum would be unity, for the (100) planes, but, if the structure factor, Eq. (2.2.2.3.3),should be zero for the (100) reflcctioii the planes of the next smaller spacing, probably the (110) planes, would correspond t o d,. For wavelengths longer than this cutof wavelength 110 Bragg scattering occurs an d the only remaining processes that remove neutrons from a beam are incoherent scattering and neutron reactions (usually neutron capture). For certain materials the capture and incoherent scattering are extremely low, and these materials are therefore suitable for neutron filters. The materials that have been used most extensively are graphite, beryllium, and beryllium oxide, with cutoff wavelengths 6.7, 4.0, and 4.5 A. The neutrons of wavelength beyond the cutoff are transmitted through

2.2.

DETERMINATION OF MOMENTUM A N D E N E R G Y

569

a block of filter mat erial with practically no loss in intensity, whereas the shorter wavelength neutrons are removed by Bragg scattering. Although the resultant beam is not monoenergetic, it has an extremely sharp cutoff wavelength, which for some purposes, particularly mirror reflection, is as useful as a monoenergetic beam. The accurately known cutoff wavelength in a filtered spectrum is often used for accurate calibration of slow neutron velocity selectors. Although filters are simple in operation and produce high neutron fluxes, the fixed cutoff and wide wavelength spread of the transmitted beam limit their use greatly. The usual type of wavelength selection is performed with a large single crystal, oriented so th a t the neutrons of a particular wavelength are scattered a t the Bragg angle and detected. The equipment used is usually called a crystal spectrometer, or crystal monochromator, and it consists of a single crystal mounted on a stand so that it can be rotated a t half the angular velocity of the arm on which the neutron detector is mounted. The crystal monochromator is extensively used to select a narrow band of wavelengths from a thermal spectrum for use in study of crystal structure by a second scattering a t the sample under study. The crystal monochromator is also used for energies other than thermal. For wavelengths longer than thermal, the monochromatic beam is contaminated with neutrons of shortcr wavelength, reflecting with values of n higher than unity. These higher order reflections make the use of the crystal for long wavelengths very difficult At the most intense part of the Maxwell distribution a crystal can give an intensity of about lo4 neutrons per second with a resolution in wavelength of about 3%. The wavelength resolution in terms of the collimation of the incident beam, A@, follows from Eq. (2.2.2.3.1),

-x_ - cot eae

(2.2.2.3.5)

thus the AA/X of 3 % just mentioned corrcsponds to a A0 of 0.5 deg. The intensity of neutrons reflected by the monochromating crystal is larger for increased mosaic spread of the crystal. As the mosaic spread is usually less than the beam collimation for slow neutrons, a few minutes compared with 30 min, increase of the former will increase the reflected intensity without affect,ing the resultant wavelength spread appreciably. A good crystal monochromator for thermal neutrons is thus one with a large mosaic spread and the metal crystals, for example lead, are advantageous for this reason. Some recent investigations of McReynolds* show the wide variation in mosaic spread among different crystals; for example

* Privatr communication (1958).

570

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Ge has a spread of only 3' whereas that for natural magnetite (Fe304) is 1.3". Crystal monochromators are usually used for neutron energies in or near the thermal region. In order to obtain reflections for higher energies it is necessary to use beams of high collimation and crystals of small mosaic spread. For a typical crystal LiF, the value d for the (111) planes is 2.32 A hence for neutrons of 1-ev energy the glancing angle will be 3.5". Whereas this value of 0 is a practicable one, although small, it becomes 0.35" for 100-ev neutrons, of the same order of magnitude as the angular collimation of many monochromators in use today. Even for a collimation of 0.1' the conditions just described would imply a n energy resolution AE/E (-2AXIX) of about 50% a t 100 ev. The angular requirements just mentioned are important factors in reducing the range of utility of present crystal monochromators to energies well below 100 ev, in practice below about 20 ev. Another effect that restricts the instruments to low energy is the rapid decrease in reflected intensity with increasing energy. This decrease is a result not only of the decreasing incident flux ( l / E ) but of the reflectivity of the crystal itself, which varies as 1/E. The resulting flux a t the neutron detector drops off rapidly with energy, as 1 / E 2 ,and counting rates are prohibitively low for the high collimation associated with good resolution.

2.2.2.4. Determination of Momentum and Energy of Neutrons with Hes Neutron Spectrometer.* 2.2.2.4.1. INTRODUCTION. The helium-3 spectrometer was originally developed to measure neutrons of energy in the range 100 kev to 1 Mev.1 It has subsequently been extended in application t o measure neutrons of energy u p to several M ~ v . ~ At- the ~ time of its development fast neutron energies were usually measured by methods based on the scattering of neutrons by hydrogen. Such methods suffer from the disadvantage that incident monoergic neutrons are converted into a continuum of energies of the recoil protons. Thus a suitable nuclear transformation induced by neutrons was sought, in which one energy release corresponds unambiguously to one neutron energy. R. Batchelor, R. Aves, and T. H. R. Skyrme, Rev. Sci. Znstr. 26, 1037 (1955). G. C. Morrison, AERE Report NP/R 2076 (revised), p.76 (1958). a A . R. Sayres, I<. W. Jones, and C. S. Wu, Bull. Am. Phys. SOC.3, 365 (1958). a* A. R. Sayres, Thesis, Columbia University, New York, Cu (PNPL)-200 (1960). D. West, private communication (1960). 2

2

~

* Section 2.2.2.4 is by G. C.

Morrison.

2.2.

DETERMINATION OF MOMENTUM A N D E N E R G Y

57 1

I n order t o be useful the cross section of such a reaction should be fairly large and its variation with neutron energy should be smooth and free from resonances. I n addition the energy release for zero neutron energy while positive should be small and there should he no low-lying excited state of the residual nucleus leading to the emission of particles of reduced energy. Because of these requirements the choice is limited to reactions with light nuclei and of the various possibilities (BID,Li6, He3), the most satisfactory is based on the disintegration of He3:

+ on'+

zHe3

lH1

+ 1H3+ 764 kev.

The total energy of the reaction, th at is the kinetic energy of the neutron plus 764 kev, is shared between the triton and proton. The neutron energy is therefore obtained with no ambiguity from the measurement of the total energy released. Measurements of the reaction cross section have been carried out by several workers. The reaction cross section varies smoothly with energy from a cross section for thermal neutrons of about 5000 barns6s6to a value for 1-Mev neutrons of about 0.8 barn1-3a.7 which is comparable with the scattering cross section of neutrons of this energy by hydrogen. A summary of the experimental results obtained in direct measurements and also those inferred from the inverse reaction T(p,n)He3using reciprocity'" are shown in Fig. 1. From the viewpoint of neutron spectroscopy the smoot,h behavior of the cross section is ideal although the rapid rise a t low energies favors the detection of thermal and epithermal neutrons which can produce a background extending into the fast neutron region. The major disadvantage of the helium-3 reaction arises from the competing effect of elastic scattering of neutrons by helium3 which has a cross section approximately three times that for the He3(n,p) reaction 013r the energy range 0-2.5 M e ~ . ~ j ~(See * s *Fig. 1.) Since the recoil spectrum from elastic collisions between neutrons of energy E nand helium-3 nuclei extends from zero to a maximum energy E = :En, a recoil of maximum energy for E , = 1 Mev cannot be distinguished from a slow neutron-induced helium-3 disintegration on the basis of energy alone. However even if the incident neutron spectrum contains neutrons of energy greater than 1 MeV, there is always a region of the spectrum extending from ($En - 764 kev) to the pulse height (En &) of the

+

L. D. P. King and L. Goldstein, Phys. Rev. 76, 1366 (1949). J. H.Coon and R. A. Nobles, Phys. Rev. 76, 1358 (1949). 7 J. H.Coon, Phys. Rev. 80, 488 (1950). ?nJ. H.Gibbons and R. L. Macklin, Bull. Am. Phys. Soc. 3, 365 (1958). EL. Cranberg, R. L. Mills, and T. R. Roberts, U.S. Atomic Energy Commission Rept. LA-1853 (1955). 5

6

572

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

maximum neutron energy E n which can be examined without ambiguity. Furthermore a neutron spectrum consisting of distinct groups can in principle be analyzed since a low-energy group can be separated from a continuous recoil spectrum arising from a group of energy greater than 1 Mev by the successive subtraction of monoergic spectra.

* 0

~

SAYRES JONES,AND WU, BULL. AM. PHYS. SOC, 3. 3 6 5 , / 9 5 8 CRANBERG, E T AL , LA-1853 BATCHELOR, ET AL., R S . I . , S , 1037. 1955

-

SIMMONS, CRANBERG, AND SEAGRAVE, BULL AM PHYS. SOC 3 338,1958

- BRANSDEN ET AL., PROC. PHYS. SOCh 5A 6T9 *

-

877, GIBBONS AND MACKLIN. BULL AM. PHYS SOC 3, 365, 1958

2

4

6

8 10 12 14 16 NEUTRON ENERGY (Mev)

FIG. 1. The variation of the n neutron energy.

18

20

z

+ He8 total, elastic and reaction cross sections with

The use of helium-3 in a neutron spectrometer is very convenient. The good resolution which can be obtained with a proportional counter or gridded ionization chamber can be exploited ‘with helium-3 introduced as a partial filling. The work reported so far has been restricted to the use of gas filled counters of this type although it has also been s u g g e ~ t e d ~ ~ ~ ~ and showngb that a gas scintillation counter could be used with a partial filling of helium-3. J. A. Northrop and R. Nobles, NucleoTiics 14(4), 36 (1956). A. Sayres and C. S.Wu, Rev. Sci. IT&-. 28, 758 (1957). 9’’ A. Sayres and C. S. Wu, Cu (PNPL)-199 (1960).

LJ

‘Ja

2.2.

DETERMINATION O F MOMENTUM A N D E N E R G Y

573

2.2.2.4.2. THE SPECTROMETER. 2.2.2.4.2.1. Design Considerations. If the best performance is to be obtained with a helium-3 spectrometer considerable attention must be paid to the design of the counter. Ideally we require the counter t o record proportionally the total energy released in each reaction with optimum resolution. The extent to which a practical device can be expected to depart from the ideal can be discussed under three headings, namely, rcsolution, wall effect, and linearity. Resolution. For a n energy EL released in the counter there will be a n intrinsic spread in pulse height due tJothe st,atistics on the mean number of ions initially liberated in the counter. For a n energy release of greater than 764 kev which is of interest to the helium-8 counter, the intrinsic spread (full width a t half-height) becomes quite small ( < 2 % ) and the resolution is mainly determined by other effects not of fundamental origin.I0 For a proportional counter they include variations in the anode wire diameter, variations in the applied voltage and counter end effects; the latter requires the use of end plate assemblies including field and guard tubes. l 1 For an ionization chamber the resolution becomes almost entirely dependent on noise in the associated preamplifier. * I n addition t o these factors, the purity of the gas filling must also be considered. This requires not only that the filling must be free from electron capturing impurities such as oxygen but also the tritium content of the helium3 must be as low as possible. Extremely small traces of tritium give rise to many P-disintegration pulses-10-8 cc tritium corresponds to about 5 X lo4pulses per second-which “pile u p ” and produce an additional spread in pulse height. Provided that the number of tritium decays are kept below lo4per second it is reasonable to expect anobserved spread of between 3 and 5%. Wall Efect. The wall effect in the counter arises as a result of some of the reaction products being cut off by the wall of the counter or escaping from the sensitive region of the counter. Thus a fraction of the reactions, p , have tracks cut off b y the walls and contribute a continuum of pulses between energies E = 0 and E’ = E,;the remainder, (1 - p), expend their full energy in the counter and appear in the peak centered a t E = El. A theoretical calculation of the magnitude and shape ofthe wall effect for a cylindrical counter is given by Batchelor et aZ.l I n a conventional counter the wall effect can only be kept small by keeping the proton and triton ranges small compared to the size of the counter. This iequires the use of a large counter and the introduction of a * See a180 Part 1 of this volume and Chapter 6.2 of Vol. 2. U. West, in “Progress in N u c l e ~ rI‘hysim” (0. It. Frisoh, pd.), Vol. 3, p. 29. Pwgamon, New York, 1953. l 1 ii. 1,. Cockcroft and S. C. Curran, Rev. Sci. Instr. 22, 37 (1951). 1”

574

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

heavy gas into the counter in addition to the helium-3. For most applications it is inconvenient to use too large a counter and the emphasis is on high pressure. However, the employment of high pressure requires the use of high counter voltages and hence consideration of the special problems thereby engendered in the design of end plate assemblies. Krypton is recommended as the heavy gas since it has a negligible cross section for neutrons, has good collection characteristics, and can be easily separated from helium-3.

36% N I C K E L IRON

QUARTZ

FIG.2. (a) Proportional counter used by Batchelor el al.' ( b ) The end plate insulator assembly.

Linearity. Since the total energy expended in the counter may be several Mev care must be t,aken in the operation of a helium-3 proportional counter to avoid saturation of the ionization pulse. Hanna and coworkersI2 have shown th at for a proportional counter saturation sets in when the multiplication factor X the energy (in electroil volts) is greater than about lo*. Thus a helium-3 proportional counter must be operated with a low multiplication fact,or; a value of about 10 is recommended which is still sufficient to ensure that the pulse is independent of the position of the initial ionization. 12

G. C. Hanna, D. W. H. Kirkwood, and B. Pontecorvo, Phys. Rev. 76,985 (1949).

2.2.

DETERMINATION OF MOMENTUM AND E N E R G Y

575

Linearity requires no special consideration with a gridded ionization chamber and the screening of the collecting electrode ensures th a t the pulse is position independent. 2.2.2.4.2.2. Details of Existing Counters. The development of a proportional counter containing helium-3 as a device for measuring fast neutron spectra was first carried out by Batchelor et at.I a t Harwell. It will be appreciated that in view of the high cost of helium-3 at th a t time considerable technical development was involved. Figure 2 shows the proportional counter design which was adopted. The end plates carrying the insulator and electrode assemblies were screwed and soft soldered into the ends of the counter body. The inside diameter was 2; in., the active length 4.8 in., and a stainless steel wire of 0.004 in. was used. The

1 0

-

0

n 01 z

I L

u

n

0

10

20 30 40 Pulse Height (Volts)

50

60

FIG. 3. Pulse-height distribution with neutrons of 2.05 Mev observed with the spectrometer of Batchelor el. a/.'

counter was filled with 27 cm helium-3, purified over heated calcium, 164 cm spectroscopically pure krypton, and 1.8 cm carbon dioxide. Carbon dioxide was added to stabilize the multiplication process in preference to methane to avoid pulses from proton recoils in the case of the latter. For neutron spectroscopy the counter is operated a t a voltage of 2600 v. The pulse distribution when neutrons of energy 2.05 Mev are incident perpendicular to the axis of the counter are shown in Fig. 3. To decrease the efficiency of the spectrometer for detection of thermal reactions the counter is shielded by a thin sheath of cadmium. The main features of the distribution are the peak corresponding to neutrons of full energy, the continuous background resulting from wall effects and pulses arising from the helium-3 recoils. The lowest peak is due to background epithermal neutrons. Below this peak the distribution rises very rapidly due to the effect of y rays. This discrimination against y rays is, of course,

576

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

very useful in many applications. I n the calibration of the counter pulse distributions of monoergic neutrons were taken at different, energies, and these can be used as a basis for analysis of more complex spectra. The pulse-height resolution of the spectrometer was found t o be about 7% for neutron energies up to 1.2 Mev which was the highest energy a t which initial measurements were made. Of course the energy resolution for neutrons is greater than the spread in pulse height by a factor ( E n Q ) / E n .That the resolution of 7 % is larger than expected was ascribed to tritium impurity which gave rise to 1.5 X lo5low-energy background counts per second. The calculated value of the wall effect for this particular filling is about 40% a t 1 MeV. Beyond 1 Mev the background now includes helium-3 recoils so that a knowledge of the wall effect itself at high energies is less important. At higher energies the value ( E(1) - p) relative to that at 1 Mev can be directly of the product C ~ , ~ X obtained from the number of counts recorded in the full energy peak for t,he same number of monoergic neutrons incident on the counter. The measured values of the He8((n,p) cross section a t different energies have been shown in Fig. 1. Assuming an average value of 0.8 barn for the cross section, the detection efficiency is about, 4 X for neutrons incident perpendicular to the counter axis. About the same timc Bloom et a1.13 designed a high-pressure proportional counter about 10 cm in diameter, 14 cm sensitive length which was operated a t voltages of 6000-8000 v and gas pressures between 6 and 10 atmospheres. I n view of the small amount of helium-3 a t their disposal they only carried out measurements with thermal neutrons since the A36(n,a)S38 reaction in argon, the heavy gas used completely swamped all other effects a t high energies, To improve the resolution for thermal neutrons they found i t necessary to line the counter walls with a graphite liner, since they did not use COz or CH4. Since then helium-3 has become available in larger quantities and attention has been given toward improvements in the performance of the spectrometer with three objectives in view: increased resolution, decreased wall effect, and increased efficienry. As has been previously noted an improvement in resolution requires a reduction in the concentration of tritium present in the helium-3. Thc method adopted hy the author a t Harwell was to condense out the tritium by prolonged circulation of the helium-3 through a cold trap maintained a t liquid helium temperatures. Using a proportional counter of design similar to t ha t developed b y Batchelor a resolution for thermal neutrons

+

I3S. D. Bloom, E. Redly, and B. J. Toppel, Brookhaven National Lab. Report, RNL 358(T-66) (1955).

2.2.

577

DETERMINATION O F MOMENTUM A N D E N E R G Y

of between 3 and 4 % was obtained with counter fillings of 30 cc purified helium-3 and pressures of krypton between 1 and 5 atmospheres. The concentration of tritium was estimated to be about 4 parts in 1O1O parts of helium-3. A final filling of 103 cm helium-3, 340 cni krypton, and 0.5 cm methane was adopted, the latter being preferred to carbon dioxide a t high pressures since it considerably reduces the collecting voltage required on the wire. For neutron spectroscopy the counter is operated at a voltage of 4400 v. Under these conditions the pulse height resolution of the spectrometer was 5;% for thermal neutjrons.2 Sayres and a s ~ o c i a t e s ~ a t~ Columbia "~ University have constructed a proportional counter with an insidc diameter of about 2 in. and active length 9 in. The counter has been operated with pressures up to 1 atmosphere helium-3, again purified by condensing out the tritium a t liquid helium temperatures, up to 4 atmospheres krypton and about 3 cm CO,. It was estimated that the purification had reduced the tritium content to less than 1 part in 10" parts of helium-3. The wall effect in the counter was considerably reduced by irradiating the counter with its axis parallel He3 to the direction of the incident neutrons and restricting the rb reactions in the counter to a region along the axis by the use of collimators. Under these conditions a pulse-height resolution of 5% was obtained for neutrons of 8.0 MeV. The spectrum of 8.0 Mev includes the peak due to the He3(n,d) reaction (see Fig. 4). Recently West4 a t Harwell has designed a high-pressure helium-3 spectrometer which utilizes a cylindrical gridded ion chamber instead of a proportional counter. The inside diameter was 4+ in., active length 9$ in. It contains a cylindrical grid surrounding a central collecting electrode. The counter was filled with 205 cm (9 liters in the total volume) helium-3, purified over pyrophoric ~ r a r i i u m ' ~t,o, ~remove ~ the tritium, 290 cm krypton and about 5 cm methane. The final concentration of tritium was estimated to be 3 parts iii lo1' parts of helium-3. The pulse-height resolution for thermal and 2.5-Mev neutrons is about 6 % and 2&%respectively, with noise in the preamplifier contrihut,ing about 40-kev spread. At 1 Mev the calculat,ed value of the wall effect, is 13% and the det,ection efficiency for neutrons inciderit parallel to the axis of the c,ounter is 1.6 X lop3. O F h NEUTRON SPECTRUM FROM A X OUSEILVED 2.2.2.4.3. <~AI,CULL4TIohPULSEDIST~~IBVTIOX. Siricc there are always some epithermal neutrons in t,he vicinit,y of a fast neutron sourcc the pulse distribution ohserved in any experiment always contains a peak corresponding t o neutrons of zero energy (rf. Fig. 3). Such a peak corresponding to all energy of 764 kcv

+

11

15

W. I). Allen arid A. T. G. Ferguson, AERE Report N P / R RI2O (1!)55). M .J . W . Elliott, Rev. Sci. InstT. 31, 1218 (1960).

578

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

released in the counter serves to establish the energy scale of the pulse distribution. The calculation of the neutron spectrum from the pulse distribution observed with the counter involves first a correction of the distribution for the continuous background of wall effect and, when the spectrum contains neutrons of energy above 1 MeV, of helium-3 recoils. The com250C

He3+ n En = 80 Mev

20rx

1500

N lOOr

50 (

CHANNEL NUMBER

FIQ.4. The spectrum of neutron interactions with He3 observed a t Columbia University for E , = 8.07 MeV. The calculated wall effect distribution for the Hea(n,p)H3 reaction is shown as the dashed curve.

bined effect can be conveniently estimated by dividing the observed distribution into sections of equal width, say 200 kev. The distribution obtained with monoergic neutrons of energy corresponding to the mean energy of the highest section is normalized to the observed distribution according to the number of neutrons present in this section. The background appropriate to neutrons of this energy can then be subtracted

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

579

from the distribution a t lower energies. In a similar way successive corrections for the helium-3 recoils and counter wall effect produced by neutrons in all other sections can be carried out. The resultant curve obtained in this way yields the pulse distribution corrected for background effects. Since the number of neutrons of energy Enrecorded by the counter is dependent, on the He3(n,p)cross section and counter wall effect, the incident neutron spectrum can be obtained from the corrected distribution on division by u,,=(E)x (1 - p ) . The relative errors in the resultant neutron spectrum are those arising from counting statistics and uncertainties in the counter wall effect and the cross section of the He3reaction with energy. Below 100 kev the rapid increase of cross section with decreasing neutron energy, and the finite resolution of the counter do not permit an accurate calculation of that part of the neutron spectrum. Correction for the finite resolution of the counter is not usually practicable. I n principle the above procedure could be applied to the analysis of complex pulse distributions containing neutrons of any energy. I n practice, however, the application of the helium-3 spectrometer is limited to the measurement of spectra containing neutrons of only a few Mev by the magnitude of the background effects a t higher energies. It will be appreciated that the calculation of the neutron spectrum and the determination of the associated error involve considerable routine analysis. It is therefore an advantage in the widespread application of a particular spectrometer to analyze the observed pulse distributions with the aid of a computer. OF THE SPECTROMETER. The helium-3 spec2.2.2.4.4. APPLICATIONS trometer has considerable advantage in the range 100 kev to a few Mev over any devices of comparable resolution based on hydrogen recoils. I n the !ow-energy field, and in cases where it is not possible to pulse the source of neutrons, investigations have been carried out which have hardly been feasible with other instruments. A good example of its use in these circumstances is the work of Batchelor and Hyder on the spectra of delayed neutrons from the fiss.on of uranium-235.16 The spectrometer has also been applied in the determination of the spectrum of neutrons in the envelope of a fast reactor and has been considered a s a means of measuring the energy spectrum of neutrons emitted from a thermonuclear device. The spectrometer in use a t Columbia University has been utilized in a determination of the total elastic, the He3(n,p)H3and the He3(n,d)d reaction cross sections for neutrons incident on helium-3 a t energies from 0.95 t o 17.5 I n addition the differential elastic scattering cross 16

R. Batchelor and H. R. McK. Hyder, J . Nuclear Energy 3, 7 (1056).

580

2.

DETERMINATION O F FUNDAMENTAL Q U A N T I T I E S

sections were obtained from the observed shape of t,he helium-3 recoil spectra through the relationship between the scattering angle of the neutron and the observed energy of the helium-3 recoil and compared with the theoretical distributions of Bransden el al. 16* At the present time the instrument has been applied most extensively in the study of the inelastic scattering of neutrons in the energy range 500 kcv to 2 Mev.17,18~18n The method adopted was to examine the spec.trum of neutrons produced inside a spherical shell of the element under investigation on bombardment with monoergic neutrons. From the observed distribution of neutrons the spectrum of inelastic neutrons and hence the position of levels in the target nucleus can be obtained as is discussed by B a t ~ ~ h e l ofrom r , ~ the ~ ~ observed ~~~ transmission of the shell the total cross section for nonelastic scattering and the partial cross sections for excitation of the levels in the target nucleus can he obtained on application of the sphere transmission analysis developed by Bethe et al. l 9 Although a shell experiment has the advantage of yielding a high intensity of inelastically scattered neutrons a t the counter, the geometry of an individual scattering event is not well defined. Thus the spectrum of neutrons inelastically scattered is spread out in energy as a result of center-of-mass effects. Furthermore, multiple elastic scattering in the shell previous to the inelastic event produces further broadening of the spectrum. The improved geometry offered by a n arrangement in which the neutrons scattered from a cylindrical bar are detected in the spectrometer placed parallel to the axis of the bar and shielded from the direct neutron source has also been utilized. The advantage of the arrangement, however, is offset by the reduction in the number of iieutrons recorded by the spectrometer. Thus the low efficiency of the spectrometer can severely limit its application in the study of inelastic neutron scattering. When the source of neutrons can be pulsed methods involving time-of-flight possess considerable advantages over those using the helium3 spectrometer in the investigation of inelastic neutron scattering. The helium-3 spectrometer has also been applied to the study of B. H. Bransden, H. H. Robertson, and P. Swan, Proc. Phys. Soc. (London) A69, 877 (1956). l 7 R. Batchelor, Proc. Phys. SOC. (London) A69, 214 (1956). 18 C:. C. Morrison, Amsterdam Nuclear Reactions Conference. l’hysica 22, 1135 (1956); and to be published. 18n R. Batchelor and G. C. Morrison, in “Fast Neutron Physics” (J. B. Marion and J. T,. Fowler, cds.); Part 1, p. 431. Interscience, New York, 1960. 19 H. A. Bethe, J. R. Beyster, and R. E. Carter. U.S. Atomic Energy Commission Rept. LA-1429 (1955).

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

58 1

+

the following (p,n) reactions: H 2 ( p , n ) p Li7(p,n)Be7;2t,z2and V61(p,n)Cr51.23 After the allowances for sources of background the spectrum of neutrons from the reaction under investigation is obtained directly from the observed pulse distribution. In this work the problems of intensity which can limit the study of inelastic neutron scattering are no longer severe. It would appear therefore that the spectrometer may successfully compete with time of flight techniques in the study of (p,n) reactions. The highly endothermic ( p , n ) reactions which are now within the range of a tandem Van de Graaff appear t o be particularly suited for study with the helium4 spectrometer. 2.2.2.4.5. F U R T H E R DEVELOPMENTS O F T H E SPECTROMETER. Despite the improvement in resolution which has resulted from the reduction of tritium concentration in the helium-3, the optimum resolution to be expected using a proportional counter has not yet been realized. However, the difficulties of obtaining good resolution with a proportional counter a t high pressures are well known. In this connection the results obtained using a high-pressure ionization chamber are very encouraging, particularly in view of the emphasis on the development of very low noise preamplifiers for use with solid state counting devices. Further improvements in the performance of the spectrometer are still possible. A further increase in the efficiency of the spectrometer should be sought. At the present time considerably more helium-3 is available than at the time of the original development of the spectrometer. Thus the construction of a large-volume, high-pressure counter containing several atmospheres of helium4 can be envisaged. With such a counter the wall effect, can be further reduced by limiting the reactions to a while preserving central region of the counter as done by Sayres et al.3*3a the angular resolution of the spectrometer. Finally, a method of overcoming the fundamental disadvantage of the recoil effect would be most desirable. A. T. G. Fcrguson and G. C. Morrison, Nuclear Phys. 6, 41 (1958). R. Batchelor, Proc. Phys. SOC.(London)A68, 452 (1955). 22R. Batchelor and G. C. Morrison, Proc. Phys. SOC.(London) A68, 1081 (1955). 23 A. T. G. Ferguson and G. C. Morrison, Conference on the Neutron Interaction with the Nucleus, Columbia University, New York, September, 1957, p. 178. 20

21

582

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

2.2.3. Gamma-Rays 2.2.3.1. Internal and External Conversion Lines.* t 2.2.3.1.1. INTROAs early as 1910 von Baeyer and associates’ observed the discrete lines in the beta-ray spectrum. A few years later Rutherford et al. found t ha t the secondary electrons expelled by X-rays impinging on a thin target show a spectral distribution of homogeneous groups of electrons closely resembling the electron spectra of radioactive elements. Moreover, it was found3 that the energy difference between certain lines corresponds t o the difference in binding energies of the electronic shells, known at th at time from X-ray spectra. The spectra obtained for the electron secondaries of X-rays and, later, nuclear gamma rays were immediately interpreted as due to photoelectric conversion of electromagnetic energy into kinetic energy of the electron secondaries. The electron energies E , are given by DUCTION.

E,

=

w,- W A

(2.2.3.1.1)

where W , is the energy of the X-ray or the nuclear gamma ray and WA is the binding energy of the electronic shell. Hence, each gamma ray will produce a number of lines in the electron spectrum corresponding to the K , LI, LIr, LIII, MI, MI,, etc. shells. When Meitner and Hahn4 found that such electron spectra of discrete lines are obtained also when radioactive sources known to be pure alpha emitters are used in the spectrograph they drew the important conclusion that the emission of electrons is a consequence-not the cause--of the radioactive decay. The appearance of homogeneous groups of electrons was, by analogy, to the external photoelectric conversion, interpreted as due t o an internal photoelectric conversion. The discrete lines in the elect>ronspectrum came to be known as “internal conversion lines.” The conclusion that the emission of internal conversion electrons were emitted after the decay was later confirmed by Meitner6 and b y Ellis and Wooster,6 who showed that the energy difference of the internal See also Vol. 1, Section 7.10.1. 0. von Baeyer and 0. Hahn, Physik. 2. 11, 488 (1910); 0.von Baeyer, 0. Hahn, and L. Meitner, ibid. 12, 273, 378 (1911); 13, 264 (1912). 2E. Rutherford, H. Robinson, and W. F. Rawlinson, Phil. Mag. [6]28, 281 (1914). 3 C. D. Ellis, Proc. Roy. Soc. A93, 261 (1921); 101, 1 (1922). 4L. Meitner and 0. Hahn, 2. Physik 2, 60 (1920). 6L. Meitner, 2.Physik 34, 807 (1925). 6 C. D. Ellis and W. A. Wooster, Proc. Cambridge Phil. Soc. 22, 844 (1926). 1

___

* Section 2.2.3.1 is by T.

R. Gerholm.

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

583

conversion K and L lines equals the difference of the binding energies of the K and L shells of the daughter nucleus. On the basis of theoretical arguments, however, it was soon realized that the appearance of internal conversion electrons must be considered as a direct interaction between the radiation field of the excited nucleus and the electron and not as a two step process involving first the emission of a gamma ray and then the transfer of its energy to a n electron by means of some internal photoelectric effect. The emission of electromagnetic radiation (gamma rays) and of internal conversion electrons from the different electronic shells should be considered as competing de-excitation processes. The branching ratio, i.e., the ratio of the transition probabilities, is known as the internal conversion cosficient. Obviously, according t o this definition, the internal conversion coefficient of, for example, the K shell, becomes IK (2.2.3.1.2) QIK = I, where IK and I , are the intensities of the K conversion electrons and the corresponding gamma-rays for a given source. The total conversion coefficient becomes atot

= aK

+ a,!,~+

+

~ L I I

’

’ ’

etc.

(2.2.3.1.3)

The internal conversion coefficient defined in (2.2.3.1.2) should not be confused with the (‘internal conversion probability ”* defined as KK

= JK

+ ILI +

JK

+

~ L I I

*

*

*

+ I,

(2.2.3.1.4)

Thus KK gives the probability per nuclear transition that a K electron will be emitted. Definitions analogous to (2.2.3.1.2) and t o (2.2.3.1.4) ~ , There is a simple relation hold for a L I , a L I I etc. as well as for K L ~K, L ~etc. between K and a , namely QK KK

=

(2.2.3.1.5)

Finally, we will briefly mention a third group of discrete electron lines present in internal as well as in external conversion spectra, namely the Auger lines. These electrons are caused by atomic transitions and have nothing directly to do with the nucleus. However, they may appear as a consequence of a proceeding K conversion or photoelectric effect in the K shell. When the vacancy in the K shell is filled b y a transition of a n electron from one of the outer shells (the X shell, say) there is a certain

* In the older literature “internal conversions coefficient” is in fact used to denote what is here defined as “internal conversion probability.”

.c

5

cu >

(v

2

(u (u

%

0 o

0

0

0

584

0

(D

0

N m

P r:

Y

0

0

0 -

0

0 .

0 N

0 O

.r’

,

0

2.2.

585

DETERMINATION O F MOMENTUM A N D ENEEGY

probability that the transition energy is taken up by a second electron (from the Y shell, say). The second electron will then be emitted with an energy approximately given by where W K ,WX, and W I Tare the binding eiiergies of the corresponding atomic shells while W u xand TVsY correspond to the binding energies of the Y ( X )shell of an atom ionized in the .Y(Y)shell. Since the ionization gives rise to an increased Coulomb attraction the ionized atom may be considered as a neutral atom with a somewhat higher effective nuclear charge, i.e., d l r y x ( Z ) = W y ( Z AX) (22.3.1.7)

+

where AZ is always less than 1. FIG. 1. Identification of a 26.22 k 0.01 kev 812 transition in Ph2"5from measurements with electromagnetically isotope-separated sources of Bi206 and Bi206 in a double focusing beta-ray spectrometer (resolution about 0.4%; counter cutoff energy about 2.3 kev). As shown in the figure, the internal conversion lines of the 26.22-kev transition can be easily distinguished from the L -4uger lines by comparing the two spectra. The measured conversion-line energies correspond to the interpretations Lr 26.22, LII 26.21, LIII 26.22, MI 26.223, MII 26.217, MIII26.225, Mrv 26.22, W i 26.223, NIII26.217, and 0126.222. The niultipolarity AT2 of the transition is suggested both froin the relative L intensities and froni t h e relative M intensities. This is clear from the following table, where the experimental values are compared with the theoretical values of Sliv and Band (for L shells) and Rose ( M shells). Relative internal conversion intensities

Multipolarity

__

-

LI

Theoretical E l 560 E 2 22 E 3 12 5 E4 1 E 5 44 I 100,000 Af 2 1800 M 3 M 4 nf 5

120 27 10

-___

LII

LIII

M1

iztII

niIrr

M~~

ndv

730 1000 880 760 680 10,800

1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

720 21 15 11 8 110,000 1600 I80 56 2G

880 840 940 1030 1200 12,000 106 10 3

1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

310 20 150 1700 7000 170 22 17 17 19

380

90

5 1 0

1

18

260 2000 7700 94

I 34 300 910

~__________.___

Experimental nf 2

ltjl(J

-50

1000

1730

120

1000

-15

(By the courtesy of l h . 11. Stockendal, Nobel Institute of Physics, Stockholm, Sweden.)

2.

586

DETERMINATION OF FUNDAMENTAL QUANTITIES

2.2.3.1.2. EXTERNAL CONVERSION. The external photoelectric process presents an accurate and convenient method for the measurement of gamma-ray energies. For a given energy the photoelectric cross section increases with the atomic number as 26 and hence the converter should be made of a high Z material. Uranium is a suitable choice and special techniques have been developed for the preparation of uranium converters.' A fairly thick (1 mg/cm2 - 50 mg/cm2) and uniform layer of uranium can be obtained simply b y painting uranyl nitrate dissolved in alcohol on a suitable, low Z, backing material. A few droplets of wetting agent should be added to the solution. Th e painted layer is finally heated t o about 500°C in order t o reduce the nitrate t o oxide. Thin uranium converters ( 2 2 mg/cm2) can be made b y a special vacuum evaporation technique.8

nd AXIS

u' CONVERTER

A1 PLATE

FIG.2 . (a) Converter for lens spectrometers. ( a ) Source; (b) brass or copper capsule; (c) wax; ( d ) converter foil. [M. Deutsch. L. Elliot, and R. Evans, Rev. Sci. Znstr. 16, 178 (1944).] (b) Converter for flat inst,ruments. [A. Hedgran, Arkiv F p i k 6, 1 (1952).]

The efficiency of the converter increases with increasing thickness. However when the converter becomes too thick, the photoelectrons will be partly absorbed in the converter and the photoelectron peaks will show a low-energy tail. Because of this the resolution will be reduced. The most favorable converter thickness, therefore, is taken as a compromise between resolution and efficiency. For a resolution of about 1 % a typical figure for the converter thickness would be about 1 mg/cm2 (uranium) a t a n energy of 200 kev. For higher energies the optimum converter thickness increases approximately proportional to the energy. Energies below 200 kev, on the other hand, require considerably thinner converters than would be given by this linear rule of thumb. Hence, in order t o cover a wide range of gamma-ray energies it is necessary t o have a set of converters of different thickness. The geometrical arrangement of the converter and the radioactive source depends on the electron 7 8

K. M. Glover and P. Borrell, J . Nuclear Energy 1, 214 (1955). P. Erman and W. Parker, Nuclear Znstr. 5, 124 (1959).

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

587

optical properties of the spectrometer and also to some extent on the nature of the problem to be studied. Two typical examples of converter arrangements are shown in Fig. 2. The exact position of the peak of photoelectrons corresponding to a given gamma-ray energy is obtained by subtracting the binding energy of the atomic shell in question from the gamma-ray energy. If the converter is replaced by another converter made of a different element the distances, in the energy scale, between the K , L, M , etc. peaks corresponding to a given gamma-ray energy, will be slightly different (Figs. 3(a) and (b)).

FIG.3(a). External conversion spectrum of Ir192 and IrlQ4below 490 kev. Curves A and B were obtained with a 1.2 mg/cma Au converter, C, D, and E with a 1 8 mg/cm2 U converter. A presents the total spectrum 12 hours after the irradiation and B the Irlg* component 10 days later. [Quoted from M. W. Johns and S. V. Nablo, Phys. Rev. 98, 1599 (1954).]

588

2.

DETERMINATIOK O F FTJNDAMENTAL Q U A N T I T I E S

Land M

,6127

HP (GAUSS-CMI

FIG.3(b). The same spectrum in the energy range from 470 kev to 900 kev. I n this region a 10 mg/cm* U converter was used. The lower curve represents the IrlQ2component only, taken after the short-lived activity had disappeared. [M. W. Johns and S. V. Nahlo, Phys. Rev. 96, 1599 (1954).]

It is, therefore, alwayspossible, by a suitable choice of converter element, to separate superimposed, unresolved, lines in the spectrum whenever these lines correspond t o photoelectric conversion in different atomic shells. For instance, if for a uranium converter the K photopeak of one gamma ray is found t o nearly coincide with the L peak of another, such th a t the two lines cannot be resolved, it is possible to separate the lines if the uranium converter is replaced by a lead converter. For precision energy measurements magnetic beta-ray spectrometers (cf. Section 2.2.1.1) are t o be preferred. In these measurements one determines the momentum-or rather the ‘‘Bp value” of the photoelectrons (Fig. 4).The corresponding energy is then calculated from the relation ~

y,

=

m,,c2

( J(2

_ 5)l

mo c

_

_

+ 1 - 1)

(2.2.3.1.8)

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

580

25 rng U-CONVERTER

I3 rng Pb- CONVERTER

0

6500

6wO

6 700

6800

6900

8) Gausrcrn

FIQ.4. (a) IrI84 decay as observed in photoelectron spectrum. 25 mg/cm* U converter. Note that the 1671K line is masked by the Compton edge due to intense transitions of higher energies. (b) 13 mg/cma P b converter. Here the 1671K line appears a t higher energy and for this reason it will not interfere with the Compton edge and is clearly seen in the photoelectron spectrum.

590

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

where Bp is expressed in gauss cm units and

Eo = moc2 = (0.510984 L- 0.000016) Mev elm0 = (1.75888 L- 0.00005) X lo7 emu gm-’ c = (2.997929

k 0.000008) X 1O1O cm-sec-’.

Extensive tabulations of (2.2.3.1.8)may be found in the l i t e r a t ~ r e . ~ - ~ ~ * To the energy value thus obtained one adds the binding energy of the atomic shell in order to obtain the gamma-ray energy. Tabulations of binding energies are given in the literature. l 2 . l 3 Precision energy (momentum) measurements are, in fact, generally reZative measurements, i.e., one measures the ratio of the Bp values of the unknown photopeak with respect to a suitable, known calibration line. The calibration line should be chosen as close as possible in the momentum scale in order to reduce systematic errors caused by deviations from perfect instrumental linearity, i.e., deviations from a linear relation between the instrumental readings, for instance, coil current, and the Bp value of the focused electrons. There exist a considerable number of very accurately measured reference lines, both internal and external conversion lines, suitable for calibration purposes. 14-16 The exact position of a line with respect to the Bp value can be chosen arbitrarily provided, of course, that the same technique is used to define the positions of the line to be measured and the calibration line. Some experimentalists use the peak counting rate, others the extrapolated high-energy edge. The most accurate method seems to be to determine T. R. Gerholm, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), pp. 926-931. North Holland Publishing, Amsterdam, 1955. l o T . R . Gerholm, in “Handbuch der Physik-Encyclopedia of Physics” (S. Flugge, ed.), Vol. 33, pp. 678-684. Springer, Berlin, 1956. l l A . H. Wapstra, G. J. Nijgh, and R. van Lieshout, “Nuclear Spectroscopy Tables,” pp. 31-36. North Holland Publishing, Amsterdam, 1959. lI8L. Marton, C. Marton, and W. G. Hall, Electron Physics Tables. Nutl. Bur. Standards Circ. No. 671 (1956). l 2 R. D. Hill, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), pp. 914-919. North Holland Publishing, Amsterdam, 1955. l 3 A. H. Wapstra, G. J. Nijgh, and R. van Lieshout, “Nuclear Spectroscopy Tables,” pp. 76-79. North Holland Publishing, Amsterdam, 1959. l4 A. C. G. Mitchell, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 227. North Holland Publishing, Amsterdam, 1955. j 6 T. R. Gerholm, in “Handbuch der Physik-Encyclopedia of Physics’’ (S. Flugge, ed.), pp. 648-649. Springer, Berlin, 1956. l6 A. H. Wapstra, G. J. Nijgh, and R. van Lieshout, “Nuclear Spectroscopy Tables,” p. 127. North Holland Publishing, Amsterdam, 1959.

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

591

the “position of maximum overlap.” The spectrum is reproduced in “divided form,” i.e., counting rate divided with Bp values versus Bp (Fig. 5). The peak t o be measured is normalized to the same area as the nearest calibration peak. Finally, one determines by which factor the Bp readings of the unknown peak have to be multiplied in order to bring the areas of the two lines to cover each other a s precisely as possible. Using high-resolution beta-ray spectrometers it is possible t o determine gamma-ray energies to within an accuracy of a few parts in 106.17,’7a There are in nuclear spectroscopy two special situations when the external conversion method is particularly useful. One of these is when the internal conversion lines are superimposed on an intense background of electrons belonging to the continuous spectrum. In such a case weak internal conversion lines are difficult, or even impossible, t o observe in the spectrum. If one uses a strong source, however, and a sufficiently thick, low 2, absorber to stop all the beta rays, the background will be substantially reduced and the external photopeaks may be seen. The other situation where external conversion is particalarly suitable is when the radioactive source can be obtained only with a low specijic activity. Specific activity = total activity/weight. In this case it is not possible t o study the internal conversion spectrum. Rut external conversion measurements may still be feasible. It should be observed, however, that the total converter efficiency is generally much smaller than the corresponding internal conversion probability. Therefore, it becomes necessary t o use strong sources, from several millicuries and upwards. For this reason coincidence experiments with external conversion electrons are hardly feasible. While the energy measurements with the external conversion method are straightforward, the measurement of gamma ray intensities by this technique is a far more intricate problem. Of course, the relative intensities of the photopeaks is a measure of the relative intensities of the initial gamma rays. A rough figure for the absolute converter efficiency and its variation with the gamma-ray energy can be calculated from the photoelectric cross secti0ns~8~~9 and the geometry of the converter and source arrangement. But in order to derive more accurate values for the absolute and relative intensities of the initial gamma rays it is necessary to know not only the photoelectric cross sections but also the angular distribution of the ejected photoelectrons. These problems have been studied theoK. Siegbahn and K. Edvarson, Nuclear Phys. 1, 137 (1956). R. L. Graham, G. T. Ewan, and J. S. Geiger, Nuclear Instr. 9, 245 (1960). 18 C. M. Davisson, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), pp. 857-874. North Holland Publishing, Amsterdam, 1955. 19 G. White Grodstein, Natl. BUT.Standards C ~ T C NO. . 683 (1957). 17

1’8

592

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

gas . so, . oio . s i 2 ' sir SPECTROMETER READINGS

R. . . . . . , . . . SPECTROMETER REhOINOS

>

R

'.

3s'OO 3dlO 3520 3g30 SPECTROMETER READINGS

R

h G 7 T x T & R SPECTROMETER REAOINGS

A

ThL

YhX

5

W z

SPECTROMETER READINGS

Fro. 5. To illustrate the different procedures used to determine the positions of electron lines on the Bp scale. (a) Peak position; (b) extrapolated high-energy edge; (c) overlap or folding method. Resolution 0.17%. [Quoted from A. Hedgran, Arkit, Fysik: I, 1 (19521.1

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

593

IHIERNAL CONVERSlON 1176K

I

I

5050

4 900

1184K

I

5 i50 Gauss cmBP

5 100

4 950

5bO

Gaurscm ‘r

FIG.6. (a) Comparison between internal and external conversion spectra from I r l S 4 . Due to the intense beta-ray background the two weak K conversion lines corresponding to the transitions 1176 and 1184 can not be seen above the background. (b) I n the external conversion spectrum, however, the two K photolines are clearly seen above the background. A 6 mg/cm* U converter was used (by the courtesy of G. Bkckstrom and J. Kern).

retically by HeitlerZO and by Hulme et aLzl and have also been experimentally investigated by several experimentalists. 22-z7 An excellent review article on the photoelectric process has been published by Davisson and Evans.26The geometrical arrangement of the converter, straggling, and absorbtion of the electrons in the converter introduce additional complications to the problem. Finally, the electron optical properties of 2 ” W. Heitler, “The Quantum Theory of Fhdiation,” Oxford Univ. Press, London and New York, 1936; 3rd ed., 1954. 2 1 H. R. Hulme, J. McDougall, H. A. Buckingham, and R. H. Fowler, Proc. Roy. SOC.A149, 131 (1935). 22 L. H. Gray, Proc. Cambridge Phzl. SOC.27, 103 (1931). zsG. D. Latyshev, Revs. Modern Phys. 19, 132 (1947); G. l). Latyshev, A. F. Kompaneetz, N. D. Borisov, and I. M. Gusak, J . Phys. U S S R 3, 251 (1940). . * 4 C. M. Davisson and R. I). Evans, Phys. Rev. 81, 404 (1951). 2 6 s . A. Colgate, Phys. Rev. 87, 592 (1952). 2 6 C. M. Davisson and R. D. Evans, Revs. Moderia Phys. 24, 79 (1952). 27 S. Hultberg, Arlciv Fgsik 16, 307 (1959).

594.

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

the beta-ray spectrometer should be taken into consideration. Recently, Hu1tbergz7has made a careful study of the problems involved in experimental measurements of gamma-ray intensities using magnetic beta-ray spectrometers to analyze the photoelectron spectrum. Using Hultberg’s results it seems to be possible, a t least in favorable cases, to determine the gamma-ray intensity to within an accuracy of about 5%. The relative intensities of the photopeaks in the spectrum can be taken directly as the ratios of the peak counting rates. However, for accurate measurements it is preferable to compare the areas under the line profiles. For such area comparisons it is necessary to keep in mind that the momentum band pass of a magnetic spectrometer is proportional to the Bp value of the focused electrons. The intensity of a given line is therefore proportional to the area under the line profile times the Bp value. In order to obtain the true intensity distribution-intensity per unit momentum interval-the spectrum should be reproduced in “divided form,” i.e., counting rate divided with Bp value versus Bp (cf. Section 2.2.1.1.). Just as in the case of external con2.2.3.1.3. INTERNAL CONVERSION. version, the internal conversion electron energy is given by

E’,

=

W

- WA.

(2.2.3.1.9)

Here, however, WA is the binding energy of the atomic shell in question for the daughter nucleus and W is the nuclear transition energy. Since the transition energy almost exactly, the difference being the generally quite negligible nuclear recoil energy, equals the energy of the corresponding gamma ray, the latter can be conveniently and accurately measured by means of the internal conversion electrons. The methods used for energy (momentum) measurements are exactly the same as already described for external photoelectrons (see above). The only difference in the experimental methods is in the source conditions. Experiments on internal conversion electrons require, for obvious reasons, sources of high specijic activity. The radioactive material should be deposited in the form of a very thin, uniformly distributed layer on a thin backing support. The backing should be electrically conducting to avoid distortions of the special distribution due to electrical charge built up at the source as a consequence of the decay and emission of electrons. A low 2 material is preferable for the backing from the backscattering point of view. These requirements are not easily met with in practice, but a wide variety of experimental methods of source preparation have been dkveloped, in particular vacuum evaporation and electrodeposition. The high specific activity required can be obtained from cyclotron-

2.2.

DETERMINATION OF MOMENTUM AND E N E R G Y

595

produced activities or from samples irradiated in high-flux reactors. It is frequently necessary to use special radiochemical methods, such a s ion exchange, t o get carrier-free active materials. The electromagnetic isotope separator has also proved to be a very useful instrument for the preparation of suitable beta-ray spectrometer sources. As already mentioned, gamma-ray emission and internal conversion constitute competing de-excitation processes. The relative transition probabilities, i.e., the internal conversion coefficient, depend on the multipolarity and the character of the nuclear transitions. Abso1ut)e measurements of internal conversion coefficients, therefore, reveal COUNTS PER UNIT TIME

22.960 ,970 980 990 23.aw) Dl0 ,020 .030 .Om .050 ,060 SPECTROMETER CURRENT (amps)

FIG.7. Separation of the K , LI, and L I Iconversion lines of the 238.6-kev transition in Pbalz (ThB, F, I, and I a lines in Ellis notation). The relative intensities of the conversion lines indicate a pure M1 transition. [Quoted from E. Sokolowski, K. Edvarson, and K. Siegbahn, Nuclear Phys. 1, 160 (1956).]

important nuclear characteristics and can be used a s a tool t o determine the spins and parities of the excited nuclear states a s well a s the multipolarity and character of the nuclear transitions. I n the case of mixed electric and magnetic transitions the mixing ratios can be determined from a comparison between the experimental conversion coefficients and the theoretical coefficients for pure transitions. (See Section 2.4.2.3.) Moreover, it is not always necessary to determine the absolute values of the conversion coefficients; in certain cases the conversion ratios, i.e., the ratio of K t o the sum of the L conversion intensities (LI LrI L U I ) , are sufficiently sensitive measures of the multipolarity and character of the transitions and for this reason the relative intensities of the conversion lines in the spectrum can be used t o decide about the multipolarities and

+ +

596

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

the characters of the corresponding transitions. Combining this information with the theoretical values for the absolute conversion coefficients it is then possible to estimate the absolute transition rates and to draw conclusions about the branching ratios of between different nuclear transitions. may be used for the Alternatively, the L subshell ratios (Ll:LIl:L~ll) same purpose. These ratios are generally very sensitive t o character and multipolarity. However, for most transitions the L lines come very close together and a high-resolution beta-ray spectrometer is required to resolve the L complex. It is sometimes possible to determine the absolute values of conversion coefficients directly from the relative intensities by making use of a previously known conversion coefficient in the decay as a reference. The unknown conversion coefficients (or, more precisely, the product of the transition intensities and the conversion probabilities) are then related to the already known standard. The method, of course, requires that there is a t least one already measured or theoretically known conversion coefficient in the decay. This is the case, for instance, in the decay scheme of even-even nuclei where the transition from the first excited state t o the ground state normally is a (pure) E2 transition and, hence, it is possible t o use the theoretical conversion coefficient for tlhis traiisitioii as a reference. However, it is also possible to perform absolute measurements of conversion coefficients with a beta-ray spectrometer. The most precise and straightforward method is based on a comparison between the area under the conversion line profile and the area under the continuous betaray spectrum. The spectrum should, of course, be reproduced in “divided form” (cf. above). The latter is a measure of the total number of nuclear decays. If the decay is complex, involving several beta branches to different levels in the daughter nuclei, it is necessary to know the decay scheme beforehand and to resolve the beta spectrum into its components by means of a Kurie plot analysis. For positron emitters the additional difficulty of competing electron capture decay arises if it is present. The method is therefore primarily limited t o fairly simple negatron decays. An alternative m e t h ~ d ,which ~ ~ , has ~ ~ the advantage of being independent of the details of the decay scheme, is based on a comparison between the relative intensities of internal and external conversion lines, originating from the same nuclear transition. The same sourye-or two sources 2N J. J. Murray, P1t.D. Thesis, Culiforiiiu Ilistitute of Trrhnology, I’nsadr~i;~, California, 1954. z9 S. Hultberg and Et. Stockendal, A ~ k i vFysik 14, 565 (1959).

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

597

ctmin 200

150 EXTERNAL CONVERSION

loo

50 1040

1060

1080

1100

llZC

1140

POTENTIOMETER OHMS

FIG. 8. Absolute determination of the K-conversion coefficient of the 662-kev transition in Cs137. Upper curve: internal conversion line. Lower curve: external convenion line obtained with a 2.19 mg/cm2 U converter. The same source was used for both internal and external measurement. From the relative intensities and b y making use of Hultberg’s method for calculating the absolute converter efficiency a conversion coefficient (YK = 0.093 zk 0.006 was obtained in good agreement with the theoretical values for a n M4 transition. The theoretical values are LYK = 0.092 (Sliv and Bandas) and 0.093 (RoseS2).[Quoted from S.Hultberg and R. Stockendal, Arkiv Fysik 14, 565 (l959).]

of known relative strengths-have to be used in both experiments (Fig. 8). The main difficulty stems from the above-mentioned problems connected to the absolute converter efficiency. Since the converter efficiency is generally much lower than the conversion coefficient, the counting rate becomes considerably lower in the external conversion measurements than in the internal conversion runs. The method is limited to activities available in the form of strong sources with high specific activity. A powerful method for the measurement of conversion coefficients in

598

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

beta-ray spectrometers is based on coincidence experiments. In this case the beta-ray spectrometer is supplied with a second detector operated in coincidence with the beta channel. The second detector may be a NaI(T1) scintillation spectrometer or it may in itself be a magnetic beta-ray s p e ~ t r o m e t e r (electron-electron ~~ coincidence spectrometer). If this second channel (channel 11, say) is set to record events proceeding or following the transition under investigation and whose conversion electrons are focussed in channel I, the number of coincidences per unit time (C) divided by the counting rate in channel I1 (NIT)becomes* C/NII = A q B 6 B

’

*

XB

’ WI

(2.2.3.1.10)

where the product A q B d B may be interpreted as the probability that the preceding (or following) transition A is followed (preceded) by the nuclear transition B which is the transition under investigation. Whenever B corresponds to a transition from the first exited state to the ground state, A Q B ~ Bis equal to 1. WIis the effective transmission in the spectrometer. The transmission can be accurately determined in various ways, for instance, by means of the same coincidence method and a second source which has a known conversion coefficient. In addition to these methods for the determination of conversion coefficients there exists a number of possibilities to utilize the beta-ray spectrometer in combination with scintillation spectrometers, crystal diffraction spectrometers, 47r counters, etc. A survey of various experimental methods for the determination of conversion coefficients has been given by A I b ~ r g e r . ~ ~ Theoretical conversion coefficients including screening and finite nuclear size effects have been published by Rose32and by Sliv and Band.33 The error in the theoretical calculation (for the K shell) are supposed to be of the order of a few per cent a t least for the K shell and are therefore comparable or superior to the most precise experimental measurements. There is, in general, a good agreement between the theoretical and the experimental values. However, there are a few exceptions, in particular

* We neglect for simplicity directional correlations and assume that the efficiencies of the coincidence circuit and the detector in channel I are both 100%. T. R. Gerholm, rlrkiu Fysik 11, 55 (1956). 31 I). E. Alburger, in “Handbuch der Physik-Encyclopedia of Physics” (S.Flugge, ed.), Vol. 42, p. 1. Springer, Berlin, 1957. 3 a M. E. Rose, Internal Conversion Coefficients.” North Holland Publishing, Amsterdam, 1958. 3 3 L. A. Sliv and I. M. Band, “Tables of Internal Conversion Coefficients of Gamma Rays,” Parts 1 and 2. Academy of Sciences of the U.S.S.R., Moscow-Leningrad, 1956. (Issued in the United States as Reports 571 C C K l and 581 CCLI, Physics Department, University of Illinois, Urbana, Illinois.)

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

599

among certain El and a few M 1 transitions, where the theoretical values are definitely outside the limits of experimental error. This is, in fact, to be expected as pointed out by Church and Weneser as a consequence of nuclear structure-dependent “dynamic ” effects not taken into account in the theoretical calculations. These dynamic effects, however, are generally too small to obscure character and multipolarity assignments. On the other hand, the presence of nuclear structure dependent factors in the theoretical expressions implies that there exists a possibility to get further information about the nuclear structure from precision measurements of internal conversion coefficients.

2.2.3.2. Determination of Momentum and Energy of Gamma Rays with the Curved Crystal Spectrometer.* 2.2.3.2.1. ESSENTIAL ELEMENTS

CURVEDCRYSTAL FOCUSING SPECTROMETER IN Two ARRANGE2.2.3.2.1.1. Arrangement I, the Conventional or Cauchois Photographic Arrangement, Long Used for the Study of X-rays. The range OF THE

MENTS.

Q2

ARRANGEMENT

I

RADIATION (ION CHAMBER OR ‘COUNTER) ARRANGEMENT II:

FIQ.1. Illustrating two ways of using the transmission type curved crystal spectrometer.

of the transmission type curved crystal spectrometer, a n instrument originally conceived‘ and developed2-6 about 1930 for measurement of X-ray wavelengths, was extended in 1947 to permit the study of nuclear gamma-ray spectra over a wavelength range from 500 milli1 J. W. M. DuMond and H. A. Kirkpatrick, Rev. Sci. Znstr. 1, 90 (1930). ZY. Cauchois, Compt. rend. 196, 1479 (1932); Ann. phys. [Ill 1, 215 (1934). 8 T. Johannsson, Z.Physik 82, 507(1933). 4 R. Bozorth and F. E. Haworth, Phys. Rev. 63, 538 (1938). 5 A. Guinier, Ann. phys. [ll]12, 161 (1939). 6 H. H. Johann, 2.Physik 69, 185 (1931).

* Section 2.2.3.2 is by J. W. M. DuMond.

600

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

angstroms to about 9 milliangstroms corresponding to quantum energies from 25 kev to 1.3 Mev.7 Figure 1 shows schematically two ways of using the transmission-type curved crystal spectrometer. Arrangement I is the familiar conventional way, developed by Cauchois in 1932 for X-ray spectroscopy, in which radiation from a source a t -4 outside the focal circle falls on the convex side of the bent crystal lamina and the various wavelengths are selectively reflected b y internal atomic planes of the crystal to form a spectrum of focused lines on a cylindrically bent photographic emulsion conforming to the focal circle. The crystal lamina, usually a plane-parallel thin slab in its initial unstressed condition, is bent t o cylindrical form to a radius of curvature (of the neutral axis of the slab) equal to the diameter of the focal circle. The focal circle is tangent to the cylindrical neutral axis of the bent slab at the center of the arc of crystal. The slab is usually cut so th at its chosen atomic reflecting planes are normal to its initially flat surfaces. Bending the crystal then makes the chosen atomic planes converge so that, if produced, they would 7 J. W. M. DuMond, Rev. S c i . Inslr. 18,626 (1947);J. W. M. DuMond, I). A. Lind, and E. R. Cohen, ibid. 18, 617 (1947); J. W. M. DuMond, D. A. Lind, and B. H. Watson, P h y s . Rev. 73, 1392 (1948); B. B. Watson, W. J. West, D. .4. Lind, and J. W. M. DuMond, ibid. 76, 505 (1949); J. W. M. IhM ond, 1). A. E n d , and B. B. Watson, ibid. 76, 1226 (1949); D. A. Lind, J. Brown, 13. Iclein, U. Muller, and J. W. M. DuMond, ibid. 76, 1544 (1949); 1). A. Lind, J. 11. Brown, .)1 E. Muller, and J. W. M. UuMond, ibid. 76, 1633 (1949); D.A. Lind, W. J. West, and J. W. M. DuMond, zbid. 77, 475 (1950); D. E. Muller, H. C. Hoyt, D. J. mein, and J. W. M. DuMond, ibid. 88, 775 (1952); J. W. M. DuMond, Physics T o d a y 6, 13 (1952); H. C. Hoyt and J. W. M. I h Mo n d , I’hys. Rev. 91, 1027 (1953); P. Snelgrove, J. El-Hussaini, and J. W. M. DuMond, ibzd. 96, 1203 (1954); E’. Boehm, P. Marmier, and J. W. M. DuMond, ibid. 96, 864 (1954); J. J. Murray, II‘. Boehm, P. Marmier, and J. W. M. DuMond, ibid. 97, 1007 (1955); P. Marmier and F. Boehm, ibid. 97, 103 (1955); L. L. Baggerly, P. Marmier, F. Boehm, and J. W. M. I hM ond, ibid. 100, 1364 (1955); F. Boehm and P. Marmier, ibid. 103,342 (1956); E. N. Hatch, F. Boehm, P. Marmier, and J. W. M. DuMond, ibid. 104, 745 (1956); F. Boehm and P. Marmier, ibid. 106, 974 (1957); E. N. Hatch, P. Marmier, F. Boehm, and J. W. M. DuMond, Bull. .4m. Y h y s . Soc. [2] 1, 170, Abstract El (1956); E. N. Hatrh, F. Boehm, I-’. Marmier, and J. W. M. DuMond, P h y s . Re,. 104, 745 (1956); F. Boefrm, Bull. Am. P h y s . SOC.121 1, 245, Abstract B l (1956); F. Boehni and P. Marmier, P h y s . Rev. 106, 974 (1957); E. N. Hatch and F. Roehm, Bull. .4n~.Phyls. SOC.[2] 1, 390, Abstract M-ti (1956); F. Boehm and E. N. Hatch, ibid. [2] 1, 390, Abstract M-7 (1956); F. Boehm and E. K. Hatch, ibid. [2] 2, 231, Abstract W-3 (1957); E. N. Hatch and F. Boehm, ibid. [2] 2, 212, Abstract QA-8 (1957); E. N. Hatch and F. Boehm, P h y s . Rm. 108, 113 (1957); J. W. M. DuMond, zn “Proceedings of the International Conference on Nuclear Structure, Rehovoth, Israel” (H. J. Lipkin, cd.), North Holland Puhlishing, Amsterdam, 1058; J. W. M. J)uMond, Ann. Rrv. Nicrlrnr Sci. 8, 163 (1958); E. N. Hatch and F. Boehm, 2. P h y s i k 166, 609 (1959); C. J. Gallagher, Jr., Nuclcar P h y s . 14 (1959/60); C. J. Gallagher, Jr., W. F. Edwards, and G. Manning, Transitions and levels in Re1*’. (To appear in Nuclear P h y s i c s ) ; A. H. Muir and F. Boehm, B u l l . Arrr. P h y s . Soc. [2] 4, 367 (1959).

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

60 1

intersect iii :I common poilit p on the focal circle diametrically opposite the center of the bent crystal arc. Radiation of a particular wavelength X will then hc selectively reflected by the crystal planes in a direction making the grazing angle e wit>hthose planes according t80Bragg’s law

nX

=

2d sin 0

(2.2.3.2.1)

where d is the grating constant of the planes and n the order number. For any region of the bent crystal lamina, it can readily be shown th a t this produces focusing of the radiation of wavelength a t a point R on the focal circle separated from the point 0by an arc 28. The focal point R may be on either side of fl depending on the position of the source A.* 2.2.3.2.1.2. Arrangement 11, as a Variable Monochromator with Source at Real Focus for the Study of Very Short Wavelengths.One of the innovations which permitted the extension in 1947 of the range of this instrument t o include the much shorter gamma-ray wavelengths mentioned above, consists in placing the source, in very concentrated form, at the real focus R, so that the radiation propagates into the entire solid angle defined by the aperture of the curved crystal lamina. This is the arrangement shown at I1 in Fig. 1. Because of the familiar property of the circle, insuring constancy of an inscribed angle which subtends a constant arc of the circle, the radiation from R will fall on all the atomic planes of the curved crystal a t essentially the same angle and that wavelength in the source for which this happens to be the Bragg angle will be selectively reflected by the curved crystal over its entire aperture. After selective reflection this radiation will propagate in a diverging beam directed as though it had come from the virtual image point V on the focal circle. The selectively reflected monochromatic radiation is received and its intensity is measured b y a detector, usually a sodium iodide scintillation crystal and associated photomultiplier tube. With this arrangement the instrument becomes essentially a monochromator and the spectrum must be explored by causing the source t o scan successively through a series of minutely spaced settings 011 the focal circle. 2.2.3.2.1.3. The Collimator-Bafle Required with Arrangement I I . It is necessary t o interpose a set of fan-shaped absorbing baffles or partitions, as shown, between the crystal and the detector to shield the manyorders-of-magnitude-more-intense heterochromatic direct beam of gamma rays (transmitted straight through the crystal without diffraction) from Cauchois was the first to point out that, if the crystal bends like a beam elastically following Hooke’s law, the change in grating constant on the concave and convex sides of the neutral cylinder of the bent lamina should be just such as t o give focusing of radiation from these regions a t t h e same point of the focal circle as radiation reflected in the neutral region of the crystal.

602

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

reaching the detector and completely masking the much weaker monochromatic reflected beam which is the object of measurement. This diverging baffle system, whose partitions must clearly point at the virtual image point V for the wavelength under study, is usually called the “collimator.” It has, however, nothing whatever to do with the spectral resolving power of the instrument. Its angular resolving power merely determines the radiation of shortest wavelength (highest quantum energy) which can be studied with adequate suppression of the direct beam from reaching the detector. In the Mark I, 2-meter focal length instrument a t the California Institute of Technology, for example, the collimator affords a theoretical geometrical angular resolving power (if we ignore scattering and penetration of the gamma rays in the collimator partitions) of 8 minutes of arc. This instrument uses the (310) atomic planes of a bent quartz crystal lamina having a grating constant, d = 1.18 angstroms, and therefore, from Eq. (2.2.3.2.1), the angular dispersion in first-order reflection a t short wavelengths is 0.7 milliangstrom per minute. This collimator therefore would permit an upper working limit of quantum energy of about 3 MeV, but in actual practice it is found that the background from scattering of the direct beam on the collimator partitions rises so fast at small angles as to place a practical working limit somewhere around 1.3 and 1.5 MeV. With the Mark I instrument using the (310) planes of quartz we have been able to record and measure quite accurately the C060 line a t 1.3 Mev which occurs in first-order reflection a$ a Bragg angle of one quarter of one degree. Very recently it has been found possible to observe a y-ray line a t 1.692 Mev in the y decay of Te1Z4. t and Their 2.2.3.2.1.4. The Three Essential Elements in A r r u n g ~ m e n If Geometrical Relationships. M a r k I and M a r k 111 Designs. The essential elements then in this arrangement I1 for gamma-ray spectroscopy are the gamma-ray source, the bent crystal, and the collimator-detector system. As one scans the source R over the various spectral positions on the focal circle, the virtual image point V must also scan through equal and opposite angular displacements along the circle and the partitions of the collimator-baffle system must remain always pointed at V . Clearly then, any one of the three elements of the system may be stationary if the appropriate motion is provided for the other two so as to maintain correct geometry during the scanning of the spectrum. In the Mark I instrument, which is perhaps the most inexpensive design, we have chosen to maintain the collimator-detector system stationary since it involves a great deal of heavy lead shielding. Accordingly, provision is made in this instrument to scan the spectrum by rotating the crystal (and its focal circle) around a pivot at the center of the neutral axis of the crystal lamina and simultaneously rotating the source twice as fast

2.2.

D E T E R M I N A T I O N OF MOMENTUM A N D E N E R G Y

603

on the extremity of a variable radius-vector from the same center in such a way that it always remains on the focal circle. This is accomplished automatically in single discrete steps by a “sine-screw ” linkage system which is designed t o provide a linear scale of wavelengths (in accord with Bragg’s law) proportional to the turning of a precision driving screw.g One turn of this screw which advances the wavelength carriage about 1 mm corresponds (with a proportionality factor quite close to unity)

FIG.2. Line drawing of the Mark I design of curved crystal diffraction spectrometer for the study of gamma-ray spectra up to 1.3 MeV. The distance from the crystal C t o the source inside its spherical lead shield a t R is about 2 meters.

to a change in wavelength of one milliangstrom. The exact proportionality factor of the instrument is readily calibrated by observing wellknown X-ray lines whose Bragg angles and wavelengths have already been determined by absolute methods (the “ two-crystal” X-ray spectrometer and comparison of crystal wavelengths in “x units” with “grating wavelengths” in milliangstroms using ruled gratings in grazing incidence)-methods whose accuracy is amply adequate for the precision a t present obtainable in measuring gamma-ray wavelengths. Figure 2 is a line drawing of the Mark I instrument. I n contrast t o 9 It is a n interesting fact t h a t the correction for refractive index vanishes for the case applicable t o this instrument. There is, however, a small but negligible geometrical aberration if the bent crystal lamina is initially plane in the unstressed state. The reader is warned that it is beyond the scope of this article to treat many such instrumental or technical details, a knowledge of many of which is nonetheless absolutely essential for precision in this technique. For complete information the reader is referred to an article by J. W. M. DuMond in Ergeb. esazt. Naturw. 28,232 (1955). A more abridged account by the same author will be found in Chapter I V of the book, “ Bcta- and Gamma-Ray Spectroscopy ” (K. Seigbahn, ed.). Interscience, New York, 1055.

604

2.

DETERMINATZON OF FUNDAMENTAL QUANTITIES

this, a 7.7-meter focal length instrument a t the Argonne National Laboratory (the Mark III)I0 is designed for study of radiation from a source situated inside a neutron reactor, a source which must therefore remain stationary. In this instrument then the spectrum is scanned by slowly rotating the curved quartz crystal about a pivot centered on its neutral axis and the entire collimator-detector system with an aggregate weight of nearly 13 tons is simultaneously caused to rotate about the same center at just twice the rate of the crystal by a servo system controlled by an optical lever. 2.2.3.2.1.5. Importance of a Design in WhichReJlections from Both Sides of the Crystal Planes Can Be Studied with Continuous Scanning I’hrough the Intermediate Zero WavelengthPoint. The writer considers it of particular importance to provide means so that, in scanning the spectrum, the real and virtual image points ( R and V of Fig. 1) can cross over the point of zero wavelength p and exchange places. In this way the entire spectrum can be explored by diffraction from either side of the bent crystal’s atomic reflecting planes with equal facility. The travel of the wavelength carriage (in terms of turns of the wavelength “sine screw”) requisite to move from the position for reflection of a given line on one side to that of the same line on the other side is then a measure of twice the sine of the Bragg angle. This permits precise wavelength determinations with use of the method of “superposition of profiles” which is t o be recommended for its superior precision. In this method the quantity determined is the carriage displacement, in terms of the number of screw turns (and fractions thereof), requisite to establish best mutual superposition of the pair of line profiles obtained by reflection on the two sides throughout the entire proJile of each line. I n this way all points on the line profile contribute to the precision of the measurement. Usually the lines from a properly prepared bent quartz lamina have very satisfactorily symmetric profiles. An asymmetric profile can be the result of (1) nonuniform distribution of intensity across the width of the emitting source, or departures from perfect focusing of the bent crystal, in which case the asymmetry faces in the same direction in the two superposed profiles and introduces no systematic error, if the method of superposition of profiles is employed, or (2) unresolved multiplet or satellite structure in the spectral character of the line, in which case the asymmetry of one line profile is a mirror image of that of the other profile since the dispersion is of opposite sign on the two sides of the B point. This means of distinguishing two quite different sourres of line nsymmetry is obviously an extremely valuable feature. 2.2.3.2.1.6. Comparison of Advantages and Drawbacks of Arrangements 1). Rose,

H. Ostrander, and B. Harnermesh, Rev. Sci. Znstr. 28, 233 (1957).

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

605

I and 11. The Cauchois arrangement, shown a t I in Fig. 1, has the advantage that all the lines in an extended spectral region can be photographically recorded a t one time and that fluctuations a t primary source intensity need not be monitored. For gamma-ray spectroscopy, however, it has the following two serious limitations. Firstly, the solid angle of radiation flux of a given wavelength which the bent crystal will accept from each atom or nucleus in the source A is extremely restricted in the Cauchois arrangement. It is of the order, in the horizontal plane, of the Darwin diffraction pattern width, that is to say a few seconds of arc only. Secondly, with this arrangement it is extremely difficult t o prevent the intense undiffracted beam, transmitted straight through the crystal, from completely masking the spectrum over the part of the focal circle where the shorter wavelength lines occur, e.g., the part in Fig. 1 beyond Q going clockwise. Severtheless the Cauchois photographic arrangement, because of its simplicity, is not to be ignored for gamma-ray spectroscopy in the lower energy regions up to 100 kev or so, especially if thick nuclear emulsions are used t o increase the fraction of radiation absorbed in the emulsion. It has in 1958 been successfully used with the reactor at Livermore, California to record and measure the neutron-capture y rays (approximately 2 MeV) emitted when protons capture neutrons to form deuterium, and it thus furnishes a reliable method of determining the binding energy of the deuteron. 2.2.3.2.1.7. Typical Sources, Their Sizes and Strengths. The geometry of the source itself usually plays the role of the defining slit when working with a n instrument of the type shown in arrangement 11, Fig. 1. A typical neutron-activated source for the Mark I instrument might consist of a small quantity of the required substance enclosed in a sealed quartz capillary of order 0.05 mm in diameter and 2 or 3 cm long, which, in this form, has been irradiated in a reactor. The Mark I 2-meter spectrometer permits a source volume consistent with good focusing of dimensions 30 mm high, 5 mm wide in the direction of propagation from source t o crystal and 0.05 mm thick in the direction of dispersion, but so great a depth a s 5 mm is rarely used unless the available specific activity per unit volume offers a serious limitation because it requires large and somewhat uncertain corrections t o th e relative line intensities for self-absorption in the source. Source strengths varying from 50 mc up to several curies have been used as eacbh individual case ma y happen to require. 2.2.3.2.1.8. Considerations Rearirq O I L Ihc Choice of the Scalr of Size in a Dwign. If highest attainable resolution is desired, consistent with the character of a given crystal such as quartz, this sets an upper limit to the angle a t the crystal subtended by both the height and width of the

606

2.

D E T E R M I N A T I O N O F FUNDAMENTAL Q U A N T I T I E S

source and also t o the ratio of source depth t o focal length. Therefore the permissible source volume varies as the cube of the linear dimensions of the instrument, an important consideration which tends to make a large scale instrument attractive if limited specific activity per unit volume of the sources to be studied is likely to be a serious problem. Also, experience has shown that it is scarcely safe t o try to bend quartz lamina that are thicker than one-thousandth of the radius of curvature, and it has been observed th at the reflected gamma-ray intensity obtainable increases in direct proportion to the thickness so long as the gammaray absorption in the quartz is negligible. This then introduces one additional way in which an increase in linear scale yields proportionally increased intensity. On the other hand, the area of the detecting scintillation crystal which must intercept the diverging beam from the diffracting crystal must increase as the square of the scale and with this increase the background from local and cosmic radiation may become a limiting factor. With a quartz crystal a focal length gf at least 2 meters seems desirable and lengths as great as 10 meters do not seem out of the question. 2.2.3.2.1.9. The Curved Crgstal; Two Methods of Bending. Two general crystal bending methods have been used for elastic (as contrasted with plastic) crystals such as quartz: (1) the method of the freely sprung lamina” in which equal and opposite torques are simply applied to the two ends of the quartz strip; and (2) the “method of imprisonment” in which the quartz slab with initially optically flat parallel faces is compressed between two hardened blocks of stainless steel provided with appropriately curved convex and concave cylindrical surfaces. The convex steel surface is precision ground and lapped so as to define the curvature of the crystal very accurately while an elastic gasket is placed between the concave steel surface and the quartz lamina. The writer is a proponent of this method of imprisonment because it has yielded very stable, satisfactory, and reproducible results over long periods of time. The details of the technique are considerable and of great importance for its success. The entire procedure has been adequately described in the literature t o permit any skillful person to succeed with it.12Because of the smallness of the Bragg angles involved in gamma-ray spectroscopy and the fact that the whole crystal aperture is utilized J. B. Borovskii, Doklady i i k a d . h’auk. SSSR 72, 485 (1950); A. B. Gilvarg, ibid. 72, 489 (1950); G. Brogen, ilrkiv Fgssik 3, No. 30, 515 (1951); Thomas C. Furnas, Jr., Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1952. Patents have been assigned to the General Electric Co. by David Harker on a n X-ray spectrometer in which the crystal curvature is variable as the spectrum is explored. None of these references on the freely sprung lamina unfortunately give adequate technical details of the method or quantitative measures of its results. l 2 J. W. M. DuMond, D. A. Lind, and E . R. Cohen, Rev. Sci. Instr. 18, 617 (1947).

2.2.

DETERMINATION O F MOMENTUM A N D ENERGY

607

simultaneously in arrangement I1 of Fig. 1, the crystal must focus with excellent precision. B y the method of imprisonment we have repeatedly been able t o produce bent crystals which by actual test focus to within a width of least confusion of 0.05 mm a t 2 meters distance for all parts of the utilized window area of the crystal of 20 cm2. The test is similar in principle t o the Hartmann test for large telescope mirrors. 2.2.3.2.2. APPLICATIONS AND SCOPEOF THE CURVED CRYSTAL SPECTROMETER, 2.2.3.2.2.1. Illustration with a Typicat Spectrum of Nuclear Gamma- and X - R a y Lines from a Sample of Neutron-Activated Tantalum. Figure 3 is a bird’s-eye view of the entire spectrum of X-ray and gammaray lines, for reflection from one side of the crystal planes only, obtained with a neutron-activated source of metallic tantalum. Three abscissa scales are shown, the linear one giving wavelengths in milliangstroms, the reciprocal scale showing energies in Mev,I3 highly compressed near zero wavelength, and finally a nearly linear scale, the Bragg angle in degrees. The rising background at short wavelengths from increasing scattering of the direct beam on the partitions of the collimator is clearly evident. The lines comprise nuclear gamma rays from both WIa2and WIg3,the which in turn have daughter products after /3 decay of Ta1S2and TaLS3 been formed from the stable natural isotope Talsl b y single and double neutron capture respectively in the reactor. There appear automatically in this spectrum also all the K series X-ray lines from both tantalum and ~’~ tungsten. Since many of these have already been m e a ~ u r e d ’ ~with 13 The author, at, first singly, later in collaboration with E. R. Cohen, has devoted considerable study to the evaluation of “best” values of the constants and conversion factors of atomic physics by least-squares fitting of a wide selection of prerisiori data. As a result of our latest study the conversion from wavelengths A, in cm t o energies E in electron volts is given by the following equation:

EX, = (12397.67 k 0.22) X

lopgev cm

or

EX,

= 12372.44

k 0.16 kilovolt 2 units.

E. R. Cohen, J. W. M. DuMond, T. W. Layton, and John S. Rollett, REVS. Modern Phys. 27, 363 (1955); J. W. M. DuMond and E. R. Cohen, Phys. Rev. 103, 1583 (1950); E. R. Cohen and J. W. M. DuMond, Fundamental constants of atomic physics (especially Sections 12, 30, 34 and Addendum, p. 86). I n “Handhuch der Physik-Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 35. Springer, Berlin, I957 (in English); E. R. Cohen, K. Crowe, and J. W. M. DuMond, “Fundamental Constants of Physics.” Interscience, New York (in press). (See also Vol. 1, Section 2.4.1.4 of the present work.) l 4 E. Inglestam, Inaugural Dissertation, IJppsala, Sweden 1937; also quoted in Y. Cauchois and H. Hulubei “X-Ray Wavelength Tables.” Hermann, Paris, 1947. 16 B. B. Watson, W. J. West, D. A. Lind, and J. W. M. DuMond, Phys. Rev. 76, 505 (1949); P. Snelgrove, J. El-Hussaini, and J. W. M. DuMond, ibid. 96, 1203 (1954).

n

n

FIG.3. Bird's-eye view of a complete spectrum of gamma-ray and X-ray lines from a source of neutron activated tantalum. The gamma-ray wavelength's are given numerically. The X-ray lines are indicated by their spectral designations. A complete mirror image of this spectrum is also obtained by reflection from the other side of the crystal planes.

2.2.

DETERMINATION OF MOMENTUM A N D E N E R G Y

609

high precision on the Siegbahn crystal scale in “x units” they serve as excellent reference points to calibrate the linear wavelength scale of this instrument. 2.2.3.2.2.2. Method of “Superposition of Profiles.” The method of superposition of profiles alluded to under Section 2.2.3.2.1.5 has the great advantage for the precise determiiiation of wavelengths th a t it does not make use of any one local feature of a line profile such a s its peak or the center of a single chord across the line at half-maximum height but uses t o optimum extent the entire statistics collected concerning the entire profile of the line. It has been shown by Muller et al. lSa that if the imprecision in wavelength determination (due only to the statistics of counting associated with determining the points of the line profile) is + a then l / u 2 = (4HT/W2)log[(H B ) / B ] (2.2.3.2.2)

+

where W is the base width of the line profile (assumed with sufficient accuracy for this analysis to be a triangle, see Fig. 5) and H is the peak height above the background B, while T is the total time of counting requisite t o collect all the data on all points across the line profile. The errors fa, calculated b y this formula can be easily made less than 0.001 milliangstrom in=reasonably favorable cases, but these are only the errors of location due to limited statistics of counting with very symmetric line profiles. The individually observed points are too close together to be depicted in the “bird’s-eye view” of Fig. 3 and the lilies are drawn somewhat broader than their true instrumental profiles for better visibility. A detailed plot of the three highest energy lines, the triplet a t about 10 milliangstroms or 1.24 MeV, is therefore shown in Fig. 4 as they were delineated using sources of two different widths, 0.004 inch and 0.011 inch. Two of the numbered abscissa divisions in Fig. 4 correspond to 1 milliangstrom change in wavelength or a motion of 1 mm of the screw-driven wavelength carriage. The statistical counting uncertainty is indicated on a few of the points. When lines are as closely adjacent and with a s unfavorably large and steep a background a s these, a n absolute wavelength determination by the method of superposition of profiles t o better than 8 or 10 % of the line width a t half-maximum is about all th a t can be expected. Figure 4 is about the most unfavorable example from this point of view. Figure 5, on the other hand, gives a n idea of about the best t ha t can be done-a typical pair of line profiles for the 412-kev line ill HgLyS.The size of the dots represents approximately the statistical D. E. Muller, H. C. Hoyt, D. J. Klein, and J. W. M. I h M o n d , Phus. Rev. 88,

1 5 ~

781 (1952).

m

m.

I

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

611

counting uncertainty. The method of superposition of profiles is reliable here t o 2 or 3% of the line width at half-maximum, i.e., to about fO.O1 milliangstrom, a relative precision of about a part in 6000 a t this energy. The results actually reproduce somewhat better than this. 2.2.3.2.2.3. X-Ray and Gamma-Ray Line Widths, Resolution, and Precision of Energy Determination. The line widths with a 2 mm thick crystal in the 2-meter Mark I instrument using the (310) planes of quartz in first order can be made somewhat less than 0.25 milliangstrom wide a t half-maximum, roughly a constant width independent of wave-

FIG.5. Profiles obtained on both sides of the spectrometer for the 412-kev gamma ) . method of superposition of profiles Iine in Hg1V8 (following the decay of A U ’ ~ ~The for precision wavelength determination consists in measuring, in terms of turns and fractions of the precision wavelength screw, the precise carriage displacement requisite t o superpose one of these two profiles exactly upon the other.

length. This width corresponds to a change in Bragg angle of 21 seconds of arc. The W K a lX-ray line, for example, has a natural spectral breadthI6 of 0.15 mA and this natural breadth is clearly visible with the Mark I instrument. The natural breadths of gamma-ray lines, on the other hand, are far too narrow to be perceptible with the Mark I in comparison with its instrumental breadth. It is therefore quite easy to distinguish X-ray lines from gamma-ray lines by their breadths. If AE is the full energy width of a gamma-ray line a t half-maximum and E is its energy, each in kev, then the resolution AEIE is given by AEIE = 0.3 E X We usually assign for safety an uncertainty of energy determination corre18 Measured with the two-crystal spectrometer. See A. H. Compton and S. K. Allison “X-Rays in Theory and Experiment,’’ Table IX-21, p. 745. Van Nostrsnd, New York, 1934.

612

2.

DETERMINATION O F FTJNDAMENTAL Q U A N T I T I E S

sponding t o go of this resolutioii or f 10 ev, whichever is thc litrgcr. Tlw uncertainty of energy coming from statistical uncertainty of counting can easily be made half this or less. At, 100 kev the above rule gives for the resolution, AEIE = 0.3% and for the uncertainty of energy determination, f a x = 15 volts. 2.2.3.2.2.4. Comparison of Advantages and Disadvantages of Crystal Difraction Spectrometry and Magnetic Spectroscopy of Conversion Electrons. In addition to its self-evident value for precision determination of gamma-ray wavelengths and quantum energies on the absolute cgs scale, diffraction spectroscopy of gamma-ray spectra has been shown to be a very valuable exploratory method whenever sufficiently intense sources are feasible. Its advantage over magnetic spectroscopy using conversion electrons comes about because in the case of crystal diffraction spectra each nuclear transition results in only one gamma-ray line, thus greatly simplifying the interpretation and avoiding the confusion and inaccuracy resulting from overlapping and superposed conversion lines of quite different origin. The combination of the crystal diffraction instrument with the magnetic instrument, however, constitutes the most powerful tool of all, not only because each technique complements and aids the other in its virtues and defects as regards sensitivity and resolving power in different ranges of energy, but also and very importantly because in combination they furnish a very powerful method of determining conversion coefficients which in turn yield the multipolarities of the transitions and permit indubitable assignments of spins and parities to the nuclear energy levels and permit unique interpretations in terms of the complete gamma-ray decay scheme. To do this it suffices to establish a single normalization factor between the intensity scales of (a) the gammaray lines in the diffraction instrument and (b) the corresponding conversion lines in the magnetic instrument. For examples of this the reader is referred to two of our recent studies." It is a great mistake which many physicists at present commit, to think of the technique of crystal diffraction spectroscopy of nuclear y rays as a competitor to magnetic spectroscopy of conversion electrons (both externally and internally converted) as though for some reason one were compelled to make a mutually exclusive choice between the two, adopting either one or the other only. The two methods measure different

*

l7 J. J. Murray, F. Boehm, P. Marmier, and J. W. M. IhMond, Phys. Rev. 97, 1007 (1955); W. F. Edwards, Conversion coefficients in some deformed nuclei. Thesis, California Institute of Technology, Pasadena, California, 1960. In this study Edwards has measured the exponent in the law, E-" expressing the reflectivity of the (310) planes of the bent quartz crystal and has found TZ = 1.987 0.22 for 60 kev 5 E 5 400 kev. He has also shown that true relative intensities of -pray lines can be measured with the bent crystal spectrometer with an uncertainty of order & 1 %.

2.2.

DETEItMINATION OF MOMENTUM AND ENERGY

613

things, the intensities of y rays on the one hand and the intensities of conversion electrons on the other, so that two such appropriately matched instruments in the same laboratory serve to yield immediately a wealth of data of tremendous value in unraveling decay schemes. Largely for these reasons the Mark I instrument, since its inception and with the help of two companion magnetic instruments*8a19has served to determine the decay schemes of more than 25 nuclear species and in particular has played an important role in verifying with considerable precision the theoretical predictions of the “collective model” of the nucleus developed b y the Copenhagen school of physicists under Aage Bohr. With the Mark I instrument the first spectrum of the annihilation radiation (from recombination of positrons from Cu64 with structure electrons) was observed and shown to consist of a quite sharp line of wavelength, h/(moc),slightly broadened b y Doppler effect with a breadth and structure consistent with the momentum distribution of the structure electrons in the copper. (See the fifth and ninth paper cited in reference 7.) 2.2.3.2.2.5. Applications to Study of Neutron Capture Gamma-Rays and Coulomb Excited Gamma-Rays. For an account of two other applications of crystal diffraction spectroscopy, (1) to the study of neutron capture gamma rays (with the 7.7 meter Argonne Laboratory instrument), and (2) t o the study of Coulomb excited gamma rays (from the A 4 8 linear proton accelerator a t Livermore using a 2-meter photographically recording spectrometer in the Cauchois arrangement) the reader is referred to recent publications.20-26 H. E. Henrikson, U.S. Atomic Energy Commission, Rept. S T R 24, Contract AT(04-3)-63. California Institute of Technology, Pasadena, California, October, 1956. l o J. W. M. DuMond, Ann. Phys. 2, 283 (1957). so D. Rose, H. Ostrander, and B. Hamermesh, Rev. Sei. Insir. 28, 233 (1957). 21 E. L. Chupp, A. F. Clark, J. W. M. DuMond, F. J. Gordon, and H. Mark, Report, University of California Radiation Laboratory (Livermore, California Site) UCRL 4871 (1957); UCRL 5142 (Abstr.)(February, 1958); UCRL 5171 (1958) ;UCRL 5172 (1958); UCRL 5180T (April, 1958); UCRL 5241 (Abstr.)(May, 1958); UCRL 5271 (June, 1958); E. L. Chupp, J. W. M. DuMond, and H. Mark, UCRL 53381’ (1958); Phys. Rev. 107, 745 (1957). 22 E. L. Chupp, J. W. M. DuMond, F. J . Gordon, R. C. Jopson, and H. Mark, Phys. Rev. 109, 2036 (1958). ZaE. L. Chupp, J. W. M. I h M o n d , F. J. Gordon, It. C. Jopson, and H. Mark, Phys. Rev. 112, 518 (1958). z4E. L. Chupp, J. W. M. IhiMond, F. ,J. Gordon, R. C. Jopson, and H. Mark, Phys. Rev. 112, 532 (1958). zL E. I,. Chupp, J. W. M. IhMorid, F. J . Chrdori. R . C. Jopson, and H. Mark, Phys. Reu. 112, 1183 (1958). 26E. L. Chupp, J. W. M. IhMond, and H. Mark, Reu. Sei. Instr. 29, 1153 (1958).

614

2.

DETERMINATION O P FUNDAMENTAL QUANTITIES

Figure 6 is a photograph of the very simple spectrometer used in this work. I t s focusing curved crystal was prepared a t the California Institute of Technology. The A-48 linear proton accelerator of Livermore was operated a t 3.7 Mev with a beam of 25 to 30 milliamperes of protons incident on a water-cooled target, usually of copper, upon which the target material for study had been deposited b y vaporization. Exposures of tens of hours were required. For the photographic emulsion on which the spectral lines were recorded, special plates with BOO-micron thick nuclear emulsions mounted on 0.030-inch thick glass were prepared and a

FIG. 6. Two-meter photographic bent quartz crystal spectrometer as used a t Livermore for the study of Coulomb-excited gamma rays produced by proton bombardment of targets in the A-48 linear accelerator and of neutron-capture gamma rays from the Livermore reactor.

whole technique of slow development for these thick emulsions was worked out. Thin glass was required t o permit bending t o conform t o the 1 m radius focal circle for which the plate holder was profiled. This is the first time t ha t Coulomb-excited y-ray spectra have ever been resolved by crystalline diffraction and recorded. Intense X-ray lines also appear in these spectra which serve to calibrate the wavelengths of the y rays with high precision. More than 42 Coulomb excited y lines from 12 different elements have been recorded and measured. I n addition 12 lines from G isotopes excited by (p,ny) and ( p , p y ) reactions and six lines from three isotopes excited b y deuteron beam activation have been recorded. Although it does not fall strictly in the category of curved crystal

TABLE I. List of Crystal Diffraction Gamma-Ray Spectrometers

No. 1 1 1 1

2

2 cn

1 1 1

Location Cal. Tech. Cal. Tech. Cal. Tech. Argonne N.L. Livermore R.L. U.C.R.L. Berk. Chalk River Chalk River

1 Chalmers Inst.

Goteborg, Sweden 1 Stockholm 1 U. Uppsala 1 Munich 1 C.E.N. Belgium

1 U.S.S.R. 1 Mass. Inst. of Tech. 1 Mass. Inst. of Tech.

Authors

Distance from crystal t o source (meters)

Boehm and DuMond Boehm and DuMond Boehm and DuMond B. Hamermesh et al. W. John et al. K. Crowe et al. J. W. Knowles et al. J. W. Knowles et al.

2 2 2

Ryde and Andersson

2

Ryde and Andersson 0. Beckman, P. Bergvall, and B. Axelson H. Maier-Leibnitz H. Pollak J. B. Borovskii and A. B. Gilvarg H. Mark and Rasmussen H. Mark and Rasmussen

Crystal .Bent quartz Bent quartz Bent quartz Bent quartz Bent quart,z Bent quartz Bent quartz Calcite

Type C-means Cauchois geometry D-means DuMond geometry M.S.-Moving Source S.S.-Stationary Source

Bent quartz

D M.S. D S.S. C Photographic 1) S.S. C Photographic D S.S. 1) M.S. Knowles 2-crystal transmission type D M.S.

2

Bent quartz Bent quartz

D D

4.65 2 1 6 2

Bent Bent Bent Bent Bent

D S.S. C Photographic ? C Photographic C Photographic

7.7 2 7.7 2

?

quartz quartz quartz quartz quartz

?

S.S.

61 6

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

inst,rument,s t o which the t,it>leof the present section limits us, it, would be unfortunat,e not, to make mention here of the beautiful work in two(flat)-crystal diffraction spectroscopy of nuclear y rays by J. W. Knowles and his associates at8 Chalk River. This method has permitted pushing t,he study of y-ray spectra (mostly neutron capture y rays) all the way up t o 6 MeV-the record so far for crystal diffraction methods. As is well known the extremely high resolution of the two-crystal spectrometer is obt,ained a t the cost of extremely low luminosity, the solid angle usable from each emitting nucleus being of order steradian or less but Knowles compensates for this in part by the use of very thick crystals (optimum thickness in fact, i.e., that thickness beyond which the attenuation by absorption outweighs the benefit of increased reflect,ing power or about 5 cm of calcite for 1-Mev radiation reflected from the (211) planes) and in part b y the use of extremely strong sources. A t Chalk River where one of t,he world’s most intense neutron fluxes is available, Knowles can have a t his disposal source strengths of a thousand curies or more. His technique is, however, limited t o locations affording such intense neutron flu x e~ .~ 7 Gamma-ray spectroscopy utilizing the methods of crystal diffraction has been slow t o achieve wide-spread adoption, probably because of the rather specialized and exacting t,echniques and instrumentation required which have frightened many workers. Kevertheless now that 13 years have elapsed since the first, successful operatioii of the Mark I instrument a t California Institute of Technology, it is possible to list in Table I some fifteen crystal diffraction y-ray spectrometers in use or under construction in various parts of the world.

2.2.3.3. Gamma-Ray Scintillation Spectrometry.* Extensive use of the scintillation method for measuring complex gamma-ray spectra was made possible by the development of end-window photomultiplier tubest with low noise and uniform photocathodes of large surface area, and b y the production of large single crystals of alkali halide scintillators. Recent. improvements in pulse amplifiers and pulse-height analyzers$- have greatly 27

J. W. Knowles, Atomic Energy of Canada Ltd., Chalk River Report KO. 6P1-42

(1857).

t See also Vol. 2, Section 11.1.3, Vol. 4, A, Sections 2.1.2 and 2.3.1, and this volume, Chapter 1.4. 1See Vol. 2, Chapter 9.6. __

* Section 2.2.3.3 is by G. D.

O’Kelley.

2.2.

DETERMINATION OF MOMENTUM AND EN ERG Y

617

speeded up the recording of pulse-height distributions from a scintillation spect’rometer. A precise technique for the study of complex gamma-ray spectra has evolved, which has made possible a number of studies not previously amenable to experimental investigation. For information on gamma-ray scintillation spectrometry beyond that included here, the on the subject. reader is referred to several SCINTILLATORS. An effective scintillator for 2.2.3.3.1. GAMMA-RAY gamma-ray spectrometry should have as many as possible of the following characteristics : 1. It should be of high density and effective atomic number, for high detection efficiency. 2. The emitted light should have a frequency near that for optimum response of available photomultiplier tubes. 3. The crystal should be transparent to its own light. 4. The index of refraction should be low enough to permit transmitting the light to the photomultiplier with a minimum loss due to critical reflections. 5 . A short decay time is useful where reliable performance is desired a t high counting rates and in fast coincidence circuits. 6. Because the energy resolution of the scintillation spectrometer improves as the number of photoelectrons produced by the photocathode is increased, the light output per Mev of energy dissipated in the crystal should be large. Only the inorganic scintillators have the high density required for efficient detection of gamma rays. Activated alkali halides are of greatest interest, since they are available in large cryst’als transparent to their own light, and have a high light output in the frequency range of a cesium-antimony photocathode. Other inorganic scintillators such as ZnS, CdS, and CaW0 4 are available only as very small crystals, which are generally not very transparent to the emitted light and have large indices of refraction. Thallium-activated sodium iodide is the most useful scintillator for gamma-ray studies. It has a very high efficiency and comparatively short fluorescence decay time (0.25 psec). The light is emitted in a band a t 1

W. H. Jordan, A n n . Rev. Nuclear Sci. 1, 207 (1952).

2

J. B. Birks, “Scintillation Counters.” McGraw-Hill, New York, 1953.

3 S. C. Curran, “Imninesccnce arid the Scintillation Counter.” Academic Press, New- York, 1!)53. 4 R. K.Swank, Ann. R e v . Nuclear Sci. 4, 1 1 1 (1954). 6 1 ’. It. Bcll, in “Beta- and Gamma-ltay Spectroscopy” (K.Siegbahn, ed.), Chapter 5. Interscience, New York, 1955. 0 W. E. Mott and R. B. Sutton, in, “Handbuch der Physik-Encyclopedia of Physics” (S. Flugge, ed.), Vol. 45, pp. 86-173. Springer, Berlin, 1958.

618

2,

DETERMINATION OF FUNDAMENTAL QUANTITIES

4100 angstroms about 800 angstroms wide and is well suited to photomultipliers with Cs-Sb photocathodes and glass windows (8-11 response). While large crystals of NaI(T1) can be grown with excellent transparency, they are hygroscopic and must be worked and used either under oil, in a dry atmosphere, or in a vacuum, to prevent the yellow discoloration resulting from iodine released in the presence of moisture. The index of refraction of NaI(T1) (1.77) is higher than desirable. When crystals are optically coupled to photomultiplier windows having a refractive index of about 1.5, much of the light is critically reflected back into the crystal volume. Rough grinding the crystal surfaces increases the probability that light, once critically reflected, can ultimately be directed by diffuse reflection onto the exit face of the crystal within the critical angle. Any light escaping the other surfaces of the crystal should be returned by the most effective reflector. Diffuse reflectors such as magnesium oxide and a-alumina are superior to specular surfaces. Recent work by Van Sciver' indicates that unactivated NaI also may be useful in some applications. Under gamma-ray excitation pure NaI emits light in a band centered at 3030 t o 3100 angstroms, depending on the temperature. Photomultiplier tubes with windows of fused silica (e.g., the RCA 6903) are suitable. The characteristic decay time of the scintillations from pure NaI varies between -lo-* second at room temperature t o 3 X 10V second at -190°C, a t which temperature the scintillation efficiency reaches a maximum value which is approximately equal to that of thallium-activated NaI at room temperature. Pure NaI is not very transparent to its own light-a 1-in. thickness reduces the pulse height by half. In spite of its opacity and the inability to employ optical seals at low temperatures, it has been possible to attain a resolution of 10% full width at half-maximum counting rate at 0.66 Cesium iodide also has interesting possibilities as a gamma-ray scintillator. I t is nonhygroscopic and can be prepared in large, colorless crystals. The high average atomic number and high density lead to a favorable absorption coefficient for photons. The scintillation efficiency of CsI(T1) has been reported as 0.28 that of NaI(T1).eThis is probably a minimum value, in view of the work of Knoepfel et al.,1° who demonstrated that the major portion of the CsI(T1) emission spectrum lay in the red, beyond the optimum response of conventional photomultiplier tubes having an 8-11 response. Although there is little information on the response of 7 W. Van Sciver, I.R.E. Trans. on Nuclear Sci. NS-3(4), 39 (1956); Nucleonics 14 (4), 50 (1956). 8 W. Van Sciver, private communication (May, 1958). W. Van Sciver and R. Hofstadter, Phys. Rev. 84, 1062 (1951). l o H. Knoepfel, E. Loepfe, and P. Stoll, Helv. Phys. Acta 29, 241 (1956).

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

619

CsI(T1) to gamma rays, the fluorescence decay time for excitation by electrons from 0.66-Mev gamma rays is 0.70 Ksec,l1and the pulse height is approximately proportional to gamma-ray energy. l 2 Exposure to light or intense gamma radiation induces a phosphorescence having a decay time of about 220 sec,15 which may be removedI4 by heating to 150°C; thereafter the crystal should be kept in total darkness. Unactivated CsI has been studied by Hahn and Rossel,16who found that, at 100°K, the pulse height for gamma-ray excitation was nearly twice that of NaI(Tl), and the absolute scintillation efficiency was larger than for NaI(T1). Pure CsI was found to yield a linear pulseheight-energy response from 40 kev to 1.3 MeV. 2.2.3.3.2. CRYSTALMOUNTING. Because sodium iodide crystals are very hygroscopic, they must be prepared and mounted in a dry atmosphere. A dry box, fitted with gloves and an airlock, is generally used. Dry air should be circulated through the box until the relative humidity is less than 5 % . The crystal may then be removed from its package and the outer surface removed by grinding with No. 80 grit emery paper. If the crystal retains its rough-ground appearance and no chalk-like formation occurs at the surface after 15 minutes, the surface water has been removed. Final grinding of all surfaces can be made with No. 120 grit emery paper. Before mounting the crystal it is advisable to measure and record its final dimensions, which may be needed for geometry calculations. The crystal may be attached t o the photomultiplier tube using a small quantity of Dow-Corning DC-200 silicone oil ( lo6 centistokes viscosity) or opthalmological white petrolatum* as the optical seal. Excess fluid should be removed from the vicinity of the optical joint, and any fluid on the exposed sides of the crystal should be removed by further grinding. An enclosure described by Bells for use when both crystal and photomultiplier have the same diameter is shown in Fig. 1. The 0.005-in. thick can serves to protect the crystal from the atmosphere and to support the diffuse reflector. The can is sealed to the photomultiplier tube by a ring of Apiezon Q or settable rubber, and a hypodermic needle is imbedded in the seal as shown. The can is then evacuated, which forces the crystal

* Opthalmological white petrolatum, obtained from Burroughs Brothers Manufacturing Company, Baltimore, Maryland, is suitable. 11 R. S. Storey, W. Jack, and A. Ward, Proc. Phys. Soc. (London)A72, Pt. 1, 1 (1958). 1 2 M. L. Halbert, Phys. Rev. 107, 647 (1957). 1s J. E. Francis and P. R. Bell, Oak Ridge National Laboratory Rept. ORNL-1975 (1955)-unpublished. 1 4 W. Beusch, H. Knoepfel, E. Loepfe, D. Maeder, and P. Stoll, Nuovo cimento [lo] 6, 1355 (1957). 15 13. Hahn and J. Rossel, Helv. Phys. Acta 26, 803 (1953).

A20

2.

D E T E R M I N A T I O N O F F UNDAME KTA L Q U A N T I T I E S

aiid phot omultiplier together hy atmospheric pressure. While pressing on the vacuum seal thc hypodermic. needle is removed. If the vacuum is lost during this operation, the small folds of the enclosure will have a slight resiliency, lacking in an evacuated can, which will signal the loss of vacuum. The arrangement just described is very stable mechanically and can be used either vertically or horizontally. -,ALL

A

SURFACES ROUGH-GROUND

\TINCUP FOLD SEALED WITH R-313 OR CIEA RESIN

Nol(T1) PHOSPHOR

Aa

ALUMINA REFLECTOR

r, 0.005-in.ALUMINUM CAN c -3 1 n

I ~

DC-200 OPTICAL JOINT

PHOTOMU LT I P L I E R

APIEZON-Q SEAL HYPODERMIC NEEDLE-

i TO VACUUM SYSTEM

FIG. 1. NaI(T1) crystal mounting arrangement for use when both crystal and photomultiplier have the same diameter. From Be11.6

Both the container and diffuse reflector surrounding the crystal should be thin, since scattering in these materials can produce spurious responses in the crystal from Compton electrons and degraded gamma rays. Containers made from 0.005-inch aluminum sheet are satisfactory. The can is conveniently formed on a mandrel about & in, larger in diameter than the crystal, as the reflector will require about in. and the

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

62 1

remaining space will provide a loose fit. A cylinder is formed from sheet aluminum using a “tin can” seam secured with Biggs’ bonding agent R313.” A cap is then made from an aluminum disc, carefully forming a tin can seam a t the edge which is cemented with R313 (see Fig. 1). The diffuse reflector will only stick to a clean surface. If any R313 cement has hardened inside the can, its surface must first be roughened with abrasive, following which the inside of the can is scrubbed with a brush, using hot water and detergent. After rinsing, the surface is dried using a heat lamp or oven. Four to six coats of a suspension of 10 gm of Linde A-5175 a-aluminat in 15 ml of sodium silicate and 25 ml of water are sprayed onto the inner surface, and then two to four coats of a mixture containing 10 gm of a-alumina, 8 ml of sodium silicate, and 25 ml of water are applied. It is important that each coat be thoroughly dried with a heat lamp before proceeding with the next one. The reflector performance may be improved by dusting dry a-alumina on the sprayed coat before placing the can over the crystal. While the container fabrication just described is eminently suited to packaging crystals of nearly the same diameter as the photomultiplier, other crystal and photomultiplier configurations can be enclosed using 0.005-inch aluminum sheet and the same reflector technique. For example, Davis et al.16 mounted three photomultiplier tubes on a 9+in. diameter crystal using an adaptation of the technique just described. 2.2.3.3.3. RESPONSEOF THE N A I ( T ~ )SPECTROMETER. The changes in the shape of the pulse-height distribution from a gamma-ray scintillation spectrometer as the incident gamma energy is increased are due principally to the energy dependences of the absorption coefficients for the three processes by which gamma rays interact with matter: the photoelectric effect, Compton scattering, and pair production. Partial absorption coefficients in NaI for these effects are shown in Fig. 2, which presents the absorption coefficients calculated by White,” with the coherent scattering removed.6 In a photoelectric interaction the gamma-ray energy is converted into kinetic energy of the ejected photoelectron and X-ray energy from the residual atom. At low gamma energies the photoelectric events occur near the crystal surface and the X-ray may thus be lost, but at higher energies the gamma ray undergoes the photoelectric effect at such a depth that the probability for capture of both photo*Obtained from Carl H. Biggs Co., 11616 West Pic0 Blvd., Los AngeIes 64, California. t Obtained from Linde Air Products Company, 30 East 42nd Street, New York 17, New York. 18 R. C. Davis, P. R. Bell, G. G. Kelley, and N. H. Lazar, I.R.E. Trans. on Nuclear S C ~NS-S . (4), 82 (1956). 17 G. R. White, Natl. Bur. Standards Rept. No. 100s (1952)-unpublished.

622

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

electron and X-ray is very large. The Compton e$ect divides the incident gamma-ray energy between an electron and a degraded photon. This scattered photon may have an energy nearly as high as the incident gamma-ray energy, and thus the degraded photon may be lost from a small crystal. If the scintillator is large enough, such degraded photons will be stopped either by photoelectric absorption or by further Comptoil scatters ending in photoelectric absorption. Multiple interactions of this

--Ex

PHOTOELECTRIC

ENERGY, Kev

FIG.2. Gamma-ray absorption coefficientsof NaI, corrected for coherent scattering. From Bell.6

sort transfer all of the gamma-ray energy to the scintillator, which stresses the importance of large crystals for gamma spectrometry. In pair production, the electron-positron pair share a total kinetic energy equal to the incident gamma-ray energy less the 2m& (1.02 MeV) required for producing the pair. When the pair is absorbed by the crystal, this kinetic energy is captured, but either or both of the two 0.511-Mev photons from the positron annihilation may be lost; therefore it is necessary to absorb the two 0.511-Mev quanta if all of the incident energy is to be detected. This requires crystals of rather massive proportions.

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

623

FIG.3. Response of a 3 X 3-in. NaI(T1) crystal to Sc” gamma rays, illustrating the loss of iodine X-rays from the crystal.

624

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

5

2

io'

5 0

W

cn \

0

w

w, 0

2

z

5

2

0.1

0

200

400

600

800

(000

PULSE HEIGHT

FIG. 4. Pulse-height distribution from a CrS1 source, 10 cm above a 3 X 3411. NaI(T1) crystal. From Heath.20

The practical importance of the effects just described are shown in the pulse-height distributions of Figs. 3-9, which present data for a range of gamma-ray energies and NaI (Tl) crystal sizes. A pulse-height distribution from the 0.155-Mev gamma ray of 3.4-day Sc4' incident on a 3 X 3-in. NaI crystal is displayed in Fig. 3. At such a low energy, the photoelectric effect in iodine is the principal absorption mechanism. The full-energy peak was fitted to a Gaussian shape, and the small peak on the low-energy side was subtracted as shown. This low-energy peak is due to loss of iodine K X-rays (-28 kev), and must be included as part of the

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

625

5

2 t o2

eY

5

W 0

-z

G ; 2

40’

5

’0

200

400

600 800 1000 1200 PULSE HEIGHT

Fro. 5. Response of various sizes of NaI(T1) crystals to gamma rays from a source. From Heath.10

Cs187

photopeak area in the quantitative determination of gamma-ray intensities described below (Section 2.2.3.3.4). It will be noted that the Compton distribution peaks rather sharply a t about 190 pulse-height divisions and is due largely to 180’ gamma-ray scattering from the environment. As the incident gamma-ray energy is increased, the photoelectric events occur deeper in the crystal, and the escape-peak intensity becomes very low relative to the full-energy peak. For example, the spectrum due to the 0.320-Mev gamma ray of CrS1shown in Fig. 4 exhibits no detectable escape peak, as this effect is small and falls within the energy resolution of the main peak. The influence of multiple Compton processes at higher energies is illustrated by Fig. 5, a comparison of the response of various crystal sizes to the 0.662-Mev gamma ray from a Cs13’ source. While only 33% of the recorded events result in complete energy absorption using a 1Q X l-in. detector at 0.662 MeV, 55% undergo complete absorption in a 3 X 3-in. scintillator. At the energy of the Znes gamma ray, 1.11 MeV, the fraction of total events in the full-energy peak (“peak-to-total ratio”) is 0.39 for the 3 x 341-1. crystal and 0.21 for the 16 x I-in.

626

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

1000

500

200

I

I in. X 1 Vz in.

CRYSTAL;7 100

50

20

10 - 3in.X 3in. - CRYSTAL5

k

1

2 PULSE HEIGHT

1

600

000

1

0

200

400

FIG. 6 . Pulse-height distributions from 1 X l*in. trometers using a Zn65 source. From Be11.5

1000

1200

and 3 X 3-in. NaI(TI) spec-

detector. The practical effect of this improved response for the larger crystal is shown in Fig. 6, which demonstrates that, in spite of a systematic decrease in peak-to-total ratio with increasing energy, the 3 X 3in.' crystal still produces a very useful response. Because about 1.5% of the Zn65decay events involve positron emission, a small peak due to the

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

627

FIG. 7. Response of 1; X 1-in. and 3 X 3-in. NaI(T1) crystals to the gamma rays of Na44.

0.511-Mev annihilation radiation appears on the Compton electron distributions of Fig. 6. Although the threshold for pair production is 1.02 MeV, this mechanism is not significant below about 1.5 MeV; thus, while the gamma-ray spectra of NaZ4of Fig. 7 show a very pronounced pair production effect at 2.76 MeV, none is seen from the gamma ray a t 1.38 Mev. Responses for 2.76-Mev gamma rays on both 3 X 3 and 1+ X 1-in. crystals show, in addition to the full-energy peak and Comp-

628

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

to3

5

2

io2

5

2

to

5

2

!

PULSE HEIGHT

FIG.8. Pulse-height distribution from a 3 X 3-in. NaI(T1) spectrometer, using the gamma rays from:the nuclear reaction N15(p,ay)CI2.

ton distribution, peaks corresponding t o pair production with loss of one or both annihilation photons, i.e., at 2.25 and 1.74 MeV. The 1.38-Mev gamma ray produces only a full-energy peak and Compton electron distribution. A t low energies the distribution rises because of environmental scattering. The superiority of the 3 X 3-in. spectrometer over small ones is illustrated by its peak-to-toal ratio of 0.21 at 2.76 MeV, which is about 3 times larger than that of a 14 X l-in. crystal. Figure 8 shows the

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

629

performance of a 3 X 3-in. NaI(T1) spectrometer for the 4.43-Mev gamma ray from the excited state of C'2 produced in the nuclear reaction N16(p,ay)C12.The pair peak from escape of a single annihilation photon (3.92 MeV) is more intense, and the pair peak from escape of both annihilation photons (3.41 MeV) is almost as intense, as the full-energy peak. A dotted line was drawn to indicate the expected Compton distribution, and the structure at low energies is due to low-energy gamma rays produced in the target assembly, annihilation radiation from pair production in the environment, and gamma scattering. A 3 X 3-in. NaI(T1) crystal is a good choice for general gamma-ray spectrometry up to several Mev. Uniform crystals of this size are commercially available, and when mounted on a 3-in. diameter photomultiplier tube (e.g., DuMont type 6363) are capable of as good resolution as smaller spectrometers. Loss of resolution and considerable increase in cost make it less desirable a t present to employ very large crystals for general use. However, a large NaI(T1) crystal may be very useful as a total absorption spectrometer. When a crystal is very large, and the gamma-ray source is located at its center, almost every interaction of a gamma ray with the crystal results in total absorption of the energy. If several gamma rays are in prompt coincidence, the high peak efficiency and essentially 47r geometry will cause all of the energy in the cascade to be absorbed and recorded as a single event. The pulse height of this sum peak is just equal to the pulse heights of the gamma rays in the cascade. An example of the measurement of Na24gamma rays with the source at the center of a 9+in. NaI(T1) crystal is shown in Fig. 9. The summing feature of large, total absorption spectrometers is very valuable in augmenting coincidence data, and frequently as a sensitive method of assay. It is desirable in the investigation of complex spectra t o employ a spectrometer which records only a single peak for each gamma ray. In this connection the two-crystal Compton spectrometer'* and the threecrystal pair ~pectrometer'~ have been employed. The Compton spectrometer records only those events in which a Compton electron is absorbed in a small NaI(T1) crystal and a scattered photon is detected in coincidence by a second detector placed at about 135' to the initial beam. The pair spectrometer employs three scintillation crystal spectromers in a straight line, with the incident gamma rays collimated onto the small, center crystal. A coincidence circuit is provided such that the events in the center crystal are analyzed only if the other two scintillation spectrometers each register the detection of a 0.511-Mev annihilation photon. The energy recorded will then be E, - 1.02 MeV. Above about 1*

R. Hofstadter and J. A. McIntyre, Phys. Rev. 78, 619 (1950).

1Q

J. K. Bair and F. C. Maienschein, Rev. Sci. Znslr. 22, 343 (1951).

630

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

2 MeV, the pair spectrometer is to be preferred in spite of its low efficiency, as its response is more nearly that of a single peak. Table I presents a brief comparison between the peak-to-total efficiencies for the 1+ X 1-in., 3 X 3-in., and 9+in. NaI(T1) crystals, and some results on the Compton and pair spectrometers. I 03

5

2

102

5

2

10

5

2

10

0

200

400

600

800

(000

1200

I400

FIG.9. Response of 9gin. “total absorption” spectrometer to Na*4 gamma rays, showing the intense coincidence sum line.

It is usually necessary to enclose the detector in a radiation shield, and the design of a shield which does not also introduce severe spectral distortions can be a serious problem. Spurious responses from the shield can arise from any of the processes by which gamma rays interact with matter, i.e., the photoelectric effect, Compton scattering, and pair production.

2.2.

63 1

DETERMINATION O F MOMENTUM AND ENERGY

The photoelectric effect with its attendant X-ray production is especially bothersome in a shield of high atomic number, because its cross section is high for low-energy gamma rays, whose spectra may be greatly distorted by X-rays, e.g., 72-kev Pb X-rays. Spurious radiation from this source can generally be reduced by the use of “graded” shields, in which TABLE I. Peak-to-Total Ratios for Some NaI(T1) Scintillation Spectrometers Peak-to-total ratios Type

Crystal size (in.)

Source geometry

Small crystal

1+ X 1

Uncollimated

Medium crystal 3 X 3 Large crystal

9i

Compton 1 X 1 Pair

1X 1

0.28 0.662 1.11 Mev Mev Mev

2.76 Mev

6.1

Mev

Efficiency

0.83 0.33 0 . 2 1 0.074

-

High

Uncollimated

0.85 0 . 5 5 0.39 0 . 2 1

0.08

High

Source inside well

1.0

0.65

High

0.88 0 . 7 4 0.69

Collimated onto 0.88 0 . 8 1 0.72 0 . 3 8 initial crystal Collimated onto center crystal

-

-

-

0.84

-

0.55 -5

-10-3 X

a high-Z material is covered with one or more materials in decreasing order of 2. These materials are chosen so as to efficiently absorb fluorescent radiation from the preceding one. The usual method is to cover Pb surfaces first with 0.03 to 0.06 in. of Cd, and then with 0.005 in. of Cu. Heathz0 has studied the effect of various shielding configurations, and Fig. 10 shows his data on the reduction of fluorescent X-rays in a Pb shield by graded liners. In a Pb shield with 6 X 6 X 18-in. internal dimensions and walls 4-in. thick, a CrS1source produced an intense Pb X-ray line, which was considerably reduced when the shield was lined with 0.030 in. of Cd. The dotted line shows the spectrum obtained inside a 32 X 32 X 32-in. Pb shield with the exposed surfaces covered with 0.06 in. of Cd and then 0.005 in. of Cu. The combination of reduced solid angle for scattering and detection of secondary radiation and the graded lining have reduced the fluorescent radiation below the limit of detection; the “backucatter” peak has also been reduced. Compton scattering is the principal cause of spurious radiation from *O It. 1,. Heath, Scintillation spectrometry gamma-ray spectrum catalogue. U.S. Atomic Energy Commission Rept. IDO-16408 (July 1, 1957)-unpublished.

632

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

the shield and is the most difficult to eliminate, as it occurs for all gammaray energies and its energy distribution is sensitive to the particular geometrical arrangement of source, shield, and detector. Figure 11 illustrates the effect of the shield configuration upon the scattered spectrum. The two upper curves show the scattering obtained when a MnS4source was measured in an unlined shield 6 X 6 X 18 in. made of Fe and Pb. 40

5

I

W

l

l

1 1 1

I

!

I

I

I

I

16"x 6",Pb SHiELD I I I

-z 0

10

5

2 \32"x 32"Pb SHIELD

PULSE HEIGHT

FIG. 10. Response of a 3 X 3-in. NaI(T1) spectrometer to CrS1 gamma rays in different shielding arrangements, showing reduction of fluorescent X-rays by a graded ahield. From Heath.20

It is apparent that the lower-2 material contributes much more scattered radiation, while the h i g h 3 Pb shield contributes the expected fluorescent X-ray line in addition to a broad backscatter peak, containing both single and multiple scattering components. The lowest curve again demonstrates the advantages of a shield with large inside dimensions. A 32 X 32 X 32in. Pb shield with graded lining was used, resulting in the elimination of fluorescent X-rays and a reduction in magnitude of the scattered spectrum, which then assumed more nearly a line shape ("backscatter peak"). These improvements in the scattered photon spectrum are a consequence

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

633

of the reduced solid-angle subtended by the detector for a point source on the shield wall. If high-energy gamma rays are measured in a scintillation spectrometer, pair production in the shield or other material in the vicinity of the I

1

-

!

1

I

I

5

2 to 0

1

5

\ 0

w T)

- 2 W

1-1T-I-

I$

IIT I

10

5

FIG.11. Effect of shielding size and atomic number on the response of a 3 X 3-in. NaI(TI) spectrometer t o Mns4gamma rays. From Heath.*O

source will contribute a low-intensity peak from the annihilation radiation. A gamma spectrum of Ca49is shown in Fig. 12,21 in which a prominent peak at 0.511 Mev is due to the 3.10-, 4.05-, and 4.68-Mev gamma rays in the sample. It is frequently necessary to remove the beta-ray contribution to the gamma spectrum by use of an absorber. Low-2 materials, such as polystyrene or beryllium, are satisfactory. Besides the attenuation of the gamma-ray intensity the presence of an absorber will influence the spectrometer response by production of electron bremsstrahlen and by gamma scattering. Bremsstrahlen production becomes especially important if the ratio of beta rays to gamma rays is very high, and is mania G.

D. O’Kelley, N. H. Lazar, and E. Eichler, Phys. Rev. 101, 1059 (1956).

634

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

fested as ail almost exponentially decreasing photon energy distribution from zero energy to t,he beta end point. The usual absorber geometry favors small-angle Compton scattering; hence, scattered photons reach the detector reduced only slightly in energy. This excess of counts near the full-energy peak tends to fill the valley between the full-energy peak

I

I

0

200

1.0

I

I

I

400 600 000 PULSE HEIGHT

3+

I

(000 1200

FIQ. 12. Annihilation radiation (0.511 Mev) from pair production in the walls of a lead shield by high-energy gamma rays of Ca49. From O’Kelley et aLe1

and the normal Compton distribution, so standard spectra for use in spectrum analysis should be measured with the same beta absorber used in measuring the unknown. The relationship between energy resolution and gamma-ray energy is important to the analysis of multicomponent gamma-ray spectra. Workers in this field have long been aware that the resolution is somewhat better a t low energies and worse at high energies than expected from statistical considerations of the photoelectric and electron multipli-

2.2.

DETERMINATION OF MOMENTUM A N D E N E R G Y

635

cation processes within the photomultiplier tube. Kelley et a1.22reported that the resolution characteristic of a photomultiplier is about 4% full width a t half-maximum counting rate, when a cathode-ray light flasher is employed, whose light pulses are equivalent to scintillations from a 0.66-Mev energy loss in NaI(T1). The observed gamma-ray line width W can be attributed t o a combination of the measured photomultiplier resolution Wp and an intrinsic scintillator resolution Wr as follows : W 2 = W p 2 W12.Thus, for a typical scintillation spectrometer resolution of 7.70/,, there is a resolution of 6.6% inherent in the crystal itself. The experiments indicated that neither crystal inhomogeneities, defects in the linearity of the electron energy response, nor light collection difficulties were responsible. A recent investigation by Leyteyzen and c o - w ~ r k e r s using , ~ ~ a gas-discharge tube light source and a large number of photomultiplier tubes, obtained an average resolution Wp = 5.3%. The average gamma-ray line width a t 0.66 Mev was about 9%, which implies an intrinsic resolution of about 7.3 %, in reasonable agreement with the result of Kelley et ul. I n practice, it is helpful to measure the resolution function for a particular detector over the entire energy range of interest. Such a resolution function is shown in Fig. 13 for two typical 3 X 3-in. spectrometers, in which uy is the half-width of the measured gamma line at l / e of the maximum counting rate, and h’, is the gammaray energy in the same units as a,. The ratio a,/E, is a dimensionless quantity characteristic of the detector only, and independent of the gain of the amplifier and photomultiplier tube. Because the line shape is very nearly a Gaussian distribution, the resolution expressed as full width a t half-maximum counting rate is simply 1.67 ( a , / E y ) . There has been widespread disagreement in the early literature between results obtained by various workers who have sought to measure the pulse height-energy relationship for NaI(T1). Pringle and StandilZ4 observed a nonlinear dependence of pulse height on photon energy below 150 kev, but reports by a number of other failed to disclose any nonlinearity between 1 and at least 660 kev. New data are available which demonstrate that the pulse height for

+

22

G. G. Kelley, P. R. Bell, R. C. Ihvis, and K.H. Lazar, I.R.E. Trans.on NztcZeur

Sci. NS-S(4), 57 (1956). a 8 L . G . Leyteyzen, B. M. Glukhovshoy, and I. Ya. Breydo, Kristallograjya 2(2), 290 (1957). 2 4 R.. W. Pringle and S. Standil, Phys. Rev. 80, 762 (1950). 2 6 H . I. West, W. E. Meyerhof, and R. Hofstadter, Phys. Rev. 81, 141 (1951). 2 0 C. J. Taylor, W. K. Jentsohke, M. E. Remley, F. S. Eby, and P. G . Kruger, Phys. Rev. 84, 1034 (1951). 2 7 V. 0. Eriksen and G . Jenssen, Phys. Rea. 86, 150 (1952). 2 8 R. C. Bannerman, G. M. Lewis, and S. C. Curran, Phil. M a g . [7] 44, 1097 (1951).

636

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

gamma rays is not proportional to gamma-ray energy. BernsteinZ9 observed a nonlinearity below 200 kev, using five NaI(T1) crystals of various sizes. The direction of curvature indicates that the slope of the pulse height-energy curve, AVlAE,, is greater a t low energies than at high energies. Extrapolation of the high-energy, linear portion of the curve intercepts the energy axis a t -25 kev. The work of EngelkemeirSO

M ev

FIG. 13. Resolution functions for two 3 X 3-in. NaI(Tl) spectrometers. KEY: X, 7.8%; 0, 8.6% full width a t half-maximum counting rate for 0.662-Mev gamma rays. The dashed line represents a resolution function which follows a l/E>Pdependence, normalized to the upper curve a t 0.662 MeV.

in the range of 10-1500 kev not only disclosed a pronounced nonlinearity below about 300 kev, but also showed a significant nonlinearity over the entire region investigated, with an energy intercept of - 16 kev. ManaganSoarepeated the measurements of Engelkemeir and obtained approximately the same nonlinearity and energy intercept. In all of these measurements, both the zero and linearity of the pulseheight analysis system are established by using precisely known pulses 29 W.Bernstein, Nucleonics 14(4), 46 (1956); I.R.E. Trans. on Nuclear Sci.NS-S(4), 143 (1956). ao D. Engelkemeir, Rev. Sci. Instr. 27, 589 (1956). W. W. Managan, unpublished data (communicated May, 1960); and Proceedings of the Sixth Tripartite Instrumentation Conference, Part 5 : Radiation Detectors. Atomic Euergy of Canada Ltd., Rept. AECG805 (August, 1959)-unpublished.

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

637

from a pulse generator; however, it is difficult to prove that the analysis system responds to the test pulses in the same way that it responds to the pulses from the photomultiplier tube. For this reason, Peelle and Love30bemployed a method which is independent of the zero of the pulseheight analyzer, and which does not depend very sensitively upon the linearity of the analysis system if energy calibration standards are available over the energy range of interest. In their technique, radioactive sources were positioned very close to a NaI(T1) crystal, and the apparent energy corresponding to each ‘‘sum’’ peak arising from simultaneous detection of cascade gamma rays was compared with the known sum of the individual members of the cascade. An average of the results from 3 experiments yielded an energy intercept of -30 k 6 kev, in good agreement with the nonproportionality observed by other methods. The nonproportionality of the pulse height-energy curve for gamma rays is consistent with recent experiments which indicate that the scintillation efficiency of NaI(T1) for electrons is low, being about 0.7 that for protons. This result is predicted by Meyer and Murray3OCfrom their model for the process of energy transfer from the incoming particle to activator sites, in which the low scintillation efficiency for electrons arises as a consequence of the low recombination probability for particles of very low dE/dx (see Section 2.2.1.2). 2.2.3.3.4. MEASUREMENT OF GAMMA-RAY INTENSITIES. One of the most important applications of scintillation spectrometry is in the measurement of gamma-ray intensities. Because the gamma spectrum emitted from most nuclei contains more than one gamma ray, the observed spectrum will be a summation of the responses to the individual gamma rays. If a set of standard spectra from monoenergetic gamma-ray sources have been measured under the same conditions as the unknown, the complex spectrum can be broken down into its components by first drawing in the shape of the most energetic gamma component, and then by successive subtractions the lower-energy components may be obtained. Should the gamma rays in the unknown spectrum prove to have energies which do not match the standards exactly, a Gaussian peak shape computed from the detector resolution function (cf. Fig. 13) may be fitted to the subtracted full-energy peak, as a check. The other features of the gamma-ray response (the “response function”) can be drawn in, using standard spectra in the same energy region as a guide. To test this quantitative method, Heath20 performed an analysis of a composite source made up of three activities, each emitting a single 80b R. W. Peelle and T. A. Love, U. S. Atomic Energy Commission Rept. ORNL2801 (October, 1959)-unpublished; Rev. Sci. Instr. 31, 205 (1960). 800 A. Meyer and R. B. Murray, I.R.E. Trans. on Nuclear Sci. NS-7(2-3), 22 (1960).

G38

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

gamma ray: Sc4’ (0.155 Mev), Be7 (0.478 Mev), and h4ti64(0.835 Mev). The results are shown in Fig. 14. The highest-energy component at 0.835 Mev was normalized at the full-energy peak to the standard spectrum shape for a gamma ray of that energy, which was then subtracted from the gross spectrum. The residual spectrum exhibited a peak at 0.478 MeV, to which the proper spectrum was fitted, and the difference

’0

200

400

600

800

1000

1200

PULSE HEIGHT

FIG. 14. Analysis of a composite gamma-ray spectrum by successive subtraction of standard spectral shapes. From Heath.lo

obtained. It is seen that the subtracted points fit the standard shapes very well except in the valleys where the statistical errors are large; however, for the determination of gamma-ray intensities only the peak area is needed. In Table I1 is shown the excellent agreement between the intensities of the individual sources and those obtained from analysis of the composite source. This analysis was idealized since the “response functions ” for the individual components were directly measurable. Generally, the accuracy of such an analysis depends on the number of components, their energies and relative intensities, and the extent to which the required response functions have been established.

2.2.

639

DETERMINATION O F MOMENTUM AND ENERGY

To obtain either the relative or absolute gamma-ray intensity, it is necessary to know the intrinsic peak efficiency, e p ( 7 ) , which is the probability that a gamma ray of given energy will cause a pulse falling in the full-energy peak if it strikes the crystal. The intensity of gamma radiation of energy E7 is related to the area under the full-energy peak by the peak efficiency and the solid angle which the source subtends at the detector. TABLE 11. Quantitative Analysis of Composite Gamma-Ray Sourcen Gamma rays emitted/sec Source MnK4 Be7 SC”

Per cent of total gamma rays emitted

Single spectra

Composite spectrum

Single spectra

Composite spectrum

Per cent error

54,731 4256 3633

4393 3567

87.4 6.8 5.8

87.3 7.0 5.7

0.1 3.2 1.8

a R. L. Heath, Scintillation spectrometry gamma-ray spectrum catalogue. U.S. Atomic Energy Commission Rept. IDO-16408 (July 1, 1957)-unpublished.

Because of the multiple processes which contribute to the full-energy 7 ) However, Lazar et dal and peak, it is difficult to calculate ~ ~ (directly. 7 ) e T ( 7 ) , the total efficiency for a Heathz0have obtained values of ~ ~ (using gamma ray, which is readily calculable, and R, the ratio of the area under the full energy peak to the area of the total spectrum, for sources that emit only one or two gamma rays; then, e p ( y ) = Re,(y). This method of determining e p is independent of the experimental conditions, provided the face of the crystal is fully illuminated by the source. Variations in resolution and environmental scattering do not affect the efficiency as defined here. Lazar et al. measured values of R, the “peak-to-total” ratio, using sources with single gamma rays to 1.11 MeV, but at 1.85 and 2.76 Mev only sources with two gamma rays were available. It was necessary to correct for the contribution to the total area from the lower energy gamma rays of Yes and NaZ4by fitting a Gaussian shape to the full-energy peaks for these radiations, and using the value of R from the low-energy data. Above 2.76 MeV, gamma rays from nuclear reactions were used. Values of ~ ~ (from 7 ) the data of Lazar et aZ. are presented in Fig. 15 for 14 X 1-in. and 3 X 3-in. NaI(T1) cylinders, and a 3 X 3-in. cylinder with 6 in. beveled off of the radius and height. These values are believed accurate to 7 % to 2.76 MeV. 31 N. H. Lazar, It. C. Davis, and P. R. Bell, I.R.E. rl’rons. OTL Nuclear Sci. NS-3 (4), 136 (1956); Nucleonics 14 (4), 52 (1956).

2.

640

DETERMINATION OF FUNDAMENTAL QUANTITIES

LL

W

r 0.40

i 2 a 0

2a 0.05

k

z

0.02

0.01

0.4

0.2

0.3 0.40.5

1.0

2.0 3.0 4.05.0

10.0

GAMMA-RAY ENERGY. Mev

FIQ. 15. Measured intrinsic peak efficiency of several NaI(T1) crystals. From Lazar et dS1

The absolute gamma-ray emission rate N ( y ) may be obtained from the area of the full-energy peak of a gamma ray P ( y ) by (2.2.3.3.1)

where A is a factor to correct for absorption in the source and in the beta absorber, and 0 is the solid angle subtended by the detector from the source. If the gamma ray whose intensity is required is coincident with another gamma ray, the area of its full-energy peak will be reduced by coincident summing of the cascade gamma rays. Adopting a nomenclature similar to that of Lazar and Klema13*the equation for the absolute emission rate then becomes

where P(rl)is the area of the full-energy peak due to yl,e P ( y l )and e-(yz) denote the peak efficiency for y1 and the total efficiency for yz, respec32

N . H. Lazar and E. D. Klema, Phys. Rev. 98,710 (1955).

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

G4 1

tively, v(0”) is the angular distribution funct)ioii of the two gamma rays integrated over the face of tlhe cxrystal evaluated as described by R ~ s e , ~ : ~ and q2,1is the number of y z i l l coincidence wit,h yl. Values of cT computed at Oak Ridge and the values of +It computed by Wolicki m d c o - ~ o r k e r s ~ ~ have been compiled in a review article by Mott and Sutton.6These calculations were extended by Heathz0and by Vegors et al.36to include the cases of point, line, and disk sources located on the axis of several sizes of NaI(T1) cylinders. If two intense cascade gamma rays are in prompt coincidence, measurement of the crossover gamma-ray intensity must involve a correction for the coincident sum peak from simultaneous detection of both gamma rays, with the total energy of the cascade transferred to the crystal. The expression for the area under the coincident sum peak, also derived by Lazar and Klema,32is

N , represents the contribution to the coincident sum peak by random summing, i.e., by addition of full-energy pulses from y1 and y z within the resolving time of the amplifier. If T is the clipping time of the pulse amplifier, (2.2.3.3.4) Nr = 2 T p ( Y 1)p(”2) where P(v) and P ( y z ) are the areas of the full-energy peaks due to y1 and 7 2 .

2.2.3.4. Determination of the Momentum and Energy of Gamma Rays with Pair Spectrometers.* Positron-electron pair production’ constitutes one of the basic interactions of gamma rays with matter a t energies above the threshold of 1.022 Mev. External pair formation can occur in the vicinity of a nucleus lying in the path of the gamma ray, the probability for the process increasing both wit,h the gamma-ray energy and with the M. E. Rose, Phys. Rev. 91,610 (1953). E. A. Wolicki, R. Jastrow, and F. Brooks, U.S. Naval Research Laboratory Rept. No. 4833 (1956)--unpublished. 35 S. H. Vegors, Jr., L. M. Marsden, and R. I,. Heath, Calculated efficiencies of cylindrical radiation detectors. U.S. Atomic Energy Commission Rept. IDO-16370 (September 1, 1058)-unpublished. 1 For a theoretical discussion see W. Heitler, “The Quantum Theory of Radiation.” Oxford Univ. Press, London and New York, 1936. 38

34

* Section 2.2.3.4 is by

D. E. Alburger.

642

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

square of the atomic number of the nucleus. Two other types of positronelectron pair emission which may take place are the formation of internal pairs in the field of the same nucleus responsible for the transition-a process in competition with both gamma-ray emission and internal conversion-and nuclear pair formation, a special case of internal pair conversion in which the emission of gamma radiation is completely forbidden-the so-called 0-0 transitions. A common feature of all types of pair formation is the continuous distribution, between the two components, of the energy available. While the total kinetic energy of the pair is a discrete quantity equal to ( E , - 1.02) Mev either of the particles can have any energy from zero up to the full energy. Another feature is the continuous angular distribution which the pair components have with respect to each other or with respect to the incident quantum in the case of external pair conversion. In the latter instance the number of positrons or electrons will be maximum a t an angle of -moc2/E radians with respect to the incident quantum, where E is the electron energy and moc2= 0.51 MeV. The NaI (Tl) 3-crystal scintillation pair spectrometer is a relatively simple device whose efficiency is not particularly affected by the angular dependence or energy division characteristics of external pair production. It consists of three crystals, with associated photomultiplier tubes and other circuitry, placed in line and a collimator shield which allows gamma radiation from a source to fall on the center crystal but not on the side crystals.* When a pair is produced in the center crystal the total kinetic energy is absorbed and the positron will subsequently annihilate and result in two oppositely directed 0.5-Mev gamma rays. If both of these quanta escape from the center crystal and are absorbed in the side crystals, the pulse-height spectrum of the center crystal in triple coincidence with the side crystals (each channeled on the 0.5-Mev full-energyloss peak) will display only the so-called “2-escape” peak. From a calibration of pulse-height versus energy for the center crystal using either the “singles” photopeak or the 3-crystal pair peak of a known gamma ray, the energy of a n unknown gamma ray may be derived from the 2-escape peak energy by adding 1.02 MeV. Relative intensities of a number of gamma rays in a 3-crystal pair spectrum may be obtained with an accuracy of about 10% by correcting the area under each peak by the pair production cross section for iodine.2 The absolute efficiency of the instrument depends not only on the pair cross section as a function of energy but on the source to center crystal distance and the sizes, shapes,

* For a discussion of some components used here see also Chapter 1.4 of this volume, as well as Parts 6 and 11 of Vol. 2. ZL. G. Mann, W. E. Meyerhof, and H. I. West, Jr., Phys. Reo. 92, 1481 (1953).

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

643

and exact geometrical arrangement of the three crystals. Bent and Kruse3 have calculated the absolute efficiency for a typical NaI(T1) %crystal pair spectrometer arrangement for gamma-ray energies from 1 to 10 MeV. A t 3 Mev it yields 1 pair count per 106gamma rays emitted from a source. Because of the various geometrical factors the accuracy in determining the absolute gamma-ray intensity is probably not better than 25%. An effective gamma-ray energy resolution4 of between 2 and 4% is possible for gamma rays of 1 to 3 MeV. At higher energies the escape of bremsstrahlung broadens the lines and results in a low-energy tail on each line. Aside from the somewhat better effective gamma-ray resolution than for detection with a single crystal the 3-crystal pair spectrometer gives a “one-to-one” correspondence between peaks and gamma rays. I t eliminates the continuous Compt,on distributions and its spectrum is clear and easy to interpret. Figure 1 shows the 3-crystal pair spectrum4 of the 1.38- and 2.75-Mev gamma rays of NaZ4compared with the singles spectrum, both recorded with a gray wedge pulse-height analyzer. An alternative form of the three-crystal pair spectrometer described by Bent and Kruse6 uses a plastic scintillator for beta-ray detection as the center crystal. If a source of pairs, such as a thin Van de Graaff target, is placed at the bottom of a well drilled into the plastic crystal one may observe pulses due to the pairs in triple coincidence with two NaI(T1) side crystals channeled on 0.5-Mev annihilation radiation. This device is especially useful for the study of nuclear pair emission from a target. In magnetic spectrometers* the energy division and angular dependence characteristics of pair production make it technically difficult to generate spectral lines, one for each gamma ray present. T o do so requires a combination of magnetic analysis and coincidence measurements with the attendant loss of a very large fraction of the events. The very first magnetic pair spectrometer6 consisted of a cloud chamber containing a pair converter foil and placed in a uniform magnetic field. Coincidence observation amounted to seeing the two pair tracks originating from the same point on the foil in a photograph of the chamber and the energies of the quanta incident on the foil could be deduced by measuring the radii of curvature of the two tracks. Data collection with this device is very laborious and the gas scattering of the electrons results in poor energy resolution. The modern version of the uniform field pair spectrometer was first

* See also Section 2.2.1.1.1. R. D. Bent and T. H. Kruse, Phys. Rev. 108, 802 (1957). D. E. Alburger and B. J. Toppel, Phys. Rev. 100, 1357 (1955). 5 R. D. Bent and T. H. Kruse, Phys. Rev. 109, 1240 (1958). 6L. A. Delsasso, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 61, 391 (1937).

a 4

50

-

20

> 10-

t

-

z

-

I n W

c

5-

-

2-

I

I

I

I I

0

I

I

2

I 3

GAMMD, ENERGY (MEV)

50

-

20

-

10.

>

-

I-

z W

c

z

. 5-

.

2.

I.

I I02

I

I

I

I

1

2

3

GAMMA

ENERGY ( M E V I

I 4

FIG.1. See opposite page for descriptive legend 644

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

645

proposed by McDariiel et al.? arid was developed further by Walker and McDanie18 and others.go10The principle of operation is illustrated in Fig. 2. Gamma rays from a distJantaccelerat>ortarget or reactor sample are collimated and allowed t o strike a thin converter foil located in the center of a uniform field between the poles of a n electromagnet. Pairs

u SCALE: CM

FIG. 2. Uniform field semicircular focusing pair spectrometer of Walker and McDaniel. The magnetic Geld direction is normal t o the plane of the paper. [Reproduced from Phys. Rev. 74, 315 (1948) by permission.]

produced in the foil will bend in opposite directions and can enter slits and counters. A coincidence between a given pair of opposite counters corresponds to a positron-electron pair whose sum of H p values, or momenta, is determined by the slit separation and the magnetic field setting. To a first approximation the sum of radii of curvature of positron and electron B. D. McDaniel, G. von Dardel, and R. L. Walker, Phys. Rev. 72, 985 (1947). Walker and B. D. McDaniel, Phys. Rev. 74, 315 (1948). BB. B. Kinsey, G. A. Bartholomew, and W. H. Walker, Phys. Rev. 77, 723 (1950). lo J. Terrell, Phys. Rev. 80, 1076 (1950).

* R. L.

FIG.1. A roinparison of the singles pulse-height spectrum of the 1.38- and 2.75-Mev gamma rays of Naa4(upper figure) with the 3-crystal pair spectrum (lower figure) taken a t the same gain setting and recorded on a gray wedge analyzer.

646

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

reaching the slits, a t a given gamma-ray energy, is the same for all points over a rather extended converter foil. As the field H is varied a coincidence peak will be observed when the sum of orbit diameters matches the slit separation. Because of the continuous nature of the energy division only a small fraction of the pairs produced in a given region of the foil will reach the counters. I n the spectrometer of Walker and McDaniel four counters are used on either side allowing 7 neighboring momentum points to be recorded simultaneously. The width and shape of the pair coincidence line obtained with the uniform field instrument are determined not only by the counter slit widths but by the finite divergence of the incident gamma-ray beam, electron scattering in the converter foil and by the fact that in higher orders of approximation the sum of positron and electron orbit diameters varies with their ratio. The converter size thus introduces a line spread. Because of the large dimensions of the converter foil the spectrometer can have no defining baffles and so to minimize the effects of electron scattering one must use thin foils of light elements such as beryllium or copper to achieve good resolution. The line obtained with the semicircular pair spectrometer generally has a steep high-energy edge which can serve as an accurate measure of the total pair momentum according to a method developed by Bartholomew and Kinsey.” Although their pair lines have a full width at half-maximum of -2% they have determined transition energies with an accuracy of 0.1 % using their analysis of the extrapolated end point. The efficiencyof this instrument can be expressed as the number of pair counts recorded at the peak of the line, using two counters only, per gamma ray emitted by the source. Terrell’O has studied the efficiency using a source-to-converter distance 1.5 times the counter slit separation. The full width at half-maximum of the coincidence line is -13% for gamma rays of between 2.6 and 4.4 Mev using a tin radiator 31 mg/cm2 in thickness. Under these conditions the efficiency at 2.6 Mev is 1 count per 8 X lo9 gamma rays and it increases by a factor of 110 in going from 2.6 to 4.4 MeV. The efficiency of Bartholomew and Kinsey’s instrument is of course very much less than Terrell’s both because of the better resolution used and because of the large distance between the converter foil and the reactor sample. They have found that the efficiency at 3 Mev increases with the fifth power of the gamma-ray energy. Amodification by Hornyak (cf. Siegbahn12)consists of curving the conl1 G. A. Bartholomew and B. B. Kinsey, Phys. Rev. 86, 605 (1952); Can. J. Phys. 31, 537 (1953). l2 K. Siegbahn, ed., “Beta and Gamma-Ray Spectroscopy,” p. 792. North Holland Publ., Amsterdam, 1955.

2.2.

DETERMINATION O F MOMENTUM A N D ENERGY

647

verter foil in such a way that under the divergent gamma-ray conditions employed by Terrell the resolution would be considerably improved. I n this way the spectrometer might be useful at good resolution with gamma rays from accelerator targets. A limitation of the uniform field pair spectrometer is th a t it is not practical for the study of internal or nuclear pairs. If one were to place a target at the converter position and attempt to study such pair emission one would observe lines only if the target were small and a series of baffles were located so as to define the positron and electron paths. However a very small coincidence yield would result because of the inherently low transmission associated with semicircular focusing. Another approach to the measurement of pairs has been the development of axially focusing pair spectrometers. Siegbahn and Johansson13 were the first t o suggest and demonstrate that a n axially focusing instrument could be used for this purpose. Their scheme consisted of moving the source of pairs off the axis of a thin lens, depending on the 90"rotation of the image to separate the positrons and electrons. Counters connected in coincidence were located a t the two image positions and recorded a coincidence line as the field was varied through the energy value 4 X ( E , - 1.02) MeV. It is only a t this energy t h a t both the positron and electron have about the same energy and can therefore be focused simultaneously through the spectrometer. Thus the spectrometer selects and detects only those positrons and electrons in a small energy slice a t the center of the continuous distribution of each. The disadvantages of the Siegbahn-Johansson method include the requirement of point focusing and the asymmetry of the source and detectors with respect to the baffles. An alternative idea suggested by Bame and BaggettI4and by Alburgerl5 was to leave the source on the axis and to record the arrival of a pair a t the common focal position by means of a statistical separation detector. If the image circle of confusion or final detecting area is split in half and two semicircular detectors are placed in these areas a coincidence count will occur a t a focusing energy of X ( E , - 1.02) Mev when the positron and electron happen t o enter opposite counters. Evidently only half of the pairs passing through the instrument will be counted provided the points of arrival over the detecting area are independently at random. I n spite of the factor of two loss in detecting the number of pairs one can then use in place of point focusing other axially focusing instruments which have very much higher transmission than the thin lens.

+

K. Siegbahn and S. Johmsson, Rev. Sci. Instr. 21, 442 (1950). S. J. Bame and L. M. Baggett, Phys. Rev. 84, 891 (1951). 16 D. E. Blburger, Rev. Sci. Znstr. 23, 671 (1952). 18 14

648

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

The most recent development along this line is the application of int,ermediate-image focusing16 to pair spectroscopy. Sliitis and SiegbahnI7 discovered the int,ermediate-image focusing principle using a n ironjacketed beta-ray spectrometer and found that a maximum transmission of 8 % of 4a occurred at, a momentum resolution of 4%. Lat,er Daniel and BotheI8 were able to obtain intermediate-image focusing with iron-free coils. Their basic design has been modified and improved by AlburgerI6 whose instrument has a resolution variable from 0.5 to 4% with corresponding transmissions of 1 to 8% of 4~ for internal conversion electron lines. The installation of a double-crystal detector a t the final focus converts the instrument to a pair spectrometer. This is illustrated in Fig. 3 which includes a recently modified and improved light-piping and coincidence detecting system. Because of the strong magnetic field it is necessary to pipe the light out to magnetically shielded photomultipliers.* Using Piiot-B scintillattion crystals the pulse-height spectrum a t the photo tube output, shows a line whose full width a t half-maximum is 14% when l-Mev electrons are focused by t,he spectrometer. This corresponds t,o a n improvement in light collection efficiency by a factor of 10 over a previous arrangement using 1P21 phot,omultipliers and smaller light pipes. Reasonably good pulse-height resolution is necessary in order t'o minimize or eliminate the coincidence background caused by the multiple scattering of single electrons from one crystal to the other. Direct scattering is prevented by a tungsten absorber placed between the crystals. The standard fast-slow coincidence technique is used where the fast portion of the circuit, having a resolving time of a few millimicroseconds, detects all coincidences regardless of pulse size and the slow part imposes pulseheight conditions by means of single-channel analyzers. When integral biases set t o accept >+ of the maximum pulse height are used in the channels i t is then not possible for a single electron of the energy being focused to multiply scatter from one crystal out to the walls or spectrometer baffles and back into the other crystal giving enough energy to trigger both discriminators. The only background observed is the random coincidence rate. Transmission and resolution tests of the intermediate-image pair spectrometer have shown that in favorable cases it, can yield pair lines, i.e., coincidence yield versus magnetic field current,, whose full width a t half-maximum is only O.G% in momentum. Expressing the pair transmission as t,he number of count,s por pair emit,t,edfrom n source t,he number * See d a o Vol. 2, Section l6

1 . 1 . 1 .:3.1.

D. E. Alburger, Rev. Sci. Instr. 27, (195(i). . H.Sliitis and K. Siegbahn, Arkiv Fysik 1, 339 (1949); P h p . Rev. 76, 1055 (1'349). H. Daniel and W. Bothe, Z. Naturforsch. 9a, 402 (1954).

-

0 2 4 6 8 012

INCHES

FIG. 3. Intermediate-image pair spectrometer. The light piping and coincidence detecting system ihas been improved over the design described in Rev. Sci. In&. 27, 991 (1956); see D. E. Alburger, Phys. Ref).111, 1586 (1958).

650

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

at 2.5% resolution is 6 counts per lo6source pairs, and is lower by a factor of 100 a t 0.6% resolution. In external pair conversion using 2.76-Mev gamma rays from NaZ4and a uranium converter foil the yield at 3 % resolution is 1 count per 7 X lo8 gamma rays emitted by the source. Comparing with the figures mentioned above for Terrell’s uniform field instrument the yield per gamma ray of the intermediate-image pair spectrometer at about the same energy is 10 times greater even though the resolution is better by a factor of 4. However, comparing with the efficiency of 1 pair count per lo6 gamma rays for the 3-crystal NaI(T1) 250

I

I

I

I

I

I

RELATIVE MOMENTUM

FIG. 4. Nuclear and internal pair coincidence lines occurring in the F19(p,a)016 reaction as observed with the intermediate-image pair spectrometer. The numbers labeling the three peaks are the transition energies in MeV.

scintillation pair spectrometer discussed above it is seen that the intermediate-image spectrometer is less efficient than the 3-crystal instrument at 3 Mev by a factor of 7000, when the resolution of both is approximately 3 %. The usefulness of the intermediate-image pair spectrometer lies both in its ability to achieve higher resolution and in its capability of studying nuclear and internal pair emission. The latter types of transitions observed with the 3-crystal spectrometer using a plastic center crystal give rise to pair lines6 -30% wide a t 3 Mev and -17% wide a t 6 MeV. The spectrum’6 of nuclear plus internal pairs in the F19((p,a)Ol6 reaction taken with the intermediate-image instrument at 1.7 % resolution is shown in Fig. 4. investigation^'^ of nuclear reaction gamma rays have been made a t l9 See, for example, R. D. Bent, T. W. Bonner, and R. F. Sippel, Phys. Rev. 98, 1237, 1325 (1955); R. D. Bent, T. W. Bonner, J. H. McCrary, W. A. Ranken, and R. F.

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

651

the Rice Institute using a similar instrument. It should be pointed out that in nuclear reaction work the Doppler shift of lines associated with fast transitions in recoiling nuclei must be corrected for and that the Doppler broadening of the lines may limit the resolution obtainable. In addition to the above-mentioned semicircular instrument proposed by Hornyak, several other pair spectrometer designs have been proposed which have not yet been tried. Jungerman20has suggested a multichannel homogeneous field pair spectrometer. This is essentially an axially focusing solenoid beta-ray spectrometer which uses a statistical-separation detector but which is also equipped with two sets of annular baffles which accept electrons over two angular ranges with respect to the axis. The possibilities that the positron and electron can pass through the same or different exit rings onto the statistical-separation detector increases the transmission over the use of one acceptance ring alone. Owing to the large distance along which the focused electrons cross the axis in a solenoid spectrometer the detectors must be either very large diameter flat crystals or small diameter long concentric cylindrical crystals. In either case the light piping problem is difficult. The iron-jacketed Slatis-Siegbahn intermediate-image spectrometer can also be used for pair measurement by piping the light from two statistical-separation crystal detectors out through a hole in the pole piece. A design of this type has been constructedz1at the Nobel Institute of Physics. Nielsen and Kofoed-HansenZ2have suggested that the multisection (6 orange ” beta-ray spectrometer could be applied as a pair spectrometer if the pole piece shapes were modified. The operation of each gap is analogous to that of the uniform field instrument. A novel pair spectrometer recently proposed by MalmforsZ3uses the time of flight of electrons in a magnetic field. The field produced in a ring-shaped magnet falls off as l/P. If the field is strong, i.e., 15,000 gauss, the electrons emerging from a source placed in the field will follow trochoidal paths and will drift away from the source with a velocity which depends on their energy but which is relatively independent of the emission angle. The time required to drift halfway around an annular Sippel, 99.710 (1955); R. D. Bent, T. W. Bonner, J. H. McCrary, and W. A. Ranken, 100, 771 (1955). 20 J. A. Jungerman, Rev. Sci. Instr. 27, 322 (1956). 31 J. Kiellman and B. Johansson, Arkiv Fysik 14, 17 (1958). 22 0. B. Nielsen and 0. Kofoed-Hansen, Kgl. Dansk. Videnskab. Selskab Mat.-fys. Medd. 29. No. 6 (1955). 2aK. G. MaImfors, in “Proceedings of t,he Rehovoth Conference on Nuclear Structure,” p. 506. North Holland Publ., Amsterdam, 1958.

653

2.

I)ETEH.MIS.\TION OF FUNDAMENTAL QUANTITIES

magnet will be -1 psec for electrons of a few MeV. Supposing that a pair is produced a t the source the two components will drift in opposite directions and if the energy is equally divided beti! een the two they will arrive a t the same time a t their respective detectors which are placed back to back halfway around from the source. The occurrence of a coincidence between the two vounters establishes the equal energy division and a measuremeiit of the time o f flight determines their energy. Evidently the mode of producing the pairs must be pulsed if a timing measurement is to be made. Using millimicrosecond timing techniques it should be possible t o derive the energy of the pair to an accuracy of better than 1%. The bunched beam of a cyclotron or a pulsed beam from a Van de Graaff accelerator could be used to induce nuclear reactions and the internal pair spectrum from a target would consist of coincidence pulses as a function of time following the bombarding pulse. No estimates of the pair transmission have been made although for single electrons the transmission is expected to be > 15% of 48.

2.2.3.5. Shower Detectors.* 2.2.3.5.1. INTRODUCTIOS. High-energy gamma rays or electrons may be detected and their energies measured by means of large scintillation or Cerenkov counters. It is also possible to use absorbers and radiators in combination with such large counters to utilize their capabilities most fully. In the case of gamma rays the quanta are made to materialize and the magnitude of their ensuing electromagnetic showers can be measured. If the counter is properly made the efficiency of detection can be as high as 100% and the energy of the gamma ray may be measured to *15% or better. This type of gammaray detector cannot at present compete with the well-known pair spectrometer as far as energy resolution is concerned but its efficiency greatly exceeds that of the pair spectrometer. A diagram of a typical practical shower detector is shown in Fig. 1. The counter shown in the figure may be used to detect either electrons or gamma quanta incident on the left and entering the porthole of the counter. Light produced in the crystal is detected by t,he photomultiplier a t the right. 2.2.3.5.2. NEEDFOR SHOWER DETECTORS. Why is such a detector needed? There are at least two answers to this questtion. (a) The efficiency of other methods of detection is low compared to that obtainable with shower detectors. I n high-energy experiments where the cross section is very small, e.g., in the Compton effect of the proton,

*Section 2.2.8.5 is by R. Hofstadter.

A-SILICONE

OIL

‘-PHOTOMULTIPLIER

GLASS WINDOW SCALE

INCHES

FIG. 1. This figure shows a typical large shower detector. In this wsr the detector itself is a NaI(T1) scintillation counter. The scintillation counter may easily be replaced by a Cerenkov light emitter with very few other changes. Between the NaI(T1) crystal and the photomultiplier lies a Lucite light coupling lens. The conical end shape of the scintillation counter is determined by the present crystal growing techniques and is known to be not optimum for light collection purposes. This counter has been used to detect electrons up to energies of 600 Mev with the results shown in Figs. 7, 12, and 13 of this paper. The counter was developed and used hy Knudscn and Hofstsdter.’!

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DETERMINATION O F FUNDAMENTAL Q U A N T I T I E S

no loss of efficiency can be tolerated. Hence the shower detector forms an important element in this experiment.’ (b) As the energies of man-made machines become higher and higher, the gamma rays and electrons appearing as bremsstrahlung, pairs, reaction products, etc., will also have very high energies. It will become increasingly difficult, if not impossible, to employ magnetic means to measure the energies or momenta of such particles. For example it requires something of the order of 100 tons of iron to provide a magnet which can bend 1.0-Bev electrons and still have a suitable solid angle and collection efficiency. If the energy is raised to 10 Bev the magnet will scale somewhere between the second and third power of the energy and its weight becomes of the order of the heaviest battleships ever built. To go on beyond this extreme value seems to be unreasonable at the present time. On the other hand a shower detector, of the kind we envisage, increases in size only logarithmically with the energy of the gamma ray or electron. For example, if a transparent material such as PbF2 could be developed in the form of a cylinder 80 cm long and 40 cm in diameter it would successfully contain a shower of energy 10 Bev. If a sodium iodide (Tl) scintillator is used for this energy its size would be about 1.2 m long and 1.0 m in diameter. Ninety per cent of a 10-Bev shower would be retained in such a crystal. These dimensions are rough and are merely presented to show the order of magnitudes involved. 2.2.3.5.3. METHOD OF OPERATION. When an energeticelectron orgamma ray* is permitted to enter a large crystal it develops a “cascade electromagnet,ic shower.”2 This shower is produced because the electron radiates electromagnetic quanta in any material it strikes. The quanta then produce electron-positron pairs. The electrons and positrons radiate again and so the chain is continued. The higher the energy of the incident electron the more intense is the subsequent shower. Thus the number of electrons and positrons will become greater and greater. The shower consists of electrons, positrons, and gamma rays all of which will be eventually degraded into very low-energy electrons and positrons (1.0 ev to 1.O MeV) . Now each of these electrons or positrons produces light, either by scintillation processes, or by cerenkov radiation; and the sum total of the light produced may be measured in conventional ways by a single photomultiplier or by a combination of photomultipliers. * Let us confine our attention to electrons, since gamma-measuring devices ultimately reduce t o electron detectors, and in fact, both detectors are very similar. ’L. B. Auerbach, G. Bernardini, I. Filosofo, A. 0. Hanson, A. C. Odian, and T. Yamagata, CERN Symposium, Geneva p. 291 (1956); see also I. Filosofo and T. Yamagata, ibid. p. 85. B. Rossi, “High Energy Particles.” Prentice-Hall, New York, 1952.

2.2.

DETERMINATION OF MOMENTUM A N D ENERGY

655

The total amount of light produced is expected to be proportional to the total track length2 of the particles appearing in the shower and hence proportional to the energy of the shower itself. In this way the height of the pulse produced by the photomultiplier will be proportional to the energy of the incident particle. We see that the basis of the method is the transformation, by electromagnetic means, of the kinetic energy of a single particle into that of a great many less energetic particles of smaller penetrating power. The less energetic particles can easily be contained in a relatively small volume. In actual practice the crystals will be surrounded by a shield carrying a small opening in one end permitting the entrance of the incident particles. The weight of this shield should not be underestimated although, of course, it does not approach the weight of a large magnet. The method outlined above was first employed by Kantz and Hofstadter3and by Fregeau and H ~ f s t a d t e rIt . ~has since been used by many authors, such as Cassels el CZZ.,~ Filosofo and Yamagata,6 Knudsen and Hofstadter.7 This method will not apply to heavier particles such as protons because heavy particles radiate very little compared to electrons and are stopped mainly by collision losses and nuclear reactions. 2.2.3.5.4.SIZEOF SHOWER DETECTOR. An important practical question arises immediately: how large does the detector have to be to measure the energy of the incident electron or gamma ray in order to obtain thereby a given accuracy? This type of question can be answered by reference to a diagram such as that shown in Fig. 2. In this case the data are taken from reference 3 and represent the amount of energy lost in a given ring of material in a tin* absorber when struck by electrons of 185 Mev initially traveling along the direction of the axis labeled “depth.” This axis represents the long axis of the specimen and the figure is considered to be a section of a cylinder obtained by rotation about the “depth” axis. The figure is then to be interpreted according to the following example. Consider a ring corresponding to an interval of depth in the specimen between 2 and 3 radiation lengths and lying between l and 2 radiation lengths in radius. (For

* Experiments have been carried out in tin and other metallic absorbers. Since a transparent material has to be used for an actual counter, tin is of course not suitable. NaI(T1) would give very similar results, however (see text). a A. Kants and R. Hofstadter, Nucleonics 12 (3), 36-43. 4 J. Fregeau and R. Hofstadter, see reference 3. 6 J. M. Cassels, G . Fidecaro, A. M. Wetherall, and J. R. Wormald, Proc. Phys. SOC. (London) A70,404 (1957). 6 T. Filosofo and T. Yamagata, CERN Symposium, Geneva p. 85 (1956). 7 A. Knudsen and R. Hofstadter (to be published).

656

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DETERMINATION OF FUNDAMENTAL QUANTITIES

tin, one radiation length equals 1.22 cm.) This ring will absorb on the average 2.40 % of all the energy of the incident electxon of 185 MeV. Other rings correspond to other amounts of absorbed energy. By adding together all the energy absorbed in various rings one arrives a t the fractional amount of energy absorbed in a figure of revolution of whatever size one chooses in the diagram. If it is desired to select 90% of the original energy as a typical absorption figure, then this implies, e.g., a total depth of a tin cylinder of 10.5 radiation lengths and 12 radiation lengths in diameter. Other combinations are possible, of course, to secure 90% absorption. Such calculations are assisted materially by diagrams of t,he type indicated in Figs. 3 and 4 which apply to tin and lead respectively a t 185 MeV. Such “isoenergetic” curves permit one to choose the dimensions needed to absorb a certain amount of energy from the initial energy of

FIG.2. A block diagram showing the amount of energy lost in a given ring of material in a tin absorber for 185-Mev electron showers. The details of the figure are explained in the text.

185 MeV. For example it is not possible to choose a cylinder of tin less than 9 radiation lengths long, or one less than about 9 radiation lengths in diameter if one is to obtain a 90% absorption figure. I t is to be noted that these figures apply to tin (which incidentally is quite similar to NaI(T1) except for a density allowance which alters the scale in absolute centimeter units) at a certain energy, namely 185 MeV. At a different energy the behavior will be different. For example, as the incident energy is increased the peak of the shower absorption moves towards greater depths in the cylindrical specimen. However, from shower theory the qualitative features are probably valid a t even much higher incident energies. I t is unfortunate that, more data of the type shown in Figs. 2, 3, and 4 arc not available. However, atj 185 Mev data are given by Kantz and Hofstadter for several materials including (*arboil,nlumiiium, copper, tin, and lead. The size of the crystal, as we have seen above, is intimately related to the amount of energy absorbed in the cryst,al. The amount of energy is,

2.2.

DETERMINATION OF MOMENTUM AND E N E R G Y

657

FIG.3. Isoenergetic curves for 185-Mev electron showers in tin. Any point on a n isoenergetic curve corresponds to the same fraction of the energy of a shower absorbed by the material with the given outside dimensions.

I

Depth (Radiation lengths)

I

FIG.4. Isoenergetic curves for 185-Mcv clertron showers in Pb. This figure is sirnilnr to Fig. 3.

(358

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

in turn, related to the question of the resolution of the crystal with respect to energy and to the linearity of the response of the shower detector. These two considerations can be understood easily from the discussion that follows. It is clear that if 100% of the energy is absorbed then the response of the detector will not suffer from straggling difficulties. In other words, if the transformation of light into pulse height is reasonably good (this point will be discussed later) every absorbed electron or gamma ray of a given energy will give rise to a pulse of the same magnitude. This means that the response function of the detector provides a unique pulse height for a given energy. I n actuality, due to straggling losses, the crystal detector is never as good as this. Instead, when the detector absorbs on the average only 90% of the incident energy, the 10% average loss results in a finite width of the pulse distribution. Occasionally a large quantum escapes from the back end of the crystal before interacting with it. For example, there is a finite probability that a 50-Mev quantum can escape, and if the incident energy is, say, 200 MeV, this escaping energy can produce a very large fluctuation in the pulse-height distribution. An approximate method of computing the pulse-height distribution has been given by Kantz and Hofstadter3and further calculations of this kind for NaI(T1) have been made by Knudsen and Hofstadter.7 We shall indicate below how such calculations are made. The statistical fluctuations in the “lost energy” will be approximated by a Poisson distribution. Take an example: consider a cylinder of lead fourteen radiation lengths long and fourteen radiation length‘s in diameter. A transparent Cerenkov crystal of PbF2 will behave in much the same way as lead. This specimen will capture approximately 90%, on the average, of a 200-Mev shower produced by an electron. This can be determined from Fig. 4. Thus a fraction of 10% of the incident energy escapes from the cylinder. We will assume that the radiation escaping from the greatest depths of the crystal will have the greatest penetrability. The energy of such radiation will lie at the minimum of the absorption cross-section curve.2 For this curve the abscissa is proportional to the gamma-ray energy and the ordinate to absorption coefficient. For lead the minimum occurs a t about 3.5 MeV. Thus the escaping 20 Mev will be made up in approximately 5.7 parcels of gamma rays, each corresponding to 3.5 MeV. Any actual distribution will not be as good as this because sometimes higher energy quanta will escape, since the absorption curve has a rather flat minimum. Now the distribution can be computed by applying Poisson statistics to the 5.7 gamma rays of 3.5 Mev. Such a calculation gives the curves in Figs. 5 and 6. Figure 5 shows the spread in energy in the response curve for Pb and PbF2for an incident energy of 185 In this case the peak of the curve is labeled 185 Mev and provides a nominal calibration figure. This

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

659

Energy in Mev (Pulse height)

FIG. 5. This figure shows the results of a crude theory of the shape of the pulseheight distribution expected in lead and in PbF2for 185-Mev electrons. Because of the small nuyber of photoelectrons a t the emitting surface of the photomultiplier used with the Cerenkov material, PbF2, the statistical fluctuations of emission produce a width greater than t h a t expected from pure straggling in Pb. I n fact due to the occasional escape of gamma rays with energy larger than 3.5 Mev the actual pulse-height distribution may be slightly broader and will have a tail on the low-energy side.

explains why higher energies than the incident energy are apparently present in the beam. Figure 6 shows the corresponding calculations in NaI(T1) when 200 Mev and a 20% figure for the escaping energy are used and when different choices near the minimum of the gamma-ray absorption curve are employed as the quanta of escaping energy. It is observed that shapes vary but are not widely different. Using the minimum of the absorption curve the calculations give the solid line of Fig. 7.7 The dashed line shows the actual data obtained experimentally.? As expected, the actual curve shows a low-energy tail corresponding to the escape of a few energetic gamma rays. For a rough theory the check can be considered rather satisfactory. *

* Note added in proof: Recently a very important contribution to the technique of obtaining high resolution in large NaI spectrometers has been made by B. Ziegler, J. M. Wyckoff, and H. W. Koch, by choosing a combination of crystals in a suitable manner and selecting coincident events so t h a t a single annihilation photon escapes from the ’ front surface of the main spectrometer crystal. In this way a spectacular figure of 3; %

660

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

I40 W

z 0 V

II.

40

0 0

=

20

n -

100 I20 140 160 180 200 ENERGY ABSORBED BY C R Y S T A L ( M E V )

70

FIG.6. The calculated line shapes are presented for straggling losses of 20% corrcsponding to twenty 2-Mev gamma rays and eight 5-Mev gamma rays for 200-Mev electron showers in NaI(T1). The calculations were made using Poisson statistics. This figure probably shows the extremes to be expected, except for the low-encrgy tail described in the legend t o the previous figure. The appropriate widths are shown in the figure and actually straddle the experimentally observed width, except for the tail. 2 . 2 . 3 . 5 . 5 . SIMILARITY RULE.Unfortunately the present detailed shower data have been obtained only a t 185 Mev and at lower energies. However, we may use shower theory t o help extrapolate the data to higher energies. For example, in shower theory2 (approximation B ) the behavior of all materials is the same if the lengths are expressed in radiat,ion unit,s and the energies are expressed in the units of “critical energy” = Eo.We define the critical energy as the collision loss per radiat,ion length of material for electrons of energy h’o. In actual practice this energy loss is the same as the collision loss per radiation length for electrons a t the minimum of the energy loss curve. If we apply this rule it is easy to convert the shower data of Kantz and Hofstadter to higher energies. We can use tin, as before, to represent the

energy resolution was obtained for 17.6 Mev gamma rays with a 9-in. diameter main crystal. The detection efficiency for 17.6 Mev gamma rays is stated to be about 7%. If this technique can be applied to high energies, which seems possible, qiiite good resolution can he achieved with crystal spectrometers.

2.2.

DETEIIMINATION O F MOMENTUM A N D ENERGY

140

66 1

c

CALIBRATION ENERGY ( M E V ) I I I I I 1 100 120 140 160 180 200 ENERGY ABSORBED BY CRYSTAL ( M E V )

I 80

Fro. 7. This figure is similar to Fig. 6 except that elcven 3.65-Mev gamma rays are cmploycd to arcount for the average energy loss. The energy 3.65 Mev lics a t thc minimum of the gamma-ray ahsorption curve for NaI(T1). The experimental curve is also shown In t h e figrirr.

iodine of NaI(T1) and can do similar things for carbon and carbon-bearing scintillators or Cerenkov materials. Thus NaI(T1) at 300 Mev is equivalent, to P b a t 185 by the similarity rule. Also 2.4-Bev electrons in carbon correspond to 6he Pb data at, 185 Mev. TTnfort,unately, we cannot extrapolate the data for heavy elements any further. I n any case this type of extrapolation must be considered only as a rough approximat,ion to the truth. 2.2.3.5.G. OTHERT H E o R E T I C i l L AIM. 1%.It. Wilson' has performed Some interesting Monte Carlo calculat,ioris OH t.he development of photon and electron-inhiated showers in lead. He has chosen to investigate incident, energies in the range 50-500 Mev. From these calculations rough est,imates can be made of the lengt,h of ,z shower detector only if the diameter of tmhecounter is very large. For pract,ic:aldiameters of Cerenkov or scintillat.ioii counters the Wilson estimates of the percentage of the shower retained in a sample cannot be made wit,h good accuracy. The reason is BR. R. Wilson, Phys. Rev. 86, 261 (1952).

662

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

that radial spreading of the shower is considered in the theory only in the roughest approximation, by including effects of multiple scattering. Radial, i.e., sidewise, propagation of annihilation radiation and low-energy electrons is not considered in the theory. For information regarding the core of the shower, Wilson’s calculations may be very useful. An example of Wilson’s results is given in Fig. 8. This figure shows the number of electrons that would be counted in the core of a shower with paths in the

tFIG.8. Wilson’s* calculations for 20-500-Mev showers in lead. The figure shows the number of electrons expected in the core of a shower with paths in the direction of the shower. The abscissa gives the depth in lead of the detector when expressed in radiation lengths.

direction of the shower. The abscissa is the depth in lead in radiation lengths. The incident radiation consists of photons of the appropriate energies in Mev given in the figure. Other types of useful “transition”2 curves are also given in Wilson’s paper. A simple shower theory, neglecting radial spreading of the shower is also included in Wilson’s paper. The shower calculations of other authors have been compared by Wilson with the results of his own theory. For example the results of Rossi and Greisen9 and Arley’” are quoted by him. Some older shower calculations may still be useful to the reader and B. Rossi and K. Greisen, Revs. Modern Phys. 18, 240 (1941). Niels Arley, “On the Theory of Stochastic Processes and Their Application t o the Theory of Cosmic Radiation.” G.E.C. Gad, Copenhagen, 1943. lo

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

663

for convenience, they are quoted For air showers radial spreads are estimated by Molihre, l6 Blatt,16 and Roberg and Nordheim.17 Experiments on radial spreads of copper (y,r]) threshold gamma rays were

gL L

3 2 MEV

79 MEV

> m a a

5

iL 5 0 MEV

r

9 5 MEV

r

1 2 5 MEV

V

c -I > 0 a

??A!%-b 3

8

0

8

16

24

32

0

8

16

24

32

40

VOLTS

FIG.9. The actual response, measured by Cassels et a1.6 of a eerenkov shower detect,or, to positrons in the energy range 32-125 MeV.

made by J. W. Rose1*in two materials: copper and lead. Transition curves for 330 Mev bremsstrahlung were also measured in various materials by Blocker, Kenney, and Panofsky.l9 2.2.3.5.7. EXAMPLES OF DATAOBTAINEDWITH SHOWER DETECTORS. Many investigators have employed large shower detector~’.~-~ and it is posH. J. Bhabha and W. Heitler, Proc. Roy. SOC.A169, 432 (1937). J. F. Carlson and J. R. Oppenheimer, Phys. Rev. 61, 220 (1937). 13 H. S. Snyder, Phys. Rev. 76, 1563 (1949). l 4 I. B. Bernstein, Phys. Rev. 80, 995 (1950). 16 G. MoliBre, Naturwissenschaften 30, 87 (1942); also W. Heisenberg, “Cosmic Radiation.” Dover, New York, 1946. 1.3 J. M. Blatt, Phys. Rev. 76, 1584 (1949). 17 J. Roberg and L. W. Nordhein, Phys. Rev. 76, 444 (1949). l 8 J. W. Rose, Phys. Rev. 82, 747 (1951). 1 9 W. Blocker, R. W. Kenney, and W. K. H. Panofsky, Phys. Rev. 79, 419 (1950). 11 12

064

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

sible to give practical examples of the type of data that can now be obtained. Figure 9 t,aken from reference 5, shows the response of a &mnkov shower detector to positrons in i8heenergy range 32-125 Mcv. In this case Chssels et aL6 constructed their gamma spectrometer of “two right cylinders of Chance EDF 653335 glass, cach 5 in. in diameter and 4 in. long. These were placed in optical contact, to make a cylinder 8 in. in total length. All surfaces were polished and a 5-in. EM1 photomultiplier with a photocathode sensitivity of 37 pA/lumen was placed in optical contact. with one end.” An external lead convertor was used at the input end of the spectrometer and the pairs produced in the lead were detected by a thin count,er placed in coincidence with the shower detector. The purpose of

TRANSMISSION

3000

4000

5000

6000

WAVELENGTH (A)

FIG. 10. Physical properties of the Cassels shower detector. Details are given in the text.

this coincidence counter was to guarantee that the showers in the glass were generated near the axis of the counter. In this way the straggling is reduced and the resulting light pulses are more homogeneous. The glass was surrounded by a coaxial white shield and the photomultiplier was shielded magnetically by coaxial soft iron and mu-metal shields. The radiation entered the shower detector through a thin polished light-tight aluminum cap. Their results, as shown in Fig. 9, were observed with a 50-channel pulse analyzer and exhibit increasing pulse height as a function of increasing energy, as expect,ed. The glass used had a density of 3.9 grams/cm3 and a refractive index of 1.69. The radiation length of this material was 2.56 cm and its critical energy 16 MeV. Figure 10 shows some physical properties of the Cassels shower detector. The figure shows the transmission of light through 10 radiation

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

665

lengths of glass as a function of wavelength. The response (quantum efficiency) of the photocathode and the spectrum of the Cerenkov radiation, are all given in arbitrary units. A type of glass possessing a higher response a t ultraviolet wavelengths would give still better performance. The data in Fig. 10 have been analyzed by Cassels et aL6 and plotted in Fig. 1 I . The mean pulse height of the spectrometer is proportional to the energy and the half-width is approximately proportional to the square root of the energy. The data in Fig. 9 clearly show that the gamma rays are detected with very nearly 100% efficiency. Similar results have been obtained by Knudsen and Hofstadter' who used a NaI(T1) scintillation counter. The dimensions of their counter are shown in Fig. 1. In Figs. 12 and 13 we see the various pulse-height spectra 30 1

10

c

/

0.15

Y 50

100 MeV

150

0

0.01

0.02 0.03 0.04 (MeV)-'

FIG.11. The spectra given in Fig. 9 have been analyzed to give the mean pulse heights and line widths plotted in this figure. The data are taken from Cassels et d 6

observed a t several energies for electrons between 125 and 600 Mev and also the pulse heights a t the maximum of the peaks as a function of the incident energy. Here again we see that practically 100% detection efficiency is realized and a very good linear response is obtained. Energy measurements are therefore quite simple after a calibration of the spectrometer has been made. The nearly 100% efficiency of the detector proves to be a great advantage. Other examples of line shapes are given in reference 6. 2.2.3.5.8. LIMITATIONS OF THE METHOD. One of the important advantages of the shower det,ect,or is the relative absence of systematic errors in obtaining an energy spectrum, since the efficiencyisnearly loo%, and since the whole spectrum is obtained simultaneously. Other advantages of a single counter in taking a spect
666

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

(1) The presently obtained resolution in energy is of the order of 10 to 20%. This may be improved with the development of better cerenkov materials and highly transparent glasses or crystals [PbF, (?)I. In the case of sodium iodide the material is clear and transparent and the presently used optical mountings can probably be greatly improved. It seems likely that the conical surfaces of the NaI(T1) crystal shown in Fig. 1 produce a relatively poor pulse-height distribution a t lower gamma-ray energies (Co60 radiation),20and these surfaces are also harmful at the 160 125 MEV

200 MEV

350 MEV

500 MEV

600 MEV

I40 ul I-

4 U 0 LL

0

lY m W

3z

60 40

20

ELECTRON

ENERGY

FIG.12. This figure shows the various pulse-height spectra obtained by Knudsen and Hofstadter' for electron showers between 125 and 600 Mev in NnI(T1). The peak of each curve has been placed a t the energy of the incident electrons even though the actual energy absorbed by the crystal must be less than the incident energy. This assignment is of course part of the procedure of calibration. An arbitrary observed peak would be associated with the energy given by the calibration procedure. It is to be observed that, e.g., in the 600-Mev curve, no shower pulses extending up t o energies of 700 Mev are expected. Nevertheless, due to straggling, a n unknown spectrum might be so analyzed. When the half-widths become smaller the possible errors of this type will be correspondingly reduced.

higher energies. We may expect that this limitation will be removed as time goes on, but it is not now known whether the crystal performances will ever mat,ch the resolution of pair spectrometers. * (2) Large shower detectors are likely to have background difficulties unless they are well shielded. In the case of experiments carried out on new large machines such as synchrotrons, synchrocyclotron, and betatrons this difficulty is not likely to be very serious, since the beams can be made to spill out over times of the order of 10-1000 psec. Hence the chances of obtaining background due to accidental coincidences arising from stray radiations are small if the shielding is at all moderate. On the

* See, however, note added in proof, p. aa

659. E. Stewart, Harshow Chemical Co., Cleveland, Ohio (private communication).

I

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

667

other hand in applications with machines such as linear accelerators the beam comes out of the machine in intervals on the order of microseconds and the background may pile up, even when the detector is well shielded. Of course the shielding may always be increased, at least in principle, until the pileup is essentially zero but this may require very heavy shields. When pileup of undesired events come from the target itself this limitation must be carefully weighed in the design of a new experiment. I

I

'-

PULSE

I

HEIGHT VS.

ELECTRON

ENERGY

I

6

W J

a 0 m 5 0: >.

a a t

m

4

a -

3

5

'3

w I

2

W v)

A

'

5)

n 0 0

100

2

ELECTRON

I0

ENERGY, MEV

FIG. 13. The figure shows the pulse-height peak values plotted against incident energy for the curves shown in Fig. 12. The relationship observed experimentally is linear within experimental error. This is a very desirable result from the point of view of calibration.

(3) The effective solid angle at the entrance of a shower detector will usually not be as large as one might think at first. This is because the incoming radiation should be kept near the axis of the counter in order to reduce geometrical discrepancies in the average amounts of low-energy radiations leaking out of the side walls of the counter. A diaphragm or a coincidence counting arrangement similar to that of Cassels and coworkers6 may be used to keep the desired radiation near the axis. (4)If the size of a shower detector is not large enough to retain a whole shower, it may still be used successfully when properly calibrated. In this case the pulse height will not increase as rapidly as demanded by the linear or proportional behavior. This feature of the detector is not neces-

668

2.

DETERMTNATI ON O F FUNDAMENTAL Q U A N T I T I E S

sarily a real limitation or disadvantage of this type of counter, but it is a point that, should be considered when choosing among various possible detectors. I t seems highly likely that a t very high energies 2.2.3.5.9. CONCLUSION. of electrons, positrons, or gamma rays the shower detector will prove to be one of the only economical types of counter available. It is desirable to search for new crystalline or glassy substances of high density and high optical transparency in the hope th at very good energy resolution with efficiencies of 100% can be realized without requiring excessively large volumes of the detecting material.

2.2.3.6. Gamma-Ray Telescopes.* Counter telescopes have been used successfully for the detection of gamma rays on many o ~ c a s i o n s . ~A- ~ typical arrangement is shown in Fig. 1. Sl, Sz, Sa, and Sq are countersfor instance, scintillation counters; C is a converter; and A l and A t are

A9

s4

FIG. 1 . A typical gamma-ray telescope. S1, S2, S3 and Sa are counters. C is a converter; A , and .4* absorbers. For their function see text..

absorbers. To identify the particle as a gamma ray one demands a coincidence between Sz, S s , and S d in anticoincidence with S1.Thus a neutral particle is converted into a charged particle between Sl and Sz, usually in C. This, of course, could also be a neutron which gives rise to a recoil proton between S1 and Sz, but the neutron background can be checked

* W. K. H. Panofsky, J. N. Steinberger, and J. S. Steller, Phys. Rev. 86, 180 (1952).

* G. Coeconi and A. Silverman, Phys. Rev. 88, 1230 (1952). L. J. KoeRter and F. E. Mills, Phys. Elev. BB, 651 (1956).

* Section 2.2.3.6

is by A. Silverman.

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

669

easily by observation of the counting rate variation as the coiivcrter is varied. For instance, the vonverter is usually lead. If the lead converter is replaced by a n appropriate thickness of carbon or polyet hyleiie, the expected counting rate for gamma rays can be changed by a n order of magnitude without appreciably affecting the expected neutron rate. The absorbers -4,arid A , are not, in principle, necessary. I n general, their main function is to determine the efficiency of the gamma-ray detector

--

0.5

-

0.4

-

W

r

-

0.3 Q ..-u

r

w' 0.20.1 ..

I 01 0.

I

I

50

100

I

I50

I

200

2 0

E(MeV)

FIG. 2 . Efficiency ?(A')for the gamma-ray trlcscope shoun in Fig. 1. The curve nT refrrs to coincidenrcs hetarrn S? and S1 with St in anticoincidrnce. The rurvc n p is for (8: S , S , - 8,) The points arc measured cfhcicncirs using monorncrgetir gamma-rays of rnrrgy 100, 140, and 200 MeV. The two curves arc calculated from thr rxprrssions

+ +

as a function of gamma-ray cnergy. It is clear that, to detect a gamma ray in the above arrangement one of t,he pair of electrons formed between S1 and Sz must traverse S2A,S3A,S4. The total amount of material determines the minimum energy elect,ron required and thus determines a minimum energy gamma ray which can he detect.ed. In most applicat>ions,thc absolute efic*ic.iicy of t,he telescope must, tw determined for all energy gamma rays. This efIiciency may be determincd experimentally if monoenergetic gamma rays arc available or call be determined by cnlculabion. Figure 2 shows the efficiency versus gamma-ray

670

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

energy for the particular telescope shown in Fig. 1.2In this telescope A 1 = s i n . Al; A z = 25in. Al; S1,S2, and S3 are 1-cm stilbene; and 5 4 is 1-cm NaI. The converter C = 1.5 rad lengths of Pb. The curve labeled nT is for coincidences between S2 and Sa in anticoincidence with S1. The curve labeled n F includes Sq in coincidence. The points at 100 MeV, 140 MeV, and 190 Mev are measured using monoenergetic gamma rays obtained by the technique described by Weil and M ~ D a n i e l The . ~ curves are fitted to the points using the empirical relationship V T ( E )= (0.50)[1 - e--(E-26)/40] V F ( E )= (0.45)[1 - e--(E--81)/43].

One can see that both curves have similar characteristics. They both rise rapidly from a threshold determined primarily by the absorbers and then

FIG.3. Efficiency of gamma-ray telescope of reference 5 as a function of converter thickness.

tend to saturate a t an efficiency approximately equal to 0.5. It is also possible to calculate the efficiency of such a telescope. Silverman and Stearnss calculated the efficiency of such a counter using the Monte Carlo shower calculations of Wilson.6 Koester and Mills3 used a Monte Carlo calculation programmed for the Illinois computer to make a similar calculation. J. W. Weil and B. D. &Daniel, Phys.Rev. 86, 582 (1952). A. Silverman and M. Stearns, Phys.Rev. 88, 1225 (1952). 8 R. R. Wilson, Phys. Rev. 86, 261 (1952).

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

67 1

It is clear that such a telescope does not yield much information about the spectrum of gamma rays striking the telescope. In fact, for energies several times the threshold energy, the efficiency becomes independent of energy. A total absorption Cerenkov or scintillation counter whose output is proportional to the gamma-ray energy is very often a much more satisfactory detector. The efficiency of the counter is also influenced by the thickness of the converter C. The efficiency usually varies linearly with converter thickness for C 0.3 rad lengths, reaches a broad maximum a t 1-2 rad lengths, and then decreases as the converter becomes many radiation lengths thick. Qualitatively this behavior can be understood quite simply. Figure 3 shows the efficiency versus converter thickness for the counter telescope used by Silverman and st earn^.^ While the gamma-ray telescope is often a convenient and reasonably efficient instrument for gamma-ray detection, it has the following serious difficulties: (1) the efficiency of the counter is difficult to determine accurately; and (2) it gives very little information about the spectrum of the gamma ray under observation.

<

2.2.3.7. Measurement of y R a y Energy b y Absorption.* Gamma-ray absorption methods, or more exactly “attenuation” methods, are useful for estimating the energy hv of photons when an accuracy of the order of 5% of hv is acceptable, and when the photons are either homogeneous in energy or consist of two or more groups with widely spaced energies. Occasionally a relatively crude and quick attenuation measurement may be valuable or even sufficient in the identification of a radioactive sample. The attenuation of a beam of photons is due to the combined action of several independent and competing processes. The cross section for each process depends in a different way on the photon energy hv and on the atomic number Z of the attenuator. The photoelectric process predominates in the case of high 2 and low hv. The Compton process predominates for intermediate energies, and the pair production process predominates at very high energies, especially in high-2 materials. Therefore the choice of attenuator may often depend upon the estimated value of hv. Figure 1’ shows the domain of Z and hv in which each of these three principal attenuation processes is most important. 1

R. D. Evans, “The Atomic Nucleus,” p. 712. McGraw-Hill, New York, 1955.

-

* Section 2.2.3.7 is by

Robley D. Evans.

072

2.

DETERMINATION O F F U N D A M E N T A L Q U A N T I T I E S

Each of these priiieipal interactions is capable of removing a primary photon from a beam of y rays, but each process also yields a secondary radiation. After photoelectric absorption by any atom one or more characteristic X-ray photons is emitted. After a Compton collision there always remains a scattered photon. After pair production there is always the annihilation radiation from the positron. Other secondary radiations include bremsstrahlung produced in the attenuator by the secondary photo-, Compton, and pair electrons. I

I I 1 I IIII

I

I

I I IIIII

I

I I Illlll

I

I

I I l I T

120 -

-

-

-

00 -

-

hv. in Mev

FIG.1. Relative importance of the three major types of -pray interaction. The lines show the values of 2 and hv for which the two neighboring effects are just equal [From Evans1].

Any reasonably accurate attenuation experiment therefore requires a geometrical arrangement of baffles which will prevent these scattered and secondary radiations from reaching the detector. This gives rise to the so-called “narrow-beam geometry” or “good geometry” arrangements, the essential features of which are illustrated in Fig. 2. Energy-selective detectors, such as scintillation counters, are also helpful in reducing the unwanted effects of scattered and secondary photons, but generally if these detectors are available they will be used directly for the photon energy measurement. Therefore we assume in this section th a t the

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

673

detector is a Geiger-Muller counter, an ionization chamber, or any other device which is not strongly energy-selectivo. I n very narrow-beam geometry, the unwanted effects of small-angle coherent scattering by bound electrons (Itayleigh scattering) in the attenuator may become significant. Rayleigh scattering is important relative to Compton scattering only for small-angle scattering, high-2 attenuators, and low-hv photons. For example, corrections for Rayleigh scattering become significant when the scattering angle is less than 0.5", if Pb attenuators are used, and hv < 1.5 MeV. But for A1 or Cu attenuators, scattering angles greater than 0.3", and hv > 0.6 MeV, the Rayleigh scattering is negligible compared with Compton scattering. Leod Detector Housing

Leod Source Housing

43

I

125cm.

FIQ. 2. Arrangement of collimators (C), source, attenuators, and detector for minimizing secondary effects in narrow-beam ?-ray attenuation experiments down to transmission factors of -0.0002. Additional collimation of the primary beam is needed in the case of very dense attenuators, such as Ta (16.5 gm/cmg) [From Davisson and Evans, Phys. Rev. 81, 404 (1951)l.

Figure 3 shows the total mass-attenuation coefficients of Al, Cu, and Pb attenuators for photon energies from 0.01 Mev to 100 MeV. Each curve is the sum of the photoelectric, Compton, and pair-production coefficients, hence represents the total attenuation experienced by a wellcollimated beam of photons in narrow-beam geometry. The units of the p ) cm2/gm, and the attenuator total mass-attenuation coefficient ( ~ ~ / are "thickness" is to be expressed as its physical thickness times its density p, hence in units of (zp) gm/cm2. Then the fractional transmission 1/10of a homogeneous beam is I

- =

I"

exp[-((Cco/p)(w)I.

(2.2.3.7.1)

If the y-ray beam is a mixture of n1 photons having energy hvl, and mass 2

S. A. Colgate, Phys. Rev. 87, 592 (1952).

674

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

FIG. 3. Total mass-attenuation coefficients (sum of photoelectric, Compton, and pair-production coefficients) for photons in Al, Cu, and Pb [From E ~ a n s 8 ~ ~ . . ]

attenuation coefficient ( p o / p )1, plus n2 photons having energy hvz, and mass attenuation coefficient ( p o / p )2, then the fractional transmission will be

where €1 and € 2 are the efficiencies of the detector for photons having energies hvr and hvz. Interpolation for other attenuators should not be made directly from Fig. 3 because each individual mass-attenuation process has a different

* R. D. Evans, “The Atomic Nucleus,” Chapters 23, 24, 25. McGraw-Hill, New York, 1955. 4 R. D. Evans, in “Handbuch der Physik-Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 34, pp. 218-298. Springer, Berlin, 1958.

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

675

functional dependence on 2. For attenuators other than All Cu, or P b accurate values of the mass-attenuation coefficients for the individual photoelectric, Compton, and pair-production interactions must first be calculated, and then summed to obtain the total mass-attenuation coefficient. Graphs and interpolation formula^,^,^ and tabulations6 of y-ray interaction coefficients are readily available. In narrow-beam geometry the detector should respond only to primary photons which pass through the attenuator without experiencing any collision. The elaborate experimental precautions indicated by Fig. 2 are required to minimize the response of the detector to singly scattered photons from the attenuators, from the edges of the defining apertures, and from the environs of the experimental setup, as well as to bremsstrahlung photons from recoil electrons, to coherent Rayleigh scattering, and t o multiply scattered photons. The chemical purity of the attenuators must be controlled also. Degradation of the primary photon spectrum by self-absorption within the source must be minimized by using sources of the highest available specific activity and smallest feasible mass. Narrow-beam experiments usually require a relatively strong source, of the order of 1 or more millicuries for nuclides which emit one photon per disintegration. If only weaker sources are available, the solid angle subtended by the detector as seen from the source may be increased either by enlarging the collimating apertures or by reducing the over-all source-to-detector distance. When this is done an increasing number of unwanted scattered photons will reach the detector, and the measured attenuation coefficient will be smaller than the true value of the total mass-attenuation coefficient. The magnitude of the error introduced by wider angle geometry can be estimated by making comparative or calibration measurements using photons of known energy. Convenient sources of essentially monoenergetic y rays include Hg203(0.279 MeV) , Aulg8 (0.411 Mev), CsI3’ (0.662 Mev), Zn@ (1.11 Mev), Coao(1.17 and 1.33 Mev), and Na24(1.37 and 2.75 Mev). Critical absorption edges can be utilized in some cases to obtain exceptionally accurate measurements of photon energy. The K edge of Pb is a t 0.0880 MeVJ and Fig. 3 shows that ( p ~ / p )changes discontinuously by a factor of 7.5 at this energy. Then a 0.0890-Mev y ray is very much more strongly attenuated by Pb than is a 0.0870-Mev y ray. The K edges of the elements adjacent to szPb are 0.0855 Mev for *IT1 and 0.0905 Mev for *3Bi, and there are corresponding discontinuities in their attenuation coefficients at these critical energies. Attenuators of T1, Pb, and Bi can then be used to bracket a photon energy which lies in the 2-3 kev domain between these critical absorption edges. For example, 6 Gladys White Grodstein, Null. Bur. Stundurds (U.S.) Circ. No. 685 (1957).

676

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

a 0.0870-Mev y ray is strongly absorbed by T1, but only weakly by Pb or Bi; a 0.0890-Mev y ray is strongly absorbed by T1 and Pb, but only weakly by Bi. The K edges of the elements are shown in Fig. 2 of Section 2.1.1.2, and in selected cases the critical absorption method can be used, even with weak sources in broad-beam geometry, for photons whose energy is Iess than the K edge of uranium (0.115 Mev).

2.2.3.8. Detection and Measurement of Gamma Rays in Photographic Emulsions.* Gamma radiation, contrary to particle radiation, does not produce a specific effect in nuclear emulsions, but causes in the emulsion a general blackening, similar to visible light. I n gamma-ray exposures of low intensity it is possible to distinguish single beta-ray tracks, emitted ill secondary processes, and thus recognize the presence of gamma radiation. This method is applied in biological problems, but is more of a qualitative nature; it becomes important for problems of physics only for high-energy gamma radiation when electron pair production sets in and electromagnetic cascades start to develop. The study of these cascades, their multiplication, and the energy of individual electron pairs are used for the determination or, at least, estimation of the primary photon energy causing these phenomena. 2.2.3.8.1. ENERGY MEASUREMENT OF ELECTRON PAIRS.The most obvious method for the measurement of electron pairs is the multiple scattering method applied to both partners of the pair. The conditions for the scattering method are not very favorable if the pair energy is below 10 MeV, because of the frequency of large angular deviations which often cause the disappearance of one or both electron tracks from the emulsion layer. Energies of electron pairs in the region from 10 to 100 Mev can be determined with errors smaller than 30%, in general, if both tracks have path lengths exceeding 1 mm. The limitation of track length is a serious problem in the case of pairs in the Bev range, where large cell lengths in the scattering measurements should be used for good results. In these cases one tries to replace the ordinary method by the relative scattering method.' At high energies the two tracks of the pair are nearly parallel, and the separation distance of the tracks, measured a t regular intervals (cells), serves as a parameter for the energy determination of the pair. I n this type of measurement the stage noise is negligible, and M. Koshiha and M. F. Kaplon, Phys. Rev. 97, 193 (1955).

* Sect,ion2.2.3.8 is by M. Blau.

2.2.

DETERMINATION OF MOMENTUM A N D E N E R G Y

677

only grain and reading noise are present if occasional irregularities in t>hc emulsion, leading to spurious scattering, can be avoided. Iielative scattering measurements similar to ordinary scattering measurements can also be made with noise elimination, using two sets of cell lengths. The mean relative scattering angle in degrees per 100 p , ~~2 is given by E12 =

K

[(b3;;;)2+ (-')I*

P2P2C

(2.2.3.8.1)

where p$, and p!& refer to the two individual tracks. Variation of K , the scattering constant, with cell size and particle velocity has to be considered in the usual way. The relation between the individual values and p12/3~2 of the pair can be written in the form: 24

2 p12812c 5 PlPlC + p2P2c.

Relative scattering measurements can also be made on more than two tracks; for instance, electron pairs can be scattered with respect to tracks of fast mesons in highly collimated nuclear interactions. I n such cases p & h is found from the combination of ~ ~ $ 3and ~ 2paps. The distance between tracks used for relative scattering should be smaller than 20 p . Under favorable conditions relative scattering measurements give reliable results up to 20 t o 40 Bev pair energy.' 2.2.3.8.2. OPENINGANGLEOF PAIRS.The pair energy can be determined from the opening angle, assuming equipartition of energy for electron and positron. Borsellino2 describes the relation between opening angle w and pair energy E by the equation: (2.2.3.8.2)

Here m is the electron mass and cp.(a) the partition function, which depends only slightly on pair energy and atomic number Z of the atom in the field of which the pair is formed; a is the partition ratio; and cpl(a) % 1 for a = 0.5. cpz(a)deviates not very much from unity for values a # 0.5. According to relation (2.2.3.8.2)w is the most probable opening angle for a pair, with electron and positron having each the energy E / 2 . The validity of Eq. (2.2.3.8.2) has been confirmed up to energies of about 20 Bev, provided that deviations due to multiple scattering are considered in the measurements of the opening angle. The methods used in the determination of opening angles are essentially the same as in scattering measurements; one determines the separation between the two tracks in regular distances (cells) from the pair origin.

* A. Borsellino, Ph98. Rev. 88, 1023 (1953).

678

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

Both measurements cease to give significant results when the separation distance becomes of the order of a grain diameter. METHOD.Fortunately there exists another 2.2.3.8.3. IONIZATION method, based on ionization measurements, which can be used at higher energies when scattering measurements fail to give reliable results. Observations made first by King3 and later confirmed and thoroughly investigated by perk in^,^ have shown that the grain density in highenergy pairs, near the origin of the pair, is smaller than twice the plateau value. The latter ionization density would be expected if both particles were to ionize independently from each other. The effect can be explained as a kind of “dipole effect,” and the phenomenon was treated mathematically by various authom6-9 In pairs of very high energy, and hence with extremely small opening angles, positron and electron move SO closely together that their separation is smaller than the impact parameter of distant collisions for relativistic singly charged particles. Thus a large number of electrons in the medium along the first few 100 p of the common path will receive energy from the mutual electromagnetic field of the pair, the latter being weakened by the destructive interference of the two fields of particles of equal but opposite charge. The grain density increases slowly with the distance from the origin and reaches the saturation value when the separation distance of the two partners is larger than twice the maximum impact parameter of a singly charged relativistic particle (-2.10W em). The effect is measured by the ratio R = ~[(dE/dR),,i~]/[2(dE/dR).in,l,]), where (dE/dR),in,l,is the energy loss of a singly charged relativistic particle. R can be calculated theoretically as a function of x, the distance from the pair origin, for pairs of various energies; in accurate calculations the influence of multiple scattering on the separation distance must be considered. A disadvantage of the method is the short extension of the dipole effect-100-200 p for pairs of ~ 1 0 ev-which ~ 1 impairs the statistical accuracy of the method. Weill et al. l o and Weill” have recently made an interesting observation which also might be useful for the energy determination of pairs. They found that the grain density in high-energy pairs D. T. King, unpublished data (1950). D. H. Perkins, Phil. Mag. [7] 46, 1146 (1955). A. E. Chudakov, Zzvest. Akad. Nauk S.S.S.R. 19, 650 (1955). 6 I. Mito and H. Ezawa, Progr. Theoret. Phys. (Kyoto) 18, 437 (1957). G . Yekutieli, Nuovo cimento [lo] 6, 1381 (1957). * H. Burkhardt, Nuovo cimento [lo] 9,375 (1958). J. Iwadare, Phil. Mag. [8] 3, 680 (1958). lo R. Weill, M. Gailloud, and Ph. Rosselet, Nuovo cimento [lo] 6, 413, 1430 (1957). R. Weill, Helv. Phys. Acta 31, 641 (1958). a

2.2.

679

DETERMINATION OF MOMENTUM AND ENERGY

remains below the saturation value (twice the plateau value) a t separation distances a t which the dipole effect should have ceased to exist. For pair energies of about loll ev the region of reduced grain density extends to distances of about 1 mm from the pair origin. The authors explain this phenomenon as a kind of geometrical effect: the two partners of the pair move so closely together that occasionally both cross the same grain of the emulsion, which would result in an apparent reduction of developable grains. This effect should cease to exist a t separation distances of the order of a grain diameter. The authors seem to find good agreement between experimental and calculated values, taking into account deviations due to multiple scattering. 2.2.3.8.4. GAMMARAYSBROM THE DECAY OF ?yo MESONS.The energy determination of gamma rays, electron pairs, respectively, is of great importance in connection with the problem of zro-mesonproduction and, energy distribution in high-energy interactions. The earliest data on &meson mass and half-life were derived from emulsion experiments by Carlson et a1.12 The authors searched the immediate neighborhood of stars (nuclear interactions) with minimum tracks for the presence of electron pairs and determined their energies by multiple scattering methods. The energy of the two gamma rays belonging to the same ?yo meson of total energy W has values between EI = &(V d W z - eo2) and Ez = $(W - d W z - E 0 2 ) , where eo is the rest energy of the ?yo meson. The angle 0 between the two gamma rays is given by

+

(2.2.3.8.3) where p is the velocity of the PO meson and r is the ratio of the two gamma-ray energies: r = EI/E2. The rest mass of the ?yo meson was found from €0 = 2 dm2 and the half-life from measurements of the finite path length of the ?yo meson before its decay into two gamma rays. The latter is given by the distance between the star centrum and the intersection of the two bisectors of the electron pairs. The search for electron pairs and the measurements necessary to correlate electron and y-ray pairs are quite lengthy and laborious. At higher energies, however, and hence in highly collimated shower cones, the experimental side of the problem becomes easier to handle, although correlation between electron pairs and individual T O mesons is not always possibIe if the shower density is very high. I n the shower cone immediately below the original interaction no electron pairs will be seen because of the large Lorenta time dilatation factor of the decaying high-energy d' mesons. Later, individual electron pairs will be visible, and still further l2

A. Carlson, I. Hooper, and D. T. King, Phil. Mag. [7] 41, 701 (1950).

680

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

down in the emulsion stack the electromagnetic component starts to multiply and forms well-marked cores of electron showers. The experimental procedure for locating high-energy events consists in surveying the emulsion deep down in the stack for electronic showers, and tracing these showers back into the emulsion nearer the origin of the primary event. Such procedure presumes that the energy of the primary event is large, of the order of 5 x 10l2ev, and that the stack is well aligned and large enough to comprise more than 1 radiation unit; a radiation unit in a pure emulsion stack is 2.9 cm. A more effective procedure, first applied by Kaplon et aE.13and Kaplon and Ritsonl4 is the so-called emulsion-cloud chamber method, which uses instead of a pure emulsion stack a combination of emulsion and metal sheets, the emulsion being sandwiched between consecutive metal layers. The radiation length in an emulsion-metal combination is shorter, especially if metals of high atomic number are used. IEThe multiplication starts nearer the original event, the lateral spread is smaller, and hence the cores of electronic showers are more visible above the background of random tracks. According to Kaplon and RitsonI4 it is possible to detect showers in lead sandwiches from primary events of only 1Olo ev. A further advantage of a sandwich stack is the possibility of following the cascade down to lower energy values than in the case of a pure emulsion stack. The complete analysis of high-energy events (jets), however, often is not possible in combination stacks. Even in cases where the primary interaction takes place in an emulsion layer, large corrections have to be applied for secondary interactions occurring in the metal foils. Recently Duthie et al.'s have employed a n arrangement which combines the advantage of a pure emulsion stack with the improvements of sandwich stacks, in which case a multiple sandwich of lead and emulsion is placed underneath a large stack of pure emulsion. The number of T O mesons emitted in high-energy jets can be inferred from the number of electron cores, provided that each core is initiated by photons emitted from the decaying ?yo mesons. In the majority of experiments, corrections will have to be applied for photons arriving from outside the stack, for bremsstrahlung pairs, and for cascades from ?yo mesons produced in secondary interactions. M. Kaplon, B. Peters, and D. Ritson, Phys. Rev. 86, 900 (1953). M. Kaplon and D. Ritson, Phys. Rev. 88, 386 (1952). 0. Mmakawa, Y. N. Shimura, M. Tsuzuki, H. Yamanouchi, H. Aizu, H. Hasegawa, Y. Ishii, S. Tokunaga, Y. Fujimoto, 5. Hasegawa, J. Nishimura, K. Niu. K. Nishikawa, K. Imaeda, and M. Kazuno, Nuovo cimento [lo] 11, Suppl. No. 1, 125 (1959). l*J. Duthie, C. Fisher, P. Fowler, A. Kaddoura, D. H. Perkins, and I<. Pinkau, Intern. Union Pure and Appl. Phys. Conf. on Cosmic Rays, Moscow, 1959. la

l4

2.2.

DETERMINATION O F MOMENTUM A N D ENERGY

68 1

The energy determination of individual r o mesons is relatively easy if the photon cascades related to each ?yo meson emitted in the interaction can be distinguished clearly. The energy, respectively velocity, of each z' meson can be found from the opening angle of the two related photons [Eq. (2.2.3.8.3)]. In more complicated cases energy measurements of individual electron pairs will have to be made in order to determine the total energy contained in the electromagnetic cascades. Another less difficult and laborious approach to the problem is the measurement of the longitudinal and lateral spread of the cascade. 13,14*17*18 More r e ~ e n t l y 'the ~ total energy content of the electromagnetic component in high energy showers was determined by photometric measurements, rather thaniby counting individual electron pairs. The authors found a simple relation between the maximum value of the photometric density D,, and the energy E of the cascade

E=

-

3 * 10' XE D,,,

A

(Bev).

Here the energy E is measured in Bev and the radiation length AH in cm; A is the grain area blackened by each electron (including the effect due to &rays). The formula is valid for pure electromagnetic cascades and cascades produced by very high energy nuclear interactions, where all emitted gamma rays are nearly parallel to the shower axis. In lower energy collisions the situation is more complicated due to the transverse momenta with which the various gamma rays are emitted relative to each other; in these latter cases correction terms have to be introduced. Detection and energy measurement of r0 mesons by their electromagnetic radiation is of great importance for the whole problem of multiple meson production in high-energy interactions. At very high energies it is often impossible to measure or even to identify the charged shower particles emitted in the narrow cone. Therefore the ratio of emitted r to k mesons or baryons would remain unknown if it were not for the number of observed a" mesons, which allow an estimate of the total amount of r mesons produced in the interaction (charge independence). Furthermore, the energy distribution of charged ?r mesons can be deduced from the rO-mesonspectrum. A very interesting problem, connected with high-energy jets, is the K. Pinkau, Phil. Mag. [8] A!, 1389 (1957). 1*B. Edwards, J. Losty, K. Pinkau, D. H. Perkins, and J. Reynolds, Phal. Mag. [8] 3, 237 (1958). 19 H. Mi, J. Duthie, A. Kaddoura, D. H. Perkins, and P. H. Fowler, Pror. A m . IZochester Conf. on High Energy Phys. 829 (1960). 17

682

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

search for gamma rays not originating from T O ' S , but rather from the decay of other neutral particles. To solve this problem careful energy determination of electron pairs is necessary as well as measurements by which the geometrical relation of these pairs with the origin of the jet is established.

2.2.4. Neutrino* 2.2.4.1. Neutrino Reactions.' The neutrino can be studied from two points of view. In the first it is assumed that momentum and energy are conserved and the characteristics of the neutrino are inferred from measurements made on the other particles associated with the neutrino in the process of production. The second approach is to observe a reaction caused by a neutrino subsequent t o its production and then infer the neutrino energy and momentum from those observations. These reactions are typified by the equation: antineutrino reactions : neutrino reactions:

+ + + A' + + + A'.

As+ A'+' pYA6+I S p+ AS+ Ad-' p+ v+ A*-' S 6-

+

+

Y-

Y+

(2.2.4.1a) (2.2.4. lb) (2.2.4.2a) (2.2.4.2b)

Equation (2.2.4.1a) represents the decay of a free neutron or a neutron bound in a nucleus with the simultaneous emission of a negative electron and an antineutrino. Equation (2.2.4.110-3) represents the process of inverse beta decay induced by a free antineutrino with the consequent lowering of the nuclear charge by one unit and the emission of a positron. Equation (2.2.4.lbt) depicts the capture of a positron by a nucleus with the emission of an antineutrino and the raising of the nuclear charge by one unit. Equation (2.2.4.1.2a,b) represents reactions similar to those shown by (2.2.4.la,b) except that neutrinos rather than antineutrinos are involved and the process (2.2.4.2bt) corresponding t o (2.2.4.1b~) is the well-known process of electron capture in which an atomic electron is captured by the nucleus. A complete experimental study of these reactions would exhaust the list of measurements involving low energy nuclearly produced 1 For a discussion of the history of the neutrino concept and the properties of the neutrino see F. Reines, Ann. Rev. Nuclear Sci. 10, 1-26 (1960).

* Section 2.2.4 is by F. Reines.

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

683

neutrinosa and would serve t o determine all of their theoretically required characteristics. In addition to neutrinos involved in nuclear beta decay, evidence is that the neutral particles associated with the decay of mesons may also be neutrinos. As examples we cite the two most carefully studied reactions: Tf --+ pf Vf (2.2.4.3a) (mean life T = 2 X lo-* sec) (2.2.4.3b) Pf-+ P* v+ v(T = 2 X sec).

+

+ +

Judging from the half-lives and energy spectra of the decay products in the muon decay reactions (2.2.4.3a) it is concluded that the coupling constant is about the same as that for nuclear beta decay. The neutral particles required by the conservation laws are known not t o be gamma rays. Most recently, experiments on the nonconservation of parity4 have shown angular distributions of the electron from muon decay which can be explained by applying the same kind of neutrino theory to ?r and p decay that is used to explain the observed nonconservation of parity in nuclear beta decay. For these reasons and considerations of simplicity the neutrinos in Eqs. (2.2.4.2) are assumed6 to be the same as the neutrinos involved in meson decays as typified by Eqs. (2.2.4.3). As discussed below for neutrinos from nuclear beta decay the properties of the meson decay neutrinos can be inferred from observations of the decay process using the laws of momentum and energy conservation. The direct observation of the neutrinos produced in meson decay has not yet been accomplished, the main limitation being the weakness of meson sources. It now appears hopeful that the next few years will see the development of sufficiently intense sources of energetic pions with multi Gev electronuclear machines.& 2.2.4.2. Determination of Neutrino Energy. 2.2.4.2.1. BETADECAY. Reactions (2.2.4.1a), (2.2,4.2a), and (2.2.4.2bc) can be used to determine the energy of the emitted neutrino (or antineutrino) via the laws of energy and momentum conservation. Consider first (2.2.4. la) in which nucleus Ad decays at rest from an initial state of energy Ei to a final state We omit discussion of double beta decay and the elastic scattering of neutrinos by electrons. a J. Tiommo and J. A. Wheeler, Revs. Modern Phys. 21, 144 (1949). 4 R. L. Garwin, L. M. Lederman, and M. Weinrich, Phys.Rev. 106, 1415 (1957). 5 C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys. Rev. 106, 1413 (1957). 6 5 T. D. Lee, Invited Paper N.Y. Meeting of the American Physical Society (January, 1960); T. D. Lee and C. N. Yang, Phys.Rev. Letters 4,307 (1960).

ti84

2 . DETERMINATION OF FUNDAMENTAL QUANTITIES

of energy E f with /3 and v emission. This energy is shared by the three reaction products and the energy E , of the Y is calculated to be

E , = E,

-

Ef - EB -

R~z4.i

(2.2.4.4)

where Ed - E, is the energy available for the decay, Ep is the total and E A z + I is the recoil electron energy, including rest mass energy m2, 8

6

D4 2

0

ENERGY IN KILOVOLTS

F I Q . ~Fermi . plot of an allowed 8- decay electron spectrum, neutron decay [J. M. Robson, Phys. Rev. 83, 349 (1951)l. N ( p ) is the number of decay electrons in the dp. momentum range p , p

+

kinetic energy of the product nucleus Ax+1.However, is usually neglected in such a calculation because it is very small relative to the other terms. This is so because the product nucleus is 22000 times as massive as the electron and hence can provide the monientuni necessary to conservation with low velocities and herice small kinetic energies. The remaining terms are determined by a measurement of the energy distribution of the decay electrons. Figure 1 shows a typical /3 spectrum.R (I

J. M. Blatt and V. F. Weisskopf, “Theoretical Nuclear Physics.” Wiley, New York

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

685

The upper limit of the spectrum, Emax - mc2 corresponds to the condition in which the electron has taken all of the available energy leaving none to the neutrino, i.e., Emax= E; - E f . In terms of the beta spectrum alone, then E , = Em, - EB. (2.2.4.5) We have omitted the subscript on the v and the superscript on the 9, because the same relationship applies to positron decay (2.2.4.2a). E,, can also be determined from measurements of the masses of Az, and information on the decay scheme which makes it possible to assign the energy available to beta decay as opposed to gamma-ray emission, for example. So, for neutron decay the n-p mass difference is 1.293 Mev leaving 782 kev as the maximum energy available to the neutrino. This method is applicable to the determination of the energy of the neutrino in Eq. (2.2.4.2bt-). Reaction (2.2.4.lbc) has not been observed. The determination of Eg,and hence E , in a given beta decay is accomplished in many ways: most precisely by p r a y spectrographs which employ static magnetic and electric fields to select the p energy for detection; less precisely by scintillation detect.ors, cloud chambers, electron sensitive photographic emulsions, etc. EXPERIMENT. The energy of an antineu2.2.4.2.2. FREENEUTRINO trino, E’, can be determined directly by employing a reaction of type (2.2.4.lb-t), specifically the aiitineutrino bombardment of protons V-

+p

-+

8+

+ n.

(2.2.4.6)

In this case the energy of the antineutrino, Ev-, is used in overcoming the and in kinetic energy of the positron, reaction threshold, 3.53 m2, (Eg+- 1) and product neutron, En.

E,-

=

3.53

+ (Eb+ - I ) + En.

.4s in the case of ,f3 decay the resultant nucleus-here a neutron-takes a small part of the available energy, only a few kev, so that we can neglect En and write to good accuracy in mc2 units,

E‘,

=

2.53

+ EB+.

(2.2.4.7)

A measurement of the energy of the resultant p+ gives the incident antineutrino energy through Eq. (2.2.4.i). Studies of this nature are characterized by minute cross sections. Reaction (2.2.4.6), for example has a (1952). The subject of beta decay and spectral shapes is treated, including the effects on the spectrum of nuclear electrostatic attraction, (0-decay) or repulsion (@+decay). See also E. Fermi, 2. Physik 88, 161 (1934); C. S. Wu, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), pp. 314-356. Interscience, New York, 1955.

686

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

cross section,7 u, for antineutrino energy E,, ~(g,)=

1.0 x 10-44(E, a) d ( E , - a)z - 1 (cmz) (2.2.4.8)

where E , is in units of the rest energy of the electron and a = 2.53. Because of this tiny cross section only the most intense neutrino source and largest detectors have been suitable for the study of such reactions.* Only antineutrino sources are available with the strength required, and only reaction (2.2.4.6) had yielded to experimental investigation thus far. The antineutrino fluxes now available from the decay of fission fragments in nuclear reactors are cm2/sec or lo7 times the fluxes of v- or v+ from other artificial source^.^ In addition, the energies of many of these reactor antineutrinos are well above the 1.8-Mev ( 3 . 5 7 ~ ~threshold ) of (2.2.4.6) so that the cross section, averaged over the reactor spectrum is em2. Because of the low rates, about events per hour in a about liter of water, the detection system must be designed to discriminate strongly against backgrounds. In order t o render the task of compiling data surmountable, the detector must contain many hundreds of liters of sensitive volume and have a reasonably high detection efficiency for the v-,p reaction. The second requirement was met by using a giant liquid scintillation detector of 1.4 X 103 liters sensitive volume in which the protons comprising the scintillation solvent (triethylbenzene) are themselves the antineutrino targets. The background problem is made manageable by several experimental features which will be described later. Figure 2 is a schematic diagram of the experiment designed to detect reaction (2.2.4.6) and measure the antineutrino spectrum from fission fragments. The essential idea of the experiment is to require a delayed coincidence of the proper character and then t o analyze the energy spectrum of the first pulse. The sequence depicted in the diagram is as follows. (1) An antineutrino from the reactor transmutes a proton in the scintillation solution producing a neutron of a few kev energy and a positron and kinetic energy ranging from 0 to about 8 MeV. (2) The positron slows down delivering its kinetic energy to the scintilAccording to the ideas of T . D. Lee and C . N. Yang [Phys. Rev. 106, 1671 (1957)l this cross section is twice the value quoted here. However, since the factor-of-two does not appear to be energy-dependent, it will not affect the neutrino energy determinations treated here. * F. Reines and C . L. Cowan, Jr., Phys. Rev. 113, 273 (1959). We rule out the intense if short-lived flux from a nuclear explosion if only because of the extreme complexity of experimentation with such a source vis a vis a steadystate nuclear reactor. Actually the number of events even in a very large detector, say 100 tons, is too few for experiments.

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

687

lator and annihilates with an orbital electron producing two, 0.511-Mev annihilation gamma rays. This process takes about 3 X second and so is instantaneous from the point of view of our system which has a resolving time of 2 X 10-7 second. The scintillator is so large-a rectanguAntineutrino (vJ Giant Liqufd

Capture gamma rays

Scintillation Detector

AMihilation gamma rays

FIG.2. Schematic of antineutrino detector. This 1.4 X loa liter detector is filled with a mixture which consists primarily of triethylbenzene with small amounts of terphenyl (3 gm/liter), POPOP wavelength shifter (0.2 gm/liter), and cadmium (1.8 gm/liter) as cadmium octoate. An antineutrino is shown transmuting a proton producing a neutron and positron. The positron slows down and annihilates producing annihilation radiation. The neutron is moderated by the hydrogen of the scintillator and is captured by the cadmium-producing capture gamma rays.

lar parallelopiped 2 X 1.5 X 0.7 meters-that there is very little leakage of the positrons and a fair probability'O for absorption of the annihilation radiation. (3) The neutron slows down and diffuses about in the scintillator until it is captured by the cadmium, emitting capture gamma rays. Figure 3 shows a schematic block diagram of the electronics and detectors employed in the experiment. The antineutrino detector and anticoincidence shield detector are shown enclosed in a lead and paraffin 10 A 4 M e v electron has a range of 2.3 cm in TEB. An 0.5-Mev gamma ray has a total mean free path for scattering and absorption in triethylbenzene of 13.5 cm. The edge effects due t o events taking place near the surface of the scintillator will distort the positron spectrum, lowering it by about 0.5 MeV. A detector with no edge effects can be built by adding cadmium to a central section only and using a peripheral region to catch the 0.6-Mev gamma which is headed outward.

Amplifier8

Coincidenceanticoincidence

Scalers

Anticoincidence shield detector Antineutrino m W

W

Oscilloscope sweep gate

FIQ.3. Schematic of antineutrino experiment. The 110 photomultiplier tubes of the antineutrino detector were split into two banks of 55 tubes each and a prompt coincidence w w required in order to discriminate against tube noise. This detail is omitted from the figure for simplicity.

2.2.

DETERMINATION OF MOMENTUM AND E N E R G Y

689

shield. The signals are sent through preamplifiers and amplifiers to gating circuits which select the pulse heights, and hence energies, characteristics of the positron and neutron: whenever a pulse in the energy range 1 to 8 Mev occurs it is registered on the positron scaler and opens a 25-microsecond gate on the neutron coincidence unit; a second pulse in the range 3 to 10 Mev occurring in this time gate registers on the delay coincidence scaler and triggers the sweep of the oscilloscope. In addition, any pulse in the 3 to 10 Mev range registers on the scaler as a neutron-like capture pulse. During the 25 microseconds the electronics requires to decide whether it has seen an acceptable delayed coincidence, the signals have been stored in 30 microsecond delay lines for presentation on the oscilloscope and photographic recording once the decision is made via the scope sweep trigger. If a pulse greater than 0.5 Mev occurs in the anticoincidence guard detector coincidentally with a pulse in the antineutrino detector, an anticoincidence veto prevents the occurrence of a scope sweep. The raw data that result from such an experiment are the following. From the scalers: The rates of events satisfying the positron gate np+, the neutron gate, n,, and delayed coincidences, ndcJsatisfying both positron and neutron gates and occurring with a time interval 25 microsecond. From the oscilloscope films: The pulse heights of the positron-like pulses and of the neutron-like pulses and the time intervals between these pulses. These data for the reactor on and off together with the detection efficiencies of the system for the positrons and neutrons and a shielding experiment to rule out reactor radiations other than neutrinos suffice to determine the cross section for the reaction. We assume that the neutrino flux is known from information about the fission fragments (6.2 antineutrinos/fission), the fission rate in the reactor, and the distance of the detector from the reactor. Knowledge of the energy resolution for the positron makes it possible to determine the antineutrino spectrum above 1.8 MeV, the reaction threshold. To make the background manageable the following experimental steps were taken. (a) The neutrons and positrons are detected in delayed coincidence, so raising the effective signal count rate by the factor l/&,where t d is the mean time between the positron signal due to slowing down and annihilation and the capture of the neutron. For a reasonable value of t d = 25 microseconds, l / t d = 4 X lo4. The delay interval t d is determined by the moderation and capture time of neutrons in a cadmium, hydrogen mixture. Cadmium was chosen because of the high thermal neutron capture cross section (3300 X cm2 for the natural isotopic mixture) and the energetic gamma rays it emits (total energy 9 MeV, average multiplicity 4)

690

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

on neutron capture. The capture times were dictated by the maximum amount of the cadmium compound, cadmium octoate, which we could add to the scintillator without undue depression of the scintillation efficiency." 12

1

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

Reactor

01

I

I

FIRST PULSE

ENERGY (MeV)

FIQ.4. Spectra of first pulses in free neutrino experiments with reactor on and off (Reines and Cowan').

(b) An anticoincidence shield set above the detector discriminated against those charged cosmic rays which penetrated the underground location. (c) A heavy lead, paraffin, water shield was set up against reactor-associated gamma rays and neutrons and radioactivity in the environment. A procedure for data reduction with a view to the determination of the shape of the antineutrino spectrum only, is as follows. A plot is made of the first pulse spectrum with the reactor on. A second plot, normalized 11

Scintillators for free neutrino experiments are described by A. R. Ronzio, C. L.

Cowan, Jr., and F. Reines, Rev. Sci. In&. 29, 146 (1958).

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

691

to the same run length, is prepared of the first pulse spectrum with the reactor off. If the accidental background is reactor-independent, and the only reactor associated signal is due to antineutrinos, a subtraction of these two curves would result in the spectrum of positrons produced in

ANTINEUTRINO ENERGY, Ev (Mev)

FIG..5.Antineutrino spectrum from fission fragments deduced, using Eq. (2.2.4.7), from the difference between the two curves of Fig. 4 (Reines and Cowan').

the antineutrino reaction (2.2.4.6). A calculation of the accidental background rate A can be made using the relationship

A

= fng+n,td.

(2.2.4.9)

The factor f is close to unity for our case even though the energy gates overlap somewhat because most of the pulses occur in the nonoverlapping end of the positron gates. Figure 4 shows two such experimental curves. If the system required no correction for energy resolution the difference curve could readily be interpreted in terms of the antineutrino spectrum. Such corrections are very likely less than 1 MeV, but the resultant smearing of the curve in the vicinity of the threshold makes the antineutrino spectrum totally unreliable there. Figure 5 shows the antineutrino energy spectrum above 4 Mev deduced

692

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

from the positron difference curve and Ey. (2.2.4.8). 30 corrections for energy resolution were applied in preparing this figure. For orientation, the signal rate in these experiments was 36 f 4 per hour with a correlated background from cosmic-ray produced neutrons some 2.5 times this figure and an accidental background also 2.5 times the signal. 2.2.4.3. Determination of Neutrino Momentum. As with the measurement of neutrino energy the neutrino momentum, both linear and angular (spin) can be determined indirectly and directly. The neutrino energy E , and linear momentum p, are simply related by the equation

E,

=

pyc

(2.2.4.10)

where c is the velocity of light and the neutrino rest mass is taken as negligible relative to the energies under consideration. l 2 Therefore, a knowledge of one quantity serves to determine the other and the momentum determination may be termed direct or indirect according to the way in which the energy was measured. I n addition, the neutrino momentum may be determined from so-called recoil experiments in which the momenta of the other particles participating in beta decay are measured. 2.2.4.3.1. RECOIL EXPERIMENTS.’~ Although the kinetic energy of the product nucleus in beta decay is small, it can be measured in certain favorable cases, as can the direction of the recoiling product nucleus and decay electron. For a vanishingly small initial momentum of the system, the net momentum of the decay products must also vanish. A measurement of the electron and product nucleus momenta pa and p N , respectively then determines the neutrino momentum, py. These momenta are shown in Fig. 6. In the ca6e of K-electron capture (2.2.4.2bt), p~ = 0 and the neutrino momentum is equal and opposite to that of the recoiling product nucleus. The first example of indirect determination of neutrino momentum by the recoiling technique was an experiment using K-capturing Be7 by Allen.I4He used a windowless electron multiplier as did most subsequent le The neutrino rest mass may be identically zero according to recent theoretical developments (Lee and Yang’) although K. M. Case [Phys. Reu. 107, 307 (1957)] points out that experimental evidence on asymmetries in 6 decay merely require the mass to be small. Consideration of the shape of the beta spectrum near the upper energy limit makes it possible to place an upper limit on the neutrino mass of H$oa the electron mass. See L. M. Langer and R. J. D. Moffat, Phys. Rev. 88, 689 (1952). l a A comprehensive review of neutrino recoil experiments by 0. Kofoed-Hansen may be found in the book “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), pp. 357-372. Interscience, New York, 1956. For a Iess elaborate review, see the article on the neutrino by B. W. Ridley in Progr. in Nuclear Phys. 6, 188-251 (1956). J. S. Allen, Phys. Rev. 61, 692 (1942).

2.2.

DETERMINATION O F MOMENTUM A N D E N E R G Y

693

workers, and detected the recoiling Li7nucleus in coincidence with Auger electrons ejected from the excited product atom. The Li7 recoil energy (57 ev) was measured by use of a retarding field. Beta decay experiments of type (2.2.4.1a), (2.2.4.2a) have been observed with much the same type of apparatus used for the K-capture studies, except that the coincidence is taken between the recoil nucleus and the decay electron rather than the Auger electrons. The velocity of the recoil nucleus has also been

FIG. 6. fomentum triangle for the beta decay. pp PN, and pv are the momenta a the electron, product nucleus, and neutrino respectively.

determined by using the time of flight of the recoiling nucleus from the source to the first dynode of an electron multiplier. The time fiducial in these time-of-flight experiment8 is obtained from the Auger or decay electron pulse. As an example, a schematic of the neutrino recoil apparatus of Rodeback and Allenl6 in which the process of orbital electron capture by Ar37was observed, is shown in Fig. 7. In this experiment the gaseous source is distributed throughout the system and the effective source volume, defined by various baffles is the extended region shown shaded in the figure. The pressure in the system was maintained a t mm Hg, corresponding to a mean free path many times the dimensions of the detector. The entire system of baffles, shields, .and grids was maintained at ground except for the accelerating potential of -4.5 kev on grid 3. The energy spectrum of the recoil C137product nuclei determined from the distribution of time intervals between the pulses due to the Auger electrons and the recoiling nuclei is shown in Fig. 8. The rise in the curve at short delay times is attributed to decays near the recoil detector which are associated with electrons that reach the electron detector despite the bafffes. The maximum occurs at a velocity Is

G. W. Rodeback and J. S. Allen, Phys. X ~ P86, . 446 (1953).

694

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

of 0.71 f 0.04 cmlpsec as compared with the predicted maximum of 0.71 f 0.06 cm/psec. Some uncertainty resulted from the finite source volume and hence the uncertainty in the distance of about 30 % from the source to grid 2. Corrections are necessary for the thermal velocities of the A r 3 I atoms in the source, the acceleration time of the C13' between

FIG. 7. Neutrino recoil apparatus (Rodeback and Allen's). 1, 2, 3 are grids. The detectors are Allen-type electron multipliers [J. S. Allen, Rev. Sci. Znstr. 18, 739 (1947)l.

grids number 2 and 3 and the penetration of electrical fields into the region between the source volume and grid 2. The dotted line on the figure shows the spectrum predicted for a unique recoil energy allowing for the distortion of the spectrum due to the thermal motion. More recently the Ar3' problem has been studied with increased precision and in greater detail by Snell and PleasontonI6 using electric and LEA, 8.Snell and F. Pleasonton, Phys. Rev. 100, 1396 (1955).

2.2.

695

DETERMINATION OF MOMENTUM AND ENERGY

magnetic deflection. The use of both electric and magnetic fields enabled the determination of e l m for the ions counted. In addition the use of the electric field had the practical feature that it provided additional focusing and made possible the reduction in scattering of the product Cla7 by allowing the introduction of another stage of differential pumping through an enlargement in the system. Figure 9 is a schematic of the recoil spectrometer used by these investigators. A 3-meter magnetically shielded,

200

160

Resolving Time 0.6 pSec 120

d

2

80

40

0

2

4

6

8

10

TIME OF FLIGHT M @EC

FIQ. 8. Measured and predicted time-of-flight distribution for Cls7 recoils from ArJ7 decay (Rodeback and Allen16).

stainless steel conical source volume containing Ara7at a pressure 5 2 X 10-6 mm Hg passed of the recoil product nuclei out through a inch diameter hole to the analyzing and detecting system. The beam was first analyzed by a 96.5" deflection in a wedge-shaped, focusing magnetic analyzer and then was focused and analyzed again by an electrostatic deflector having two spherical sector plates f 3 cm about a mean deflection radius of 20 cm. The ions leaving the electrostatic analyzer were accelerated through 4600 volts and detected by a modified Allen-type secondary electron multiplier. Because of the prohibition against the use of source defining windows or foils imposed by the require-

+

ADJUSTABLE LEAK

A" STORAGE AND PURIFICATION

I 1

TOEPLER PUMP

- I J Q D Nz TRAP

_I

I

Hg DIFFUSION PUMP r 2 0 0 liter/sec

\

/

MAGNET POLE\ INSULATOR

MAGNETIC SHIELD

INSULATOR

RAY" 2 liter/sec

FOREPUMP

ION SOURCE FOR TESTING

OIL DIFFUSION PUMP 120 liter/sec

I TO PREAMPLIFIER

I

-5000 v

ELECTRIC DEFLECTION

FIG.9. Neutrino recoil apparatus (Snell and Pleasonton"). The source volume is insulated from ground enabling the application of a voltage for predeflection acceleration, a parameter on which the energy resolution of the system depends.

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

697

RECOIL ENERGY (ev) 1 2 1 0 8 I

I

I

I 'I

6

I

4

2

0

I l l

I

'

As' D E C A Y CHARGE 1 C1" RECOIL IONS MAGNET SETTING: 19,531 Bp

400

300

C

'8

\

3

8

200

100

t

BACKGROUND

0 470

480

490

500

PRE-DEFLECTION ACCELERATION (Volts)

FIG.10. The singly charged recoil peak as a function of the positive predeflection voltage applied to the source volume. Also shown is a recoil energy scale. The magnet was set at 19,531 = Bp (gauss-cm) with p = 16.75 cm (Snell and Pleasontonle).

ment t h a t the low energy (9.65 f 0.05 ev, predicted) recoil nucleus be free from scattering, other measurements were taken t o define the source volume. As indicated above, Snell and Pleasonton employed diff erentinl pumping to isolate the source from the detector. A study of the singly charged recoils, Fig. 10, showed a recoil energy for the C13' product of 9.63 f 0.06 ev, as compared with the value pre-

698

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

dicted from the C137(p,n)Ar37threshold. This experiment fixes the 2 kev and sets an upper limit via the energy neutrino energy a t 815 and momentum conservation laws of 5 kev on the neutrino mass. 2.2.4.3.2. SPINOF THE NEUTRINO. The neutrino spin is fixed by consideration of reactions (2.2.4.1) and (2.2.4.2) t o the nhj2, where n is an odd integer. Arguments based on the shapes of the beta-decay spectra and the relationship between half-life and the endpoint of the betaspectra limit n to the value unity.17 A more direct approach to the problem is possible via the principle of microscopic reversibility applied to an observation of reaction (2.2.4.6)and the decay of the free neutron. This argument has been complicated by the recent two-component theory which assigns one unique spin direction (relative to its linear momentum) to the antineutrino and the opposite spin direction to the neutrino. However, the cross section (2.2.4.8) is uniquely given for a monoenergetic antineutrino by the reversibility principle and a quantitative comparison of the results of the free neutrino experiment described in Section 2.2.4.2.2 and the cross-section average over the antineutrino spectrum from fission fragments should check the composite factor which depends on the spin and the validity of the two-component theory. The antineutrino spectrum from fission fragments has recently been determined1*with the result that reasonable agreement is found with expectations. For permission to use figures associated with their names, the author wishes to thank Dr. Allen, Dr. Robson, and Dr. Snell. l7

S. Kusaka, Phys. Rev. 60, 61 (1941). R. E. Carter, F. Reines, J. J. Wagner, and M. E. Wyman, Phys. Rev. 113, 280

(1959).

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 his name does not appear on that page.

A Abashian, A., 142, 321 Adair, R. K., 539 Adams, C. D., 87 Adams, R. V., 200 Adelson, H. E., 462, 464(5), 479 Ahearn, A. J., 268 Airapetiants, A. V., 269 Aizu, H., 680 Alburger, D. E., 369, 374, 598, 643, 647, 648, 649, 650(16) Alexander, G., 26, 35, 36, 242, 249 Alikanov, A. I., 71 Allen, A. J., 526 Allen, G., 81 Allen, J. R., 9 Allen, J. S., 692, 693, 694, 695 Allen, W. D., 577 Allison, B., 412, 413(8), 415(8), 416(8), 418(8), 419(8) Allison, S. K., 22, 41, 43, 343, 412, 413, 611 Alvial, G., 263, 306 Aly, H., 681 Alyea, E. D., 205 Ambler, E., 3, 683 Ames, O., 474 Ammar, G., 306 Anderson, C. D , 200 Anderson, E. C., 143, 158(32) Angus, J., 117 Annis, M., 388 Arley, N., 662 Armenteros, R., 437 Arnold, W. A., 523, 525 Aron, W. A., 2, 12, 13(47), 21(47), 44, 45, 46, 233 Arroe, H., 334 Arzan, P. E., 205 Ashkin, A., 87

Ashkin, J., 3, 19, 22, 41, 42(3), 44, 53, 54(3), 58, 59, 73, 74, 77, 79, 82, 84, 85, 86, 89(216), 144, 158(33) Askarian, G. A., 40, 204 Askey, C. M., 24(72), 31 Atkinson, J. H., 189, 455 Auerbach, L. B., 654, 663(1) Aves, R., 570, 571(1), 573(1), 574(1), 575(1) Avignon, Y., 246 B Bacher, R. F., 335, 555 Bacon, G. E., 566 Backstrom, G., 593 Baggerly, L. L., 600 Bagget, L. M., 369, 647 Baicker, J. A., 550 Bainbridge, K. T., 358, 360 Bair, J. K., 629 Baker, C. P., 555 Baker, W., 320 Bakker, C. J., 11, 12, 13, 30, 32, 104 Baldwin, E., 181 Baldwin, G. C., 527 Bame, S. J., 369, 463, 647 Band, I. M., 598 Bannerman, R. C., 411, 635 Barber, W. C., 24, 25, 32, 33, 104, 528, 530 Barbour, I., 299 Barish, B. C., 455 Barkas, W. H., 36, 45, 54, 222, 227, 228, 230,231,232,234,235,236,238,243 Barker, K. H., 380, 381, 382, 384, 386 Baroni, G., 45, 232, 246 Barrett, 0. E., 196 Barrett, P. H., 45, 222, 227(56), 228(56), 235(56) Barschall, H. H., 462, 539 699

700

AUTHOR INDEX

Bsrtholornew, G. A., 645, 646 Bartlett, A. A., 358 Bartlett, J. H., 70 Barton, G. W., Jr., 297 Bashkin, S., 415, 416, 419 Baskin, R., 27, 37 Batchelor, R., 570, 571 (l),573, 574, 575, 579, 580, 581 Battell, W. J., 557 Batty, C. J., 222, 317 Baxter, P., 388 Beach, L. A., 87 Beall, E., 283 Becker, J., 31 Becker, R. L., 414 Beckurts, K. H., 516 Beiduk, F. M., 353 Beiser, A., 208, 214(5), 215(5), 216(5) Bell, P. R., 142, 411, 417(1), 425(1), 427, 617, 619, 620, 621, 622, 626, 6:35, 639, 640(31) Bell, R. E., 440 Benczer-Koller, N., 66, 71 (165) Bender, R. S., 360 Beneveniste, J., 269(31), 270 Bennett, W. R., Jr., 137, 138, 433 Bent, R. D., 643, 650, 651 Berenson, R. E., 108 Berger, M. J., 396, 399, 405 Bergkvist, K. E., 343 Berman, A. I., 87 Bernardini, G., 654, 663(1) Bernstein, I. B., 663 Bernstein, W., 636 Berriman, R. W., 212 Hethe, H. A., 3,4, 11, 19,22 34,41, 42(3), 44, 53, 54, 57, 58, 59, 60, 73, 74, 75, 77, 79, 82, 83, 84, 85, 86, 87, 89 (216), 144, 158(33), 230, 237, 313, 317, 330, 335, 385, 300, 394, 542, 580 Beusch, W., 619 Beyster, J. R., 580 Bhabha, H. J., 8, 56, 663 Bianchi, L., 364 Bichsel, H., 12, 13, 21, 15, 46, 233 Biggerstaff, J. A., 414 Bincer, A. M., 64, 65, 66 Bird, J. R., 544, 564 Birge, R. W., 199, 233

Birkhoff, R. I)., 39,343, 423,424, 428(37) Birks, J. B., 125, 128, 417, 421(22), 422, 617 Birnbaum, W., 54, 227 Bishop, A. S., 222, 231 Biswas, S., 399, 401, 402, 403 Bitter, F., 335 Bizzeti, P. C., 264 Bjornerud, E. K., 449 Bjorklund, F., 316, 317(20), 318 Blackett, P. RI. S., 44, 377, 381, 382, 385 Blair, J. M., 523 Blair, J. S., 309, 312, 321 (2) Blanc, D., 168 Blankenship, J. L., 266, 269, 271 BIatt, J. M., 247, 248, 253, 263(136), 663, 684 Blau, M., 208, 209, 211, 257, 264 Bleuler, E., 72 Blevins, M. E., 204 Blinov, G. A., 41, 204, 207, 451 , 45:5(7) Bloch, F., 4, 11, 233, 484 Bloch, S. C., 250, 257 Block, C., 542 Blocker, W., 663 Bloernbergen, N., 11 Bloom, S. D., 546, 576 Blum, J., 229 Blumberg, L., 487 Blumberg, S., 542 Blunck, O., 17, 39 Bobone, R., 71 Bockelman, C. K., 539 Bodmer, A. R., 334 Boehm, F., 600, 612 Bogart, L., 370 Bogdanov, G . F., 546 Boggardt, M., 212 Bohr, .4., 9, 12, 335 Bohr, N., 4, 41, 42, 43, 52, 236 Boicourt, G. P.,432 Boley, F. I., 371 Uollinger, L. M., 531, 532, 534, Ii35, 544 Bolotin, H. H., 544 Bonetti, A , , 217, 218(35), 263 Bonner, T. W., 650, 651 Booth, N. E., 319 Borisov, N. D., 593

701

AUTHOR INDEX

Borkowski, C. J., 266, 269 Borovskii, J. B., 606 Borrell, P., 586 Borsellino, A., 37, 57, 85, 677 Bortner, T. E., 32, 43 Bostick, H. A., 462, 464(5), 479 Bothe, W., 374, 648 Bowen, T., 27, 28, 37, 38 Bowey, E. M., 537 Bozorth, R., 599 Brabant, J. M., 190, 191 Bradner, H., 222, 231, 382 Bradt, H. L., 300, 301 Brandt, W., 12 Bransden, B. H., 580 Breit, G., 322, 334, 463 Breydo, I. Ya., 635 Bridge, H. S., 93, 100, 102, 388 Brinkman, H. C., 41 Brisbout, F., 400, 401(33), 405 Brisson, J. C., 441 Brix, P., 334, 339(78) Brockhouse, B. N., 514, 515 Brode, R. B., 23, 29, 387 Brogen, G., 606 Brolley, J. E., Jr., 128, 412, 432, 472 Bromley, D. A., 266, 269 Brooks, F., 641 Brooks, F. D., 121, 128, 146, 421, 422, 425, 537, 551 Brosi, A. R., 411 Brown, G. E., 317 Brown, J. L., 204 Brown, J. R., 600 Brown, L. M., 36 Brown, R. H., 210 Brown, W. L., 269(29), 270, 271(29), 274(29) Browne, C. P., 343 Brugger, R. M., 514, 515 Bruner, J. A., 354 Brussard, P. J., 312, 313 Buck, W. L., 124, 135 Buokingham, R. A., 78, 593 Budini, P., 8, 9, 29(22), 30, 35 Bumiller, F., 323, 325(51), 328(51 I , 335(51) Burcham, W. E., 417 Burge, R. W., 221 Burhop, E. H. S., 198

Burkhardt, H., 678 Burkhart, L. E., 296 Burkig, V. C., 12, 13(48), 21, 45, 46 Burrowes, H. C., 458 Burwell, J. R., 384 Butslov, M. M., 159 Byfield, B., 320

C Caldwell, D., 181 Caldwell, D. C., 233, 241 Caldwell, D. O., 11, 12, 13, 30, 46, 458 Camac, M., 360 Camerini, U., 210, 231 Cameron, A. G. W., 335 Carlson, A., 679 Carlson, B., 486 Carlson, B. C., 333 Carlson, J. F., 663 Carlson, R. R., 415, 416(17), 419(17) Carter, C. F., 257 Carter, D. S., 143, 155(30), 156(30), 157(30), 158(30) Carter, R. E., 580, 698 Carter, R. S., 22, 23, 510, 511, 5L2 Case, K. M., 692 Casseis, J. M., 190, 655, 663, 664, 665 Casson, H., 343, 412, 413 Castagnoli, C., 45, 232, 246, 247, 253, 254, 255 Cauchois, Y., 297, 599, 607 Causey, C. W., 317 Cavanagh, P. E., 71 Ceccarelli, M., 264 cerenkov, P. A., 163 Chadwick, J., 209, 349, 430 Chamberlain, O., 142, 172, 182, 456 Chang, J. K., 205 Chang, W. Y., 329 Chanson, P., 31 Chase, C. T., 71 Cheka, J. S., 343, 423, 428(37) Chen, J. J. L., 39 Chen-Niang, C., 207 Cherry, R. D., 332 Cheston, W. B., 31 1 Chou, C. N., 37 Choyke, W. J., 20'2 . Chrien, R.. E., 544

AUTHOR INDEX

Chudakov, A. E., 678 Chupp, E. L., 613 Church, E. L., 297 Chynoweth, A. G., 269 Clark, A. F., 613 Clark, D. D., 32 Clementel, E., 493 Cocconi, G., 668, 670(2) Cocconi, V. T., 441 Cochran, W., 514, 515 Cocking, S. J., 513 Cockroft, A. L., 117, 469, 573 Cockroft, J. D., 527 Cohen, B. L., 314 Cohen, E. R., 600, 606, 607 Cohn, H. O., 481 Coleman, C. F., 71 Colgate, S. A., 89, 593, 673 Colli, L., 100 Collins, G. B., 163 Compton, A. H., 611 Condon, P. E., 160 Conversi, M., 120, 282 Cook, C. S., 354 Cool, R., 142, 321 Coombes, C. A., 281 Coon, J. H., 473, 474(25), 525, 571 Coor, T., 319 Cooper, L. N., 329, 333(61) Corak, W. S., 513 Cork, B., 281, 283, 317, 460 Corson, D. R., 23, 29, 76, 404, 405 Cortini, G., 45, 232, 247, 253(137), 254(137), 255(137), 256, 257 Cosyns, M. G. E., 220, 228 CotB, R. E., 531, 532, 544 Cotton, E., 364 Courant, H. W. J., 199 Cowan, C. L., 143, 148, 154, 155(30), 156(30), 157(30), 158(30), 686, 690 Cowan, G. A., 499 Craggs, J. D., 89 Craig, D. S., 343 Craig, H., 348 Cranberg, L., 442, 462,476, 546, 547, 548, 551, 552, 556, 571 Crane, H. R., 71 Cranshaw, T. E., 282 Critchfield, C. L., 313 Cronin, J. W., 142, 321

Cross, W. G., 360 Crowe, K., 607 cudakov, A. E., 36 Cuer, P., 45, 211, 222, 227(56), 228(56), 235(56), 237 Curran, S. C., 89, 117, 125, 146, 411, 417, 421(23), 469, 573, 617, 635 Curtis, R. B., 64, 68(162)

D Dabbs, J. W. T., 266, 269(3, 4, 5) Dahanayake, C., 400, 401 (33), 405(33) Dainton, A. D., 218, 302, 303 Dalitz, R. H., 75 d’Andlau, C. A., 23(69), 29, 30(69), 197, 446 Daniel, H., 648 Daniel, R. R., 26, 35, 232, 242 Danos, M., 340 Danysz, J., 348 Danysz, M., 242 Das Gupta, N. N., 195 Davidson, G., 337 Davies, H., 87 Davies, J. H., 26(94), 35, 242 Davies, T. H., 395 Davis, G., 247, 248 Davis, R. C., 142, 621, 635, 639, 640(31) Davisson, C. M., 77, 78, 79(206, 207, 215), 83, 88(215), 89(206, 2071, 591, 593, 673 Day, P. P., 362, 363(48) Day, R. B., 462 Dayton, I. E., 314, 315 Dearnaley, G., 269 DeBeer, J. F., 282 deBoisblanc, D. R., 558 Debye, P., 89 De Carvalho, H. G., 233 Dedrick, K. G., 176 Della Corte, M., 247, 250, 257, 261, 264 Delsasso, L. A., 643 DeMarco, A., 250 Demers, P., 208, 211(2), 214(2), 215(2), 216, 221, 264, 302 Denney, J. M., 269, 536 Dennison, D., 358 De Pasquali, G., 66, 67(164), 71 der Mateosian, E., 411, 413

703

AUTHOR INDEX

de-Shalit, A., 71, 336 Desjardins, J. S., 498, 520, 529, 540, 541, 542,543, 544 D’Espagnat, B., 399, 402(29) Detoef, J., 441 Deutsch, M., 360, 369, 371, 586 Devlin, T. J., 455 de Vries, C., 358 De Waard, H., 71 DeWire, J. W., 87 Dicke, R. H., 167 Dickinson, W. C., 470 DiCorato, M., 402, 407 Dienes, G. J., 280 Dietrich, C. F., 348 Dilworth, C. C., 209, 217, 218(34, 35), 263, 299, 403 Diven, B. C., 143, 158(31) Dodd, C., 40, 204, 382, 451 Dodder, D. C., 470, 482, 484, 493 Dolmatova, K. A., 374 Donovan, P. F., 269(29), 270, 271(29), 274(29) Douglas, R. A., 415, 416(17), 419(17) Driggers, F. E., 102 Duckworth, J. C., 527 Dudeiak, W., 365 Duffield, R. B., 527 DuMond, J. W. M., 335, 368, 370, 599, 600, 603, 606, 607, 609, 612, 613 Dunning, J. R., 501, 538 Duthie, J., 680, 681 Dzhelepov, V. P., 319

E Ebel, M., 322 Eby, F. S., 128, 412, 413, 415, 423(6), 635 Eckert, J. P., 563 Edelstein, R., 320 Edgar, M., 218 Edvarson, K., 358, 359, 591, 595 Edwards, B., 681 Edwards, W. F., 600, 612 Egelstaff, P., 511, 513 Eggler, C., 137, 432 Ehrenberg, H. F., 324, 325, 326(55) Eichler, E., 633, 634(21)

Einbinder, H., 486 Eisberg, R. M., 32 Eisenberg, Y., 402, 408 Eisenbud, L., 463 Eisenhauer, C. M., 509, 510, 511, 512 Eisinger, J., 335 El-Hussaini, J., 600, 607 Eliseiev, G. P., 31, 71 Elliott, L., 360, 369, 371, 586 Elliott, M. J. W., 577 Ellis, C. D., 297, 436, 582 Elmore, W. C., 105 Elwyn, A. J., 546 Emigh, C. R., 87 Engelkemeir, D., 636 Engler, A., 400, 401(33), 405(33) Enstein, K., 250 Erickson, K. W., 523 Eriksen, V. O., 635 Erman, P., 586 Ershler, B. V., 71 Evans, J. E., 504, 514, 515 Evans, R. D., 77, 79(206), 89(206), 289, 292, 298, 313, 360, 369, 371, 586, 593, 671, 672, 673, 674, 675(3, 4) Ewan, G. T., 358, 591 Eyeions, D. A., 24(71), 30 Eeawa, H., 36, 678

F Facohini, U., 100 Fairbank, W. M., 204 Falk-Vaivant, P., 441 Fan, H. Y., 269 Fano, U., 9, 32, 43, 52, 105 Farwell, G. W., 309 Fay, H., 231, 399, 401(31) Fazzini, T., 441 Feather, N., 72 Feenberg, E., 335 Feld, B. T., 109, 144, 145(35), 527, 528 Feldman, L., 370 Felthauser, H. E., 473, 474(25) Ferguson, A. T. G., 577, 581 Fermi, E., 5, 7, 9(17), 11, 36, 42, 74, 231, 502, 685 Fernbach, S., 316, 317, 318, 319 Fertel, C. E. F., 209

704

AUTHOR INDEX

Feshbach, H., 67, 70, 313 Fichtel, C., 405, 406 Fidecaro, G., 441, 655, 663(5), 664(5), 665(5) Figgler, C., 103 Filosofo, I., 190, 654, 655, 663(1, 6) Fine, S., 297 Fink, G., 501 Firk, F. W. K., 513, 544, 564 Fischer, J., 205 Fisher, C., 680 Fitch, V. L., 141, 183, 329, 330(60), 456 Flammersfeld, A., 72 Fleishman, H., 486 Fleming, J. R., 26, 35, 36, 242 Fluharty, R. G., 504 Focardi, S., 282 Ford, G. W., 64 Ford, K. W., 329, 334 Foreman, B. M., 269(40), 270 Forstat, H., 103, 104(19) Forster, H. K., 204 Forster, T., 124 Forte, M., 425 Fowler, E. C., 40, 41(123) Fowler, G. N., 9 Fowler, J. L., 472, 481, 523 Fowler, P. H., 76, 208,210,211,212,235, 243, 247(123), 248, 250, 251, 252, 253, 263(123), 302, 303, 397, 400, 401(33), 402, 405(33), 680, 681 Fowler, R. H., 78, 593 Francis, J. E., 619 Frank, I., 8, 163 Prankel, S., 367 Franr, W., 89 Franeinetti, C., 45, 232, 282, 299 Fraser, J. S., 274, 412, 417(10) Frauenfelder, H., 66, 67, 71 Freedman, M. S., 362, 363(48), 428 Fregeau, J., 655, 663(4) Freier, G., 523 Freier, P., 300, 301 Fretter, W. B., 23(69), 29, 30(69), 197, 201, 380, 386, 446, 447(2), 449(2) Friedland, S. S., 269, 270, 278(35), 280, 536 Friedlander, M. W., 45, 233, 235, 405, 406

Friedman, J. I., 3, 233 Friesen, E. W., 380, 386, 446, 447(a), 449(2) Frisch, D. H., 458 Froman, D. K., 100 Frost, R. H., 43 Fujimoto, Y., 680 Fukui, S., 282 FuImer, C. B., 415 Funt, B. L., 537 Furnas, T. C., Jr., 606 Furst, M., 134 Furth, H. P., 299

G Gaerttner, E. R., 527 Gailloud, M., 227, 678 Galbraith, W., 281 Gallagher, C. J., Jr., 600 Gallagher, L. R., 205 Galonsky, A., 415 Gammel, J. L., 482, 484, 488, 493 Gamow, G., 313 Gard, G. A., 71 Gardner, D. G., 427, 429, 430 Garlick, G. F. J., 417, 421(24) Garwin, R. L., 3, 683 Gatti, E., 557 Gattiker, A. R., 218 Gauthe, B., 229 Gavrilorskii, B. V., 319 GBgauff, C., 229 Geiger, J. S., 358, 591 Geoffrion, 353, 368, 369 George, E. G., 402, 403 George, E. P., 302 George, G. G., 232 George, W. D., 528, 530 Gerholm, T. R., 342, 343, 374, 590, 598 Germain, L. S., 196 Getting, I. A., 167 Ghose, A., 509, 510 Ghosh, S. K., 15, 16, 22, 23, 195 Gibbons, J. H., 553, 554, 571 Gibson, W. M., 231 Gigli, A., 205 Gillespie, A. B., 105 Gilly, L., 457

c.,

705

AUTHOR INDEX

Gilvarg, A. B., 606 Ginther, R. J., 534 Glaser, D. A., 40, 203, 204, 382, 451 Glasoe, G. N., 546 Glrtssgold, A. E., 311, 313, 316, 317, 319 Glendenin, L. E., 72, 427, 429 Glover, K. M., 586 Glukhovsky, B. hl., 635 Godfrey, T. N. K., 142 Goldenberg, J., 527 Goldhaher, G., 221 Goldsack, S. J., 299, 317, 403 Goldschmidt-Clermont, Y., 208, 210, 214(7), 215(7), 216(7), 218, 299, 390, 395, 396, 404 Goldsmith, G. T., 269 Goldstein, L., 571 Goldstein, M., 486 Goldwasser, E. L., 28, 39, 337 Golovin, B. M., 319 Good, W. M., 553, 554 Goodman, B. B., 513 Gordon, F. J., 613 Gordon, G. F., 269(30), 270, 273(30) GOBS&,B. R., 266, 269 Gottstein, K., 76, 231, 397, 400, 401, 403(21, 34), 405 Goudsmit, S. A., 74, 75(193), 390, 394, 395 Gould, F. T., 507, 509 Goward, F. K., 222 Gozsini, A., 120, 282 Graham, R. L., 358, 591 Grainger, R. J., 269(39), 270 Graves, A. C., 100 Graves, G. A., 297 Gray, L. H., 593 Green, A. E. S., 333, 335, 336(92) Green, R. E., 440 Gregory, B., 437 Greisen, K., 61, 62, 73, 390, 662 Grimaldi, L., 306 Grismure, R., 546 Grodstein, G. W., 591, 675 Groendijk, H., 519 Groenewold, H. J., 519 Guinier, A., 599 Gunter, W. D., 335, 336(93)

Gursky, J. M., 434, 435(60) Gumk, I. M., ,593

H Haddad, E., 463 Hahn, B., 205, 327, 328, 340(56), 619 Hahn, J., 562, 563 Hahn, O., 348, 582 Hain, K., 231 Halbert, M. L., 266, 269, 415, 416, 619 Hall, H., 8, 38, 77, 78 Hall, T., 239 Hall, W. G., 342, 500 Halliday, D., 372 Halpern, J., 412, 413(8), 415(8), 416(8), 418(8), 419(8) Halpern, O., 8, 38 Hamermesh, B., 604, 613 Hamilton, J. H., 431 Hamilton, S., 181 Hanna, G. C., 28(109), 37, 574 Hanna, S. S., 237 Hansen, L. F., 23(69), 29, 30(69), 75, 197, 446 Hanson, A. O., 28, 39, 66, 67(164), 71 (164), 524, 525, 654, 663(1) Happ, M. W., 247,248 Harrison, F. B., 132, 142, 143, 146, 147, 148, 155(30), 156(30), 157(30), 158 (30) Harth, E. M., 204 Harvey, F. E., 303 Harvey, J. A., 542 Hasegawa, H., 680 Hasegawa, S., 680 Hashimoto, K., 71 Hatch, E. N., 600 Havens, W. W., Jr., 498, 504, 509, 520, 528, 529, 538, 540, 541, 542, 543, 544, 555, 562, 563 Haworth, F. E., 599 Haworth, J., 555 Hayes, F. N., 149 Hayward, E., 23, 29 Hayward, R. W., 3, 683 Hazen, W. E., 23, 29, 93, 100, 102(6), 196 Healey, R. H., 92, 93 Heanny, C. H., 227

706

AUTHOR INDEX

Heath, R. L., 624, 625, 631, 632, 633, 637, 638, 639, 641 Heckman, H. H., 45, 222, 228(56), 235(56) Hedgran, A., 358, 586, 592 Heiberg, E., 189 Heinz, O., 233 Heisenberg, W., 663 Heitler, W., 57, 58, 60, 77, 83, 87, 209, 593, 641, 663 Hendee, C. F., 297 Hendel, A., 437 Henderson, J. E., 163 Henley, E. M., 329, 330, 331, 333(61) Henrikson, H. E., 613 Hereford, F. L., 25, 34, 72 Herman, R., 323 Hernandez, H. P., 207 Herrlander, C. J., 374 Hertz, R. H., 229 Herz, A. J., 218, 247, 248, 302 Herz, R. H., 303 Hess, W. N., 317, 455 Higginbothem, W., 441 Higinbotham, W. A., 556, 558, 560, 561, 566 Hildebrand, R. H., 204 Hill, D., 181 Hill, D. A., 319,458 Hill, D. L., 329, 334, 338 Hill, R. D., 297, 590 Hincks, E. P., 28(109), 37 Hines, K. C., 32 Hirschberg, D., 402, 407(37) Hirschberg, L., 403 Hirschfelder, J. D., 230 Hodgson, P. E., 246 Hoffman, B. G., 2, 44(2), 45(2), 46(2) Hoffmann, M. M., 421, 424 Hofstadter, R., 27, 38, 67, 189, 190, 269, 323,324,325(49,51, 53, 55), 326(55), 327, 328, 330, 335(51), 340(56), 618, 629, 635, 653, 655, 658, 659(7), 663(3, 4,7) Holland, R. E., 443 Hood, R. F., 414 Hooper, I., 679 Hopkins, J. I., 423, 428(35) Hopper, J. E., 250 Hoppes, D. D., 3, 683

Hornyak, W., 319 Horton, C. W., 343 Howell, W. D., 557 Hoyt, H. C., 600, 609, 613(7) Hubbard, E. L., 11 Hubbel, H. H., Jr., 423, 428(37) Hubbs, R. A., 335 Hubert, R., 367, 368 Huby, R., 323 Huddleston, C. M., 103, 137, 432 Hudes, I., 25, 34 Hudson, A., 27, 38 Hudson, A. M., 28(121), 29, 39 Hudson, R. P., 3, 683 Hug-Bousser, F., 34 Huggett, R. W., 384 Hughes, D. J., 186, 377, 509, 510, 511, 512, 533, 544, 566 Huizenga, J. R., 527 Hull, T. E., 247, 248(134) Hulme, H. R., 77, 78, 87, 593 Hultberg, S., 593, 596, 597 Hulubei, H., 607 Hungerford, E. T., 39 Hunter, G. T., 523 Huq, M., 182 Hurst, G. S., 32, 43 Hutchinson, G. W., 182,319, 559 Huybrechts, M., 9 Huzita, H., 306 Hyder, H. R. McK., 579

I Igo, G. J., 32, 309, 310 Imaeda, K., 680 Inglestam, E., 607 Inglis, D. R., 237 Irvine, J. W., 421 Ishii, Y., 680 Iwadare, J., 678 Iwata, G., 364

J Jaccarino, V., 335 Jack, W., 414, 619 Jackson, J. D., 64, 68(160)

707

AUTHOR INDEX

Jacobs, J. A., 415, 416(17), 419(17) Jacobsohn, B. A., 341 Jacrot, B., 503 Jaeger, J. C., 87 Jaffe, G., 268 Jaffey, A. H., 364 Jakobsen, M. J., 141 Janco, M., 421, 424(32) Jancovici, B. G., 333 Jarmie, N., 486 Jastrow, R., 641 Jauch, J. M., 57 Jauneau, L., 34 Jelley, J. V., 10, 167, 168, 183, 191(11) Jennings, B., 526 Jenssen, G., 635 Jentschke, W. K., 128, 412, 413, 415, 423(6), 635 Jesse, W. P., 32, 41(83), 43, 103, 104, 105 (19), 410 Jester, M. H. L., 190 Jobes, F., 321 Johann, H. H., 599 Johannsson, T., 599 Johansson, B., 651 Johansson, S., 369, 647 Johns, M. W., 587, 588 Johnson, C. H., 108, 415, 469 Johnsrud, A. E., 434, 435(60) Johnston, L. H., 420 Johnston, L. W., 423, 428(37) Johnston, R. H. W., 26, 35, 36, 242, 247, 248(131), 249, 407 Johnstone, C. W., 442, 556 Jones, G. M. D. B., 9, 15, 16(51), 22(51), 23(51) Jones, H., 377 Jones, K. W., 550, 570, 571(3), 577(3), 579(3), 581(3) Jones, L. W., 162 Jones, P., 400, 401(33), 405(33) Jongejans, B., 27, 35, 36 Jopson, R. C., 613 Jordan, W. H., 617 Joseph, J., 57 Joshi, M. C., 343 Judd, D. L., 360 Jung, J. J., 45 Jungerman, J. A., 650

K Kaddoura, A., 680, 681 Kageyama, S., 39 Kahn, H., 81, 144, 158(33) Kalibjian, R., 162 Kalil, F., 39 Kalinin, S. P., 546 Kallmann, H., 134, 137 Kantz, A., 189, 190, 655, 658, 663(3) Kaplon, M. F., 676, 677(1), 680, 681(13, 14) Karlsson, E., 350 Katz, L., 71, 72, 527 Kayas, C., 264 Kazuno, M., 680 Keefe, D., 45, 233, 235(93) Keepin, G. R., 500 Keller, J. M., 371 Kelley, G. G., 621, 635 Kelly, E. L., 230 Kelly, G. C., 142 Kelman, V. M., 353,360, 361, 374 Kennett, T. J., 544 Kenney, R. W., 663 Kent, D. H., 302, 303(16) Kepler, R. G., 23, 29, 30, 197, 446 Kerlee, D. D., 309 Kern, J., 593 Kerr, V. N., 149 Kerth, L. T., 221 Ketelle, B. H., 411, 426, 428 Keuffel, J. W., 120, 142 Keywell, F., 269, 536 Kilburn, T., 563 King, D. T., 395, 396(15), 404(15), 678, 679 King, L. D. P., 571 Kinoshita, S.,208 Kinsey, B. B., 645, 646 Kirkpatrick, H. A., 599 Kirkwood, D. W. H., 574 Kisslinger, L. S., 319 Kjellman, J., 651 Klein, D. J., 600, 609, 613(7) Klein, O., 80 Klema, E. D., 640, 641 Knipp, J., 237

708

AUTHOR INDEX

Knoepfel, H., 414, 618, 619 Knowles, J. W., 616 Knudsen, A., 653, 655, 658, 659(7), 663 (7)

Kobayashi, Y., 343 Koenigsberg, E., 371 Koester, L. J., Jr., 337, 668 Kofoed-Hansen, O., 332, 333, 361, 362, 363(45), 651

Kohl, J. L., 370 Koller, E. L., 190 Kolyunov, V. A., 361 Kompaneetz, A. F., 593 Konopinski, E. J., 353 Kopfermann, H., 334, 335, 339(78) Korsunsky, M., 353, 360 Koshiba, M., 303, 676, 677(1) Kosmachevsky, V. K., 31 Kotani, M., 364 Kramers, H. .4., 41 Kraybill, H. L., 24, 31 Krestnikov, I. S., 41, 451, 453(7) Kristiansson, K., 264, 303, 304, 305 Kruger, P. G., 128, 412, 413(6), 423(6), 526, 635

Kruse, H. W., 148, 151(39) Kruse, T. H., 643, 650(5) Kulchitzky, L. A., 390 Kumar, R. C., 250 Kupferman, S., 71 Kupperian, J. E., 31 Kurie, F. N. D., 358 Kusaka, S., 698 Kuznetsov, E. V., 207

Ladu, M., 263 Lagarrigue, A., 437 Lal, D., 228 Lamb, W. E., 57, 85 Lambertson, G. R., 281 Lampi, E., 523 Landau, L. D., 3, 17, 18(53), 38, 39, 64(10), 68(10), 468

Landon, H. H., 5-16 Lang, J. M. B., 54'2 Langer, TJ. M., 297, 354, 426, 431, 692 Langsdorf, A., Jr., 462, 464(6)

Lannuti, J. E., 221 Lanou, R. E., 24, 31, 199 Lanzl, L. H., 75 Lapalme, T., 216 Lark-Horowitz, K., 269 Larsson, K. E., 503, 505, 506 Laslett, L. J., 428 Lassen, N. D., 104 Latter, R., 81, 144, 158(33) Lattes, C. M. G., 210, 211, 231 Lattimore, S., 395 Latyshev, G. D., 78, 390, 593 Lauritsen, C. C., 643 Lavatelli, L., 360 Lawson, J. L., 87 Layton, T. W., 607 Lazar, N. H., 142, 131, 621, 633,634(21), 635, 639, 640, 641

LeCouteur, K. J., 542 Lederman, L. M., 3, 683 Ledley, B., 319 Lee, T. D., 3, 64(9), 683, 686, 692 Lees, L., 44 Lee-Whiting, G. E., 356, 359 Lefevre, H. W., 548, 550 Legros, M., 441 Leighton, R. B., 377 Leisegang, S., 17, 39 Leks, J. E., 337, 338(98), 339(98) Leontic, B., 457 Lepore, J. V., 484 Lepri, F., 556 Leprince-Ringuet, L., 437 Levin, J. S., 546, 551 Levine, N., 66, 67(164), 71 Levintov, I. I., 484 Levi-Setti, R., 395, 397, 399 Levy, F., 299 Lewis, G. M., 411, 635 Lewis, H. R., 71 Lewis, H. W., 52, 75, 236, 390 Lewis, R. R., 64, 68(162) Leyteysen, L. G., 635 Li, Y. Y., 176 Lind, 11. .4., 600, 606, 6Oi, 613(7) Linden, €3. R., 159 Lindenbaum, &I., 264 Lindenbaurn, 8. J., 142, 166, 172, 186, 188, 456

Lindgren, I., 374, 375(82)

AUTHOR INDEX

Lindhard, J., 12, 54, 236, 361, 362, 363(45) Lipkin, H. J., 71, 408 Lipman, N. H., 441 Livesy, D. L., 239, 242(110) Livingston, M. S., 11, 19(36), 44, 54 (36), 230, 237 Lloyd, J. C., 387 Lloyd, P. E., 200 Locatelli, B., 402, 407(37) Lock, W. O., 218, 242, 395 Loepfe, E., 414, 618, 619 Lofgren, E. J., 300, 301(9) Logan, A. V., 269 Lohrmann, F., 399, 401(32) Lomanov, M. F., 41, 204, 207, 451, 453(7) Lonchamp, J. P., 229, 237, 238, 239 Lord, J. J., 26, 35, 222, 242, 396, 405(18) Losty, J., 681 Louisell, W. H., 71 Love, T. A., 269(34), 270, 277, 637 Lovera, G., 263 Lubimov, V. A,, 31, 71 Luebke, E. A., 555 Lukoff, H., 563 Lundley, A., 457 Luzzatto, G., 256, 257(146) Lyman, E. M., 75 Lynch, F. J., 443

M Mabboux, C., 397 McClellan, L. W., 515 McClure, G. W., 25, 34 McCollum, K. A., 558 McCrary, J. H., 650, 651 McDaniel, B. D., 645, 670 McDiarmid, I. B., 35, 397 McDougall, J., 78, 593 McEllistream, M. T., 414 McGuire, A. D., 148 McIntosh, J. S., 321 McIntyre, J. A., 321, 330, 629 Mack, J. E., 334 McKay, K. G., 268, 269 McKenzie, J. M., 266, 269, 270

709

MacKenzie, K. R., 11, 12, 13(49), 21, 45, 46 McKinley, W. A., 67, 70 Mackintosh, I. M., 269(29), 270, 271(29), 274(29) Macklin, R. L., 571 McNeill, K. G., 523 Madey, R., 44 Madgwick, E., 39 Madsen, C. B., 12, 13(48) Mady, R., 144 Maeder, D., 619 Magee, J. L., 230 Maienschein, F. C., 629 Mallmann, C. A., 362, 364 Malmfors, K. G., 651 Managan, W. W., 636 Manfredini, A., 45, 232, 247, 253(137), 254( 137), 255(137), 256, 257 (146) Manley, J. H., 555 Mann, H., 269(32), 270, 276 Mann, L. G., 642 Manning, G., 600 Mano, M. G., 232 Mark, H., 613 Mark, J. W., 207 Marmier, P., 600, 612 Marquez, L., 365 hfarsden, L. M., 641 Marshall, J., 72, 163, 168(5), 170, 172, 189, 502 Marshall, L., 502 Martin, H. C., 143, 158(31) Marton, C., 342, 590 Marton, L., 342, 590 Massey, H. S. W., 55, 63(146), 68, 70(168), 71(168), 462 Mateosian, E., 128 Mather, R. L., 11, 183, 233 Mathiesen, O., 303, 305 Mathieu, R., 264 Maximon, L. C., 87 -May, A. N., 209 Illayer, J. W., 266, 269, 270, 272(2), 278(35), 536 Mazzetti, L., 556 Meers, J. T., 24(72), 31 Meinke, W. W., 427, 429(46), 430(46) Meitner, L., 297, 348, 582 Melkonian, E., 509, 537

710

AUTHOR INDEX

Menon, M. G. K., 45, 76, 233, 235(93), 247, 248(130), 397, 400, 403, 405(21, 22), 453 Merlin, M., 299 Merrison, A. W., 441 Merrison, E. R., 527 Meshkovskii, A. G., 37 Messel, H., 9, 35 Metskhvarishvili, R. Ya., 361 Meunier, R., 457 Meyer, A., 416, 637 Meyer-Berkhout, U., 324, 325(55), 326 (55) Meyerhof, W. E., 635, 642 Michaelis, R. P., 35, 242 Miesowicz, M., 37 Mihelich, J. W., 297 Mileikowsky, C., 360, 364 Millar, C. H., 28, 37 Millburn, G. P., 54 Miller, A. V., 484 Miller, G. L., 269(29), 270, 271(29), 274(29) Millington, G., 39 Millman, J., 555 Mills, F. E., 28, 39, 337, 668 Mills, R. L., 571 Milton, J. C. D., 274, 412, 417(10) Minakawa, O., 680 Mitchell, A. C. G., 590 Mitchell, D. P., 501 Mitchell, T. W., 211 Mito, I., 36, 678 Miyamoto, G., 364 Miyamoto, S., 282 Mladjenoric, M., 358 Moak, C. D., 415 Mobley, R. C., 549 Mdler, C., 19, 63 Moffat, R. J. D., 297, 692 Mohr, C. B. O., 70 MoliBre, G., 74, 75, 76(194), 390, 394, 399, 663 Montalbetti, R., 527 Moody, N. F., 657 Morand, R., 212 Morellet, O., 264 Morgan, J., 263 Morrish, A. H., 26, 34, 242, 247, 248(134)

Morrison, G. C., 570, 571, 577(2), 580, 581 Morton, G. A., 127 Moskalev, V. I., 319 Motley, R., 141, 183, 456 Mott, N. F., 55, 63(146), 68, 70, 71, 211, 301,462 Mott, W. E., 421, 425, 617, 641 Moion, M. C., 544 Moyal, J. E., 399 Moyer, B. J., 190, 191(31), 462, 464(5), 479 Mozley, R. F., 12, 13(47), 21(47), 45(47), 46(47), 233 Muehlhause, C. O., 546 Mueller, D. W., 146 Muir, A. H., 600 Muirhead, E. G., 412, 413(8), 415(8), 416(8), 418(8), 419(8) Muirhead, H., 210, 395, 396(15), 404(15) Muller, D. E., 370, 600, 609, 613(7) Muller, F., 437 Mullin, C. J., 64 Mullins, J. H., 205 Mulvey, J. H., 26(94), 35, 76, 242, 397, 400, 403(21, 34), 405(21) Murphy, P. G., 283 Murray, J. J., 596, 600, 612 Murray, R. B., 269(34), 270, 277, 416, 417, 637 Murtas, P., 282 Myers, F. E., 71

N Nablo, S. V., 587, 588 Nageotte, E., 31 Nagle, D. E., 204 Nakagawa, S., 306 Nechaj, J. F., 526 Nedzel, V. A., 471 Neidigh, R. V., 314 Neiler, J. H., 553, 554 Nelms, A. T., 81 Nemirovsky, P., 57, 85 Neuendorffer, T. A., 237 Newby, G. A., 500 Newton, T. D., 542 Ney, E. P., 300, 301(9)

711

AUTHOR INDEX

Nichols, R. T., 371 Nielsen, C. E., 43, 196, 202, 446 Nielsen, 0. B., 361, 362, 363(45), 651 Nigam, B. P., 75, 394 Nijgh, G. J., 342, 343(2), 590 Nishikawa, K., 680 Nishimura, J., 680 Nishimura, I<., 39 Nishina, Y., 80 Niu, K., 680 Nobles, R. A., 103, 137, 139, 140(18), 432, 433(53), 434, 571, 572 Nonaka, I., 71 Nordberg, E., 266, 269 Nordhein, L. W., 663 Nordling, C., 359 Northrup, J. A., 137, 139, 140, 432, 433, 434, 435(60), 572 Novey, T. B., 428

0 O’Brien, B. T., 253, 254, 263(145) Occhialini, C. P. S., 209, 210, 217, 218 (34, 35) Occhialini, G., 263 O’Ceallaigh, IC., 76, 246, 247, 248, 397, 400(22), 403, 405(21, 22), 453 O’Dell, F. W., 218, 219, 221 Odian, A. C., 654, 663(1) O’Kelley, G. D., 431, 633, 634 Okudaira, K., 306 Olbert, S., 388 Oleksa, S., 546 Oliver, A. J., 222, 223, 227(53) Oliver, J. W., 269(39), 270 Onai, Y., 39 Oppenheimer, F., 300, 301(9) Oppenheimer, J. R., 663 Orman, C., 269 Orman, P. R., 556 Osborne, L., 181 Osoba, J. S., 358 Ostrander, H., 604, 613 Ott, D. G., 149 Owen, B. G., 24(71), 30 Owen, G. E., 428, 473, 474 Owen, R. B., 425

P Page, L. A., 64, 68(161) Pal, Y., 228 Palevsky, H., 186, 503, 504, 505, 506, 509, 510, 511, 512, 533, 544 Palmatier, E. D., 24, 31 Palmer, J. P., 428 Panofsky, W. K. H., 663, 668 Papian, W. N., 562 Park, S. C., 321 Parker, W., 586 Parkinson, W. C., 546 Parmentier, D., Jr., 204, 479 Parry, J. K., 24, 30 Paskin, A,, 371 Paul, W., 28, 39, 74, 75(195) Payne, R. M., 209 Peacock, R. N., 71 Peed, W. F., 296 Peelle, R. W., 637 Peeper, G. F., 559 Pegram, G. B., 501 Pelah, I., 509, 510, 511, 512 Penfold, A. S., 71, 72 Penner, S., 337, 338(98), 339(98) Perez-Mendez, V., 189, 455 Perkina, D. H., 26(94), 35, 36, 208, 210, 212, 233, 243, 247(123), 248, 250, 251, 252, 253, 263(123), 678, 680, 681 Perkins, R. B., 485 Perl, M. L., 162, 204 Perlman, I., 297, 313 Perlmutter, A., 257 Perlow, G. A., 471 Perry, J. E., Jr., 463, 523 Persico, E., 367, 368, 369 Peters, B., 228, 232, 300, 301, 399, 401(30), 402, 403, 680, 681(13) Peterson, J. M., 539 Peterson, J. R., 233 Peterson, R. E., 500 Peterson, R. W., 421, 424(32) Petrov, B., 353 Pevsner, A., 166, 186 Peyrou, C., 388, 437 Phillips, J. A., 523, 525 Pickavance, T. C., 209 Pickup, E., 25, 34, 212, 242, 404, 405

712

AUTHOR INDEX

Pidd, R. W., 71 Pinkau, K., 680, 681 Placsek, G., 504, 513 Plakhov, A. G., 159 Plano, R. J., 204 Pleasonton, F., 694, 696, 697 Plesset, M. S., 204 Pontecorvo, B., 574 Poole, M. J., 528, 529 Poppema, 0. J., 71 Porter, C. E., 317, 321, 542 Porter, F. T., 362, 363(48), 428 Porter, N., 189 Post, R. F., 146 Powell, C. F., 208, 209, 210 Powell, C. W. F., 221 Pratt, W. W., 371 Preston, G., 523 Price, B. T., 22, 24(71), 30, 31, 34(62) Primakoff, H., 428 Pringle, R. W., 537, 635 Prowse, D. J., 231

Q Quade, E. A., 372

R Rae, E. R., 513, 537, 564 Rahm, D. C., 40, 41(123), 203, 204, 382, 451 Rainwater, J., 320, 329, 330(60), 498, 520, 529, 538, 540, 541, 542, 543, 544, 555 Rajchman, J., 562 Rama, 399, 401(30) Ramat, M., 247, 257(132), 264 Randall, R. F., 503 Ranken, W. A., 650, 651 Rasmussen, J. O., 313 Rathgeber, H. D., 24(70), 30 Rau, R. R., 200 Ravenhall, D. G., 323, 324, 325(55), 326(55), 327, 328(56), 329(52), 340 (56) Rawlinson, W. F., 582 Itaymo, C. T., 269(35), 270, 278(35) Reed, J. W., 92, 93

Reich, H., 28, 39 Reid, F. J., 280 Reiling, V. G., 163 Reilly, E., 576 Reines, F., 143, 148, 1 1(39), 154, 15 ' 2 156, 157, 168, 682, 685, 690, 691, 698 Reinganum, M., 208 Reinov, N. M., 269 Remley, M. E., 128, 412, 413(6), 423(6), 635 Renardier, M., 246 Reynolds, G. T., 132, 142, 160 Reynolds, H. L., 239 Reynolds, J., 681 Reynolds, R., 321 Ribe, F. L., 128, 412 Rich, M., 44, 144 Richards, H. T., 523 Richardson, H., 360 Richardson, R. J., 11 Richman, C., 221 Ridley, B. W., 71, 692 Riquelme, J., 306 Ritchie, R. H., 343 Ritson, D., 181, 680, 681(13, 14) Ritson, D. M., 9, 35, 210, 246, 395, 396(16), 404(15), 458 Roads, F. A., 227 Roberg, J., 663 Roberts, L. D., 266, 269(3, 4, 5) Roberts, T. R., 571 Robertson, H. H., 580 Robillard, T. R., 28(121), 39 Robinson, H., 582 Robinson, H. P., 297 Robinson, R. L., 426 Robson, J. M., 684 Rochat, O., 76, 397, 400(22), 403, 405 (21, 22) Rodeback, G. W., 693, 694, 695 Rossle, E., 387 Rogers, F. T., Jr., 343 Rohrlich, F., 39, 57, 85 Rollett, J. S., 607 Romanov, V. A., 361 Ranchi, L., 247, 257(132) Ronsio, A. R., 148, 690 Rose, D., 604, 613 Rose, J. W., 663 Rose, M. E., 598, 641

AUTHOR INDEX Roscn, J. L., 498, 520, 529, 540, 541, 542, 543, 544 Rosen, L., 462, 476, 486 Rosenblum, E. S., 87, 360 Rosendorf, S., 402, 408 Rosenzweig, N., 542 Roser, F. X., 27, 37 Ross, H., 229 Ross, M. A. S., 303 Rossel, J., 619 Rosselet, P., 678 Rossi, A., 66, 67(164), 71 Rossi, B. B., 51, 61, 62, 73, 89, 92, 93, 99, 100, 101, 102, 107, 108(3), 111, 112, 113, 114, 115, 116, 117, 390, 409, 468, 654, 655(2), 658(2), 662 Rossler, L., 559 Rotblat, J., 45, 208, 212, 214(4), 215(4), 216(4), 222, 227, 231 Rothem, T., 71 Rothwell, P., 31 Rouse, J. L., 24(70), 30 Rudin, R., 264 Russell, J. T., 548, 550 Rustad, B. M., 509 Rutherford, E., 436, 582 Rybakov, B. V., 546 Ryu, N., 71 Ryvkin, S. M., 269

S Sachs, A. M., 190 Sachs, n. C., 11 Sadaukis, J., 32, 41(83), 43, 103, 104, 105(19), 410 Safford, G. J., 504 Sagane, R., 365 Sakai, M., 365 Salam, A., 3, 64(11) Salvini, G., 132 Sangster, R. C., 125, 421 Sanna, R., 250 Satarov, V. I., 319 Satterthwaitc, C. B., 513 Saunders, B. G., 296, 423, 428(37) Saunderson, J. L., 74, 75(193) 390, 394 Sauter, F., 77, 78

713

Sawyer, G. A., 523, 525 Saxma, R. C., 200 Sawn, I). S., 314 Snyrw, A., 103, 137, 432, 433(57), 434, 435, 572 Sayres, A. R., 570,:571(3,13a), 577, 579(3, 3a), 581 Scarrott, G. G., 559 Scharff, M., 12, 54, 235, 236, 250 Schawlow, A. L., 334 Schecter, L., 54 Schein, M., 222, 303, 396, 405(18) Schenck, J., 417 Schiff, L. I., 41X2 Schiller, H., 268 Schlesinger, L., 487 Schlutcr, R. A., 458 Schmidt, F. H., 370 Schmidt, T., 339 Schoenberg, M., 9 Schottky, W., 266 Schrack, R. A., 337, 338, 330 Schrank, G., 314, 315 Schiiler, H., 339 Schultz, H. L., 559 Schulz, A. G., 141 Schumann, R. W., 561 Schwaraschild, A., 66, 71 (165) Schwemin, A. J., 204, 479 Scott, F. R., 354 Scott, J. W., 239 Scott, M. B., 75 Scott, W. T., 74, 75(196), 76(196), 390, 394, 395, 399 Seagrave, J. D., 482, 486, 493 SegrB, E., 11, 12, 13, 30, 32, 104, 142, 181, 233, 321, 456, 501 Scidl, F. G. P., 503 Sen Gupta, R. L., 23, 29 Serber, R., 317 Service, D. H., 420 Shacklett, R. L., 335 Shamos, M. H., 25, 34 Shamshev, V. N. 484 Shapiro, M. M., 26, 34, 36, 208, 211(1), 214(1), 215(1), 216(1), 218, 219, 221, 242 Shaw, D. F., 482 Shaw, P. F. D., 523 Shebanov, V. A., 37

714

AUTHOR I N D E X

Sheline, R. K., 431 Sherman, N., 68, 69, 70 Shimura, Y. N., 680 Shipnel, V. S., 364 Shive, J. H., 273 Shlaer, W. J., 129, 137(14), 139(14), 140(14), 141 Shrader, E. F., 87 Shull, C. G., 71, 566 Shull, F., 358 Shulte, G., 303 Shurman, M. B., 108 Shutt, R. P., 202 Siday, R. E., 360 Siderov, V. A., 546 Siegbahn, K., 350, 355, 358, 359, 369, 372, 373, 374(77), 591, 595, 647, 648 Silverman, A., 668, 670 Silverstein, E. A., 221, 463 Silverstone, D. A., 360 Silvia, E., 306 Simmons, D. H., 11 Simmons, J. E., 485 Sippel, R. F., 650, 651 Sjostrand, N. G., 533 Skyrme, T. H. R., 471, 570, 571(1), 573(1), 574(1), 575(1) Slack, L., 358 Slatk, H., 349, 372, 373, 374, 648 Slater, W., 221, 523 Slaughter, G. G., 204 Sliv, L. A., 598 Smith, A., 411 Smith, F. M., 45, 54, 222, 227, 228(56), 231, 235(56), 236 Smith, J. H., 44, 46 Smith, L. W., 319, 526 Smith, R. K., 463, 523 Smith, W. G., 431 SmoIiar, G., 563 Smolkin, G. E., 159 Snelgrove, P., 600, 607 Snell, A. H., 694, 696, 697 Snell, P. A., 159 Snow, G., 319 Snowden, M., 201 Snyder, H. S., 74, 75(196), 76(196), 117, 390, 394 Sobottka, S., 324, 325(55), 326(55) Sokolov, I. A., 269

Sokolovsky, V. V., 503, 528, 531 Sokolowski, E., 359, 595 Solomon, J., 455 Someda, G., 299 Sommerfeld, A., 60 Sood, P. C., 333 Sorensen, S. O., 302 Souch, A. E., 415, 419, 420 Spitzer, E. J., 296 Stack, G., 233 Standil, S., 635 Stannard, F. R., 388 Stantic, S., 306 Staub, H., 89, 92, 99, 100(3), 101, 102, 107, 108(3), 111, 112, 113, 114, 115, 116 Steams, M., 670 Stedman, R., 503, 505, 506 Steinberg, E., 427, 429 Steinberger, J. N., 141, 668 Steinwedel, H., 74, 75(195) Steller, J. S., 668 Sternheimer, R. M., 2, 3, 4(8), 8, 9, 10(20,21), 16(29), 29(20,21), 30, 31, 32(20), 33(20, 29), 35,36,38(20, 21), 46, 47(1), 49(1), 51(1), 52(1), 54, 67(8), 71(8), 232, 241, 242(111, 112) Stewart, E., 663 Stewart, L., 477 Stigmark, L., 304 Stiller, B., 26, 34, 36, 218, 219, 221, 242 Stobbe, M., 77, 78 Stockendal, R., 596, 597 Stoletov, G. D., 12 Stoll, P., 414, 618, 619 Stoppini, G., 556 Storey, R. S., 414, 619 Stork, D. H., 221, 233 Stovall, E. J., Jr., 523, 525 Stratton, W. R., 500 Street, J. C., 27, 37 Streetman, G. B., 515 Stroot, J. P., 457 Suarez-Etchepare, J., 364 Sun, K.-H., 526 Sundaresan, M. K., 75, 394 Suter, T., 364 Sutton, R. B., 421, 425, 617, 641 Svartholm, N., 355, 358, 360

715

AUTHOR INDEX

Swami, M. S., 402 Swan, P., 580 Swank, R. K., 124, 125, 129(4), 135, 417, 421(25), 422, 425, 617 Swann, C. P., 72 Swann, W. F. G., 7 Swartz, C. D., 473, 474 Swartz, C. E., 190 Sweeney, D., 486 Sweetman, D. R., 415,419,420 Swenson, D. A., 420 Symon, K. R., 17, 18(54), 468 Szeptycka, M., 457

T Taffara, L., 9 Talmi, I., 333, 336 Tamai, E., 306 Tamm, I., 8, 163 Taplin, R. H., 557 Taschek, R. F., 143, 158(31), 462, 524, 525 Tassie, L. J., 70 Taub, H., 555 Tay, C. T., 212 Taylor, A. E., 22 Taylor, C. J., 128, 412, 413, 423, 635 Taylor, E. A,, 356, 359 Taylor, T. B., 317 Taylor, T. I., 509 Teem, J. M., 205 Telegdi, V. L., 3, 227 Teller, E., 42, 237 Temmer, G. M., 338, 340 Terandy, J., 362, 363(48) Terrell, J., 143, 158(31), 645, 646 Teucher, M., 399, 401(32) Thaler, R. M., 309, 310, 493 Thaxton, H. M., 463 Thieberger, R., 336 Thomas, G. E., 531, 532, 534, 535 Thomas, H., 411 Thompson, R. W., 379, 384,386 Thompson, T., 14 Thomson, J. J., 195 Thonvenin, T., 216 Thorndike, E. H., 129, 137(14), 139(14), 140(14), 141 Thorne, W., 503

Thosar, B. V., 343 Ticho, H. K., 45, 222, 227(56), 228(56), 235 (56) Tidman, D. A., 9, 302 Tiommo, J., 683 Tokunaga, S., 680 Tolhoek, H. A., 71, 313 Tomasini, G., 250, 256, 257(146), 397, 399(19) Toppel, B. J., 576, 643 Townes, C. H., 334 Trail, C. C., 108, 469 Treille, P., 31 Treiman, S. B., 64, 68(160) Tsitovich, A. P., 566 Tsuzuki, M., 680 Tuck, J. L., 523, 525 Tunnicliffe, P. R., 471 Turkevich, A., 499 Turner, J. E., 321 Turner, J. F., 71 Tyndall, A. M., 91

U uberall, H., 60, 85 Uehling, E. A., 22

V Valladae, G., 441 Vanderhaeghe, C., 220, 228 Vander Velde, J. C., 269(37), 270 Van der Ziel, I., 266, 270(1) van Heerden, P. J., 11,268 Van Hove, L., 504, 513 van Lieshout, R., 342, 343(2), 590 Van Putten, J. D., 269(37), 270 van Rossum, L., 211, 212, 264, 441 Van Sciver, W., 618 Vegors, S. H., Jr., 641 Venkateswarlu, P., 12, 13(48) Vermassen, L., 217, 218(34) Vigneron, L. J., 45, 208, 212, 214(6), 215(6), 216(6), 227, 232 Villaire, A. E., 432 Villi, C., 493 Violet, C. E., 35, 242 Vise, J. B., 66, 71(165)

7 16

AUTHOR INDEX

Vladimirsky, V. V., 503, 528, 531 Vlasov, N. A., 546 yon Baeyer, O., 348, 582 von Dardel, G., 183, 645 von Friesen, S., 264, 301 von Goeler, E., 71 von Weisziicker, C. F., 335 Votruba, V., 57, 85 Voyvodic, L., 25, 34, 208, 212, 214(7a), 215(7a), 216(7a), 242, 302, 404, 405

W Waddell, C. N.,462, 464(5), 47cJ Wagner, F., Jr., 3Ci2, 363(48), 428 Wagner, J. J., 698 Waldeskog, B., 303, 304, 305 Walker, R. L., 87, 645 Walker, W. H., 645 Wallace, R. W., 190, 191(31), 333, 455 Waller, C., 214 Walske, M. C., 20, 46, 232 Walt, M., 462 Walter, F. J., 266, 269 Wambacher, H., 209 W-apstra, A. H., 342, 343(2) 358, 590 Ward, A,, 414, 619 Ward, A. G., 72, 471 Warner, C., 39 Warner, M., 87 Warren, J. L., 463 Warshaw, R., 239 Warshaw, S. D., 22, 39, 41, 42, 43 Wasinkynska, L., 302 Waters, J. E., 564 Watson, B. B., 600, 607, 613(7) Watson, K. M., 57, 85 Watson, R. E., 70 Watt, R. D., 207 Watts, T., 321 Webb, J. H., 211, 222(23), 230 Weber, W., 442, 556 Wegner, H. E., 546 Weil, J. W., 670 Weill, R., 227, 678 Weinberg, A., 538 Weiorich, M., 3, 683 Weinstock, R., 504 Weisskopf, V. F., 317, 335, 684 Welch, J. A., Jr., 333

Wells, F. H., 556, 564 Welton, T. A., 70 Weneel, W. A., 281, 283, 317 West, D., 31, 104, 570, 573 West, H. I., 635, 642 West, W. J., 600, 607 Wetherall, A. M., 655, 663(5), 664(5), 665 (5) Wexler, A., 513 Whaling, W., 467 Wheeler, J. A., 57, 84, 329, 341(58), 683 Whetstone, A., 412, 413, 415,416(8), 418, 419 Whetstone, H. L., 221 White, G. R., 78, 79, 83, 88, 621 White, P., 39 White, R. H., 500 Whitehead, A. B., 269 Whitehead, M. N., 199, 233 Whittemore, W. L., 22, 23, 27, 37 Wiblin, E. R., 527, 528, 529 Wick, G. C., 8 Wiegand, C., 142, 172, 181, 456 Wiener, M., 221 Wiggins, J. S., 269(35), 270, 278(35) Wigner, E. P., 538 Wilets, L., 334, 341 Wilkins, J. J., 222, 230, 237, 238 Wilkins, T. R., 209 Wilkinson, D. H., 89, 92, 94, 99, 100, 101, 104, 106, 108(4), 109, 120, 410 Williams, E. J., 17, 21, 29(59), 73, 385, 390 Williams, F. C., 2, 44(2), 45(2), 46(2) Williams, J. H., 462, 523, 524, 525 Williams, R. C., 563 Williams, R. E., 320 Williams, R. L., 269(38), 270 Williams, R. W., 93, 100, 102(6), 319, 322 Williamson, R. M., 343 Willis, W. J., 40, 41 Wilson, J. G., 15, 16(51), 22(51), 23(51), 24(71), 30, 195, 196(2), 198, 200(2), 201, 377, 380, 382, 383, 384, 385(3a), 447 Wilson, R. R., 11, 44, 62, 661, 662, 670 Wilts, .J. R., 370 Wimett, T. F., 500 Winckler, J. R., 27, 37, 192 Winzeler, H., 250

717

AUTHOR INDEX

Witcher, C. M., 367 Wolfendale, A. W., 387 Wolfenstein, I,., 64, 68(161), 464, 465(11) Wolicki, E. A., 641 Wollan, E. O., 566 Wolter, W., 37 Wood, P. P., 500 Wood, R. E., 539 Woods, A. B. D., 514, 515 Woods, R. D., 314 Wooster, W. A,, 582 Wormald, J. R., 655, 663(5), 664(5), 665(5) Wouters, L. F., 432 Wright, G. T., 129, 422, 423, $25 Wright, H. W., 266, 269(3) Wu, C. S., 3, 66, 71(165), 103, 137, 138, 141, 370, 432, 433(57), 434, 435, 486, 538, 555, 570, 571(3), 572, 577, 579(3), 581(3), 683, 685 Wu, T. Y., 75, 394 Wyckoff, H. O., 163 Wyld, H. W., 64, 68(160) Wyman, M. E., 698

Y Yagoda, H., 208, 214(3), 215(3), 216(3), 22 1

Yamagata, T., 190, 654, 655, 663(1, 6) Yamanoilchi, H., 680 Ymg, C. N., 3, 64(9), 683, 686, 692 Yearian, M., 323, 325(51), :(28(51), 335(51) Yeater, M. L., 527 Yekutieli, G., 36, 242, 408, 678 Young, D. M., 36, 45, 232 Young, 0. B., 303 Young, S. A,, 523 Ypsilantis, T., 142, 181, 456 Yuan, L. C. L., 128, 142, 172, 188, 279, 413, 141, 456

Z Zajac, B., 229, 303 Zavoiskii, E. K., 159 Zeigler, R. K., 500 Zimmerman, E. L., 269(33), 270, 27(i(33) Zorn, G. T., 226, 264 Zrelov, V. P., 12 Zuber, N., 204 Zucker, A., 239, 321 Ziinti, W., 72, 227, 372 Zurheide, F. W., 303 Zmick, N., 204

Subject Index A Aberration, spherical, 353, 354 Absorption cadmium, of thermal neutrons, 518 coefficient,gamma ray, of NaI, 621-623 counter, total, for photons and electrons, 189-19 1 cross section, 504 edges, x-ray critical, 675-676 measurement of gamma ray energy, 671-676 spectrum of scintillator, 126 total, large crystal, 629 Accelerator, van de Graaff, increased pulsed neutron yield, 548 Acceptors per unit volume, 270 Agitation energy, 92, 93 Alpha particle counters, 107 differential scattering cross section in gold, 291 image in emulsion, 237 radioactivity, 313-314 response of CsI(Tl), 415, 416 of NaI(TI), 412 of semiconductor detector, 274-276 scattering, 309-313 Amidal, 217, 218 Amplifier, ionization chamber, 105-107 Amplitudes, coherent, 567 Analyzers multichannel for high counting rates, 557-559 for low counting rates, 555-557 pulse five-hundred channel, 559-560 magnetic tape, 564-566 1024 channel with magnetic core storage, 561-562 2000 channel, 562-564 time-of-flight using digital storage techniques, 559-566

Angle between two gamma rays, 679 effective solid, 346 of ejection of delta rays, 55, 56 of emission of bremsstrahlung, 60 magnetic deviation, 299 mean absolute scattering, 394, 396, 398, 400401 scattering, 73-76, 289 screening for Fermi-Thomas potential, 392 spatial, mean of absolute value, 390 Anthracene scintillation properties 421 ff Antineutrino bombardment of protons, 685 ff detector, 687, 688 flux, 686 reactions, 682 Antiproton scattering, 321 Aperture luminosity, 346, 352, 357 Argon in ionization chambers, 92-93 Argonne fast chopper, 532 Atom ionization in cloud chamber, 444445 Rutherford nuclear model, 289 Atoms mu-mesic, formation of transient, 329-330 scattering of heavy particles by, 55-56 Attachment probability of electrons in gases, 92 electron, 100-101 Attenuation of gamma rays, 671 ff Auger lines, 583-585 Avalanche, electron-initiated, 118

B Background counts in tandem counters, 467 eliminator, time-of-flight, 549-551 in large shower detectors, 666-667 rate in antineutrino measurements, 691

718

719

SUBJECT INDEX

Baffles, ring focus, 367-368 Base spread, 344 Base1 convention, 465 Beryllium neutron production in, 525-526 neutron transmission, 507-509 Beta coincidence, double, spectrometer, 363 decay determination of neutron energy, 683-685 momentum triangle, 693 polarization of electrons from, 64 ray range in matter, 71-72 spectrometer, 342-347 magnetic, 588-590 Bethe-Bloch formula, 4-7, 231 verification at relativistic energies, 2041 Binding energy, nuclear, 335-336 Blatt fluctuation theory, 263 Blob density, 245-263 length distribution, 258 Boron disintegration reaction, 500 filtering of neutrons, 518 trifluoride chambers, 108-109 Boundary effect in proportional counters, 471472 Bragg equation, 566, 601 Breakdown, 112, 118 Breit-Wigner theory for isolated resonances, 538-541 Bremsstrahlung from electrons, 56-61 angular distribution, 60 energy distribution, 57 polarization, 60-61 Bubble chambers, 203-207 energy measurements, 436-438 ionization loss measurements, 4 0 4 1 momentum measurements, 357-388 neutron energy measurements, 478481 range measurements, 436-438 velocity measurements, 451-453 Bubble count, 4 0 4 1 Bunching system, magnetic, 548-549

C Cadmium, absorption of thermal neutrons, 518 Capacitance, junction, 270-271, 272 Carbon, electron scattering, 323 ff Cassels shower detector, 664-665 Cauchois photographic x-ray spectrometer, 599-601, 605 Cerenkov counters, 10, 162-194 velocity measurement, 454-460 radiation diffraction, 176 energy loss due to, 8-10 pulse time, 192 shower detector response to positrons, 663 Cesium iodide (Tl) scintillators, 413 ff, 618-619 Chamber bubble, 203-207 cloud, 194-201 diffusion, 201-203 discharge, 281-288 expansion, 197-201 luminescent solid, 159-162 spark, 281-288 tracke noise, 381-382 reconstruction in space, 380-381 Charge determination, 289-307 distribution of nuclei, 325, 327 Chopper fast, 529-533 Jacrat, 503 mechanical applications, 504-509 for pulsed neutrons, 499-513 neutron, speed of rotation, 500-501 slow, fast neutron background, 507508 Chronotron, 548 Cloud chambers, 194-201 drop count, 14 drop formation, 446 energy 108s measurement in gases, 22, 29-30 ionisation of atoms in, 444-445

7 20

SUBJECT INDEX

momentum measurement, 375-388 range measurement, 436-438 solid state, 280 track ionization, photometric measurement, 449-451 Coefficient, internal conversion, 583 Coincidence circuit, fast, 439 Collector electrodes, potential charge of, 90, 95 Collimator-baffle system for x-ray spectrometer, 601-602 Collision diameter, 289 radius, 290 time, 7 Collisions, ionizing, total number, 33 Composition of emulsion, 215 Compton effect, 622, 671, 672 electron kinetic energy, 80 scattering, 79-83, 631-632 cross section, 80-82 spectrometer, two-crystal, 629 Conversion external, 586-594 internal, 296-297, 594-599 ratios, 595 Converter, photoelectric, 586 ff time-to-pulse height, 440 Coordinate method, 378-379 Coulomb energy of homogeneously charged sphere, 332 excited gamma rays, 613-616 scattering deviations, 309 probability for elastic, 389 Counters alpha particle, 107 boron trifluoride proportional, 533-534 jitter time, 534 bulk, flux and energy measurement with, 4 7 1 4 7 5 Cerenkov, 10, 162-194 differential isochronous selfcollimating, 457 focusing, 166-186, 456-460 gas, 181, 184, 185, 457-460 liquid, 456-457

non-focusing, 186-189, 455-456 cloud chamber control, 200 gas-filled, 110-120 Geiger, 118-120 efficiency, 120 low pressure, 33-34 helium-3 proportional, 573 ff hydrogen-filled proportional, 4 7 1 4 7 3 proportional, 108-118, 471-473, 533-534, 573 ff ionization loss measurement in, 30-34 linearity of response, 117 space charge limitation, 111 time of response, 113-115 scintillation, 120-159 self-quenching, 119 solid for neutrons, 473-475 surface barrier, 266 tandem, background counts in, 467 telescope, 466 velocity interval selection, 183, 186 wall effect, 573-574 Counting rate, recoil, 462463 Covan, 228 Cross section absorption, 504 total, for gamma rays, 87-89 for antineutrino bombardment of protons, 686 capture, 41 Compton scattering, differential, 80-82 for D(d,n)HeS reaction, 522-523 delta ray ejection, 55 for H3(d,n)Hes reaction, 524-525 1088, 41 Mott scattering, 55 for neutral pion production, 339 neutron measurement in kev region, 553-554 plus He* reaction, 571-572 proton, 488-493 scattering, 466, 484 nuclear scattering of electrons, 67, 69 pair production, 83-87 for photoelectric effect, 78-79 Rutherford differential scattering, 291 for secondary electron production, 63, 65

72 1

SUBJECT INDEX

Thomson scattering, 77 total, of lead for 10 A neutrons, 504, 507 Crystal curved focusing spectrometer, 599-616 methods of bending, 606-607 diffraction, measurement of neutron momentum, 566-570 spectrometry, 612-613 nionochromator, neutron, 569-570 plane spacing, 567 Current ionization chamber, 109-110 photomultiplier, 129 Curvature compensation, 377-378 Curves comparison, 376 isoenergetic, 656-657 Cyclotron frequency, 367

D Debye-Waller factor, 567 Decay time fluorescence of anthracene, 421, 422 of cesium iodide (TI), 414 scintillation, in organic scintilfators, 425 Delta rays, 43 angle of ejection, 55, 56 density, 301 ff ejection cross section, 55 frequency, 301 production, 300 Density blob, 245-263 bubble, 4 5 1 4 5 3 delta ray, 301 ff effect, 7-8, 10, 13, 23-29, 231, 241 electron energy loss measurement, 38-40 emulsion, 215, 224 gap, 245-263 grain, 212, 243, 247, 300 photographic track, 449-451 photometric, relationship to cascade energy, 681

Depletion region, 266, 267 width, 270-271 Detection efficiency of BF, counter, 533 Detectors antineutrino, 687, 688 calibration, 154-159 for epithermal neutrons, 533-538 for fast time-of-flight, 548 proton recoil, 107-108 scintillation, 418, 419, 420, 537-538 semiconductor, 265-280 shower, 652-668 slow neutron, 139,. 140 solid state for neutrons, 536-537 Ileuteron response of NaI(Tl), 412 Development of thick emulsions, 217 Deviation mean scattering, 400 of multiply scattered particles, 391 ff Diameter of droplet, 447 Diff ra_ction of Cerenkov radiation, 176 measurement of neutron momentum, 566-570 phenomena at high energies, 322 Diffuseness parameter, 316, 321 Diffusion chambers, 201-203 of electrons and ions, 94, 100-101 track widening by, 381 Discharge chambers, 281-288 Discriminator, pulse shape, 537 Dispersion, 345, 352, 357 width, 177 Distortion emulsion, 228 straggling, microscopic, 228 track, 220 Donors per unit volume, 270 Drift velocity, 90, 91, 93 negative ions, 92

E EfFiciency rounting of boron-loaded liquid scintilltttion counters, 535-530 detection of boron trifluoride counter, 533 of helium-3 counter, 576

722

SUBJECT INDEX

discharge chambers, 284-285 gamma ray telescopes, 669-670 Geiger counter, 120 ionization, 103-105 quantum, 125 ff, 134 relative scintillation of gas scintillators, 433 of sodium iodide (Tl), 413 scintillation absolute, 421 vs particle mass, 416-417 scintillator, 126-128, 618 uniform field pair spectrometer, 641-652 Electrodes, guard, 99 Electron accelerator, high energy as neutron velocity spectrometer, 527 ff attachment, 92, 100-101 in oxygen, 101 backscattering on organic scintillators, 425-426 capture orbital by A r 3 7 , 693-697 transitions, 296 diffusion, 94, 100-101 -electron scattering, 63-71 energy, internal conversion, 594 loss measurement study of density effect, 38-40 initiated avalanche, 118 initiated showers in lead, 661-662 mean free path in gaaes, 113 pairs energy measurement, 676-678 ionization measurement, 678-681 opening angle, 677-678 -positron pair ionization loss, 36-37 pulse height energy relationship in anthracene, 423-425 response of semiconductor detector, 276 secondaries of x-rays, energies, 582 sensitive emulsions, 212 spectrum, mu-meson decay, 155 spin dependent cross section, 64 ff total absorption counter, 189-191 trajectories, solenoidal spectrometera, 366 volt, 497

Electrons atomic, screening of nuclear charge, 294-295 behavior in gases, 91-94 bremsstrahlung, 56-61 capture and loss a t very low energies, 41-42 elastic scattering, 322-329 ionization loss of, 18-19 longitudinal polarization, 3 mean range, 62 passage through matter, 56-72 production of secondaries, 63-71 radiation by, 56-61 Elements, identification of new, 295-296 Emission absolute gamma ray, 640-641 spectrum of scintillator, 126, 131-132 Emulsion, photographic, 208-264 charge determination of particles, 298-307 -cloud chamber method, 680 diluted, 215 electron sensitive, 212 gamma ray measurement, 676-682 grain count, 14 grain density, 35 ionization loss in, 34-37 measurement in, 240-245 nuclear momentum measurement, 388408 neutron flux and energy measurement, 475478 processing, 216-224 range-energy relation, 45 range of particles, 226-240 straggling, 54 standard, scattering constant, 405 tracks, opacity measurement, 303-305 water content, 222-224 Energy agitation, 92, 93 binding, 297-298 of characteristic x-ray quanta, 294 charged particle, 342 conversion factor, 124 to wavelength, 607

723

SUBJECT INDEX

critical, 660 determination, 341438 with ionization chambers, 409-410 distribution of neutrons scattered inelastically by aluminum, 512 of neutrons scattered by one photon, 513 in pair production, 84, 85 to form an ion pair in gas, 1, 43-44 gamma ray determination with curved crystal spectrometer, 599-616 determination with pair spectrometer, 641-652 measured by absorption, 671-676 levels in organic molecules, 123 spacing of compound nucleus, 538, 541-544 loss due to Cerenkov radiation, 8-10 electrons a t very low energies, 4 2 4 3 fluctuations, 17-18 in gases, cloud chamber measurement, 22, 29-30 per ion pair, 103-105 in liquid scintillators, 143-146 of LiTions in emulsion, 239 maximum, in single event, 448 particle, 449 smallest measureable in ionization, 409 per unit length, 230-231 measurements with cloud and bubble chambers, 436-438 of electron pairs, 676-678 by recoil methods, 461-494 neutrino, 683-692 neutron determination with helium-3 spectrometer, 570-581 instrumentation for measurement, 466481 interaction recoil, 463464 of PO meson, 681 resolution relation to momentum resolution, 345

*

relationship with gamma ray energy, 634 ff time-of-flight spectrometer, 545 response, scintillator, 128-129 spread of filtered neutrons, 509-510 transfer heavy particles to electrons, 300 kinetic, 290 maximum, 4, 8 processes, 133 Error determination in scattering measurements, 407408 in drop counting, 449 grain, 396 in range measurements, 437 in track density measurements, 451 Excitation decay-time dependence, 130 potential, mean, 10-14 Expansion adiabatic, temperature change during, 197 chambers, 197-201 time, cloud chamber, 200

F Fermi plateau, 15, 18 plot of allowed beta-decay electron spectrum, 684 -Thomas potential, screening angle, 392 type neutron velocity selector, 502-503 unit, 311 Field distribution, magnetic, 354, 355 fringing, 364 magnetic for cloud and bubble chambers, 386 Fields, inhomogeneous, axially symmetric magnetic, 370-375 Film resolution, 448 Filters boron, 518 neutron, 507-509, 568-569 Fission chambers, 108-109 counter, 140

724

SUBJECT INDEX

fragments, response of semiconductor detector, 273-274 threshold, 109 Fixing of thick emulaion, 217 Flight time, 545 measurement, 547-548 Fluctuation theory of Blatt, 263 Fluorescence decay time of anthracene, 421, 422 of cesium iodide (TI), 414 specific of sodium iodide (Tl), 413 Flux antineutrino, 686 neutron, instrumentation for, 466-481 reactor neutron, 498 Focal length, solenoidd spectrometer, 367 Focus, depth of, for cloud chamber measurements, 447 Focusing axial, 355 Cerenkov counter, 166-186 properties of magnetic fields, 370-375 Foq-ler's coordinate method, 397-399 Frank and Tamm equation, 8 Free-electron chambers, 94 Frequency of delta rays, 301 distribution of elastic waves in solids, 510

sensitivity of inorganic scintillators, 41 1 sources, 605 composite, quantitative analysis, 63!) monoenergetic, 675 size considerations, 6055606 telescopes, 668-671 width, total, 544 Gap density, 245-263 length, 246 ff Ga,s Cerenkov counters, 457-460 -filled counters, 110-120 multiplication, 110-1 18 statistics, 117 voltage dependence, 112 ff scintillator light wavelength, 433 noble, 431-435 purity, 432-433 spectrometer resolution, 435 Gases, electron mean free path in, 113 Glass, lead-loaded, 190 Grain density, 300 in high energy electron pairs, 678-679 error, 396 Grains per cluster, 259 ff

H Gamma rays, 582-682 absorption cross section, total, 87-89 attenuation, 76, 671 ff Coulomb excited, 613-616 energy determination, 599-616 by absorption, 671-676 from lead, 330 mass absorption coefficient, 88 mean free path in toluene, 144 momentum determination, 599-616 neutron capture, 613-616 penetration, 76-89 photographic emulsion measurement, 676-682 from romeson decay, 679-682 scintillation spectrometry, 616-641

H-D constant spacing model, 252 ff Helium-3 neutron spectrometer, 570-581 Helium-recoil counter-polarimeter, 484-486 Hyperfine structure magnetic, anomalies, 335 optical shift, 334

I Image base width, spectrometer, 3 5 1 3 5 2 formation, latent, 210-211 photographic of droplet, 447 pre-, speck, 211 sub-, 211 Impact parameter, 5 ff, 289 Integrating chambers, 109-110

725

SUBJECT INDEX

Intensity gamma ray, 637-641 neutron source, 522-529 relative internal conversion, 585 Interactions gamma ray photoelectric, 621 of radiation with matter, 1-89 Ion behavior in gases, 91-94 -collection chambers, high resolution with, 410 heavy, response of semiconductor detector, 273-274 negative, drift velocity, 92 neutralization, 94 pair energy loss per, 103-105 in gas, energy to produce, 43-44 thresholds, 196 Ionization amount of, 103-105 chambers, 89-110 argon in, 92-93 energy measurement with, 409-410 gas-filled, 265 parallel-plate, 96 ff pulse type, 89-91 quantitative operation, 100-103 of cloud chamber tracks, photometric measurement, 449-451 efficiency, 103-105 loss average for mu mesons, 20 ff bubble chamber measurement, 4041 charged particles, 1, 4-44 dependence on particle velocity, 7-8 electron-positron pair, 36-37 of electrons, 18-19 evaluation, 10-14 in helium, 16 most probable, 17-18 in oxygen, 15 in photographic emulsions, 34-37 in proportional counters, 30-34 protons in solids, 47-48 in a radiator, 178 relativistic rise, 7, 23-21) restricted, 14-17 in scintillator, 37-38

measurement in cloud chamber, 444-449 of electron pairs, 678-681 in emulsions, 240-245 parameters, 245-263 potential, 231, 232-233 smallest measureable energy loss in, 409 Isotope shift, 334

J Jacrat type chopper, 503 “Jitter time” of counter, 534 Junction capacitance, 270-271, 272 detector, p-n diffused, 265 ff width, 270-271

K Kinematics non-relativistic in neutron recoil experiments, 486-488 relativistic, two body problem, 487488 Kinetic energy of Compton electron, 80 Klein-Nishina formula, 80-81

L Landau effect, 17 fluctuations, 117 Lead, total cross section for 10 A neutrons, 504, 507 Length, thin down, 300 Lens spectrometers, 365-375 Lifetimes, scintillator, 129-131 Light collection uniformity, 149 output scintillation, 147-148 saturation of CsI(TI), 415 pulse time, Cerenkov, 454 wavelength of gas scintillators, 433 Line width, 351-352 Lines, conversion, 582-599 Linearity of helium-3 proportional counter, 574-575

726

SUBJECT INDEX

Lithium iodide scintillator, 417 Luminescence processes, 123 rate of, 128 Luminosity, 346, 352, 357

M Magnification of cloud chamber pictures, 447 Mass absorption coefficient for gamma rays, 88 attenuation coefficient, total, for photons, 673-674 correction factor, 51 Mean free path of electrons in gases, 113 gamma rays in toluene, 144 neutron, proton collision in toluene, 145 for scattering plus absorption in scintillation liquid, 148 Meson mu average ionization loss, 20 ff decay electron spectrum, 155 decay time spectrum, 156 7ro

energy, 681 gamma rays from decay, 679-682 Moderation of thermal neutrons, 515-517 time for fast neutrons, 519-521 Molihre theory of multiple scattering, 392 ff Momentum determination, 341-438 gamma ray with curved crystal spectrometer, 599-616 with pair spectrometer, 641-652 linear, measured by recoil techniques, 461-494 maximum detectable, 198, 384 measurement with cloud or bubble chamber, 375-388 in nuclear emulsions, 388-408 neutrino, 692-698

neutron crystal diffraction measurement, 566-570 determination with He8 spectrometer, 570-581 resolution relation to energy resolution, 345 of singly charged particle, 396 spin angular measured by recoil techniques, 461-494 Monochromator, neutron crystal, 569-570 Moseley’s law, 294-295 Mott asymmetry function, 68 ff scattering cross section, 55 of electron from heavy nuclei, 68 Multiplication factor, proportional counters, 111 gas, 110-118 Multiple scattering of charged particles, 73-76 Mu-mesic atoms, formation of transient, 329-330

N Neutralization of ions, 94 Neutrino, 682-698 energy determination, 683-4392 free, experiment, 685-692 momentum, 692-698 reactions, 682-683 rest mass, 692 spin, 698 Neutron beam cold, intensity, 511-513 counting rate, 517 ff polarized, 465 capture in cadmium, 157 cold, studies at Brookhaven reactor, 510-511 counting, 107-109 distribution, liquid nitrogen cooled Be filtered, 506 energy measurement, 466-481 with bubble chamber, 478-481

727

SUBJECT INDEX

flux measurement, 4 6 6 4 8 1 with nuclear emulsions, 475478 b y recoil techniques, 4 6 1 4 9 4 tandem counter, 466-471 helium scattering, 481-482, 493-494 interactions, 4 9 6 4 9 7 recoil energy, 463-464 time-of-flight measurement, 495-566 -nucleus scattering potential, 318 production, 495 in beryllium, 525-526 properties, 496497 proton collision mean free path in toluene, 145 cross section, 488-493 response of semiconductor detector, 273 scattering, 317-3 19 differential cross section, 466 slow, detector, 139, 140 slowing down, 145 solid state detector, 535-537 source intensity, 522-529 spectroscopy, time analyzers, 554-566 spin-angular momentum, 464-466 wavelength selection by diffraction, 568 yield from gas-containing targets, 524 ff from heavy ice target, 524 per incident electron, 528, 530 from uranium target, 527-528, 529 Neutr ons epithermal, detectors for, 533-538 filtered, energy spread, 509-510 moderated, energy spectrum shape, 521-522 thermal, 497-517 moderation, 515-517 sources, 497499 Noise chamber track, 381-382, 396, 398,399 elimination, 400-401 ionization chamber amplifier, 105-107 solid state detector, 271 Nuclear binding energy, 335-336 emulsions, 216-224 microscope procedures, 224-226 I

range-energy relation, 229-236 interaction, inelastic, in Cerenkov medium, 178 model of atom, 289 periphery, 307 radiation, response of semiconductor detector, 273 scattering of electrons, 63-71 cross section, 67, 69 size measurement, 307-341 electromagnetic, 322-325 nuclear force method, 309-322 Nuclei angular shapes, 338-341 atomic, determination of charge, 289-307 charge distribution, 325, 327 condensation, 195 mirror, 330-333 Nucleus compound, energy level spacing, 538, 541-544 -nucleus scattering, 321-322 optical model, 30 ff potential at surface, 312-313

0 Objectives, microscope, 224 Opacity measurement of emulsion tracks, 303-305 Oxygen, elastic electron scattering, 325-326 P

Pair production, 83-87, 622, 641 ff, 671, 672 energy distribution, 84, 85 total cross section, 86 Pair spectrometer axially focusing, 647 determination of momentum and energy of gamma rays, 641-652 intermediate-image focusing, 648649 multichannel homogeneous field, 651 three crystal, 629 uniform field, 643-647 Parity conservation, 3

728

SUBJECT INDEX

Particles charge detcwnination, 28!#-307 charged distrihutioti of lateral displacexnent,, 74 heavy response of NaI(TI), 412, 413 spectrum, 411 ionization loss, I, 4-44 multiple scattering, 73-76 scattering mean square angle, 7374 scintillation spectrometry, 41 1 4 3 5 detection, 1-288 energy loss, 444 Pellicles, 221 Penetration of gamma rays, 76-89 Photoelectric effect, 77-79, 671, 672 cross section from K shell, 77, 78 external, 586-594 Photomultiplier, 179 cathode spectral response, 133 circuitry, 151-154 current, 129 selection, 151-154 Photon Cerenkov, scattering, 177 initiated showers in lead, 661-662 response of semiconductor detector, 277 total absorption counter, 189-191 total mass-attenuation coefficients, 673-674 Photoproduction, coherent neutral pion, 336-338 Pion gap length distribution, 256 photoproduction, coherent neutral, 336-338 production, cross section, 339 response of semiconductor detectors, 278-279 scattering, 319-321 Plateau, 119 Plutonium critical assembly leakage neutron spectrum, 477-478 Polarizability, atomic, 9 Polarization bremsstrahlung, 60-61 effect, 241

electron from beta decay, 64 longitudinal, 3 of medium, 231 neutron beam, average, 483-486 POPOP, 149 Porter-Thomas distribution, 543-544 Positron-electron pair production, 641-642 Potassium iodide (Tl) scintillators, 417 Potential neutron-nucleus scattering, 318 nuclear a t surface of nucleus, 312-313 optical model, 310-313 proton-nucleus scattering, 316 Pressure in cloud chambers, 197-198, 202 Prism spectrometer, 359-364 Probability of internal conversion, 583 Profile symmetry of x-ray reflection lines, 604 Proportional counters, 108-118, 471473, 533-534, 573 ff Proton antineutrino bombardment, 685 ff gap length distribution, 256 high energy, number of neutrons produced, 527 ionization loss in solids, 47-48 -nucleus scattering potential, 316 range-energy relation, 44, 45-51 in aluminum, 233-234 recoil detectors, 107-108 response of cesium iodide (TI), 415, 416 of semiconductor detector, 274-27G of sodium iodide (TI), 412 scattering, 314-317 Pulse analyzers, 559-566 formation, 95-99 height, 102 distribution, 658-659 gamma rays in NaI(T1) crystals, 623 ff helium-3 spectrometer, 575 maximum, 129 from organic scintillators, 421-422, 423 semiconductor detector, 268, 269

SUBJECT INDEX

ionization chamber, 89-91, 107-109 light, very short time, 192 time, Cerenkov radiation, 192, 454 Purity of scintillator gas, 432-433

Q Quantum efficiency, 126 ff, 134

R Radiation beta, angular distribution, 60 eerenkov, energy loss due to, 8-10 by electrons, 56-61 intensity, Cerenkov counter, 163 ff length, 59 unit in pure emulsion stack, 680 Radiator Cerenkov counter, 168-170 ionization loss in, 178 Radioactivity, alpha particle, 313-134 Radius collision, 290 of curvature of tracks, errors, 381 ff nuclear, 309, 314 Random phase distribution, 543-544 Range beta rays in matter, 71-72 electron mean, 62 energy relation, 2, 44-55, 437-438 heavy ions in emulsion, 238 multiply charged particles, 237-240 in nuclear emulsions, 45, 229-236 protons, 44, 45-51, 233-234 extension, 237 in heavy materials, 436-437 mean, 44 measured with cloud and bubble chambers, 436-438 particles in emulsions, 226-240 practical, 71 straggling, 51-55, 236 Itayleigh scattering, 673 Reactions antineutrino, 682 neutrino, 682-683 Reactor Brookhaven cold neutron studies, 510-51 I

729

neutron flux, 498 nuclear, 498 pulsed, 499 Recoil counting rate, 462-463 energy, neutron interaction, 463-464 experiments, 692-698 scatterer measurements, angular distribution, 481-482 techniques, 461-494 Recombination, 100-101, 109-110 coefficient, 94 Reflectors for scintillation counters, 136-137 Refractive index, 180 carbon dioxide, 176 Rejection ratio improvement, 460 Relativistic rise in ionization, 444 Resolution Cerenkov counter, 168-170, 456 correction, k i t e instrumental, 428-429 energy relationship with gamma ray energy, 634 ff time-of-flight spectrometer, 545 film, 448 gas scintillation spectrometer, 435 helium-3 spectrometer, 573 high, with ion-collection chambers, 410 ideal velocity spectrometer, 518-519 nuclear spectroscopy, 344 semiconductor detector, 268, 275, 27G time-of-flight system, 517 wavelength, crystal monochromator, 569 Resolving power of magnetic spectrometers, 343 ff Resonance giant ( y p ) , 527 isolated, Breit-Wigner theory for, 538-541 neutron, in compound nucleus Cdll*, 538-539 levels, slow neutron in U23a, 540-541 Response function, 637 srmic*onductor detrctm to radiation, 273-279 sodium iodide (TI) spertronietrrs, 621-631

730

SUBJECT INDEX

Rutherf ord formula, 55 scattering, 289-293

S Sagitt a constant, 397, 403 mean, 402 method, 376-377 Sandwich method, 299 Scanning, emulsion, photoelectric, 264 Scattering alpha particle, 309-313 angle, 289 mean absolute, 394, 396, 398, 400-401 mean relative, electrons in emulsion, 677 anomalous, 290 antiproton, 321 in bubble chamber, 385-386 Cerenkov photons, 177 in cloud chamber, 385-386 Compton, 631-632 constantK,394,403-407 Coulomb deviation from, 309 probability of elastic, 389 cross section Mott, 55 neutron, 466, 484 neutron-He4, 493-494 differential, linear momentum measurement of neutron, 481-483 electron elastic, 322-329 -electron, 63-71 by nuclei, 63-71 heavy particles by atoms, 55-56 mean deviation, 400 measurement of trajectories of slow particles, 403 multiple, 389-395 measurement of electron pair energy, 676-678 Moli6re theory, 392 ff transition to single scattering, 392, 394 Williams theory, 390 ff

neutron, inelastic, 317-319 by liquids and solids, 504, 507 spectrum in vanadium, 512 spinning sample method, 514, 515 nucleus-nucleus, 321-322 pion, 319-321 plural, 390 process, idealized, 462 proton, 314-317 Rayleigh, 673 Rutherford, 289-293 Schottky-type potential barrier, 270 Scintillation characteristics, 125-132 counters, 120-157 boron-loaded liquid, 535-536 general applications, 141-142 large, 142-149 light pipes for, 136-137 reflectors for, 136-137 efficiency, 126-128 absolute, 421 gas scintillators, 433 us. particle mass, 416-417 sodium iodide (Tl), 413 energy response, 128-129 lifetimes, 129-131 processes, 121-125 properties of anthracene, 421 ff spectral distribution, 131-132 spectrometry of charged particles, 411435 spurious, 396, 399, 401 Scintillators absorption spectrum, 126 cesium iodide (Tl), 413 ff, 618-619 emission spectrum, 126, 131-132 gamma ray, 617-619 inorganic, 411-420 ionization loss in, 37-38 light output, 147-148 liquid, 132-134 energy loss in, 143-146 lithium iodide, 417 loaded, 137 noble gas, 137-141, 431435 organic, 420-431 electron backscattering, 425-426 pulse heights from, 421422, 423 scintillation decay time, 425

SUBJECT INDEX

plastic, 135-136 potassium iodide (Tl), 417 411 ff, 617-618 sodium iodide (Tl), transparency, 148-149 Screening coefficient, 83 of nuclear charge by atomic electrons, 63-7 1 parameter, 56 Secondary electron production by electrons, 68-71 Sector spectrometer, 359-364 Semiconductor detector, 265-280 neutron energy spectrometer, 276-277 Sensitivity inorganic scintillators to gamma rays, 411 photographic emulsions, 2 10-2 16 Sharp cut-off model, 309 Shell correction term, 4, 19-20 Shields, “graded,” 631 Shower absorption, total, Cerenkov counters, 189-1 91 detectors, 652-668 Cassels, 664465 Cerenkov, response to positrons, 663 limitations, 665-668 size, 655-660 Shrinkage factor, 216, 223-224, 226-227 Similarity rule, 660-661 “Sine-screw” linkage system, 603 Skin thickness, 327-328 Smooth model, 248 ff Sodium iodide (TI) scintillators, 411 ff, 617-6 18 gamma ray absorption coefficient, 621623 preparation, 619 Solenoidal spectrometers, 365-370 Solid state cloud chamber, 280 detector noise level, 271-273 Solutes, secondary, 134 Solvents, secondary, 134 Space charge limitation, proportional counters, 111 Spark chambers, 281-288 Spectrograph, 349

73 1

Spectrometer beta ray, 342-347 Cauchois photographic x-ray, 599-601, 605 curved crystal, 599-616 applications and scope, 607-616 transmission type, 599 ff double beta coincidence, 363 double focusing, 354-359 epithermal neutron time-of-flight, 517544 applications, 538-544 fast neutron time-of-flight, 544-554 applications, 551-553 gas scintillation, resolution, 435 helium-3 neutron, 57&581 hollow-crystal scintillation, 427428 intermediate image, 372-374 lens, 365-375 magnetic, 347-375 beta-ray, 588-590 performance parameters, 342-347 resolving power, 343 ff neutron velocity, high-energy electron accelerator, 527 ff orange, 361 ff prism, 359-364 rotating crystal time-of-flight, 514-515 sector field, 359-364 semicircular focusing principle, 348353 sodium iodide (Tl), response, 621-637 solenoidal with uniform magnetic field, 365-370 spiral orbit, 364-365 split-crystal scintillation, 426 third order focusing, 353-354 transmission, 345-346 two-crystal Compton, 629 velocity resolution of ideal, 518-519 Spectrometry charged particle scintillation, 411-435 crystal diffraction, 612-613 gamma ray scintillation, 616-641 Spectroscopy, neutron, time analyzers for, 554-566 Spectrum absorption scintillator. 126 total large‘crystal, 629

732

SUBJECT INDEX

antineutrino from fission fragments, 686 B distrihution, scintillator, 131-132 emission, scintillator, 126, 131-132 heavy charged particles, 411 leakage neutron of plutonium critiral assembly, 477-478 neutron calculation from observed pulse distribution, 577-579 from scattering on vanadium, 512 nuclear y and x-ray lines from neutron activated Ta, 607-609 time delay, 443 of fast neutron interactions, 551, 552 Spin dependent electron-electron scattering cross section, 64 ff neutrino, 698 orientation, neutron beam, 465-466 Spiral orbit spectrometer, 364-365 Stopping power at very low energies, 42-43 Straggling, 436 parameter, 53 range, 51-55 Strength function, 544 Structure fartor of unit cell, 567 Summing coincident, 430-431 random, 431 “Superposition of profiles” method, 604, 609-61 1 Supersaturation cloud chamhers, 194-196 discharge chambers, 201

T Tantalum, neutron activated, 607-609 Telescopes, gamma ray, 668-671 Temperature control bubble chamber, 205 cloud chamber, 200 Thickness, skin, 327-328 Thomson scattering cross section, 77 Time analyzers for neutron spectrometers, 554-566 burst start, 546-547

collision, 7 decay, spectrum of mu-mesons, 156 delay spectrum, 443 expansion, of cloud chambers, 200 gate, 555 moderation, for fast neutrons, 519-521 -0f-ffight analyzers using digital storage techniques, 559-566 as background eliminator, 549-551 distribution of C137 recoils from Ara7 decay, 695 measurement of neutron interactions, 495-566 method t o measure charged particle velocity, 435-443 spectrometers, 514-554 response of proportional counters, 113115, 116 rise, semiconductor detector, 272-273 sensitive, of bubble chambers, 207 spectrum of fast neutron interaction with gold, 551 with U*a*,552 -to-pulse-height converter, 440 tincertainty of BF8 counter, 533 Toluene mean free path gamma rays, 144 neutron, proton collision, 145 Track density errors in measurement, 451 photographic, 449-451 distortion in cloud chambers, 382-383 length, 226 noise, 381-382, 396, 398, 399 eliminator, 400-401 radius of curvature errors, 381 ff width in cloud chambers, 200 in emulsion, 305-307 Transfer energy, 133 radiative, 124 Transitions, electron-capture, 296 Transmission fractional, of homogeneous 7-ray beams, 673-674 spectrometers, 345-346, 599 ff Triethyl benzene purification, 149

733

SUBJECT INDEX

U Unit area function, 521 Unit cell structure factor, 567 Uranium heavy water moderated reactor, 505 target neutron yield, 527-528, 529

V Velocity charged particles, time-of-flight measurement, 438-443 interval selecting counter, 183, 186 measurement, 444-460 in bubble chamber, 451-453 using Cerenkov counters, 454-460 resolution, Cerenkov counter, 168-170, 456 selector, Fermi-type neutron, 502503 spectrometer pulsed accelerator, 519, 520 resolution of ideal, 518-519 Voltage dependence of gas multiplication, 112 ff plateau, 101

W Wall effect, counter, 573-574 Wavelength conversion t o energy, 607 determination, error in, 609 resolution of crystal monochromator, 569 shift in, 149 Weiszacker semiempirical formula, 335336 Williams theory of multiple scattering, 390 R

X X-ray critical absorption edges, 675-676 electron secondary energies, 582 fine structure measurements, 334-335 level splitting, 334 line width, 611-612 quanta, energy of, 294 from radioactive substances, 296-298 spectra, characteristic, 293-298 spectrometer Cauchois photographic, 599-601 ,605 collimator baffle system, 601-602

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