Advances in
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS VOLUME 32 A volume published in honor of Sir David Bates, Founding Editor, upon his retirement as editor of Advances in Atomic, Molecular, and Optical Physics
Sir David Bates
ADVANCES IN
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS Edited by
Benjamin Bederson DEPARTMENT OF PHYSICS NEW YORK UNIVERSITY NEW YORK, NEW YORK
Alexander Dalgarno CENTER FOR ASTROPHYSICS HARVARD-SMITHSONIAN CENTER FOR ASTROPHYSICS CAMBRIDGE, MASSACHUSETTS
VOLUME 32
A volume published in honor of Sir David Bates, Founding Editor, upon his retirement us editor of Advances in Atomic, Molecular, and Optical Physics
@
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EDITORIAL BOARD
P. R. BERMAN New York University New York, New York K. D O L D E R The University of Newcastle-upon-Tyne new castle-upon- Tyne England M. GAVRILA F. 0. M. Instituut voor Atoorn- en Molecuulfysica Amsterdam The Netherlands M. INOKUTI Argonne National Laboratory Argonne, Illinois S. J. SMITH Joint Institute for Laboratory Astrophysics Boulder, Colorado
FOUNDING EDITOR SIFt DAVID BATES
This Page Intentionally Left Blank
Contents
...
CONTRIBUTORS
Xlll
PREFACE
xv xvii
A FURTHER APPRECIATION
Photoionisation of Atomic Oxygen and Atomic Nitrogen K. L. Bell and A . E. Kingston I. Introductory Remarks 11. Photoionisation of Atomic Oxygen 111. Photoionisation of Atomic Nitrogen
IV. Conclusion References
1 2 10 16 17
Positronium Formation by Positron Impact on Atoms at Intermediate Energies B. H. Brrrnsden und C. J. Noble I. Introduction 11. Coupled Channel Equations 111. Perturbation and Distorted Wave Models IV. Conclusions
References
19 21 29
35 35
Electron- Atom Scattering Theory and Calculations P. G. Burke '
39
I. Introduction 11. Scattering at Low Energies 111. Scattering at Intermediate and High Energies
40
IV. Illustrative Results V. Conclusions Acknowledgments References
49 52 53 54
vii
44
viii
Contents
Terrestrial and Extraterrestrial H i Alexander Dalgarno I. Introduction 11. Terrestrial H,' 111. Extraterrestrial H 3 + Acknowledgments References
57 58 60 66 66
Indirect Ionization of Positive Atomic Ions K. Dolder I. Introduction 11. Ionization Processes 111. Experimental Approaches IV. Some Experimental Results References
69 71 72 74 91
Quantum Defect Theory and Analysis of High-Precision Helium Term Energies G. W. F. Drake I. Introduction 11. Quantum Defect Theory and l/n Expansions 111. Quantum Defect Analysis of High-Precision Variational Calculations IV. Comparison with High-Precision Measurements V. Summary and Discussion Acknowledgments References
93 95 103 110 113 115 115
Electron-Ion and Ion-Ion Recombination Processes M. R. Flannery I. Introduction Historical Interlude Capture-Stabilized Theory as a Basis for Future Discussion Electron-Ion Recombination Processes Ion-Ion Recombination Processes VI. Conclusions Acknowledgments References
11. 111. IV. V.
117 118 120 124 135 144 145 145
CONTENTS
ix
Studies of State-Selective Electron Capture in Atomic Hydrogen by Translational Energy Spectroscopy H. B. Gilbody
I. Introduction 11. Experimental Approach 111. Results IV. Conclusions Acknowledgments References
149 151 154 166 167 167
Relativistic Electronic Structure of Atoms and Molecules I. P. Grant I. Introduction 11. The Beginnings of Relativistic Electronic Structure Theory 111. Open-Shell Atoms IV. Basis Sets and QED of Atoms and Molecules V. Outlook Acknowledgments References
169 170 175 179 183 184 184
The Chemistry of Stellar Environments D. A . Howe, J. M. C. Rawlings, and D. A. Williams I. Introduction 11. Winds from Young Stellar Objects 111. Circumstellar Envelopes of Asymptotic Giant Branch Stars IV. Planetary Nebulae and Preplanetary Nebulae V. Novae VI. Chemistry in Supernovae Ejecta: Molecules in SN1987A V 11. Conclusions References
187 188 191 195 197 199 203 204
Positron and Positronium Scattering at Low Energies J. W. Humberston I. Introduction 11. Positron Scattering by Atoms 111. Positronium Scattering by Atoms and Charged Particles
IV. Positron Scattering by Molecular Hydrogen V. Concluding Remarks
205 207 213 217 220
Contents
X
Acknowledgments References
220 22 1
How Perfect Are Complete Atomic Collision Experiments? H. Kleinpoppen and H. Hamdy I. Introduction 11. Electron- Atom Collisions 111. Approaches to “Complete” Experiments in Heavy-Particle Atom Collisions and Photoionisation of Atoms IV. Concluding Remarks References
223 225 244 248 248
Adiabatic Expansions and Nonadiabatic Effects R. McCarroll and D. S. F. Crothers I. Introduction 11. Quantum Mechanical Approach 111. Semiclassical Formalism IV. Some Experimental Evidence V. Relevant Work of D. R. Bates (1962-1983) VI. Understanding Nonadiabatic Transitions and Effects (1971-1992) VII. Conclusions Acknowledgments References
253 255 264 269 27 1 272 274 275 275
Electron Capture to the Continuum B. L. Moiseiwitsch
I. Introduction Relativistic Theory of Electron Capture to the Continuum Cusp Asymmetry for Electron Yield Nonrelativistic Formulas for Electron Capture to the Continuum Comparisons with Experimental Data and Other Theories References
11. 111. IV. V.
279 280 284 285 286 292
How Opaque Is a Star? M. J. Seaton I. What Is Opacity? 11. What Is White? 111. Ultraviolet Radiation from Hot Stars IV. Opacities for Stellar Interiors References
296 297 298 301 305
CONTENTS
xi
Studies of Electron Attachment at Thermal Energies Using the Flowing Afterglow-Langmuir Probe Technique David Smith and Patrik Spanel
I. 11. 111. IV. V. VI.
Introduction Basic Attachment Processes at Low Energies Theoretical Description of Electron Attachment Some Experimental Techniques Used to Study Electron Attachment The Flowing Afterglow-Langmuir Probe Technique Results from the FALP Experiments; Comparisons with Results from Other Experiments VII. Recent Developments Using the FALP Apparatus VIII. Concluding Remarks Acknowledgments References
308 309 312 313 316 319 335 340 34 1 342
Exact and Approximate Rate Equations in Atom-Field Interactions S. Swain I. Introduction 11. Basic Features of the REA; the Two-Level Atom
111. IV. V. VI.
Generalization to N Atomic Levels Extensions of the REA Rate Equations in the Dressed-Atom Picture Summary Acknowledgments VII. Appendix: Derivation of the Exact Rate Equations References
345 348 356 362 368 37 1 372 373 376
Atoms in Cavities and Traps H. Walther I. Introduction 11. Review of the One-Atom Maser 111. Dynamics of a Single Atom IV. A Source of Nonclassical Light V. A New Probe of Complementarity-The One-Atom Maser and Atomic Interferometry VI. Experiments with Ion Traps VII. Order versus Chaos: Crystal versus Cloud VIII. The Ion Storage Ring IX. Ordered Structures in the Storage Ring and Comparison with the Theory References
379 380 383 385 389 39 1 393 397 399 404
xii
Contents
Some Recent Advances in Electron-Impact Excitation of n = 3 States of Atomic Hydrogen and Helium J. F. Williams and J. B. Wang I. Introduction 11. The Basic Theory
111. Coincidence Measurements IV. Radiation Trapping Acknowledgments References INDEX CONTENTS OF VOLUMES IN THIS SERIAL
407 41 1 415 418 424 425 427 437
Contr ibutors
Numbers in parentheses indicate the pages on which the authors’ contributions begin.
K. L. Bell (1), Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 lNN, Northern Ireland B. H. Bransden (19), Department of Physics, University of Durham, South Road, Durham DH1 3LE, England P. G. Burke (39), Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 lNN, Northern Ireland
D. S. F. Crothers (253), Theoretical and Computational Physics Research Division, The Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 lNN, Northern Ireland Alexander Dalgarno (57), Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138, USA
K. Dolder (69), Department of Physics, University of Newcastle Upon Tyne, Newcastle Upon Tyne NEl 7RU, England G. W. F. Drake (93), Department of Physics, University of Windsor, Windsor N9B 3P4, Ontario, Canada
M. R. Flannery (117), School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
H. G. Gilbody (149), Department of Pure and Applied Physics, The Queen’s University of Belfast, Belfast BT7 LNN, Northern Ireland 1. P. Grant (169), Mathematical Institute, University of Oxford, 24-29 St.
Giles, Oxford OX1 3LB, England
H. Hamdy (223), Physics Department, Beni-Suef Faculty of Science, BeniSuef, Egypt
D. A. Howe (187), Mathematics Department, UMIST, P.O. Box 88, ManChester M60 lQD, England
...
Xlll
xiv
Contributors
J. W. Humberston (205), Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, England
A. E. Kingston (l), Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 lNN, Northern Ireland
H. Kleinpoppen (223), Unit of Atomic and Molecular Physics, University of Stirling, Stirling FK9 4LA, Scotland R. McCarroll (253), Laboratoire de Dynamique Moleculaire et Atomique, Universite Pierre et Marie Curie, 75252 Paris, Cedex 05, France
B. L. Moiseiwitsch (279), Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 lNN, Northern Ireland C. J. Noble (19), Daresbury Laboratory, Science and Engineering Research Council, Warrington, England J. M. C. Rawlings (187), Mathematics Department, UMIST, P.O. Box 88, Manchester M60 lQD, England
M. J. Seaton (295), Department of Physics and Astronomy, University College London, Gower Street, London WClE 6BT, England David Smith (307), Institut fur Ionenphysik, Universitat Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria Patrik Spanel (307), Institut fur Ionenphysik, Universitat Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria S. Swain (349, Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 lNN, Northern Ireland
H. Walther (379), Sektion Physik der Universitat Munchen and MaxPlanck-Institut fur Quantenoptik, 8046 Garching, Germany
J. B. Wang (407), Centre for Atomic, Molecular and Surface Physics, Department of Physics, The University of Western Australia, Perth 6009, Australia D. A. Williams (187), Department of Physics, Astrophysics, Nuclear Physics Laboratory, Keble Road, Oxford OX1 3RH, England J. F. Williams (407), Centre for Atomic, Molecular and Surface Physics, Department of Physics, The University of Western Australia, Perth 6009, Australia
Preface
David Bates
David Robert Bates, later Sir David Bates, was born in County Tyrone in Northern Ireland. He attended school at the Royal Belfast Academical Institution. He entered The Queen’s University of Belfast in 1934, graduating with the degree of BSc. in 1937, and the degree of M.Sc. in 1938. Harrie Massey (later Sir Harrie Massey) was the Independent Lecturer in Mathematical Physics, a position in the University that evolved into the Chair of Applied Mathematics that David Bates was later to occupy. Sir Harrie Massey stimulated David Bates’s interest in a career of research and teaching, and when Harrie Massey left Belfast to take the position of Goldsmith Professor of Mathematics at University College London in 1939, David Bates accompanied him to become a research student in the Department of Mathematics. The war intervened, and David spent the war years from 1940 to 1945 as a research scientist at the Admiralty Research Laboratory and at the Mine Design Department. Despite the exigencies of the times, he found space in a demanding schedule for research and he published several papers in theoretical atomic and molecular physics and in atmospheric physics, the two subjects in which he was to emerge as a dominant figure. David returned to The Queen’s University of Belfast in 1951 as Professor and Head of the Department of Applied Mathematics and built an internationally renowned school in Theoretical Atomic and Molecular Physics. Its many graduate students and postdoctoral fellows have had a profound impact on the subject as they continued on to creative careers in many different countries. David made monumental original contributions across the broad range of atomic and molecular physics and its applications to atmospheric science and astrophysics. He carried out innovative studies of the mechanisms by which recombination occurs in ionized gases. He developed the theories of dissociative recombination, dielectronic recombination, and collisional-radiative recombination, and carried out calculations of their efficiencies for a diverse array of atomic systems in a wide range of physical conditions. xv
xvi
Preface
He introduced new methods for the calculation of the processes of photoionization, photodetachment, and radiative association, and he was the leader in the development of a theoretical understanding of heavyparticle collisions at low, intermediate, and high energies. As in atomic and molecular physics, his contributions to atmospheric science were profound and influential. With his deep understanding and broad interests and his eminence in the field, he was a superb choice as editor of Advances in Atomic, Molecular, and Optical Physics. The atomic and molecular physics community is greatly in his debt. Alex Dalgarno
A Further Appreciation
I would like to add an additional note of appreciation. I have been junior co-editor, along with Sir David, since 1974, which witnessed the appearance of Volume 10 of this series. Prior to that time Sir David’s co-editor was Immanuel Estermann, also an atomic beam physicist. Sir David was always the guiding light of Advances, from its first volume, which appeared in 1965. He kept a very sharp eye on the quality and timeliness of the series, and is certainly the individual most responsible for it having achieved the reputation it now commands. Of course this in itself is a tribute to his stature as a physicist. His interests spanned the full range of the editorial process, from identifying important and timely subjects worthy of inclusion in Advances, through the identification of appropriate authors for such subjects, and finally through the shepherding of articles through production, which was and continues to be rigorously monitored by Academic Press. All in all he participated in the conception and production of 31 volumes of Advances (not including this one!), containing altogether about 240 articles. Singling out any of these for special mention is a task I personally would not relish to undertake, but among the highlights in this list of 240 there would certainly appear the articles “Electronic eigenenergies of the hydrogen molecular ion” (Vol. 4), “Aspects of recombination” (Vol. 1 9 , “Ion-Ion recombination in an ambient gas” (Vol. 20), and “Negative ions: structure and spectra,” (Vol. 27), all by David Bates. The end is not yet in sight, since we anticipate a contribution on dissociative ionization in our next volume. During the approximately twenty years of our association, I have had many occasions to witness Sir David’s vast grasp of atomic physics in its many aspects and of its applications to atmospheric physics and astrophysics. The fact that the tables of contents of the Advances series virtually tracks the history of these fields over the past thirty years is no accident-it reflects Sir David’s own professional history, which is so intertwined with the general history of these fields as to be virtually inseparable. Reflecting the developments in optical physics that have so revolutionized the entire field of atomic physics, the name of our series was changed to xvii
xviii
A Further Appreciation
include optical physics in 1990. At the same time a distinguished Editorial Board, whose members are listed on the inside front leaf of this volume, was added. Among other duties, our board members are themselves responsible for the generation of special volumes reflecting their own interests and expertise. A new co-editor is now coming on board, Professor Herbert Walther, of the Max Planck Institute for Quantum Optics, Munich, and the University of Munich. At the same time as we salute Sir David and recognize the unique role he has played, and continues to play, in atomic physics, I welcome my new co-editor who will, we can all be certain, work hard to maintain the standard of excellence set thirty years ago by David Bates. Benjamin Bederson
ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS. VOL. 32
PHOTOIONISATION OF ATOMIC OXYGEN AND ATOMIC NITROGEN K . L. BELL and A . E. KINGSTON Department of Applied Mathematics and Theoretical Physics Queen's University of Berfast Bevast, Northern Ireland
I . Introductory Remarks . .
,
. . . . . . . . . . . . . . . .
11. Photoionisation of Atomic Oxygen
. . . . . . . . . . . . . .
A. Ground State . . . . . . . B. Excited States . . . . . . . 111. Photoionisation of Atomic Nitrogen A. Ground State . . . . . . . 8. Excited States . . . . . . . 1V. Conclusion . . . . . . . . . References . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2 2 7 10 10 15 16 17
I. Introductory Remarks Photoionisation of atomic oxygen by extreme ultraviolet radiation from the sun is the primary process for the production of energetic electrons and ions in the daytime thermosphere, and photoelectrons from oxygen provide the major source of electron heating in the daytime mid-latitude ionosphere. Atomic oxygen and nitrogen also play important roles in ionisation balance in HI1 regions, stellar atmospheres and planetary nebulae. For over 50 years therefore knowledge of accurate cross sections for photoionisation of these elements has been sought. In this chapter we review the progress made for total cross sections. (Attention is drawn to an excellent article on ground state oxygen by Seaton, 1987.) Before commencing, however, it is worth remarking and demonstrating the outstanding achievements of the earliest calculations. The first calculation of the cross section for photoionisation of atomic oxygen was made by Bates et al. (1939) using a Hartree self-consistent field approximation without exchange. Bates (1939) then calculated cross sections for all elements from boron to neon using the free-electron wave functions calculated in the field of 0'. These results were improved upon by Bates and Seaton (1949), who employed the Hartree-Fock approximation to evaluate the threshold cross section; the cross section as a function of energy was obtaified by normalising 1 Copyright
0 1994 by
Academic Press Inc
All rights of reproduction in any form reserved
ISBN 0-12-003832-3
K . L . Bell and A . E . Kingston
2
the results from the general formula of Bates (1946) based upon an approximation which used Slater-type orbitals for bound electrons and Coulomb functions for ejected electrons. Figures 1 and 2 compare these results for photoionisation of the ground state of oxygen and nitrogen with the most recent theoretical calculation and the latest experimental data. Whereas the older calculations were incapable of accounting for resonances, the agreement with the most recent results, in both shape and in magnitude, is truly remarkable.
II. Photoionisation of Atomic Oxygen A. GROUND STATE The early calculations considered only ejection of the 2p electron from the 2s22p43P ground state of oxygen. Dalgarno and Parkinson (1960) included contributions from the process in which a 2s electron is ejected. In 1964, Dalgarno et al. employed the Hartree-Fock formulation to calculate cross
15
h
P
E
v
v, m 0
5
0 900
800
700
600
500
400
300
200
PHOTON WAVELENGTH (A)
FIG.1. Photoionisation cross section of oxygen in the ground state. Theory: -Bell et al. (1989);- - - Bates and Seaton (1949);-. -,-, Dalgarno et al. (1964) (length and velocity values). Experiment: 0 Angel and Samson (1988).
PHOTOIONISATION OF ATOMIC OXYGEN AND ATOMIC NITROGEN
3
FIG.2. Photoionisation cross section of nitrogen in the ground state. Theory: Bell and Berrington (1991);- - - Bates and Seaton (1949). Experiment: 0 Samson and Angel (1990). ~
sections for the production of 0' in the states 2 ~ ' 2 p ~ ~ S O *Do, , 'Po and PP states, and the length and velocity formulations of the dipole 2 ~ 2 p and ~ ~' matrix element were used; as is seen in Figure 1, the length results are in good accord with the more sophisticated theory and experiment at all wavelengths and in excellent agreement for wavelengths shorter than about 500 A. Indeed these results remain the best of the calculations which d o not include resonances; there were many such calculations in the late 1960s and early 1970s, falling into two main groups: those which employed a Hartree-Fock approximation and those which utilised a central potential. The central potential calculations of McGuire (1968) and Starace et al. (1974) (HermanSkillman), T4omas and Helliwell (1970) (Klein-Brueckner), Kahler (1971) and Koppel (1971) (scaled Thomas-Fermi) are compared in Fig. 3 with the experimental data of Angel and Samson (1988). The data of Ganas (1973) (independent particle model) is not shown, but none of this work substantially improved upon that of Dalgarno et al. (1964). A similar statement holds true for the Hartree-Fock investigations of Henry (1967) and Starace et al. (1974) and the random phase approximation with exchange of Vesnicheva et al. (1986) shown in Fig. 4. All of these calculations were based upon a final state representation of '€' = A$8, where A is an antisymmetrisation operator, $ is the wave function for the ion and 8 is the ejected electron wave function.
15
F: u
10
w
v)
M 0
5
5
L
-1
O
FIG. 3. Photoionisation cross section of oxygen in the ground state. Central field approximation calculations: McGuire (1968); -Thomas and HelliweIl(l970); -.-Koppel(1971); - Kahler (1971) (Kahler data coincides with that of Koppel for I < 650A); --- Starace et al. (1974). Experiment: OAngel and Samson (1988).
15
EW
10
v)
In v)
0
!3
5
D 0
0 I n I l l l l ' l l l l ' l ' l l ' l l l l l l l l l l l l ' l l 1 900 BOO 700 600 500 400 300 PHOTON WAVELENGTH (A)
-
'
1
1
200
FIG.4. Photoionisation cross section of oxygen in the ground state. Theory: - - (length), -(velocity),Henry(1967)(HF); ... (length),-.-(velocity),Staraceet al.(1974)(HF);-.- (length),-- (velocity), Vesnicheva et al. (1986) (RPAE). Experiment: 0 Angel and Samson (1988).
4
PHOTOIONISATION OF ATOMIC OXYGEN AND ATOMIC NITROGEN
5
Significant improvement of the theory arose on abandonment of the idea that the final state could be described in terms of an ejected electron moving in the field of just one state of a product ion. The final state representation now takes the form Y = A Ci t,biOi, where the sum is over product ion states i. Not only does this permit greater accuracy but coupling between open and closed channels gives rise to autoionisation resonances. The first calculation in this close-coupling theory including autoionisation was carried out by Henry (1968a). His length formulation results are compared in Figure 5 with the experiment and the more sophisticated R-matrix close-coupling results of Taylor and Burke (1976), Pradhan (1978),and Bell et al. (1989),who included seven, seven, and eleven product ion states in the wave function expansion, respectively. Henry's results are some 30% lower than these later values but his resonance positions are in good accord, and he is in best agreement with the most recent experimental data of Angel and Samson (1988)-a point to which we shall return later. As discussed by Bell et al. (1989),good agreement exists between theory and experiment in the wavelength region 840-660 A, which includes the resonances converging on the 'Do and 2P0 thresholds. Even more remarkable is the agreement between theory and experiment for the resonance series converging on the 4P threshold. Figure 6 illustrates the structure found by Angel and Samson (1988),which had not been seen before
15
h
P
E
v
v) v)
0
5
0
900
800
700 600 500 PHOTON WAVELENGTH (A)
400
300
200
FIG.5. Photoionisation cross section of oxygen in the ground state. Theory: ... Henry (1968a); A Taylor and Burke (1976); Pradhan (1978);- Bell et a/. (1989). Experiment: x Kohl et a/. (1978);1 Samson and Pareek (1985); A Hussein et al. (1985); 0 Angel and Samson (1988).
6
K . L . Bell and A . E . Kingston
I
FIG.6. Photoionisation cross section of oxygen in the ground state; autoionising resonances converging to the 4P threshold. Theory:- Bell er al. (1989) Experiment: - - - Angel and Samson (1988).
in photoionisation experiments, and compares it with data from Bell et al. (1989) (for which the energy scale has been shifted in order to make the 4P threshold agree with the experimental value). Table I confirms numerically the exceedingly good agreement between theory and experiment for the positions of these autoionising 2 ~ 2 p ~ (np ~ P(3D0, ) 3S0, 3P0)resonances. We return now to the total cross section and the comparison between theory and experiment presented in Fig. 5. Increased sophistication of the target and free-electron-plus-ion wavefunctions should clearly lead to more accurate theoretical values and as the sophistication increases some convergence of the theoretical results should appear. This is indeed the casecomparison of the values obtained by Taylor and Burke (1976), Pradhan (1978) and Bell et al. (1989) not only show convergence but suggest that theoretically the total cross section is now known to better than a few percent at all wavelengths considered. However, comparison with experiment reveals two major discrepancies neither of which has yet been resolved. Above the 4S0 threshold, the latest experimental data (Angel and Samson, 1988) and the earlier values of Comes et al. (1968)(not shown in Fig. 5 but that agree closely with the results of Henry (1968a)) lie considerably lower than the R-matrix results (by almost 50% at threshold) and the shape of the cross section with
PHOTOIONISATION OF ATOMIC OXYGEN AND ATOMIC NITROGEN
7
TABLE I AUTOIONISING2~2p~(~P) np (3D0, 'So, 'P.) RFSONANCESERrEs BELOW THE 4P THRESHOLDIN ATOMICOXYGEN Wavelength (A) Resonance State
3p3D" 3p3S" 3p 3P0 4p 3w 4p 3s" 4p 3p. 5p 3Do 5p 3s0 5p 3p. 6p 'W lp 3Do
(1)
(2)
(3)
(4)
480.8 419.0 418.1 455.4
480.7 418.8 477.1 455.1 454.6 454.5 447.0 446.1 446.5 442.9 440.7
480.6 418.6 476.1 455.6 455.3 454.7
419.3 411.6 415.6 455.2 454.8 454.0 446.8 446.6 446.3
446.7
442.1 440.5
Columns:(1) Angel and Samson (1988);(2) Bell et al. (1989);(3) Pradhan (1918);(4)Taylor and Burke (1916).
wavelength is also at variance with theory. The second region of discrepancy lies above the 'Po threshold; Angel and Samson (1988) find a peak in the cross section and Hussein et al. (1985) also find values much larger than theory (with the exception of their very small value at 640A). Bell et al. (1988) suggested that the latter discrepancy may be due to difficulty in eliminating molecular oxygen contamination in the beam or in the work of Angel and Samson (1988) excited state oxygen contamination. In conclusion, theory suggests that the cross section for photoionisation of ground state oxygen is known to better than 5%. Further experimental work is desirable for several reasons: (1) to resolve the 4So threshold discrepancy; (2) to resolve the preceding 'Po threshold discrepancy; (3) to provide experimental results for the widths of resonances.
B. EXCITEDSTATES Little theoretical and experimental data exists for photoionisation of excited states of oxygen in comparison with the ground state case. No experimental data seem to exist for photoionisation of the excited metastable 2s22p4 'D and 'S states, and until recently the theoretical work was relatively unsophisticated. Henry (1967) employed a close-coupling approximation but
K . L . Bell and A. E . Kingston
8
included only open channels in his wavefunction expansion. Thomas and Helliwell (1970) used a central potential model based upon the KleinBrueckner potential and Koppel(l971) adopted the scaled Thomas-Fermi potential method. Wide divergence exists for both 'D and 'S states both with regard to magnitude and shape of cross section with variation in photon energy, and significantly none of these calculations permitted autoionisation. Inclusion of coupling between closed and open channels by Bell et al. (1989) in an 11-state R-matrix calculation revealed that, for both states, the cross section is dominated by a M p S 'Po Coster-Kronig resonance (see Fig. 7). Further work on these states is desirable since it is not possible from the present knowledge to give an accuracy to the currently most accurate data (that of Bell et al., 1989). Photoionisation of the higher excited states 2p33s 3*5S0, 2p33p 3.5Phave been treated using an 11-state R-matrix method by Bell et al. (1990). For both the 3s ' S o and 3s 5S0 it was found that the cross section remained relatively small until the 3s 'P and 3s 4P thresholds, when the cross sections suddenly increased, passed through a maximum and then decreased as the photon energy increased. Analysis reveals that the large cross section results from a transition in which the 2p electron is ejected into the continuum with the 3s electron behaving like a "spectator" electron.
20 h
F: u
w v)
1
l5I
FIG.7. Photoionisation cross section of oxygen in the 2s22p4'D and 'S states. Theory:Bell et ~ l(1989)-, . ID; ---,' S .
PHOTOIONISATION OF ATOMIC OXYGEN AND ATOMIC NITROGEN
9
The 3p 3*5Pstates both reveal a minimum, and we shall concentrate on the 3p3P state for which the minimum was first found by Saxon et al. (1989). Their calculation was based on a variational R-matrix approach for electron-molecule scattering and was restricted in that configurations describing e - + 0 (2P,2D) were not included in their configurationinteraction expansion so that autoionising resonances were automatically excluded. Figure 8 compares the results of Saxon et al. (1989) and Bell et al. (1990). Also shown is the value obtained by Dixit et al. (1988) as part of a study of the two-photon-resonant-three-photon ionisation of atomic oxygen using quantum defect theory to calculate the relevant atomic parameter and an experimental data point (Bamford et al., 1986). For energy values greater than about 0.12 Ry above the threshold, the Saxon et al. data falls below that of Bell et al. but the more serious discrepancy exists as one approaches the threshold where the Saxon et al. value is about 60% greater than that of Bell et al. This discrepancy may be investigated by examining the individual 3S0and 3Dopartial cross sections that contribute to the total at these low energies and making use of a continuity relationship between bound-bound and bound-free absorption. Seaton (1978) and Dubau and Seaton (1984) have +
3r T
2.5
14 FIG.8. Photoionisation cross section of oxygen in the 3p 3P state: - Bell et al. (1990); --Saxon et al. (1989); 0 Dixit et al. (1988); Bamford et al. (1986).
a
K . L . Bell and A . E . Kingston
10
shown that the differential oscillator strength per Rydberg, dflde, derived by considering bound-bound absorption and given by
joins smoothly to the differential oscillator strength derived from the photoionisation cross section and given by 1 df _ --
ds
4n2aag
t7
wheref is the oscillator strength for the bound-bound transition from state b to state a, n and p are the principal quantum number and quantum defect, respectively, for state a, t7 is the cross section and a is the fine structure ) 3 and p we constant. In the present case, state b corresponds to 2 ~ ~ ( ~ S "3P, consider as final states a 2 ~ ~ ( ~ S "3 )S n0 and s 2 ~ ~ ( ~ S "3Do. ) n dThe term dflds evaluated from the preceding two expressions is presented in Fig. 9; very accurate bound-bound oscillator strengths from Bell and Hibbert (1990) and Butler and Zeippen (1991) have been employed in the calculation of df/de below the threshold and the photoionisation cross sections of Bell et al. (1990) and Saxon et al. (1989) above the threshold. The minimum in the 3Dofinal symmetry is apparent, and for this symmetry the largest difference between the results of Bell et al. (1990) and Saxon et al. (1989) occurs. The "below threshold df/ds clearly indicates better continuity with the work of Bell et al. (1990) and would suggest that these results are more accurate than those of Saxon et al. (1989).
III. Photoionisation of Atomic Nitrogen A. GROUNDSTATE
As one might expect the history of theoretical calculations of the photoionisation cross section for ground state nitrogen closely follows that for oxygen. However, after the early work of Bates and Seaton (1949) (see Fig. 2) it took almost 30 years before a calculation was performed that included autoionisation. Dalgarno and Parkinson (1960) employed a Hartree bound state and hydrogenic continuum functions to calculate the cross section, modified the treatment of Bates (1946) to include the dipole velocity formulation and also took into account absorption due to inner shell electrons. Henry (1966, 1968b) used the Hartree-Fock approximation and
I
I
I
I
I
I
I
1
I
I
I
-T
I - -
-
\ \
\ \
.I
\ \
\ 1 -
0
1
1
1
1
1
1
1
I
.2
.1 €
I
\ '\
1
.3 -.05
0
.05
.I5
.1
.2
.25
.3
€
FIG.9. Differential oscillator strength for the transitions in oxygen ~ P ~ P - (lefthand ~ S " figure)and 3p 3P-3D"(righthand figure). The energy above the threshold, E , is in Rydbergs: - Bell et al. (1990);- - - Saxon et al. (1989); 0, derived from oscillator strengths taken from Bell and Hibbert (1990) and Butler and Zeippen (1991),respectively.
K . L . Bell and A. E . Kingston
12
included a single-state close-coupling representation of the final continuum and so did not take into account resonances. A number of model potential calculations then followed: McGuire (1968) (Herman-Skillman), Thomas and Helliwell (1970) (Klein-Brueckner), Kahler (1971) and Koppel (1971) (scaled Thomas-Fermi), and Ganas (1973) (independent particle). These calculations are presented in Figs. 10 and 11, where they are compared with the currently most sophisticated results (Bell and Berrington, 1991-note that the resonance structure has been averaged out). Kahler's data is not given since it almost coincides with that of Koppel (1971). Considerable disagreement-both quantitative and qualitative-is apparent with only the data of Koppel (1971) in accord with the most recent work (Bell and Berrington, 1991). Figure 11 also contains the results of Cherepkov et al. (1974), who employed a random phase approximation with exchange but neglected the strong final-state coupling between the 2p and 2s photoejections and failed to predict the 2s2p3 'So np4P resonances. These resonances were first considered theoretically by Hempe (1978) using a manychannel quantum defect method and later by Le Dourneuf et al. (1979) and Bell and Berrington (1991), who both employed the R-matrix method. Figure 12 presents data from these three calculations and Table I1 gives a 20
I
l
1
l
1
1
I
1
l
l
1
1
1
1
I
1
I
I
I
1
I
l
I
~
l
l
I
-
--__ - -.-.
15 -
-
l
-
\
\
\
-
\
z
Ev
10
t;] v)
v, 0 p:
V
5
0
\
\ -
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 I
I
FIG.10. Photoionisation cross section of nitrogen in the ground state. Central-field approximations:x McGuire (1968);... Thomas and Helliwell(l970); -- Koppel(1971);--Ganas (1973). R-matrix: - Bell and Berrington (1991).
-
PHOTOIONISATION OF ATOMIC OXYGEN AND ATOMIC NITROGEN
13
FIG. 11. Photoionisation cross section of nitrogen in the ground state: --(length, -.-(velocity) Henry (1966); - - (length), (velocity) Henry (1968b);.. Cherepkov et al. (1974);- Bell and Berrington (1991).
TABLE I1 THEN**(2s2p3' S o np 4P) RESONANCE PROFILES n=3
E,(Ryd) r(10-3Ryd)
r2
1.310 1.51
1.323 2.03
1.322 1.65
-1.70 ( - 1.95)#
- 2.03
OS9 (0.60)
0.52
n=4
1.316 2.02f0.22 - 1.7 kO.1 0.51k0.04
1.405 0.67
1.410 0.70 -1.85 (- 2.09) (0.62)
1.399 1.408 0.79 0.73f0.22 - 1.91 - 1.7+0.1 0.56
0.58+0.04
'(1) Hempe (1978);(2) Le Dourneuf et al. (1979);(3) Bell and Berrington (1991); (4) Dehmer et al. ( 1974).
#Values obtained for q and r2 using the velocity formulation are given in parentheses.
K . L . Bell and A . E . Kingston
14 25
20
5
0
1
1
1
1
800
1
'
1
1
1
700
'
1
1
1
'
1
1
1
1
1
1
)
1
1
1
1
600 500 400 PHOTON WAVELENGTH (A)
1
'
1
'
1
300
1
1
1
200
FIG.12. Photoionisation cross section of nitrogen in the ground state. Theory: -- Hempe (1978); Le Dourneuf et at. (1979);-Bell and Berrington (1991). Experiment F( Ehler and Weissler (1955); W Comes and Elzer (1967, 1968); 0 Samson and Angel (1990).
comparison of the 2s2p3 %"np4P resonance profiles. In Table 11, the resonance position En,width r, line profile index q and correlation coefficient p2 (see Burke and Taylor, 1975, for a definition of these parameters) are compared with the experimental results of Dehmer et al. (1974), who studied the series using a continuum light source. The energy positions used by Dehmer et al. (1974) had been determined earlier by Carroll et al. (1966), using conventional photographic absorption techniques. The agreement between theory and experiment is satisfactory,noting that Dehmer et al. set q to the value of - 1.7 to parameterize their n = 4 data. The total cross section (Fig. 12) obtained by Hempe (1978) differs in both shape and magnitude from the results of Le Dourneuf et al. (1979) and Bell and Berrington (1991). The good agreement between the two R-matrix calculations-five state of Le Dourneuf et al. and eight state of Bell and Berrington-suggests that this method has "converged" and that theoretically the cross section is known to an accuracy of a few percent. Such a statement is also supported by the recent experimental data of Samson and Angel (1990) for wavelengths shorter than 520 A. At longer wavelengths discrepancy between theory and experiment is apparent. However, Samson and Angel (1990) normalised their data in the wavelength region 520-852 A
15
PHOTOIONISATION OF ATOMIC OXYGEN AND ATOMIC NITROGEN
by employing the Thomas-Reiche-Kuhn sum rule for the total oscillator strength. Bell and Berrington (1991) have pointed out that more accurate discrete oscillator strengths give a contribution to this sum of 0.58 in comparison with the value of 0.96 used by Samson and Angel. Therefore the experimental data should be renormalised upward and so remove the discrepancy between the R-matrix results and experiment. B. EXCITEDSTATES In contrast to oxygen, sophisticated theoretical data was available for the metastable states of nitrogen in 1980. The early central potential calculations of Thomas and Helliwell (1970) and Koppel (1971) were superseded by the five-state R-matrix work of Zeippen et al. (1980). Figure 13 shows that the data of Koppel agree closely with that of Zeippen et al. and the more sophisticated 13-state R-matrix calculation of Bell et al. (1992) but of course is incapable of obtaining the 2s2p4 2P resonance that dominates the cross section for both the 2s22p32Do and 2 p 0 states. The close accord between the two R-matrix calculations suggests that for both states the cross section is known to an accuracy of a few percent. Bell et al. (1992) have also obtained data for the 2s22p33s4*2P,2s2p4 4P and 2s22p23p2S0 excited states. Again the effect of increase in the cross section due to photoejection of the 2p electron with the 3s electron behaving as a
25
l r
20
15
15
10 0
'9
l : I i ,
0 1000
800
600 A (A)
400
1200
1000
800
600
400
A (A)
and 2P" states: -FIG. 13. Photoionisation cross section of nitrogen in the Thomas and Helliwell (1970); ... Koppel (1971); 0 Zeippen et al. (1980) Bell et al. (1992). ~
16
K.L . Bell and A . E . Kingston
b
-
-1
0
4000
3500
3000 2500 A (A)
2000
1500
4000
3500
3000
2500
2000
1500
A (A)
FIG. 14. Photoionisation cross section and photoelectron angular distribution asymmetry parameter for the 3 ~ ’ s ”excited state of nitrogen. Cross section: ... Bell et al. (1992); -.Theodosiou (1988); - - - Manson (1988). Asymmetry parameter: ... Bell et al. (1992);- - - (length), _- (velocity), Nahar and Manson (1989); -.- (HS ground state), -.- (HS excited state) Theodosiou (1988). Experiment: Pratt et al. (1987).
“spectator” electron is found. Of these states, comparison with other data may be made for the 3p ’So state, and this state is also of interest because it is the only excited state of either oxygen or nitrogen for which an experimental investigation has been performed for the asymmetry parameter, p. Figure 14 compares the low-energy total cross-section data of Bell et al. with the central-field Hartree-Slater calculations of Manson (1988) (using a groundstate potential) and Theodosiou (1988) (using the excited-state potential). Clearly the central-field results are sensitive to the potential used, but all three calculations confirm the existence of a minimum. This sensitivity is more marked in the case of the b-parameter, where, as is seen in Fig. 14, wide divergenceof results exists. The single experimental point of Pratt et al. (1987) is unfortunately insufficient to discriminate between the Hartree-Slater, Rmatrix and Hartree-Fock results (Nahar and Manson, 1989).
IV. Conclusion In this short chapter we have concentrated mainly bn total cross sections. It is recognised that other theoretical data is available. The opacity project has produced total cross sections for both ground and excited states, and data are
PHOTOIONISATION OF ATOMIC OXYGEN AND ATOMIC NITROGEN
17
available for partial cross sections as well as angular distributions of photoelectrons. It is our opinion that, for both elements, the theoretical data are of sufficient accuracy for most applications. Further experimental work is clearly required particularly with a view to resolving the difficulties discussed for ground-state oxygen and for the 3p2S0 state of nitrogen. In the conclusion to his paper of 1946, David Bates made several pertinent comments-we abstract but two: “The main future development of the theory probably lies in attempting to achieve increased accuracy” and “The great computational labour involved is likely severely to limit any programme.” This review chapter clearly reflects the first statement, and without large computers (and a vast amount of computer time) the second statement would also have remained true.
REFERENCES Angel, G. C., and Samson, J. A. R. (1988). Phys. Rev. A 38, 5578. Bamford, D. J., Jusinski, L. E., and Bischel, W. K. (1986). Phys. Rev. A 34, 185. Bates, D. R. (1939). Man. Not. R . Astron. SOC. 100, 25. Bates, D. R. (1946). Mon. Not. R . Astron. SOC.106, 423. Bates, D. R., Buckingham, R. A., Massey, H. S. W., and Unwin, J. J. (1939). Proc. Roy. SOC.A170, 322. Bates, D. R., and Seaton, M. J. (1949). Mon. Not. R. Astron. SOC. 106, 698. Bell, K. L., and Berrington, K. A. (1991). J. Phys. B: At. Mol. Opt. Phys. 24, 933. Bell, K. L., Berrington, K. A., Burke, P. G., Hibbert, A,, and Kingston, A. E. (1990).J. Phys. B: At. Mol. Opt. Phys. 23, 2259. Bell, K. L., Berrington, K. A., and Ramsbottom, C. A. (1992). J. Phys. B: At. Mol. Opt. Phys. 25, 1209. Bell, K. L.,Burke, P. G. Hibbert, A., and Kingston, A. E. (1989).J. Phys. B: At. MoZ. Opt. Phys. 22, 3197. Bell, K. L., and Hibbert, A. (1990).J. Phys. B: At. Mol. Opt. Phys. 23, 2673. Burke, P. G., and Taylor, K. T. (1975). J. Phys. B: At. Mol. Phys. 8, 2620. Butler, K., and Zeippen, C. J. (1991). J. de Physique lv Coll. CI 1, 141. Carroll, P. K., Huffman, R. E., Larrabee, J. C., and Tanaka, Y. (1966). Astrophys. J . 146, 553. Cherepkov, N. A., Chernysheva, L. V., Radojevic, V., and Pavlin, I. (1974). Can. J . Phys. 52,349. Comes, F. J., and Elzer, A. (1967). Phys. Lett. U A , 334. Comes, F.J., and Elzer, A. (1968).Z. Naturf. A23, 133. Comes, F. J., Spier, F., and Elzer, A. (1968). 2. Naturg 23a, 125. Dalgarno, A., Henry, R. J. W.,and Stewart, A. L. (1964). Planet. Space Sci. 12, 235. Dalgarno, A., and Parkinson, D. (1960). J . Atmos. Terrest. Phys. 18, 335. Dehmer, P. M., Berkowitz, J., and Chupka, W. A. (1974). J. Chem. Phys. 60,2676. Dixit, S. N., Levin, D. A., and McKoy, B. V. (1988). Phys. Rev. A 37, 4220. Dubau, J., and Seaton, M. J. (1984). J. Phys. B: At. Mol. Phys. 17, 381. Ehler, A. W., and Weissler, G. L. (1955). J. Opr. SOC. Am. 45, 1035. Ganas, P. S. (1973). Phys. Rev. A 7, 928. Hemp, K. (1978). Z. Phys. A284,247. Henry, R. J. W. (1966). J. Chem. Phys. 44, 4357.
18
K. L . Bell and A. E. Kingston
Henry, R. J. W. (1967). Planet. Space Sci. 15, 1747. Henry, R. J. W. (1968a). Planet. Space Sci. 16, 1503. Henry, R. J. W. (1968b). J . Chem. Phys. 48, 3635. Hussein, M. I. A., Holland, D. M. P., Codling, K., Woodruff, P. R.,and Ishiguro, E. (1985). J . Phys. B: At. Mol. Phys. 18,2827. Kahler, H. (1971). J. Quant. Spectrosc. Radiat. nansfer 11, 1521. Kohl, J. L., Lafyatis, G. P., Palenius, H. P., and Parkinson, W. H. (1978). Phys. Rev. A18, 571. Koppel, J. U. (1971). J . Chem. Phys. 55, 123. Le Dourneuf, M., Vo Ky Lan, and Zeippen, C. J. (1979). J. Phys. B: At. Mol. Phys. 12, 2449. Manson, S. T. (1988). Phys. Rev. A 38, 126. McGuire, E. J. (1968). Phys. Rev. 175, 20. Nahar, S. N., and Manson, S. T. (1989). Phys. Rev. A 40,5017. Pradhan, A. K. (1978). J. Phys. B: At. Mol. Phys. 1 1 , L729. Pratt, S . T. Dehmer, J. L., and Dehmer, P. M. (1987). Phys. Rev. A 36, 1702. Samson, J. A. R., and Angel, G. C. (1990). Phys. Rev. A 42, 1307. Samson, J. A. R., and Pareek, P. N. (1985). Phys. Rev. A 31, 1470. Saxon, R. P.,Nesbet, R. K., and Noble, C. J. (1989). Phys. Rev. A 39, 1156. Seaton, M. J. (1978). In: Recent Studies in Atomic and Molecular Processes (A. E. Kingston, ed.), Plenum Press, New York, p. 29. Starace, A. F., Manson, S. T., and Kennedy, D. J. (1974). Phys. Rev. A 9, 2453. Taylor, K. T., and Burke, P. G. (1976). J. Phys. B: At. Mol. Phys. 9, L353. Theodosiou, C. E. (1988). Phys. Rev. A 37, 1795. Thomas, G. M., and Helliwell, T. M. (1970). J. Quant. Spectrosc. Radiat. 'Pansfer 10, 423. Vesnicheva, G. A., Malyshev, G. M., Orlov, V. F., and Cherepkov, N. A. (1986). Sou. Phys. Tech. Phys. 31,402. Zeippen, C. J., Le Dourneuf, M., and Vo Ky Lan, (1980). J. Phys. B: At. Mol. Phys. 13, 3763.
ADVANCES IN ATOMIC, MOLECULAR, A N D OPTICAL PHYSICS, VOL. 32
POSITRONIUM FORMATION BY POSITRON IMPACT ON ATOMS AT INTERnnEDIATE ENERGIES B . H . BRANSDEN Department of Physics University of Durham Durham, England
C. J. NOBLE Daresbury Laboratory Science and Engineering Research Council Warrington, England 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Coupled Channel Equations . . . . . . . . . . . . . . . . . . . . . A. Applications: e + - H . . . . . . . . . . . . . . . . . .
B.e+-Li.. . . . . . . . . . . . C.e+-Na.. . . . . . . . . . . . D.e+-He.. . . . . . . . . . . . 111. Perturbation and Distorted Wave Models . . A. The CDW and Eikonal Approximations . IV. Conclusion . . . . . . . . . . . . . References . . . . . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19 21 23 25 26 28 29 33 35 35
I. Introduction Positronium formation by positron impact on atoms is an important example of a rearrangement process, which has become of increasing theoretical interest since the development of improved positron sources has made measurement of the formation cross sections possible (Charlton and Laricchia, 1990; Charlton, 1985). The formation cross section for e + -He collisions has been measured in several experiments (Fornari et al., 1983; Fromme et at., 1986; Diana et al., 1986), while corresponding measurements for the fundamental e + - H system have been published recently (Sperber et al., 1992). It is expected that data for positronium formation in positron collisions with alkali atoms will be available soon (Stein et al., 1987; Kwan et al., 1990). 19
Copyright Q 1994 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-003832-3
20
B. H . Bransden and C. J. Noble
At low energies, for which only a small number of channels are open, it is possible to employ a parametric representation of the wave function in conjunction with the Kohn (or Schwinger) variational methods. Elaborate calculations of this kind have been made of the formation and elastic scattering cross sections for e+-H( 1s) scattering in the interval between the positronium (Ps) threshold at 6.8 eV and the threshold for excitation of the n = 2 levels at 10.2eV. In this interval, known as the Ore gap, converged cross sections have been obtained by Humberston (1986)and Brown and Humberston (1984, 1985). Earlier work (reviewed by Bransden, 1969; Humberston, 1979,1984;and Ghosh et al., 1982) on the formation cross section for both H and He targets was based on the two-state coupled equations linking the elastic scattering and formation channels. This showed that the formation Cross section near threshold was sensitive to the details of the potentials employed, the static interaction leading to very different cross sections from interactions allowing for the long-range polarization potentials in each channel. Nevertheless Basu et al. (1989) have shown that, with a larger basis allowing for polarization of the target, the coupled channel model provides cross sections in this energy region close to those of the accurate variational calculations. In this chapter, we shall discuss positronium formation at intermediate energies, which may be taken to be from the threshold for ionization up to energies of a few hundred electron volts, where the effective coupling between the rearrangement and the direct channels becomes very small. In the interval above the ionization threshold, and extending to at least 100eV, the Ps formation cross section is comparable to, or in some cases larger than, that for excitation; however, at high energies, the ratio of the formation cross section to the total inelastic cross section decreases rapidly. For example, in Fig. 1 the experimental data for e+-He collisions is shown schematically and it is seen that up to 100eV, the highest energy for which comparison is possible, the positronium formation cross section is larger than the cross section for excitation to the n = 2 level. This suggests that methods allowing fully for the coupling between channels should be employed up to energies of at least 100eV for this system and that low-order perturbation models will not be satisfactory below 100-150eV. The principal results obtained by the coupled channel method will be summarized in the next section. At energies for which the effective coupling between the formation and the direct channels is small, perturbation and distorted wave models should be accurate. Several such methods have been investigated, including first-order Born and distorted wave (DWB) models, second-order Born, the continuum distorted wave approximation (CDW), the eikonal approximation, and the impulse approximation. The ideas motivating this work and some applications are the subject of Section 111.
21
POSITRONIUM FORMATION AT INTERMEDIATE ENERGIES r
1
I
I
0.5 N 0
g
0.4
v)
c
0
0.3
W
QI
vl
: 0.2 2
U
0.1
0
50
I
I
I
100
150
200
E (eV) FIG.1. Schematic representation of the total ionization cross section, uion,the cross section for the excitation of the n = 2 levels, u " = ~and , the positronium formation cross section, ups,in e+He collisions.
II. Coupled Channel Equations Let us first consider the positron-atomic hydrogen system. Two systems of center of mass coordinates can be introduced. The boundary conditions for the arrangement e f - H can be expressed most easily in terms of the e'p coordinate 2 and the e-p coordinate 7, whereas for the rearranged system Ps p, the appropriate coordinates are 8, the e'e- coordinate and the coordinate joining the center of mass of the Ps with the proton that can be treated as being of infinite mass. In coupled channel models the total wave function is expressed approximately in the form (Bransden, 1983)
+
z,
In the first group of terms with i < N, the basis functions &@) are hydrogenic ' functions (pseudostates) bound state wave functions together with 3 representing the omitted bound states and the continuum. The channel
22
B. H . Bransden and C . J , Noble
functions, Fi(S‘x) represent the relative motion of e + with respect to the hydrogen atom in state i. Similarly in the group of terms with i > N, $i(a) represents positronium states and pseudostates and Gi(F)the relative motion of Ps with respect to the proton. By projecting the Schrodinger equation for the complete system, (H - E)Y = 0, with each of the basis functions in turn a set of coupled equations for the functions Fi and Gi are obtained. We have, in atomic units,
i= N
+ I, ..., M
where k, is the channel momentum. By expanding Fi and Gi in partial waves coupled integro-differential equations for the radial functions are obtained, which can be solved numerically subject to the usual boundary conditions. A general criticism of this approach, or of the corresponding resonating group method in nuclear physics, is that if N and M are allowed to increase indefinitely, N + 00, M + 00, the basis becomes overcomplete and even with a finite, but large, basis the system of equations may become unstable (see, for example, the discussion in Adhikari and Kowalski, 1991). In practice, when both the angular momentum of the basis states and the number of radial basis functions are restricted, no stability problems have been observed, and no problems have been encountered when the method is applied to ion-atom scattering for which basis sets of up to 100 terms have been employed. In this connection, it should be recalled that Eq. (2) follows from the variational principle using the trial function of Eq. (1) and also that with a finite number terms representing the continuum in one arrangement, (however large) of 9’ the boundary conditions in the other arrangement can never be satisfied, so that both sets of terms in the expansion of Eq. (2) are required. Several options for obtaining the numerical solutions of Eq. (2) have been employed. For example, the equations can be converted to integral equations in the radial variables and solved by standard matrix methods. Recently the R-matrix method has been developed for positron-atom systems (Higgins et al., 1990). A different approach is to write down the corresponding coupled
-
POSlTRONlUM FORMATION AT INTERMEDIATE ENERGIES
23
equations in momentum space for the half-off-shell T- or K-matrices. These are integral equations of the general form
where p, q, and n label the quantum numbers defining a channel, and k, k', and k, are channel momenta, and Vp4(k,k,) is an (off-shell) Born approximation matrix element. Equation (3) can be solved by standard linear algebra en techniques after allowing for the principal value singularity at E = 3k" (Basu et at., 1976; Ghosh et al., 1982). Whether configuration or momentum space methods are employed, the most time-consuming task is the evaluation of the matrix elements of the potentials with respect to the basis functions on different centers, and this limits the size of bases that can be used in practice. One way to compute these matrix elements comparatively rapidly is to approximate the basis functions in terms of combinations of Gaussian functions, in which case the basic integrals can be evaluated analytically. This technique is particularly useful for targets containing more than one electron (Hewitt et al., 1990).
+
A. APPLICATIONS: e+-H In the energy range 8-200eV Basu et al. (1976) solved the two-state coupled equations in momentum space with the basis set H(ls), Ps(1s). Their results (Fig. 2) indicated that the Ps(1s) formation cross section increases above the threshold to a peak of - 3 . 0 ~ ~at; 16eV thereafter decreasing rapidly, being - 0 . 0 0 3 ~ at ~ ~200eV. Subsequently Higgins and Burke (1991) confirmed these results by evaluating the two-state approximation using the R-matrix method of solution. These calculations show that, as at low energies, positronium formation at intermediate energies in the 1 = 0 partial wave is small, the principal contribution to the cross section coming from partial waves with 1 < 1 < 4. An interesting feature discovered by Higgins and Burke and later confirmed by Hewitt et al. (1991) is that the 1 = 0 partial cross section exhibits a broad resonance at 35.6eV, not connected with any threshold or with any obvious physical feature of the system. However as the 1 > 0 contributions dominate the cross section, this resonance would not be observable in the total cross section for positronium formation. Clearly the two-state approximation cannot be expected to be very accurate and does not allow the influence of Ps formation on direct excitation, or ionization, to be assessed. Accordingly Mukherjee et ul. (1990) employed two enhanced basis sets-H(ls, 2s, 2p), Ps(1s) and H(ls, 2%2p),
B. H . Bransden and C. J. Noble
24
1
5 -
4 -
N
0
e
-
3 -
v)
C
0 .t
8
ew
2 -
U
1 -
0
20
40
60
Positron Eneqy ( e V ) FIG.2. Integral cross sections for Ps(1s) formation in positron-hydrogen collisions. Six-state close-coupling results of Hewitt et al. (1990): H(ls, 2s, 2p), Ps(ls, 2s, 2p) basis, solid line; H(ls, 2s, 2p), Ps(ls, 2s, 2p) basis, long-dashed curve; coupled static H(ls), Ps{ls) basis, dot-dash line. Distorted wave polarized orbital results of Khan and Ghosh (1983a).dotted line. First Born approximation (Massey and Mohr, 1954), dash curve; modified Born approximation (Straton, 1987), triangles. Experimental results of Sperber et al. (1992), squares.
H(C)
Ps(Is),where indicates a pseudostate adjusted to give the correct dipole polarizability of atomic hydrogen, and consequently the proper long-range polarization potential in the entrance channel. In further work by Hewitt et al. (1990), the n = 2 positronium states were included in the basis, allowing the importance of capture into the excited states of Ps to be estimated and also allowing partially for polarization in the positronium channels. The cross sections for the formation of Ps(1s) calculated by Hewitt et al. are shown in Fig. 2 for the basis H( Is, 2s, 2p), Ps(ls, 2s, Zp), and for the basis H( Is, 2s, Ps(ls, 2s,2p). It is seen that the Ps(1s) cross section is considerably reduced in size when the enhanced basis sets are employed, but the cross sections for the
s),
POSITRONIUM FORMATION AT INTERMEDIATE ENERGIES
25
two six-state sets do not differ much among themselves except at the lowest energy considered (14 eV). The corresponding cross section for capture into the n = 2 level peaks in the region 20-40eV, where it is comparable in size to the 1s capture cross section. At higher energies the second Born approximation (Basu and Ghosh, 1988-see below) indicates that the ratio of the n = 2 to the n = 1 cross section slowly decreases: 0.35 at 50eV, 0.27 at 100eV, and 0.16 at 300eV. Although 2s and 2p capture are comparable in the 20-40eV region at higher energies the probability of 2p compared to 2s capture is small, 0(2p)/a(2s) steadily reducing from 0.22 at 50eV to 0.1 at 300eV and 0.065 at 500eV. With the enhanced basis sets the resonance found by Higgins and Burke is still observed in the l = 0 partial wave, but at a different energy (Hewitt et al., 1991), and it remains an open question as to whether the resonance will persist if the basis is still further enlarged. A further important question is, What effect does positronium formation have on direct scattering and excitation? Using an optical potential in conjunction with the coupled channel equations Ghosh and Darewych (1991) have shown that the calculated elastic scattering cross section is reduced from 0.402 to 0 . 3 3 6 ~ ~at6 54eV and from 0.243 to 0.226nac at 100eV, when the Ps(1s) channel is taken into account. This indicates that Ps formation will have a significant effect on direct scattering until energies of at least l00eV are reached, supporting the conclusions of Basu et al. (1989, 1990) and Mukherjee et al. (1990) based on the coupled channel equations with the H(1s, 2s, 2p), Ps(1s) basis. The total formation cross section has been measured over the energy range 13eV to 205eV by Sperber et al. (1992). Surprisingly, these results agree rather well near the cross section maximum with the first Born approximation cross sections of Massey and Mohr (1954) and with the modified Born cross sections of Straton (1987). The coupled channel cross sections are considerably smaller in this region (see Fig. 2). Although it is difficult to be confident that the basis sets employed in the coupled channel calculations are sufficient to provide converged cross sections the magnitude of the disagreement with the experiment is unexpected. B. e+-Li For positron scattering by alkali atoms experimental total cross section data are available for Na, K, Rb, and Cs (Stein et al., 1987; Kwan et al., 1990), and it is expected that positronium formation cross sections and also cross sections for Li targets will be measured soon. Since the ionization potentials for alkali atoms are less than the binding energy of ground state positronium,
26
B. H . Bransden and C . J . Noble
the positronium formation reaction is exothermic and is expected to have a large influence on direct scattering at both low and intermediate energies. For positron impact on Li, two-state coupled channel calculations with a Li(2s), Ps(1s) basis have been performed by Guha and Ghosh (1981) (for E < lOeV), who also investigated the effect of adding polarization potentials in each channel. Two-state calculations have also been made, over an extended energy range up to 100eV, by Abdel-Raouf (1988). Since the coupling between the 2s and 2p states of Li is strong and accounts for 97% of the polarizability of Li, it is important to include the Li(2p) state in the basis. The first calculation of this type with a Li(2s, 2p), Ps( 1s) basis was reported by Basu and Ghosh (1991) (for 2 < E < 100eV), which confirmed the importance of including the 2s-2p coupling. Since the n = 2 states of Ps lie close in energy to the Li(2s) ground state, it may be expected that these states should be included in an acceptable basis. Accordingly Hewitt et al. (1992a) have made a number of calculations with various basis states, including the set Ps( Is, 2s, 2p), Li(2s,3s, 2p, 3p). Over the energy range investigated (0.5 < E < 40eV) the cross section for capture into the n = 2 levels equalled or exceeded capture into the ground state, indicating that it is likely that further positronium states are required to obtain converged cross sections. Another important conclusion is that the inclusion of the positronium channels has a marked effect on the direct channels, for example reducing the 2p excitation cross section by a factor of two at the cross section maximum. This is illustrated in Fig. 3. It follows that, up to energies of 40 or 50eV, realistic coupled channel calculations for positron scattering by the alkalis must include all the significant rearrangement channels. C. e+-Na
The scattering of positrons by sodium atoms has also been studied using the have calculated cross sections for close-coupling method. Hewitt et al. (1992~) energies up to 50eV using a single-active-electron model and a seven-state Na(3s, 4s, 3p, 4p), Ps(ls, 2s, 2p) basis. The total cross section obtained is compared in Fig. 4 with experimental measurements by Kwan et al. (1991). The good agreement between experiment and theory for energies between 10 and 50eV confirms the expectation (Walters, 1976) that the ionization cross section is small in this energy range. There is also good agreement in this range with both the five-state close-coupling results of Ward et al. (1989) and with the four-state results of Sarkar et al. (1988). Both of these calculations neglect the effect of Ps formation. At lower energies there are two significant effects, which have been discussed by Ward et al. First, there is an increased sensitivity to the target
0
30
20
10
40
I
Positron Energy ( e V 1
-.
FIG. 3. Integral cross sections for the excitation Li(2s 2p) in positron lithium collisions. Seven-state results, Li(2s, 3s, Zp, 3p), Ps( Is, 2s, 2p), of Hewitt et al. (1992b),solid curve; four-state results, Li(2s,2p, 3s, 3p), omitting positronium formation, dotted curve. Five-state results of Ward et al. (1989),also omitting positronium formation, dash curve. The total Ps formation cross section into the n = 1, 2 levels given by Hewitt et a/. (1992b) are shown for comparison by the dot-dash curve. 1
200
1
I
I
\
z
I
I. 0
6
28
B. H . Bransden and C. J . Noble
wave functions and to the core potentials used in the close-coupling calculations. Second, the experimental results do not discriminate against positrons that are scattered elastically through small forward angles. This effectively reduces the total cross section at small energies (Kauppila and Stein, 1990). The seven-state calculations of Hewitt et al. show that capture into the n = 2 levels is larger than ground state capture for positron energies greater than 7 eV. At lower energies the ground-state capture cross section rises steeply and is dominant. As in the case of positron scattering by lithium, Ps formation has a significant effect on the direct channels, for example, reducing the 3p excitation cross section almost 50% at the cross-section maximum. Cross sections have also been calculated by Abdel-Raouf (1988), using the coupled static approximation and a single-valence electron model for scattering energies between 0.1 eV and 1OOOeV. D. ef-He
A further target for which coupled channel calculations have been carried out is helium, a case of particular interest in view of the existing experimental measurements of cross sections for excitation of the n = 2 levels and of positronium formation. Calculations of the excitation cross sections for transitions to the 2lS and 2lP states have been carried out by Varracchio (1990), using the random phase approximation. The apparently good agreement obtained with the experiment must be regarded as fortuitous, as more recent work indicates that the effect of positronium formation, which was neglected in these calculations, is significant. Apart from a two-state calculation by Mandal et al. (1975, 1976), the only coupled channel calculations including positronium channels are by Hewitt et al. (1992a),using a 2p, G),Ps(ls, 2s, 2p) one active-electron model and a basis with He( Is, 2s, IG, states. The states and 6 are pseudostates with energies lying just above the ionization threshold. Calculations were carried out in the energy range 31-200 eV. Here the relevant target and positronium energy levels are further apart than is the case for hydrogen or the alkalis. As a consequence in positron-helium collisions the contribution to capture into the n = 2 levels of positronium is fairly small over most of the energy range, decreasing from 42% at 31 eV to 6% at 200 eV. The calculated total capture cross section is in reasonable accord with the experimental data near the cross section maximum, which is around 40eV, but lies somewhat below the data at higher energies (see Fig. 5). However, it has been suggested by Schultz and Olson (1988) that the measured formation cross section is an overestimate because
POSITRONIUM FORMATION AT INTERMEDIATE ENERGIES
29
0.6
0.5
-& -
; r
-2
0.4
0
z9( 0.3 ln
VI
L!
U
0.2 0.1
0
0
50
100
150
200
Positron Energy ( e V ) FIG. 5. Ps formation cross sections for positron collisions with helium. Eight-state closecoupling results, He&, 2s, ks,2p, @), Ps(ls, 2s, 2p), of Hewitt et al. (1992a), solid curve. Distorted wave polarized orbital results of Khan and Ghosh (1983b), inverted open triangles. Experimental results of Fornari et al. (1983),filled triangles; Fromme et al. (1986),crosses; Diana et al. (1986), solid circles.
of inefficiencies in detecting positrons scattered through large angles following ionization. The effect of positronium formation on direct excitation is significant for E < IWeV, but less marked than for the more weakly bound targets of H and Li. The calculated cross section for excitation of the n = 2 levels of He is about 40% larger than the measured values (Sueoko, 1989),but the energy variation of the cross section is well represented.
111. Perturbation and Distorted Wave Models At energies for which the probability of positronium formation is small, it is reasonable to look for perturbation models. The first-order Born approximation for the Ps formation amplitude is (Massey and Mohr, 1954) T/i = <@/
I v/I@i>=
(4)
B. H . Bransden and C. J . Noble
30
where, for the e+-H system, the initial and final interactions are
v = K+p(z)+
v, = I/,+,(?) + K-,(7)
(5)
The unperturbed wave functions in the initial and final states are
mi = exp(Zi*?)c$i(7), mf = exp(iEf-$)$f(R)
(6) where ki, k, are the initial and final relative momenta (in atomic units). In ion-atom collisions these first-order expressions (Eq. 4) provide remarkably accurate total (but not differential) cross sections for charge exchange in proton-hydrogen collisions, and indeed, when corrected to allow for the overall Coulomb interaction, accurate total cross sections are obtained for electron capture by ions of higher charge from atomic hydrogen (Bransden and McDowell, 1992).As emphasized by Shakeshaft and Wadehra (1980) the first-order amplitude cannot be expected to provide an accurate description of a positron collision because even at high energies the positron will be strongly deflected by the e+-p interaction. This is in contrast to heavy particle scattering in which the relative motion of the ions is well described by a classical straight-line motion and the internuclear interaction plays no role in determining the transition probability. To allow for this effect, either higher Born approximations can be used or distorted waves x' , x: can be introduced, satisfying 4
-
+ %-,(7) + U,(?) - E)X' (T + K+&) + U , @ ) - E)x; (T
(7)
=0 =0
(8) where T is the kinetic energy operator for the system and U i and U , are potentials that represent the relative motion in the initial and final states but that cannot induce the rearrangement. The first-order distorted wave amplitude is then (Massey and Mohr, 1954)
Both the post and prior forms of the amplitude containing(V, - U,) and (V;. - U J , respectively, are identical if no other approximations are made. In the simplest approximation U , and U, can be taken to be the static interactions in the initial and final states. For e+-H, we have
V, = ,
U , = (+/lvJ- I$/>
0
(10)
This approximation was first computed for Ps(1s) formation in e + - H collisions by Mandal et al. (1979) and has been extended for capture into excited states by a number of authors, the most complete investigation being that of Nahar (1989). DWB cross sections have also been calculated for e + Li, e t -Na by Nahar and Wadehra (1987) (see also Mazumdar and Ghosh,
POSITRONIUM FORMATION AT INTERMEDIATE ENERGIES
31
1986). It should be noted that capture from inner shells is expected to dominate the high-energy capture cross section, as in the corresponding case of charge exchange in ion-atom collisions. As we have seen, polarization of both the target atom and the positronium in the final state is important. This can be taken into account by adding polarization potentials to either y, U,, or both. Alternatively the unperturbed target and positronium wave functions can be replaced by polarized orbitals. A number of distorted wave polarized orbital (DWPO) calculations have been carried out along these lines for H and for He targets (Khan and Ghosh, 1983a, b; Khan et al., 1985). For Ps formation in the 1s state by e + impact on He the DWPO cross sections agree well with those from the coupled channel model above 80eV, however the DWPO cross sections for capture into the n = 2 states are significantly smaller than those predicted by the coupled channel model. It is interesting to note that at high energies the ratio of cross sections for capture into the 1s state given by the DWB and first-order Born approximations tend to a finite limit, both behaving like E - 6 . The experimental data for e+ + He -,Ps + He' seems to show an energy variation of between E-'.' and E - ' . 5 above 100eV, in contrast to the rapid E - decrease suggested by perturbation theory. The classical trajectory Monte Carlo method has been applied to this system by Schulz and Olson (1988) and Schulz et al. (1989), and this predicts that the cross section should decrease like E - 3 . 5 . As noted earlier these authors have suggested that the experimental formation cross sections may be overestimated for E 100eV, and the formation cross section may decrease more rapidly than the measurements would suggest. It is well known that in heavy-particle charge exchange the high energy limit of the cross section is provided by second-order terms in the Born series and consequently at high energies (and possibly at intermediate energies) the first-order DWB model is expected to be inadequate since only part of the second-order terms in the Born series is included. Of the four second-order terms containing T / , + p G O K +T/,-pGoT/,+p, p, T/,+pGOT/,+e-, V,-,C,V,+,-, the last two are the most important, being peaked about an intermediate state corresponding to a classical double scattering (Thomas scattering). Shakeshaft and Wadehra (1980) showed that, for e' + H + Ps + H + if the final state of positronium is of even parity, the two terms cancel at high energies and the cross section is proportional to E - 6 . In contrast, if the final state of positronium is of odd parity, the two terms are of the same sign and dominate the cross section in the high-energy limit, which is E - 5 . 5 . As a consequence capture into the 2p state is predicted to be the dominant process as E + co. This is certainly not the case at intermediate energies and even up to 8,000 eV, for which explicit second Born calculations (Basu and Ghosh, 1988) show
=-
B. H . Bransden and C. J. Noble
32
that capture into the n = 2 levels is small compared with ground state capture, and capture into the 2p state is less important than into the 2s level. For positronium formation into the n = 1 and n - 2 levels in e+-H collisions Basu and Ghosh (1988) have presented approximate second Born cross sections based on the target Green’s function over an energy range from 50 to _ _500eV (see also Barman and Sural, 1974).In their first model (SB) Is, 2s, 2p, 3s, and 3 hydrogenic intermediate states were employed whereas in a second model (CCSB) the exact two-state close coupling amplitudes were corrected by constructing second-order terms with 2s, 5,%, and 5;1 hydrogenic intermediate states. In view of the importance of the 1s intermediate state, the CCSB model would seem to be the more accurate. Second Born cross sections for positronium formation by capture from helium have also been reported by Sarkar et al. (1992). A different approach has been followed by Deb et al. (1987a, b) and Deb (1989) (see also McGuire et al., 1986), who used a method (DMS) of approximating the full Green’s function that includes all second-order and some higher order terms, but the evaluation involves a number of approximations that are difficult to assess. Some cross sections for ground-state capture from hydrogen calculated from these second-order calculations are shown in Table I, compared with those arising from the two-state coupled channel approximations. Also included in Table I are cross sections from a second-order Fadeev-Watson (FW2) multiple scattering expansion (Roberts, 1989) and from an eikonal approximation (Tripathi et al., 1989). It is noteworthy that at 50eV the cross sections arising from four- and six-state coupled channel calculations are smaller than those of the two-state approximation while the second-order cross sections are larger, which suggests that higher order terms are important, at least until much larger energies are TABLE 1 TOTALCROSSSECTIONS (IN UNITS OF naf) FOR Is- 1s CAPTURE IN e+ -H SCATTERING E(eV)
FB
50 100 200 300
0.464 0.458-’ 0.251-’ 0.367-’
cc 0.546 0.505-’
0.282-2 O.4Ol-’
SB
CCSB
DMS
0.615 0.527-’ 0.313-’ 0.488-3
0.562 0.46-’ 0.257-’ 0.388-3
-
-
-
0.451-’ 0.240-3 0.368-3
1.21 0.124-I 0.168-’
0.389-’ 0.190-* 0.266-3
FW2
EIK
FB = first Born approximation; CC = two-state close coupling; SB = second Born model; CCSB = two-state close coupling with second Born terms (all from Basu and Ghosh, 1988). DMS = second Born with additional terms (Deb et al., 1987a). FW2 = second-order FadeevWatson model (Roberts, 1989). EIK = eikonal model (Tripathi et al., 1989).
33
POSITRONIUM FORMATION AT INTERMEDIATE ENERGIES
reached. At lOOeV and above the CCSB and DMS cross sections are fairly close to each other and smaller than those of the two-state approximation, pointing to the importance of the 1s contribution to the intermediate states in the higher order terms. Closely related to the approximation of Deb et al. is the impulse approximation of Cheshire (1964a,b), which however produces cross sections for Ps(1s) formation in e + - H collisions that appear to be too large in comparison with the coupled channel cross sections over the intermediate energy range. Over the intermediate energy range the FW2 cross sections for an H target also appear to be too large, and the same is true for He targets, the calculated cross section being much too large compared with the experimental data at energies below 275 eV. A. THECDW
EIKONAL APPROXIMATIONS
AND
The DWB and DWPO approximations take into account the effective interactions in the initial and final states of the system, but in the CDW approximation of Cheshire (1964b) distorted waves are introduced that attempt to represent the complete three-body wave function in more detail. The CDW wave functions are, in the e+-H arrangement,
where I , , ?fare the relative velocities before and after the collision, and w*(u,$,?)
= e x p ( f m ) r ( l k iu),F,(f
ia, 1 , k i p y - G . T )
(13)
In this approach the Coulomb potentials between each pair of particles are reflected in the Coulomb functions, w * , and the perturbation is part of the kinetic energy operator. The post and prior forms of the distorted wave amplitudes T + and T - , respectively, are not the same and are given by Ti+ = <@fI~'Ix' >, T i = (xiI V @ i > (14) In the eikonal model the Coulomb functions, w * , are replaced by their eikonal approximations, G*,where
w*(u, 7j, 5;) = exp(T ia log(py T ji
a?))
(15)
The corresponding post and prior amplitudes are denoted by T i and T i .
34
B. H . Bransden and C. J . Noble
Using either the CDW or eikonal wave functions symmetrical approximations can be derived of the form
T = ( X i W - EIx,) (16) but no applications of this have been reported as yet. In post form Chen, Wen, and Xi (1992) have calculated the CDW cross section for hydrogen targets, and Bransden et at. (1992) have calculated both the post and prior forms of cross sections for both hydrogen and helium targets, using a one-active-electron approximation for the case of helium. In the eikonal approximation Tripathi et al. (1989) have calculated the post form of cross section for hydrogen targets, whereas Liu et al. (1992) have calculated the prior form for helium and Tripathi and Sinha (1990) have investigated lithium targets. Both the post and prior forms have been calculated by Bransden et al. for hydrogen and for helium targets. In Fig. 6 the total cross sections of Bransden et al. are compared with the coupled channel cross sections of Hewitt et al. (1992a) and with the DWPO cross 0.5
0.4
-d"
N
P. 0.3 v)
c
0 .c
*8 0.2
t
U
0.1
I
n v
-
40
60
80
100
120
140
160
lea
200
Positron Energy ( eV 1 FIG.6. Total cross sections for e + + He(ls2)-,Ps + He+(ls). Results of Bransden et al. (1992) using these approximations:Born (solid curve]; CDW post (dash curve]; CDW prior (dotdash curve); eikonal post (dot curve); eikonal prior (long-dash curve).Close-coupling results of Hewitt et al.(1992a). solid triangles. Experimental results of Fromme et al. (1986),crosses;Diana et al. (19861, solid cirvles.
POSITRONIUM FORMATION AT INTERMEDIATE ENERGIES
35
sections of Khan and Ghosh (1983b). It is seen that the post form of matrix element provides CDW and eikonal cross sections that merge with the Born approximation from above at E 2 200eV. However, there is a large post prior discrepancy, and the prior forms of cross section approach the Born approximation from below. A similar phenomenon is seen in the VPS approximation (McDowell and Coleman, 1970) applied to electron scattering. Above 200 eV the CDW and eikonal cross sections are very close to each other in both the post and prior versions. For E < lOOeV where coupling effects are large, it would be unwise to expect realistic cross sections from these essentially high energy methods. Above lOOeV the post eikonal and to a lesser extent the post CDW total cross sections are close to the results of coupled channel calculations. Although in the forward direction the differential cross sections (and hence the total cross sections) are rather close to those of the Born and DWPO approximations, the large-angle scattering shows an interesting structure that differs considerably between the distorted wave and Born approximations at higher energies and is not yet properly understood. All the theoretical models exhibit a rapid decrease of cross section at high energies, not shown by the data, including a modified CDW model of Deb et a/. (19901, which shows a fairly flat cross section over a limited energy region between 160eV and 300eV.
IV. Conclusions In the intermediate energy range up to energies of several hundred electron volts, the use of coupled channel models is feasible in principle and can be expected to provide accurate cross sections, including the formation cross section. It is likely that larger basis sets than those used hitherto should be employed to be certain of obtaining converged cross sections. At the higher energies although most approximations lead to similar values for the groundstate capture cross section the differential cross section is very sensitive to the details of the approximation employed and in contrast to the comparatively well-understood situation in charge exchange, there are still large gaps in our understanding of high-energy positronium formation.
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B. H . Bransden and C . J. Noble
Barman, S. K., and Sural, D. P. (1974). Phys. Rev. A 10, 1161. Basu, D., Banerji, G., and Ghosh, A. S. (1976). Phys. Rev. A 13, 1381. Basu, M., and Ghosh, A. S. (1988).J. Phys. B 21, 3439. Basu, M., and Ghosh, A. S. (1991).Phys. Rev. A 43,4746. Basu, M., Mukherjee, M., and Ghosh, A. S. (1989). J. Phys. B 22, 2195. Basu, M., Mukherjee, M., and Ghosh, A. S. (1990). J. Phys. B 23,2641. Bransden, B. H. (1969). Case Stud. At. Collision Phys. 1, 171. Bransden, B. H. (1983). Atomic Collision Theory, 2d ed., Benjamin/Cummings, Reading, Mass. Bransden, B. H., Joachain, C. J., and McCann, J. F. (1992). J. Phys B 25,4965. Bransden, B. H., and McDowell, M. R. C. (1992).Charge Exchange and the Theory of lon-Atom Collisions, Oxford University Press, Oxford. Brown, C. J., and Humberston, J. W. (1984). J. Phys. B 17, L423. Brown, C. J., and Humberston, J. W. (1985).J. Phys. B 18, L401. Charlton, M. (1985). Rep. Prog. Phys. 48, 737. Charlton, M., and Laricchia, G. (1990).J . Phys. B 23, 1045. Chen, X. J., Wen, B., Xu, Y. W., and Zhang, Y. Y. (1992). J Phys. B 25, 4661. Cheshire, I. M. (1964a). Proc. Phys. Soc. 83, 227. Cheshire, I. M. (1964b). Proc. Phys. Soc. 84, 89. Deb, N. C. (1989). J. Phys. B 22, 3755. Deb, N. C., Crothers, D. S. F., and Fromme, D. (1990).J. Phys. B 23, L483. Deb, N. C., McGuire, J. H., and Sil, N. C. (1987a). Phys. Rev. A 36,3707. Deb, N. C., McGuire, J. H., and Sil, N. C. (1987b). Phys. Rev. A 36, 1082. Diana, L. M., Coleman, P. G., Brooks, D. L., Pendleton, P. K., and Norman, D. M. (1986).Phys. Rev. A 34, 2731. Fornari, L. S., Diana, L. M., and Coleman, P. G. (1983). Phys. Rev. Lett. 51, 2276. Fromme, D., Kruse, G., Raith, W., and Sinapius, G. (1986).Phys. Rev. Lett. 24, 3031. Ghosh, A. S., and Darewych, J. W. (1991). J. Phys. B 24, L629. Ghosh, A. S., Sil, N. C., and Mandal, P. (1982). Phys. Rep. 87, 313. Guha, S., and Ghosh, A. S. (1981).Phys. Rev. A 23, 743. Hewitt, R. N., Noble, C. J., and Bransden, B. H. (1990).J. Phys. B 23,4185. Hewitt, R. N., Noble, C. J., and Bransden, B. H. (1991).J. Phys. B 24, L635. Hewitt, R. N., Noble, C. J., and Bransden, B. H. (1992a).J. Phys. B 25, 557. Hewitt, R. N., Noble, C. J., and Bransden, B. H. (1992b).J. Phys. B 25,2683. Hewitt, R. N., Noble, C. J., and Bransden, B. H. (1993).J Phys. B 26, in press. Higgins, K., and Burke, P. G. (1991).J. Phys. B 24, L343. Higgins, K., Burke, P. G., and Walters, H. J. R. (1990).J. Phys. B 23, 1345. Humbertson, J. W. (1979). Adu. At. Mol. Phys. 15, 101. Humberston, J. W. (1984).J. Phys. B 17, 2353. Humberston, J. W. (1986). Adv. in At. Mol. Phys. 22, 1. Kauppila, W. E., and Stein, T. S. (1990). Adv. in At. Mol. Opt. Phys. 26, 1. Khan, P., and Ghosh, A. S. (1983a). Phys. Rev. A 27,1904. Khan, P., and Ghosh, A. S. (1983b).Phys. Rev. A 28,2181. Khan, P., Mazumdar, P. S., and Ghosh, A. S. (1985).Phys. Rev. A . 31, 1405. Kwan, C. K., Dababheh, M. S., Kauppila, W. E., Lukaszew, R. A., Parikh, S. P., Stein, T. S., Wan, W.J., and Zhou, S. (1990). In Abstracts o f t h e XVI 1st. Con$ on the Physics 5fEIect. and Atomic Collisions (D. A. R. Freund, M. Lubell, and T. Lucatorto, Eds.), A.I.P., New York, Kwan, C. K., Kauppila, W. E., Lukaszew, S. P., Parikj, S. P., Stein, T. S., Wan, Y. J., and Dababneh, M. S. (1991). Phys. Rev. A 44, 1620. Liu, X. W., Sin, Y., Zhou, 2. F., and Watanabe, T. (1992). Phys. Rev. A, in press. Mandal, O., Basu, D., and Ghosh, A. S. (1976). J. Phys. B 9, 2633. Mandal, P., Ghosh, A. S., and Sil, N. C. (1975).J. Phys. B 8, 2372.
POSITRONIUM FORMATION AT INTERMEDIATE ENERGIES
37
Mandal, P., Guha, S., and Sil, N. C. (1979). J. Phys. B 12, 2913. Massey, H. S. W., and Mohr, C. B. 0. (1954). Proc. SOC. Phys. SOC. London A 67, 695. Mazumdar, P. S., and Ghosh, A. S. (1986). Phys. Rev. A 34, 4433. McDowell, M. R. C., and Coleman, J. P. (1970). Introduction t o the Theory of Ion-Atom Collisions, North-Holland, Amsterdam. McGuire, S. H., Sil, N. C., and Deb, N. C. (1986). Phys. Rev. A 34, 685. Mukherjee, M., Basu, M., and Ghosh, A. S. (1990). J. Phys. B 23, 757. Nahar, S. N. (1989). Phys. Rev. A 40,6231. Nahar, S. N., and Wadehra, J. M. (1987). Phys. Rev. A 35,4533. Roberts, M. J. (1989). J. Phys. B 22, 3315. Sarkar, K. P., Basu, M., and Ghosh, A. S. (1988). J. Phys. B 21, 1649. Sarkar, N. K., Basu, M., and Ghosh, A. S. (1992). Phys. Rev. A 45,6887. Schulz, D. R., and Olson, R. E. (1988). Phys. Rev. A 38, 1866. Schulz, D. R., Reinhold, C. O., and Olson, R. E. (1989). Phys. Rev. A 40,4947. Shakeshaft, R., and Wadehra, J. M. (1980). Phys. Rev. A 22, 968. Sperber, W., Backer, D., Lynn, K. G., Raith, W., Schwab, A., Sinapius, G., Spicher, G., and Weber, M. (1992). Phys. Rev. Lett. 68, 3690. Stein, T. S., Dababheh, M.S., Kauppila, W. E., Kwan, C. K., and Wan, Y. J. (1987). In Atomic Physics with Positrons (J. Humberston and E. Armour, eds.), Plenum, New York, p. 251. Straton, J. C. (1987). Phys. Reo. A 35, 3725. Sueoka, 0. (1989). Quoted by Charlton and Laricchia (1990). Tripathi, S., and Sinha, C. (1990). Phys. Rev. A 45, 5743. Tripathi, S., Sinha, C., and Sil, N. C. (1989). Phys. Rev. A 39, 2924. Varracchio, V. F. (1990). J. Phys. B 23, L779. Walters, H. R. J. (1976). J . Phys. B 9, 227. Ward, S. J., Horbatsch, M., McEachran, R. P., and Stauffer, A. D. (1989). J. Phys. B 22, 1845.
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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 32
ELECTRON-A TOM SCATTEMNG THEORY AND CALCULATIONS P . G . BURKE Department of Applied Malhematics and Theoretical Physics Queen's University of Berfat Beljhst, Northern Ireland 1. Introduction . . . . . . . . . . . . . 11. Scattering at Low Energies . . . . . . A. Basic Scattering Equations . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39 40 40 B. Coupled Integrodifferential Equations . . . . . . . . . . . . . . . 42 C. Relativistic Effects for Heavy Atoms and Ions . . . . . . . . . . . . 44 111. Scattering at Intermediate and High Energies . . . . . . . . . . . . . 44 A. Modified Low-Energy Methods . . . . . . . . . . . . . . . . . . 45 B. Optical Potential Methods . . . . . . . . . . . . . . . . . . . . 46 C. Born Series Methods . . . . . . . . . . . . . . . . . . . . . . 47 IV. Illustrative Results. . . . . . . . . . . . . . . . . . . . . . . . . . 49 A. Low Energies . . . . . . . . . . . . . . . . . . . . . . . . . . 49 B. Intermediate and High Energies . . . . . . . . . . . . . . . . . . 50 V. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 53 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 54 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I. Introduction This chapter reviews recent progress in the theory of electron-atom and electron-ion scattering and describ6s some recent calculations that can be compared with experiments. The review is restricted to theories that describe elastic scattering and excitation for incident electron energies that range from zero energy to many times the ionization threshold. The important ionization process will not be considered here. An enormous amount of work has been carried out in this field in recent years driven mainly by the many applications in astronomy, laser physics, plasma physics and upper atmosphere physics, which require accurate scattering data for their interpretation. Recent experimental advances have also provided an important stimulus to theory. These advances include the use of coincidence techniques, polarized beams and targets and very high energy-resolution electron beams. On the theoretical side, many of the new developments have been stimulated by the increasing availability of powerful
39 Copyright 0 1994 by Academic Press lnc AII nghts of reproduction in any form reserved ISBN 0-12-003812 1
40
P. G.Burke
computers, which are allowing detailed calculations to be carried out for the first time for complex atoms and ions where electron correlation and relativistic effects are important. The review commences by discussing scattering at low energies. In this case the theoretical and computational methods are now well established and a number of major computer program packages have been written allowing accurate calculations for light atoms and ions to be carried out routinely. The following section then addresses the much more difficult problem of how to describe theoretically scattering at intermediate and high energies. A brief review is given of a number of approaches that have been developed to describe this situation, where an infinite number of channels are open. In the next section a few illustrative calculations are presented and compared with experiment. Finally in the last section conclusions are drawn and some future directions of research are mentioned.
11. Scattering at Low Energies We consider first the scattering of electrons by atoms and atomic ions at low energies. That is we consider the process e- +A,-+e-
+ Aj
(1)
where A, and A, are the initial and final states of the target atom or atomic ion and where by low energies we mean that the velocity of the scattered electron is of the same order or less than that of the target electrons playing an active role in the collision. Hence in this section we will be concerned mainly with the range of energies where only elastic scattering and excitation processes are possible.
A. BASICSCATTERING EQUATIONS We assume initially that all relativistic effects can be neglected. This restricts our discussion to light atoms and ions. We will relax this restriction later. The Schrodinger equation describing the scattering of an electron by a target atom or ion containing N electrons and having nuclear charge 2 is
ELECTRON-ATOM SCATTERING THEORY AND CALCULATIONS
41
where E is the total energy of the system and the (N + 1)-electron Hamiltonian H N + i is given by
In this equation and in later equations we use atomic units where h = h/2nn, me, and e are taken as unity, h being Planck’s constant, me the mass of the electron and - e its charge. Also we have written rij = (ri- cjl where and cj are the vector coordinates of the ith andjth electrons and we have taken the origin of coordinates to be the target nucleus which is assumed to have infinite mass. We now introduce the target eigenfunctions Q iand the corresponding eigenenergies wi by the equation (QiJHNI@j)
= wi6ij
(4)
where HN is defined by Eq. (3) with N + 1 replaced by N. We will assume in what follows that sufficiently accurate target states can be calculated. The solution for Eq. (2) corresponding to the process defined by Eq. (1) then has the asymptotic form
and xlllmj are the spin eigenfunctions of the incident and where x~/~,,,, scattered electrons, where the direction of spin quantization is usually taken to be the incident beam direction, and hi(& 4) is the scattering amplitude corresponding to scattering angles 0, 4. The wave numbers ki and k j are related to the total energy of the system by the equation E = wi +fkjZ = wj + $ k j
(6) The outgoing wave term in Eq. ( 5 ) contains contributions from all target states that are energetically allowed, i.e., for which kj’ 2 0. The remaining states for which k j 0 can occur virtually only in the collision. Finally we note that when the target is an atomic ion, logarithmic phase factors must be included in the exponentials in Eq. (5). Since these factors do not introduce any essential difficulties we shall not include them explicitly in what follows. The differential cross section for a transition from an initial state li) = lki,Qi, x ~ , ~ , , , to ~ ) a final state l j ) = 1kj, Q j ,x ~ / ~ ,can , , ~be) obtained by calculating the incident and scattered flux in Eq. (5). We obtain
-=
42
P . G . Burke
and the total cross section is obtained by averaging over initial spin states, summing over final spin states and integrating over all scattering angles. B. COUPLED INTEGRODIFFERENTIAL EQUATIONS
To obtain an explicit expression for the scattering amplitude and cross section most approximation methods describing low-energy scattering start from the following expansion of the total wave function:
Here zi= t y i represents the space and spin coordinates of the ith electron and the channel functions 6;are obtained by coupling the target eigenstates or pseudostates Oi with the spin-angle function of the scattered electron to form eigenstates of r = LSM,M,z, which are conserved in the collision where L is the total orbital angular momentum, S is the total spin, and x is the parity operator. The FS are reduced radial functions describing the motion of the scattered electron, the operator d antisymmetrizes the first summation with respect to interchange of any pair of electrons in accordance with the Pauli exclusion principle and the square integrable (L2)correlation functions XF allow for electron correlation effects not adequately represented by the first expansion, which in practice is limited to a finite number of terms. If the expansion over the correlation functions is omitted, Eq. (8) is referred to as the close coupling expansion, which is discussed in greater detail by Seaton in another chapter in this book. Coupled equations for the radial functions F ; and the coefficients u; can be obtained by substituting Eq. (8) into Eq. (2) and projecting onto the channel functions 6; and onto the Lz functions x;. Eliminating the coefficients a; between these equations yields the following set of coupled integrodifferential equations satisfied by the functions F::
where Ii is the orbital angular momentum of the scattered electron and V L , Wf, and Xf are the partial wave decompositions of the local direct, nonlocal exchange, and nonlocal correlation potentials. These potentials are too complicated to write down explicitly except for the simplest atom, hydrogen, first discussed by Percival and Seaton (1957). Instead they are determined by general computer programs mentioned later.
ELECTRON-ATOM SCATTERING THEORY AND CALCULATIONS
43
The scattering amplitude and cross section can be obtained by solving Eqs. (9) subject to the following K-matrix asymptotic boundary conditions:
FL
-
k; ''2(sin Oidij + cos f l i K L ) , open channels
r-rm
FC
--*
0, closed channels
r-a
where 1
I
Oi = k i r - A lin 2
+ .L In(2kir) + oi ki
+
with z = Z - N, the residual charge on the ion, and oi = arg r(li 1 - iz/ki), the Coulomb phase shift. The S-matrix is related to the K-matrix, defined by Eqs. (lo), by the matrix equation
where the dimension of the matrices in this equation is n, x n,, where n, is the number of open channels at the energy under consideration. The hermiticity and time reversal invariance of the Hamiltonian ensures that K r is real and symmetric while Sr is unitary and symmetric. The scattering amplitude defined by Eq. ( 5 ) and hence the differential and total cross sections can be readily related to the S-matrix. The total cross section is given by
for a transition from an initial target state denoted by aiLiSito a final target state a j L j S j , where ai and uj represent any additional quantum numbers required to define the initial and final states. It is also useful in applications to define a collision strength by
Q ( i , j ) = k?(2Li
+ 1)(2Si+ l)olot(i- j )
(14)
which is dimensionless and symmetric with respect to interchange of the initial and final states denoted by i and j . Most of the recent results emerging from this approach have been obtained using one of four computational methods to solve Eqs. (9). These are the Rmatrix method (Burke et al., 1971),the reduction of Eqs. (9) to linear algebraic equations (Seaton, 1974), the noniterative integrals equations method (Smith and Henry, 1973a, b), and the matrix variational method (Nesbet, 1980).
44
P. G . Burke
These methods and the associated computer program packages that implement them have been reviewed by Burke and Eissner (1983). c . RELATIVISTICEFFECTS FOR HEAVY ATOMSAND IONS As the nuclear charge Z of the target increases, relativistic effects become important even for low-energy electron scattering. There are two ways in which relativistic effects play a role. First, there is a direct effect corresponding to the relativistic distortion of the wave function describing the scattered electron by the strong nuclear Coulomb potential. Second, an indirect effect is caused by the relativistic change in the charge distribution of the target. For atoms and ions with small or intermediate Z values, relativistic eEects are small so that the scattering calculation can first be performed in LS coupling using the nonrelativistic Hamiltonian defined by Eq. (3); the corresponding K-matrices are subsequently recoupled to yield transitions between fine-structure levels (Saraph 1972, 1978). On the other hand, for atoms and ions with high 2 values, relativistic terms must be retained in the Hamiltonian both for the target and for the electron-target system. One way of doing this is to retain terms from the Breit-Pauli Hamiltonian (Jones, 1975; Scott and Burke, 1980). A computer program package that implements this approach, within the R-matrix framework, has been written by Scott and Taylor (1982). An alternative way of including relativistic terms in the Hamiltonian is to use the Dirac Hamiltonian (Walker, 1974; Chang, 1975). A computer program package that implements this approach, again within the R-matrix framework, has been written by Norrington and Grant (198 1). Further details of methods appropriate for the scattering of electrons from heavy atoms is given by Grant in another chapter in this volume.
III. Scattering at Intermediate and High Energies We now consider elastic and inelastic electron-atom and electron-ion scattering at electron impact energies greater than the ionization threshold of the target. In this case an infinite number of channels are open, and we shall examine a number of theoretical methods that have been proposed to obtain accurate results in this situation. These include extension of low-energy methods based on expansion (8), the use of optical potentials that can take account of the loss of flux into the infinity of open channels and extension of the Born approximation by the inclusion of higher-order terms. We shall discuss these approaches in turn where for simplicity we restrict our attention to nonrelativistic scattering.
ELECTRON-ATOM SCATTERING THEORY AND CALCULATIONS
45
A. MODIFIED LOW-ENERGY METHODS The first approach we consider is to extend expansion (8) by including a few well-chosen pseudostates as well as the target eigenstates of interest. These pseudostates are not eigenstates of the target Hamiltonian, but instead are chosen to represent in some average way the high-lying Rydberg states and continuum states of the target that cannot be included explicitly. An approach of this type was introduced by Burke and Schey (1962), who suggested that a pseudostate basis should be chosen to represent both short and long range correlation effects, and by Rotenberg (1962), who suggested an expansion in Sturmian functions would be appropriate. Later, Damburg and Karule (1967) introduced a 2p pseudostate which represented the complete polarizability of the atomic hydrogen ground state. Intermediate energy e - - H scattering calculations using this approach were carried out by Burke and Webb (1970), Burke and Mitchell (1973) and Callaway and Wooten (1973, 1974, 1975), which demonstrated its usefulness. Recently, accurate results for 1s-2s and 1s-2p excitation have been obtained by Callaway et al. (1987), who used a Slater basis to represent the pseudostates, Burke et al. (1987), Scott et al. (1989) and Scholz et al. (1991), who used an intermediate-energy R-matrix (IERM) basis for the pseudostates, and Bray and Stelbovics (1992a,b), who used a Sturmian basis. In the last calculation, a momentum-space coupled channel formalism was used, which is equivalent to solving the coupled integrodifferential Eqs. (9) in configuration space. One difficulty with the pseudostate approach that arose in the early work was the appearance ofunphysical pseudoresonances at intermediate energies. It was shown by Burke et al. (1981) that physical information can be obtained by performing a suitable averaging of the T-matrix elements over these pseudoresonances. However, in the more recent studies, Bray and Stelbovics (1992a,b) have found that as the pseudostate basis is extended toward convergence the pseudoresonances became less pronounced and eventually disappear. In spite of these very encouraging developments some major difficulties still persist. First, there is still no preferred way of introducing a pseudostate basis in the case of electron scattering by complex atoms. Second, the number of pseudostates that must be retained to obtain converged cross sections for excitation to highly excited Rydberg states becomes prohibitively large. Recently an extension of the IERM approach, mentioned earlier, whereby a two-dimensional R-matrix is propagated out to large distances, has given encouraging results for the Temkin-Poet e - -H s-wave model for principal quantum numbers up to n % 5 (Le Dourneuf et al., 1990). However, further work is necessary to examine its usefulness in the general atom case.
46
P . G. Burke
B. OPTICAL POTENTIAL METHODS The Schrodinger equation describing the scattering process can be rewritten in terms of Feshbach projection operators P and Q , which partition the (N + 1) electron space into two orthogonal parts (Feshbach, 1958, 1962). We define P+Q=1 P2 = P,
Q2 =
Q
(15)
PQ=QP=O
where P is usually chosen to project onto a few target states of interest. Equation (2) can then be formally rewritten as
We then eliminate Q Y from the first equation in (16) by substituting for it from the second equation. We obtain
The second term in this equation, which we write as
describes scattering out of P space into Q space, propagation in Q space and then scattering back into P space. The optical potential is defined by
v,,, = PVP + Y
(19)
where V is the electron atom (ion) interaction potential appearing in H , , 1. To solve Eq. (17), which is completely equivalent to the original Schrodinger equation (2), some approximation must be made for V,,,. We have already mentioned an approximation, defined by expansion (8), where Q is approximated by a projection onto a discrete set of L2 functions This results in a real separable approximation for the optical potential that gives accurate results at low energies. Unfortunately this approach gives rise to unphysical pseudoresonances at intermediate energies. As already mentioned, physical information can be obtained by a suitable averaging over these pseudoresonances, but the accuracy of this approach is difficult to quantify. On the
xr.
ELECTRON-ATOM SCATTERING THEORY AND CALCULATIONS
47
other hand, at higher energies it is appropriate to make a perturbation expansion of V,,, of the form
v
opt
=
V(1)
+
+
V(2)
y(3)+
...
(20)
where the first-order term Y c l )= PVP. The second and higher order terms are in general nonlocal, complex operators. However at sufficiently high energies local approximations to these can be obtained. For example, Byron and Joachain (198 1) converted the lowest order terms of perturbation theory for elastic scattering into an ab initio local complex optical potential that has been used to describe the elastic scattering of electrons and positrons from a number of atoms. The optical potential calculated in the second order has also been used by Bransden and Coleman (1972) and Bransden et al. (1982) to describe e - - H scattering where the Is, 2s, and 2p states are retained in P space. This has enabled 1s-2s and 1s-2p excitation cross sections to be determined at intermediate energies, although the results indicate that not all important effects are included in the second order at 54.4eV. More recently, McCarthy and Stelbovics (1983), Bray et al. (1989), and McCarthy (1990) have studied an approximation to the optical potential going beyond the second order and also making allowance for exchange. This method, called the coupled-channel optical model, has recently given remarkable agreement with experiment for e - - Na scattering (Bray et al., 1991), but there are still some discrepancies with the pseudostate methods, discussed earlier, in the case of e - - H scattering. Finally we mention interesting work by Callaway and Oza (1985), who have constructed an optical potential using a set of pseudostates to enable the evaluation of the sum over intermediate states to be carried out. This approach has yielded results for the important n = 1 + 3 and n = 2 + 3 transitions in e - - H scattering (Callaway et al., 1987) but further work needs to be carried out to obtain a firm indication of their reliability. C. BORNSERIESMETHODS
In the “high-energy” domain, which extends from several times the ionization threshold upward, methods based on the Born series are appropriate. The Born series for the direct scattering amplitude can be written as
where the nth Born term,
fBn,
contains the interaction I/ between the
48
P. G. Burke
scattered electron and the target atom or ion n times and the Green’s function GJ describing the propagation of a free electron and a noninteracting target n - 1 times. The first term in Eq. (21), fBl, is the familiar first Born amplitude. Its evaluation and predictions have been discussed in detail by Bell and Kingston (1974). The second Born term, fB2, describing a transition from an to a final state l j ) = lkj,Qj), can be written as initial state li) = 1ki,ai)
where the sum over the intermediate state involves an integration over the continuum. To evaluate this summation some approximations must be made. A useful approximation suggested by Massey and Mohr (1934) is to replace the energy difference w, - wi in the denominator by an average excitation energy iir that enables the summation to be performed by closure. An improvement is to calculate the first few terms in the summation exactly and to treat the remaining terms using closure. A review of various techniques that have been developed was given by Holt and Moiseiwitsch (1968a,b). An important development in recent years has been the recognition of the need to retain consistently all the terms in the Born series with similar energy and momentum transfer A = Iki- kjl dependence to obtain accurate results. Byron and Joachain (1974,1975) showed that it is necessary to include Ref,, as well as the first and second Born terms to obtain the high-energy limit correct through order k;’. Since the third Born term is very difficult to evaluate directly, Byron and Joachain suggested that to the third order, the scattering amplitude should be calculated using the eikonal-Born series (EBS) defined by fEBS = f B l + f B Z + f G 3
+ goch
(23)
wheref,, is the third-order term in the expansion of the Glauber amplitude (Glauber, 1959; Byron and Joachain, 1977) in power series of K The amplitudef,, is easier to calculate thanf,,, and on the basis of an analysis of potential scattering, Byron and Joachain have conjectured that fG3 gives the same result as Ref,, in the limit of a large k for all A. The last term retained in Eq. (23) is an approximate electron exchange contribution gochproposed by Ochkur (1964), which is also of order k;’ for large ki and A and hence must be included for consistency. The EBS method is very successful when perturbation theory converges rapidly; that is, for fast e* scattering from light atoms at not too large scattering angles. If such conditions are not met, some improvements are necessary. These can be obtained by constructing methods that include terms from all orders of perturbation theory to ensure unitarity. One successful
ELECTRON-ATOM SCATTERING THEORY AND CALCULATIONS
49
approach that achieves this, called the unitarized eikonal- Born series (UEBS), has been developed by Byron et al. (1981, 1982, 1985). A comprehensive review of perturbation methods in electron-atom and positron-atom scattering that includes a discussion of the distorted-wave Born series as well as the Born series considered here has been given by Walters (1984).
IV. Illustrative Results A. Low ENERGIES
Over the last 10 to 15 years many results have been obtained by solving the integrodifferential Eqs. (9) using the general program packages mentioned in Section IIB. For light atoms and ions, reliable cross sections can now be obtained in the low-energy region. As an example we show in Fig. 1 the e--He l'S-23S excitation cross
1
5M c
m
06
c
-5
.I .I
u
04
02
t
n=2 15
155
1.6
165
17
175
Electron energy (Ryd)
18
185
19
FIG.1. The e--He, 1'S-2% excitation cross section. Full curve: 29-state calculation; dotted curve: 19-state calculation; chain curve: 11-state calculation; 0: 5-state calculation; 0; experimental data (Brongersma et al., 1972). (From Sawey et al., 1990.)
50
P. G. Burke
section calculated using the R-matrix method by Sawey et al. (1990). Results obtained including the first 5,11,19, and 29 target eigenstates in expansion (8) are shown. The calculations agree close to the threshold but diverge at higher energies. The 29-state calculation, which includes all target eigenstates up to and including the n = 5 states of He, is expected to be accurate, with errors not more than about lo%, up to and including the n = 5 thresholds. These calculations exhibit resonances converging to each of the excited thresholds, which are in excellent agreement with the measurements of Buckman et al. (1983) and Bass (1988). The 29-state calculation also yields cross sections between all excited states of He up to n = 5, which are important in applications. There have also been many calculations of electron-ion excitation cross sections. A survey of many hundreds carried out over the last 15 years using the R-matrix packages has recently been given by Burke and Berrington (1993). Here we just mention a recent calculation of e - - Fe' scattering by Pradhan and Berrington (1993). Fe' is of considerable importance since its lines are observed in many astrophysical spectra. However, the target ion consists of large numbers of closely spaced levels, making it difficult to perform realistic calculations including all relevant states in expansion (8). Recently, using a CRAY Y-MP supercomputer, Berrington and Pradhan have been able to include many more states than earlier work in an attempt both to examine the effect of truncating the expansion and to obtain results for new transitions. Two calculations were carried out: one with 38 states in LS coupling including all 3d7, 3d64s, and 3d64p quartet and septet spin states yielding collision strengths for 703 transitions between LS states; the second using the Breit-Pauli R-matrix program with 41 fine-structure levels, including the lowest four states together with all 3d6 ( 5D)4s, and 4p states, gave a total of 820 fine-structure transitions. These new calculations are the first to include the coupling between the low-lying even parity states to the higher-lying odd parity states. Results show that the extra coupling and resonance structures included have a dramatic effect on the energy dependence of the collision strength for many transitions, even for low-lying levels. This had not been predicted by earlier distorted wave and R-matrix calculations, which had ignored these couplings. AND HIGHENERGIES B. INTERMEDIATE
In this section we shall briefly review recent work on e - - H scattering at intermediate energies. In this case the target states are known exactly, and hence any deviation between theory and experiment must be due to inadequacies in the scattering approximations or experiments.
ELECTRON-ATOM SCATTERING THEORY AND CALCULATIONS
51
We show in Fig. 2 the integrated ls-2s excitation cross section for a range of incident electron energies from the threshold to 4 Ryd. The figure shows the results of calculations carried out using several of the approximations discussed in Section I11 compared with experiment. We see that the ls-2s-2p close coupling calculation overestimates the cross section by more than a factor of two at low energies, while the IERM results and the pseudostate results are in very good agreement with experiment over the whole energy range. This illustrates the critical importance of allowing for loss of flux into highly excited and ionizing channels at these intermediate energies. A more stringent test of theory is to look at the angular distribution of the ;i and R parameters for ls-2p excitation. These parameters are obtained by measuring the scattered electron in coincidence with the decay Lyman-cr photon enabling a test of the magnitude and relative phases of the excitation amplitudes involved in the process to be made (see the review by Andersen et al., 1988). We show in Figs. 3 and 4 the angular distribution of these parameters at 54.42 eV (4 Ryd). The theoretical results, with the exception of the ls-2s-2p
FIG.2. The e - - H , ls-2s excitation cross section. Full curve: IERM results (Scott et al., 1989); open squares: pseudostate (Callaway et al., 1987); six-point stars: ls-2s-2p close coupling (Kingston et al., 1982); full pentagons: experiment (Kauppilaet al., 1970; Long et al., 1968).(From Scott et al., 1989.)
P . G . Burke
52 1.2
1
x .G
.2
t I 0
I A 1
I
I
I
I
1
1
50
I
I
I00
1
I
1
1
'
1
150
Angle FIG.3. The e - - H I parameter for ls-2p excitation at 54.42eV. Theory: full curve, IERM results (Scholz et al., 1991); short dash-dot line, ls-2s-2p close coupling (Kingston et al., 1982); long dash-dot line, pseudostate (Bransden et al., 1985);dotted line, pseudostate (van Wyngaarden and Walters, 1986).Experiment: solid circles (Williams, 1981);solid squares (Williams, 1986); five-point open stars (Hood et al., 1979); triangles (Weigold et a/., 1980); seven-point stars (Slevin et al., 1980). (From Scholz et al., 1991.)
close coupling results, are in good accord with experiment for angles less than 90" but diverge noticeably from experiment for larger scattering angles. Recent work by Bray and Stelbovics (1992b) using a Sturmian expansion, not shown in these figures, also disagree with experiment at large scattering angles, pointing to the need for further experimental work as well as theoretical work in this area.
V. Conclusions In this short review we have presented an overview of electron-atom and electron-ion scattering theory and calculations. At low energies, the development of major computer packages over the last 10 to 15 years based on expansion (8) has meant that reliable results can now be calculated for light atoms and ions. However, more work is necessary before a similar statement
ELECTRON-ATOM SCATTERING THEORY AND CALCULATIONS
0
100
50
53
150
Angle
FIG.4. The e - - H R parameter for 1s-2p excitation at 54.42eV. Symbols in common with Fig. 3 refer to the same data. (From Scholz et al., 1991.)
can be made about collisions involving heavy atomic systems, where relativistic effects are important. This important area of work is now being given considerable attention. Major advances have also been made in recent years in the theory and calculations at intermediate and high energies. In particular, within the last few years accurate results have been obtained for e--H elastic scattering and 1s-2s and 1s-2p excitation at intermediate energies. However, there is still much more work to be done to obtain reliable elastic scattering and excitation cross sections at intermediate energies for complex atoms and ions and for atoms in highly excited states. In conclusion, the continuing need for accurate electron-atom and electron-ion scattering data in many applications will ensure that this field remains a very lively one for many more years to come.
Acknowledgments The author would like to take this opportunity of thanking David Bates for his continued support and friendship and for the stimulating environment he created in the Department of Applied Mathematics at Queen’s University where the author has spent the last 25 years.
54
P. G . Burke REFERENCES
Andersen, N., Gallagher, J. W., and Hertel, I. V. (1988). Phys. Rep. 165, 1. Bass, A. (1988). Ph.D. thesis, University of Manchester. Bell, K. L., and Kingston, A. E. (1974). Adv. Atom. Molec. Phys. 10, 53. Brongersma, H. H., Knoop, F. W. E., and Backx, C. (1972). Chem. Phys. Lett. 13, 16. Bransden, B. H., and Coleman, J. P. (1972). J. Phys. B (At. Mol. Phys.) 5, 537. Bransden, B. H., Scott, T., Shingal, R., and Raychoudhury, J. (1982).J. Phys. B (At. Mol. Phys.) 15, 4605. Bransden, B. H., McCarthy, I. E., Mitroy, J. D., and Stelbovics, A. T. (1985).Phys. Rev. A32,166. Bray, I., Konovalov, D. A., and McCarthy, I. E. (1991). Phys. Rev. A 44 7830. Bray, I., McCarthy, I. E., Mitroy, J., and Ratnavelu, K. (1989). Phys. Rev. A 39, 4998. Bray, I., and Stelbovics, A. T. (1992a). Phys. Rev. Lett. 69, 53. Bray, I., and Stelbovics, A. T. (1992b). Phys. Rev. A 46, 6995. Buckman, S . J., Hammond, P., Read, F. H., and King, G. C. (1983). J . Phys. B (At. Mol. Phys.) 16, 4039. Burke, P. G., and Berrington, K. A. (1993). Atomic and Molecular Processes: An R-Matrix Approach, IOP Publishing Ltd., Bristol. Burke, P. G., Berrington, K. A., and Sukumar, C. V. (1981). J . Phys. B (At. Mol. Phys.) 14, 289. Burke, P. G., and Eissner, W. (1983). In: Atoms in Astrophysics (P.G. Burke, W. B. Eissner, D. G. Hummer, and I. C. Percival, eds), Plenum Press, New York. Burke, P. G., Hibbert, A., and Robb, D. W. (1971). J . Phys. B 4, 153. Burke, P. G., and Mitchell, J. F. B. (1973). J . Phys. B (At. Mol. Phys.) 6, 665. Burke, P. G., Noble, C. J., and Scott, M. P. (1987). Proc. R. SOC. A 410, 289. Burke, P. G., and Schey, H. M. (1962). Phys. Rev. 126, 147. Burke, P. G., and Webb, T. G. (1970). J . Phys. B (At. Mol. Phys.) 3, L131. Byron, F. W., Jr., and Joachain, C. J. (1974). J . Phys. B (At. Mol. Phys.) 7, L212. Byron, F. W., Jr., and Joachain, C. J. (1975). J. Phys. B 8, L284. Byron, F. W., Jr., and Joachain, C. J. (1977). Phys. Rep. 34, 233. Byron, F. W., Jr., and Joachain, C. J. (1981). J . Phys. B (At. Mol. Phys.) 14, 2429. Byron, F. W., Jr., Joachain, C. J., and Potvliege, R. M. (1981). J. Phys. B ( A t . Mol. Phys.) 14, L609. Byron, F. W., Jr., Joachain, C. J.,and Potvliege, R. M. (1982).J. Phys. B ( A t . Mol. Phys.) 15,3915. Byron, F. W., Jr., Joachain, C. J., and Potvliege, R. M. (1985).J. Phys. B (At. Mol. Phys.) 18,1637. Callaway, J., and Oza, D. H. (1985). Phys. Rev. A 32, 2628. Callaway, J., Unnikrishnan, K., and Oza, D. H. (1987). Phys. Rev. A 36, 2576. Callaway, J., and Wooten, J. W. (1973). Phys. Lett. A 45, 85. Callaway, J., and Wooten, J. W. (1974). Phys. Rev. A9, 1924. Callaway, J., and Wooten, J. W. (1975). Phys. Reo. A 11, 1118. Chang, J. J. (1975). J . Phys. B (At. Mol. Phys.) 8, 2327. Damburg, R., and Karule, E. (1967). Proc. Phys. SOC.90, 637. Feshbach, H. (1958). Ann. Phys. (NY) 5, 357. Feshbach, H. (1962). Ann. Phys. (NY) 19, 287. Glauber, R. J. (1959). In: Lectures in Physics, Vol. I (W. E. Brittin, ed), Wiley Interscience, New York. Holt, A. R., and Moiseiwitsch, B. L. (1968a). Adu. Atom. Molec. Phys. 4, 143. Holt, A. R., and Moiseiwitsch, B. L. (1968b). J . Phys. B. (At. Mol. Phys.) 1, 36. Hood, S. T.,Weigold, E., and Dixon, A. J. (1979). J. Phys. B (At. Mol. Phys.) 12, 631. Jones, M. (1975). Phil. 7hans. Roy. SOC.A277, 587. Kauppila, W. E., Ott, W. R., and Fite, W. L. (1970). Phys. Rev. A 1, 1099.
ELECTRON-ATOM SCATTERING THEORY AND CALCULATIONS
55
Kingston, A. E., Liew, Y. C., and Burke, P. G. (1982). J. Phys. B (At. Mol. Phys.) 15, 2755. Le Dourneuf, M., Launey, J. M., and Burke, P. G. (1990). J. Phys. B (At. Mol. Opt. Phys.) 23, L559. Long, R. L., Jr., Cox, D. M., and Smith, S. J. (1968). J.R.N.B.S.A. 72, 521. Massey, H. S. W., and Mohr, C. B. 0. (1934). Proc. R o y . Soc. A 146, 880. McCarthy, I. E. (1990). Comm. Atom. Molec. Phys. 24, 343. McCarthy, I. E., and Stelbovics, A. T. (1983). Phys. Reu. A 28, 2693. Nesbet, R. K. (1980). Variational Methods in Electron Atom Scattering Theory, Plenum Press, New York. Norrington, P. H., and Grant, I. P. (1981). J. Phys. B (At. Mol. Phys.) 14, L261. Ochkur, V. I. (1964). Sou. Phys. J E T P 18, 503. Percival, I. C., and Seaton, M. J. (1957). Proc. Camb. Phil. Soc. 53, 654. Pradhan, A . K., and Berrington, K . A. (1993). J. Phys. B ( A t . Mol. Opt. Phys.) 26, 157. Rotenberg, M. (1962). Ann. Phys. (NY) 19, 262. Saraph, H. E. (1972). Comp. Phys. Commun. 3, 256. Saraph, H. E. (1978). Comp. Phys. Commun. 15, 247. Sawey, P. M. J., Berrington, K. A,, Burke, P. G., and Kingston, A. E. (1990). J. Phys. B (At. Mol. Opt. Phys.) 23, 4321. Schotz, T. T., Walters, H. R. J., Burke, P. G., and Scott, M. P. (1991). J. Phys. B (At. Mol. Opt. Phys.) 24, 2097. Scott, M. P., Scholz, T. T., Walters, H. R. J., and Burke, P. G. (1989). J. Phys. B (At. Mol. Opt. Phys.) 22, 3055. Scott, N. S., and Burke, P. G. (1980). J . Phys. B. (At. Mol. Phys.) 13, 4299. Scott, N. S., and Taylor, K. T. (1982). Comp. Phys. Commun. 22, 467. Seaton, M. J. (1974). J . Phys. B. ( A t . Mol. Phys.) 7 , 1817. Slevin,J., Eminyan, M., Woolsey, J. M., Vassiler, G., and Porter, H. Q. (1980). J. Phys. B (At. Mol. Phys.) 13, L341. Smith, E. R., and Henry, R. J. W. (1973a). Phys. Rev. A 7, 1585. Smith, E. R., and Henry, R. J. W. (1973b). Phys. Rev. A 8, 572. van Wyngaarden, W. L., and Walters, H. R. J. (1986). J. Phys. B (At. Mol. Phys.) 19, 929. Walker, D. W. (1974). J. Phys. B. (At. Mol. Phys.) 7, 97. Walters, H. R. J. (1984). Phys. Rep. 116, 1. Weigold, E., Frost, L., and Nygaard, K. J. (1980). Phys. Rev. A 21, 1950. Williams, J. F. (1981). J. Phys. B (At. Mol. Phys.) 14, 1197. Williams, J. F. (1986). Aust. J. Phys. 39, 621.
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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 32
TERRESTMAL AND EXTRATERRESTRIAL
H3+
ALEXANDER DALGARNO Hurvurd-Smithsonian Center for Astrophysics Cambridge, Massachusetts
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Terrestrial H S f . . . . . . . . . . . . . . . . . . . . . . . . . 111. Extraterrestrial H,+ . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Jovian Planets B. Jupiter Magnetosphere . . . . . . . . . . . . . . . . . . . . C. Interstellar Space . . . . . . . . . . . . . . . . . . . . . . . D. Starburst Galaxies . , . . . . . . . . . . . . . . . . . . . . E. Supernova 1987A . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
. . .
. . . . . .
.
51 58 60 60 62 62 65 65 66 66
I. Introduction The molecular ion H3+ is the simplest stable polyatomic molecular system. It consists of two electrons and three protons. In the ground state, the protons occupy the vertices of an equilateral triangle. It is the major ion produced by high-pressure electric discharges in hydrogen gas. Because of the simplicity of its structure, it has been a testing ground for theoretical methods of calculating potential energy surfaces and rotation-vibrational energy levels of polyatomic systems. The development of a quantitative picture of the structure of H 3 + has occurred through a dynamic interplay of experimental and theoretical studies. These basic studies were crucial to the identification of H 3 + infrared emission in the spectra of the Jovian planets and must lead eventually to the detection of H 3 + in absorption in the interstellar medium, where it occupies a central role in interstellar chemistry. They have led to a tentative identification of H 3 + in the envelope of supernova 1987a. 57 Copyright 0 1994 by Academic Press, Inc. All righls of reproduction in any form reserved. ISBN 0-12-W3832-3
A . Dalgarno
58
11. Terrestrial H3+ The early history of experiments on H3+ beginning with its discovery by J. J. Thomson (1911, 1912) has been described by Oka (1983, 1992). In the discharge, the H3+ ions are produced by the conversion of the initial H 2 + ions by the fast ion-molecule reaction: Hz++H2+H3++H
(1)
(Hogness and Lunn, 1925; Stevenson and Schissler, 1958; Varney, 1960 Barnes, Martin and McDaniel, 1961; Albritton et al., 1968; Miller et al., 1968). Martin, McDaniel and Meeks (1961) and Stecher and Williams (1969, 1970) pointed out that the same process operates in interstellar clouds so that H3+ will be present there to the near exclusion of H2+.Reaction (1) is similarly effective in the ionospheres of the Jovian planets (Bauer, 1973; Altreya, Donahue and McElroy, 1974; Atreya, 1986; Cravens, 1987; McConnell and Majeed, 1987; Majeed and McConnell, 1991; Kim, Fox and Porter, 1992). The laboratory studies on ion mobilities in hydrogen (Albritton et al., 1968; Miller et al., 1968) also pointed to the three-body reactions
+ + + H, H+ + H, + H + H 3 + + H
€I+ H2 H 2 + H 3 +
(21
(3)
as additional sources of H3+ ions. The reaction of H + with vibrationally excited H, molecules in vibrational levels v 2 4,
H+
+ Hz(v
4) + H
+ HZ
+
(4)
followed by reaction (l), may have contributed to the conversion of H + to H3+. McElroy (1973) suggested that reaction (4) plays a similar role in the Jovian atmospheres in addition to reactions (2) and (3) (Dalgarno, 1972). H, has been the subject of repeated quantum-mechanical calculations. Anderson (1992) listed 40 references to theoretical studies of the electronic potential energy surfaces published since the first analytic variational calculations of Hirschfelder (1938). The availability of an accurate potential energy surface stimulated theoretical efforts to develop methods for the determination of the energy levels of complex molecules. Carney and Porter (1974, 1976) obtained a value of 2516cm-' for the frequency of the v,-fundamental band of H3+, which is within 5 cm- of the experimental value. Calculations with increasing +
'
TERRESTRIAL AND EXTRATERRESTRIAL H,'
59
accuracy have been presented since the studies of Carney and Porter, most recently by Spirko et al., (1985), Tennyson and Sutcliffe (1986), Whitnell and Light (1989), Tennyson and Henderson (1989), Bartlett and Howard (1990), Carter and Meyer (1990), Henderson, Miller and Tennyson (1990), Day and Truhlar (1991), Henderson, Tennyson and Sutcliffe (1992), Carter and Meyer (1992) and Lie and Frye (1992).The frequency predicted by Lie and Frye is 2520.7cm-', which differs by -0.6cm-' from the experimental value of 2521.31 cm-'. The discrepancy may be due to adiabatic corrections to the potential energy surface. There remain formidable difficulties in predicting accurately the positions of the high-lying rotation-vibration levels and in determining the density of states, but considerable progress has been made using quantum-mechanical (Henderson and Tennyson, 1990; BaEiC and Zhang, 1991, 1992) and semi-classical methods (Berblinger et al., 1992). Accurate calculations of the partition function and equilibrium constant of reaction (1) have been carried out (Chandra, Gaur and Pande, 1991; Sidhu, Miller and Tennyson, 1992; Gaur, Pande and Chandra, 1992). The intense theoretical activity was stimulated by laboratory investigations. The infrared absorption spectrum of H 3 + was detected by Oka (1980) using a tunable laser infrared source and analyzed by J. K. G. Watson (see Oka 1992). Because of the large interaction between the rotational and vibrational motion, the identification of the spectral lines was far from obvious. There have been many subsequent investigations (see Oka 1992)and an extensive table of absorption lines has been presented by Kao et al. (1991). Additional levels are given by Dinelli, Miller and Tennyson (1992). The measurements of absorption lines have been extended to the higher lying bands (Bawendi, Rehfuss and Oka, 1990; Xu, Gabrys and Oka, 1990; Lee et al., 1991; Xu et af., 1992). Transition probabilities and line strengths have been presented by Pan and Oka (1986), Kao et al. (1991) and Dinelli et al. (1992). Information about high-lying levels of H, near the dissociation limit has been obtained in a remarkable series of experiments by Carrington, Buttenshaw and Kennedy (1982), Carrington and Kennedy (1984) and Carrington and McNab (1989). They observed an infrared spectrum of H3+ between 872cm- and 1094cm- that contains nearly 27,000 lines. They measured the fragment H + ions and attributed the spectrum to transitions from discrete and quasi-bound rotation-vibrational levels of H, + into predissociating levels that separate to H + + H,. The H 3 + ion has also been seen in emission in the laboratory in the fundamental and overtone bands (Majewski et al., 1987,1989) not long before its observation in the atmosphere of Jupiter. +
'
60
A . Dalgarno
111. Extraterrestrial H3' A. THEJOWANPLANETS Emission lines of H,+ in the 20,(2) + 0 overtone band of H, are present in the spectrum of Jovian aurora taken by Trafton, Lester and Thompson (1989) on September 21, 1987, although the identification of them was not made until later. They are present and correctly identified in the spectrum of Drossart et al. (1989).The history of the identification has been summarized by Oka (1992).The intensity distribution of Drossart et al. (1989)is consistent with an effective rotational temperature of 1100 f 100 K (Kim et al., 1990). Oka and Geballe (1990) discovered the fundamental band of H,' in the spectrum near 4p and derived a rotational temperature of 670 f 100K. Maillard et al. (1990) also measured the emission lines of the fundamental band near 4p and derived rotational temperatures of 1100 & 40K and 835 f 50K for the southern and northern zones, respectively. Miller, Joseph and Tennyson (1990) measured the emission in the 2.1p and 4.0~ regions in the overtone and fundamental bands and obtained a vibrational temperature of 1100 & 100K. Maillard et al. and Miller et al. noted that the level populations were close to thermal equilibrium at a temperature near 1000K. Recently Drossart et al. (1992) have obtained a translational temperature of 1150 f 60K from the widths of the H 3 + lines. The appearance of emission features over a broad range of infrared wavelengths at which imaging with an infrared camera can be carried out offers a marvelous opportunity for the observational study of auroral morphology in space and time (Kim et al., 1991; Baron et al., 1991; Drossart, PrangC and Maillard, 1992; Billebaud et al., 1992). Emission from H 3 + has also been detected from Uranus (Trafton, Geballe and Miller, 1992) and Saturn (Geballe, Jagod and Oka, 1992). The physics of the H 3 + emission has been discussed in detail by Kim, Fox and Porter (1992). During the auroral bombardment, H2 and H + ions are produced by the impact of energetic electrons. They are transformed into H 3 + ions by reactions (1)-(4). The H 3 + ions are removed by dissociative recombination +
+
H3++ e + H ,
+H
+H+H+H The process is critical to determinations of the abundance of H 3 + in the Jovian atmospheres, in interstellar clouds and in the envelopes of supernovae.
TERRESTRIAL AND EXTRATERRESTRIAL H 3 +
61
The rate coefficient for dissociative recombination of H 3 + has had an interesting history. The earliest measurements (Leu, Biondi and Johnsen, 1973) of recombination in a stationary afterglow yielded a rate coefficient of 2.3 x 10-'cm3s-' at 300K. Similar values were obtained in measurements using inclined beams (Peart and Dolder, 1974), merged beams (Auerbach et al., 1977; McGowan et al., 1979) and ion traps (Mathur, Khan and Hasted, 1978). The rate coefficient was measured again in a pulsed afterglow over a wide range of temperatures and a rate coefficient varying between 1.6 x 10-7cm3s-' at 240K and 1.2 x 10-7cm3s-' at 50K was obtained (Macdonald, Biondi and Johnsen, 1984). Then Adams, Smith and Alge (1984) employed a flowing afterglow Langmuir probe apparatus and were unable to detect dissociative recombination. They gave for the rate coefficient an upper limit at 300K of 2 x 10-*cm3s-', later reduced to 10-"cm3s-' (Adams and Smith, 1987, 1988). Their low vaIue received support from theoretical calculations of the potential energy surfaces of H3 (Kulander and Guest, 1979; Michels and Hobbs, 1984),which showed that none exists that would facilitate dissociative recombination of H, ions in the ground vibrational state. The discrepancy between the previous measurements and those of Adams, Smith and Alge was attributed to the presence of vibrationallv excited H, or to contaminant ions like CH,+ (Johnsen, 1987). A dependence on vibrational level popution was found in merged beam experiments of Hus et al. (1988). However, Amano (1988,1990), exploiting the growing understanding of the absorption spectrum of H, +,carried out an experiment using an infrared absorption technique, in which the recombination of H 3 + ions in the u = 0 level was measured. He obtained a rate coefficient of 1.8 x lo-' cm3s-l at 300K. The interpretation of his data was subject to some uncertainty because of possible contributions from collisional-radiative recombination at high electron densities, but the objections (Adams and Smith, 1989) have been answered by Bates, Guest and Kendall (1993). New experiments using a modified flowing afterglow technique have been carried out by Canosa et al. (1992), who obtained large coefficients in agreement with the results of Amano (1988, 1990) and earlier experimenters. Finally, what seems to be overwhelming confirmation of a rate coefficient on the order of 10-7(300/T)'/2 cm3 s-' has come recently from merging beam cross-section measurements using a storage ring (Larsson et al., 1993), the long lifetime of the mass-selected ions before recombination ensuring that only the u = 0 level is populated. The basic question of the mechanism responsible for the rapid dissociative recombination remains to be answered. It does not proceed through a crossing of potentials. However, Bates (1992) has suggested a mechanism in which curve crossing is not mandatory. In it the electron is captured into a +
+
62
A . Dalgarno
Rydberg state, followed by a sequence of transitions into a higher lying vibrational level of some Rydberg state from which predissociation to the repulsive potential of H, is effective. A more detailed model of Bates, Guest and Kendall(l993) offers some support for the mechanism, but the story is not yet over. B. JUPITERMAGNETOSPHERE
The first detection of extraterrestrial H3+ ions was not by observation of its electromagnetic spectrum but by discovery of an energetic mass 3 system in Jupiter's magnetosphere by the low-energy particle telescope carried on Voyager 2 (Hamilton et al., 1980,1981).The discovery was a clear indication of the presence of H3+ in the Jupiter ionosphere, providing a source for the magnetospheric ions. The mechanism that accelerates the ions presents a challenging problem in plasma physics.
SPACE C. INTERSTELLAR The H3+ ion occupies a central place in the ion-molecule chemistry of interstellar clouds (Herbst and Klemperer, 1973). Following its production by cosmic ray ionization to produce H2+ and the conversion of H2+to H,+ by reaction (l), H3+ may react with heavy atoms like oxygen and carbon to initiate reaction sequences such as
+ Hz O H + + H2 +H,O+ + H H 2 0 + + H, + H 3 0 + + H H30++ e-, H20+ H H3+ + 0 +OH+
+OH+H2 and H3+ + C
CH+ + H,
CH++Hz+CHz++H CH2+ + H,
+H +H
+ CH3+
CH3+ + e + C H ,
+CH+H,
(7)
63
TERRESTRIAL AND EXTRATERRESTRIAL H,'
The laboratory data have enabled searches to be made for H, in interstellar clouds by looking for its absorption of radiation from infrared sources, but it has so far escaped detection (Oka, 1981; Geballe and Oka, 1989; Black et al., 1990). A measurement of its abundance would provide a measure of the cosmic ray ionizing flux in interstellar clouds. Thus if Cs-l is the ionizing flux, the production rate of H,' is 1.7[n(Hz) cm-'s-', where n(H,) is the number density of H, molecules and the additional 0.7 takes account of secondary ionization. The H, ions are removed by reactions with neutral constituents of the gas, but mostly with 0 and CO, +
+
H3+ + O + O H +
+ H,
H3++CO+HCO++Hz We define an effective loss rate coefficient by
(9) (10)
where n(x) is the number density of constituent x and k, is the rate coefficient for the reaction of H 3 + with x. The H 3 +ions are also removed by dissociative recombination (5) and (6) at a rate ane s- where c1 is the rate coefficient for dissociative recombination. Then in equilibrium
',
n (H3 +) = C/(E + mne/n(HJ)
(11)
Black et al. (1990) have shown how estimates may be made of the fractional electron density and the depletion of oxygen and carbon so that useful limits to C can be obtained from the upper limits to the H 3 + column densities. If H z D + and H 3 + were detected in the same location, the uncertainties in formula (1 1) could be removed. The deuterated ion H z D + is produced by the reaction
H,+
+ HD + H z D + + H,
and removed by the reverse reaction
H z D + + H, 3 H 3 + + H D and the analogs of (9) and (10)
H z D + + 0 + OH+
+HD + O D + + H, H z D + + CO HCO" + H D + DCO+ + H, -t
(12)
A . Dalgarno
64
and by dissociative recombination
+D
(16)
+HD+H
(17)
HzD+ + e + H 2
+H+H+D (18) If we adopt the same rate coefficients for H2D+as for H 3 + ,we may write for the equilibrium
where k,, and k,, are the rate coefficients of the corresponding reactions. Thus
n(H2D+) =-n(HD)
W,+)
n(H2)
where the factor f is an enhancement factor,
f = k 1 3 + E +k12un,/n(H2) a result first given by Watson (1976). From it, we derive the relationship
so that expression (1 1) gives for
5,
The rate coefficients k12 and k I 3 are known functions of the temperature (Smith, Adams and Alge, 1982; Herbst, 1982; Sidhu, Miller and Tennyson, 1992). The rate coefficient k,, for the endothermic reaction (3) becomes very slow at low temperatures and may be ignored in very cold clouds, where the simple formula C = k,,n(H,+)/f is applicable. The chemistry must be modified to take into account the atom exchange reaction (Dalgarno and Lepp, 1984)
D
+ H 3 + + H + HzD'
because of the supply of deuterium atoms from the dissociative recombinations (16) and (18) of the enhanced H2D+ions. A further source is
DCO'
+ e + D + CO
TERRESTRIAL A N D EXTRATERRESTRIAL H3+
65
A tentative detection of H 2 D + has been reported toward NGC 2264 (Phillips et al., 1985) but not confirmed (van Dishoeck et al., 1992; Pagani et al., 1992). An absorption feature toward the infrared source IRC2 in Orion has been found by Boreiko and Betz (1993) and attributed by them to para H,D+. Upper and lower limits to the fractional abundance of H3+ in several clouds have been estimated by van Dishoeck et al. (1992). It seems that the actual detection of interstellar H3+ cannot be long delayed.
GALAXIES D. STARBURST
NGC 6240 is an ultraluminous infrared-bright galaxy, apparently undergoing a period of intense star formation. Very strong emission is seen from excited rotation-vibrational levels of H,. Draine and Woods (1990) have suggested the emission originates in molecular clouds subjected to transient X-radiation and have pointed out that absorption of the X-rays may lead to significant emission from the rotation-vibrational levels of H, +.The X-rays heat the gas and ionize it. Their calculations assumed that dissociative recombination of H 3 + is slow so that their calculated intensities are overestimates. However, they raise the intriguing possibility that H, emission may be a unique diagnostic of X-ray irradiated molecular gas. +
E. SUPERNOVA 1987A There are two strong unidentified emission features at 3.41 pm and 3.53 pm in the infrared spectrum of Supernova 1987A at day 192 after the explosion of a blue supergiant star in the Large Magellanic Cloud (Meikle et al., 1989). They can be matched by a thermal emission spectrum of H,+ at a temperature of about 1000-2000 K (Miller et al., 1992). No other convincing identification has yet been advanced. The envelope chemistry is similar to that of the early universe, supplemented by reactions
+e H3++ e
H(n = 2) + H -+ H H(n
= 2)
+ H, -+
+
involving hydrogen atoms in the n = 2 excited states. Reactions involving more highly excited states may also contribute to the formation of H 3 + .The associative ionization of H(2s) colliding with H( 1s) has been investigated experimentally and theoretically (Urbain et al., 1991) and a value of the rate coefficient is available.
66
A . Dalgarno
A detailed model of the chemistry of the envelope (Miller et al., 1992; Yan, Lepp and Dalgarno, 1993) in which clumping has occurred appears to be successful in producing the inferred amount of H 3 + at day 192, but the temperature of less than 2000K is lower than the model predicts.
Acknowledgments I am greatly indebted to the National Science Foundation, Division of Astronomical Sciences, and to the National Aeronautics and Space Administration for their support of my research. I am grateful to T. Oka and L. J. Lanzerotti for valuable criticism of an earlier draft.
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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 32
INDIRECT IONIZATION OF POSITWE ATOMIC IONS K . DOLDER Departmen1 of Physics The University of Newcastle upon Tyne Newcastle upon Tyne. England
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Investigations of Isoelectronic Series . . . . . . . . . . . . . . . C. Investigations of Isonuclear Series . . . . . . . . . . . . . . . D. Multiple Ionization . . . . . . . . . . . . . . . . . . . . . E. Measurements with Energy-Resolved Electrons . . . . . . . . . F. Measurements of the Photoionization of Positive Atomic Ions . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 11. Ionization Processes .
. . . . . . . . . . . . . . 111. Experimental Approaches . . . . . . . . . . . IV. Some Experimental Results. . . . . . . . . . . A. Experiments with Very Highly Charged Ions. .
. . . . . . . .
. . . . . . . .
. . . .
. .
69 71 72 74 74 77 79
80 84 87 91
I. Introduction At first sight it might seem that ionization is a relatively uncomplicated collision process. A projectile strikes an atomic target and one or more electrons are ejected. The topic has been addressed for at least 80 years, and in 1913 J. J. Thomson devised a simple, two-dimensional classical theory. With the advent of quantum mechanics, approximations of increasing sophistication were developed but experiments have revealed that the ionization of many targets is truly awesome in its complexity. The wealth of minute detail already uncovered is beyond the reach of contemporary theory, and one can be sure that studies of ionization will continue to occupy experimentalists and theoreticians for at least the next two generations. This review will consider ionization from the experimental standpoint, and we will not attempt to mention all the measurements that have been made. Instead, our discussion will be confined to experiments in which the targets were positive atomic ions, and it will be focussed mainly on recent results 69 Copyright 0 1994 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003832-3
70
K. Dolder
obtained by four different experimental techniques that are, to an appreciable extent, complementary. The methods all employ intersecting beams. The first crossed beam measurement of ionization of a positive ion by electron impact was made more than 30 years ago (Dolder et al., 1961), but it was an experiment with Ba+ (Peart and Dolder, 1968) that revealed that indirect processes can sometimes dominate an ionization cross section to a surprising extent. Results obtained for Be' (Falk and Dunn, 1983), Mg+ (Martin et al., 1968) and for Ca', Sr', and Ba+ (Peart and Dolder, 1975) are illustrated in Fig. 1, where it can be seen that cross sections for the three heavier ions all display sudden, steep jumps. These were attributed (Hansen, 1975) to np + nd inner shell transitions that decayed, not by radiation, but by ejecting a more weakly bound electron from the outer shell. These were early examples of indirect ionization, and it has since been shown that these processes are diverse and extremely complicated. It is with these complications that we will be concerned.
Electron energy IeW
FIG.1. Measured cross sections for electron impact ionization of Be', Mg+, Ca+, Sr+ and Ba'. Cross sections of the three heavier ions show sudden increases due to the onset of indirect ionization.
INDIRECT IONIZATION OF POSITIVE ATOMIC IONS
71
11. Ionization Processes Let us start by considering a relatively simple example: the ionization of Mg+ by electron impact. Mg' has a Na-like structure that is susceptible to direct outer-shell ionization, e + Mg+(2p63s) -+ Mg2+(2p6)+ e
+e
(1)
and we must also consider inner shell ionization e
+ Mg+(2p63s)
-+
Mg2+(2p53s) + e + e
(2)
We have already noted the possibility of excitation autoionization (EA), e
+ Mg+(2p63s)
4
Mg+(2p53snf) + e -+ Mg2+(2p6)+ e + e
(3)
but we should also include two processes in which the incident electron initially combines with the target ion. If it is to combine, a fast projectile must first lose energy by exciting the target, and since it is to become trapped in a discrete quantum state, the process must be sharply resonant. This leads to the formation of neutral magnesium in a doubly excited state that may then decay by resonant-excitation-double-autoionization (REDA), e.g., e
+ Mg+(2p63s)
+
Mg(2p' 3s nln'l') -+ Mg+(2p53s 3p) + e
1
+
Mg2+(2p6) e
(4)
or in a single stage by resonant-excitation-auto-double-ionization (READI), e.g., e + Mg+(2p63s) -+ Mg(2pS3s2nI) -+ Mg2+(2p6)+ e + e
(5)
There is some debate about the extent to which these two processes are truly distinct, but they are both related very closely to dielectronic recombination (DR), in which a doubly excited neutral atom is also formed but it then decays radiative1y. We see that for a simple ion such as Mg+ at least five channels could lead to ionization, and they may compete and interfere. Even this list may be incomplete. But we have, as yet, considered only single ionization. A number of experiments have investigated multiple ionization, where more than one electron is ejected. These produced evidence of other mechanisms. For example, in the case of the double ionization of Ba+ we expect the direct ejection of two electrons, e.g., Ba+(4dIo5s2 5p6 6s) + e -+ Ba3+(4d" 5s2 5p5)
+ 3e
(6)
72
K. Dolder
or autoionization following ionization of the 4d subshell, Ba'(4d''
5s2 5p66s)
+ e -+
+ 2e Ba3+(4d1'5s' 5p5) + e
Ba2+(4d95s2 5p66s)
1
(7)
Alternatively, double ionization might proceed by the two-stage process, Ba+(5sZ5p6 6s) + e -P Ba2+(5s5p66s)
+ 2e
+
Ba3+(5s25p5)+ 3e
(8)
There is also experimental evidence for reactions of the type, Ba'(4d''
5s2 5p66s)
+ e + Ba+(4d95s' 5p66s nl) + e 4
Ba2+(4d1'5s 5p66s) + e
1
(9)
+
Ba3+(4d" 5s' 5p5) e and also by 4d excitation followed by auto-double ionization, where excited Ba+(4d95s' 5p66s nl) decays to Ba3+(4d1' 5s' 5p5)and two electrons. Miiller et al. (1988) have already seen narrow peaks in the multiple ionization cross sections of several heavy metal ions that they attributed to triple (RETA), quadruple (REQA) and even pentuple (REPA) autoionization. Even more channels will no doubt be identified in due course. This is sufficient to illustrate the complexity of ionization, and it is high time to turn to experimental matters.
111. Experimental Approaches Our discussion will be confined to four techniques that have been developed to study indirect ionization. All employ intersecting beams; three of the methods were developed for electron-ion collisions and the fourth for photon-ion interactions. Details of crossed electron-ion beam experiments have been reviewed so frequently (e.g., Dolder and Peart, 1976, 1986; Dunn, 1985; Miiller, 1991a) that they need not be repeated here. Briefly, well-collimated beams of ions and electrons are made to intersect and collision products are separated from their parent ions in electric or magnetic fields. The three developments of this technique that concern us originated principally in the laboratories of Oak Ridge, Giessen and Newcastle. The photon-ion experiments (e.g., Lyon et al., 1986a) were somewhat similar to those with electrons but the target ions moved in confluence with a beam of resolved VUV synchrotron radiation from the Daresbury storage ring. Oak Ridge Laboratory has pioneered the use of large, sophisticated ion sources to extend measurements to very highly charged ions such as Fe'" and U16+. This is an important development because this type of ion is
INDIRECT IONIZATION OF POSITIVE ATOMIC IONS
73
encountered in thermonuclear and astrophysical plasmas and it also enables one to trace indirect ionization processes along isoelectronic (e.g., Li, Be+, B2+, C3+)and isonuclear (e.g., Fe, Fe’,. . . ,Fe15+)series. Much of this work is at the very limit of experimental technique, and it naturally has its limitations. Many ions cannot be produced exclusively in their ground states so that interpretation of the results may sometimes be a little ambiguous and the accuracy and detail of the results are limited because only weak beams of very highly charged ions can be produced. Moreover, it is, for example, much more difficult to resolve say, Fe16+ from Fe15+ than Fez+ from Fe+. Nevertheless, the results are extremely impressive and techniques continue to advance. The experimental storage ring of the GSI at Darmstadt should soon operate, and this might even facilitate the ultimate experiment-the ionization of Ugl +. We saw in Section I1 that some indirect ionization processes are sharply resonant, and so they are revealed as very narrow structures in the ionization function. Resolution of this structure throws light on the nature of the transitions, and the chief obstacle to interpreting these measurements is the limited resolution, due mainly to the spread of energies in the electron beam. Space charge within electron beams causes the potential at the axis to be less than at the periphery and this potential gradient is undesirable for three reasons. First, it causes the beam to spread so that its geometry is ill defined; second, the electrons act as a lens for the ion beam and may therefore interfere with its collection but, more relevant to the present discussion, is the loss of energy resolution in the cross-section measurements. The energy spread is typically between 1 and 2eV. An ingenious remedy was developed by Salzborn and his colleagues at Giessen (e.g., Muller et al, 1985; Tinschert et al., 1988). Their apparatus uses unusually high electron currents (z15 mA) in which the potential gradients would normally be unacceptably large, but they inject slow K r + ions into the beam to partially neutralize the space charge. This permits resolution of structures only 400meV wide, although the energy spreads in the beams are significantly larger. The technique has three valuable advantages. The ability to work with large electron currents enables data to be collected rapidly, and experiments can be performed with highly charged ions that can be produced only in tenuous beams. Moreover, extremely small structures with amplitudes little more than 0.01% of the total cross section have been detected (Muller et al., 1989). Limitations of the method appear to be the presence of metastable ions in some of the target beams, although these can be minimized by using a “cold” Penning ion source (e.g., Tinschert et al., 1991). There are also uncertainties about the profile of the electron energy distributions and the absolute energy scale. Tinschert et al. gave some numerical values typical of the method and stated that the “energy calibration is within deviations of not more than about 1 eV.”
74
K . Dolder
A different approach has been developed in this laboratory. The first energy-resolved measurements for a positive ion were made by Peart et al. in 1973. They replaced the electron gun in a conventional crossed beam apparatus by a 127”electrostatic energy selector. More recently (e.g., Peart et al., 1989) a double selector has been used since this gives a “cleaner,” near Gaussian energy distribution some 100meV wide (FWHH). The method has unique advantages. The profile and width of the energy distribution are known and the energy scale is absolute to within 40meV. The cross sections are absolute and it has been possible, so far, to work only with ions that are initially unexcited. These advantages permit the closest comparisons with the theory, and the method was designed with this in mind. But a heavy price is paid. The electron currents are only about lo-’ A, so that data collection is slow and tedious, and the method does not lend itself to experiments with weak beams of multiply charged ions. We see that, compared with the Giessen method, a fivefold gain in resolution has cost five orders of magnitude in current or signal. But the ability to define the energies of transitions very accurately and, by deconvolution, to examine their profiles, provides the closest comparisons with theory. With present techniques it is difficult to envisage electron-ion experiments with resolution much better than 100meV and one must turn to the photoionization of ions where optical methods are available. These experiments have given resolutions as high as 4meV. Only a limited number of ions have so far been studied and there is clearly enormous scope for more experiments. As more intense synchrotron sources become available it will be possible to make measurements in the grazing incidence region (S300 A) and to study ions that are relevant to astrophysics or even those that have more than one initial charge. C + and Mg+ are obvious candidates for future study.
IV. Some Experimental Results It is impossible to discuss all the published measurements but we will endeavor to cite the most recent work, and this will indicate where references to earlier experiments can usually be found. A. EXPERIMENTS WITH VERYHIGHLYCHARGED IONS
More details on this topic can be found in recent reviews by Miiller (1991a, b, c), who noted that ion source developments already point toward the production of beams of U g l +that may eventually be adequate for crossed beam experiments, but the most highly charged ions for which results are presently available are Fe’” and UI6+ (Gregory et al., 1987; 1990).
INDIRECT IONIZATION O F POSITIVE ATOMIC IONS
75
Relativistic effectsassume much greater importance in the atomic structure and collision process of highly charged ions. For example, elementary, nonrelativistic classical scaling predicts the same scaled cross sections for all hydrogenlike ions (He', Li2+,. . . , U91+), but Moores and Pindzola (1990) calculated that relativistic effects increase the scaled cross section of U91 about fivefold. Great interest centered on Fe15+ following a calculation by LaGattuta and Hahn (1981) that predicted the existence of REDA and indicated that it should substantially increase the ionization cross section. More elaborate calculations have now been made by Chen et al. (1990), and it can be seen in Fig. 2 that, when contributions from EA and REDA are added to those for direct ionization, the results agree very well with the measurements by Gregory et al. (1987). An example of a more complex, highly charged ion is U16+,for which cross sections are illustrated in Figs. 3(a) and 3(b). The target ions in these experiments were believed to be mainly in the metastable state 4f14 5s' 5p6 5d7 6s and the dashed curve shows results of a semiempirical Lotz estimate of single ionization from that configuration. The disparity +
600
800 Energy (eV)
1000
FIG.2. Ionization of FeI5+.Measurements by Gregory et al. (open circles) compared with calculationsby Chen et al. The dashed curve is a theoretical estimate of direct ionization and the lower solid curve shows the effect of adding EA. The upper solid curve includes the contributions from direct ionization, EA and REDA.
K. Dolder
76 4
/--.3.5
m
E
0
3
OD I
9
2.5
v C
.-0
2
c
U Q
1.5
v) v)
1
v,
e
V
0.5 0 200
1000 Electron Energy (eV)
(a)
1 i
I
+ u'6+
...
u'7' +
2e
2.7 510
520
530
540
550
Electron Energy (eV)
(b)
FIG. 3. (a) Experimental and theoretical cross sections for single ionization of U16'. The dashed curve is an estimate of direct ionization and the solid curve includes EA. (b) Further evidence of indirect ionization is structure observed benveen 510 and 540eV.
with experiment suggests the presence of strong contributions from indirect ionization. This is confirmed by the structure observed between 510 and 540eV,shown in Fig. 3(b). The continuous curve in Fig. 3(a) is the result of a distorted wave calculation by Pindzola and Buie (1989) that included excitation autoionization.
INDIRECT IONIZATION OF POSITIVE ATOMIC IONS
77
B. INVESTIGATIONS OF ISOELECTRONIC SERIES
Useful insights can be gained by studying the development of indirect ionization along isoelectronic series, and the He-like (Li', Be2+, B 3 +,... ) and Li-like (Be', B2+,C3+,...) ions have been studied most intensively. Early work was reviewed by Crandall (1981) but measurements have subsequently been considerably refined and extended. Hofmann et al. (1990) recently made very elegant measurements of the ionization of BZ+, C 3 + ,N4*, 0 ' ' and F6+. They were able to resolve structures only 400 meV wide, and statistical uncertainties in their cross sections were as low as 0.1%. Numerous resonances due to REDA and READI (Eqs. (4) and (5)) were resolved, and to illustrate the quality of this work, we reproduce their results for B2+ in Figs. 4(a) and qb). The larger E.
160
/ Zz I e V 1
E
7
IEO
200
9
220
10
240
11
260
ElecCron energy E, rev1
(a)
-
E, / zZ rrVl 6 5
6 0
67
6.8
6 9
70
7.1
7.2
FIG.4. (a) Some of the structure seen in the ionization cross section of B2+.The larger peaks above 180eV are due to REDA. Smaller peaks between 165 and 175eV were attributed to READI. These are shown on an enlarged scale and in greater detail in Fig. qb).
K . Dolder
78
peaks above 180eV in Fig. 4(a) were attributed to REDA and the smaller peaks between 165 and 175eV to READI. These are shown in greater detail in Fig. qb). Similar patterns were observed for the other ions, and the energies of the transitions for each member of the sequence could then be expressed as scaled energies, E / Z 2 ,where Z is the atomic number of the ion. Corresponding REDA and READI transitions had roughly the same scaled energies although they tended to increase with Z for reasons that were discussed in the paper. Zhang et al. (1992) recently described measurements of the Na-like ion, Ar7+. Their paper included a discussion of scaling laws for the Na-like sequence that indicates that the relative importance of indirect ionization initially increases with ionic charge (n). However, when 11 ;5 n ;5 17 the autoionization branching ratio decreases so that, for the more highly charged ions in the sequence, the relative contribution of indirect processes remains roughly constant. A somewhat similar pattern was noted for the Si-like ions (Sataka et al., 1990). The most refined measurements on He-like ions are those of Muller et al. (1989), who obtained cross sections for Li' with statistical uncertainties of only 0.01%. This revealed the structure illustrated in Fig. 5, which was interpreted in terms of the nonresonant excitation, Li+(ls2)+ e + Li+(2121')+ e -,Li2+(ls)+ e
+e
(10)
5
-
N
b
4 U
0
".
9-
b
I
-5
n b
130
135
140
145
150
155
160
Electron energy lev1 FIG. 5. Extremely small structures observed in the ionization cross section of Li'. These correspond to excited states of Li and Li' that have two K-shell vacancies. Energies of these states are indicated by arrows labeled with probable configurations.This is evidence of direct K to L shell excitation,followed by autoionization; and resonant capture to form triply excited Li that decays by emitting two correlated electrons.
79
INDIRECT IONIZATION OF POSITIVE ATOMIC IONS
and resonant capture processes, e.g., Li+(ls2)+ e+Li(2s22p)+ Li2+(Is)+ 2e
(1 1) The ordinate in the figure was obtained by subtracting direct ionization from the measurements, and the arrows indicate energies of Li and Li+ that have two K shell vacancies. The dips at about 137 and 140eV were attributed to the formation of resonant intermediate states of Li and the peak at 157.2eV was probably due to resonant capture with double excitation to form triply excited neutral lithium, Li (2s 3p2). C. INVESTIGATIONSOF ISONUCLEARSERIES Much of this work has concerned impurity ions in thermonuclear plasmas, and measurements have been made for Fen' (n = 1,2,5,6,9, 11, 13, 15) (e.g., Gregory et al., 1986; 1987),Ni"+ (n = 1, 3, 5, 6, 7,8, 12, 14) (e.g., Wang et al., 1988; Pindzola et al., 1991), Cr"' (n = 6, 7, 8, 10, 13) (e.g., Sataka et al., 1989; Gregory et al., 1990b),Ti"' (n = 3, 5, 11) (e.g., Gregory et al., 1990)and U"+ (n = 10, 13, 16) (e.g., Gregory et al., 1990a).Ions that might be suitable for Xray lasers have also been studied. These included Ar"' (n = 7) (e.g., Zhang et al., 1992), Kr"' (n = 1, 2, 3, 8) and Xe"' (n = 8) (e.g., Bannister et al., 1988). Figure 6, which shows the results of measurements of various Fe ions, is an example of the results obtained. Three conclusions stand out from the mass of experimental data. Indirect ionization is frequently very influential; interpretation of the results is often complicated by large metastable components in the target beams; distorted 1000
r;
. . .. ;
10
a
100
I000
Electron energy (eV) FIG.6. Measured cross sections for the ionization of Fen+( n = 0,1,. . . ,15).
K . Dolder
80
wave calculations are often quite successful in predicting the general magnitudes of cross sections. Three examples will illustrate these conclusions and show the quality of the measurements. Figure 7 illustrates results of Sataka et al. (1989) for Cr"+, where the steep rise in the cross section due to the onset of autoionization (which is typical of many other measurements of highly charged ions) can clearly be seen. The continuous curve is a Lotz estimate of direct ionization. Other manifestations of indirect ionization are the resonance structures detected by Gregory et al. (1990) in the ionization of uranium ions. An example was illustrated by Fig. 3(b). The presence of metastable contamination was unavoidable in many of these experiments, and it was revealed by the near-threshold measurements of Ara+ ionization (Zhang et al., 1991). These results are shown in Fig. 8, and they clearly indicate ionization below the ground-state threshold due to the presence of about 3% of metastables in the target beam.
D. MULTIPLEIONIZATION Experimentalistsare now paying considerable attention to collisions in which the projectile removes more than one electron from the target, and it has been demonstrated that indirect processes can be very influential. For example, if ( T ~ represents , ~ the cross section when i and! are the initial and final charges one might expect oi,l to become smaller as i increases. But this is not
c 0 ._ *V
In
Electron Energy (eV)
FIG.7. Measured cross sections for the ionization of Cr'O'. The conthuous curve is a Lotz estimate of direct ionization, and its deviation from the measurements at higher energies indicates the presence of indirect ionization. This behavior is typical of most of the other chromium ions investigated and also of many ions of other metals.
INDIRECT IONIZATION OF POSITIVE ATOMIC IONS
81
FIG. 8. Near-threshold measurements of the ionization of Ar*+. The failure of the cross section to fall to zero at the ground-state ionization energy indicated the presence of metastable contamination in the target beam.
necessarily so. Figure 9 illustrates measurements for Ar2 and Ar3+ from Muller and Frodl (1980), where it can be seen that 03,5approaches 02,4at the higher energies. The dashed curves are estimates of direct ionization and sudden increases in the cross sections can be correlated to L-shell thresholds included in the figure. In the case of bismuth there is a surprise in the single ionization cross sectionssince 02,3exceeds u1,2above 200eV. Miiller (1991a)explained this by considering the double ionization cross sections. Essentially, oIv3is several times larger than 02,4because excitation autoionization from a 5d electron is possible for Bi+ but not for Biz+. As a result some ionization of Bi+ is diverted to produce Bi3+ with a consequent reduction in 01,2.This is a nice example of interplay between single and multiple ionization. Similar behavior was found in antimony. Miiller also reviewed other aspects of multiple ionization, where most attention has been paid to multiple ionization of Arn+ (n = 1,2,. ..,7) (Tinschert et al., 1989),Ban+(n = 1,2, 3 ) (e.g., Tinschert et al., 1991; Peart et al., 1992) and to a variety of other heavy metal ions (Miiller et al., 1988). We will therefore concentrate on a few recent developments. Figure 10 is taken from the paper of Muller et al. (1988) and illustrates parts of their results for single and multiple ionization of several ions. The +
82
K. Dolder
2
z 0
2
0 c a N
5
2
0
2
10-20
5 200 500 ELECTRON ENERGY lev)
100
4000
FIG.9. Measured cross sections ul., for double and triple ionization of Ar2+ and Ar3+. Suffixes iand f denote the initial and final numbers of charges of the ions, and at higher energies, double ionization from Ar2+ and Ar3 are almost equally probable. Dashed curves are estimates of direct ionization, and sudden increases in cross sections correlate to L-shell thresholds included in the figure. +
narrow peaks resulted from resonant capture of the incident electron followed by REDA, RETA or REQA. Barium and its ions exert a special fascination on account of the "giant resonances" that arise from 4d-Ef excitation. Double ionization of Ba" has been measured by Hirayama et al. (1987) and Peart et al. (1992). Both results showed a broad maximum in the cross section associated with this resonance but with an onset at about 85eV. Peart et al. used the method outlined in Section 111 to resolve the steplike structure illustrated in Fig. 11. This was most probably due to 4d excitation followed by double ionization, although alternative mechanisms were discussed. Another interesting feature of Ba"" is the major change that occurs in the two important dipole-allowed excitation channels, 5p6 45ps kd and 4d" 5s' 5p" 4 4d95s' 5p" kf (where u = 1, 2,. ., 6 ) as the ionic charge n increases from 0 to 3. For II I the excited kd and kf states have only a small overlap with the core orbitals but, as n increases, the excited orbitals collapse inward, and the overlap becomes much
-=
83
INDIRECT IONIZATION OF POSITIVE ATOMIC IONS .
15-. r?
.
.
.
.
.
.
-1
1 .o
E V I!
-
.5
C
0.
V
.4
b u
0 .*-
"1
W - La-
0)
lo
.2
0. 74
78
76
85
80
95
90
100
loo
95
90
Electron energy CeVl FIG. 10. Structures observed in the cross sections for single and double ionization of Cs+, Ba2+ and La2+.The peaks were attributed to resonant capture followed by REDA, RETA or REQA.
I
/
1111
2.0-
*"*.%i
l,#,il
41 *,,G*f I
I
1
Interaction
1
1
I
1
J
energy (eV 1
FIG. 11. Steplike structure in the double ionization cross section of Ba' revealed by measurements with high-energy resolution. The structure is probably due to 4d excitation followed by the ejection of two electrons. The continuous curve illustrates measurements by Hirayama et al.
84
K . Dolder
greater. This motivated Tinschert et al. (1991) to measure the single and double ionization of Ba" and Ba3'. Their paper includes a detailed discussion of these effects, and they were able to resolve numerous sharp resonances due to excitation ionization. Results for triple and quadruple ionization of Ba' were reported by Hofmann et al. (1991), and evidence was found of capture, followed by subsequent emission of four or even five electrons. E. MEASUREMENTS WITH ENERGY-RESOLVED ELECTRONS In Figs. 2 and 4 we saw that distorted wave calculations can successfully reproduce the general magnitudes and some features of indirect ionization, but this may have given an unduly favorable impression of the accuracy of theory. The effects of indirect ionization are so profuse and complicated that a detailed theoretical interpretation is an immensely formidable task. Some appreciation of the current situation can be gleaned by comparing results of calculations and experiments for the relatively simple ion, Ca'. The dotted line in Fig. 12 represents measurements with energy-resolved electrons (Peart et al., 1989a)and curves labeled 4CC, 13CC and DW are results of 4-state and 13-state close coupling calculations (Badnell et al., 1991)and a distorted wave calculation (Griffin et al., 1984). None of these calculations can successfully reproduce the experimental details. The scale of the task of interpreting the measurements is seen by looking at them in greater detail. Figure 13 shows the measurements between 24 and 31 eV; the upward pointing arrows denote energies of resonances in photoionization of Ca' that will be discussed in the next section. The figure also indicates energies of autoionizing levels calculated by Griffin et al. (GPB) (1984) and Pindzola et al. (PBG) (1987); the downward pointing arrows show these energies deduced from the ejected electron spectrum of neutral calcium. In an attempt to offer some assistance to the evolution of the theory, a program of experiments was undertaken to study the ionization of a series of alkali-like ions of increasingcomplexity, Mg', Ca', Sr' and Ba' (Peart et al., 1991; 1989a, b, c). Similar measurements were also made (Peart and Underwood, 1990, 1991) for Ga' and Zn', which also exhibit indirect ionization. Unique features of these measurements (which are particularly useful when making comparisons with theory) are that the energy scale is absolute to 40meV (Peart et al., 1989c) and the widths and profiles of the energy distributions are known. In the case of Mg', contributions from indirect ionization are relatively small but the experiments of Muller et al. (1990) and Peart et al. (1991) were able to reveal them in some detail. Figure 14 shows the latter measurements and compares them with calculations of excitation autoionization by Griffin
INDIRECT IONIZATION OF POSITIVE ATOMIC IONS
I
25
30
35
LO
85
45
Energy (eV 1 FIG.12. Single ionization of Caf. Measurements (dotted curve) compared with results of 4state (4CC)and 13-state(13CC)close coupling calculationsby Badnell et al. and with a distorted wave calculation (DW) by Griffin et al.
et al. (G) and Henry and Msezane (H). Curve Y is an estimate of direct ionization. Another example of the state of contemporary theory is illustrated by Fig. 15. The continuous curve in this figure is the result of an R-matrix calculation by Tayal (1991) for the single ionization of Mg’. These calculations were performed with a very fine energy mesh (0.0136eV) to resolve the resonance structure, but in the figure, the calculations have been convoluted with the electron energy distribution of the measurements by Peart et al. (1991), which are shown as solid points. We see a significant measure of agreement between the experiment and the theory, although the energy scale of Tayal’s calculations has been reduced by 2.5eV. This was done to bring his values of resonance energies into line with those calculated by Griffin et al. (1982) and Miiller et al. (1990) and to make major features of the measured and calculated cross sections coincide.
86
K . Dolder
,
I
t t t 1 ' '# -
06-
I
I
I
I
Excitation Zp 3s Threshold 1
5.5
4.0
X 4
t
40
I
I
I I
I
4s 3 Lp 4d
Y 1 1
50 60 Interaction energy (eV)
70
FIG. 14. Structure seen in the ionization cross section of Mg'. Arrows indicate calculated energies of autoionization thresholds. Curves G and H are theoretical estimates of excitation autoionization that have been added to an estimate of direct ionization (curve Y).
INDIRECT IONIZATION OF POSITIVE ATOMIC IONS
87
IT
FIG.15. Comparison of measurements of the ionization of Mgf (solid points) with results of an R-matrix caiculation (curve T ) .
We have seen copious evidence of indirect ionization in all the ions mentioned in this section, and in an attempt to simplify the unravelling of these structures, some measurements have been made of the photoionization of atomic ions.
F. MEASUREMENTS OF THE PHOTOIONIZATION OF POSITIVE ATOMICIONS Photons are simpler probes than electrons. They excite only dipole-allowed transitions; they cannot combine with the target; they perturb it less; and there are no complications from postcollision interactions. It therefore seemed that absolute photoionization cross sections could offer tests for theory before it advanced to embrace the additional complications of electron collisions. A further advantage is that photoionization can be measured with an energy resolution of only a few meV, and the energy scale can be defined with similar precision. Measurements have therefore been made of absolute photoionization cross sections of C a + , Sr' and Ba' (Lyon et al., 1987a, b, 1986); Ga' and Zn' (Peart et af., 1987); and K + (Peart and Lyon, 1987). All these ions, except K ', displayed strong evidence of inner shell ionization. The most prominent features of the spectra of Ca', Sr' and Ba' were large resonances with cross sections greater than lo-'' cm2.The largest resonances in Baf are illustrated
K . Dolder
88
by Fig. 16. There were also many smaller peaks, and more than 70 were resolved in the spectrum of Ba' at wavelengths between 420 and 775 A (29.516.0 eV). It was possible to assign configurations to the major resonances and compare their energies with calculations by Hansen (1975) in Hartree-Fock (HF) and multiconfiguration Hartree-Fock (MCHF) approximations. The results for the energies of the major resonances observed in Ca', Sr+ and Ba' are shown in Table I. Agreement is remarkably good (perhaps fortuitously in view of the assumptions inherent in the theory), and there is even consistency between the observed and predicted splittings. The largest resonance seen in G a + is shown by Fig. 17. It was centered at 21.88eV, which is similar to the energy of 21.9 eV calculated by Pindzola et al. (1982) for the 3d" 4s' -+ 3d94s' 4p 3P,transition. Comparisons of ionization by electrons and photons revealed a general Energy lev1 21.L
21. 5
I
i 0
I I
'
21.6
I
1
I
\
Ihi
4
I
I
FIG.16. Large resonances observed in the photoionization cross section of Ba'.
89
INDIRECT IONIZATION O F POSITIVE ATOMIC IONS
1 'I
AZ 1
FYHH
21.6
21.7
21.8
21.9 Energy IcV)
22.0
22.1
FIG. 17. Large resonance structure seen in the photoionization cross section of Ga', attributed to 3d1'4s' --* 3d94sZ4p3P, excitation.
TABLE I. ENERGIES AND SPLITTINGS OF AUTOIONIZING STATESOF Ca', Sr+ AND Ba+ BY HANSEN USINGHARTREE-FOCK (HF) AND MULTICONFIGURATION HARTREECALCULATED FOCK(MCHF) APPROXIMATIONSCOMPARED WITH EXPERIMENT
Ion
Configuration
HF Energy (eV)
MCHF Energy (eV)
Splitting
Ca+
(3p53d 'P)4s 'P (3ps 3d 'P)4s 'P
33.68 24.95
33.20 24.87
0.001 0.12
33.190
(4p54d 'P)5s 'P
27.13
26.77
0.02
26.972 26.950
(4pS4d 3P)5s ZP
21.60
21.31
0.3
(5p55d lP)6s 'P
2 1.44
2 1.09
0.04
21.194 21.234 21.320 21.480
(5p55d 'P)6s 'P
16.26
15.96
0.40
-
Sr+
Ba
+
Experimental energy (eV)
-
-
K.Dolder
90
result that was, at first sight, surprising. The very large photoionization resonances had no comparable counterparts in the electron impact cross sections. An example of this general result is shown in Fig. 18, which illustrates electron impact ionization cross sections of Sr measured between 19 and 28 eV (Peart et al., 1989). Vertical lines attached to the abscissa denote energies and relative magnitudes of resonances in the measured photoionization cross sections, and in this scale, lines representing the two largest resonances should be increased by a factor of two and the broken lines reduced by a factor of five. There is no obvious correspondence between the electron and photon spectra, but energies of autoionizing levels, deduced from ejected electron spectra of neutral strontium (denoted by arrows), show some correlations. In some cases it was possible to assign configurations to levels, and these are included in the figure. When comparing these two types of measurement it must be remember that electron-impact cross sections could be seen as interference patterns arising from competing transitions (“permitted” and “forbidden”) and from the formation of compound states. Photoionization cross sections are therefore much “cleaner”; they seem to offer particularly good ground from which to tackle the complexities of the theory indirect ionization. +
i
i
7
--.,
1.5
I
+P I 0.5
10
Fro. 18. Measured cross sections for electron impact ionization of Sr’. Lines attached to the abscissa denote energies and relative magnitudes of resonances in the photoionization spectrum (in this scale lengths of the two longest lines should be increased by a factor of two and the broken lines reduced by a factor of five). There is little obvious correlation between electron impact and photoionization spectra. Arrows denote energies of autoionizing levels deduced from ejected electron spectra of neutral strontium.
INDIRECT IONIZATION OF POSITIVE ATOMIC IONS
91
REFERENCES
Bannister, M. E., Mueller, D. W., Wang, L. J., Pindzola, M. S., Grimn, D. C., and Gregory, D. C. (1988) Phys. Rev. A 38, 38. Badnell, N. R.,Griffin, D. C., and Pindzola, M. S. (1991). J . Phys. B 24, L275. Chen, M. H., Reed, K. J., and Moores, D. L. (1990) Phys. Rev. Lett. 64, 1350. Crandall, D. H. (1981) Physicn Scripta 23, 153. Crandall, D. H., Phaneuf, R. A., Gregory, D. C., Howald, A. M., Mueller, D. W., Morgan, T. J., Dunn, G. H., Griffin, D. C., and Henry, R. J. W. (1986) Phys. Rev. A 34, 1757. Dolder, K., Harrison, M. F. A,, and Thonemann, P. C. (1961) Proc. Roy. SOC.A 264, 367. Dolder, K., and Peart, B. (1976) Reports on Prog. in Physics 39, 693. Dolder, K., and Peart, B. (1986) Ado. in Atom and Mol. Phys. 22, 197. Dunn, G. H. (1985) In Electron Impact Ionization (T. D. Mark and G. H. Dunn, eds.), SpringerVerlag, New York. Falk, R. A., and Dunn, G. H. (1983). Phys. Rev. A 27, 754. Gregory, D. C., Huq, M. S., Meyer, F. W., Swenson, D. R., Sataka, M., and Chantrenne, S. J., (1990a) Phys. Rev. A 41, 106. Gregory, D. C., Meyer, F. W., Muller, A., Defrance, P. (1986). Phys. Rev. A 34, 3657. Gregory, D. C., Wang, L. J., Meyer, F. W., and Rinn, K. (1987). Phys. Rev. A 35, 3256. Gregory, D. C., Wan& L. J., Swenson, D. R., Sataka, M., and Chantrenne, S. J. (1990b). Phys. Rev. A 41,6512. Griffin, D. C., Pindzola, M. S., and Bottcher, C. (1984). J. Phys. B 17, 3183. Hansen, J. E. (1975). J . Phys. E 8, 2759. Hirayama, T., Kobayashi, S., Matsumoto, A,, Ohtani, S., Takayanagi, S., Wakiya, K., and Suzuki, H. (1987). J. Phys. Soc. Japan 56,851. Hofmann, G., Miiller, A., Tinschert, K., and Salzborn, E. (1990). Z. Phys. D 16, 113. LaGattuta, K. J., and Hahn, Y. (1981). Phys. Reu. A 24, 2273. Lyon, I. C., Peart, B., Dolder, K., and West, J. B. (1987a). J , Phys. B 20, 1471. Lyon, I. C., Peart, B., and Dolder, K. (1987b). J. Phys. B 20, 1925. Lyon, I. C., Peart, B., West, J. B., and Dolder, K. (1986). J . Phys. B 19, 4137. Martin, S. O., Peart, B., and Dolder, K. (1968). J . Phys. B 1, 537. Moores, D. L., and Pindzola, M. S. (1990). Phys. Rev. A 41, 3603. Miiller, A. (1991a). Springer Series in Chemical Physics 54, 13. Miiller, A. (1991b). Z. Phys. D 21, S39. Muller, A. (1991~).Comments in At. Mol. Phys. 27, 1. Muller, A., Hofmann, G., Tinschert, K., Weissbecker, B., and Salzborn, E. (1990). Z. Phys. D 15, 145.
Miiller, A., Hofmann, G., Weissbecker, B., Stenke, M., Tinschert, K., Wagner, M., and Salzborn, E. (1989). Phys. Rev. Lett. 63, 758. Miiller, A., Huber, K., Tinschert, K., Becker, R., and Salzborn, E. (1985). J . Phys. B 14, 2993. Miiller, A.,Tinschert, K., Hofmann, G., Salzborn, E., and Dunn, G. H. (1988). Phys. Reo. t e t t s 61, 70.
Peart, Peart, Peart, Peart, Peart, Peart, Peart, Peart,
B., and Dolder, K. (1968). J . Phys. B I, 872. B., and Dolder, K. (1975). J . Phys. B 8, 56. B., Green, S. J. T., and Thomason, J. W. G. (1992). J. Phys. B, in press. B., and Lyon, I. C. (1987). J . Phys. B 20, L673. B., Lyon, I. C., and Dolder, K. (1987). J . Phys. B 20, 5403. B., Thomason, J. W. G., and Dolder, K. (1991). J. Phys. B 24, 4453. B., and Underwood, J. R. A. (1990). J . Phys. B 23, 2343. B., and Underwood, J. R. A. (1991). J . Phys. B 23,489.
92
K. Dolder
Peart, B., Underwood, J. R. A., and Dolder, K. (1989a). J. Phys. B 22, 2789. Peart, B., Underwood, J. R. A., and Dolder, K. (1989b). J. Phys. B 22, 4021. Peart, B., Underwood, J. R. A,, and Dolder, K. (1989~).J. Phys. B 22, 1679. Pindzola, M. S., Bottcher, C., and Griffin, D. C. (1987). J . Phys. B 20, 3535. Pindzola, M. S., and Buie, M. J. (1989). Phys. Rev. A 39, 1029. Pindzola, M. S., Griffin, D. C., and Bottcher, C. (1982). Phys. Rev. A 25, 211. Pindzola, M. S., Griffin, D.C., Bottcher, C., Buie, M. J., and Gregory, D. C. (1991). Physica Scripta T37,35. Rim, K., Gregory, D. C., Wang, L. J., Phaneuf, R. A., and Miiller, A. (1987).Phys. Rev. A 36,595. Sataka, M., Ohtani, S., Swenson, D., and Gregory, D. C. (1989). Phys. Rev. A 39,2397. Tayal, S. S. (1991). J. Phys. B 24, L219. Tinschert, K., Miiller, A., Hofmann, G., Huber, K., Becker, R., Gregory, D. C., and Salzborn, E. (1988). J . Phys. B 22, 531. Tinschert, K., Miiller, A., Phaneuf, R. A., Hofmann, G., and Salzborn, E. (1989). J. Phys. B 22, 1241. Tinschert, K., Miiller, A., Hofmann, G., Salzborn, E., and Younger, S . M. (1991). Phys. Rev. A 43, 3522. Wang, L.J., Rinn, K., and Gregory, D. C. (1988). J . Phys. B 21, 2117. Zhang, Y.,Reddy, C. B., Smith, R. S., Golden, D. E., Mueller, D. W., and Gregory, D. C. (1992) Phys. Rev. A 45,2929.
ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 32
QUANTUM DEFECT THEORY AND ANALYSIS OF HIGH-PRECISION HELIUM TERM ENERGIES G . W . F. DRAKE Department of Physics University of Windror Windror, Canada
I. Introduction
............................
. . . . . . . . . . . . . . A. Hartree’s Proof . . . . . . . . . . . . . . . . . . . . . . . . .
11. Quantum Defect Theory and l/n Expansions
B. l/n Expansions . . . . . . . . . . . . . . . . . . . . . . . . . C. Comparisons with Perturbation Expansions . . . . . . . . . . . 111. Quantum Defect Analysis of High-Precision Variational Calculations . . A. Summary of Asymptotic Expansions . . . . . . . . . . . B. Quantum Defect Fits for L < 6 . . . . . . . . . . . . . IV. Comparison with High-Precision Measurements. . . . . . . . . . . . V. Summary and Discussion. . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
.
93 95 95 95 97 103 104 105 110
113 115 115
I. Introduction The quantum defect method (Edlen, 1964; Seaton, 1983) dates back to the earliest days of quantum mechanics and the analysis of atomic spectra. Ritz (1903) first pointed out that the term energies of a single Rydberg series of states for a quasi-hydrogenic ion are well represented by the formula En = -2R,ZZff/(2n*2)
(1)
where R , = ( p / M ) R , is the reduced mass Rydberg constant for an atom with nuclear mass M and reduced electron mass p = m M / ( M + m), and Zcff is the screened nuclear charge experienced by the Rydberg electron. The quantity n* is an effective principal quantum number defined by n*
= n - &I*)
(2)
93 Copyright 0 1994 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003832-3
G. W . F. Drake
94
with
S(n*) = 60
64 + ... + (n - 6)2 + (n 62
~
4 4
(3)
6 called the quantum defect, and the Siare constant expansion coefficients. Equation (3) contains two important features. The first is that the full 6 appears in the denominators on the righthand side, so that the equation must be solved iteratively to find S for each n. The second is that only the euen inverse powers of (n - 6) appear. Both are essential aspects of the Ritz expansion for the quantum defect. A theoretical justification of the Ritz expansion was given by Sommerfeld (1920), based on the old quantum theory. Hartree’s (1928) proof based on wave mechanics is discussed further in Section 1I.A. The quantum defect method, and in particular the Ritz expansion, is now well established as the method of choice for the analysis and compact representation of term energies in a Rydberg series. However, recent calculations (Drake, 1993a,b; Drake and Yan, 1992) and measurements of transition frequencies of unprecedented accuracy (Lichten et al., 1991; Sansonetti and Gillaspy, 1992) raise new questions concerning the ultimate limits of validity of the Ritz expansion. For example, if the odd terms in Eq. (3) are small, but not exactly zero, then omitting them in a least squares fit to data may give an unrealistically small estimate of the uncertainties in quantum defect extrapolations (Drake and Swainson, 1991). This chapter has two main goals. The first is to apply the quantum defect method to the analysis of high-precision calculations of energies for the Rydberg states of helium and to compare with recent measurements. The second is to study the form of the Ritz expansion itself and to identify physical effects that might contribute an additional S,/(n - 6) term to Eq. (3). Since such a term is not part of the Ritz expansion, the coefficient dl is the leading term in what might be called the Ritz defect. Expansions of this form were tried long ago by Hicks (1910, 1919). Section I1 briefly reviews quantum defect theory for a single isolated sequence of Rydberg states. In particular, the role of Hartree’s proof of the Ritz expansion is emphasized, and the information that can be deduced from it concerning the coefficients in a l/n expansion of the energy. Particular examples are considered in an attempt to identify terms that would generate a “Ritz defect.” The results suggest that Hartree’s proof can be extended to include exchange terms and nonspherical potentials resulting from core polarization. The conclusion is that the Ritz expansion remains a valid functional form, even at the current very high levels of precision. Not included is the vast literature on multichannel quantum defect theory and applications to multiple sequences of interacting states. Section 111 applies the Ritz form of the quantum defect formula to the
QUANTUM DEFECT ANALYSIS O F HIGH-PRECISION TERM ENERGIES
95
analysis of high-precision variational calculations for the states of helium up to L = 6. These results, together with asymptotic expansion methods for L 3 7, cover the entire singly excited spectrum of helium, with the exception of S-states. All known finite mass, relativistic, and quantum electrodynamic corrections are included. These effects require important adjustments to the Ritz formula in order to avoid the introduction of a Ritz defect. Section IV reviews the comparison of the theory with recent high-precision measurements on transitions to the Rydberg states of helium, and in particular their use to determine the Lamb shift in the ionization energy of the ls2s ' S o state. A revised analysis of the experimental data is presented, using both empirical quantum defect fits and theoretical values for the highlying Rydberg states as points of reference. Finally, Section V presents a summary and suggestions for future work.
II. Quantum Defect Theory and 1/n Expansions A. HARTRFE'SPROOF A quantum mechanical justification for the Ritz expansion expressed by Eq. (3) was first given by Hartree (1928). He proved that, if the motion of an electron is describable by a Hamiltonian of the form
H
= Hc
+ 1v
(4)
where H, is the Harniltonian for a purely Coulombic potential and J.V is a local, short-range, spherically symmetric correction potential, then the eigenvalues of H are given exactly by Eq. (1) with only the even powers of l/(n - 6) appearing in Eq. (3). The proof is nonperturbative, and so applies for arbitrary values of the strength parameter A. As will be shown in the following sections, this fact can be used to advantage in calculating the general n dependence of the terms in a perturbation expansion containing powers of 1. B. l/n EXPANSIONS As discussed by Drake and Swainson (1991), the mathematical implications of the Ritz expansion are made evident by expanding Eq. (1) as a straight
power series in l/n of the form
E , = -2R,,,Zzff
(-2n2 + C 1 0 3
i=3
ai/ni)
G . W . F. Drake
96
The result is
The influence of the &-expansionin the denominators of Eq. (3) first appears in the terms of order l/na. For example, replacing 6 by 6, in the denominators of Eq. (3) reduces the coefficient of 6; from 7/2 to 3/2, and the coefficient of 6,6, from 9 to 3, while leaving the other terms unchanged. One can draw important conclusions from the fact that, for terms up to a given highest order, Eq. (6) contains twice as many terms as Eq. (3), with the coefficients of the even powers of l/n in Eq. (6) being determined once the coefficients of the odd powers of l/n are known. In terms of the general a, coefficients in Eq. (9,one can conclude that a3 = a, and for odd i 2 5, a, = Also, to the extent that the Ritz expansion is valid, the even order coefficients are related to the odd ones by (74
a4 = $a3 a6
= 5a3a,
+ O(69
+ 2a3a,) + 0(6;6;) = 9(a3a, + a5a7)+ 0(6$),
a8 = $(a:
(7b) (74
a,, etc. (74 Suppose now that the actual coefficients in a l/n expansion of the En do not satisfy these equations. For example, if some other value ii, is correct, then the difference Aa4 = a4 - 2,
(8) is the leading term in the Ritz defect. It represents the degree to which the data cannot be represented by the Ritz expansion of the quantum defect containing only even powers of 1/(n - 6). As an illustration, the error introduced when 6 is replaced by 6, in the denominators of Eq. (3) corresponds to a Ritz defect of
Aa8 = (7/2 - 3/2)ag = 26;
(9)
(see the discussion following Eq. (6)). The equations a, = 6,- etc. lead to physical identificationsof the quantum defect coefficients in terms of multipole moments of the core, as discussed by Edl6n (1964), and extended by Drake and Swainson (1991) to include nonadiabatic terms. A semiclassical picture, including core penetration effects, is discussed by Curtis (1981).
QUANTUM DEFECT ANALYSIS OF HIGH-PRECISION TERM ENERGIES 97
C. COMPARISONS WITH PERTURBATION EXPANSIONS Further insight into the nature of the Ritz expansion and the physical significance of the ai can be obtained by treating the short-range term AV(r)in Eq. (4) as a perturbation. The unperturbed wave function $ 3 ( r ) for the Rydberg electron satisfies the Coulomb Schrodinger equation
The AV(r) term represents a complex hierarchy of dynamical interactions between the Rydberg electron and the core electron@),including exchange. However, when all is said and done, an effective spherically symmetric AV(r) may still exist that is local and that reproduces the actual spectrum to an extremely high degree of accuracy. Even though the actual AV(r) may not be known, all that is required for the moment is that it exist in principle. Then the exact energies can be expanded in the form
where
and
$:!L
satisfies the perturbation equation (Hc
+
$)$:!m +
VW::m =J w w ( T ) -
(14)
Each of the E a can now be expanded as a power series in l/n. It can be shown that, for any V(r)that decreases as l/r2 or faster as r + co,E!!L has the expansion (Drake and Hill, 1993)
while, for i 2 2, the more general form
applies in which both the even and odd terms are present. Substituting
98
G. W. F. Drake
expansions (15) and (16) into Eq. (11) and comparing with Eq. (6) then shows that
6 , = R & y + p & f ' o ) + 13&f.O' + ... 6 , = , I & c s 2 ) + R2EP.2) + A3EL(3.2) + ...
(174 (17b)
etc. Futhermore, the validity of the Ritz expansion requires that the coefficients of the even powers of l/n be related to the odd powers according to Eqs. (7). By expanding the a, in powers of 1 and equating coefficients of A2, one obtains
-&p = 3( E y ) 2
( W
and, similarly, from (7b), (7c), and (7d), - & ~ 2 . 3 )= 5 E f f . 0 ) E c . 2 )
( W ( W
+ 2Ei1.0)E294))] = 9[&Ll.O)(1.6)+ &(L1.2) (1.4)
-,$.5)=$[(@-2))2 -&$.7)
11
EL
EL
(184
(the signs are negative because of the minus sign in Eq. (5)). To the extent that Hartree's proof is valid, Eqs. (18a) to (18d) provide a ) coefficients for the second-order trivial way of calculating the ~ ( L z ' jexpansion energies (see Eqs. (13) and (16)). An example that can be worked out analytically is provided by taking R V(r) to be the long-range polarization potential - a l p 4 ) . Then a
E t l = -f
<$:w 4
(0)
l$".L>
=-[
-24n2 + 8L(L + 1) n5 (2L - 1)(2L)(2L + 1X2L + 2)(2L a1
+ 3)]
is the first-order energy, so that
&y =
- 2401, (2L - 1)(2L)(2L + 1X2L
&(1.2) =
+
+ 2X2L + 3)
8a14L 1) (2L - 1)(2L)(2L 1X2L 2X2L
+
+
+ 3)
Then Eqs. (18a) to (18d) give &f."
=
- 25-27at [(2L - 1)(2LX2L+ 1K2L 2)(2L
+
+ 3)12
(19)
QUANTUM DEFECT ANALYSIS OF HIGH-PRECISION TERM ENERGIES
+ + + 3)12 -25 . ~ & L ( L + 1)i2 = [(2L - lX2LX2L + 1X2L + 2X2L + 3)12
&(2.3) =
42.5)
99
1) 2’ * 3Oa:yL [(2L - 1)(2L)(2L+ 1)(2L 2X2L
Defining j-1
f3
+ 1XL + 2) = ( L - 2XL - 1)L(L+ 1)(L + 2)(L + 3) =y L
+ l),
f2 =
( L - 1)yL
(26)
the preceding &I2#” agree with the last three terms in the general expression
+ 623 + 3640f2 + 560f3) +n-4f:f2(3 + 40f1 + 240f2)) + 27n-’
(45
fl
- 30n-3f,
+ 7n-’(:]
(27)
obtained by Swainson and Drake (1992) for the second-order energy. Thus there is no Ritz defect up to the second order for an r-4 potential. Hartree’s proof of the Ritz expansion is clearly valid to all orders for any such AV(r) if exchange effects are neglected. However, exchange effects correspond to a nonlocal effective one-electron potential, which could in principle lead to a Ritz defect. The following example shows that Hartree’s proof remains valid even for the exchange terms, at least up to the second order. As discussed by Bethe and Salpeter (1957), Heisenberg’s method for excited states consists of writing the Hamiltonian in the unsymmetric form
H
= Ho
+ IV
(28)
where 1
Ho = - - (v:
2
+ v:)
2
2-1
I1
12
----
and 1 v=---
1
r12
12
(30)
The zero-order wave function (including exchange) is 1 r2) = - Cuo(rl)vo(r2)f ~ 0 ( ~ 2 ) ~ 0 ( ~ 1 ) 1(3 1)
Jz
G. W. F. Drake
100
where, for brevity,
The zero-order energy is
J , is the direct energy and K O is the exchange energy. Performing the integration over r-, in Eq. (36) allows J , to be written as an expectation value: Jo =
with
s
dr2J(rz)l~o(rz)12
(
~ ( r , )= - Z
+):r
exp(-2Zrz)
(38)
(39)
but there is no similar effective local potential corresponding to KO.One therefore might look for a Ritz defect due to exchange effects when higherorder perturbation corrections are included. As a test case, consider a model problem where the energy contains the terms (for 2 = 2):
+
E,, = -2 - 1/(2n2) J o k K O
where a1 P, = -(uo(r-4~uo) 2
a: (ul(r-41uo) P, =4
+ Po + J , k K , + P,
(40)
QUANTUM DEFECT ANALYSIS OF HIGH-PRECISION TERM ENERGIES 101
and u1(r4) satisfies the perturbation equation
Po and P , correspond to the terms E!;) and Eii) respectively, already discussed in Eqs. (19) to (27). J , and K , are corrections to J , and KO due to the effect of the long-range dipole polarization potential - a1/(2r4)on the outer electron. This potential is itself the asymptotic limit of terms of the second order in AI/:Hence, a1 and 1 are not really independent, but they can be treated as independent in the following argument to keep track of the terms of different order. Further short-range and nonadiabatic corrections do not affect the argument because they just change the effective local potential. All the terms included can be rigorously derived from a detailed perturbation expansion. The next step is to introduce l/n expansions for all the above quantities. J,, KO,and Po all have l/n expansions containing only the odd inverse powers
1, = n - 3 ( 4 0 ’ + n-ZJ$JZ’+
...)
(46)
while J , , K , , and P , all have expansions of the form J 1 -- ,-3(J‘o’1
+ n-”‘” 1 + n-2Jf’
+ ...) (47) with both even and odd inverse powers appearing. The terms of O(n-3) in Eq. (5) are then a3 =
-(J‘O’
f KLo) + Pbo)+ J\”) & K\’)
+ Pio))
(48)
and the terms of O ( ~ I -are ~)
+
a4 = -(,I K\’) \’)Pcl) 1 )
(49)
Substituting expansions (48) and (49) into a, = 3&2 and collecting terms of O(a:) yields Pi1)= - 3(P’p))2/2 (50) which has already been verified in Eqs. (19) to (27). The terms of O(all)yield J\U = -3~(o)pv~) 0 0 (51) and ~ \ 1= )
-3
~0 w 0p 0 )
(52)
G . W.F. Drake
102
These equations are satisfied to within the numerical accuracy of the expansion coefficients listed in Tables I and I1 for the P, D and F states of helium. The numbers in Table I were obtained from analytic formulas (Drake and Hill, 1993), while those in Table I1 for J y ) and KY) were obtained by least squares fits to direct evaluations of the integrals for a range of n. The corresponding equations for terms of O(l/n6) from
are and These equations are also satisfied to within the more limited numerical accuracy in Table 11. It seems reasonable to assume that similar results will hold to all orders in l/n2. The conclusion is that there is no Ritz defect, at least up to the second order in the direct and exchange interactions. This result is to be expected for the direct terms from Hartree’s proof, but his proof does not treat explicitly the exchange terms. The evident absence of a Ritz defect for the exchange terms therefore represents an extension of Hartree’s proof. As a consequence, singlet and triplet energies should not first be averaged before performing a quantum defect fit. Otherwise, large Ritz defects appear (Drake and Swainson, 1991). TABLE I l/n EXPANSION ~ E F F I C I E N T SFOR Jo, KO, AND Po Coefticient
P States
D
J6“’
- 1.0445867280 x lo-’ 7.9948541165 x lo-’ 1.9253597796x lo-’ 4.1310250243x
- 1.7683327875x
36’ Jb4)
Jp Jb“’ JL10)
K(0) K(2)
Kh4’ K(6)
Kf) KLIO)
Pp
PO P:’
8.8159742966x 1.9033642812x lo-’ 3.5144776254x lo-’ - 1.4868019747x lo-’ - 1.2039146630~ lo-’ -5.3463849117 x lo-’ - 1.9543788408x lo-’ -6.4810506509 x -91160 3/80 0
States
8.1806304078x -3.9558102520 x - 1.7544O131OOx -5.2011575013 x lo-’ - 1.3676318289x lo-’ 6.4974298581x -2.7988851181 x lo-’ 5.5577482779x 8.4819540479x 4.5774967676x 1.8770748937x - 311 120 31560 0
F States - 1.3328796677x
1.7983844555~ lo-’ - 5.6077388148x
lo-’
1.8081463559x lo-’ 1.3725627565x lo-’ 5.3028608581 x 5.0685646949x -6.6845972674 x lo-’ 1.9287877285x lod4 -9.6027182107x -5.6518825335 x -3.7377542438 x - 112240 1/560 0
QUANTUM DEFECT ANALYSIS OF HIGH-PRECISION TERM ENERGIES
TABLE I1 l/n EXPANSION COEFFICIENTSFOR .I\’), K\’), .I\’), AND K\’) PREDICTED FROM THE RITZ EXPANSION IN EQS. (51) TO (55)
C o M P h R I W N OF THE VALUES
Coefficient
P States - 1.76274(1)x
- 1.76214~ 5.9307q1) x 5.93068 x 4.206(2)x 10-3 4.20715 x lo-’ 1.0763(4) x 1.07713x
WITH THE
F States
D States - 1.42094(3)x - 1.42098 x 5.22102(4)x 5.22115 x 1.572(1) x lo-’ 1.56928 x -5.496(2) x lo-’ - 5.48889 x
103
- 1.78519(3)x -1.78511 x lo-’ 6.7869(1) x 6.78826 x 5.201(2)x lo-’ 5.20432 x lo-’ - 1.957(5)x l o e 7 - 1.94465 x 10-
Numbers in parentheses denote uncertainties in the final figures quoted.
The preceding discussion still leaves open the question of whether or not a Ritz defect might appear due to higher order interactions. To the extent that the outer electron can be viewed as moving in an effective central potential, the answer is no. The polarization of the inner electron wave function due to the electric field of the Rydberg electron of course makes the instantaneous potential noncentral, but in the adiabatic approximation, the polarization smoothly tracks the motion of the Rydberg electron. The result is still an effective central potential. If exchange effects are neglected, then the nonadiabatic corrections modify the radial dependence of the central potential (Drachman, 1982, 1993a; Drake, 1993b),but this by itself cannot produce a Ritz defect. Even the combined effects of exchange and nonadiabatic corrections do not produce a Ritz defect, at least up to the second order in the interactions. This is not a proof that the Ritz defect is exactly zero, but all the results obtained so far for the l/n expansions of various contributions have failed to reveal one. More work is necessary to identify what terms, if any, would lead to a failure of the Ritz expansion for the Rydberg states of helium. Further numerical tests using variationally calculated eigenvalues are discussed in the following section.
111. Quantum Defect Analysis of High-Precision
Variational Calculations Extremely high-precision variational calculations for all states of helium up to n = 10 (or n = 12 for the F, G, and H states) and t = 7 are now available from the work of Drake (1993a,b) and Drake and Yan (1992). A detailed discussion of the variational methods can be found in these references. The
104
G . W. I;. Drake
various correction terms to the nonrelativistic energy are summarized in Section III.B.2. These results are important for two reasons. First, they provide a precise assessment of the accuracy of the asymptotic expansion method further discussed in Section 1II.A. Second, quantum defect extrapolations of the variational results for a low L, combined with asymptotic expansion results for a high L, cover the entire singly excited spectrum of helium with essentially exact predictions for the energies, at least up to quantum electrodynamic corrections of 0 ( a 3 ) a.u. and higher. This section first summarizes the asymptotic expansion method and then presents the quantum defect extrapolations for a low L. A.
SUMMARY OF
ASYMPTOTICEXPANSIONS
The asymptotic expansion method begins with a model in which the Rydberg electron is regarded as a distinguishable particle moving in the effective potential generated by the polarizable core consisting of the nucleus and the tightly bound inner electron. The result is an asymptotic potential of the form r
where r is the radial coordinate for the Rydberg electron. The first few terms are c4 = a l , c5 = 0, and c6 = a2 - 6/31, where al is the dipole polarizability, is a nonadiabatic correction to al. a2 is the quadrupole polarizability, and The leading i = 4 term is the one discussed in connection with Eqs. (1 9) and (40).All of the multipole moments and nonadiabatic corrections for the hydrogenic core can be calculated analytically as rational fractions. This topic has a long history, culminating with the work of Drachman (1993a,b). He has obtained the coefficients of all terms up to ~ ~ ~(Ther numerical - ~ ~ value of cl0 obtained by Drake (1993b) by a different method is not quite the same, but the difference is not numerically significant.) The corresponding asymptotic expansions for the relativistic corrections have been extended by Drake(1993a,b) and by Hessels (1992)to the necessary accuracy. Asymptotic expansions for the quantum electrodynamic corrections are discussed by Goldman and Drake (1992). The major strength of the asymptotic expansion method is that it provides a simply evaluated analytic expression for the energy of an arbitrary nl-state that rapidly improves in accuracy with increasing L. The major limitation is that for a low L, only terms up to i = 2L 2 in Eq. (56) can be included before the expectation value of V(r)diverges. Also, exchange and short-range effects become important. A further limitation follows from the fact that the terms in Eq. (56) are proportional to [(Z - 1)/Zli. The rate of convergence is therefore substantially worse for the higher-Z heliumlike ions than for
+
.
QUANTUM DEFECT ANALYSIS OF HIGH-PRECISION TERM ENERGIES
105
helium. In the low-L region, there is therefore no substitute for direct variational calculations. A comparison with the variational calculations (Drake and Yan, 1992; Drake 1993a, b) shows that at L = 7, the asymptotic expansions for the energies of helium come into agreement to within a few Hz. There is therefore no need to extend the variational calculations beyond L = 7, because for L > 7 the asymptotic expansions can be expected to be more accurate still. Thus the asymptotic expansion method is capable of completely covering the region L 2 7 with essentially exact predictions for the energies (i.e., well within experimental accuracies), not counting QED uncertainties. For example, Casimir-Polder modifications to the retarded electron-electron interaction are predicted to shift the energies by several hundreds of Hz for L = 7 (Babb and Spruch, 1988; Au and Mesa, 1990; Spruch, 1993). This has recently been an area of active experimental study for transitions among the n = 10 states of helium (Hessels et al., 1992).
B. QUANTUM DEFECT FITSFOR L < 6 Quantum defect fits to the high precision variational calculations of Drake and Yan (1992) provide a means to extend the results to a higher n and thereby cover the remainder of the singly excited helium spectrum in the lowL region where the asymptotic expansion method is not sufficiently accurate. 1. Nonrelativistic Energies
Quantum defect fits to the nonrelativistic variational energies for infinite nuclear mass have been discussed by Drake and Swainson (1991). The numerical fits obtained there seemed to indicate the presence of a statistically significant Ritz defect (i.e. u4 # 3.5). However, the failure to find a theoretical basis for Ritz defects (see Section 1I.C)large enough to affect the data suggests that those found previously are probably an artifact of the numerical fitting procedure with too few data points. Further fits to the data with just a single non-Ritz 6, term have yielded significantly smaller values that are consistent with zero. It therefore seems safe to assume that the Ritz expansion provides a valid means of representing the nonrelativistic energies. The results based on the input data tabulated by Drake and Yan (1992) are listed in Table 111. They were obtained by a “bootstrap” least squares method in which the fit is repeated many times with deviations being selected at random from the data set and added to each point. The results are then averaged, and the statistical deviations in the fitting parameters computed. The results are consistent with, but slightly different from, those tabulated by Drake and Swainson (1991).
TABLE I11 QUANIUM DEFECT PAUMETEM6;x 106FOR THE NONRELATIV~TIC: ENERCIE~ OF HELIUM, A W J ~ ~ JI Gm m - NUCLEAR ~ MASS
Parameter
Value
'P 12114.19211q27) 7507.8878(39) 13959.1y 19) 4876.7(3.6) 910.q26.6) 685.q60.6) 'F 434.3214072(17) - 1677.46421(40) -68.659(29) 17.29(8 1 1
79.q7.41 IIf 43.54740024(31) - 433.242713(74) - 7.6834(51) - 6.9q 10)
Value
3P 682Y3.6 142q9) - 1 8636.057( 12) --12316.76(50)
Value 'D
2101.880710(10) - 3085.7821(15) 9.297(65)
Va Iue 3D
2880.5O9914415) -6361.899323) 336.04(11) 839.7tI.7)
- 8073.!(S.S)
- 32O.8(1.1)
-4508.5(62.0) - 19384133.1) 3F
- 308.0(5.4)
'G
3G
439.04412233(74) - 1739.33043(16)
120.7239096q28) - 795.988484(35) - 14.2707(14) - 7.024(17)
120.7411929q35)
105.54441I)
24.8q29) 20.q2.6) 3H
43.54744182(13)
-433.244958(30) -7.6434 19) -7.209(36)
-
-
'I 18.6976389251(14) - 261.07602535(28) - 3.894965(18) - 3.39484(39)
383.9(H.5) -
- 796.489459(41) - 10.0642( 15)
- 16.861(18) 31
18.6976390132(9) - 261.07603589(18) -3.894454(12) - 3.40441(26)
QUANTUM DEFECT ANALYSIS OF HIGH-PRECISION TERM ENERGIES 107
It is clear from the results in Table I11 that, with increasing L, 8’ rapidly becomes the coefficient of largest magnitude in the quantum defect expanas given by sion. This is because for large L, 6, + --d1*’)and 6, + -$*’), Eqs. (20) and (21). Thus the ratio tends to 62/&, + -L(L + 1)/3. Asymptotic expansions for the quantum defects are further discussed by Drake and Swainson (1991). 2. Total Energies A detailed discussion of the many corrections required to obtain total energies that can be compared with experiment is given by Drake (1993a)and Drake and Yan (1992). In summary, the terms included are
AEL.2
The first-order mass polarization correction due to the term p/Mp, *pz in the full three-body Hamiltonian. The second-order mass polarization correction of O ( p 2 / M z ) Relativistic corrections of O(a’) from matrix elements of the Breit interaction. The singlet-triplet mixing correction between states with the same n, L, and total angular momentum J . Relativistic reduced mass corrections due to the mass scaling of the Breit interaction terms and transformation to centerof-mass coordinates. Relativistic recoil cross-terms between the Breit interaction and the mass polarization operator. Finite nuclear size correction. Electron-nucleus Lamb shift (QED) terms, calculated from the asymptotic expansion of Goldman and Drake (1992). Araki-Sucher electron-electron QED corrections.
This includes all terms of orders a2, p / M , (p/M)’, a2(p/M), and a3. The dominant sources of uncertainty are the Bethe logarithm in AELVland relativistic and QED corrections of order a4 and higher. Also not included are Casimir- Polder retardation corrections to the electron-electron interaction of orders a2 and a3 contained in AE,,, and AEL.2 (Babb and Spruch, 1988; Au and Mesa, 1990). These would reveal themselves as small discrepancies between measured and calculated transition frequencies. For example, the predicted shift for the 101-10K transition is -0.304kHz. However, agreement with the experiment of Hessels et al. (1992) is at present not satisfactory. A more detailed discussion can be found in Drake and Yan (1992). Before presenting the final quantum defect fits, it is necessary to discuss some important adjustments that must be made to the Ritz formula. First, Eq. (3) for the quantum defect is based on the assumption that all corrections
G. W . F. Drake
108
to the Rydberg energies decrease at l/n3 or faster. However, the term AE@ has the asymptotic n-depencence
-
A E ~- [ ( p / ~ + ) ~(~p.4)~ + .-I(2 - 1)2/(2n2)+ o(nand the term (AE& 1992)
(57)
has the asymptotic n-dependence (Drake and Yan,
-
-da2(p/M)2Z2(Z- 1)’/(2n2) + O [ a Z ( p / M ) n - 3 ]
(58)
Although the leading term of (AERR)Xis nominally smaller than the next term by a factor of p / M , it is in fact the dominant contribution for the Rydberg states of helium down as far as 4F. Since the leading terms scale only as n-’, the quantity
6Ez(n) = -2R,{[(p/M)’
+ ( P / M )+~ -**](z - 1)2/(2n2)
- &t2(p/M)ZZZ(Z- 1)~/(2n~))
(59)
must be subtracted from the data to obtain the desired n - 3 scaling for the remainder. Second, since the relativistic energy for a screened hydrogenic electron is given by (in LS coupling)
the second term proportional to C4should also be subtracted to avoid a Ritz defect of order a’. A small Ritz defect of order a’6, remains since the remaining relativistic corrections violate the Ritz condition a4 = 2.: when they are added to the nonrelativistic parts. However, attempts to correct for this do not significantly improve the quality of the quantum defect fits. A direct relativistic version of quantum defect theory appropriate for high-2 ions is discussed by Johnson and Cheng (1979). Applications to fine and hyperfine structure are also discussed by Pendrill (1983). In summary, the procedure is to first subtract the quantity 6E(n) = 6Ez(n)
+ 3a2(Z - 1)4RM/(4n4)
(61)
from the input data, and then add it back after the quantum defect fit has been made. If this is not done, the quality of the fits (as measured by x2) deteriorates by several orders of magnitude. The final energies are then given by E, = -RM/n*’ 6E(n) (62)
+
The values of the quantum defect coefficients 6, required to calculate n* from the Ritz expansion in Eq. (3) are collected together in Table IV. Together with asymptotic expansions for L > 7, these results cover all singly excited states of helium with the exception of the S states. Results for the S
QUANTUM DEFECT ANALYSIS OF HIGH-PRECISION TERM ENERGIES
109
states up to n = 10 will be presented in a future publication. The necessary physical constants are a = 1/137.0359895(61)
and, for 4He, R , = 109722.273515(18)cm-' = 3289391007.44(54)MHz p / M = 1.370933540(30)x
TABLE IV QUANTUMDEFECTPARAMETERS 6j X lo6 FOR THE TOTAL ENERGIESOF HELIUMWITH TERMSUBTRACTID(SEEE~Q. (62)) Parameter
Value 'PI
- 12141.80536(12)
7519.082(14) 13977.78(40) 4837.y3.7) 1228.(9) 'D2
21 13.37842(9) - 3090.051(12) 8.25(51) - 309.1(7.7) - 401.8(36.8)
Value 3p0
Value "1
68357.85568(88) 68328.0002(9) - 18630.46(10) - 18641.98(10) - 12330.4(3.0) - 12331.y3.0) - 7951.7(27.2) -7953.1(27.1) - 5448.3(65.4) - 5444.3(64.4) 9, 3D, 2890.941469(58) 2885.580249(57) - 6357.185380) -6.357.6017(82) 337.84(34) 336.7q36) 838.1(5.4) 839.q5.8) 437.q26.1) 381.7(28.5)
IF3
'F2
3F3
440.29433(80) - 1689.45(9) - 118.2(3.0) 325.3(27.6)
444.86993(14) - 1739.278(16) 104.83(56) 33.1(5.4)
448.59490( 17) - 1727.236(20) 152.48(66) - 249.q6.3)
IG4
124.734488(22) - 796.2292(35) - 12.1q16) - 12.9(2.1)
IH5 47.100906(50) - 433.2276(83) - 8.14(29) 'I6
21.8689311) - 261.071(14) - 3.95(42)
3G3
125.707422(45) - 796.4966(70) -9.88(31) - 1844.1) 3H4 47.79707315) -433.2321(24) - 8.075(82) 31s
22.390775(70) -261.069(9) - 4.01(28)
3G4
128.713165(61) - 796.246(9) - 11.89(41) - 14.1(5.2)
3H5 49.757627(16) -433.2280(26) - 8.112(89) 316
23.768503(72) -261.068(9) -4.02(28)
THE
6E(n)
Value 3p2
68360.28261(84) - 18629.23(9) - 12332.7(2.9) - 7953.2(25.9) - 5449.8(62.3) 3D3
289 1.328793(63) - 6357.7045(89) 336.72(38) 839.1(5.9) 382.9(28.6) 3F4 447.3793q13) - 1739.218(16) 104.81(52) 32.9(5.0) jG5
127.141674(47) - 796.4841(73) -9.86(32) - 19.1(4.3) 3H6
48.729853(14) -433.2285(23) -8.088(77) jl,
23.04763(9) - 261.069(12) - 3.99(36)
110
G . W. F. Drake
IV. Comparison with High-Precision Measurements Two recent series of measurements of unprecedented accuracy provide particularly significant comparisons with theory. Both experiments start from the 1s2s'S0 metastable state of helium and measure transition frequencies to the Rydberg states by laser resonance. The experiment of Lichten et al. (1991) measures the two-photon transitions ls2s ' S o - lsnd ' D o with n in the range 7 < n < 20 to an accuracy of about f 100 kHz. The experiment of Sansonetti and Gillaspy (1992) measures the transitions ls2s ' S o - lsnp ' P o with n in the range 7 < n < 74. The accuracy of each measurement is about f 1.7 MHz, but a quantum defect fit to all the data determines the ionization energy of the ls2s ' S o state to an accuracy of about & 200 kHz. A large number of older, less accurate measurements are reviewed by Martin (1987), and more recent work by Drake (1993a). These measurements are important for two reasons. First, they provide a high-precision test of theory for the energy spectrum of Rydberg states. Second, they determine the absolute ionization energy of the ls2s ' S o state to an accuracy that is within an order of magnitude of the best Lamb shift measurements in one-electron He+ (Dewey and Dunford, 1988; van Wijngaarden et al., 1991). They therefore provide a significant test of twoelectron QED effects. The total QED shift of the 1s2s'S0 state is about 2806.0 MHz [relative to He+(ls)]. The determination of the ls2s ' S o ionization energy can be done in one of two ways. The first method (Method I) is to use a quantum defect fit to the measurements to extrapolate to the series limit, as done by Sansonetti and Gillaspy (1992).This provides a direct determination of the ionization energy that is independent of the theory (other than the validity of the Ritz expansion). The second method (Method 11) is to take the theoretical energies for the Rydberg states as correct and subtract them from the measured transition energies, as done by Lichten et al. (1991). Since the estimated theoretical uncertainty for the 1OP and 1OD states is only f17 kHz (Drake, 1993a; Drake and Yan, 1992), and decreases as l/n3, this is substantially less than the experimental error. Omitted Casimir-Polder retardation corrections are only on the order of 10kHz or less at n = 10 (Au and Mesa, 1990) and therefore do not affect the analysis. Beginning with the P-state measurements, a direct quantum defect fit to the experimental data (Method I) yields an ionization energy of 32033.228859(5)cm- '. This is slightly larger than the value 32033.228855(5)cm-' reported by Sansonetti and Gillaspy (1992) in part because of the 6E(n)terms in Eq. (61),which are first subtracted from the data, and in part because of the bootstrap fitting procedure used in place of an ordinary least squares fit. The fit.includes an additional term of the form Sn' to allow for small Stark shifts in the high-n states (Bethe and Salpeter, 1957,
QUANTUM DEFECT ANALYSIS OF HIGH-PRECISION TERM ENERGIES
111
Sect. 57). Exact diagonalizations of the Stark Hamiltonian in each n = const. manifold of states show insignificant deviations from the n7 scaling at the small background electric field strength of -14mV/cm, at least up to n = 74. Thus the full formula for the transition frequencies is a(2 ' S - n ' P ) = E m - R,/(n - 6,)'
+ 6E(n) + Sn7
(63)
The values of the fitting coefficients are given in Table V. The final uncertainty for E m is + 7 in the last figure quoted due to a f2.5 x lO-'cm-' uncertainty in their laser frequency calibration. The alternative Method I1 fit to the theoretical P-state energies (and their quantum defect extrapolations for n > 10) involves just two adjustable parameters E m and S. This yields the values Em = 32033.228852(4)cm-l and S = 8.06 x lO-"cm-', with x2 = 0.97. The result for E , is slightly smaller than, but statistically consistent with, the value just obtained by the first method. There is therefore no observable difference between the calculated and measured P-state ionization energies. For the D states, Method I1 is statistically the more accurate. However, it yields the value E m = 32033.2288303(8)cm- which is significantly smaller than either of the values from the P states. The result obtained by Method I, as shown in Table V, is smaller by only (1.2 f 1.3) x 10-6cm-'; i.e., 32033.2288291(13). Since the two methods agree with each other for both the P states and the D states, and theoretical uncertainties from QED terms are smaller for the D states, it does not seem likely that the difference can be blamed on the use of the theoretical D-state energies in the analysis. The detailed comparison in Table VI shows that the agreement between the theory and experiments is as good as can be expected. The final uncertainty in E , is dominated by a f 5 x uncertainty in the realization of the meter (Lichten et al., 1991). Table VII shows the influence of including the 6E(n)correction term in the
',
TABLE V QUANTUM DEFECT FITS(METHOD 1) TO THE EXPERIMENTAL 2's - n'P AND 2 'S - n 'D TRANSITION FREQUENCIES Parameter Em S 60 62
64
x2
'P States" 32033.228859(5)cm-' 7.93(14)x lO-"cm-' - 0.0121414(4) O.O0748(5) 0.0153( 19)
0.85
"Sansonetti and Gillaspy, 1992. bLichten et al.. 1991.
'D Statesb 32033.2288291(13)m-' -
0.002113367(18) - 0.0030899(9) -
0.93
G . W. F. Drake
112
TABLE VI COMPARISON OF THEORY AND EXPERIMENT FOR E , - 4 2 ‘ s - n ID), WHEREu Is THE TRANSITION FREQUENCY, AND E , = 32033.2288303(8)cm-’ IS THE AVERAGE2 ’ s IONIZATION ENERGY DETERMINED FROM THE DATA.UNITS ARE Cm-
’
n
Experiment”
Theoryb
Difference
7 8 9 10 11 12 13 14 15 16 17 18 19 20
2240.5405873(40) 1715.294923q23) 1355.2202274(36) 1O97.6794427(33) 907.1396882(35) 762.2257642(35) 649.453397l(23) 559.9751984(35) 487.79 10179(47) 428.71 51748( 14) 379.7557386(36) 338.7281384(56) 304.0071 377(66) 274.3634253(72)
2240.54059336(6) 1715.29492426(5) 1355.22022628(4) 1097.67944605(6) 907.13968947(5) 762.22576424(5) 649.45339659(4) 559.97520308(4) 487.79 102017(4) 428.71517239(4) 379.75573685(4) 338.728 13339(3) 304.00713992(3) 274.36342990( 3)
- O.OoOOO61(40) - 0.0000013(23) 0.000001l(36) -O.OoO0034(33) - 0.0000013(35) -0.0000001(35) O.OOO0@35(23) -0.0000047(35) -O.oOOoO23(47) 0.0000024( 14) 0.0000017(36) 0.000005q56) -0.0000022(66) -O . m 6 ( 7 2 )
‘Lichten et al. (1991). bDrake (1993a) for n < 10, and quantum defect extrapolations fromTable IV for n > 10.
TABLE VII COMPARISON OF THE THEORY AND EXPERIMENT FOR THE QUANTUM DEFECTPARAMETERS. THE 6E(n) = 0 DOES NOTINCLUDE THE TERM 6E(n) IN THE QUANTUM DEFECT COLUMN LABELED FIT(SEEEQS.(61) AND (63)) Parameter
SE(n) included
6E(n) = 0
Theory
32033.228859(5)cm-’ 7.93(14)x lO-”m-’ -0.0 121414(4) 0.00748(5 ) 0.0153( 19) 0.85
‘P States 32033.228857(5)cm- * 7.9q14)x lO-”cm-’ -0.012 142q4) 0.00733(5) 0.0175( 18) 0.85
32033.2288291( 13)cm0.0021 13367(18) -0.0030899(9) 0.93
‘ D States 32033.228803q25) cm0.0021 12107(33) -0.003 1671(17) 2.4
-0.012 14180536(12) 0.0075 19082(14) 0.01397778(40)
’ 00021 1337842(9) -0.OO309OO5 1(12)
QUANTUM DEFECT ANALYSIS OF HIGH-PRECISION TERM ENERGIES
113
quantum defect fit. For the P states, the value of E m is affected only slightly because the measurements extend to such a high n value; but for the D states, the effect is quite pronounced. In both cases, the values of the 6, are generally in better agreement with the theoretical values when the 6E(n) term is included. The one exception is the value of 6, for the P states, which appears to be slightly anomalous. The anomaly, if real, would indicate a systematic discrepancy between theory and experiment in the high-n states. Despite this discrepancy in E m , the two experiments taken together represent a measurement of the ionization energy of the ls2s ' S o state of unprecedented accuracy. Using the Method I1 analysis and converting to MHz, the two results are 960332041.52(21)MHz E m = { 960332040.87(15)MHz
from the P states from the D states
Both are in agreement with the theoretical value 960332039.43 k 0.18 f 1 MHz obtained as shown in Table VII. The electron-nucleus Lamb shift term AEL,' contains the recent high-precision calculation of the Bethe logarithm by Baker et al. (1993). This reduces E , by 93.54(18) MHz relative to the 1/Z expansion calculation of the Bethe logarithm by Goldman and Drake (1984) and removes what would otherwise be a significant discrepancy. The kO.18 MHz uncertainty represents only that due to the convergence of the Bethe logarithm calculation. There are additional uncertainties of at least f 1 MHz due to uncalculated QED corrections of order a4Z4a.u. For example, relativistic corrections of 0(a4Z5)to the lowest-order Lamb shift contribute - 51.99 MHz, calculated in a one-electron approximation, but using the correct electron density at the nucleus (see Drake and Yan, 1992; Drake, 1993a, for further details). The renormalization of the electron density at the nucleus is expected to account for the majority of the two-electron effects, but this is by no means exact. Further progress in the comparison of theory and experiment will require a full calculation of higher-order twoelectron QED corrections (Table VIII). However, the present level of experimental accuracy is within an order of magnitude of the best Lamb shift measurement in one-electron He' (van Wijngaarden et al., 1991), and the agreement with theory already represents a triumph of both the theory and experiments.
V. Summary and Discussion The main purpose of this short review is to summarize a large body of theoretical calculations for the singly excited states of helium in terms of quantum defect representations. The crucial issue then becomes whether or
114
G . W . F. Drake TABLE VIII CONTRIBUTIONS TO THE 2 'S-STATEENERGY OF 4HE, RELATI%TO HE+(~s), IN MHz. THENOTATION Is AS DEFINED IN THE TEXT Contribution AEnr AEG' A E ~ AErd
WRR)M (AERll)X AEnuc AEL.1 AEL.2 Higher-order QED Total
Value -960331428.82 8570.43 - 16.72 - 11969.81 - 14.83 9.84 2.00 3136.3418)" - 330.35 2.49 & lb -960332039.43(18) f 1
"Includes a shift of 93.5418) MHz due to the Bethe logarithm calculation of Baker et a/. (1993) relative to the l/Z expansion value of Goldman and Drake (1984). bDrakeet al. (1993).
not the Ritz expansion for the quantum defect is sufficiently general. A careful analysis of possible theoretical sources of a "Ritz defect" failed to reveal one large enough to affect the data, even when exchange effects are included. Also, revised fits to the nonrelativistic energies for infinite nuclear mass were made in which only one non-Ritz term at a time was included. These gave much smaller coefficients for the non-Ritz term than found previously (Drake and Swainson, 1991), indicating that even with the very high-precision energies now available as input data, the Ritz defect is statistically consistent with zero. On this basis, it now seems safe to say that the Ritz expansion is an adequate functional form, provided that second-order mass polarization and relativistic corrections are not included in the energies. The leading l/n dependence of these terms, represented by SE(n) in Eq. (62), must first be subtracted from the input data for the functional form of the Ritz formula to be correct. The final quantum defect fits to the calculations for a low L, together with asymptotic expansions for a high L ( L 2 7), cover the entire singly excited spectrum of helium with essentially exact predictions for the energies. The higher lying S states have not yet been included in the quantum defect fits, but here, QED uncertainties remain large. Measurements of the 21s3S-n'93S two-photon transition frequencies analogous to the experiment of Lichten et al. (1991) on the D states would be of considerable interest in determining the QED shifts for the S states.
QUANTUM DEFECT ANALYSIS O F HIGH-PRECISION TERM ENERGIES
115
The agreement between theory and experiment for the relative positions of the P states and D states is as good as can be expected, down to the +l00kHz level of accuracy in the case of the D states. However, a discrepancy of about 700 200 kHz in the apparent location of the 2 ' S o state relative to the P states and D states remains to be resolved. For future work, the question of whether or not the Ritz expansion is exact in principle for the Rydberg states of helium (in the nonrelativistic infinite nuclear mass limit) remains to be solved. No specific contribution to the energy that would generate a Ritz defect has yet been identified, and there is no numerical evidence that one exists. The analysis of Section 1I.C shows that Hartree's proof can be extended beyond local spherically symmetric potentials to include exchange terms, at least up to the second order in the interactions. Nonadiabatic effects make the polarization potential effectively nonspherical, but this also does not produce a Ritz defect, at least in the asymptotic limit. It remains to be seen whether some subtle higher order interaction involving nonadiabatic corrections or exchange effects would lead to odd powers of l/(n - 6)in the Ritz expansion.
Acknowledgments I am grateful to John Gillaspy, William Lichten, and Craig Sansonetti for helpful conversations and correspondence concerning quantum defect fits to their experimental results and to Robert Nyden Hill for his contributions to an understanding of the l/n dependence of various contributions to the energy. This work was supported by the Natural Sciences and Engineering Research Council of Canada. REFERENCES Au, C.-K., and Mesa, M. A. (1990). Phys. Rev. A 41, 2848. Babb, J. F., and Spruch, L. (1988). Phys. Rev. A 38, 13. Baker, J., Forrey, G . C., Hill, R. N., Jerziorska, J. D., Morgan, J. D., 111, and Shertzer, J. (1993).To be published. Bethe, H. A., and Salpeter, E. E. (1957). Quantum Mechanics of One- and Two-Electron Atoms, Section 28, Springer-Verlag, Berlin and New York. Curtis, L. J. (1981). J . Phys. B 14, 1373. Dewey, M. S., and Dunford, R. W. (1988). Phys. Rev. Lett. 60,2014. Drachman, R. J. (1982). Phys. Rev. A 26, 1228. Drachman, R. J. (1993a). In: Long-Range Casimir Forces: Theory and Recent Experiments in Atomic Systems (F. S . Levin and D. A. Micha, eds.), Plenum Press, New York. Drachman, R. J. (1993b). Phys. Rev. A 47, 694. Drake, G. W. F. (1993a). In: Long-Range Casimir Forces: Theory and Recent Experiments in Atomic Systems (F. S . Levin and D. A. Micha, eds.), Plenum Press, New York.
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Drake, G. W. F. (1993b). Adv. At. M o f . Opt. Phys. 31, in press. Drake, G. W. F., and Hill, R. N. (1993). J . Phys. B, in press. Drake, G. W. F., Khriplovich, I. B., Milstein, A. I., and Yelkhovski, A. S. (1993). Phys. Rev. A, in press. Drake, G. W. F., and Swainson, R. A. (1991). Phys. Rev. A 44,5448. Drake, G. W. F., and Yan, Z.-C. (1992). Phys. Rev. A 46,2378. Edlbn, B. (1964). In: Encyclopedia of Physics, Springer-Verlag. Berlin and New York. Goldman, S. P., and Drake, G. W. F. (1984). J . Phys. B 17, L197. Goldman, S. P., and Drake, G. W. F. (1992). Phys. Reo. Lett. 68, 1683. Hartree, D. (1928). Proc. Cambridge Phifos. SOC.24, 426. See also Langer (1930). Hessels, E. A. (1992). Phys. Rev. A 46, 5389. Hessels, E. A., Arcuni, P. W., Deck, F. J., and Lundeen, S. R. (1992). Phys. Rev. A 46, 2622. Hicks, W. M. (1910). Phil. Trans. Roy. SOC. A 210, 517. Hicks, W. M. (1919). Phil. Trans. Roy. SOC. A 220, 335. Johnson, W. R., and Cheng, K. T. (1979). J . Phys. B 12, 863. Langer, R. M. (1930). Phys. Rev. 35, 649. Lichten, W., Shiner, D., and Zhou, Z.-X. (1991). Phys. Rev. A 43, 1663. Martin, W. C. (1987). Phys. Rev. A 36, 3575. Pendrill, L. R. (1983). Phys. Scr. 27, 371. Ritz, W. (1903). Ann. Phys., Lpz. 12, 264. Sansonetti, C. J., and Gillaspy, J. D. (1992). Phys. Rev. A 45, R1. Seaton, M. J. (1983). Rep. Prog. Phys. 46, 167. Sommerfeld, A. (1920). Ann. Phys. (Leipzig) 63, 221. Spruch, L. (1993). In: Long-Range Casimir Forces: Theory and Recent Experiments in Atomic Systems (F. S. Levin and D. A. Micha, eds.), Plenum Press, New York. Swainson, R. A., and Drake, G. W. F. (1992). Can. J . Phys. 70, 187. van Wijngaarden, A., Kwela, J., and Drake, G. W. F. (1991). Phys. Rev. A 43, 3325.
ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 32
ELECTRON-ION AND ION-ION RECOMBINATION PROCESSES M . R . FLANNERY School of Physics Georgia Institute of Technology Atlanta, Georgia I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Historical Interlude . . . . . . . . . . . . . . . . . . . . . . . . . 111. Capture-Stabilized Theory as a Basis for Future Discussion. . . . . . . . IV. Electron-Ion Recombination Processes . . . . . . . . . . . . . . . . A. Radiative and Dielectronic Recombination Stabilized by Radiation and Three-Body Collisions . . . . . . . . . . . . . . . . . B. Dissociative Recombination . . . . . . . . . . . . . . . . C. Collisional-Radiative and Collisional-Dissociative Recombination V. Ion-Ion Recombination Processes. . . . . . . . . . . . . . . . . . . A. Mutual Neutralization . . . . . . . . . . . . . . . . . B. Termolecular . . . . . . . . . . . . . . . . . . . . . C. Tidal Recombination . . . . . . . . . . . . . . . . . . VI. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117 118 120 124 124 127 133 135 135 136 143 144 145 145
I. Introduction Recombination processes have historically been distinguished by the mechanism responsible for disposal of the excess energy of recombinationvia radiative decay, dissociation or collision with third bodies. When intermediate resonant states become involved, the processes are further distinguished either by the mechanism for production of an intermediate, collisionally delayed energy-resonant complex of superexcited states if it exists, as in dielectronic recombination; or by the mechanism subsequently responsible for stabilization of this complex, as in dissociative recombination. A more robust nomenclature based on a combined capture-stabilized notation can be proposed for resonant processes. The customary electronatomic ion dielectronic recombination and electron-molecular ion direct dissociative recombination are then assigned as dielectronic-radiative and 117 Copyright
CI 1994 by Academic Press, Inc.
All rights of reproduction in any form reserved. ISBN 0-12-003832-3
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M . R . Flannery
dielectronic-dissociative recombination, respectively, since electron capture initially occurs in both cases via resonant formation of a doubly excited atomic or molecular state, which is subsequently stabilized either by radiation or by dissociation. Collisions with third bodies not only can stabilize dielectronic recombination to give dielectronic-collisional recombination, but it can also enable dissociative recombination to yield collisional-dissociative recombination. Recombination processes are exothermic, with the cross sections 0 exhibiting a monotonic decrease with increasing energy E except in the cases where specific windows of opportunity open up, with appearance of (high) energy thresholds for accessing these superexcited resonant complexes. For electron-atomic ion dielectronic recombination, 0 manifests a series of characteristic Breit- Wigner resonance profiles characteristic of autoionization from doubly excited states. Recombination processes span a wide range of physical conditions, the ranges 10 < T < lo6 in temperature T("K) and 1 < N < 10'' in particle density N(cm-3) of the components. They assume significance in astrophysical plasmas, interstellar chemistry, comets, shock waves, laboratory and laser produced plasmas, in electron cooling of heavy ion beams in storage rings, in discharges, in magnetic and laser confined fusion devices, and in electron-beam ion-trap experiments. Apart from their obvious applied interest, their study provides much fundamental knowledge as to the nature of dynamic interactions between multicomponent atomic and molecular species, the three-body problem, the formation of transient superexcited molecular complexes not normally accessible to experiment but common to many exit channels or energy pathways, and the subsequent formation of neutralized fragments. Some of the current interesting basic problems are concerned with the nature of these superexcited states, the mechanism for their stabilization and the partitioning into the various products of fragmentation, as in dissociative recombination. Three-body ion-ion recombination in a gas represents the study not only of the simplest chemical reaction involving three bodies in particular, which is well suited to a textbook case study, but also serves, as the gas density is increased, as a prototype study of transport-influenced reactions in general.
11. Historical Interlude Apart from the classic case of three-body ion-ion recombination in a gas, pioneered by Langevin (1903) at high gas densities, by Thomson (1924) at lower gas densities, and subsequently placed on a firm microscopic basis in the low gas density limit by Bates and Moffett (1966) and by Bates and
119
ELECTRON-ION AND ION-ION RECOMBINATION PROCESSES
Flannery (1968), and then by Flannery (1982a) for all gas densities, investigation of the Earth’s ionosphere and the solar corona resulted (see Bates, 1988) in the discovery and subsequent explanation of dielectronic recombination (DIR) by Massey and Bates (1943), mutual neutralization (MN) by Bates and Massey (1943) and dissociatiue recombination (DR) by Bates (1950). In an effort to explain the large effective rates aefffor electron recombination in the ionosphere, Massey and Bates derived an expression for the rate aDIRof the process e - + O f ( i )-,O * * ( j , nl) -+ 0 + hv, which they termed dielectronic recombination (since both incident and bound electrons participate), but found MDIR << a e f f Bates . and Massey then proposed a negative ion theory of recombination and introduced in 1943 the new process 0’ 0--, 0 O* of mutual neutralization, which they recognized as occurring via avoided crossings of the adiabatic potentials and estimated its rate aMN lo-’ cm3s-’ to be large. Due to other compelling evidence, they discarded the negative ion theory of the ionosphere in 1946. In 1947 Bates and Massey then inferred from laboratory data on associative ionization, Cs(n2P)+ Cs + C s z + e - , that its inverse, dissociative recombination, was sufficiently rapid in general to account for recombination in the atmospheric E-layer. Bates, however, waited until the stimulus of measurements of Biondi and Brown (1949) on the rate of decay of electrons in a helium afterglow before he discovered and succinctly explained in 1950 (in a two-page Physical Reoiew letter) the beautiful mechanism underlying dissociative recombination. Exactly 40 years later Bates and Morgan (1990), in an explanation of why XeCI* is the dominant species produced in Xe:-Clrecombination in a gas, discovered yet another beautiful mechanism called tidal ion-ion recombination, again succinctly explained in a three-page Physical Review letter. Within that time span Bates placed recombination on a firm microscopic basis. Bates (199la-c) has recently discovered and thoroughly investigated a new class of dissociative recombination, which he termed superdissociative cm’ srecombination, characterized by much higher rates a (Incidentally, the original inference (see Bates, 1988)that e - + Cs; was fast was later invalidated by an essential curve crossing feature of Bates’s theory; there is no curve crossing between the ion Cs: and neutral repulsive potentials. Also, the inferred atypical value aDR(T) lo-’ em's- for He: of Biondi and Brown, which was obtained from too small a range of electron density data to be accurate, was published just six months before the more 7. 10-’cm3 s - l at 300K. As for dielectronic, typical e - Ne: rate a,,(T) Unsold suggested to Seaton in 1961 (see Seaton and Storey, 1976) the possibility that the 1943 dielectronic recombination process of Massey and Bates might be responsible for the discrepancy between temperatures of the solar corona obtained from Doppler profiles and ionization equilibrium. This
+
-
-
-
+
’.
-
120
M. R. Flannery
discrepancy was greatly reduced by Burgess (1964), who recognized at very high temperatures lo6K that the sum of small contributions from a very large number of doubly excited autoionizing states provided rates ~t!,,,-5*10-'~ cm3s-' much larger than the rate u R - 10-'4cm3s-' for pure radiative recombination-a significant advance.) N
111. Capture-Stabilized Theory as a Basis for Future
Discussion Recombination processes are best explained by a theory that tracks the detailed history of the recombining species as they undergo various energyand angular-momentum-changing collisions and interaction with radiation, as exemplified by the collisional-radiative models of Bates, Kingston and McWhirter (1962) and of Bates and Khare (1965) for electron-ion recombination in a plasma and a gas, respectively, by the developments of Seaton and Storey (1976) and Bell and Seaton (1985)for dielectronic recombination, by the unified theory of Giusti (1980) that couples direct and indirect dissociative recombination,and by the microscopic theory of Flannery (1982a, 1991)for ion-ion recombination in a (high-density)gas. It is only then that a macroscopic theory, with its reliance on artificial parameters as a trapping or capture radius or on macroscopic physical parameters as stabilization probability or frequency, can be intelligently exercised. These parameters are in general detailed averages over nonequilibrium distributions and can be assigned once the detailed history of the recombining species are known. They are, however, useful for discussion purposes. A strong energy-reducing collision i +A even if unrealistic, is taken in macroscopic argument to mimic the effect of many weak i ej e k -tf energy-changing collisions. The following macroscopic two-stage view of recombination,
between A and B to form a specific product channel, Ci and Di ( i = 1,. . .,N), has a microscopic basis (see Flannery, 1987a). Here recombination between A and B (which may be electrons, ions or neutrals) occurs by stabilization with frequency v,,-via emission of radiation or collision with third bodies M-of a complex AB* temporarily formed at capture rate k, and subject to redissociation (or autoionization) at frequency v,. The complex AB* denotes a long-lived atomic or molecular species with a total internal energy above its
ELECTRON-ION AND ION-ION RECOMBINATION PROCESSES
121
dissociation limit. A steady-state macroscopic analysis of (1) yields the overall two-body rate k(cm3s-') at temperature T as
where is the defined probability of stabilized production of channel i. When vSi >> v, then Pf x 1 and k = k,, the initial rate of capture as for dissociative recombination and for dielectronic recombination involving highly excited Rydberg levels n >> 50 at high T 2 lo7K. The capture rate k, in (2) may be recast in terms of the reaction (complex) volume (cm3),
K ( T ) = iiTB/fiAiiB= kc/va
(3)
where fis are the equilibrium number densities of species S, so that,
Equation (3) is a macroscopic detailed balance relation, i.e., K is not an equilibrium constant in the usual sense since nfB includes only those states that satisfy energy and angular momentum conservation above the dissociation limit. It is given in usual notation by
where q and o are the internal partition functions and electronic statistical weights associated with each reactant A and B and complex AB* of reduced mass MAE. (a) For electron-atomic ion B' collisions, K ( T )=
k3 0* exp[ - E(B**)/kT] ( 2 ~ m k T ) 2~0' ,~
the Saha-Boltzmann distribution, where E** is the energy of the autoionizing state B** relative to the original ion B + and h3(2nmkT)-3'2 = 4.1212. 10- ' 6T-3 /2 ~ m3When . the stabilization in (1) is by radiative decay at frequency v, = v,, then (4)with (6) provides the rate for dielectronic recombination in the isolated resonance approximation, and is the basis of a large body of work by Nussbaumer and Storey (1984). There is an intimate and instructive connection between the terms in square brackets in (4)standard in chemical kinetics with detailed BreitWigner resonance scattering theory (cf. Flannery 1990).
122
M . R . Flannery
(b) For processes like termolecular ion-ion recombination, which do not proceed in general via a time delayed resonance, the connection of Eq. (4) with microscopic theory is as follows. The rate for positive ion-negative ion ( A + - B - ) recombination stabilized by collisions, at frequency v, with third bodies M for A-B separations R < R , is, K(T)
b) k, = K(T)v, = K(T)/T(E,
P'(E, b; Ro)
where V = (8kT/nMA,)'/' and where nbi = nRi[1 - V(R,)/E] is the capture cross section for formation under mutual interaction V(R) and energy E of ( A + - B - ) complexes with R < R,. The probability for ( A + - B - ) complex-M reactive collisions of frequency v and cross section CT along the ( A + - B - ) trajectory of elemental length ds, of energy E = E/kT and impact parameter b is
Ps(&,b; R,)
=
$1:
v(t)dt =
$1:
ds/1
where Ri is the orbit's pericenter and ,I= (No)-' is the microscopic path length toward reactive collision with M . Since the redissociation R > R , occurs after a time Z(E, b) = f dt from first entry at R, into the reaction zone, the frequency v, for redissociation is 7 - ' , as in (7). For constant 1 (hard sphere, s-wave and charge-transfer collisions), P = .Y(E, @/ where ,I, 9 is the length of the orbit segment enclosed by the sphere of radius R,. The ( b , s ) order of integration is reversed to give, for Coulomb attraction Vc(R), between the ions, the rate
kc
This expression for Ps(Ro) is the exact generalization (Flannery, 1991) to hyperbolic orbits of Thomson's straight line collisional probability 4 R o l l valid at low gas densities N. For ion-neutral spiraling collisions under polarization attraction at constant frequency v,, then Ps = V,T(E, b) = v,/v, so that (7) directly yields a = K(T)v, = K(T)v,Ps. A classical variational treatment (Flannery, 1991) assigns R , to be 0.4(e2/kT)= 0.4Re,in agreement with
123
ELECTRON-ION AND ION-ION RECOMBINATION PROCESSES
more sophisticated calculations (Flannery and Mansky, 1988a). The rate for termolecular (volume) recombination at low gas densities N is then aTER = 0.32ij ($xRJ3 a,N
For ion-ion termolecular recombination, Ua, aTER
-
2' 10-25(300/T)2.5N
-
(10) lo-' cm3 s - l , so that
cm3 s-'
(11)
For electron-ion termolecular recombination in an electron gas of density n,, 6, = 4nR; is the Coulomb cross section for energy changes 2 3kT of the recombining electron (i.e., for scattering angles 8 2 42), and a,,(T) = 2.7- 10-26(300/T)4.5necm3s-l
(12)
which agrees with the more elaborate analysis of Mansbach and Keck (1969). For electron-ion termolecular recombination in an atomic gas M, only a small fraction 6 = (2m/M)of the electron's energy is lost on collision so that the &-limitsof integration must be restricted to between 0 and E , = 6e2/R(see Bates, 1980a). Expression (7) for constant 1 is then recast as (Flannery, 1991) aTER =
[Jr
1
4 x R 2 d R JoEm(R)fi(R, E)u(R, E ) d E (6,N)
(13)
where fi(R, E) is the Maxwell-Boltzmann equilibrium distribution and u is the relative ( A + - B - ) speed. The rate for ( e - A + ) collisional recombination is then (Flannery, 1991)
-
[10-26/M(a.m.u.)](300/T)2.5Ncm3 s - I (14) which agrees with the energy diffusion result of Pitaevskii (1964) when Ro = $ R e , the Thomson radius, and which is linear (Bates, 1980a) in the trapping radius Ro in contrast to (9), which varies as RG for ion-ion recombination. Most recombination processes may be discussed in terms of these simplified arguments once the basic mechanisms are treated at the microscopic level so that artificial parameters such as R , and physical parameters such as Ps, which represent results of detailed averaging over nonequilibrium distributions, can be assigned, Recombination in dense gases is an example of transport-influenced reactions when the rate of transport aTR(RO) to R,, say, becomes comparable to and eventually becomes much less than the rate a(R,) of the reaction within R,.
124
M . R . Flannery
IV. Electron- Ion Recombination Processes A. RADIATIVEAND DIELECTRONIC RECOMBINATION STABILIZED BY RADIATIONAND THREE-BODY COLLISIONS Direct radiative recombination (RR),
the excess energy being carried off by radiation hv spread over the recombination continuum is normally regarded as time-reversed (or the detailedbalance inverse of) photoionization without autoionization (with no intermediate collisionally delayed complex formed). Scaling behavior for all ions and angular distributions of hv have been discussed by McLaughlin and Hahn (1991) and by Pajek and Schuch (1992), respectively. Detailed calculations for (e- - 0') have been performed recently by Escalante and Victor (1992). Dielectronic (radiative) recombination (DIR),
an example of a recombination process occurring via an intermediate resonant state (k, nl), can be regarded as the inverse of autoionization stabilized by radiative decay at discrete frequencies vkj. In (16) the electron excites the electronic configuration ( i -+ k) of the target ion and is thereby captured into an excited state nl that lies below the state k of the excited ion. This doubly excited (dielectronic) state is embedded in the ionization continuum of the original ion state i and other accessible states of the ion. It can therefore autoionize back to all these states at probability v, = A, per unit time, or else be stabilized to lower, fully bounded statesf by radiative decay with probability v, = A, per unit time. This radiation appears as satellites, usually on the lower frequency side of the resonance transition k + j of the recombining ion and is observed in solar and in high temperature fusion plasmas. This radiation is a valuable diagnostic of electron temperature, electron densities and the various stages of ionization. At high temperatures T lo7K 1 keV characteristic of the solar corona, the full Rydberg series of autoionization levels nl must be included as Burgess (1964) realized, and core relaxation is the main radiative decay mechanism. For n >> 50, the autoionization probability A, << A,, the radiative probability, so that (4) yields k(T) = K ( T ) A , , which is autoionization limited, and k simply reduces to the initial rate k, of capture. The separation AE
-
-
ELECTRON-ION AND ION-ION RECOMBINATION PROCESSES
125
-
between the resonances of the Rydberg series Z2/n3becomes less than the radiative width r, = hA, as n increases, the detailed resonance structure is then smeared out by interaction with the radiation field, and the presumed isolated resonance approximation loses validity. Radiative stabilization can also occur via nl+ n’l’ transitions of the Rydberg electron, a mechanism increasingly effective at much lower temperatures 7‘ < lo4 K, characteristic of planetary nebulae. For n << 50, A, >> A, and (4) is radiatively limited to give k(T) = K(T)A,, the original result of Massey and Bates (1943) and the basis of extensive calculation by Nussbaumer and Storey (1984) on low-energy dielectronic recombination. Although the cross section for radiative recombination (15 ) decreases with increasing energy, a characteristic of direct i +f energy-reducing transitions, and is effective mainly at lower temperatures T < lo3K, while the cross section for dielectronic recombination is energy selective, exhibits a series of Breit-Wigner profiles, a characteristic of intermediate i + k +f transitions, and is effective mainly at much higher T and for highly charged ions, consideration of (15) and (16) in isolation is artificial. The two mechanisms are coupled and interference may occur. Interaction with the background RR continuum and overlapping resonances-important for lower 2 and lower T-is achieved by a unified treatment. Dielectronic recombination at low T was first explained by Bates and Massey (1943). Bell and Seaton (1985) then developed a precise theory of dielectronic recombination in terms of multichannel electron-ion theory and quantum defect theory, which, because of its close connection with Rydberg series, is ideally suited to dielectronic recombination at a high n and a high 7: Chen (1986), Badnell and Pindzola (1989), Hahn (1985),Hahn and LaGattuta (1988), Haan and Jacobs (1989) and Nahar and Pradhan (1992) have made significant contributions to DIR. The theory appears to be effectively complete. Good agreement between the theory and experiments (Ali et al., 1991) exists for heliumlike argon. Photoionization cross sections apand electron-ion collision cross sections can now be calculated by ab-initio close coupling and R-matrix codes (Berrington et al., 1987) as used in the large-scale opacity project of Seaton (1987).Hence, o pwith its rich series of autoionization resonances may be used to extract the cross section or for the superimposed effects of radiative and dielectronic recombination via use of the detailed balance (Milne) relation 0
h2v2 rJp(v)
a,(v) = 2-2wi mc2 +mu2
where 0,and m iare the electronic statistical weights of the recombined and recombining ions, respectively. The microscopic forward and reverse routes must be exact inverses.
126
M . R . Flannery
n a
&0 M'
-
-11.5
0
FIG. 1. Rates for uncoupled radiative (RR) and dielectronic (DR) recombination calculated separately. Total represents combined closely coupled rate. See text for details. (From Nahar and Pradhan, 1992.)
Figure 1 illustrates various calculations of
for a Maxwellian distribution f(u) of electron speeds u. The unified treatment (solid line) of Nahar and Pradhan (1992) couples both radiative (RR) and dielectronic (DR) in a unified manner via close coupling of all states. The dotdash curve (- .-) represents the effective D R and RR contribution implicit from integration over detailed photoionization cross sections that include autoionizing resonances for all states with n < 10. The dash-long-dash curve is the D R contribution arising from higher levels 10 < n < co obtained from the precise Bell and Seaton (1985)theory. The isolated radiative contribution is RR, the radiatively limited dielectronic rate of Nussbaumer and Storey (1984) is DR(NS), and * denotes high-temperature dielectronic isolatedresonance rates of Badnell and Pindzola (1989). The figure shows that the sum of RR and D R considered uncoupled results in an underestimation of the overall rate, particularly at intermediate temperatures. See, however, Pindzola et al. (1992). Not only can the dielectronic captured complex A('-')+ be stabilized by radiation, but at sufficiently high electron densities n, 2 loi6~ m - colli~ ,
ELECTRON-ION AND ION-ION RECOMBINATION PROCESSES
127
sional de-excitation by electrons becomes relevant (Bates and Dalgarno 1962, Jacobs and Davis 1978).The rate of dielectronic recombination stabilized by radiation and three-body collisions is then
where the effective rate for electronic de-excitation into fully bound levels is a,, and the full rate for e - A('-')+ collisional excitation, de-excitation and ionization is ax.This rate (19) can be important in ion-beam driven interial confinement fusion and in XUV laser experiments. B. DISSOCIATIVE RECOMBINATION
1. A Direct Process As Bates explained in 1950, dissociative recombination for diatomic ions
occurs via a crossing at R , between the bound and repulsive potential energy curves V + ( R )and V**(R)for AB' and AB**, respectively. Here, DR involves the two-stage mechanism e-
+ AB+(oi)$(AB**),
--t
A
+ B(*)
--t
A
+ B + hv
(20)
wherein the free electron of energy E = V**(R)- V + ( R )excites an electron of the diatomic ion AB+ with internal separation R and is then resonantly
". , 1
1.5
2
2.5
3
3.5
4
R(a@
FIG. 2. Diabatic potential energy curves involved in the low-energy dissociative recombination of H i . The broken curves symbolize a few (lsa,nla,)' Z: Rydberg states, with 1 = 0 or 2. (From Schneider et al., 1991.)
128
M . R . Flannery
captured by the ion to form a repulsive state AB** of the doubly excited molecule, which in turn can either autoionize at probability frequency v, or predissociate into various channels at probability frequency vd (see Fig. 2). This competition continues until the (electronically excited) neutral fragments accelerate past the crossing at R,. Beyond R , the increasing energy of relative separation has reduced the total electronic energy to such an extent that autoionization is essentially precluded and the neutralization is then rendered permanent past the stabilization point R,. Bates's interpretation has remained intact and robust in the current light of ab initio quantum chemistry and quanta1 scattering calculations for the simple diatomics (O:, N:, Ne:, etc.) Observation of emitted radiation hv yields information on the excited products (Malinovsky et al., 1990). Characteristics
(a). This process is dielectronic in the sense that the electron is indeed captured into a doubly excited state. The stabilization (dissociative) mechanism in DR is much more secure than in DIR, in that the product of dielectronic-radiative recombination (16) may be subsequently reionized by interaction with its environment. The doubly excited state AB** is resonant with respect to resonant capture at fixed internuclear separations R, i.e., E + V + ( R )= V**(R),but motion of the nuclei permit the satisfaction of this energy matching over a continuous range of E, in contrast to discrete windows for E in electron-atomic ion dielectronic recombination. The recombination is fully stabilized at the crossing point R,, where V+(R,) = V**(R,). For favorable crossings, electrons with near zero energy E have access to doubly excited molecular (predissociative) states in contrast to the high E windows for doubly excited (autoionizing) atomic levels ( E = 35.2 eV for He and 0.50 eV for 0 above the ion ground state). (b). The beauty of DR is that the kinetic energy E of the electron in the field of A B + is effectively transferred to motion of the nuclei, not by direct collision but via a rearrangement of the whole electronic cloud-the double electron capture transition into two antibonding molecular orbitals is accompanied by change of the vibrational wavefunction for the nuclei from an initial bound to a final free state. DR is reactive in that reactant and product species are different, thereby resulting in a combination of electron-ion and neutralneutral scattering technologies with a quantum chemical description for the excited complex. Thus, in spite of its resonant character, the cross section uDR for (20) will not exhibit sharp resonance structure, as for (16), but will decrease initially as E - with broad superimposing oscillations arising from vibrational overlap. Stronger autoionization at higher E will then cause a more rapid variation.
ELECTRON-ION AND ION-ION RECOMBINATION PROCESSES
129
2. 7he Direct and Indirect (Curve-Crossing) Mechanisms When the two diabatic potential energy curves V + ( R )and V**(R) cross at R,, the rate for dissociative recombination given by a two-state semiclassical treatment (Flannery, 1993a) is
(21) where V,/,,(R)is the electronic energy, (4,,(R,r)lHel(R,r)14e(R, r)), coupling the electronic wavefunction 4 d for the doubly excited repulsive (A@,** intermediate state to the electronic wavefunction 4efor the scattering system e-AB+. The molecular functions f$d,E are diabatic in that they are not pure eigenstates of the full electronic fixed-nuclei Hamiltonian H,,(R, r), and the continuum functions are energy normalized according to ( d E1 4,.), = a(& - E') with consequent unit density P(E) of states for the scattered electron of energy E. The bound vibrational wavefunction for the original ionic state is $ + ( R ) . From the time ti for initial electron capture into AB** at nuclear separation Ri to separation R(t) < R,, the probability for survival of classical A - B motion dissociating under the potential V**(R) in the two state treatment is
where Tc = hv, defines the energy resonance width for electron capture in terms of the frequency v, for autoionization. Since
-
then the observed rates, aDR 2.10- 7(300/T)'/2cm3 s-', for normal diatomic ions (with large dissociation energies D): imply that the dimensionless quantity in the curly brackets is 1/8. A first-order quanta1 treatment (via Fermi's golden rule for v, combined with detailed balance to give the capture cross section oc and the delta function representation, rl/**(R) = (aV**/dR)- '126(R - R,) for the dissociating vibrational wavefunction) 1, and agrees with the working yields (23) with aI/+/aRomitted and P , formula of Bates (1990a). The omitted term, being zero only for crossing at the potential minimum, would become important for ions initially in
-
-
130
M . R . Flannery
vibrationally excited levels and does distinguish between crossings on either side of the potential minimum. Thus (23) predicts the tendency for crossings to the repulsive side of the minimum to contribute more than those at the attractive side, all else being equal. Bardsley (1968) has previously pointed out that the three-stage mechanism,
e-
+ AB+(u')
+ [ A B + ( u f )- e-1,
-,(AB**), + A + B*
(24)
the so-called indirect process, might be important. Here the accelerating electron loses energy by vibrationally exciting (0' + u,) the ion and is then captured into a Rydberg orbital of the bound molecule AB*, which then interacts (via configuration mixing) with the doubly excited repulsive molecule AB** (see Fig. 2). The capture initially proceeds via a small effectthe breakdown of the Born-Oppenheimer approximation-at certain resonance energies E, = E(uf) - E(u2) and, in the absence of the direct channel, would therefore be manifest by a series of characteristic Lorentz profiles in the cross section. Uncoupled from (20) the indirect process could augment the rate. Capture proceeds more easily when uf = uf + 1 so that energy conservation stipulates that Rydberg states with n = 5 - 9 be involved. Recent ab initio treatments (Schneider et al., 1991; Takagi et al., 1991) that couple both (20) and (24) in a unified manner have, however, shown that the indirect process interferes destructively with the direct process for (e-H:) and (e-CH') recombination. The cross sections in Fig. 3 exhibit a series of dips falling below the E - ' I 2 continuous variation for the direct process. Here the direct + indirect coupling interrupts the recombination at specified
80
100
120
RmV) FIG.3. Comparison between theory and merged beam measurements for the rate of e--H,+ dissociative recombination.(From Schneider et al., 1991.)
ELECTRON-ION AND ION-ION RECOMBINATION PROCESSES
131
energies. For heavier diatomics and polyatomics the indirect process is of little significance (Bates 1991a)when the direct curve crossing mechanism is operative.
3. Modern Quanta1 Methods: Diatomics There exist two main quantal treatments for dissociative recombinationone (CM) based on configuration mixing (Bardsley, 1968; Bottcher, 1976; O'Malley, 1981), and the &her (MQDT) based on multichannel quantum defect theory (Lee, 1977;Giusti, 1980). Giusti (1980)removed some of the inconsistencies in the unified MQDT approach of Lee (1977).CM has been applied to H: by Bottcher (1976),Giusti-Suzor, Bardsley and Derkits (1983) and Hickman (1987).MQDT has been applied to H,: (Nakashima, Takagi and Nakamura, 1987,Van der Donk et al., 1991,1992,and Schneider et al. 1991,to NO+ (Sun, Nakashima and Nakamura, 1993),to 0: (Guberman and Giusti-Suzor, 1991),to N: (Guberman, 1991)and to CH' (Takagi et al., 1991). This quantal theory for diatomic ions is considered to be essentially complete for cases involving favorable crossings between the ion and doubly excited neutral states. Takagi (1993)has recently modified MQDT to see if rotational nuclear motion results in an observable structure. A generalization of the nonadiabatic K-matrix method has recently been applied (Sarpal et al., 1993)to HeH+, a case of dissociative recombination without curve crossing, to yield good agreement with merged beam results (Yousif and Mitchell, 1989). Evaluation of branching ratios for the case of several interacting dissociative channels needs to be developed by a combination of close coupling and MQDT methods. Agreement with the merged beam experiment (Van der Donk et al., 1992) exists for the (e-H:) system (Fig. 2). A recent flowing afterglow measurement (Canosa et al., 1991) of (e-N:(v = 0)) yields ct 2.6.10-7cm3s-' to be compared with the ab initio MQDT result of 1.6.10-7cm3s-'of Guberman (1991)and the merged beam result (cf. Mitchell, 1990)of 3.6.10-'cm3 s - l at 300K.Dissociative recombination of 0; is a source of the oxygen green line ('S + 'D) and red lines ('D -+ 3P)at 5577 A and 6300A, respectively, in the atmospheres of Earth, Mars and Venus. By analyzing two sets of observational data on the nightglow, Bates (1992b) found an inconsistency, as yet unexplained, with ab initio 0; calculations of Guberman and Giusti-Suzor (1991). The large difference (cf. Canosa et al., 1992) between various measurements (2. 10- cm3s - l ) for H i (significant to the Jovian atmosphere) has now been reduced to 10-7-10-8cm3s - l . The recent conference proceedings of Rowe and Mitchell (1993)provide interesting details of the present status of the quantal theory and experiment.
-
-
132
M. R. Flannery
4. Normal and Super Dissociative Recombination
-
Bates (1991b) has recently distinguished between two classes of dissociative recombination: normal, characterized by rates aDR(T) 2. 10-7(300/T)1/2,as for ions O:(v+ = 0) and NO+ with large dissociation energies DZ 7eV and 1 1 eV, respectively; and super, characterized by much larger rates a,,(") 2 . 10-6cm3s-'. Two families are associated with these super rates: (a) dimers of atomic and molecular species, such as Xe: and N;, with relatively low dissociation energies 1eV, so that many Rydberg levels of the dissociated neutral products are reached by increased curve crossings between V' and V**, and (b) proton-bridge ions, such as H 2 0 * H' * H 2 0 and NH3 * H + * NH,, where dissociative recombination occurs in the absence of curve crossings (V* lies below V') with a rapidity attributed to singleelectron transitions (rather than dielectronic as in (a)), combined with a multistep indirect dissociative recombination process, recently introduced by Bates (1992~). Although all the modern quantum (scattering and chemistry) technology has been brought to bear for the simpler diatomics in a way considered in general correct, simple expressions as (21) are invaluable in that they reveal tremendous insight into the essential physics, general characteristics and workings of recombination for various distinct systems. With remarkable ingenuity Bates has again demonstrated how expressions such as (21) may be utilized to promote new insight. His recent five papers (Bates 1991a-c, 1992a,c) remain a powerful testament to this approach. As Bates (1991b) explains, normal DR systems have dissociation energies D: 6.66eV for 0; and 10.85 for NO', high compared with those (1.31)eV for those super DR systems that are dimer ions of rare gas atomic or molecular species as Ne:, Ar:, Kr;, Xe: or 0: .02,NO' .NO and N i .N2. This is due to the stronger valence bonds in the normal systems versus the weaker electron-exchange (delocalization) bonds in the dimers. When the difference between DZ and the smaller of the ionization energies I, and I, of the products A and B is small, then A and B can remain only in their ground states. Larger differences (e.g., -,11 eV for Xe:) imply that A and B can be left in various Rydberg states. The electronic weight ratio (w**/w+) in (21) is therefore increased. Smaller DZ also implies that the ions AB' can be distributed among excited vibrational levels v: at room temperature (e.g., u+ 5 for Xe:), with increased access to the increasing number of crossings with Rydberg products. Moreover, the gradient la(V** - V')/aRI becomes smaller for systems with lower D: and higher u+, although d V + / a R will eventually offset the unlimited decrease as dV**/aR tends toward zero. All of these factors, increased statistical weight a**, increased number of crossings at various separations R,, and reduced gradients, conspire in (21) to render
-
-
-
-
-
ELECTRON-ION AND ION-ION RECOMBINATIONPROCESSES
133
super DR rates -2. 10-6cm3s-' for various systems large compared with normal DR rates 2 . lo-' cm3 s- '. This reasoning of Bates for population of the Rydberg levels n in e - + N l . N, + N, + N,(n) was soon confirmed by Cao and Johnson (1991) in their spectroscopic observations of photoionized afterglow plasmas. Bates (1992a) has also discovered a second family (the dimer ions being the first) of systems with super DR rates: cluster ions such as H,O+(H,O), and NH:(NH,),. Here, not enough energy is available to populate the Rydberg states by dissociation, and the actual capture mechanism is not dielectronic but occurs via a single electron transition, as in
-
+
H 2 0 * H + * H 2 0 e - +2H,O
+ H(1s)
(25)
where the relevant potentials V,* and I/+ not only do not cross, but I/: for a singly excited state always lies below V + .The super nature of this DR process arises from the increased single-electron attachment frequency LOL6s-' in contrast to that, 3 * 1014s-1, for the dielectronic process (24). For these systems with no crossing Bates (1992~)has introduced multistep indirect DR that proceeds via several intermediate states N by a sequence of Au = uf - u: = 1 transitions. Bardsley's indirect process for one intermediate Rydberg level is based on second-order perturbation theory. Since Bates invokes N intermediate states, the required formulation demands cumbersome proapplication of (N + 1)-order perturbation theory-a cedure. A simplified approach concluded that multistep indirect dissociative recombination not only accounts for the HCO+ measurements, but is probably as common for the no crossing-single electron transition case as direct DR is in all other instances. When crossings exist, however, predissociation is normally so rapid that the multistep process loses significance.
-
C. COLLISIONAL-RADIATIVE AND COLLISIONAL-DISSOCIATIVE RECOMBINATION The classic collisional-radiative model of Bates et al. (1962) for electron-ion recombination, e- + A +
+ e- + A
+e-
(26)
in an electron gas takes detailed account of the whole array of mechanisms in operation: radiative recombination, radiative cascade, three-body recombination to a specific state n, electron-impact excitation, de-excitation and ionization and self-absorption of (Lyman-a) emitted radiation by groundstate atoms. At low electron temperatures T, and high electron densities a,,
134
M . R . Flannery
the recombination is mainly collisional governed therefore by the rate (12).At higher T, and lower n,, the highly excited levels collisionally formed within kT of the ionization limit become increasingly stabilized by radiative transitions. A working formula that agrees with experimental data for collisional recombination is (Stevefelt, Boulmer and Delpech, 1975) uCR= 3.8. 10-9T,-4.5ne + 1.55. 10-10Te-0.63 + 6 . 10-9T-2.98 0.37 e ne
(27)
Cm3s-l
where the first term is the collisional rate (12), the second term is the radiative contribution and the third term arises from collisional-radiative coupling. Analogous collisional-radiative models for e-
+ A + + M -+ A + M
(28)
were also developed by Bates and Khare (1965) and by Bates, Malaviya and Young (1971) for atomic and molecular gases M, respectively. These models have received widespread application and are well documented. At lower gas temperatures and higher electron densities (28) proceeds mainly by e-M binary collisions at the classical rate (14).As the gas density N is increased, AM' is however formed and then undergoes dissociative recombination. The DR is enhanced by the formation, via ( e - M ) collisions, of Rydberg (nl) electron-molecular ion pairs (e-AM+),, . This process is collisional-dissociative recombination (Bates, 1981, 1982). At even higher N, the rate of (28) becomes limited by the mobility K for e- in M by the Langevin rate 4nKe N - l . Monte-Carlo simulations for (28) have been performed (Morgan, 1987) as a function of N and compared with experiments for A = M = COz, NH3 and HzO. When M E A, recombination of electrons to ions in their parent gas not only proceeds via (e-A) binary collisions as in (28), but also via
-
e-
+ A + + A + e - + A:
+At* + A
+ A*(n)
(29)
Here A + + A forms a quasi-molecular complex that is a superposition of the (gerade) ground and (ungerade) first excited electronic states of A:, as in the theory of symmetrical resonance charge transfer. Since the probability that A + A possesses A: ground state character is 4, the recombination can then proceed via dissociative recombination (Mihajlov et al., 1992). This DR is also enhanced by collisional-dissociative recombination (e-A:)nl. The overall process (29) is effective mainly at small ion-atom (A' + A) separations, in contrast to the larger separations where (e- A) binary collisions effect the recombination. +
135
ELECTRON-ION AND ION-ION RECOMBINATION PROCESSES
V. Ion-Ion Recombination Processes A. MUTUALNEUTRALIZATION As introduced by Bates and Massey (1943), mutual neutralization, A+
+ B-
-+ A
+ B*
(30)
occurs via curve crossing of the diabatic potential energy curves associated with the ionic and covalent molecular systems (A'B-) and AB*, respectively. Rates uMN 2 . 10-7(300/T)'iZcm3s-' are in general typical at thermal temperatures T for most systems. Agreement between various approximate curve-crossing Landau-Zener theories and measurement remained elusive even for the simplest (H+-H-) system until revised measurements agreed in 1984 with detailed close-coupling calculations that included couplings to states neglected in previous simplified versions. The recent cross-section measurement (Peart and Hayton, 1992), the full curve in Fig.4, exhibits a broad minimum of about 5.5.10- l 5 cmz around 8 eV and a substantial peak
-
I
01 2
I
I
4
1
I
6
I
I l l
I
8 1 0
20
1
I
, I
LO 60 90 100 Centre of moss energy IeVl
I
200
400
!
600
I I ' I
1000
FIG.4. Comparison between theory and merged beam measurementsfor the cross sections of H+-H- mutual neutralization. (From Peart and Hayton, 1992.)
136
M . R . Flannery
of 1.4. 10-'4cmZclose to 300eV is in general accord with previous measurements (Szucs et al., 1984), the closed and open squares in Fig. 4 for H + - H and D +-H - recombination, respectively. One-electron molecular-orbital quanta1 calculations with translational factors (Fussen and Kubach, 1986; Sidis, Kubach and Fussen, 1983),two-electron semiclassical calculations with exchange (Shingal and Bransden, 1990) and two-electron molecular orbital results without translational factors (Borondo, Macias and Riera, 1983) are denoted in Fig. 4 by F, S, D and B, respectively, and are in general agreement. The (H+-H-) calculations of Ermolaev (1988) with a large basis set of atomic pseudostates with translation factors pertain to higher center-of-mass energies 310eV-40 keV. B. TERMOLECULAR A+
+ B- + M - t AB + M
When a positive ion A + approaches a negative ion B - to within a distance R, = (2e2/3kT)z 370 A at room temperature 17; its kinetic energy of relative motion has doubled. If collision with gas M reduces the superthermal kinetic energy to its original thermal value To = 2kT then bound ion pairs with energy E = $kT - eZ/R 0 can be formed (Thomson, 1924). The thirdbodies M therefore utilize very effectively the strength of the Coulombic attraction even at large separations R, where the Landau-Zener probability of curve crossing is very weak. This is the key reason why termolecular recombination rates are so large in comparison to mutual neutralization (30), which occurs mainly at curve-crossing separations R, (10-2O)A with rates - 1 O - ' ~ m ~ s - ~in the absence of gas. Typical rates a 2 . 10-6(300/T)2.s-3.0 cm' sK1 for (31) are much greater than that for (30) in isolation. The rate of (30) is, however, considerably enhanced (Bates and Morgan, 1983) by third-bodies M,
-=
-
-
A+
+ B- + M +
[A' - B - ]
+M + A +B +M
(32) which cause the formation of bound ion pairs ( A + - B - ) so that the probability of curve crossings at R, is greatly promoted. The variation of (32) with gas density N is quite different from that exhibited by (31). Since termolecular recombination (32) involves the formation of bound pairs ( A + - B - ) by repeated collision with M , a molecular gas, by absorbing energy via rotational and vibrational excitation, is more efficient than an atomic gas. When one of the ions is molecular, then in addition to the termolecular channel,
Xe:
+ C1- + Xe
-t
Xe,CI
+ Xe
(33)
ELECTRON-ION AND ION-ION RECOMBINATION PROCESSES
137
another channel, Xe:
+ C1- + Xe .+ XeCl + 2Xe
(34)
is possible. For the (ArZ-F-) system, Rokni et al. (1977)observed that ArF* production similar to channel (34) was more probable than Ar,F* production. This was attributed (Flannery, 1979) to dissociative electron-transfer in the quasi-bound triatomic system [Ar: -F-]* produced by collision with third bodies. Mezyk, Cooper and Sherwell (1989) for the Xe2-Cl- system also observed that channel (34) was more probable than (33). Bates and Morgan (1990) then recognized that the molecular ion can be excited rotationally and vibrationally via the two body process which reduces the two-body orbital energy Ei to E,. This energy conversion occurs mainly at the orbit's pericenter at the frequency of orbital motion until dissociation of Xe: results, and C1- then recombines with Xe' to form XeCl. Bates and Morgan (1990) termed this mechanism (34) tidal recombination. Morgan and Bates (1992)illustrated that the rates for (34)are not only greater than for (33) but exhibit a different dependance with gas density (see Section V.C). 1. Status
The underlying principles and historical development of termolecular recombination (3 1) have been well reviewed (Flannery, 1972,1976;Bates, 1974).Two further reviews (Flannery, 1982; Bates, 1985) cover up to 1985 with additional information provided on electron-ion and ion-ion recombination by Flannery (1990). The process (31) at low gas densities is now well understood theoretically. Theoretical generalization to all gas densities N has recently been accomplished (Flannery, 1991) by the development of the required set of transport-collisional master equations. Measurements have appeared (Mezyk et al., 1989, 1991; Lee and Johnson, 1989, 1990).The results for XeCl* production in (34) were recently explained successfully by the tidal recombination mechanism of Morgan and Bates (1992)within a modification of the Monte Carlo method of Bardsley and Wadehra (1980). 2. Collisional Reaction at Low Gas Densities Here the process is accomplished via the rate limiting step of reaction-a sequence of energy-changing collisions between the ion pairs ( A+ - B - ) in various stages of binding E and the gas M. The transport of A toward B - is essentially uninhibited, being much faster than the reactive phase. There is a +
138
M . R. Flannery
block d of highly excited levels of [ ( A + - B - ) ] * with (orbital) energy E , within 5kT below the dissociation limit that has an nonequilibrium distribution n(E)in energy. This block, sandwiched between a block V of dissociated states (E, > 0) and a block Y of low-lying bound levels (- S 2 Ei 2 - D)that are stabilized against gas collisions with the thermal gas, acts as the essential conduit for the recombination flow between the end blocks V and Y . Contrary to the previous recombination processes, (31) is a closed timedependent system, wherein equilibrium between recombination and association will eventually be established. Additional consideration is therefore required to extract the rate. Recombination proceeds at a rate a(cm3s - ') that is coupled with the reverse process of dissociation that occurs at frequency k,(s- I). N
The coupling between both processes appear via the dependence of the cm-3 concentrations N A , N , and NAB of species A, B and AB with time t. The rate constants a and kd become decoupled and can be determined in isolation by a time-independent procedure (Flannery, 1985). Recombination then emerges as if ion pairs with equilibrium energy distributions gAB(E > 0) are collisionally transferred downward through block d to block Y maintained at zero population. Dissociation emerges as if pairs in block Y with a thermal concentraion #AB are collisionally transferred upward through block 6 to block V maintained at zero population. The rate constants so deduced satisfy the detailed balance relation a#A fiB = k, N A B . The steady state (recombination) rate of flow from dissociated block V through the highly excited block 6 to block Y of stabilized states provides I
aRAfiB=
J:
dE,
lo
[jtivis - n , ~ , , ] dE,
-D
(37)
where the nonequilibrium distribution n, = n(E,) among the highly excited &-block of states E, must be given by the QSS solution of the collisional input-output master equation. dn,
:1
- = n,
dt
1
m
vifdE, -
n,v,,dE,
x0
0 2 Ei2 - S
-S
(38)
where, for energies E , > 0 in block V, n, = iii and n, = 0 for block Y + dE, is v,dE,. Because of the long range of the Coulomb attraction vif = vi"," vt', where for example v $ ~is the frequency resulting from A - M
( - S 2 Ei 3 -D). The frequency for collisional transitions E, + E,
+
ELECTRON-ION AND ION-ION RECOMBINATION PROCESSES
139
collisions alone with B treated as a spectator. Since viJ varies linearly with the gas density N, so too does the rate (37). Recombination at low gas densities has been considered exactly (the earlier QSS method (see Flannery and Mansky, 1988a) and an equivalent variational treatment (see Flannery, 1988)) and by various approximate methods, based on energy diffusion (Flannery, 1987), bottleneck and electrical analogies (Flannery and Mansky, 1988a, b). A set of rate tables are available for the full range of component masses over three classes of ion-neutral A - M interactions: symmetrical resonance charge transfer (A -A), hard sphere and polarization (Flannery and Mansky, 1988a). See also Bates and Moffett (1965), Bates and Flannery (1968), Bates and MendaS (1982b) and Flannery (1980, 1981) for earlier work. An averaged separation R , within which recombination occurs with probability P”(R,) given by (8) can be assigned (Flannery and Mansky, 1988a) by setting the exact numerical rate equal to tl of (9) with A;’ = :;A + A&. The important basic advance has been the recognition of the existence of block B nonequilibrium highly excited states with a nonequilibrium distribution n(E,), the essential feature in all the collisional-radiative models pioneered by Bates. This is in contrast to transition-state or bottleneck treatments that ignore the existence of block 8,and set the rate to be a one-way equilibrium rate, +
a(E*)gA&
=
j:
dEi
iiiviJdEj
(39)
across the transition (or bottleneck) level at energy E* chosen to minimize (Flannery and Manksy, 1988aJ The transition level separates the reactant block, where n, = Gi, from the product 9,where ni = 0. Another fundamental advance is that an expression (Flannery, 1985) more basic than the steady-state current (37) must be used for the rate when approximate distributions n(Ei), which are inconsistent with QSS, Eq. (38), are adopted. Monte Carlo simulation is difficult at low gas densities.
c1
3. Reaction at Higher Gas Densities As the density N of the gas M is raised, the recombination rate a increases initially as N to such an extent that there are increasingly more pairs n;(R, E) in a state of contraction in R than there are those n:(R,E) in a state of expansion, i.e., the ion-pair distribution densities n f ( R , E) per unit interval dEdR are not in equilibrium with respect to R in blocks V and 8.Those in the highly excited block B in addition are not in equilibrium with respect to energy E . Basic sets of coupled master equations have been developed
140
M . R. Flannery
(Flannery, 1991) for the microscopic nonequilibrium distributions n*(R, E, L2) and n*(R, E ) of expanding (+) and contracting (-) pairs with respect to (A-B) separation R, orbital energy E and orbital angular momentum L. The rate for termolecular recombination (31) may then be expressed as
and
where 0 = ( ~ ~ T / A M , , ) ~ E '= ~ E/kT, , b2 = L2/(2kf,,E), bi = Ri[l - V(R)/E] and b:ax = Ri[1 - V(R,)/kT]. The distributions n(*)/fi(*)normalized to their thermodynamic equilibrium values are p i . The probability P - ( E ,b), and the appropriate b- and (b, &)-averagedvalues P-(E)and P - , respectively, that a unit distribution of ion pairs in state i = (E, L2) contracting at Ro, will recombine by collision within R , is determined from the solutions n'(R) of the master equations. The association probability P - ( R , ) increases with gas density to a unit saturation value, while p - ( R , ) decreases monotonically from unity. As N is increased, the rate ci will therefore increase, reach a maximum and then decrease. In the absorption limit when the back-coupling collisional terms vfi(R) are neglected, then
(42)
= 1 - exp[ -$vi(t)dt]
where ,Ii is the microscopic free path length, ui/vi = (dsi/dt)/vi, toward reactive collisions within the path si(t) of the orbit, with pericenter R i , enclosed by the sphere of radius R,. For constant ,Ii = A, then P,: = 1 - exp[-9(E, b; RO)/J],where 9 is the length of the enclosed segment. At low gas densities J >> R,, then it has recently been shown (Flannery, 1991) that the required b and (E,b) averages are
P - ( E , R , ) = -43- Ro 1
( ;%;"I)( 1
1 --% f o ) ) - l
(43)
and P - ( R , ) = P-(kT, R,), respectively, for exact hyperbolic trajectories under Coulomb attraction Vc(R)= - e 2 / R . Numerical evaluation of (42) and (40) is required for higher gas densities N.
ELECTRON-ION AND ION-ION RECOMBINATION PROCESSES
141
-
Figure 5(a) illustrates the variation with gas density N Ro/Aof P - ( E ,R,) for various energies E = EJkT. As E increases, P decreases monotonically from the envelope associated with the parabolic orbits (Ei<< I V(R)I)to the Thomson envelope, = Ro/l) =
P(X
1 [l 2x2
1 --
-
e-2X(1
+ 2X)]
(44)
for straight-line orbits (Ei>> 1 V(R)().Since 2R, and 4Ro/3 are the lengths 9 enclosed in each limit, the envelopes initially increase linearly as the collision probability 2R0/1 and 4R0/31 in the parabolic and straight-line limits. The numerical &-averagedprobability P(R,), in Fig. 5(b) is graphically indistinguishable from P(E= k?: R,) and lies closer to the parabolic limit than it does to the straight-line limit. For Coulombic attraction, the pairs are collisionally maintained in L2equilibrium. The absorption solutions of the master equations for n*(R, E) (Flannery, 1991) yields the probability of recombination for constant 1 and all gas densities as
0
1
0 Ial
1
I
2 X=R,/h
I
3
l
I
0
4 Ibl
1
I
I
8
3
4
X=R&
FIG. 5. (a) Probability P - ( R , , E; A), for collision of an ion in a hyperbolic (Coulomb) trajectory with the gas within a sphere of radius R , = 0.4Re as a function of reduced gas density X = R,/A at various reduced Coulomb-orbit energies E = E/kT; (b) dashed curve: E-averaged probability P-(R,; A) = P-(R,, kT; A). Solid lines are parabolic ( E = 0) and rectilinear ( E -P m) envelopes to the collision probabilities, respectively. Inset figure: E-variation between envelopes at low gas density. Re = eZ/kTis the natural unit of length. (From Flannery, 1991.)
142
M . R. Flannery
where, X = R,/1 and
2 = b,/1
so that
The &-averagedprobability is P - ( R o ) = P - ( E = k?; Ro). Probability (45) varies between the appropriate parabolic and rectilinear envelopes, 1 4x
P - ( E+ 0,X) = 1 - - [I - e - 4 x ]
-
P-(E
(47)
1 8X2
co,X ) = 1 - -[4X - 1 - e - 4 x ]
respectively, as a function of X or gas density N. They are similar in shape but are in general lower than those portrayed in Fig. 5, since here the collisions that establish L2-equilibrium do not result in absorption. Analytical result (45) is exact and is the exact generalization of the Thomson densitydependent probability (44) to curved trajectories and all gas densities under the assumption of L2-equilibrium.As the gas density N increases, the rate (40) will therefore follow the increase of P - with N until the monotonic decrease in p - ( R o ) from unity eventually begins to offset the P - versus N variation which is asymptotically approaching unit saturation. The N-variation of p ultimately controls the variation of u with N. Numerical solution of the microscopic master equations for p-(R,) represents a boundary-value problem at the forefront of research in mathematical methods.
4. Reaction and Pansport Under the assumption that recombination proceeds by the sequence of diffusional drift to Ro at rate a,(R,) during which the kinetic energy gained from the Coulomb field between collisions is lost upon collision, and of collisional reaction within R , at rate URN(&), the overall rate is the wellknown result (cf. Bates and Flannery, 1968; Flannery, 1991)
where the transport rate
for Coulombic attraction is known in terms of the relative mobility
ELECTRON-ION AND ION-ION RECOMBINATION PROCESSES
143
+
K = K , K , of each species A, B in the gas M , and therefore decreases as N - I. The macroscopic ion-pair distribution,
therefore decreases monotonically from p(c0) = 1 at low N where ffRN/ffTR << 1, to zero at high N where UTJffRN << 1. Rate (49), characteristic of two mechanisms proceeding in sequence, is controlled by the rate limiting step, i.e., u is URN N at low N and u is ffTR N - ' at high N . At low N , u $nR;iiNa, where r~ is the cross section for ( A + - B - ) - M collisions. Since R , varies with temperature T as T - I , then u(T) varies as T - 2 . 5for hard sphere collisions and as T - 3 for the more realistic spiralling ion-neutral collisions under a polarization interaction, where the collisional frequency v is constant. The mobility K and am, which vary as TI'* * T2" for interactions V ( R ) R-", are therefore independent of T for polarization attraction.
--
-
-
5. Bates's Universal Curve Monte Carlo simulations provide u versus N . Bates (1980) has shown from dimensional considerations that a plot of l a x versus qNx for an unknown case X should coincide with the a, versus N , plot for a standard case S provided the parameters are assigned as
where A and B are assigned by the low and high density limits, uL = A N and uH = B/N, respectively. Moreover, the plot of l(T/300)3'2ux versus q(300/77",, where t = 1 or 3/2 for hard-sphere or polarization ion-neutral collisions, respectively, is universal with respect to T and also N (Bates and Menda6, 1982a). The measurements of Lee and Johnsen (1990) on Xe+ + F-
+ He + XeF* + He
(53)
at 300K at gas pressures between 140 and 500 Torr, are in satisfactory agreement with this procedure and with the Monte Carlo calculations. RECOMBINATION C. TIDAL Figure 6 shows the impressive agreement between the measurements of Mezyk et al. (1991) and the recent Monte Carlo simulations of Morgan and Bates (1992) for the tidal recombination (34) that proceeds via (35). The
M . R . Flannery
144 10.0
'I
1
CD
I
0 c
W
a"
0.02
0.10 1 .oo Ns (in units of Loschmidt's number)
10.00
FIG. 6. Universal curve (solid line) for termolecular recombination between structureless atomic ions: measurement and 0 calculation for process (34); 0,0measurements for process (54) with M = He and Ar, respectively.All data points are scaled by I and q as in text. (From Morgan and Bates, 1992.)
variation with gas density for tidal recombination is quite different from that of termolecular recombination (33) as illustrated by the universal curve for termolecular recombination (full line) of Bates and MendaB. The measurements of Lee and Johnson (1989) for
H,O+(H,O),
+ NO; + M -+
neutrals
(54)
are also shown for M = He(.) and Ar(0). All the data points are scaled by the factors (52). Since the universal rates for He and Ar should be equal for termolecular recombination, the difference may be due to tidal recombination.
VI. Conclusions Although it is difkult to do full justice within the allotted pages for a subject so vast as recombination, it is obvious from the foregoing that the contributions of Bates have been seminal, profound and enduring. There is apparently no recombination process of significance to which Bates has not contributed significantly. I would like to be able to say that Bates has forgotten more about recombination processes than most of us will ever know, but I know this not to be true. As this chapter portrays, his elucidation
ELECTRON-ION AND ION-ION RECOMBINATION PROCESSES
145
of recombination mechanisms and his education of us continues and has remained intact throughout all these years.
Acknowledgments This research is supported by the US. Air Force Office of Scientific Research under grant no. AFOSR-89-0426. I thank Annick Giusti-Weiner in particular for influential discussion on dissociative recombination. I am also indebted to E. J. Mansky I1 for his invaluable contribution to the production of this manuscript typeset with TeX.
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Bates, D. R., and MendaS, I. (1982a). Chem. Phys. Lett. 88, 528. Bates, D. R., and MendaS, I. (1982b). J. Phys. B: At. Mol. Phys. 15, 1949. Bates, D. R., and Moffett, R. J. (1966). Proc. Roy. SOC.(London) Series A 291, 1. Bates, D. R., and Morgan, W. L. (1983). Chem. Phys. Lett. 101, 18. Bates, D. R., and Morgan, W. L. (1990). Phys. Rev. Lett. 64, 2258. Bell, R. H., and Seaton, M. J. (1985). J . Phys. B: At. Mol. Phys. 18, 1589. Berrington, K. A., Burke, P. G., Butler, K., Seaton, M. J. Taylor, K. T., and Yu Yan. (1987). J. Phys. B: At. Mot. Opt. Phys. 20, 6379. Biondi, M. A., and Brown, S. C. (1949). Phys. Rev. 76, 1697. Borondo, F., Macias, A., and Riera, A. (1983). Chem. Phys. 81, 303. Bottcher, C. (1976). J . Phys. B: At. Mol. Phys. 9, 2899. Burgess, A. (1964). Astrophys. J . 139, 776. Canosa, A., Comet, J. C., Rowe, B. R., and Queffelec, J. L. (1991). J . Chem. Phys. 94, 7159. Canosa, A., Comet, J. C., Rowe, B. R., Mitchell, J. B. A., and Queffelec,J. L. (1992). J . Chem. Phys. 97, 1028. Cao, Y. S., and Johnsen, R. (1991). J . Chem. Phys. 95, 7356. Chen, M. H. (1986). Phys. Rev. A 34, 1079. Ermolaev, A. M. (1988). J. Phys. B: At. Mol. Opt. Phys. 21, 100. Escalante, V., and Victor, G. A. (1992). Planet. Space Sci. 40, 1705. Flannery, M. R. (1972). In: Cases Studies in Atomic Collision Physics (E. W. McDaniel and M. R. C. McDowell, 4s.). 2, chap. 1. Flannery, M. R. (1976). in: Atomic Processes and Applications (P. G. Burke and B. L. Moiseiwitsch, eds.), North-Holland, chap. 12. Flannery, M. R. (1979). Int. .J. Quan. Chem.: Quant. Chem. Symp. 13, 501. Flannery, M. R. (1980). J . Phys. B: At. Mol. Phys. 13, 3649. Flannery, M. R. (1981). J. Phys. B: At. Mol. Phys. 14, 915. Flannery, M. R. (1982a). Phil. Pans. Roy. SOC.(London) Series A 304,447. Flannery, M. R. (1982b). In: Applied Atomic Collision Physics (E. W. McDaniel and W. L. Nighan, eds.), Academic Press, New York, 3, Chap. 5. Flannery, M. R. (1985). J. Phys. B: At. Mol. Phys. 18, L839. Flannery, M. R. (1987a). In: Recent Studies in Atomic and Molecular Processes (A. E. Kingston, ed.), Plenum Press, New York, p. 167. Flannery, M. R. (1987b). J. Chem. Phys. 87, 6847. Flannery, M. R. (1988). J . Chem. Phys. 89, 214. Flannery, M. R. (1990). In: Molecular Processes in Space (T. Watanabe, I. Shimamura, M. Shimiza and Y. Itikawa, eds.), Plenum Press, New York, Chap. 7. Flannery, M. R. (1991). J . Chem. Phys. 95, 8205. Flannery, M. R. (1993a). Preprint. Flannery, M. R. (1993b). In: Dissociative Recombination: Theory, Experiment and Applications (B. R. Rowe and J. B. A. Mitchell, eds.), Plenum Press, New York. Flannery, M. R., and Mansky, E. J. (1988a). J . Chem. Phys. 88, 4228. Flannery, M. R., and Mansky, E. J. (1988b). J. Chem. Phys. 89,4086. Fussen, D., and Kubach, C. (1986). J. Phys. B: A t . Mol. Phys. 19, L31. Giusti, A. (1980). J. Phys. B: At. Mol. Phys. 13, 3867. Giusti-Suzor, A., Bardsley, J. N., and Derkits, C. (1983). Phys. Rev. A 28, 682. Guberman, S. L. (1991). Geophys. Lett. 18, 1051. Guberman, S. L., and Giusti-Suzor, A. (1991). J . Chem. Phys. 95, 2602. Haan, S. L., and Jacobs, V. L. (1989). Phys. Rev. A 40,80. Hahn, Y. (1985). Ado. At. Mol. Phys. 21, 123. Hahn, Y., and LaGattuta, K. J. (1988). Phys. Rep. 166, 195.
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Hickman, A. P. (1987). J. Phys. B: At. Mol. Phys. 20, 2091. Jacobs, V. L., and Davis, J. (1978). Phys. Rev. A 18,697. Langevin, P. (1903). Ann. Chem. Phys. 28,433. Lee, C . M. (1977). Phys. Rev. A 11, 1692. Lee, H. S., and Johnsen, R. (1989). J. Chem. Phys. 90, 6328. Lee, H. S., and Johnsen, R. (1990). J. Chem. Phys. 93, 4868. McLaughlin, D. J., and Hahn, Y. (1991). Phys. Rev. A 43, 1313. Malinovsky, L., LukaE, P., Trnovec, J., Hong, C. J., and Talsky, A. (1990).Czech. J . Phys. 40,191. Mansbach, P., and Keck, J. (1969). Phys. Rev. 181, 275. Massey, H. S. W., and Bates, D. R. (1943). Rep. f r o g . Phys. 9, 12. Mezyk, S. P., Cooper, R., and Sherwell, J. (1989). J. Phys. Chem. 93. 8187. Mezyk, S. P., Cooper, R., and Sherwell, J. (1991). J. Phys. Chem. 95, 3152. Mitchell, J. B. A. (1990). Phys. Rep. 186, 215. Morgan, W. L. (1987).In: Recent Studies in Atomic and Molecular Processes (A. E. Kingston, ed.), Plenum Press, New York, p. 149. Morgan, W. L., and Bates, D. R. (1992). J. Phys. B: At. Mol. Opt. Phys. 25, 5421. Nahar, S. N., and Pradhan, A. K. (1992). Phys. Rev. Lett. 68, 1488. Nakashima, K., Takagi, H., and Nakamura, H. (1987). J. Chem. Phys. 86, 726. Nussbaumer, H., and Storey, P. J. (1987). Astron. Astrophys. Supple. Series 69, 123. OMalley, T. F. (1981).J. Phys. B: At. Mot. Phys. 14, 1229. Peart, B., and Hayton, D. A. (1992). J. Phys. B: At. Mol. Opt. Phys. 25, 5109. Pajek, M., and Schuch, R. (1992). Phys. Rev. A 45, 7894. Pindzola, M. S., Badnell, N. R., and Griffin, D. C. (1992). Phys. Rev. A 46, 5725. Pitaevskii, L. P. (1962). Sou. Phys. J E T P 15, 919. Rokni, M., Jacob, J. H., and Mangano, J. A. (1977). Phys. Rev. A 16, 2216. Rowe, B. R., and Mitchell, J. B. A,, eds. (1993). Dissociative Recombination; Theory, Experiment and Applications, Plenum Press, New York. Sarpal, B. K., Tennyson, J., and Morgan, L. A. (1993). Preprint. Schneider, I. F.. Dulieu, O., and Giusti-Suzor, A. (1991).J. Phys. B: At. Mol. Opt. Phys. 24, L289. Seaton, M. J. (1987). J. Phys. B: At. Mol. Opt. Phys. 20, 6263. Seaton, M. J., and Storey, P. J. (1976). In: Atomic Processes and Applications (P. G. Burke and B. L. Moisewitsch, eds.), North-Holland, p. 135. Shingal, R., and Bransden, B. H. (1987). J. Phys. B: At. Mol. Opt. Phys. 23, 1203. Sidis, V., Kubach, C., and Fussen, D. (1983). Phys. Rev. A 27, 2431. Stevefelt, J., Boulmer, J., and Delpech, J. F. (1975). Phys. Rev. A 12, 1246. Sun, H., Nakashima, K., and Nakamura, H. (1993). In: Dissociative Recombination: Theory, Experiment and Applications (B. R. Rowe and J. B. A. Mitchell, eds.), Plenum Press, New York. Szucs, S., Karema, S., Terao, M., and Brouillard, F. (1984). J. Phys. B: At. Mol. Phys. 17, 1613. Takagi, H. (1993). J . Phys. B: At. Mol. Opt. Phys., in press. Takagi, H., Kosugi, N., and Le Dourneuf, M. (1991). J. Phys. B: At. Mol. Opt. Phys. 24, 711. Thomson, J. J. (1924). Phil. Mag. 47, 337. Van der Donk, P., Yousif, F. B., Mitchell, J. B. A., and Hickman, A. P. (1991). Phys. Rev. Lett. 67, 42. Van der Donk, P., Yousif, F. B., Mitchell, J. B. A., and Hickman, A. P. (1992).Phys. Rev. Lett. 68, 2252. Yousif, F. B., and Mitchell, J. B. A. (1989). Phys. Rev. A 40, 4318.
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ADVANCES IN ATOMIC, MOLECULAR, AND OF'TICAL PHYSICS. VOL. 32
STUDIES OF STATE-SELECTIVE ELECTRON CAPTURE IN ATOMIC HYDROGEN BY TRANSLATIONAL ENERGY SPECTROSCOPY H . 3. GILBODY Department of Pure und Applied Physics Queen's University of Belfast Beljhrt, Northern Ireland
I. Introduction . . . . . . . . . . . . . . . . . . . . . . 11. Experimental Approach . . . . . . . . . . . . . . . . . A. Basic Principles . . . . . . . . . . . . . . . B. The Belfast TES Apparatus . . . . . . . . . . 111. Results. . . . . . . . . . . . . . . . . . . . . . . . . A. Collisions of C3' Ions with H . . . . . . . . . B. Collisions of Cz' Ions with H . . . . . . . . . C. Collisions of Ar4', Ar5+ and Ar6' Ions with H . . D. Collisions of 0 3 +Ions with H . . . . . . . . E. Collisions of 0 ' ' Ions with H . . . . . . . . F. Collisions of S3+ Ions with H . . . . . . . . . IV. Conclusions . . . . . . . . . . . . . . . . . . . . . . Acknowledgments. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
149 151 151 152 154 154 157 157 161 163
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 . . . . . . 166 . . . . . . 167 . . . . . . 167
I. Introduction Electron capture in collisions between multiply charged ions and hydrogen atoms in reactions of the type
X 4 ++ H(ls) -+X ( 4 - ' ) + (n,1) + H
+
(1)
leading to product ions in specified excited states n,l is of considerable current interest. At low velocities V < 1 a.u. (corresponding to 25 keV a.m.u.- ') such collisions may take place very effectively in reactions of moderate exothermicity through pseudocrossings of the adiabatic potential energy curves describing the initial and final molecular systems. These 149
Copyright 0 1994 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003832-3
150
H. B. Gilbody
crossings occur at internuclear separations R, given by (neglecting polarization) R, N (q - l)/AE a.u., where AE is the energy defect for a reaction channel corresponding to the formation of a particular X ( q - l ) +product ion in a specified excited state n, 1. In many reactions, the total cross section for one-electron capture is often dominated by transitions through a limited number of such crossings. A proper understanding of the electron capture process therefore requires identification and a quantitative assessment of the excited product channels. Apart from the fundamental interest, cross sections for state-selective capture in collisions with H are relevant to controlled thermonuclear fusion research both in the understanding of energy-loss mechanisms in the neutral beam heating of magnetically confined plasmas and in schemes for plasma diagnostics (cf. Gilbody, 1979; Winter, 1982). In astrophysics these processes are also important in the understanding of the interaction of low-energy cosmic rays with interstellar hydrogen and as a possible explanation for the observed underabundance of some ions in astrophysical plasmas (cf. Steigman, 1975). In 1983 we demonstrated (McCullough et al., 1983) for the first time that studies of state-selective electron capture by multiply charged ions in collisions with H atoms are feasible using the technique of translational energy spectroscopy (TES)with a tungsten tube furnace to provide a target of highly dissociated hydrogen. The method has since been used both in this and other laboratories to study a number of important collision processes in atomic hydrogen. In 1985 the Amsterdam-Groningen group (CiriC et al., 1985) showed that the technique of photon emission spectroscopy (PES) was also feasible in measurements with atomic hydrogen targets. In this approach, the photon emission spectra arising from the spontaneous decay of excited products in a crossed beam arrangement are analyzed using optical spectroscopy.The TES and PES methods are complementary. While the PES method provides a much higher energy resolution, so that the contributions to different collision channels are more easily distinguished, it is less sensitive and the indirect nature of the calibration procedure usually leads to greater overall uncertainties in calibration than the TES approach. The TES method, unlike the PES method, has an additional advantage in that it provides an unambiguous indication of the presence and influence of any metastables species present in the primary beam. In the case of collisions between slow fully stripped ions and H, electron capture is found to be highly selective and generally dominated by a single pseudocrossing, leading to product ions in one excited state (cf. Kimura et al., 1987). For example, in the case of C6+-H and N7+-H collisions, the
TES STUDIES OF STATE-SELECTIVE ELECTRON CAPTURE
151
dominant products are C 5 + (n = 4) and N 6 + (n = 5). For partially ionized primary ions, a number of curve crossings may be important, and the product state distributions are less easy to predict. Experimental data provide an important check of the range of validity of theoretical models of the electron capture process. In this short review, the TES approach used in this laboratory will be described together with a few representative results obtained with different partially ionized projectiles at keV energies.
11. Experimental Approach A. BASICPRINCIPLES In the TES approach, the primary ion beam of well-defined kinetic energy Tl is passed through the target. If the kinetic energy of the forward scattered X ( 4 - I ) ' ions formed as products of single collisions is measured as T2 then the change in kinetic energy is given by AT
= T2 -
TI = AE - AK
where AK is the recoil energy of the target given by
where m is the projectile mass, M is the target mass and 4 is the CM scattering angle. Provided that AEIT, << 1 and the scattering is confined to small angles, then the observed change in translational energy AT
= AE
A rough estimate of the CM scattering angle 4 may be obtained from the expression given by Olson and Kimura (1982):
-
2
a at
+ 4)'"
-
(3)
where a = Z,Zbe2/Eb in which Z , and Z , are the charges of the product ions and b is the impact parameter at a CM energy E. A rough although slightly high estimate of the minimum scattering angle can be obtained by taking b = R,. When necessary, these values of may be used in Eq. (2) to obtain the recoil correction AK although, for many cases, this is small enough to neglect.
+
152
H.B. Gilbody
B. THEBELFAST m APPARATUS Figure 1 shows a schematic diagram of the latest rersion of the translation 1 energy spectrometer used in the measurements in Belfast within the energy range 1.5-18 keV. Suitable beams of primary ions X4' for these measurements have been derived from both oscillating electron and recoil type ion sources but, very recently, an electron cyclotron resonance (ECR)ion source has been installed to provide more intense beams with a wider range of primary ion charge states q. The momentum analyzed keV energy beam of Xq+ ions enters the electrostatic lens system LZ where it is focused and decelerated to an energy typically between 20 q and 120 q eV before passing through the two hemispherical electrostatic analyzers EAZ and EA2 to select ions with an energy spread as low as 0.17 q eV FWHM. After focusing and acceleration by lens L2 to the required final energy, the ions pass through two diametrically opposed apertures (1mm dia. entrance and 2mm dia. exit)
FIG.1. Schematic diagram of the Belfast TES apparatus. (From Wilkie et al., 1986.)
TES STUDIES OF STATE-SELECTIVE ELECTRON CAPTURE
153
midway along a 9mm diameter tungsten tube furnace, F. Hydrogen gas flows into the furnace at a constant rate low enough to ensure single collision conditions. The furnace tube is heated by passing a.c. directly through it but the potential drop across the beam entrance and exit apertures typically does not exceed 0.15V. The furnace is maintained at a potential V, while the surrounding heat shield H S and walls of the vacuum chamber housing are biased at a potential V, negative with respect to V, to prevent ions formed by electron capture in the residual gas outside the furnace from contributing to the measured signals. The furnace can be operated either at room temperature as a simple gas cell containing H, or at about 2500K to provide a target of dissociated hydrogen with an estimated dissociation fraction of at least 92%. ions formed as products of single collisions Forward scattered X ( 4 within a mean half-angle of 0.5" are focused and decelerated by lens L3 and energy analyzed by the third hemispherical electrostatic analyzer EA3 and recorded by a position-sensitive detector. In this detector, product ions at the exit plane of EA3 are accelerated to about 2.5 (q - 1) keV onto a two-stage microchannel plate array M C P . The resulting electrons are accelerated onto a phosphor screen PS and an optical image of this is then focused by a wide aperture lens on to a charge coupled device CCD. After a preset time, the accumulated charge from each pixel in the CCD is sequentially transferred as a series of pulses of height proportional to image intensity and routed to a multichannel analyzer. In order to account for any nonuniformity in response, the product ion spectrum is tracked across the face of the detector. The entire product ion analysis, detection and data acquisition system is computer controlled. Figure 2 shows typical energy change spectra obtained (Wilkie et al., 1986) for 15 keV C 3 + ions with the furnace of 300K (when the hydrogen was undissociated) and at 2500 K when the hydrogen was highly dissociated. In the spectrum at 2500 K the four main peaks can be identified with particular C2+excited products arising from electron capture by C3+ in collisions with H atoms; this is discussed in Section 1II.A. The energy change spectrum observed at 300K corresponding to C3+-H, collisions can be seen to be quite different. Analysis of both these spectra shows that the spectrum obtained at 2500 K contains a negligible contribution from the small amount of undissociated hydrogen molecules present. The rich structure in the energy change spectrum for H, corresponds to various excited C2+ product channels, which may involve the formation of either H: ' C i ions in vibrationally excited states or the H i 2pc, state, which dissociates to H(1s) + H + (Wilkie et al., 1986).
154
H.B. Gilbody .8
Energy change (eV)
FIG.2. Energy change spectra for one-electron capture by 15 keVC3+ in collisions with hydrogen contained within a tungsten tube furnace at 2500 K when the dissociation fraction is. high and at 300 K when the hydrogen is undissociated. (From Wilkie et al., 1986.)
111. Results A. COLLISIONS OF C 3 + IONS WITH H
Studies of one electron capture by C 3 + ions in collisions with H have been carried out in this laboratory in the range 1.5-18keV using TES (McCullough et al., 1984; Wilkie et al., 1986) and by the AmsterdamGroningen group (CiriC et al., 1985) using the PES technique in the
TES STUDIES OF STATE-SELECTIVE ELECTRON CAPTURE
155
overlapping energy range 9-60 keV. In the TES energy change spectrum shown in Fig. 1, the position of peak A is consistent with possible contributions from the collision channels C3+(ls22s)2S+ H(1s) + C2+(ls22s3d)'D+H+-0.02 eV
+ 0.78 eV with R, + C 2 + ( 1 ~ 2 2 ~ 3 p ) 3 P o ++H2.05 + eV with R, + C 2 + ( 1 ~ 2 2 ~ 3 p ) 1 P o ++ H 2.15 + eV with R, -+C2+(ls22s3d)3D +H
+
(a) N
69.7 a.u.
(b)
N
26.5 a.u.
(c)
-N
25.3 a.u.
(d)
which are not separately resolved. The main peak B can be correlated with the product channels +C2+(1s22s3s)'S+H+ +3.61 eV with R,= 15.1 a.u.
(4
+ C 2 + ( 1 ~ 2 2 ~ 3 ~ ) 3 S +4.72 + H + eV with R,? 11.5 a.u.
(f)
of which the contribution of (e) appears to be relatively small at the energies considered. Peak C is correlated with the product channel +C2+(ls22p2)'S+H++ 11.63 eV with R,=4.7 a.u.
(g)
while peak D can be identified with the channel +C2+(1~22p2)1D+Hf+16.17 eV with R,=3.4 a.u.
(h)
Calculations by Bienstock et al. (1982)based on a fully quanta1 quasimolecular description of the collisions predicted strong adiabatic coupling for the collision channels (f), (g), (c) and (h), which is consistent with the TES observations. Subsequent calculations by Opradolce et al. (1988) using a molecular model modified to take account of translation effects to obtain cross sections for the triplet product channels (c) and (f) are significantly different from those calculated by Bienstock et al. (1982). In the most recent and detailed theoretical study by Errea et al. (1991), cross sections for all channels (a)-(h) were determined using an expansion in terms of 22 molecular states modified with a common translation factor. We limit this discussion to the main collision channels (f), (g) and (h) associated with the peaks B, C and D, which were adequately resolved in the TES measurements of Wilkie et al. (1986). Cross sections for the separate product channels were determined from observed energy change spectra by normalization to known total cross sections for one electron capture measured in several different laboratories. In the energy range of interest, these measured total cross sections can be seen to be better described (Fig. 3) by the theoretical estimates of Bienstock et al. (1982) rather than by the more recent calculations of Errea et al. (1991).
H.B. Gilbody
156
In Fig. 3 the TES measurements of Wilkie et al. (1986) can be compared with the PES measurementsof CiriCet al. (1985) for the (f), (g) and (h) product ~S, channels corresponding to the formation of C 2 + in ( 1 ~ ~ 2 s 3 s )(ls22p2)'S and ( ls22p2)'D states respectively. The theoretical predictions of both Bienstock et al. (1982)and Errea et al. (1991)are also shown. In the case of the 3S and 'S product channels, agreement between the TES and PES measurements is generally within the maximum combined uncertainties. However the agreement is less satisfactory for the 'D product channel, where the PES cross sections are much smaller than the TES values in the energy range of overlap. Cross sections calculated by Errea et al. (1991) for the 3S product channel can be seen to be in reasonable accord with the high-energy PES measurements but agreement with the lower energy TES measurements is less satisfactory than the calculations of Bienstock et al. (1982). In the case of the ' S product channel, the cross sections calculated by Errea et al. (1991) provide a better description of experiment than the results of Bienstock et al. (1982) but both calculations fail to predict the rapid fall in the high-energy I
'
I
" " " "
10 :
1:
3 3
-5
0.1 :
5
v)
VI
e
U
0.01
7
ii 0.001 0 001 0.1 0 1
1
10
100
Energy (keV)
FIG.3. Data for electron capture by C3+ ions in collisions with H. Total cross sections for electron capture: Phaneuf et al. (1978, 1982); V, Crandall et al. (1979); A,Yousif and Geddes cited by Wilkie et al. (1986); A, Gardner et al. (1980); ---, theory by Bienstock et al. (1982);-, theory by Errea et al. (1991). Cross sections for electron capture into ( 1 ~ ~ 2 s 3 s )(ls22pz)'S ~S, and (ls22p2)'D states of C2+: 0, V;TES measurements by Wilkie et al. (1986). 0,0 ,V; PES measurements by Cirii: et al. (1985). Curves B, theory by Bienstock et al. (1982). Curves E, theory by Errea et al. (1991).
.,
+,
TES STUDIES OF STATE-SELECTIVE ELECTRON CAPTURE
157
PES values. In the case of the 'D product channel, the results of the two calculations differ substantially. However the values calculated by Errea et al. (1991)are in somewhat better accord with the TES experimental values. The degree of accord between many of the data in Fig. 3 for the three main collision channels is encouraging. At the same time, the discrepancies, which are also evident, illustrate the difficulty of accurate studies of this type from both the experimental and theoretical standpoints.
B. COLLISIONS OF C2 IONSWITH H +
TES studies of one-electron capture in C'+-H collisions (McCullough et al., 1984) provide energy change spectra that show collision product channels associated with both ground state C2+(2s2)'Sand metastable C2+(2s2p)3Po primary ions. At energies between 2 and 8 keV, between 60% and 65% of the total C+ product yield arises from exothermic channels involving metastable C2+ primary ions. This is a severe complicating feature, and a quantitative analysis of the observed spectra is not possible unless the initial fraction of metastables in the primary beam is known. The observed energy change spectra show that the dominant collision channels are
C * + ( 2 ~ 2 p )+~ H(1s) P ~ +C'(2~~3p)~P +'H + + 3.54 eV for metastable primary ions and Cz+(2sz)'S+ H(1s) + C+(2s2p2)'D + H + + 1.48 eV for ground state C2+ impact. Theoretical studies of C2'('S)-H(ls) collisions by Heil et al. (1983a) using ab initio configuration-interaction methods show that the reaction takes place through the three 'X+ states of CH". Avoided crossings between the incoming molecular state and the outgoing state leading to C+(2D)products are predicted to occur at both 3 and 24 a.u. At 8.1 eV, the highest energy they consider, their calculations predict that C+('D) formation through the innermost crossing provides the main contribution to electron capture. This prediction is in accord with the TES observations for C2+ground state ions. C. COLLISIONS OF Ar4+, Ar5+ AND Ar6+ IONSWITH H
Electron capture in collisions of Arq' ions with H is an interesting case for TES studies since it is known (cf. Crandall et al., 1980) that for q > 3, at velocities u < 1 a.u. total cross sections are large, increase with q and exhibit
H. B. Gilbody
158
only a weak dependence on v. This is typical of the behavior of manyelectron partially ionized ions, where electron capture takes place through a number of curve crossings leading to different excited Ar(q-')+ product states. TES studies have been carried out by McCullough et al. (1987) for q x 1 keV and q x 2 keV Ar4+,Ar5+and Ar6+ ions. Analysis of the results for Ar4+ and Ar6 is complicated by the possible presence in the primary beams of an unknown fraction of metastable species (Mi)as well as ground state (G) ions as follows: +
Species Ar4+ Ar4+ Ar4+ Ar5+ Ar6+ Ar6+
State G M1 M2 G G M
Configuration (3s23p2)3PO (3s23p2)'D (3s23p2)'S (3s23p)ZP (2p63sZ)'S (3s3p)3PO
Energy (eV) 0 2.0 4.7 0 0 14.1
Figure 4 shows observed energy change spectra for Ar4+,ArS+and Ar6+ ions at energies of q x 2 keV. The possible collision channels corresponding to ground (G)and metastable (M) primary ions are also shown. For Ar4+ in H, the spectrum is dominated by peak A centered near four product channels in the G series and three channels in the M1 series corresponding to the formation of Ar3+(3p24p)ions. These involve curve crossings at distances R, between about 8.1 and 7.4 a.u. There is also evidence of smaller contributions from other channels corresponding to larger values of AE. For Ar5+in H, the major peak A in the energy change spectrum is centered near five channels in the G series with R, between 10.0 and 8.9a.u. corresponding to the formation of Ar4+(3p4d) ions. The smaller peak B embraces 10 channels corresponding to the formation of Ar4+(3p4p)ions with R, between 6.5 and 5.5 a.u. In the case of Ar6+ in H, the energy change spectrum exhibits four incompletely resolved peaks A, B, C and D. The large number of M series product channels precludes a detailed analysis. However contributions from four G series channels leading to Ar5+in (3s25p)'P, (3s25s)'S, (3s24f)'F and (3s24d)'D states are possible with R, ranging from 12.0 and 6.5 a.u. In the absence of detailed theoretical studies of these processes, McCullough et al. (1987) have calculated cross sections for individual excited product channels using the multichannel Landau-Zener (MCLZ) model (cf. Janev and Winter, 1985).In this model the probability P , for formation of a
Energy change teV) FIG.4. Energy change spectra for electron capture by 4 x 2 keV h4', Ars+ and h6* ions in collisions with H. Positions of possible collision channels correspondingto ground state ( G )and metastable (M)primary ions are indicated. Also shown as vertical lines are results of MCLZ calculations for G channels. (From McCullough et al., 1987.)
160
H. B. Gilbody
product ion in state n through a system with N curve crossings assuming no coupling between adjacent channels is pn=P1P2..
.pn(1 -Pn)[1
+ bn+ 1Pn + 2 . . .PN
-I
+~:+1(1
1Pn+ 2 *.
+@n+
-p.+J2
) ~ 1( -
.PNl2
+ @n + 1 - .P N
PN)'
*
+(1 - P , , + ~ ) ~ ]
for 1
-
2)2(1- P N - d2
where p,, for n = 1 to N is the single-crossing Landau-Zener transition probability at each pseudocrossing R, given by Pn = ~ X ~ ( - ~ R H : Z / V ~ A F ) R = R ~
In this expression, the radial velocity is U, = ~
( -l b2/R:)11z
for a collision velocity u and impact parameter b. The parameter H i , = AE12/2, where AElZ is the potential difference between the adiabatic curves at the pseudocrossing. This was calculated from the following expression given by Taulbjerg (1986), which allows for the 1 dependence of H12: H I , = 9.13f,,q-'/2 exp(- 1.324R,/lq-1/2) where
f,, = (- 1)"+'-~(21+ i)1/2r(n)[r(n + I + l)r(n - I ) ] - ~ / ~ with /l = (It/13.6)'l2 and I, is the ionization potential of the target. The cross section a, for the formation of products in a particular state n is then CT,,
= 271
JoRc
P,bdb
and the total cross section for one-electron capture is a = &,a,,. MCLZ cross sections were calculated taking into account all the G series channels within the range of the observed energy spectra. Possible M series channels were not considered since metastable populations of the primary beam were unknown. Relative MCLZ cross sections calculated for all significant G channels may be compared with the observed energy change spectra in Fig. 4. To facilitate comparison the value of the largest calculated cross section in each case has been normalized to the peak value observed experimentally. In the case of Ar4+,the calculations can be seen to predict the dominance of crossings at rather greater values of R, than are observed experimentally. For Ar5+ (where only ground state species are present) the MCLZ cal-
TES STUDIES OF STATE-SELECTIVE ELECTRON CAPTURE
161
culations can be seen to correctly predict the positions of the major peaks A and B although the relative magnitude of B is underestimated by the calculations. In the case of Ar6+ impact, the MCLZ calculations are in reasonable accord with the observed peaks A and B but the contributions of peaks C and D are overestimated. The sum of the calculated cross sections for individual channels also provides total cross sections = Z,, on,which are in reasonable agreement with total cross sections measured by Crandall et al. (1980). While there are discrepancies between the MCLZ cross sections and the TES measurements, this simple theoretical approach does provide a useful rough prediction of some of the main product channels. OF 03+ IONSWITH H D. COLLISIONS
TES studies of one-electron capture in collisions of 03+ ions with H have been carried out by Wilson et al. (1988) in the energy range 4.2-12 keV. Figure 5 shows the energy change spectrum observed at 9 keV. Possible collision channels associated with either ( 2 ~ ~ 2 p ) ~ground PO state (C) or ( 2 ~ 2 p ~ ) ~metastable PO (M) primary O 3 ions, respectively, are indicated in this spectrum. The three peaks A, B and C can be correlated with C channels, and the role of M channels is believed to be minor. The G channels that can contribute to the observed peaks are summarized as follows: +
02+ Product State
Observed Peak
(2s22p)3p'D (2s22p)3p3P (2s22p)3p% (2s22p)3p3D (2s22p)3p' P
A
(2s22p)3s' PO (2s22p)3s3PO
AE(eV) 3.32 4.10 4.44 4.88 5.26
R,(a.u.) 16.39 13.27 12.25 11.15 10.34
B
7.47 8.18
7.28 6.65
C
15.24 16.90
3.57 3.21
The minor peak C involves excitation of the primary ion core to (2s2p3)'Po and ( 2 ~ 2 p ~ ) ~states. P O However throughout the energy range 4.2-12 keV, the dominant electron capture contribution arises through peak B, which is associated with two O2'(2s22p)3s product channels. A smaller contribution arises through peak A, which is associated with five O2+(2s22p)3pproduct channels. The results of MCLZ calculations carried out by Wilson et al. (1988) in the way described in Section 1II.C are included in Figure 5. Again, to facilitate
162
H.E. Gilbody (a. u.1
R,
1 I
2
7
5
4
I I I
Ill I
llllll I I 1111 I 1 IIII I I Ill
M G
B 9keV
a
-I W H
>-
+
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>
H
k
a
-1 W
lx
*: .; . . .; . ."*.:.-
.
.: ......_C-::.*. .:..*.
t. .>.a
.*:.
-111 *.,' .-*
..y..*.**.
.*
2 4 6 u 10 12 14 16 ENERGY CHANGE AT' f e V )
0
FIG.5. Energy change spectrum for electron capture by 9 keV 0 ' ' ions in collisions with H. Positions of possible collision channels corresponding to ground state (G) and metastable (M) primary ions are indicated.Vertical lines are results of MCLZ calculations for G channels.(From Wilson et al., 1988.)
comparison with experiment the largest calculated cross section has been normalized to the maximum observed peak height. The MCLZ calculations can be seen to correctly identify the main 3s and 3p product channels, and while the relative cross sections differ from those observed, the dominance of the 3s channels corresponding to peak E is also correctly predicted. The results of both the TES measurements and the MCLZ calculations are at variance with calculations based on a fully quanta1 approach by Heil et al. (1983bl and Rienstock et al. (1983), which predict the dominance of 3p rather than 3s 02+product channels in the energy range considered. However subsequent measurements by Hoekstra et al. (1989) using the PES technique
TES STUDIES OF STATE-SELECTIVE ELECTRON CAPTURE
163
for 24keV03+ ions have confirmed the correctness of the TES measurements of Wilson et al. (1988). In addition, calculations carried out by Gargaud and McCarroll (1989) using a fully quantal molecular approach also provide cross sections for the 3s and 3p 0 2 + product channels that are in good accord with the relative cross sections obtained from the TES measurements. Total electron capture cross sections, which are roughly twice the MCLZ values, have been measured by Phaneuf et al. (1982), and these values could be used in conjunction with the TES relative cross sections to derive separate cross sections for 0 2 + formation in 3s and 3p states. OF 02+ IONSWITH H E. COLLISIONS
TES studies of electron capture by 0 2 + ions in collisions with H have been carried out by McLaughIin et al. (1990) at energies in the range 2-8 keV. Analysis in this case is complicated by existence of the metastable (2s22p2)'D and (2s22p2)'Sstates of 02+ as well as the ( 2 ~ ~ 2 p ' )ground ~ P state. Although the fractions of each state present in the primary beam were unknown, if the populations were roughly in accord with the statistical weights of the states, the 3P ground state would account for 60% of the beam and the 'D and ' S metastable components 33% and 7%, respectively. The observed energy change spectra exhibit two peaks. The main peak, ' product yield, can be identified with which accounts for about 70% of the 0 the ground state primary ion channel: 02+(2s22p2)3P + H(1s) + O+(2s2p3)2p4P+ H + + 6.64 eV which involves 2p capture simultaneous with core excitation; this corresponds to a pseudocrossing at R, N 4.1 a.u. A small contribution to this peak from the 'S metastable primary ion channel
+
02+(2s22p2)'S+ H(1s) -+ 0+(2s2p3)2p2D H + + 6.29eV cannot be excluded. A second peak in the observed spectra can be correlated with the 'D metastable primary ion channel: 02+(2s22p2)'D+ H + O+(2s2p3)2p2D+ H + + 3.45 eV corresponding to a pseudocrossing at R, N 7.88 a.u. Calculations by Butler et al. (1980) based on a close-coupling quantal approach predict that in collisions of ground state 0 2 +ions with H, at thermal energies, only the 0 +(4P)product channel is significant. The TES
H. B. Gilbody
164
measurements indicate that this prediction is also valid at much higher energies.
F. COLLISIONS OF S3+ IONS WITH H The TES measurements of Wilson et al. (1990) provide data on state-selective electron capture by S 3 +ions in collisions with H at energies in the range 2.46.0 keV. In the energy change spectra shown in Fig. 6 the possible collision channels associated with primary S3+ ions in both the (3s23p)’P0 ground state and ( 3 ~ 3 p ~metastable )~P states are indicated G and M, respectively. These channels are based on published tabulations of the energy levels of S 2 + by Bashkin and Stoner (1978) and by Kelly (1987). The spectra exhibit the three peaks A, B and C, which are not completely resolved. Peak A, which accounts for about 40% of the S 2 + product yield over the energy range considered, cannot be correlated with any colIision channel based on published tabulations of S’+ energy levels. However, Wilson et al. (1990) show that the energy defect of 6.20 deduced from the position of peak A is consistent with the channel: S3+(3s23p)’P0+ H(1s) -+ S’+(3~’3p3d)~F~ + H + + 6.20eV The S’+(3~’3p3d)~F~ state, which is missing from published tabulations, had previously been suggested by Dynafors and Martinson (1978) on the basis of both calculations and beam-foil studies. The latter indicate a 3F term energy of 15.25 eV, which would provide an electron capture channel with an energy defect of 5.98 & 0.01 eV, which is in satisfactory agreement with the 6.20 & 0.2 eV observed value. The peak B can be correlated with the following two G channels:
+
S3+(3s23p)’P0 H(1s) -+ S2+(3s3p2)3p3Po + H + + 8.98 eV -+
S2+(3s3p2)3p’D+ H + + 8.31 eV
corresponding to pseudocrossings at R, 1: 6.05 and 6.54 a.u., respectively. It is also evident in Fig. 6 that three M channels may also contribute to peak B, although there is no clear evidence of M channel contributions elsewhere in the spectra. Peak C, which can be seen to increase in relative importance with increasing energy, is correlated with the G channel: S3+(3s23p)2Po + H(1s) + S’+(3s3p2)3p3Do+ H + + 10.81 eV corresponding to a pseudocrossing at 5 a.u. In the absence of detailed theoretical studies, Wilson et al. (1990) have
9 7 I
5
9 I
4
I
I
II Ill II I
I lit
Ill
6 5
4
I
I
I
II
Ill
II I
A:.
..
.. . ..-\.:.--... c . . . -, .
4. 2keV
-3 -1 i 3
s
i
9 ii i3
is
ii
1 1
3
:% .
5
I
I
7
9
I
-.
I
I
1113151
0 2
4 6 8 10 12 14 16 18 i
ENERGY CHANGE teV) FIG.6. Energy change spectra for electron capture by 2.4, 4.2 and 6.0 keV S 3 + ions in collisions with H. Positions of possible collision channels correspondingto ground-state (G)and metastable (M) primary ions are indicated. Vertical lines are results of MCLZ calculationsfor G channels. (From Wilson et al., 1990.)
166
H. B. Gilbody
carried out MCLZ calculations of the type described in Section 1II.C to predict cross sections for electron capture by ground-state S 3 + ions into all F ' The results of relevant states of S 2 + including the missing ( 3 ~ ~ 3 p 3 d ) ~state. these calculations are again shown as vertical lines in Fig. 6, with the largest calculated cross section normalized to the largest peak height observed experimentally. These MCLZ calculations can be seen to correctly predict the participation of the G channels that give rise to peaks A and B, but the theory fails to account for the G channel associated with peak C. There have been no measurements of the total electron capture cross section to which the TES data could be normalized. The MCLZ total cross sections which increase from 1.5 x lo-'' cm2 at 2.4 keV to 1.98 x lo-'' cm2 at 9.0 keV are likely to be rather smaller than the true cross sections.
IV. Conclusions The use of a tungsten tube furnace to provide a highly dissociated target of hydrogen in the Belfast TES apparatus has proved to be an effective way of studying state-selective electron capture by slow multiply charged ions in collisions with atomic hydrogen. Within the limits of the available energy resolution, the main collision product channels can be identified and relative cross sections determined from an analysis of the observed energy change spectra. Cross sections for specific product channels may then be determined by normalization of the energy change spectra to known total electron capture cross sections. The reactions considered, which all involve partially ionized primary ions, illustrate that collision channels corresponding to curve crossings at moderate internuclear separations are generally dominant. In not all cases are the TES results in good accord with theoretical predictions, including those based on a fully quanta1 molecular approach. Such discrepancies illustrate the difficulty of carrying out accurate calculations. Rough calculations based on the MCLZ approach can provide a useful indication of some of the main product channels, but the quantitative agreement with experiment is generally poor. The TES measurements provide an unambiguous indication of the presence of metastable ions in the primary beam. However, a detailed quantitative analysis of the TES data is impossible unless the fraction of metastable ions present in the primary beam is known. For many reactions this is a major complicating feature. The use of double translational energy spectroscopy (DTES) is an attractive possible solution to this problem that is planned for future work in this laboratory. The feasibility of DTES has been
TES STUDIES OF STATE-SELECTIVE ELECTRON CAPTURE
167
demonstrated by Huber et al. (1984) for gas targets, and the same general principles could be applied to measurements in atomic hydrogen. In the measurements planned, primary ions in a particular ground or metastable state will be prepared by electron capture collisions in a suitable gas target and then selected and identified by TES. These selected ions will then be used as primary ions in a second-stage TES experiment in atomic hydrogen. DTES measurements require primary beam intensities between lo3 and lo4 times greater than a single TES experiment, but the yields available from an ECR ion source are expected to be adequate for studies of many reactions.
Acknowledgments The work described in this review has been supported by a Rolling Research Grant from the United Kingdom Science and Engineering Research Council.
REFERENCES Bashkin, S., and Stoner, Jr., J. 0. (1978). Atomic Energy Levels and Grotrian Diagrams, North Holland, Amsterdam. Bienstock, S.,Heil, T. G., Bottcher, C., and Dalgarno, A. (1982). Phys. Rev. A 25, 2850. Bienstock, S., Heil, T. G., and Dalgarno, A. (1983). Phys. Reo. A 27, 2741. Butler, S. E., Heil, T. G., and Dalgarno, A. (1980). Astrophys. J . 241,442. Cirik, D., Brazuk, A,, Dijkkamp, D., de Heer, F. J., and Winter, H. (1985). J. Phys. B: A t . Mol. Phys. 18, 3639. Crandall, D. H., Phaneuf, R. A., and Meyer, F. W. (1979). Phys. Rev. A 19, 504. Crandall, D. H., Phaneuf, R. A., and Meyer, F. W. (1980). Phys. Rev. A 22, 379. Dynafors, B., and Martinson, I. (1978). Phys. Scr. 17, 2883. Errea, L. F., Herrero, B., Mendez, L., and Riera, A. (1991). J . Phys. B: At. Mol. Opt. Phys. 24, 4061.
Gardner, L. D., Bayfield, J. E., Koch, P. M., Sellin, 1. A., Pegg, D. J., Pederson, R. S., and Crandall, D. H.(1980). Phys. Rev. A 21, 1398. Gargaud, M., and McCarroll, R. (1989). J. de Physique, Colloque CI 50, C1-127. Gilbody, H. B. (1979). In: Advances in Atomic and Molecular Physics (D. R. Bates and B. Bederson, eds.), Academic Press, New York, 15, p. 293. Heil, T. G., Butler, S. E., and Dalgarno, A. (1983a). Harvard Smithsonian Center for Astrophysics, preprint series no. 1732. Heil, T. G., Butler, S. E., and Dalgarno, A. (1983b). Phys. Rev. A 27, 2365. Hoekstra, R., Boorsma, K., de Heer, F. J., and Morgenstern, R. (1989). J . de Physique, Colloque CI 51, C1-349. Huber, B. A,, Kahlert, H. J., and Wiesemann, K. (1984). J. Phys. B: At. Mol. Phys. 17, 2883. Janev, R. K.,and Winter, H. (1985). Phys. Rep. 117, 265. Kelly, R. L. (1987). J. Phys. Chem. R d Data 16, Suppl 1, 343.
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Kimura, M., Kobayashi, N., Ohtani, S., and Tawara, H. (1987). J. Phys. B: At. Mol. Phys. 20, 3873. McCullough, R. W., Lennon, M., Wilkie, F. G., and Gilbody, H. B. (1983).J . Phys. Bc At. Mol. Phys. 16, L173. McCullough, R. W., Wilkie, F. G., and Gilbody, H. B. (1984).J . Phys. B: At. Mol. Phys. 17, 1373. McCullough, R. W., Wilson,S. M., and Gilbody, H. B. (1987).J. Phys. B: At. Mol. Phys. 20,2031. McLaughlin, T. K., Wilson, S.M., McCullough, R. W., and Gilbody, H. B. (1990).J . Phys. B: At. Mol. Opt. Phys. 23,737. Olson, R. E., and Kimura, M. (1982).J. Phys. B: At. Mol. Phys. 15,4231. Opradolce, L., Benmeuraiem, L., McCarroll, R., and Piacentini, R. D. (1988).J. Phys. B: At. Mol. Opt. Phys. 21, 503. Phaneuf, R. A., Alvarez, I., Meyer, F. W., and Crandall, D. H. (1982). Phys. Reo. A 26, 1892. Phaneuf, R. A,, Meyer, F. W.,and McKnight, R. W. (1978). Phys. Rev. A 17.434. Steigman, G. (1975). Astrophys J . 195, L39. Taulbjerg, K. (1986). 1. Phys. B: At. Mol. Phys. 19, L367. Wilkie, F. G., McCullough, R. W., and Gilbody, H. B. (1986).J . Phys. B: At. Mol. Phys. 19,239. Wilson, S . M., McCullough, R. W., and Gilbody, H. B. (1988).J . Phys. B: At. Mol. Phys. 21, 1027. Wilson, S. M., McLaughlin, T. K.,McCullough, R. W., and Gilbody, H. B. (1990).J . Phys. B: At. Mol. Opt. Phys. 23, 1315. Winter, H. (1982). Comments At. Mol. Phys. 12, 165.
ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 32
RELA TWISTIC ELECTRONIC STRUCTURE OF ATOMS AND MOLECULES I. P . GRANT Mathematical Institute University of Oxford Oxford, England
I. Introduction . . . . . . . . . . . . . . . . . . . . . 11. The Beginnings of Relativistic Electronic Structure Theory A. Hydrogenic Atoms . . . . . . . . . . . . . . . . B. Many-Electron Atoms . . . . . . . . . . . . . . . 111. Open-Shell Atoms. . . . . . . . . . . . . . . . . . . IV. Basis Sets and QED of Atoms and Molecules. . . . . . V.Outlook.. . . . . . . . . . . . . . . . . . . . . . Acknowledgments. . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 169 . . . . . . . . 170 . . . . . . . 170 . . . . . . . 172 . . . . . . . 175 . . . . . . . . 179 . . . . . . . 183 . . . . . . . 184 . . . . . . . 184
I. Introduction The idea of a volume to honour Sir David Bates’s long and distinguished career in atomic, molecular and optical physics prompts many reflections, especially the manner in which these disciplines have grown over this period. The position has changed dramatically in the field of relativistic electronic structure calculations, driven by a combination of developments in experimental techniques, in the mathematical and computational tools available, and in the explosive growth of computing power in the last two or three decades. The story is worth telling as background to the state of the art of relativistic electronic structure calculations and to provide a basis for conjecture on future developments.
169
Copyright 0 1994 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003832-3
I. P. Grant
170
11. The Beginnings of Relativistic Electronic Structure Theory A. HYDROGENIC ATOMS The beginnings of relativistic electronic structure theory can be traced back to Sommerfeld (1916), who combined the theory of special relativity with the ''old'' Bohr quantum theory to treat line spectra. This led to the first appearance of the fine structure formula
in which E is the energy of a hydrogenic level in an atom with nuclear charge Z, R is Rydberg's constant, a is the fine structure constant, and the integers n and k are the principal and inner quantum numbers, respectively. The relativistic correction, the terms of order aZZ4and higher, appears as a small correction in the spectrum of atomic hydrogen and He+. Dirac's (1928) wave equation for the electron,
a*
[ciji *sj + PmcZ + V ] + = ih at
in which ? and i B are Dirac four-component matrices, rn is the electron rest mass, and c is the velocity of light' reproduces (1) when
is the interaction energy of an electron at distance I from the nucleus, with the substitution of j + 4 for k. Here j is an odd multiple of 1/2, representing the total angular momentum of the electron, obtained by vector coupling its orbital angular momentum with its intrinsic spin 4h. Dirac's new theory correctly predicted the intrinsic spin of the electron postulated by Goudsmit and Uhlenbeck (1926) to account for the doublet structure in alkali spectra and the anomalous Zeeman effect. Attempts to confirm the predictions of Dirac's theory by spectroscopic methods continued throughout the 1930s and 1940s, but it was the classic experiment of Lamb and Retherford in 1947, which revealed that the 2p1,2 level in hydrogen lies 1057 Mhz below the 2s,,, level (with which the Dirac theory predicts it to be degenerate), that stimulated a major elaboration of the model. The story is documented in the book by Bethe and Salpeter (1957), 'We shall use Hartree's atomic units throughout. In these units, m 137. In general, we adhere to the notation of Grant (1970).
c = ti-' x
= h = e2/4ns, = 1
and
RELATIVISTIC ELECTRONICSTRUCTURE OF ATOMS AND MOLECULES
171
and more up-to-date accounts will be found in the books edited by Series (1988) and Kinoshita (1990). Today, experiments involving beam-foil spectroscopy, lasers, and ion traps have made it possible to produce sources of hydrogenic ions with nuclear charges, 2, as large as 92. The modern covariant form of quantum electrodynamics (QED) was developed to explain the deviations from Dirac's theory and has successfully explained a widening range of experiments of increasing precision since it was introduced. From the point of view of atomic structure, the most important effect is the contribution of the electron's self-energy to hydrogenic energy levels due to its interaction with the quantized radiation field. This shift can be expressed in the form ESE
a (Za)' F(Za)mc2 n3
= -R
(4)
The main variation with Z is due to the leading factor; the function F ( 2 a )is a slowly varying number of order unity. For values of Z less than about 20, F(Za) can be represented by a perturbation expansion of the form F(Za) = A40
+ A,,
In(Za)-2
+ A,o(Za) + A , ~ ( Z ~ ) '
+ A61(Za)2In(Za)-, + A,,(z~)'
In2(Za)-2+ ...
(5)
(6)
This expansion is slowly convergent, so that the interaction between the electron and the nucleus should be treated in a nonperturbative manner. Most predictions are now based on the results for the self-energy of the lq,,, 2s1,,, 2p,,, and 2p3,, levels for atomic numbers Z = 10,20,. . . , 110 obtained by Mohr (1974, 1982). The next most important correction comes from the polarization of the vacuum by the charged nucleus; this can be expressed by a formula similar to Eq. (4) with F(Za) replaced by a new function H(Za). For large values of 2, the finite size of the nucleus and also of its motion (which depends on the small ratio m/M of the electron rest mass to that of the nucleus) must also be taken into account. The most comprehensive tabulation of hydrogenic energy levels including all these contributions has been reported by Johnson and Soff (1985). So far, measurements of hydrogenic energy levels and Lamb shifts have revealed no systematic differences between the experimental and theoretical level energies tabulated by Johnson and Soff (1985) for elements up to uranium. A convenient summary has been given by Pipkin (1990), who concluded that no deviations are likely to be found that cannot be accounted for within a QED, electro-weak, QCD framework. Further progress at low Z depends on an independent determination of the charge radius of the proton. At high Z (measurements in P14+, S',', C116+, A"+ and U9'+), the experimental Lamb shifts are roughly one standard deviation smaller than
172
I . P . Grant
the theoretical values, and more precise measurements and calculations are still needed.
B. MANY-ELECTRON ATOMS The modelling of many-electron atoms, either nonrelativistically or relativistically,introduces many new features, not the least of which is the fact that exact analytic solutions are no longer available. Methods of approximation, in particular the use of variational methods for bound states of atoms, and methods of perturbation early made their appearance in atomic structure calculations. The self-consistent field method (without exchange) is due to Hartree (1928), and Fock's method, using antisymmetric wave functions, followed in (1930). A number of self-consistent field calculations (for simple configurations of elements as large as Cs' or Hg) were carried out by hand during the 1930s, the first multiconfiguration Hartree-Fock (MCHF) calculations (for O"', n = 0-3) being by Hartree et al. (1939). The Ritz variational method was first applied to the ground state of He by Kellner (1927) and was then developed extensively by Hylleraas (1928, 1929), who exploited trial functions that contained the interelectron distance explicitly. This technique was later much developed by Pekeris (1958)and collaborators in papers such as Accad et al. (1971) and by many others. It is still the most accurate method available for the states of two- and three-electron systems in nonrelativistic quantum mechanics; recent work (e.g., Drake 1988, 1989) achieves a precision of lo-'' Rydberg for the n1s3S states of helium, n = 2 , 3 ,..., 6. It was realized very early that relativistic effects must be taken into account in order to accurately model fine structure in many-electron atoms as well as in one-electron atoms. This means that the noncovariant electrostatic interaction between electrons must be replaced by a relativistic expression taking account of the fact that electromagnetic disturbances propagate with the velocity of light. The interaction introduced by Breit (1929) in his study of the He atom bears a close resemblance to the classical relativistic Lagrangian for interacting charged particles of Darwin (1920). As a function of the interparticle vector, a, Breit's interaction can be written
l 1 and c7ii is the Dirac matrix operator representing the spacelike where R = a components of the current of the ith electron. The first term is the instantaneous Coulomb interaction of nonrelativistic quantum mechanics,
RELATIVISTICELECTRONICSTRUCTUREOF ATOMS AND MOLECULES 173
and the remainder describes the interaction of the electron currents. In combination with the Pauli approximation, which leads to an expansion of the relativistic correction terms in powers of the parameter crZ, the free twoelectron atom has an effective Hamiltonian (Bethe and Salpeter, 1957, Section 39):
H = Ho
+ H , + H , + H , + H4 + H5
(8)
where H , is the nonrelativistic Hamiltonian for the helium atom; H , represents the classical “variation of mass with velocity” predicted by special relativity; H 2 is a classical current-current interaction (the “orbit-orbit” term); H 3 is the interaction between the spin magnetic moments of the electron and the magnetic field it perceives in motion (the “spin orbit” and “spin other orbit” interactions); H4 is the “Darwin term” proportional to the scalar product 5;*$ of the electron’s momentum with the electric field acting on it (giving a delta function interaction when 2 is the field due to a point charge); H 5 is the interaction energy of the electron spin dipole magnetic moments. Further terms can be added to account for the interaction with external fields. The extension to systems with more than two electrons took a long time to get established. Gaunt’s (1929) paper on the relativistic theory of many-electron atoms was followed by a formulation of relativistic selfconsistent field theory by Swirles (1935). The first Dirac-Hartree calculation (of Cu’) was done by Williams (1940). Further progress was hampered by two problems: the first was that Gaunt’s angular integrals were expressed as lengthy algebraic formulae that were difficult to evaluate numerically; the second was that worthwhile problems were too large to be computed manually. The angular integral problem was essentially solved with the development of the theory of angular momentum in quantum mechanics by Racah (1942, 1943) and its exploitation in atomic and nuclear physics in the 1950s. And a Dirac-Hartree calculation for Hg was one of the early tasks tackled by the primitive Cambridge EDSAC computer (Mayers, 1957). The pace began to quicken in the 1960s following the publication of a new formulation of the problem by Grant (1961). This exploited Racah’s methods to write the relativistic formulae for matrix elements of the Dirac-Coulomb Hamiltonian:
in a simple and tractable form with clear analogies to the conventional nonrelativistic theory of atomic structure expounded in such treatises as that of Condon and Shortley (1935). Computers grew sufficiently powerful that the coupled integro-differential equations of Dirac-Hartree-Fock theory
174
1. P. Grant
could be solved by finite difference iterative methods without too much manual intervention, although computer memories still consisted of at most a few thousand words. By the end of the decade a review (Grant, 1970)recorded Dirac-Hartree calculations for some 24 ground states, Dirac-Hartree-Slater calculations(with a local exchange potential) for the ground configurations of all elements with 8 < 2 < 95 as well as for the hypothetical superheavy elements 104 ,< Z < 132, and a small number of full Dirac-Hartree-Fock calculations, mainly for the inert gases and superheavy elements. Desclaux (1973) has since tabulated ground, LS-average of configuration, DHF results for all elements with Z < 120. Attempts to use methods in which the Dirac radial spinor components were expanded in a set of L’ analytic functions, first proposed for nonrelativistic atomic and molecular calculations by Hall (1951) and Roothaan (1951), were less successful. Synek (1964) formulated some equations, but the first real calculations using the matrix Dirac-Hartree-Fock method were made by Kim (1967). He computed ground states of He, Be, and Ne and was able to optimize the total energy to about five decimals. He encountered a number of difficulties, in particular in representing the small component of the Dirac spinors, and noted with some alarm that the variational calculation did not necessarily generate upper bounds to the total energy as in nonrelativistic SCF calculations. The computer memory requirements were considerably greater than for equivalent nonrelativistic calculations, and Kim concluded that the computers of the day did not have the necessary muscle to treat atoms with Z 2 50 or to treat molecules. Another approach which was particularly appropriate for positive ions with 2-10 electrons was Z-expansion theory. Using the scaled variable x = Zr/a,, the nonrelativistic Hamiltonian can be written
where 1= 1/Z, and pi is conjugate to the scaled position vector Zi. Energies and other properties of the ion can be expanded in powers of the coupling parameter R about the uncoupled hydrogenic problem with 2 = 0. The relativistic generalization by Layzer and Bahcall(l962) requires a relativistic expansion parameter p = a2Z2as well as the nonrelativistic parameter 1.The expansion coefficients for matrix elements of the Hamiltonian within a complex’ were obtained by Layzer and Bahcall(l962) and extended by Doyle (1969). Goldsmith (1974) later included terms that accounted for nonrelativistic correlation, improving the agreement with experiment. Because the formulae can be applied to a complete isoelectronic sequence, the method ?States based on products of hydrogenic orbitals with a common principal quantum number.
RELATIVISTIC ELECTRONIC STRUCTURE OF ATOMS AND MOLECULES
175
continues to be used to predict properties of highly ionized atoms of astrophysical interest, especially in the Russian literature (Safronova and Senashenko, 1984).
111. Open-Shell Atoms Relativistic atomic structure theory was still exotic in 1970, though it was beginning to make its mark. By the early 1970s, the power of machines such as the IBM360/195 and CDC 7600 were sufficient to make more ambitious atomic models a practical proposition. The arrival of CRAY and CYBER supercomputers with both fast scalar and vector processing opened up further opportunities later in the decade. However, it was first necessary to extend the simple models of the 1960s. The options were considered in some detail at a workshop at CECAM, Orsay, in 1970. Although it was known that the fundamentals of relativistic atomic structure theory needed examination, it was generally agreed that most rapid progress would be made by constructing codes capable of handling atomic configurations containing several open shells. This workshop led first to the MCDF code of Desclaux (1975) and later to the more comprehensive package of Grant et al. (1980), whose modular construction reflected a wish to make the interface between human and machine as simple as possible. There were essentially two choices for the machinery to handle open-shell configurations. One possibility was to generate sets of Slater determinants, each describing some antisymmetric assignment of N electrons to a set of orbitals, and then to construct suitable linear combinations with prescribed angular properties using techniques based on the methods of Condon and Shortley (1935).This procedure (see Chapter 1 of Judd, 1963), was embodied, for example, in the n6nrelativistic SUPERSTRUCTURE code (Eissner et al., 1974). The alternative, which exploited Racah’s methods in conjunction with the theory of fractional parentage, was adopted for the codes of Grant et al. (1980). The formalism has been described in a review (Grant, 1988).It was strongly influenced by the lectures of Judd (1967), which explored the use of second quantization (Fock space) methods to atomic shell theory, following Brink and Satchler (1956),who had earlier applied these techniques in nuclear shell theory. Judd described the use of second quantization techniques to discuss configuration interaction, the use of the graphical methods of Yutsis et al. (1962) and the topological equivalence of Jucys and Feynman diagrams, and the theory of quasispin and its connection with fractional parentage, all of
I . P. Grant
176
which have been utilized in the component modules of Grant et al. (1980). Dirac central field orbitals are characterized by quantum numbers n, K , m, where, the integer n is the principal quantum number, K = A ( j + 1/2) is the angular quantum number, and j, m = - j, - j + 1 , . . . ,j are the total angular momentum quantum number and its projection on the axis of quantization. The sign of K is negative when the “large” component has orbital angular momentum 1 = j - 1/2, and positive when I = j + 1/2. The 2 j + 1 states with the same values of n, K have the same energy and transform into one another under spatial rotations. These states are said to be equivalent and constitute a subshell in the usual terminology. Subshell states ( n ~with ) ~q-electrons can be labeled by total angular momentum, J , parity P = (- l)q’,and may require an additional quantum number y, usually just the seniority number v, to distinguish degenerate states. The projection of the total subshell angular momentum can be ignored. After establishing a dictionary ordering of subshells, their angular momenta are vector coupled in the order of their appearance in the list. A configurational state (CSF) of N = Ei n,qi electrons can therefore be specified in terms of an ordered set ((nilci)q‘yi, J i , Pi, i = 1,2,. . and a coupling scheme (...{[J1Jz)X1]J3}X2...Jn)J . input to the codes thus requires a simple list of numbers identifying the CSF to be considered. Although this is easy if only a small number of CSF are needed, it can become a considerable chore to list all the CSF for a complex atom. In nonrelativistic quantum mechanics, subshells with the same nl values but different j = l f. 1/2 are degenerate, so that it is often easier to specify a “nonrelativistic”configuration set and let the code generate all the relevant jjcoupled CSFs, of which several hundred may be required in contemporary applications. Configuration generating utilities now form a part of both relativistic and nonrelativistic codes (Sturesson and Fischer, 1992). Matrices of quantum mechanical operators between CSFs are expressible as linear combinations of integrals over the radial components of the Dirac central field spinors (Slater integrals). The coefficients of the Slater integrals, which reflect the angular momentum coupling structure of the CSFs, may be generated from a knowledge of the CSF parameter lists using straightforward Fortran programs: MCT (Pyper et al., 1978) for one-electron operators; MCP (Grant, 1973) for the matrices of the Coulomb interaction between electrons; MCBP (Beatham et al., 1979) for the rest of the relativistic electron-electron interaction in the transverse photon gauge. The matrix of the N-body Hamiltonian constructed in this way has eigenvalues and eigenvectors that approximate the physical atomic energy levels and states. It remains to construct the radial parts of the Dirac orbital wave functions, for which the MCDF module (Grant et al., 1980) (or its equivalent in Desclaux, 1975) can be used. The MCDF module allows for a wide variety of models: parametrized model potentials (screened) Coulomb fields, and self-
.>
RELATIVISTICELECTRONICSTRUCTURE OF ATOMS AND MOLECULES 177
consistent fields of various types. For a general multiconfiguration Dirac(Hartree)-Fock (MCDF) problem, the potentials can be constructed knowing the angular coefficients generated by the MCP, MCBP, and MCT module^.^ There are several ways in which the MCDF method can be employed (Grant et al., 1976). The O L (optimal level) scheme applies the variational method to a linear superposition of CSFs for a single level or, in its extended form, a small subset of levels of the same symmetry. This is the way in which the method was first defined by Hartree et al. (1939). It is relatively expensive, as each level is computed in a separate iterative calculation; also the SCF orbitals from different calculations are constructed from different potentials and are therefore not orthogonal, which complicates the calculation of transition rates. As pointed out by Grant et al. (1976), most physical applications require information about a number of atomic states, and a method that generates such information relatively cheaply is therefore extremely useful. The AL (average level) scheme was therefore designed with this consideration in mind. It assumes that the same orbitals will be used to construct all CSFs, so that the variational procedure is applied to a weighted mean of subshell average energies. The angular coefficients are then not needed for determining the orbitals. The AL scheme is much cheaper to use than OL, but the employment of the same orbitals for all levels means that they are not optimal for any one level. If O L calculations of the same levels show that some orbitals are significantly changed, then the AL results will be less accurate. However, the orbitals are automatically orthogonal, which simplifies transition rate calculations. The simplicity of the AL method has encouraged its use especially for highly ionized atoms of interest to the fusion program and to astrophysics, where the dominance of the nuclear Coulomb potential means that there is little difference between the OL and AL solutions for a large number of levels. It is usual to use a finite size model of the nuclear charge distribution, especially for high 2. The BENA module (McKenzie et al., 1980) computes the matrix of the magnetic and retarded parts of the relativistic electronelectron interaction in the CSF basis as well as screened hydrogenic estimates of the QED corrections based on the tables of Johnson and Soff (1985), and the perturbed Hamiltonian is rediagonalized to give the final estimates of energy levels and wave functions for the atom. Radiative transition rates can also be calculated. A typical example of the use of these codes for spectrum identification is described in the papers of Ekberg et al. (1989, 1991). They studied 3-3 transitions in laser-produced EUV spectra of magnesium-like 'In practice, the MCBP coeficients, which define the Breit or transverse potentials, have been rarely used in self-consistentcalculationsbecause they generate a large number of small integrals; their effect is usually taken into account as a perturbation.
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-I3
t 20
ATOMIC NUMBER
25
30
35
40
ATOMIC NUMBER
FIG. 1. Difference, AE, between observed and calculated energies for (left) 3pz lD,, 'So, 3s3d ID, and (right) 3d2 ID2. 'G, states of magnesiumlike ions. (From Ekberg et a[., 1989.)
ions with 22 < 2 < 55. Only the spectroscopically relevant configurations 3s2, 3s3p, 3p2, 3s3d, 3p3d, and 3d2 were included in the AL calculation, a minimal CSF basis for this problem. As shown in Fig. 1, the difference between ab initio and experimental level energies is a smooth function of 2 and can be used to interpolate and extrapolate the energies of unobserved levels. The relative error is of order 0.1%. Relativistic effects are also important for inner shell transitions in heavy neutral atoms, and here again MCDF calculations have proved invaluable. Thus Mokler et al. (1990)discuss the processes leading to the generation of Xrays in heavy ion-atom collisions. In the case of RTE processes? the collision produces a doubly excited state that stabilizes radiatively with the emission of two X-rays. The capture process is the inverse of an Auger transition, and the cross section for X-ray/X-ray coincidences can therefore be obtained by using detailed balance to obtain the capture cross section along with the energy levels and radiative rates predicted by the codes. There is generally good agreement between measured and predicted resonance structure. X-ray and 4Resonant transfer and excitation.
RELATIVISTIC ELECTRONIC STRUCTURE OF ATOMS AND MOLECULES 179
Auger transition rates are needed in general to understand the decay processes by which an atom or ion in an excited state relaxes, and a number of people have constructed modules that use MCDF wave functions to compute these rates. A review of the state of the art, especially for highly ionized atoms, was given by Chen (1 990). Code developments have continued in response to demands for improved performance, largely for dealing with spectroscopic problems in complex atoms. The number of CSF needed to model the low-lying states of transition metals, lanthanide, or actinide spectra with any attempt at realism can easily run into hundreds or thousands. Self-consistent field calculations are not too effective in such problems, and large-scale superposition of CSFs is the preferred route. The new version of the Oxford codes, GRASP (Dyall et al., 1989), addressed some of the limitations of the (Grant et al., 1980) package, but the files of angular coefficients and the size of the matrices in these more demanding problems means that further code development is necessary. A GRASPZversion, which incorporates substantial technical improvements, is in preparation.
IV. Basis Sets and QED of Atoms and Molecules The early work of Kim (1967) revealed problems with matrix methods based on nonorthogonal basis sets of the sort popular in nonrelativistic quantum chemistry. Kagawa (1975, 1980) extended Kim’s work by developing a MCDF code, but the tendency to “variational collapse”-convergence of trial eigenvalues to limits below the exact eigenvalues-remained a problem. Several attempts to devise procedures for relativistic molecular structure generated highly unphysical results, which damped enthusiasm for such methods. At the same time, Sucher (1980), following Brown and Ravenhall (1951), pointed out that an unquantized many-particle relativistic wave equation needs to be protected from “continuum dissolution,” so generating a vigorous debate that is only now beginning to die down. Sucher maintains that some explicit mechanism is required to prevent electrons from “collapsing” into negative energy states in unquantized theories. In the hydrogenic case, Dirac’s hole theory (which appeals to Pauli’s exclusion principle to prevent collapse by filling all vacancies in negative energy states), or the equivalent positron theory stabilizes the electron distribution. The same mechanism operates in many-electron systems, and the situation is quite transparent as long as the theory is formulated properly within a QED framework (Grant and Quiney, 1988). However, Sucher asserts that unquantized relativistic Hamiltonians are “unhealthy” and will be restored to
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“health” only by explicitly inserting projection operators to restrict physical interactions to electrons in positive energy states. This 1980 critique brought to a head the uncertainties that had so far been laid (consciously) to one side and focussed attention on the need to demonstrate the consistency of relativistic atomic structure theory with quantum electrodynamics. There was an obvious, though at that stage apparently impractical,way to proceed. Basis sets generate finite-dimensional approximations to quantum-mechanical operators, and the idea of using them to compute quantities within the Furry (1951) bound interaction picture of QED was highly attractive. The continuum integrations of conventional field theory could then be replaced by matrix algebra, which by 1980 was becoming easy to implement on modern vector- and array-processing computers, and QED perturbation theory would then become amenable to methods familiar from quantum chemistry. However, the well-known impossibility of representing unbounded operators such as position and momentum components by finite matrices’ serve as warnings that such proposals have to be carefully justified. The first stage of this investigation was to understand why existing basis set methods failed (Grant, 1986). The answer lay mainly in the boundary conditions in the highly singular region near the nucleus, where it is essential that variational trial solutions must perform well. In the nonrelativistic limit, the large and small components of a Dirac wave function must satisfy Pauli’s relation
I,P a (?i-$)$L
(11)
to recover the Schrodinger equation. This implies that the basis functions used for large and small component expansions should be paired so that
FS + (Z-jj)FL (12) in the mathematical limit c + co. This is sometimes called strict kinetic balance. For radial components with angular symmetry u the limiting relation takes the form d
u
dr
r
ascjco. It is straightforward to construct basis sets with strict kinetic balance, and even to devise a relation that is valid for j n i t e values of c in the case of a point charge nuclear model. The Lspinor (Quiney et al., 1988) and S-spinor (Grant ’It is necessary to observe only that no finite Hermitian matrices a, b or real number c exist such that commutation relations of the form [a, b] = ic can be satisfied.
RELATIVISTIC ELECTRONIC STRUCTURE OF ATOMS AND MOLECULES 181
and Quiney, 1988; Quiney, 1990) basis sets described later satisfy such a relation, which is consistent with the small r power expansions built into finite difference algorithms such as those of Desclaux (1975) and Grant et al. (1980). For a finite nucleus, strict kinetically balanced Gaussian functions, Gspinors (Ishikawa et al., 1985; Ishikawa and Quiney, 1987), appear the best choice. It has been shown (Grant, 1986) that when the lowest eigenvalue of the matrix of the potential is greater than -2mc’ there is an effectively Jixed lower bound to the bound state part of the Dirac spectrum. This means that the positive energy part of the spectrum is bounded below as in the nonrelativistic case, and variational calculations for relativistic bound states will converge as they do nonrelativistically. We have thus eliminated variational collapse from both molecular and atomic calculations. The next problem is that for the variational process to converge to the correct mathematical limit in one-electron problems, the basis set must have linear independence and completenessproperties in a suitable function space. This is a nontrivial problem; while definitive answers to all the questions have not yet been obtained, we now know enough to be confident that the results obtained so far are mathematically valid (Grant, 1989). The main difficulty is to ensure that a linearly independent (nonorthogonal) basis set can be constructed that gives sufficiently accurate results for the observables of interest. This is easy to arrange in atoms, though the desire to keep the dimensionality of the problem as low as possible in molecular problems means that many calculations are far from converged. We concentrate here on methods for atoms. The key here are the L-spinors, the relativistic analogues of Coulomb Sturmian functions. Their functional form is
The apparent principal quantum number is given by Nnr,K= ,/n:
+ 2n,y + K’
and y =.-/, The integers n, start from 0 for K < 0 and from 1 for K > 0. NnrK is a normalizing constant, and the L::(21r) are Laguerre polynomials. The real, positive number, A, is at our disposal; when A = Z/N,+, then the Lspinor reproduces the Dirac radial components for the (n, K) orbital in a hydrogenic ion of charge 2 where n = n, + 1x1 is the
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principal quantum number. Of course, the states (n’,x ) with d # n are then represented as linear combinations of Lspinors. In general, we can use ;1as a “tuning” parameter. The matrix of the Dirac hydrogenic problem is easy to construct; its elements are simple algebraic expressions, and it is possible (though not usually necessary) to take the maximum value of n, as high as 1000 without encountering problems. However, the calculation of electron-electron interaction integrals presents major problems, and the S-spinors, obtained from Eq. (13) by setting n = 0 (for K < 0) or n = 1 (for IC 0), are more useful for most self-consistent or MBPT applications. The definition (Quiney, 1990) is
=-
f;fi(r)= (A:
+ IZirB:)r~e-“r
(15 ) where the label T takes the values L, S and where A: and B,T are defined by comparison with Eqs. (13) and (14). For S-spinors, the set is defined by choosing the exponents li.Quantum chemists usually optimize basis sets in an ad hoc manner based on experience, but we have found that a set of exponents in geometric progression (Schmidt and Ruedenberg, 1979) has many computational advantages. Thus we consider families of basis sets with
Aj = aI p-1, I
i = 1...1
(16)
where at -,0 PI > 1, PI + 1 as I + co. In the nonrelativistic limit, Eq. (14) becomes a Slater type basis set, and it is easy to optimize at and PI for each symmetry in a nonrelativistic Hartree-Fock calculation for relatively small I. The same exponents work well in the relativistic case, and a suitable rule (Klahn and Bingel, 1977)for changing the parameters as I increases is all that is required to get eight or nine decimal accuracy in a self-consistent field for a large atom without further expensive optimization (Quiney et al., 1989, 1990; Grant 1990). This process works also with G-spinors (Mohanty et al., 1990), which are appropriate to models with finite size nuclei, and can be taken over into molecular calculations. Alternative methods of developing matrix representations of the Dirac spectrum in atomic calculations have been proposed, for example, by Johnson et al. (1988a), and by Salomonson and Oster (1989). The former use B-splines as their basis functions, while the latter use a finite difference representation. Johnson et al. in particular have applied their techniques to relativistic many-body theory of the energy levels of ions having one valence electron outside closed shells in the Li isoelectronic sequence (Johnson et al., 1988b),the Na sequence (Johnson et al., 1990a), and most recently in the Cu sequence (Johnson et at., 1990b). The calculations include Coulomb and Breit correlation corrections (as independent perturbations) to the third order, with some estimates of neglected fourth-order terms. The error due to
RELATIVISTIC ELECTRONIC STRUCTURE OF ATOMS AND MOLECULES 183
truncation of the perturbation series is said to be less than the numerical error of the calculated terms. The new matrix methods open up a wide range of applications in relativistic atomic and molecular structure theory that were not previously accessible using code based on finite difference algorithms. Basis sets afford an economical representation of the whole Dirac spectrum; the bound states are represented as accurately (to eight or nine decimals) as in a finite difference calculation with GRASP, and the continuum states are represented by L2wave packets that approximate the analytic solutions near the nucleus. Finite difference calculations require the solution of additional coupled equations for each new orbital. The unretarded Breit interaction can be included self-consistently with virtually no overhead (Quiney et al., 1987) because the additional radial integrals are necessarily generated with those for the Coulomb interaction. This would be prohibitively expensive in a GRASP calculation. Diagrammatic perturbation theory calculations (Quiney et al., 1989, 1990) exploit the efficiency of matrix algebra codes on modern computers using the basis set representation of the Dirac spectrum. Coupled cluster calculations (Quiney et al., unpublished) can be done the same way. And at long last, it is possible to achieve the objective of calculating radiative corrections including Q E D renormalization (Quiney and Grant, 1993; Persson et al., 1993), so demonstrating that the relativistic quantum theory of atoms and molecules can be sited firmly within perturbative QED.
V. Outlook Relativistic atomic structure calculations have become the standard tool for a variety of atomic physics applications in 1992. On the other hand, ab initio relativistic molecular structure is still in its infancy, with most molecular studies based at present on the use of effective core potentials derived from atomic calculations. Groups involved in a b initio relativistic molecular calculations have mostly used extended kinetic balance, in which they have sought to satisfy Eq. (11) by adding additional functions to the small component basis set, relying on the variational process to satisfy (12) or (13) approximately. Apart from unnecessarily inflating matrix dimensions, the matrix then acquires eigenstates with no physical meaning. Codes built to exploit strict kinetic balance avoid both problems as well as being more economical, and progress is likely to be slow until they are introduced. Codes like GRASP will continue to be used, and their range of application expanded as new modules come into service; for example, Fritzsche et al.
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(1992a,b) have recently completed a new GRASP-compatible module to compute Auger transition rates. However the new atomic basis set methods are potentially much more powerful and can be made highly efficient on modern parallel and distributed computer architectures. The study of relativity-correlation effects in complex atoms is possible in either scheme using CI or many-body procedures. So far this has been demonstrated only for problems with one active electron (Johnson et al., 1988b, 1990a,b); calculations with more than one active electron await the adaptation of the GRASP open-shell machinery. It only remains to calculate the residual QED radiative corrections; the recent calculations of Grant and Quiney (1992) and Lindgen et al. (1992) demonstrate that the renormalized selfenergy for a many-electron atom can be computed accurately ab initio using matrix methods. This will do away with the present need to rely on scaled hydrogenic estimates of the self-energy in many-electron calculations, a source of one of the major uncertainties for applications to the spectra of highly ionized atoms.
Acknowledgments The authors and publishers of Physica Scripta are thanked for permission to copy Figures 2 and 3 from the paper of Ekberg et al. (1989). I am indebted to H. M. Quiney for a critical reading of the typescript.
REFERENCES Accad, Y., Pekeris, C. L., and Schiff, B. (1971). Phys. Reo. A 4, 516. Beatham, N., Grant, 1. P., McKenzie, B. J., and Pyper, N. C. (1979). Comput. Phys. Commun. IS, 245. Bethe, H. A., and Salpeter, E. E. (1957). Quantum Mechanics of One- and "0-Electron Atoms, Springer-Verlag, Berlin. Breit, G . E. (1929). Phys. Reo. 34, 553. Brink, D. M., and Satchler, G. R. (1956). Nuooo Cimento 4, 549. Brown, G. E., and Ravenhall, D. G. (1951). Proc. Roy. SOC.A 208, 552. Chen, M. H. (1 990). In: AIP Conference Proceedings 215: X-ray and Inner-Shell Processes (T. A. Carlson, M. 0. Krause, and S. T. Manson, eds.) American Institute of Physics, New York, p. 391. Condon, E. U., and Shortley, G. H. (1935). The Theory of Atomic Spectra, Cambridge University Press, Cambridge. Darwin, C. G. (1920). Phil. Mag. 39, 537. Desclaux, J. P. (1973). At. Data Nucl. Data Tables 12, 311. Desclaux, J. P. (1975). Comput. Phys. Commun. 9, 31; (1977) 13, 71.
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Dirac, P. A. M. (1928). Proc. Roy. SOC.A 117, 610; 118, 351. Doyle, H. T. (1969). Adv. At. Mols. Phys. 5, 337. Drake, G. W. F. (1988). Nucl. Inst. Meth. B 31, 7. Drake, G. W. F. (1989). in: AIP Conference Proceedings 189: Workshop on Relativistic, Q E D and Weak Interaction Eflects in Atoms (P. J. Mohr, W. R. Johnson, and J. Sucher, eds.), American Institute of Physics, New York, p. 146. Dyall, K. G., Grant, I. P., Johnson, C. T., Parpia, F. A,, and Plummer, E. P. (1989). Comput. Phys. Commun. 55, 425. Eissner, W., Jones, M., and Nussbaumer, H. (1974). Comput. Phys. Commun. 8, 270. Ekberg, J. O., Feldman, U., Seely, J. F., and Brown, C. M. (1989). Physica Scripta 40,643. Ekberg, J. O., Feldman, U., Seely, J. F., Brown, C. M., MacGowan, B. J., Kania, D. R., and Keane, C. J. (1991). Physica Scripta 43, 19. Fock, V. (1930). 2. Physik 61, 126; 62, 795. Fritzche, S., and Fricke, B. (1992a). Physica Scripta T 41, 45. Fritzche, S., Fricke, B., and Sepp, W.-D. (1992b). Phys. Rev. A 45, 1465. Furry, W. H. (1951). Phys. Rev. 81, 115. Gaunt, J. A. (1929). Proc. Roy. SOC.A 124, 163. Goldsmith, S. (1974). J. Phys. B 7, 2315. Goudsmit, S. A,, and Uhlenbeck, G. E. (1926). Nature 117, 264. Grant, I. P. (1961). Proc. Roy. SOC.A 262, 555. Grant, I. P. (1970). Adu. Phys. 19, 747. Grant, I. P. (1973). Comput. Phys. Commun.5, 263; (1978) 13,429; (1978) 14, 312. Grant, I. P. (1986). J. Phys. B. 19, 3187. Grant, I. P. (1988). Methods in Comput. Chem. (S. Wilson, ed.) Plenum Press, New York 2, 1. Grant, I. P. (1989). In: AIP Conference Proceedings 189: Workshop on Relativistic, Q E D and Weak Interaction Eflects in Atoms (P. J. Mohr, W. R. Johnson, and J. Sucher, eds.), American Institute of Physics, New York, p. 235. Grant, I. P. (1990). In: AIP Conference Proceedings 215: X-ray and Inner-Shell Processes (T. A. Carlson, M. 0.Krause, and St. T. Manson, eds.), American Institute of Physics, New York, p. 46.
Grant, I. P., Mayers, D. F., and Pyper, N. C. (1976). J . Phys. B 9, 2777. Grant, I. P., McKenzie, B. J., Norrington, P. H., Mayers, D. F., and Pyper, N. C. (1980). Comput. Phys. Commun.21, 207. Grant, I. P., and Quiney, H. M. (1988). Adv. At. Mol. Phys. 23, 37. Hall, G. G. (1951). Proc. Roy. SOC.A 205, 541. Hartree, D. R. (1928). Proc. Camb. Phil. SOC.24, 89, 111, 426. Hartree, D. R., Hartree, W. R., and Swirles, B. (1939). Phil. Pans. Roy. SOC.(London) A 238,229. Hylleraas, E. A. (1928). 2. Physik 48, 469. Hylleraas, E. A. (1929). 2. Physik 54, 347. Ishikawa, Y., Baretty, R., and Binning, R. C. (1985). Chem. Phys. Lett. 121, 130. Ishikawa, Y., and Quiney, H. M. (1987). Int. J. Quant. Chem., Quantum Chem. Symp. 21, 523. Johnson, W. R., Blundell, S. A., and Sapirstein, J. (1988a). Phys. Rev. A 37, 307. Johnson, W. R., Blundell, S. A., and Sapirstein, J. (1988b). Phys. Rev. A 37, 2664. Johnson, W. R., Blundell, S. A,, and Sapirstein, J. (1990a). Phys. Rev. A 41, 1698. Johnson, W. R., Blundell, S. A,, and Sapirstein, J. (1990b). Phys. Rev. A 42, 1087. Johnson, W. R., and Soff, G. (1985). At. Data and Nucl. Data Tables 33, 405. Judd, B. R. (1963). Operator Techniques in Atomic Spectroscopy, McGraw-Hill, New York. Judd, B. R. (1967). Second Quantization and Atomic Spectroscopy, Johns Hopkins Press, Baltimore.
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Kagawa, T. (1975). Phys. Reo. A 12, 2245; (1980) 22, 2340. Kellner, G. W. (1927). Z. Physik 44, 91. Kim, Y.-K. (1967). Phys. Rev. 154, 17 Kinoshita, T., ed.(1990). Quantum Electrodynamics, World Scientific, Singapore. Klahn, B., and Bingel, W. A. (1977). Theor. Chim. Acta 44, 9, 27. Layzer, D., and Bahcall, J. N. (1962). Ann. Phys. (New York) 17, 177. Mayers, D. F. (1957). Proc. Roy. SOC.A 241, 93. McKenzie, B. J., Grant, I. P., and Norrington, P. H. (1980). Comput. Phys. Commun. 21, 233. Mohanty, A. K., Parpia, F. A., and Clementi, E. (1991).In: Modern Techniques in Computational Chemistry: MOTECC-91, ESCOM Science Publishers, Leiden, p. 167. Mohr, P. J. (1974). Ann. Phys. (NY) 88, 26. Mohr, P. J. (1982). Phys. Rev. A 26, 2338. Mokler, P. H., Stohlker, T., Kozhuharov, C., Ullrich, J., Reusch, S., Stachura, Z., Warczak, A,, Miiller, A., Schuch, R., Livingston, E. A., Schulz, M., Awaya, Y., and Kambara, T. (1990). In: AIP Conference Proceedings 215: X-ray and Inner-Shell Processes (T. A. Carlson, M. 0. Krause, and S. T. Manson, eds.), American Institute of Physics, New York, p. 352. Pekens, C. L. (1958). Phys. Rev. 112, 1649. Persson, H., Lindgren, I., and Salomonson, S. (1993). Physica Scripta T 46, in press. Pipkin, F. M. (1990). In: Quantum Electrodynamics (T. Kinoshita, ed.), World Scientific, Singapore, p. 696. Pyper, N. C., Grant, I. P., and Beatham, N. (1978). Comput. Phys. Commun. 15, 387. Quiney, H. M. (1990). In: Supercomputational Science (R. G. Evans and S. Wilson, eds.), Plenum, New York, pp. 159, 185. Quiney, H. M., and Grant, I. P. (1993). Physica Scripta T 46, 132. Quiney, H. M., Grant, 1. P., and Wilson, S. (1987). J . Phys. B: At. Mol. Phys. 20, 1413. Quiney, H. M., Grant, I. P., and Wilson, S. (1988). In: Lecture Notes in Chemistry, No. 52 (U. Kaldor, ed.), Springer-Verlag, Berlin, p. 307. Quiney, H. M., Grant, I. P., and Wilson, S. (1989). J. Phys. B 22, L15. Quiney, H. M., Grant, I. P., and Wilson, S. (1990). J . Phys. B 23, L271. Racah, G. (1942). Phys. Rev. 62, 438; 63, 367. Racah, G. (1943). Phys. Rev. 76, 1352. Roothaan, C. C. J. (1951). Rev. Mod. Phys. 23, 69. Safronova, U. I., and Senashenko, V. S. (1984). Theory of Spectra of Multicharged lons, Energoatomizdat, Moscow [in Russian]. Salomonson, S., and Oster, P. (1989). Phys. Rev. A 40,5548. Schmidt, M. W., and Ruedenberg, K. (1979). J . Chem Phys. 71, 3951. Series, G. W., ed. (1988). The Spectrum of Atomic Hydrogen: Adoances, World Scientific, Singapore. Sommerfeld, A. (1916). Ann. Phys. Lpz. 51, 1, 44, 125. Sturesson, L., and Fischer, C. F. (1992). Coyput. Phys. Commun., to be published. Sucher, J. (1980). Phys. Reo. A 22, 348. Swirles, B. (1935). Proc. Roy. SOC. A 152, 625. Synek, M. (1964). Phys. Rev. 136, A1552. Williams, A. 0.(1940). Phys. Rev. 58, 723. Yutsis, A. P., Levinson, I. B., and Vanagas, V. V. (1962). Mathematical Apparatus of the Theory of Angular Momentum (A. Sen and R. N. Sen, trans.). Israel Program for Scientific Translations, Jerusalem.
ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 32
THE CHEMISTRY OF STELLAR ENVIRONMENTS D. A . HOWE Mathematics Department UMIST Manchester, England
J . M . C. RAWLINGS Department of Physics Astrophysics, Nuclear Physics Laboratory Oxford, England
D . A . WILLIAMS Mathematics Department UMIST Manchester. Engiand
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Winds from Young Stellar Objects. . . . . . . . . . . . . . . . . . . 111. Circumstellar Envelope of Asymptotic Giant Branch Stars . . . . . . . A. General Features . . . . . . . . . . . . . . . . . . . . . . . . B. Oxygen-Rich CSEs; For Example, OH-IR Stars . . . . . . . . . . C. Carbon-Rich CSEs; For Example, ,IRC+ 10216 . . . . . . . . . . . IV. Planetary Nebulae and Preplanetary Nebulae. . . . . . . . . . . . . . V. Novae. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Chemistry in Supernovae Ejecta: Molecules in SN1987A . . . . . . . . . VII. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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I. Introduction In 1951 Bates and Spitzer published an article entitled “The Density of Molecules in Interstellar Space.” Although other authors had considered the question of molecule formation in interstellar space, the 1951 paper of Bates and Spitzer is widely acknowledged to be the first significant discussion of the subject. It is the first to describe in some detail the range of physical and chemical processes that might play a role in the diffuse interstellar medium. Certainly, their paper was the first to make reasoned estimates of the various rates and rate coefficients of the various processes involved. One of the 187 Copyright 0 1994 by Academic Preis, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003832-3
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molecules known to be present in the interstellar medium when Bates and Spitzer were writing their article is CH'. The origin of this molecule remains a controversial topic (cf., e.g., Williams, 1992). In the intervening four decades there has been an avalanche of observational discoveries and much detailed theoretical and laboratory work. As a result of huge improvements in detectors and receivers at radio and infrared wavelengths the number of detected astronomical molecular species has grown from 3 to over 100. Most of the molecules are found not in the diffuse clouds studied by Bates and Spitzer but in much denser objects, some of which are associated with stars. The techniques developed for interstellar chemistry, which owe much to Bates and Spitzer, are now being extended to the study of situations as diverse as the early universe and supernova ejecta (cf., e.g., Hartquist and Williams, 1990). The subject is now more properly called astrochemistry or, where the emphasis is on the physical description revealed by the chemistry, molecular astrophysics. The range of topics in molecular astrophysics is too wide to be described even briefly in a chapter such as this (see, e.g., Millar and Williams, 1992a). In this chapter, therefore, we have arbitrarily selected several topics in the general area of stellar environments. We omit, however, any mention of the Solar System. In Fig. 1 we give a simplified Hertzsprung-Russell diagram, showing the luminosity vs. surface temperature (by convention increasing to the left) of the stars considered here, including their evolutionary relationships. These are discussed in Sections II-VI, while Section VII is a brief conclusion.
11. Winds from Young SteIIar Objects Molecular outflows are commonly associated with star-forming regions. The physical origins of the winds are largely unknown, and our empirical knowledge of the chemical processes that occur in primary circumstellar winds is fairly poor. These winds are usually hot, partially ionised, dense and strongly irradiated by complicated and time-varying radiation fields. In such circumstances the chemistry is restricted to small species such as H, and CO. H, is rarely the dominant form of hydrogen and often does not play such a central role as in more orthodox interstellar chemistries. The most important routes by which H, and CO can be formed are shown in Figs. 2 and 3, respectively. The information contained in these figures is relevant for a very wide range of hot circumstellar environments. Many of these reactions are highly dependent on temperature and radiation field, while three-body H, formation requires number densities of larger than 10" cmV3. Therefore
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different routes within those shown in Figs. 2 and 3 will dominate in different types of outflow and in different regions within each outflow. One general feature of these winds is the likelihood that the chemistry occurs close to the base of the wind and is essentially “frozen out” as the outflow progresses. This is particularly apparent in the lower density T-Tauri winds discussed by Rawlings et al. (1988). T-Tauri stars are young
L/solar
1
5oooO
3000
loo00
T/K FIG.1. Hertzsprung-Russell diagram (luminosity vs. surface temperature), showing approximate evolutionary tracks for stars of two different initial masses. Path a-g is for a star of about 1 solar mass, Path x-y is for a star of about 10 solar masses. At a, the solar mass star, having recently formed in an interstellar cloud, is passing through a T-Tauri phase (Section II), before joining the main sequence at b, where it remains for about 10” years. The star then expands and cools, joining the red giant branch at c; then warms up and fades a little, joining the horizontal branch at d, before going through a second red giant phase, on the so-called asymptotic giant branch (AGB) at e, during which heavy mass loss occurs as a “superwind” (Section 111). Exhaustion of the AGB atmosphere exposes the hot luminous stellar core, which moves to point 1: UV from this core (now only about 0.6 solar masses) ionises the inner part of the remnant superwind, forming a planetary nebula (Section IV). Dispersal of the remnant superwind and fading of the core star leave it as a white dwarf, at position g. A nova (Section V) occurs in some binary systems, consisting of a red giant (c) plus a white dwarf (9). where mass transferred to the latter builds up to the point where the accreted matter suddenly initiates nuclear fusion. The 10 solar mass star leaves its position on the main sequence, at x , becoming a supergiant at y, where it explodes as a supernova (Section VI).
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et
al.
H + H + H H
dH
hv
dH
+ H + H + H
FIG.2. H, formation scheme. The routes involving excited H are a recent suggestion of Rawlings et al. (1992) and Latter and Black (1991).
-
c+o
0
H, H2 0
A
lH2
.L
OH
C
C + OH
c+o
FIG.3. CO formation scheme.
( < lo6 years), low mass ( < 2 M, (solar masses)) pre-main-sequence stars in
their convective tracks. The mass loss rate from these objects is highly uncertain but is in the range of a few times lo-’ to 10-6M, yr-l. Observations also indicate that the wind is neutral, at least far from the star. Hartmann et al. (1982) proposed that the wind is heated and driven by the deposition of energy and momentum from Alfven waves. The terminal velocities and temperatures (at about 3 R, (stellar radii)) are of the order of 200 kms-’ and lo4 K respectively. Defining this to be the starting point of the chemical evolution, Rawlings et al. (1988) found that the time scales for recombination, geometrical dilution and cooling (by radiative and adiabatic effects) are all of the same order of magnitude (= a few days), while the chemical time scale (on the order of a few years) is substantially longer. The formation of H, in the recombination zone is limited by collisional dissociation and typically reaches an asymptotic fractional abundance of about 5 x 10- ’. Neutral-neutral chemistry dominates throughout the chemically
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active region of the wind and the CO fractional abundance is “frozen out” at about 3 x lo-’. This is limited purely by the effect of geometrical dilution, as the CO, once formed, is chemically very stable. It is, of course, possible that mass loading (e.g., from a preexisting molecular disk) may occur. In calculations where there is a 1 YOadmixture of molecular material it was found that the CO abundance could be raised as high as Models of a more general nature have been developed by Glassgold et al. (1989, 1991) and Ruden et al. (1990), including the SVS 13/HH7-11 source, where high-velocity CO has been detected, and other winds of a more hypothetical nature. The outflows are substantially more massive and cooler M, yr-’, T < 5000K) so than the T-Tauri winds (typically M = 3 x that three-body reactions contribute significantly to the formation of H2, and appreciable quantities of H, and CO can be formed (with fractional In addition, SiO and possibly H 2 0 abundances of the order of l o p 5to may have a large abundance in these winds. However, other than the molecules just discussed, the chemistry restricts the fractional abundances to less than about lop8.The most important requirements for the chemistry to be efficient are that the wind should have a moderate ionisation to drive the ion-molecule chemistry and that cooling must occur fairly rapidly (the temperature falling to < 2000 K within a radius of 5 R*),effectively turning off the collisional dissociation processes. A general conclusion of all models is that the chemistry is critically dependent on the outflow properties and that the chemical efficiency is primarily determined by density distribution and geometrical dilution effects.
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111. Circumstellar Envelopes of Asymptotic Giant Branch Stars A. GENERAL FEATURES When stars of between about 1-8 solar mass reach the asymptotic giant branch (AGB,see Fig. l), the gravitational pull of the star is less able to hold on to its outer atmosphere, and this begins to be lost, in the form of a “superwind,” which is much more copious than the winds of main sequence stars such as the Sun. Continued ejection of this wind gives rise to a circumstellar envelope (CSE), which has an approximately inverse square relationship between density and distance from the star. Many molecular species have been observed in CSEs, mostly in radio emission via rotational transitions, or sometimes in infrared absorption via vibrational transitions. The richest chemistry occurs in cases of a high mass loss rate. Most A G B
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stars are losing Ma yr-’ but some are losing up to M, yr-’, e.g., the nearby star CW Leonis, whose CSE is called IRC + 10216. Dust grains form close to the star, once the ejected gas cools below lo00 K. This dust, which may obscure the star almost completely at visual wavelengths, is probably responsible for the acceleration of the wind to the 10-20 kms-’, via radiation pressure and gas-dust observed speeds of coupling. The dust grains also play an important part in the chemistry, by shielding the inner regions from interstellar ultraviolet (UV) radiation (from nearby hot stars, the AGB star being too cool to produce much UV itself), and in addition may play a role as reservoirs of molecules formed in the warm inner regions or as catalysts. CSEs may either be carbon rich (C rich: C/O > 1) or oxygen rich (0rich: C/O < 1). 0-rich stars produce dust consisting largely of silicates, while the dust formed by C-rich stars is mostly carbon and Sic. Several objects are observed to have C-rich atmospheres (carbon stars), but infrared emission characteristic of silicate dust in their CSEs, which led Willems and de Jong (1988) to propose that carbon stars evolve from oxygen-rich ones, the silicate dust envelopes of the anomalous objects having been ejected just before recent convection in the star dredged up carbon to the surface, thus initiating the C-rich phase. However, this scenario is not universally accepted (see Lloyd Evans, 1991 for a brief discussion). In all CSEs the dominant molecule apart from H, is CO, which forms at local thermodynamic equilibrium (LTE) in the stellar photosphere. This means that C-rich envelopes are very deficient in other oxygen-bearing molecules, while 0-rich ones are deficient in other carbon-bearing species. Other molecules also form in or just outside the photosphere, and as a sample of gas and dust drifts from the star, these inner-envelope (relative) abundances are thought to remain fairly constant (“frozen out”), though accretion onto grains may occur with some species. However, eventually there comes a time when interstellar UV is encountered, as the more rarefied outer regions are less opaque. This initially small UV flux ionises or dissociates these “parental” molecules, creating a reactive mixture of ions, radicals, stable molecules and electrons, which begin a rich chemistry. By this time our sample of gas and dust has fallen in temperature to a few times 10 K. Further still from the star, rarefaction slows collision rates, and the increasing UV flux dissociates most of the molecules formed, though much of the dust will survive, contributing to interstellar dust. Once the mass loss has been proceeding for about lo4 years, the distributions of molecular species settle into a pattern. Those formed in or close to the star are limited at their outer radius by photolysis, becoming progressively denser close to the star; those formed via photochemistry in the outer envelope have a hollow shell distribution, limited by formation at the inner radius, and by photolysis at the
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outer radius. The chemistries of 0-rich and C-rich CSEs are very different and will be discussed separately. For a recent review of CSEs, see Omont (1991). B. OXYGEN-RICH CSEs; FOREXAMPLE, OH-IR STARS
These objects ave relatively few molecules. In Table I a list of molecules detected in 0-rich CSEs is provided. The major parental species are probably H,, CO, H,O,.,d'I The presence of several carbon-bearing species (apart from CO) is something of a puzzle, as LTE processes in the photosphere are expected to lock up nearly all carbon atoms in CO. A possibility that has been considered is that a carbon-bearing molecule, such as CH, (itself unobserved, having no dipole moment) or H,CO (recently observed) may form by grain surface reactions close to the star; this molecule would then be photodissociated upon reaching the photochemical region, providing a source of other carbon-bearing species. A widespread feature of 0-rich envelopes is the presence of masers. Oxygen-rich envelopes more commonly display maser emission than carbonrich ones. Species observed in maser emission include SiO, from just outside the stellar atmosphere (- 10'3-10'4 cm from the stellar center); H,O, from beyond this; and OH, still further out (- loi6 cm). The readiness with which OH masers occur in high mass loss oxygen-rich CSEs, has been thought to indicate that those with large dust content but no OH masers may have this OH destroyed by an internal UV source, such as a white dwarf companion star (Lewis 1992). C. CARBON-RICH CSEs; FOREXAMPLE, IRC + 10216 These objects show a rich chemistry, helped by the ability of carbon to form chains, rings and spheres; a list of detected molecules is given in Table 11. The classic example is IRC + 10216, which is close to us (-200 pc), with a large Ma yr-'). The major parental, inner-envelope mass loss rate ( - 5 x species are thought to be H,, CO, HCN, C2H2,CH,, NH,, N,, SiS. TABLE I MOLECULES DETECTED IN 0-RICHCSEs
co so2 NH3
OH H2O H,CO
so H2S
SiS
ocs
sio HCN
CN HNC
cs HCO'
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TABLE I1 MOLECULFS DETECTED IN C-RICHCSEs
co SiS HNC HCCN HC3N C5H HC,,N
CN NaCl C,H I-CpH C4H CH,CN
Sic AlCl Sic2 CSN Sic4 C,H4
CP KCI C2S
cs
c3s
C2Hz SiH, C,H
AIF
c,
CH4 HCSN
SiN H2S HCO+ c-C,Hz C, HC,N
SiO HCN NH, HZCCC H,CCCC HC9N
Notable among the observed species are the carbon chain molecules; e.g., HC2n+lN(n = 1-5), C,H (n = 2-6), C,S (n = 1-3). Sequences of reactions, involving either ions or neutral radicals, have been proposed as formation mechanisms for building these species (see, e.g., Glassgold et al., 1986; Nejad and Millar, 1987; Howe and Millar, 1990), and are shown, in much simpliEed form, in Figs. 4 and 5 for the C2.H (n = 1-3) series. In Fig. 6 theoretical abundances as a function of distance from the star are shown for some of the species involved, clearly showing the cooperation of the processes of freezeout (of C,H2), photolysis and chemical reaction. However, theoretical modelling has difficulty with the larger members of these series, owing to a predicted fall-off in abundance with increasing molecule size, larger than observed. This is due to the likelihood that gasphase formation of many of the observed large molecules occurs as a series of byproducts of a reaction sequence, rather than as a simple sequential buildup. This “siphoning-off of molecules, along with competition from photolysis (their main destruction channel), reduces the availability of reaction partners, tending to suppress the predicted abundances of larger molecules. It is therefore possible that grain processes may be necessary to explain their unexpectedly high abundances.
C2Hz
-p ypy hv
C2H
C2H2
C,H2
CIH
C2H2 b
C6H2
-
C6H
pv FIG.4. Possible neutral-radical hydrocarbon formation scheme for a C-rich CSE.
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C H
4 J
Pv
FIG.5. Ion-molecule hydrocarbon formation scheme for a C-rich CSE.
A wider astrophysical implication of chemistry in C-rich CSEs is the proposal that they may be important “factories” for the production and dispersal to the interstellar medium of large stable molecules such as polycyclic aromatic hydrocarbons (PAHs; see Latter, 1991)or fullerenes (e.g., C60; see Kroto, 1988), they or their derivatives being proposed by some as carriers of the diffuse interstellar bands or the unidentified infrared emission bands.
IV. Planetary Nebulae and Preplanetary Nebulae A planetary nebula (PN, plural PNe) is a nebula surrounding a small, hot star that partially ionises the nebula, causing it to emit radiation, notably “forbidden” lines of oxygen and neon. A PN is thought to form when an
t
-5.00
52
4.00
-7.00
r
4
-$ v
-8.00
-9.00
15.50
16.00
16.50
JogWcm)
17.00
17.50
15.50
16.00
16.50
17.00
log(r/cm)
FIG.6. Theoretical abundance distributions of selected species in a C-rich CSE.
17.50
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AGB star, discussed in Section 111, exhausts its atmosphere, bringing the “superwind” to an end and revealing a hot, degenerate core. In this picture the observed nebula consists of the inner part of the earlier superwind material that has detached and drifted away from the star. Support for the AGB + PN scenario comes from statistical arguments based on expected lifetimes and relative populations, and from several objects that appear to be in transition at the moment (“preplanetary nebulae” or rather confusingly “protoplanetary nebulae”-singular, P P N plural, PPNe). Notable PPNe are CRL2688 and CRL618, in which the superwind is still very close to the star, which is hotter than red giants (T, 7000K for CRL2688, T, 25000 K for CRL618), mass loss having largely ceased within the last 200 years. In these objects a dense, nonionised CSE still exists, with many molecular species. The exposure of a hot stellar core brings about a dramatic change in the circumstellar matter, assuming it has not had time to drift far from the star and disperse to the interstellar medium before the star reaches a high temperature. Much work is being done on the dynamical evolution of PNe from AGB envelopes. The main such model is the “interacting stellar winds” (ISW) model (see Kwok et at., 1978), which is still being refined. A simple diagram is shown in Fig. 7. The core star in PPNe is observed to produce a very fast ionised wind (-2000 kms-’) with low mass loss rate (5lo-* M, yr-’) (a). According to the ISW model, this wind experiences a stationary, inward facing shock close to the star, which heats it, forming a bubble (b) of diffuse ionised gas. This bubble expands, producing a shock moving outward through the remnant superwind material (at 20-30 kms- ’), progressively sweeping it, like a snow plough, into a dense, thin, expanding shell (c and d). The heating in such a shock, plus the presence of stellar UV, will have a major effect on the superwind remnant. For example, postshock heating will overcome reaction endothermicities. However, in the simplest form of the ISW model, the opacity to stellar UV of the superwind dust will fall within a few hundred years of the cessation of high mass loss, so that, if the central star has reached T, k 3 x lo4 K by this time, photolysis will be extremely rapid. This would tend to preclude the formation or long-term survival of most molecules after this stage. In the case of the PPNe CRL2688 and CRL618, they are young enough ( 200 yrs) for their molecular component to be a survival from the AGB stage (though some of their abundances have been thought anomalous). However, molecules are observed even in well-developed PNe; e.g., CO in the Helix nebula (NGC7293). The young PN NGC7027 has HCN, CN, OH, HCO’ as well as CO, suggesting that the real situation is more complicated than the ISW model in its basic form; requiring, e.g., dense clumps (whose UV opacity is high) or release of molecules from grains. Other differencesbetween the basic ISW model and real PNe or PPNe are the fact that they are usually
-
--
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FIG.7. Cross section of a PPN according to the simple ISW model (not to scale): * = hot, = outward-facing shock, small, luminous star; ---- = stationary, inward-facing shock; sweeping up superwind; a = fast, low density, ionised wind; b = hot, shocked, ionised “bubble”; c = ionised component of swept-up superwind remnant; d = nonionised component of swept-up superwind remnant; e = as yet unshocked superwind remnant; f = interstellar medium. ~
not spherically symmetric, but commonly show a bipolar structure, as well as the presence in some PPNe of fast (- 100 kms-’) molecular flows. The chemical consequences of such flows in conjunction with slower moving material would be far-reaching, especially in cases where the UV flux is not too destructive. One can thus see that, as with AGB envelopes (Section 111), observations of PNe and PPNe pose some difficulties for the theoretician, in that molecules exist in unexpectedly large abundances. The modelling of chemistry in planetary nebulae, as yet in its infancy (see Black, 1978; Howe et al., 1992),is likely to prove an interesting and challenging topic for the future. For more information on PNe and PPNe, see Torres-Peimbert (1989).
V. Novae Novae are eruptive binary systems that provide us with a unique opportunity to study the chemical development of hot, expanding winds. Unlike the stellar winds discussed previously, novae occur as one-off events. The characteristic time scales for chemical and physical changes in the wind are on the order of tens to hundreds of days. About a third of all novae produce optically thick dust shells so that in novae we might observe the characteristics of dust grain nucleation and growth. The classification of novae is confused by the very wide variety of
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characteristics exhibited by individual novae. Typical parameters for a classical nova follow: Outburst energy Mass loss Bolometric luminosity Ejecta velocity Composition
ergs 10-5-10-4 M, 2 x lo4 Lo (solar luminosities)
300-10,000 kms-' Metals (e.g., C, N, 0)enhanced over cosmic relative abundances by 10-1000 times
In general two main types have been identified (according to the white dwarf composition). C-N-0 types are typically associated with duller, slow, dusty novae and tend to produce an optically thick carbon dust shell. In most cases CO is the only molecule that has been detected in them. He-Mg-A1 types are typically associated with brighter, faster novae and tend to produce an optically thin silicate type dust shell. The predominant molecular lines seen in them are of Sic, SiO and other silicate features. To date, only carbonrich novae have been modelled in any detail. At first sight it would seem that the conditions in the ejecta of novae are prohibitively harsh for molecules to exist; the density at the base of the winds is greater than 1 0 ' ' ~ r n - ~ ,the temperatures (in the regions of chemical interest) are in the range 2000-10000 K and the radiation field is extremely intense and peaks in the UV. Moreover, the physical conditions are highly uncertain and in any case strongly time dependent; the nature of a nova event is such that the effective photosphere contracts while maintaining a constant bolometric luminosity (for several hundreds of days after the outburst). The resultant hardening of the radiation field results in a complex ionisation structure with a series of ionisation fronts moving out through the ejecta as it becomes progressively more diffuse. Early broadband photometric observations of novae identified the socalled 5 pm excess with CO u = l + 0 emission. This feature and the associated first overtone emission at 2.3 pm are seen in many novae some 1030 days post outburst and indicate very large CO column densities (10"cm - '). The features disappear before the onset of dust formation (typically 50- 100 days post outburst) and are probably transient chemical effects. However, the large column densities imply a very efficient C to CO conversion, with some 1 to 10% of all C being in the form of CO. Early theoretical work (Rawlings, 1988) addressed this problem and concluded that the region of emission must be relatively cool (54000 K) and carbon neutral (CI). C, Hzand CO are mutually shielding in such a region and thus are largely protected from the radiation field. The calculations also show that the chemistry and the ionisation structure of the nova ejecta are closely interrelated. As a result of the intensity of the
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radiation field and the high densities, the composition is dominated by the photochemistry and is effectively in steady state. The main H, formation routes are by three-body and H - reactions (see Fig. 2). The main loss route is collisional dissociation by atomic hydrogen. In the ionised carbon (CII) region (where the temperature is about 6000K) the H, cannot build up appreciable column densities due to the presence of the unshielded Lyman flux. In a model of the CI region limited to H, C and 0 chemistry, the only other molecules to achieve fractional abundances greater than about 10- l o are CH, OH, C, and 0,. The chemistry is also very temperature sensitive: temperatures of less than 3500K are required for H, to be optically thick. The main C O loss routes are photodissociation and collisional dissociation by H atoms. Simple ionisation models suggest that the neutral zone ceases to exist within a few days of the outburst. This implies that the region of emission must be a cool, neutral shell of ejecta of enhanced density. In addition to CO and H,, CN, SiO, SO,, Sic and PAH features have also been seen in novae. An understanding of the kinetics of dust formation in astrophysical environments requires a study of the formation of nucleation sites. The ejectae of novae are very far from LTE, and here a microscopic approach is required. The intensity of the radiation field together with the extreme inefficiency of molecule formation in ionised regions, limits nucleation to the CI ionisation zone (Rawlings and Williams, 1989), where H, and C O are the only molecules formed in appreciable abundance. At the temperatures and densities appropriate to the nucleation epoch the fraction of hydrogen in molecular form is in the range iO-4-10-2, and CO saturates in this region. This is an important point since C > 0 in the ejecta. It was found that the only viable nucleation mechanism is based on an extended hydrocarbon chemistry. The saturation of CO prevents an oxygen attack on the hydrocarbons, which would inhibit the formation of nucleation sites. The formation of the nucleation sites is fairly inefficient, and the dust composition of novae is probably described by a small population of large grains. At lower levels of ionisation (e.g., if sulphur is neutral), the ion-neutral chemistry is inhibited and nucleation sites cannot form.
VI. Chemistry in Supernovae Ejecta: Molecules in SN1987A Stars with masses larger than about 10Mo have very high internal temperatures that drive their thermonuclear reactions at a much higher rate than in low-mass stars. As with other stars, the initial fuel is hydrogen, and the first product of thermonuclear fusion is helium. However, in massive stars, successive fusion reactions can occur, larger nuclei forming nearer the
200
D.A . Howe et al.
hotter center of the core, until the formation of iron. The star then has the structure depicted in Fig. 8. The core is iron; the outermost layer contains unburnt hydrogen, and the intermediate stages are represented by separate layers in between. Since iron nuclei are the most stable, further fusion cannot occur, so the heat source and the thermal support are finally removed. The core collapses to form a neutron star, while the outer layers collapse inward, only to “bounce” at the surface of the neutron star from.which they are ejected with very high velocity. In the transient heating associated with the “bounce,” further nuclear fusion creates post-iron elements. This general picture has been largely confirmed by observations of SN1987A, a supernova detected on 23 February 1987 in the neighbouring galaxy, the Large Magellanic Cloud (cf., e.g., Arnett et al., 1989). This supernova is now a dense cloud of gas expanding hypersonically and illuminated by intrinsic y-rays from the radioactive decay of s6Co.Though this seems an unlikely environment for chemistry, some three months after observing the outburst a detection was made of infrared emission from CO. Subsequently, emission from SiO was also identified, and the ion H i has also been tentatively detected. This chemistry gives the opportunity, through modelling, of addressing various questions concerning the conditions in the ejecta (see, e.g., McCray, 1990; Lepp et al., 1990; Rawlings and Williams, 1990).Does mixing occur between the layers at the “bounce”? What are the heating and cooling processes in the ejecta? What are the physical conditions in the ejecta? Are they conducive to dust formation? In the absence of substantial mixing, the ejecta are presumed to have a
FIG.8. The chemical structure in the interior of a massive star before supernova.
THE CHEMISTRY OF STELLAR ENVIRONMENTS
20 1
hydrogen-poor inner zone and a hydrogen-rich outer zone. Parameters for these zones are suggested in Table 111. Both zones have abundances that are unlike those for any other region to which molecular astrophysics has been applied. The observations of CO in SN1987A suggest that it is formed in the H-poor inner zone. It is therefore of interest to discover by which routes a significant chemistry could arise in the dense, H-poor gas. This chemistry must operate in a gas irradiated by y-rays from radioactive 56C0.These yrays give fast electrons by inverse Compton scattering that ionise and excite He and other atoms so that a complex UV and X-ray intrinsic radiation field arises. There is also an intrinsic infrared radiation field corresponding to the temperature of the ejecta. Molecular species are subject to collisional and photoionisation and excitation. In many other astronomical situations chemistry proceeds by exchange reactions of ion-molecular or neutral-neutral type, and the initiating molecule is H,. If no mixing between zones occurs, then in the inner zone of supernova ejecta the most abundant species is He, and consequently, there is no abundant molecule with which exchange reactions may occur. It seems likely that chemistry is initiated by radiative associations of which those leading to negative ion formation are likely to be faster than the associations of neutrals. Such a scheme for C O formation is illustrated in Fig. 9. Detailed studies suggest that nearly all these formation processes play a role, but the relative importance of the several channels varies during the ejection process. Each channel operates against different impediments; for example, negative TABLE 111 CONDITIONS AND ABUNDANCES EXPECTED IN THE INNER AND OUTER AT A TIME OF ONEYEAR PARTSOF WE EIECTAOF A SUPERNOVA, AFTERTHE EXPLOSION
Velocity (kms-') Number density (m-') Temperature (K) 'XM X(He) X(C) X(0)
Inner
Outer
700 109
2000
3000
7 x 10' 7000
0.5 5 x 10-2
0.4 0.5 3 x 10-3
0.3
10-2
10-7
'X means the fractional abundance of a particle element relative to the total number density of atoms present. Thus, in both inner and outer zones, about half the atoms are helium. In the inner zone there is hardly any hydrogen, but about 30% of the atoms are oxygen. In the outer zone, however, about 40% of the atoms are hydrogen.
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D. A . Howe et al.
FIG.9. A chemical scheme for CO formation in the H-poor zone of supernova ejecta.
ions are subject to photodetachment by the infrared field. Precursor molecules such as C, and O2 are subject to photodissociation by the intrinsic UV field, to collisional dissociation by He and e-, and to dissociative chemical reaction with He+. The product molecule of this scheme, CO, is also subject to loss through UV photodissociation, collisional dissociation with He and e-, to dissociative reaction with He+ and to charge exchange with O', where the CO+ arising is lost in dissociative recombination with electrons. The results of models incorporating such networks can be compared with the interpretation of the observations that suggest that as much as of a solar mass of CO is present in the ejecta of SN1987A. This large mass of CO places constraints on the models that-in the absence of mixing between zones-have some difficulty in producing as much CO. If the formation chemistry is substantially correct, then the average loss rates for CO must be suppressed. One plausible interpretation is that some form of CO shielding against destruction is occurring in clumps of gas occupying perhaps about 10% of the total ejecta volume. Alternatively, substantial mixing of zones giving hydrogen enrichment of the inner zone may enhance the chemistry. This point is as yet unresolved. Although SiO may also be formed in the same H-poor zone as CO, H i must-if present-occur in the outer H-containing zone. In interstellar clouds, H : forms from the H: created by cosmic ray ionisation of H2, viz., H2(c.r.; e-)H:(H,;
H)H:
This channel will also be available in the outer zone of supernovae if H,
THE CHEMISTRY OF STELLAR ENVIRONMENTS
203
exists. However, since no dust is present, surface processes are not available, and H2 forms through the two-body gas-phase routes shown in Fig. 2. The total mass of Hi that can be created by these means is not too far short of that implied by the observations. Therefore, a modest degree of clumpiness in the outer zone is implied by the H i observation. We conclude that chemical studies of processes in supernova ejecta can constrain the models and help to determine conditions in the ejecta.
VII. Conclusions The approach to interstellar chemistry established by Bates and Spitzer in 1951 has now been extended in various ways to deal with the wider demands of molecular astrophysics. First, the extent of the chemistry is now necessarily very much greater. It is now routine to consider chemical networks involving a hundred or more species interacting in a thousand or more reactions. The UMIST Ratefile (Millar et al., 1991) is a database of chemical reactions of interest in astronomy; in its latest version it lists more than 3700 reactions, many of which have rate coefficients that have been measured at one or more temperatures. In any given situation, however, only a few of these reactions are likely to be significant. Second, the role of dust is now seen to be important in the chemistry of many situations (cf. Millar and Williams, 1992a). Dust formation affects the residual gas phase abundances of incorporated elements. Heterogeneous surface catalysis contributes to the gas phase network of reactions. Molecules such as H 2 0 , CO and NH, stick to dust grains, forming ices that can be returned to the gas in the same or modified form (e.g., CH,OH is supposed in some circumstances to form from the solid state chemistry hydrogenating CO). The chemical involvement of dust was a feature recognised by Bates and Spitzer (1951). Third, the steady-state approach of Bates and Spitzer is now usually set aside in favour of time-dependent studies. Time-dependent effects can be important even in the diffuse clouds considered by Bates and Spitzer (cf. Wagenblast, 1992). In all other astronomical situations chemical effects are time dependent, and this is evident in the near-stellar situations discussed here. Fourth, it is necessary to include gas kinetics and dynamics for a proper description of most astrochemical systems. This is certainly true for all of the examples discussed here. Indeed, in applying molecular astrophysical techniques to astronomical situations, the choice of the chemistry is often routine. It is the inclusion of time dependence and the appropriate dynamical formulation that allows a realistic description to be made.
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Nevertheless, the overall approach of Bates and Spitzer remains intact. Through exploration of the consequences of a set of reactions within a plausible parameter range for the physical conditions, one may arrive at a detailed and self-consistent description.
REFERENCES Arnett, W. D., Bahcall, J. N., Kirschner, R. P., Woosley, S. E. (1989). Ann. Rev. Astron. Astrophys. 27, 629.
Bates, D. R., and Spitzer, L. (1951). Astrophys. J . 113, 441. Black, J. H. (1978). Astrophys. J. 222, 125. Glassgold, A. E., Lucas, R., and Omont, A. (1986). Astron. Astrophys. 157, 35. Glassgold, A. E., Mamon, G. A., and Huggins, P. J. (1989). Astrophys. J . 336, L29. Glassgold, A. E., Mamon, G. A., and Huggins, P. J. (1991). Astrophys. J . 373, 254. Hartmann, L., Edwards, S., and Avrett, E. (1982). Astrophys. J. 261, 279. Hartquist, T. W., and Williams, D. A. (1990). Ql. J. Roy. Astr. SOC.31, 593. Howe, D. A., and Millar, T. J. (1990). Mon. Not. R. Astr. SOC.244, 444. Howe, D. A,, Millar, T. J., and Williams, D. A. (1992) Mon. Not. R. Astr. SOC.255, 217. Kroto, H. (1988). Science 242, 1139. Kwok, S., Purton, C. R., and Fitzgerald, P. M. (1978). Astrophys. J . 219, L125. Latter, W. B. (1991). Astrophys. J. 377, 187. Latter, W. B., and Black, J. H. (1991). Astrophys. J . 372, 161. Lepp, S., Dalgarno, A., and McCray, R. (1990). Astrophys. J. 358, 262. Lewis, B. M. (1992). Astrophys. J . 396, 251. Lloyd Evans, T. (1991). Mon. Not. R. Astr. SOC.249, 409. McCray, R. (1990). In: Molecular Astrophysics: A Volume Honouring Alexander Dalgarno, Cambridge University Press, Cambridge, p. 439. Millar, T. J., Rawlings, J. M. C., Bennett, A., Brown, P. D., and Charnley, S. B. (1991). Astron. Astrophys. Suppl. Ser. 87, 585. Millar, T. J., and Williams, D. A. (1992a). Sci. Prog. 75, 279. Millar, T. J., and Williams, D. A., eds. (1992b). Dust and Chemistry in Astronomy, IOP Publishing Ltd., Bristol. Nejad, L. A. M., and Millar, T. J. (1987). Astron. Astrophys. 183, 279. Omont, A. (1991). In: Chemistry in Space (J. M. Greenberg and V. Pirronello, eds.), Kluwer, Dordrecht, p. 171. Rawlings, J. M. C. (1988). Mon. Not. R. Astr. SOC.232, 507. Rawlings, J. M. C., Drew, J. E., and Barlow, M. J. (1992). In: Astrochemistry of Cosmic Phenomena, IAU Symp. 150 (P. D. Singh, ed.), Kluwer, Dordrecht, p. 387. Rawlings, J. M. C., and Williams, D. A. (1989). Mon. Not. R. Astr. SOC. 240, 729. Rawlings, J. M. C., and Williams, D. A., (1990) Mon. Not. R. Astr. SOC.246, 208. Rawlings, J. M. C., Williams, D. A,, and Cantb, J., (1988). Mon Not. R. Astr. SOC.230, 695. Ruben, S. P., Glassgold, A. E., and Shu, F. H. (1990). Astrophys. J. 361, 546. Torres-Peimbert. S., ed. (1989). Planetary Nebulae, fAU Symp. 131, Kluwer, Dordrecht. Wagenblast, R. (1992). Mon. Not. R. Astr. SOC. 259, 155. Willems, F. J., and de Jon& T. (1988). Astron. Astrophys. 196, 173. Williams, D. A. (1992). Plan. Sp. Sci. 40,1683.
ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS. VOL 32
POSITRON AND POSITRONIUM SCATTEMNG AT LOW ENERGIES J . W . HUMBERSTON Deparrment of Physics and Astronomy University College London London, England
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
205 207 A . Positron Scattering by Atomic Hydrogen . . . . . . . . . . . . . . 207 B. Positron Scattering by Alkali Atoms . . . . . . . . . . . . . . . . 209 111. Positronium Scattering by Atoms and Charged Particles . . . . . . . . . 213 A. Positronium Scattering by Atoms . . . . . . . . . . . . . . . . . '213 B. Positronium Scattering by Charged Particles . . . . . . . . . . . . 215 C. Antihydrogen Formation in Positronium-Antiproton Collisions . . . . 216 IV. Positron Scattering by Molecular Hydrogen . . . . . . . . . . . . . . 217 V. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 220 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . 220 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 11. Positron Scattering by Atoms. . . . . . . . . . . . . . . . . . . . .
I. Introduction The ever-widening range of experiments in positron collision physics made possible by recent improvements in the intensity of monoenergetic positron beams provides a continuing stimulus to further theoretical investigations of such processes. A comprehensive review of recent experimental developments has been given by Charlton and Laricchia (1990).When positron beams were first developed, little more than 20 years ago, the beam currents were so low that only total scattering cross sections could be measured, but now some partial and differential cross sections can also be measured. It is even possible to produce positronium beams using the charge exchange process e++A+Ps+A+
(1)
where A is some convenient target atom, and positronium scattering by various atoms and molecules is beginning to be investigated (Charlton and Laricchia, 1991). In addition to satisfying the need for accurate results to compare with the 205 Copyright 0 1994 by Academic Press, Inc. All rights of reproduction in any form reserved ISBN 0-12-003832-3
206
J . W; Humberston
experimental measurements, theoretical studies of such systems provide very sensitive tests of methods of approximation for use in scattering theory. The partial cancellation of the static and polarization terms in the positron-atom interaction, the absence of exchange of the incident positron with the electrons in the target, and the attraction of the positron to the electrons all contribute to making the results of a calculation of low-energy positron scattering parameters depend much more sensitively on the details of the trial wave function and the method of approximation being used than is the case for electron scattering. At low incident energies the total cross section for positron scattering from a given target atom is usually substantially less than that for electrons. For the alkali atoms, however, the positron total cross section is slightly greater than that for electrons, most probably because of a significant contribution from positronium formation. As the projectile energy is increased, the difference between the positron and electron total cross sections decreases, and a merging of the various partial cross sections for the two projectiles is expected at sufficiently high energies, where the first Born approximation is valid. Somewhat surprisingly, the two total cross sections merge at a much lower energy than do the individual partial cross sections (Kauppila and Stein, 1982; Stein et al., 1990), but this can be understood by reference to the optical theorem: 4
CJ,,,,
=-
k
Im fhp'
If a Born expansion of the forward elastic scattering amplitude is made,
2 6)
f(0)
=
n= 1
fB'"'(0)
(3)
the first Born forward amplitude is real and the first nonzero term in the righthand side of equation (2) therefore arises from the second Born amplitude, which, if exchange effects are neglected, is the same for positrons and electrons, being quadratic in the projectile-atom potential. Thus with the neglect of exchange, any difference between the total cross sections for electrons and positrons must arise from the odd orders of Born terms with n >> 3, and such terms will presumably be negligibly small at much lower energies than the region of validity of the first Born approximation. In two previous reviews in this series (Humberston, 1979, 1986) the author summarised the low-energy scattering of positrons and positronium by simple atomic systems. Here we shall again confine our attention to lowenergy scattering and consider some developments in this field since the publication of these two previous reviews, although occasional references to earlier work will be made. Positron collisions at intermediate and higher energies are reviewed elsewhere in this volume.
POSITRON AND POSITRONIUM SCATTERING AT LOW ENERGIES
207
11. Positron Scattering by Atoms A.
POSITRON
SCATTERING BY ATOMICHYDROGEN
The scattering of positrons by atomic hydrogen is at last becoming amenable to experimental investigation, and measurements have already been made of the total ionization cross section (Spicher et al., 1990) and, most recently, the positronium formation cross section as well (Sperber et al., 1992).Theoretical studies of this process have been undertaken for many years and very accurate results have been obtained for elastic scattering and positronium formation below the first excitation threshold of hydrogen (Humberston, 1986). The recent measurements of the positronium formation cross section at low energies appear to be in satisfactory agreement with these accurate theoretical values. Further investigations of s-wave positron-hydrogen scattering have recently been made by Archer et al. (1990). They used the reactive scattering method of Pack and Parker (1987) over the energy range below the n = 4 excitation threshold of hydrogen and included all energetically possible reaction channels in the formulation. The cross section for positronium formation in the ground state is shown in Fig. 1. These results are qualitatively very similar to, but approximately 15% less than, those of Humberston (1982) below kZ = 0.723, the highest energy considered in the earlier work. At higher energies considerable resonance structure is found, and this will be discussed in more detail shortly. The method employed by Archer et al. generates all the cross sections linking any one channel to any other, and they show, for example, that the cross sections for positronium formation in the 2s and 2p states are both approximately $ of the ground state formation cross section. It is well known that an infinite set of Feshbach resonances is associated with each degenerate excitation threshold in positron-hydrogen scattering. The thresholds may be those of hydrogen or positronium, since the latter system is also hydrogenic with degenerate excitation thresholds. The first such resonance, just below the H (n = 2) threshold, was found by Doolen et al. (1978). Since then several other S-state resonances associated with H ( n < 5 ) and Ps (n < 4) have been found, most notably by Ho (1989a), who used the complex coordinate rotation method with Hylleraas-type trial functions containing several hundred terms. The five resonances found by Archer et al. (1990), and seen in Fig. 1, are a subset of those of Ho (1989a), with one associated with H (n = 2), three with Ps (n = 2) and one with H (n = 3). Ho (1990a) has also investigated P-state resonances and has found one associated with the H ( n = 2) threshold and three with the Ps (n = 2) threshold.
208
J . W Humberston N o -
m
0.007
c:
Y
C .-0
0.006
c
0
0,005 In v)
2
0
0.004
7
F
/ / /
-
C
0 .-c
-
T
0.003
z
E 0.002 -
.-3 r
l2
.-
0.001
1
In 0
a
1 1 1 1 1 1 1 1 1 1 0.000 L L u L 1 0.40 0.50 0.60 0.70 ~~
d 0.80
0.90
1.00
k2 (a-2) FIG.1. The ground-state positronium formation cross section in s-wave positron-hydrogen scattering: -, Archer et al. (1990);---, Humberston (1982). (Adapted from Archer et a/. (1990) with permission.)
The positron-hydrogen system continues to attract theoretical attention because it is the simplest nontrivial system on which to test various approximation methods, even if the results are not expected to be of high accuracy. One such method is the close-coupling approximation, but the results for low energy elastic scattering are known to converge rather slowly with respect to increasing the number of target states in the expansion of the wave function. Basu et al. (1989), using the four-term expansion H( Is, 2s, 2p), Ps(1s) in the close-coupling approximation, claim to have found two s-wave resonances just above the positronium formation threshold (6.8 eV), but these authors concede that the resonances may not be real; and they have not been found by Bhatia and Drachman (1990), who also undertook a resonance search in this energy region using the stabilization and compIex coordinate rotation methods. Basu et al. (1990) have extended the formulation by adding 2p and 3J pseudostates of hydrogen to the 1s and 2s states, and also including the ground state of positronium. The presence of the positronium state improves the accuracy of the elastic scattering cross sections, and the positronium formation cross section is also in fair agreement with the very accurate results. Similar investigations have been carried out by Hewitt et al. (1990), but also including the n = 2 states of positronium, and estimates have
POSITRON AND POSITRONIUM SCATTERING AT LOW ENERGIES
209
been obtained of the cross sections for positronium formation in its n = 2 excited states. One of the most successful theoretical techniques for studying electron scattering by atoms and molecules is the R-matrix method; and it has recently been applied to positron scattering by atomic hydrogen (Higgins and Burke, 1991). The method was initially applied to elastic scattering alone, but the formulation has since been extended to include positronium formation, although results have as yet only been obtained with a restricted form of wave function that amounts to the coupled static approximation. Although at a somewhat higher energy than is normally being considered in this review, it is appropriate to mention that the s-wave positronium formation cross section exhibits a very pronounced resonance feature of width 4.2 eV at a positron energy of 35.6 eV. This had not been observed in previous studies using the coupled static approximation, but its existence has since been confirmed in an independent calculation by Hewitt et al. (1991). The mechanism for the production of the resonance, which is believed to be a genuine feature and not merely an artifact of the method of approximation, is not fully understood, but more detailed study of the eigenphases and the mixing angle show that the resonance occurs almost totally in the positronium-proton channel. It has been termed a coupled channel shape resonance. Other aspects of the positron-hydrogen system are discussed in Sections 1II.B and 1II.C. B. POSITRON SCATTERING BY ALKALIATOMS Positron scattering by the alkali atoms exhibits several unusual and interesting features. As has already been mentioned, experimental measurements reveal that, unlike for other target atoms, the total scattering cross sections at low energies are somewhat larger for positrons that for electrons. In addition, positronium formation is possible even at zero positron energy because the ionization energy of each of the alkali atoms is less than the binding energy of positronium (6.8 eV). An alkali atom consists of a loosely bound valence electron interacting with the core of tightly bound inner electrons, and this interaction may be accurately represented in terms of a local central potential. The positron interaction with the core may also be represented in this way, and the positron-alkali atom system therefore reduces to an effective three-body system rather similar to positron-hydrogen. Many theoretical studies have been made of positron scattering by alkali atoms using a variety of approximation methods, but in most cases the positronium formation channel has been neglected. Even where this channel has been included, the methods of approximation have usually been relatively
210
J . W Humberston
crude, such as the Born and coupled static approximations (Guha and Ghosh, 1981)and the distorted wave approximation (Mazumdar and Ghosh, 1986). More elaborate calculations in which the positronium channel is included have, however, recently been undertaken for lithium using the closecoupling approximation. Basu and Ghosh (1991) included three states, Li(2s,2p) and Ps(ls), and Hewitt et al. (1992) included seven states, Li(2s, 2p, 3s, 3p) and Ps(ls, 2s, 2p). The agreement between the two sets is not very good, suggesting that the close-coupling expansion converges rather slowly. The most detailed investigations in which positronium formation has been neglected are those of Ward et al. (1989) and McEachran et al. (1990), who used the close-coupling approximation with up to five states of the target atom. Elastic and excitation cross sections were calculated for lithium, sodium, potassium and rubidium over the energy range 0-100 eV. In lithium the behavior of the eigenphase sum revealed the presence of narrow resonances just below the various excitation thresholds. It should be borne in mind, however, that the positions and widths of these resonances, and even their existence, may be influenced by the inclusion of positronium formation in the scattering process. These authors also claim some evidence in support of a positron-lithium bound state, but since the positron-lithium system has a continuum of states between e+-Li (0 eV) and Ps-Li' (- 1.4 eV), a true bound state would need to have an energy < - 1.4 eV. The existence of such a state has not yet been established. A more detailed investigation of low-energy positron-alkali atom scattering, with the inclusion of positronium formation, has recently been undertaken by Watts and Humberston (1992), using the Kohn variational method in a formulation rather similar to that employed by Humberston (1982) and Brown and Humberston (1984, 1985) in accurate studies of positronhydrogen scattering. Lithium was selected as the first target atom to be considered, even though no experimental results are yet available for this atom. Initial attention has been given to s-wave scattering in the energy region below the first excitation threshold (to the 2p state at 1.84 eV) so that only elastic scattering and positronium formation need be considered: e+
+ Li -,e + + Li + Ps + Li
elastic scattering +
positronium formation
The Kohn functional for the K-matrix takes the form
(4)
POSITRON AND POSITRONIUM SCATTERING AT LOW ENERGIES
where L
= 2(H - E).
1
21 1
The trial functions are chosen to be
N
N
1 +-J471 exp[-(ar, + Br2 + y3)]1 djrtv-$yl i=l
where r,, r z , and p are the coordinates of the positron, electron, and the center of mass of the positronium relative to the core, respectively. All terms with ki Ii mi < o are included in the summations. A good representation of the electron-lithium core potential is provided by the form (Peach et ul., 1988)
+ +
1 2 V-(r) = - ~-- (1 + ur + br2)e-@" - - w(vr) 2r4 r r
(7)
where the first two terms represent the static interaction with the core and the last term arises from the polarization of the core. The values of the parameters in V- are chosen to reproduce the correct binding energy for the electron in the 2s state and to provide a good fit to the energies of the excited states. A very accurate representation of the lithium wave function aLi(r2)for the potential V _ is aLi(rz)= e - p r 2
C
ejr$
j=O
The positron-core potential V+ is derived from V- by merely changing the signs of the static terms, a procedure not entirely justified because electron exchange with the core is implicitly incorporated in V- whereas there is, of course, no exchange for the positron. Elastic scattering and positronium formation cross sections obtained with the most elaborate trial functions containing 165 terms (w= 8) are displayed in Fig. 2. For positron energies corresponding to k > 0.1 the results seem to
J . W Humberston
212
0.0
0.1
0.2
0.3
0.4
k (ao-’) FIG.2. The elastic scattering and positronium formation cross sections in s-wave positronlithium scattering (Watts and Humberston, 1992). The lowest excitation threshold of lithium is indicated by the dotted line.
be well converged with respect to systematic improvements in the trial functions (obtained by increasing w, and hence the number of correlation terms). At lower energies, however, the convergence deteriorates significantly because the trial functions only contain short-range terms, whereas longrange dipole terms should also be included. Positronium formation in positron-lithium scattering is an exothermic reaction, as also is antihydrogen formation in positronium-antiproton collisions (see Section III.C), and therefore the formation cross section is K l/k for sufficiently low positron energies, and is therefore infinite at zero positron energy. However, over most of the energy range under consideration the positronium formation cross section is several orders of magnitude smaller than the elastic scattering cross section. Similarly small positronium formation cross sections are also found in s-wave positron-hydrogen scattering, although the elastic scattering cross section is then considerably smaller than it is for lithium (Humberston, 1982, 1984). As in positronhydrogen scattering, the p- and d-wave contributions to the total positronium formation cross section are expected to be much larger than the swave contribution and may well be comparable to the elastic cross section as, indeed, the close-coupling results of Hewitt et al. (1992) already show. Uncoupling the positronium formation channel has only a very small effect
POSITRON AND POSITRONIUM SCATTERING AT LOW ENERGIES
213
on the s-wave elastic scattering parameters, as was also found in positronhydrogen scattering, but comparisons with the close coupling results of McEachran et al. reveal that the Kohn phase shifts are significantly more positive, and therefore probably more accurate. These investigations are currently being extended to higher partial waves and to other alkali atoms.
111. Positronium Scattering by Atoms and Charged Particles Although positronium is formed quite copiously in the moderation process that also yields low-energy monoenergetic positrons from the primary positron source, the energy is low and can be varied only slightly by changing the moderator material. The most convenient means of producing a monoenergetic beam of positronium for subsequent use in studying positronium-atom scattering is by charge exchange in positron collisions with atoms: e+
+A+
Ps + A +
(9)
where the energy of the positronium is determined by the energy of the incident positron beam, which can be varied at will. An alternative means of production has been proposed, but not yet implemented, in which the positronium negative ion, Ps -, is accelerated to the required energy and then undergoes photodetachment of one of the electrons (Mills, 1981). The angular distribution of the positronium formed in the charge exchange process is quite strongly peaked around the forward direction, and a reasonably well-collimated “beam” of positronium is therefore formed directly in the production process (Laricchia and Zafar, 1992). SCATTERING BY ATOMS A. POSITRONIUM By observing the decrease in intensity of the positronium signal as the pressure of the target gas in which the positronium is being formed is increased, estimates have already been made of the total cross section for positronium scattering by the target gases helium and argon (Laricchia et al., 1992). It should soon be possible to undertake experimental studies of positronium scattering by target gases other than those in which the positronium is initially formed. The experimental cross sections obtained so far are believed to relate to ground state positronium incident on the target atom, but they include contributions from excitation of the positronium and the target as well as
214
J . W Humberston
elastic scattering. Comparisons between theory and experiment are therefore difficult because most theoretical studies have considered only elastic scattering, although Peach (reported in Laricchia and Zafar, 1992), using a formulation in which the target atom is represented as a single particle, has calculated cross sections for elastic scattering and excitation of the positronium and has obtained tolerably good overall agreement with the experimental results for helium, but not for argon. Theoretical studies have been restricted mainly to the elastic scattering of positronium by atomic hydrogen and helium (Barker and Bransden, 1968). The most detailed investigations of the positronium-hydrogen system have concentrated on the electron spin singlet bound state, positronium hydride, Ps H, with a binding energy of 0.411 eV (Ho, 1986) and the numerous resonances. These arise from the rearrangement of the system into e+-H -, with the Coulomb interaction at large separations supporting an infinite set of “bound” states. But apart from the ground state, these states have energies greater than the minimum for positronium-hydrogen scattering, and they therefore manifest themselves as Feshbach resonances in the scattering process. The first such spin singlet S-state resonance was found by Drachman and Houston (1975), using the stabilization method, at a positronium energy of 4.2 eV and with a width of 0.075 eV. This finding was confirmed by Ho (1978), using the complex coordinate rotation method, and several other resonances in this system have since been found at slightly higher energies using the same technique (Ho, 1989b). Little progress has been made in the study of nonresonant positroniumhydrogen scattering beyond what was reported by Humberston (1986). Mention was made there of the rather accurate results of Page (1976) for the singlet and triplet scattering lengths and the static exchange results of Hara and Fraser (1975). A detailed study by Au and Drachman (1986) of the interaction between positronium and a hydrogenic atom has yielded the van der Waals interaction between these two systems, and the addition of this interaction to the static exchange formulation of Hara and Fraser (1975) should produce more accurate cross sections. Positronium scattering by positronium has also received theoretical attention. This system, which in some respects resembles positroniumhydrogen, also has a bound state, the positronium molecule, Ps2, with a binding energy of 0.41 1 eV with respect to dissociation into two positronium atoms (Ho, 1986). There is also an infinite set of Feshbach resonances, which is again a consequence of the Coulomb interaction at long range between a positron and Ps-, the positronium negative ion. Now, however, there are two separate series of resonances for each partial ’wave: one for scattering orthopositronium by orthopositronium (or parapositronium by paraposi-
POSITRON AND POSITRONIUM SCATTERING AT LOW ENERGIES
215
tronium) and one for scattering of orthopositronium by parapositronium. Ho (1990b) has again used the complex coordinate rotation method with trial functions containing up to 500 terms to determine the positions and widths of the first three S-state resonances in each of these two series below the energy of Ps-. A comprehensive review of resonances in positron and positronium scattering by various atoms and ions has been given by Ho (1992).
SCATTERING BY CHARGED PARTICLES B. POSITRONIUM The scattering of positronium by a single charged particle, which might be an electron, positron, proton or antiproton, or other nucleus, is another type of three-body system that has been studied quite extensively: Low-energy positronium scattering by electrons, which according to charge conjugation invariance should be identical to positronium-positron scattering, has been investigated by Ward et al. (1987) using the Kohn variational method, and Melezhik and Vukajlovic (1988) using the adiabatic method. Recently, Kvitsinsky et al. (1992) have used the Faddeev equations in configuration space to determine the singlet and triplet scattering lengths, ~ a - = 4.7a0, in good agreeand they obtained the values a + = 1 1 . 9 8 ~and ment with the extrapolated values of Ward et al. (1987). Melezhik and Vukajlovic claim to have found an S-state resonance at an incident energy of 0.6 eV, but its existence has been disputed by Bhatia and Drachman (1990), who undertook a resonance search in this energy region using the stabilization and complex coordinate rotation methods with very elaborate Hylleraas trial functions and found no such structure. Resonances certainly exist in positronium-electron (positron) scattering, but they are mainly Feshbach resonances associated with, and just below, the various excitation thresholds of positronium. Many such S- and P-state resonances have been found associated with positronium states up to n = 6 (Ward et al., 1987; Bhatia and Ho, 1990; Ho and Bhatia, 1991; Ho, 1992). In addition two 'Po and one 3P0 shape resonances have also been identified. Some features of positronium scattering by protons, and in particular the existence of Feshbach resonances, have already been considered in Section 1I.A dealing with positron-hydrogen scattering. A further important aspect of such a collision is the charge exchange process in which atomic hydrogen is produced. Alternatively, if the collision is with an antiproton, the charge exchange process yields antihydrogen. The synthesis of antihydrogen is discussed in further detail in Section 1II.C. Bhatia and Drachman (1990) have investigated resonances in the system. They claim to have found two S-state resonances positronium-He' with total energies of -9.93 eV and - 5.30 eV and they interpret the structure +
J . W Humberston
216
as the He' + being bound to positronium in its n = 2 states. This is the same mechanism as that responsible for some of the resonances in positronhydrogen scattering (see Section II.A), but because the charge on He'' is twice that on the proton, the binding is now tighter. Indeed, instead of having an energy just below that of the n = 2 state positronium ( - 1.17 eV), the lower of the two Ps-He" resonances lies below even the ground state of positronium (- 6.8 eV). Positronium scattering by a more elaborate charged system, the lithium ion, Li', has been investigated by Mukherjee and Ghosh (1991) using the Born approximation, but no results have yet been published.
c. ANTIHYDROGENFORMATION IN POSITRONIUM-ANTIPROTON COLLISIONS The simplest system consisting entirely of antimatter is the antihydrogen atom, comprising a positron orbiting an antiproton, and the possibility of forming this exotic atom is being actively investigated. Several production mechanisms have been proposed, one of the most promising of which is the charge exchange process: Ps
+ p +H + e-
(10)
The viability of this means of production depends critically on the magnitude of the formation cross section, and reasonably accurate values of this quantity are therefore required. At the time this production mechanism was first proposed, however, no calculations of the relevant cross section had been attempted, but it was found to be related in a rather simple manner to the cross section for positronium formation in positron-hydrogen collisions (Humberston et al., 1987), a process that has received extensive theoretical attention (Humberston, 1986). Starting with the process of interest, Ps
+ p -,H + e -
(cross section, on)
(1 1)
(cross section, oH)
(12)
the charge conjugate system is Ps
+p
--t
H
+ e'
and, assuming charge conjugation invariance, OH
= OH
(13)
The time-reversed form of the latter process is e+
+ H + Ps + p
(cross section, ops)
(14)
But time-reversal invariance implies the symmetry of the S-matrix, from
POSITRON AND POSITRONIUM SCATTERING AT LOW ENERGIES
217
which it follows that, if the positronium and the hydrogen (or antihydrogen) atom are both in their ground states,
where k and K are the wave numbers of the positron and positronium, respectively. They are related by energy conservation, such that
Within the Ore gap, the energy region between the positronium formation threshold (6.8 eV) and the first excitation threshold for hydrogen (10.2 eV), the most accurate positronium formation cross sections are probably those obtained by Brown and Humberston (1984, 1985)using the Kohn variational method. At somewhat higher energies the distorted wave Born approximation results of Shakeshaft and Wadehra (1980) are probably quite accurate, and at sufficiently high energies the Born approximation itself yields accurate results. When the rescaling formula, Eq. (15), is applied to these positronium formation cross sections, the antihydrogen formation cross section shown in Fig. 3 is obtained. The antihydrogen formation process is exothermic and consequently, as the positronium energy tends to zero, the antihydrogen formation cross section cc 1 / ~ and , is therefore infinite at zero energy; however the formation rate, which is proportional to K O ~ remains , finite. Estimates have been made of the cross sections for the formation of antihydrogen in various excited states from ground state positronium (Darewych, 1987; Nahar and Wadehra, 1988), and for ground-state antihydrogen formation from various excited states of positronium. Taking all these contributions into account, the total antihydrogen formation cross section in the energy range of interest is likely to be approximately 201t.a; (Humberston, 1990),and this value, when taken in conjunction with the expected currents of positronium and antiprotons, should provide a production rate of a few antihydrogen atoms per second. A progress report on antihydrogen synthesis has recently been given by Deutch (1992).
IV. Positron Scattering by Molecular Hydrogen Having two fixed centers in a diatomic molecule and, consequently, only axial symmetry about the internuclear axis in the Hamiltonian instead of the spherical symmetry in an atom introduces major complications into the
218
J . W Humberston
0
4
8 12 xz, Ps energy 16.8 eV)
16
FIG.3. The cross section for ground-state antihydrogen formation in collisions of groundstate positronium with antiprotons (Humberston et al., 1987).
formulation of the scattering of a projectile by such a target. It is therefore not surprising that many studies of positron-molecule scattering have employed relatively crude methods of approximation and rather simple model potentials to represent the positron-molecule interaction (see Armour, 1988, for a general review of the subject). Nevertheless, detailed ab initio calculations of the elastic scattering of positrons by molecular hydrogen have been attempted by Armour and his collaborators (Armour, 1988; Armour et al., 1990)using the Kohn variational method with very elaborate trial functions. Their most recent results have been obtained with trial functions containing many terms of,:C C z , nu,and n, symmetry, including several Hylleraas terms with a linear dependence on the positron-electron coordinates, and the agreement with the most accurate experimentally measured cross sections for positron energies below 5 eV is very good, as is shown in Fig. 4. Improving the trial function still further and introducing other symmetries should yield good agreement with the experimental cross sections up to the positronium formation threshold. These investigations are being extended to provide information on rotational excitation of the molecule.
POSITRON AND POSITRONIUM SCATTERING AT LOW ENERGIES
219
T
I
5 ppritron mpy
I
lo (CV)
FIG.4. Cross sections for positron scattering by molecular hydrogen. The most accurate experimental values for the total cross section are indicated by open circles (Hoffman et al., 1987) and the most accurate theoretical results for elastic scattering are indicated by open triangles (Armour et al., 1990). The positronium formation threshold is at 8.63 eV and is marked Eps. (Adapted from Armour et al. (1990) with permission.)
A further test of the quality of the trial functions used by Armour is provided by a comparison of the theoretical and experimental values of the rate for positron-annihilation, 1.This is usually written in terms of Zeff,an effective number of electrons per atom or molecule as (Humberston, 1979) 1= nrGcNZ,,,
(17)
where ro is the classical radius of the electron and N is the number density of molecules. If the total wave function representing the positron-hydrogen molecule system is "(r,,, rl, r2, R), then 2
i= 1
where Y must be normalized to unit positron density as rp + 00. This
220
J . W Humberston
expression does not constitute a variational principle and the error in Zeffis therefore only of first order in the error in Y, which makes the calculated value of Zeffa particularly sensitive test of the wave function. The experimental value of Zeff at room temperature is 14.8, whereas the latest value obtained by Armour is 10.2. This is much closer to the experimental value than any previous theoretical result and should be contrasted with the value 2, the actual number of electrons in the target molecule, which is given by the Born approximation. The R-matrix method, referred to in Section ILA, has also been used to investigate positron-molecule scattering, but the formulation is not, in its present form, very well suited to providing an adequate representation of the positron-electron correlations and virtual or real positronium formation. Nevertheless, the method has been applied to positron scattering by several diatomic molecules including H, (Danby and Tennyson, 1990), N, and CO (Tennyson and Danby, 1987), and H F (Danby and Tennyson, 1988). Evidence has been obtained of a weakly bound state of the positron-HF system.
V. Concluding Remarks The field of low-energy positron and positronium collisions has grown enormously over the past few years, and in a short review such as this it has not been possible to provide a very comprehensive coverage. Instead, a selection of topics has been made that admittedly reflects the author’s own interest, but that also includes some of the more interesting and significant developments. Some of the systems considered here are not yet accessible to experimental investigation, but so fast is the pace of development that it may not be too long before such experimental studies can be made. At the same time several systems have been investigated experimentally in some detail but have not yet received adequate theoretical attention. Much therefore remains to be done in this fascinating and flourishing field.
Acknowledgments The author wishes to thank M. S. T. Watts for his valuable assistance, particularly in the calculations of positron-lithium scattering. He also wishes to thank M. Charlton, G. Laricchia, G. Peach, and N. Zafar for several useful discussions.
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REFERENCES Archer, B. J., Parker, G . A,, and Pack, R. T. (1990). Phys. Reo. A 41, 1303. Armour, E. A. G. (1988). Phys. Reports 169, 1. Armour, E. A. G., Baker, D. J., and Plummer, M. (1990). J. Phys. B. 23, 3057. Au, C. K., and Drachman, R. J. (1986). Phys. Rev. Lett. 56, 324. Barker, M. I., and Bransden, B. H. (1968). J. Phys. B 1, 1109. Basu, M., and Ghosh, A. S. (1991). Phys. Rev. A 43, 4746. Basu, M., Mukherjee, M., and Ghosh, A. S. (1989). J. Phys. B 22, 2195. Basu, M., Mukherjee, M., and Ghosh, A. S. (1990). J. Phys. B 23, 2641. Bhatia, A. K., and Drachman, R. J. (1990). Phys. Reo. A 42, 5117. Bhatia, A. K., and Ho, Y. K. (1990). Phys. Reo. A 42, 1119. Brown, C. J., and Humberston, J. W. (1984). J . Phys. B 17, L423. Brown, C. J., and Humberston, J. W. (1985). J. Phys. B 18, L401. Charlton, M., and Laricchia, G. (1990). J. Phys. B. 23, 1045. Charlton, M., and Laricchia, G. (1991). Comments At. Mol. Phys. 26, 253. Danby, G., and Tennyson, J. (1988). Phys. Reu. Lett. 61, 2737. Danby, G., and Tennyson, J. (1990). J. Phys. B 23, 1005. Darewych, J. W. (1987). J . Phys. B 20, 5917. Deutch, B. I. (1992). Hyperfine Interactions 73, 175. Doolen, G. D., Nuttall, J., and Wherry, C. J. (1978). Phys. Rev. Lett. 40, 313. Drachman, R. J., and Houston, S. K. (1975). Phys. Rev. A 12, 885. Guha, S., and Ghosh, A. S. (1981). Phys. Rev. A 23, 743. Hara, S., and Fraser, P. A. (1975). J. Phys. B 8, L472. Hewitt, R. N., Noble, C. J., and Bransden, B. H. (1990). J. Phys. B 23,4185. Hewitt, R. N., Noble, C. J., and Bransden, B. H. (1991). J . Phys. B 24, L635. Hewitt, R. N., Noble, C. J., and Bransden, B. H. (1992). J. Phys. B 25, 2683. Higgins, K., and Burke, P. G. (1991). J. Phys. B 24, L343. Ho, Y. K. (1978). Phys. Rev. A 17, 1675. Ho, Y. K. (1986). Phys. Rev. A 33, 3584. Ho, Y. K. (1989a). Int. ConJ Phys. Electron. A . Collisions, 16th Abstracts, p. 415. Ho, Y. K. (1989b). Int. Cant Phys. Electron. A . Collisions, 16th Abstracts, p. 426. Ho, Y. K. (1990a). J. Phys. B 23, L419. Ho, Y. K. (1990b). In: Annihilation in Gases and Galaxies (R. J. Drachman, ed.), NASA Conf. Pub. 3058, p. 243. Ho, Y. K. (1992). Hyperfine Interactions 73, 109. Ho, Y. K., and Bhatia, A. K. (1991). Phys. Rev. A 44,2890. Hoffman, K. R., Dababneh, M. S., Hsieh, Y.-F., Kauppila, W. E., Pol, V., Smart, J. H., and Stein, T. S. (1982). Phys. Rev. A 25, 1393. Humberston, J. W. (1979). Ado. At. Mol. Phys. 15, 101. Humberston, J. W. (1982). Can. J . Phys. 60, 591. Humberston, J. W. (1984). J. Phys. B 17, 2353. Humberston, J. W. (1986). Adv. At. Mol. Phys. 22, 1. Humberston, J. W. (1990). In: Annihilation in Gases and Galaxies (R. J. Drachman, ed.), NASA Conference Publication 3058, p. 223. Humberston, J. W., Charlton, M., Jacobsen, F. M., and Deutch, B. I. (1987). J . Phys. B 20, L25. Kauppila, W. E., and Stein, T. S. (1982). Can. J. Phys. 60, 471. Kvitsinsky, A. A., Carbonell, J., and Gignoux, C. (1992). Phys. Rev. A 46, 1310. Laricchia, G., and Zafar, N. (1992). In: Positrons at Metallic Surfaces (A. Ishii, ed.), Trans Tech Publications, Aedermannsdorf.
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Laricchia, G., Zafar, N., Charlton, M., and Griffith, T. C. (1992). Hyperfine Interactions 73, 133. Mazumdar, P. S., and Ghosh, A. S. (1986). Phys. Rev. A 34,4433. McEachran, R. P., Horbatsch, M., Stauffer, A. D., and Ward, S. J. (1990). In: Annihilation in Gases and Galaxies (R. J. Drachman, ed.), NASA Conference Publication 3058, p. 1. Melezhik, Y. S., and Vukajlovic, F. R. (1988). Phys. Rev. A 38, 6426. Mills, A. P. (1981). Phys. Rev. Lett. 46, 717. Mukherjee, M., and Ghosh, A. S. (1991). Int. Conf Phys. Electron. A . Collisions, 17th Abstracts, p. 361. Nahar, S. N., and Wadehra, J. M. (1988). Phys. Rev. A 37, 4118. Pack, R. T., and Parker, G. A. (1987). J. Chem. Phys. 87, 3888. Page, B. A. P. (1976). J . Phys. B 9, 1111. Peach, G., Saraph, H. E., and Seaton, M. J. (1988). J. Phys. B 21, 3669. Shakeshaft, R., and Wadehra, J. M. (1980). Phys. Rev. A 22,968. Sperber, W., Becker, D., Lynn, K. G., Raith, W., Schwab, A., Sinapius, G., Spicher, G., and Weber, M. (1992). Phys. Rev. Lett. 68, 3690. Spicher, G., Olsson, B., Raith, W., Sinapius, G., and Sperber, W. (1990). Phys. Rev. Lett. 64, 1019. Stein, T. S., Kauppila, W. E., Kwan, C. K., Lukaszew, R. A., Parikh, S. P., Wan, Y. J., Zhou, S., and Dababneh, M. S. (1990). In: Annihilation in Gases and Galaxies, NASA Conferences Publication 3058, p. 13. Tennyson, J., and Danby, G. (1987). In: Proceedings ofthe N A T O Advanced Research Workshop on Atomic Physics with Positrons, UCL, 1987 (J. W. Humberston, and E. A. G. Armour, eds.), Plenum Press, New York, 1987, p. 111. Ward, S. J., Horbatsch, J., McEachran, R. P., and Stauffer, A. D. (1989). J. Phys. B 22, 1845. Ward, S. J., Humberston, J. W., and McDowell, M. R. C. (1987). J. Phys. B 20, 127. Watts, M. S. T., and Humberston, J. W. (1992). J. Phys. B 25.
ADVANCES IN ATOMIC, MOLECULAR, A N D OPTICAL PHYSICS, VOL. 32
HOW PERFECT ARE COMPLETE ATOMIC COLLISION EXPEMMENTS? H . KLEINPO PPEN Unit of Atomic and Molecular Physics University of Stirling Stirling. Scotland
H. HAMDY Beni-Sues Faculty of Science Physics Department Beni-SueA Egypt
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Electron- Atom Collisions . . . . . . . . . . . . . . . . . . . . . . 111. Approaches to “Complete” Experiments in Heavy-Particle Atom Collisions and Photoionisation of Atoms . . . . . . . . . . . . . . . . . . . . IV. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
223 225 244 248 248
I. Introduction Let us say at the outset how very much we enjoy and feel honoured to contribute to David Bates’s memorial issue of Aduances in Atomic, Molecular, and Optical Physics. The title of our chapter reflects very much our own interest in atomic collision physics and, needless to say, that in our own approach to research we were stimulated very much by theoretical atomic collision physics set up by the Massey and Bates schools in the United Kingdom. However, to draw attention to David Bates’s (1978) article, “Other Men’s Flowers,” our own stimulation from theory means that we have not been in the position to gather a bunch of colourful flowers of theoretical topics but only smelt the odour of it and got excited by it. While present-day experimentalists in atomic collision physics may be characterized as having established and extracted data from highly sophisticated experimental apparatuses, they are guided by theoretical analysis of their experimental data and relating them for comparison to theory. In a way this very close relationship or correlation between experimental and theoretical atomic collision physics is a typical characteristic and is of significance to 223 Copyright 0 1994 by Academic Press, Inc. All rights OF reproduction in any form R S ~ N ISBN 0-12-003832-3
~ .
224
H . Kleinpoppen and H . Hamdy
atomic collision physics. This correlation may be very strong compared to other fields of physics but is probably due to the fact that atomic collision processes are fully or at least approximatively describable in terms of quantum mechanics. We attempt to give critical accounts of approaches to developments and results of so-called complete or perfect collision experiments. As it has turned out, such perfect approaches are at the forefront of modern atomic collision physics and represent most sensitive tests of theories. Of course there are different ways to present and discuss the concept and results of the complete atomic collision experiments that have been discussed or summarized in recent reviews (e.g., Anderson et al., 1988; Slevin and Chwirot, 1990; Crowe and Rudge, 1988; Kessler, 1991; Lutz, 1992; MacGillivray and Standage, 1991; Hippler, 1992; and McConkey et al., 1988). Since about the beginning of the 1970s it has become possible to extract data from atomic collision processes, which provide us with a description in terms of scattering amplitudes and their phases, of target parameters such as orientation, alignment and state multipoles, of coherence parameters (e.g., degree of coherence of the excitation) and of the electron charge distribution of excited states. Some of the underlying physical concepts, such as quantummechanical interference phenomena in scattering processes were already obvious in the 1920s and the 1930s (e.g., the Bullard-Massey, 1931, experiment of electron angular distribution and the Ramsauer-Townsend effect resulting from interference effects of partial waves). Fano (1957) discussed general principles for so-called ideal atomic collision experiments. It follows from Fano’s arguments that the linearity of the Schrodinger equation describing the collision process results in a method for quantum dynamically complete experiments. An initial state of colliding atomic particles may be prepared such that it is described by a quantum mechanically pure state: I$i n ) = IP)IA), where IP), IA) are state vectors of the projectile and atomic target, respectively. Due to the linearity of the = 1P)IA) is “transferred” to the Schrodinger equation, the initial state I$i n ) final state by the scheme shown in Fig. 1. According to such a scheme the knowledge represented by the state vector [ $ o u t ) represents maximum information on the excitation process. This recipe for complete atomic collision experiments has first been applied successfully to coherent electron impact excitation of atoms. Since the electron impact process has been the most advanced one with regard to complete experiments we will first describe its development and analysis in detail, which will be useful for related applications in other areas of collision physics. Bederson (1969) and Kleinpoppen (1971) discussed and described “perfect”
HOW PERFECT ARE COMPLETE ATOMIC COLLISION EXPERIMENTS?
I
225
I
state vector
state vector before the collision
with interaction potential U(t) acting during the collisional excitation
FIG.1. Transfer of the initial state of projectile (P) and atomic ( A ) target vectors to the final state.
electron-atom collision experiments in terms of Coulomb direct and exchange amplitudes.
11. Electron- Atom Collisions An ideal example for a complete electron impact analysis is the excitation of a single 'P, state of light atoms. Experimentally the inelastically scattered electron and the photon from the deexcitation is being observed in coincidence. Electron spin effects can be neglected for the 'P, state excitation of light atoms since the total spin of the 'P, state is zero and the two spin components of the incoming electron are symmetrical with regard to the scattering plane defined by the incoming and scattered electron (Fig. 2). Accordingly the initial (before the excitation) and the final (after the excitation) states of the collision process can be represented as follows. We take helium as a light target atom:
IP,) and IPb)represent the linear momenta of the incoming and scattered electron, respectively, and IHe(1 'So) and IHe(n 'P,)) the state vectors of the helium atom in the ground and excited state, respectively. The state vector IHe(n'P,)) may be represented by the following coherent and linear superposition: IHe('P1)) =
1
fmI$m>
m
H . Kleinpoppen and H . Hamdy
226
X
7
0'
O(E-E,,,)
/ /
dE)
, c
'0
FIG.2. Scheme of electron-photon coincidence experiments. Incoming electrons (e) collide with atoms (A) and are inelastically scattered (e') at angle Be. Photons emitted from excited atoms are observed at azimuthal angle = 90". The electric vectors E , and Ex of the photons and their phase difference can be measured from a Stokes parameter polarisation analysis. For 'S --t 'P transitions the excitation amplitudes for the magnetic substates refer to the z direction (fo) and the x direction (,,hfl).
+,
with I$,) as the angular part of the wave function for the magnetic sublevel of the excited state and f, as a relevant amplitude specified by the magnetic quantum number m. This state vector can be applied in the dipole matrix element for the 'P --t ' S transition, where the amplitude f, = f,(E, 6,) is a function of the electron energy E and the electron scattering angle 6,. From this dipole matrix element follows an angular correlation formula (Macek and Jaecks, 1971; Eminyan et al., 1973, 1974) for observing the 'P, + 'So photon in coincidence with the inelastically scattered electron. Before we write this formula, we first connect the differential excitation cross section with the excitation amplitudes and their phase difference as follows:
crf& 6) = ($(He,
'P11I$(He, 'PI)>
as a differential excitation cross section,
+ 21fi12 = + 2 0 ,
(1) with uo = lfOl2 and u1 = lf1l2 as partial cross sections. We note that from reflection invariance in the scattering plane follows fl = -f-'. By setting fo d = lfo12
00
HOW PERFECT ARE COMPLETE ATOMIC COLLISION EXPERIMENTS?
227
real, fi may have a phase x so that fl = (f1leix; defining the so-called 1parameter by A = Ifo12/cr the angular correlation function becomes N ~= ,A sin28, ~
+ (1 - A) cos20, - JlfFTj
sin 8, cos
ey cos x
(2)
for a fixed electron scattering 8, angle and for observing the photons in the scattering plane at an angle Oy. An equivalent expression that makes the interference effect more transparent is given by
The physical picture following from this description of the ' S + 'P excitation implies that two coherent oscillators are excited parallel to the zaxis (amplitude f o in Fig. 2) and parallel to the x-axis (amplitude @f l in Fig. 2) in the scattering plane. During the past two decades many research groups have studied electronphoton coincidences from electron impact excitation of atoms. The most comprehensively studied sample is the excitation of the helium singlet 'P state. In this connection it is of importance to realise that the angular correlation parameters 1 and x are much more sensitive test parameters than differential and total cross sections since the latter quantities are averages over the more detailed angular correlation quantities. Before we discuss examples of the results for angular and polarisation correlation parameters in comparison to theoretical predictions we present further arguments and an experimental proof for the model of coherent excitation of the 'P, state of helium. We will also report further ways of interpreting the results of the coherence parameters. Standage and Kleinpoppen (1976) tested the model of coherent impact excitation of the He 3 'P, state as follows. The pure state vector /$('PI)) that is represented by a linear superposition of the magnetic substates Ilrn,) will "decay" into a 2lS, state with the emission of a photon hv. If the decay process conserves the coherence in the lower state both the atom and the photon should be in pure states. The decay process can then be described by the reaction IIPl)
-+
I'S0)lhv)
A critical test of these assumptions can be based upon a study of the state of the photon from the 'P + 'S transition. Being in a pure state the photon radiation is expected to be completely coherent. The coherence properties of the photon radiation can be measured by means of the classical Stokes parameters. In an electron-photon coincidence experiment, the Stokes
H . Kleinpoppen and H . Hamdy
228
parameters are related to polarised intensity components with reference to the scattering plane as follows: I = p l l + p22 = 1 (total intensity, normalized to unity) PI
= p i 1 - p 2 2 = I ( W )- I(90")
P2 = P, p1
-p21
= 1(45")-1(135")
(4)
= i(p21- p12) = I(RHC) - I(LHC)
. . are the components of the photon density matrix =
ri1p12)3 P21P22
I(oL),I(RHC) and I(LHC) are the linearly or circularly polarised components of observed photons (angle OL with reference to the beam direction of the incoming electrons; see Fig. 2). Criteria for coherence of the photon radiation (Born and Wolf, 1976) are the vector polarisation IPI = (IPi12 + IP212 -I- IP312)1/2
(5)
and the coherence correlation factor
with 1p1 as degree of coherence and p as effective phase (i.e., the phase between two orthogonal light vectors). Complete coherence is valid for IPI = 1p1 = 1. Such a complete coherence of the excitation-deexcitation l'S, + 3'P1 -+2'S, of helium could be verified by measuring the Stokes parameters of the helium I = 5016 A line radiation. Within the error bars the total polarisation IPI and the coherence correlation factor were unity, conjirming the model of completely coherent excitation of the 3'P, state. It could also be shown by Standage and Kleinpoppen (1976) that the phase difference x between the two excitation amplitudes fo and fi is identical to the macroscopically measured phase difference p of the linearly polarised components parallel to the z and x directions of the coincident photons. In other words, the identity x = p manifests that a phase diference of an atomic excitation process is directly identical to a macroscopically measurable phase diflerence of light radiation. Information on target parameters can be obtained from the angular correlation data. A complete analysis of orbital angular momentum transfer to the atom can be obtained from the knowledge of I and x data. For example, it can be shown (Eminyan et al., 1973,1974;Kleinpoppen, 1974) that
HOW PERFECT ARE COMPLETE ATOMIC COLLISION EXPERIMENTS?
229
the expectation values for the orbital angular momenta of the excited ‘P state,
x with respect to i = x,y , z directions; e.g., (L,) = 2[A(1- 1)]1’2sin x = ( L , ) = 0,
are calculable from 1 and
(L,)
-
(8)
(in units of h). It also follows (in units of h z )
(L2)
= 1,
(L;)
=
(Li) = 1-A
1,
(9)
+
with (15’) = L(L 1) = 2 for L = 1 (in units of h’). Another method of interpreting the angular correlation data is based upon the anisotropic population of the magnetic substates of the excited state. Fano and Macek (1973) connected the anisotropic population of the magnetic substates with alignment tensors (AC”’)and an orientation vector (Oc‘”)of the excited atom, which can be calculated from A and x or from the Stokes parameters P,, P, and P, (Standage and Kleinpoppen, 1976): A?’
= $(3L,
-
A?? = +(L,L,
L) = (1 - 31)/2 = -(1
+ L,L,)
=
[A( 1 - A)]
+ 3 P,)/4 cos x = - P2/2
ACOl
- r1( L : - L ; ) = g1 - 1) = (PI - 1)/4
OCOl 1-
-1 - 2(Ly) =
2+
(10)
-P,P
Following Blum and Kleinpoppen (1979), the equations describing electron-photon angular correlations become more transparent if an alternative set of parameters is used, namely, “state multipoles” or “statistical tensors,” which contain the same information as the relevant density matrix element describing the excitation process. For a detailed description of the state multipoles we refer to the above-referenced paper and give only the connection of the angular correlation parameters 1and x and the differential excitation cross section a of P state excitation to the relevant state multipoles: (~(1):~)
=
-iaJmsinx,
( ~ ( 1 ) 2 + 2 ) =+(A- 1) (1 1)
(T(1);l) = - o J m c o s x ,
(T(1);J
1 =
fi
~
a(1 - 31)
As can be seen from these equations, the state multipoles
<
7-(1);2),
( T( 1)A >,
( m)AJ
230
H. Kleinpoppen and H. Hamdy
are related to the Fano-Macek alignment tensors and the (T(l)Tl) to the orientation vector. By extracting the Stokes parameters from Eqs. (lo),we obtain the following relations to the quantities 1and x:
P,
= 21- 1
P,
=
-2[A(l
cos
- A)]1/2
P, = 2[1(1 -
sin
x
x
(124
These equations are correct for positive electron scattering angles 8, since the definition of x (followed by Eq. (1)) implicitly assumes the azimuthal angle 4e = 0. The question arises whether Eqs. (12) remain unchanged or if they have to be modified for negative scattering angles (O,, 4, = n). The general situation with Cp, # 0 can be related to the situation with q5e = 0 by a rotation around z by the angle -4c In terms of the electron momenta Po before and P, after the collision, the scattering amplitudes can be written as matrix elements of the interaction 7; which is invariant under rotations: R+(-4e)z
+ T W-4L
=T
with R and R+ as rotation operator and its adjoint, respectively. Thus, frnI(E1, O e , 4 e 5
Ed
= frn,(Pl, Po) =
= ( m l P ~R + ( GCpe)zT R ( = eirn19efml(El, 8,,
$e
<m,P,TPo)
4e)zPO)
= eim"e<m,6
= 0, Eo) = Ifrnl(El,d,,
Cp,
=O)l TISO)
(13)
= 0, E,)lei(xtrni4)
which results in
x(4e)
= X+m,4e
(14)
In the case of n 'P states of He, m, is 1 and 4, = O(n)for positive (negative) scattering angles. Since the photon detection coordinates are not affected by the transition from positive to negative scattering angles, the Stokes parameter equations (4) would in fact return x(Cpe = n) for negative scattering angles; i.e., an apparent shift of x by n. In order to obtain the scattering parameters in line with the original definition of the phase x(8,) one has to replace x in the Stokes parameter equations (12), following the equation:
P, = 21- 1 P, = - 2[A( 1 - A)]
(15 4 1/2
cos(x + 4,)
P, = 2[A( 1 - A)] 1 / 2 sin(x + 4,)
(193)
(15c)
HOW PERFECT ARE COMPLETE ATOMIC COLLISION EXPERIMENTS?
23 1
with #e = O(n) for positive (negative) scattering angles (Beyer et al., 1985, 1988). As an example we refer to the Stokes parameter analysis for the excitationdeexcitation 1 'So + 3 'P, -+ 2 'So of helium by electron impact (Figs. 3-5). The results in these figures are compared with other known polarisation correlation data for angles below Ic),I = 40". Contrary to the behaviour reported by Standage and Kleinpoppen (1976), but in line with Eq. (12b), it is found that P, changes sign between positive and negative scattering angles. The sign of the P, values of Standage and Kleinpoppen (1976) for negative scattering angles has thus been altered in Fig. 4, and the values are then in agreement with the present results. Although there is some spread of the measured values, most agree within the statistical uncertainties (one standard deviation). It would certainly be advantageous to reduce the errors much further; but in view of the unfavourable branching ratio from 3 'P to 2 'S that results in very low signal levels, long measuring times would be required. The measured Stokes parameters may be used to calculate the total polarisation P = (P: Pt P:)'", which is expected to be 1 as shown experimentally by Standage and Kleinpoppen (1976). While the majority of
+ +
1.0
Scattering ongle (deg)
FIG. 3. Stokes parameter P I , related to linear polariser angles 4 = O", 9O", for 3 'P-state excitation of He at 80eV for positive and negative electron scattering angles: 0, present results; 0, Standage and Kleinpoppen (1976); A, Ibraheim el al. (1985) and lbraheim (1986). The full curve represents calculations by Madison (1979, and private communication, 1982).
+
FIG.4. Same as Fig. 2, but for Stokes parameter P,, related to linear polariser angles = 45", 135". Note that the sign of the results by Standage and Kleinpoppen (1976) for negative scattering angles has been altered.
FIG.5. Same as Fig. 3, but Stokes parameter P3,related to circular polarisation. al. (1986, 1987).
232
.,
Beijers et
HOW PERFECT ARE COMPLETE ATOMIC COLLISION EXPERIMENTS? 233
the present points individually would not be considered inconsistent with P = 1, the points collectively fall below P = 1, indicating that the polarisation is reduced by an unaccounted systematic effect. However, this is unlikely to alter the general shape of the polarisation curves in Figs. 3-5 and, in any case, does not affect the symmetry behaviour of the Stokes parameters. The theoretical curves have been caIculated by Madison (1979; private communication, 1982). Apart from details they represent the behaviour of the experimental results fairly well. Different theoretical models have been discussed by Beijers et al. (1987). Figure 6 shows the scattering parameters II and x together with other results based on polarisation correlation measurements. The present values of x represent the average of the results obtained from P, and P,. The various measurements and the theory of Madison (1979) agree reasonably well. It may be noted that the theoretical value of x does not tend to zero for zero scattering angle and that there is no sudden change of x when 8, is varied through zero. This is supported by the measured x values for the smallest scattering angles. Measurements of x at 8, = 0 are not possible since f i should be zero there so that x is not defined. We mention at this point that, under certain circumstances, effects due to
-40
I '
-30
-20
-10
0
'
01
x)
20
30
LO
1
FIG. 6. The 1 and y, parameters for 3'P-state excitation of He at 80eV for positive and negative scattering angles. Symbols are as in Figs. 3-5.
234
H . Kleinpoppen and H . Hamdy
the finite size of the interaction region may play a significant role because they can distort the coherence parameters extracted from experiments. This type of disturbing instrumental effect has been surprisingly significant in the superelastic scattering of electrons from laser-excited 13'Ba(. . .6s 6p, 'PI) atoms (Register et al., 1983; Zenter et al., 1990). Superelastic electron-impact deexcitation of atoms studied by optical pumping (Hertel and Stoll, 1974) is the time-inverse process to the inelastic electron-impact excitation of atoms studied by the electron-photon coincidence technique. Further careful studies of the finite scattering volume on the determination of coherence parameters have been carried out by McConkey et al. (1990) and van der Burgt et al. (1991). It was shown that by keeping the interaction region and the apertures for the detection of scattered electrons and photons sufficiently small, the finite size effect may be neglected for the measurement of the three photon Stokes parameters P,, P,, P, (corrections may only be necessary for small electron scattering angles 6,) measured perpendicular to the scattering plane (Fig. 2). However, particular care is required for measurements of photon polarisations in the scattering plane (e.g., Stokes parameter P4)for which the finite size effect appears to be a more serious disturbance. We now discuss some interesting aspects of the angular dependence of angular momentum transfer ( L , ) = -P, for the n S + n" P + n'S excitation-deexcitation with special reference to the helium atom (Figs. 7 and 8). We are following the discussions of Csanak and Cartwright (1988a,b) and
FIG.7. Data for (L,) as a function of the electron scattering angle for the excitation of the 3'P, of helium at 80eV. Experimental data of Eminyan et al. (1975), Crowe et a/. (1981), McAdams and Williams (1982);theoretical data of first-order many-body theory after Csanak and Cartwright (1988a).
HOW PERFECT ARE COMPLETE ATOMIC COLLISION EXPERIMENTS?
235
I .o 0.8 0.6 I
c
L
:.
04
0.2
a I
n
o
:
-02
a A
J ' V
-0.4 -0.6 -0.8
-1.0
1
1
FIG.8. Angular momenta (Ll)transferred to the excited 2 'P, state of He by electrons of 81.6eV impact energy as predicted by FOMBT (Csanak and Cartwright, 1988a) I, = 0, l , . . .,Imax refer to the incident electron partial waves that were "distorted in the calculation.
Fano collaborators (Bohn and Fano, 1990; Pan and Chakravorty, 1990; Fano and Pan, 1991). A typical result for (L,) in the excitation process 1 'So --* 3 'PI + 2 ' S o is displayed in Fig. 7, which combines various experimental data and the theoretical first-order many-body theory (FOMBT) of Csanak and Cartwright (1988a).Several efforts have been undertaken (e.g., by Steph and Golden, 1980; Kohmoto and Fano, 1981; Beyer et al., 1982; Anderson et al., 1984) to understand the observed behaviour of (L,) on the basis of a semiclassical model by attributing the positive hump in (L,) (for electron scattering angles between 0" and 60")to attractive polarisation forces and the negative hump (between 60" and 180")to repulsive forces (presumable electron-electron correlations). However, Csanak and Cartwright argued that this interpretation cannot be considered correct since their FOMBT does not include any polarisation potential, and the effective electron-atom interaction potential included in FOMBT is attractive for all electron-nucleus distances. Following this Madison et al. (1986)suggested an interpretation based upon a distortion of the incident electron partial waves within the FOMBT. Figure 8 demonstrates their partial wave analysis of ( L , ) by indicating the distortion of an increasing number of incident electron partial waves with 1, (orbital momentum quantum number) and the associated scattered waves (i.e., scattered partial waves with 1, = 1, f 1). The analysis shows highly interesting results:
236
H . Kleinpoppen and H . Hamdy
(1) The distortion of the I , = 0, 1 incident partial waves (and their associated scattered partial waves) contribute significantly to the positive hump in (L,). (2) By distorting the I , suddenly.
= 2 incident
partial wave a large negative hum occurs
(3) Inclusion of additional partial waves does not radically change this interpretation. (4) The positive hump is the result of quantum mechanical interference of the partial waves I , = 0 and 1, = 1. In this connection it is interesting to note that the Born and Glauber approximations predict no orbital angular momentum transfer; i.e., (L,) = 0 (Eminyan et al., 1974). The Born and Glauber approximation reflects diffraction effects upon the incident and scattered electron waves. As pointed out by Csanak and Cartwright (1988?) the whole angular behaviour of (L,) is a structure due to a quantum mechanical interference effect among the electron partial waves without any background. This is a remarkable kind of a pure Ramsauer-Townsend efect without background as seen in the structure of the orbital angular momentum transfer (L,) in the electron impact excitation of P states. It is also interesting to note that this interference phenomenon holds for energies up to 500 eV, which is due to the distortion of the 1, = 0 and 1 electron waves. In addition it could be verified by applying the FOMBT that (L,) is not much affected by the principal quantum n (n = 2-8) of the excited P states. Large values of (L,) were also measured and calculated for the pure exchange excitation 1 ‘So n 3 P 4 n’ 3S,. The FOMBT shows reasonable agreement with the limited experimental data. Another aspect with regard to the analysis of extracting the dynamics from “complete” collision experiments is based upon finding out to what degree or proportion the various partial electron waves contribute to the excitation process. Following the series of papers by Lee (1986, 1987, 1989), Fano and collaborators (Bohn and Fano, 1990; Fano and Pan, 1991; Pan and Chakravorty, 1990) discussed the most detailed experimentally studied ‘PI excitation of helium. Using the earlier amplitudes fo and fl for the differential cross section o(E,e,) = Ifol’ + 21fil’, one can express it as
-
fi
,/m.
with lao12= 1 and IuJ2 = gl -A) or laOl = and lal[ = By writing these probability amplitudes as ( L BM B kJla(L, M Ak i ) , with ki and k, as the electron wave vectors for the incoming and scattered electrons, LAand
HOW PERFECT ARE COMPLETE ATOMIC COLLISION EXPERIMENTS?
237
L B as the orbital angular momenta of the atom before and after the collisional excitation and M A and ME as their magnetic components. These amplitudes can be transformed so that dynamical parameters and purely geometric factors (of scattering angles and orbital quantum numbers of the projectile) can be separated from each other as follows:
(16)
(LBolcllbl,>
The dynamical parameter G can be expressed by taking into account the orbital momentum of the incoming (1,) and scattered electron (la) with
+
( 1 OIG(1, 1 I , )
=
4n3/2 [3(21, 2)]1/2 *
(1 OIGII, - 1 1,) =
[(I,
+ + l)1/2Ao(10+ 1) - (21, + 4)1’2A1(10+ l)]
(17)
4Ir3I2 [3(21, - 1)]1’2 *
[l;’2Ao(1.
-
1)
+ (21, - 2)”2A,(1, - I)]
with
As follows from these equations and by knowing the amplitudes aHB(O,)over the full integration limits allows one to calculate the preceding dynamical parameters G. Experimental polarisation correlation parameters for electron impact excitation of the ‘P, state of helium have been used by Pan and Chakravorty (1990) for such calculations. The two different but interconvertible parametrizations {A(O,), ~ ( 0 , ) )and {LL(O,),y(0,)) of the coherence parameters can be related to the earlier probability amplitudes as follows (by choosing ao(O,) real): ao(@ =
and
Jm
(19)
238
H . Kleinpoppen and H . Hamdy
Pan and Chakravorty critically checked the experimentally available helium 'P1 data from various experimental groups (Steph and Golden, 1980; Hollywood et al., 1979; Eminyan et al., 1974; Slevin et al., 1980) to calculate the uM, and the dynamical parameters G (Figs. 9 and 10). However, there remain the following problems for this procedure. First, the experimental values of A and x data reported by Steph and Golden (1980) differ as much as 30-40% in the angular scattering range 70-1 10" from those of Hollywood et al. (1979) and Slevin et al. (1980). This results in some uncertainties in the probability amplitudes and the dynamical G parameters; however, since the two data sets of the latter two references agree fairly well with each other, they were given preference for their analysis for the dynamical parameters. In addition the error bars of these data are smaller than those of the ones of Steph and Golden (1980). The second problem is that experimental data for the scattering angles (1 30- 180") are not available. An extrapolation to these larger scattering angles remains ambiguous to an unknown degree; however, the general trend of the amplitudes aM,(0)into this scattering angular range does not imply any significant error. The dependence of the probability amplitudes a,,(@ was calculated by Pan and Chakravorty (1990) from the experimental parameters {x(e), A(@} or {Ll(e),y(e)>of the two preferred experimental data sets by using Eqs. (19), (20),(21) and (22) and is presented in Fig. 9 in comparison to predictions of the first Born approximation aE(0) and to the departures a$, between the experimental values aMs and the Born data a$; i.e., a M , - a ~ , .The actual dependence of ao(8)shows a significant variation that seems to be obscure so far with regard to an understanding or explanation of dynamical physical processes. As has been argued by Pan and Chakravorty, Im(a,,) is a generalised Hilbert transfer of Re(a,,J and accordingly roughly proportional to the derivative of Re@,,) that has a minimum at 6 60" where Im(a,,) goes through zero. From the relation laOl2 2(IRe(aJ2 IIm(a,)12}= 1 it follows that laoI2approaches unity where both IRe(al)12and IIm(a,12are small as seen in Figs. 9(b) and 9(c). It follows from Eq. (22b) that Im(a,) is directly proportional to ILL\, and it also determines the sign of L,. Both L and
+
-+
-1 .o'
0
I
60
120
180
1.0-
-
-0.5-
..-....._.. .,.*.
-1.o 0
60
120
180
(c)
-1 .o
3
60 Scattering Angle
120
180
(deg )
FIG.9. The values of the probability amplitudes a,,(O) of the experimental and theoretical Born approximation; ---data as discussed in the text: (a) a,(@, -; a:(@), ---, at(0) = no(@- a:(@; (b) Re[a,(O)], -; Re[a?(O)], ---; Re[aT(O)], ----. (c) Im[a,(8)] = Im[a:(O)], -.
239
240
H . Kleinpoppen and H . Hamdy
-
0.01 0
2
4
6
6
1
0
4,
+
+
FIG.10. Dynamical parameter moduli I( 1 OIG(l)lbl,Jl for e(80 eV) He(lsz) + e’ He(1s 2p) as function of la. Plotted are values lGBornlcalculated in the Born approximation and the for 1, = 1, + 1 (“parity unfavoured”). “residue” lGfe*1= IG-GBo‘”I. --e IG“*~, -+-IGB0‘”I, +IGreSll,-+-IGBarnl, for I , = 1, - 1 (“parity favoured”). Note how IGBornldecays more slowly than IG‘”“l and how lGresl decays roughly linearly on the logarithmic scale, reaching at 1, 6 about 10%of its peak value, a fraction comparable to its experimental error. (Adapted from Hollywood et al., 1979; Eminyan et ol., 1974; S h i n et a/. (1980);after Fano and Pan, 1991.
-
Im(a,) vanish unleks the charge cloud of the excited ‘PI state losks its cylindrical symmetry about the momentum transfer K,-k, of the electron; this is a characteristic of the first Born approximation for which Im(a,) = 0 and x = 0 are valid. Some features of the curves in Fig. 9 can be understood. At the forward and backward scattering angles the amplitudes become a,(O, = 0” and 180”) = 1 and a,(O, = 0” and 180”) = 0; this, of course, follows from the theories of impact polarisation (e.g., Percival and Seaton, 1958) since only the magnetic substate m, = 0 of the He ‘P, state is excited because no angular momentum can be transferred along the z axis. Since forward scattering is favored, most a,,(O) values are in the O < 90” range. The lack of measurements at small and particularly at large scattering angles causes an ambiguity for the fitting of a,,(@) in these scattering targets. Although the values of uM,(0) remain ambiguous to a certain degree at small and large scattering angles, Fano and collaborators considered it a minor effect on the calculation of the dynamical parameters ( L BOIGRIIbI,), except possibly for large orbital angular momentum (la, l b ) . These dynamical parameters were calculated with the “residue” amplitudes a&, = a, - uL, according to Eqs. (16) to (18). Figure 10 displays the moduli lGRl = ( 10IGRllb& 1,1,)1 versus I, for the 2 ‘P excitation of helium. The two types of curves with regard to 1, k 1 imply a “favoured” case
HOW PERFECT ARE COMPLETE ATOMIC COLLISION EXPERIMENTS?
+
241
I, = 1, - 1 and an “unfavoured” one I, = I, 1. As expected, both plots for these cases decay rapidly as I, increases. The nearly linear decay of lCRl for 1, 5 6, on the logarithmic scale, implies an approximate law IGRJa e’”’” with I, 1.5 which is approximately equal to the product of the incident electron’s wave number and the target atom’s initial radius. The projectile’s kinetic energy is larger by approximately a factor of three than that of the target $, and accordingly, penetration of electrons, which is consistent with 1, the incoming electron into the atom is not so much affected by electron exchange forces but favoured by the nuclear attraction. As seen in Fig. 10, lGBl decays less sharply as the first Born approximation. While increasing numbers of studies on complete analysis of the scattering amplitudes describing atomic collision processes are underway, the selection of appropriate atomic candidates is governed primarily by experimental feasibilities and complexities and also by theoretical approximations. For example, scattering of electrons by light one-electron atoms requires sets of amplitudes (Kleinpoppen, 1971) that have to be composed -of coherent superpositions of the Coulomb direct amplitude (e.g., direct amplitude f for elastic electron scattering, electron impact direct excitation amplitudes fo and fi for 2P excitation) and the exchange interaction amplitude (e.g., exchange amplitude g for elastic electron scattering, electron impact exchange excitation amplitudes go and g1 for 2P excitation). In principle a complete analysis of electron scattering by one-electron atoms requires a combination of electron spin and electron-photon coincidence techniques (or equivalent laser-assisted superelastic electron scattering). We hope this combination of technologies will be in a satisfactory state during this or the next decades, particularly for complete analysis of electron-atomic hydrogen scattering. Steps or approaches for one-electron atoms are already promising, and we refer to reports of such efforts (e.g., Raith, 1988; McClelland et al., 1989; Lorentz et al., 1991; Scholten and Teubner, 1991; Hanne, 1992; Kessler, 1991; Buckhari, 1991; Nickisch et al., 1992). Another line of recent developments is studies of rare gas and heavy twoelectron atoms. Particularly noticeable are elastic scattering of polarised electrons from mercury and heavy rare gas atoms in which the spin and scattering observables can be connected to the scattering amplitudes h = lhleiyl and k = IkleiY2; these amplitudes are related to collisions with spinpolarised electrons whereby the spin direction of the scattered electron is either unchanged (direct amplitude h) or flipped into opposite direction (spin flip amplitude k). While the direct scattering amplitude h approaches the usual representation in the partial-wave expansion for neglecting spin-orbit interaction in the scattering process, the spin-flip amplitude depends directly on the strength of the spin-orbit interaction for Ith partial wave. As an example Fig. 11 shows experimental and theoretical data for lhl, Ikl and their
-
-
1
270' 180'
Y1 -Y2
90'
2
0
L I
-
1
I
I
1
1
0.10
1
,t 0.05
IkllU,
0 FIG.11. Moduli of amplitudes Ih/ and Jkland differences y1-y2 between these amplitudes for elastic scattering of polarised electrons on xenon at 100 eV as a function of the scattering angle. Experimental data with error bars from Berger and Kessler (1986); for comparison are various theoretical approximations: ---- McEachran and Staufler (1986), --- Haberland et ~ l(1986) . and -Awe et al.(1983). The data for the amplitudes are in units of the Bohr radius a,; they are calibrated to measured differential cross sections u = lhI2 + Ikl'.
242
HOW PERFECT ARE COMPLETE ATOMIC COLLISION EXPERIMENTS?
243
phase differences y,-y2 for elastic electron-xenon scattering. It is notable in this example that the modulus of the direct scattering amplitude is at least one order of magnitude larger than the spin-flip amplitude. Other types of recent collision experiments with polarised electrons scattered on complex unpolarised atoms have been summarised by Hanne (1992);experimental data were obtained for left-right scattering asymmetries of initially polarised electrons and electron-photon coincidences with polarised projectile electrons. Valuable information on coherence parameters, spin-exchange effects and charge cloud distributions of excited states were extracted from such experiments. Photon Stokes parameters P,, P, and P, of photons observed perpendicular to the scattering plane and the photon Stokes parameter P,, the linear polarisation with reference to and observation in the scattering plane, have recently been reported for resonance transitions from the excitation of the spin-orbit coupled states (2 2P,,2)ns[1/21; (“singlet” ‘P, state) and (2P3,2)ns[3/2]; (“triplet” 3P, states) with n = 3,4, 5, 6 in the heavy noble-gas atoms: Ne, Ar, Kr and Xe (van der Burgt et al., 1991; Carr et al., 1991; Martus and Becker, 1989; Martus et al., 1991; Becker et al., 1992; Murray et al., 1990; Nishimura et al., 1986). If excitation of these states is carried out with unpolarised projectile electrons and without spin analysis of the inelastic electrons, four independent parameters (A, x, cos E , cos A or alternatively y, PlinrL,, poo or the FanoMacek alignment and orientation parameters) can be used to describe the electron-photon coincidence experiments with these atoms (Blum et al., 1980; da Paixao et al., 1980; Anderson et al., 1988; Fano and Macek, 1973). The analysis of the coincidence data from two different geometries determines these parameters as first verified experimentally by Zaidi et al. (1980) and McGregor et al. (1982). While the preceding independent parameters, which were extracted from polarisation correlation or angular correlation measurements, are highly important for impact excitation studies of heavy rare gases, we are still away from the kind of dynamical analysis of orbital angular momenta involved in the excitation process as described above for the helium ‘P, excitation. Characterising the quality of heavy rare-gas coherence data, it can be observed that certain parameters are more sensitive to the detailed dynamics of the collision process than others. While the alignment angle y appears to be a rather insensitive parameter, the parameters J E and ~ IAl show substantial variations as a function of the scattering angle for the Ar ‘PI excitation at various scattering energies (da Paixao, Padial and Csanak, private communication; Blum and Kleinpoppen, 1983). The parameter poo, which accounts for the relative height of the charge cloud of the excited state perpendicular to the scattering plane at the origin, is very sensitive to instrumental or geometric effects. For a more comprehensive analysis and understanding of heavy noble-gas excitation, it would be helpful to have
244
H . Kleinpoppen and H . Hamdy
complete data of the excitation of both fine structure states from all four heavy noble gases. While many applications of the electron-photon coincidence technique and the laser-assisted superelastic electron scattering technique have been reported for helium, for one-electron atoms and for heavy rare-gas atoms, data on more complex atoms are just in the process of being reported. The detailed study of excitation processes of all elements of the periodic table is, of course, a gigantic enterprise for generations of physicists; however, studies of heavier one- and two-electron atoms are already giving information on how relativistic contributions, spin effects, electric dipole polarisibilities and electron-electron correlations make important contributions to the overall excitation process. While the excitation studies of hydrogen and the alkali metal atoms are complicated by the (not yet achieved!) separation of direct and exchange interaction through spin effects (as mentioned earlier) the excitation of helium and the earth-alkaline metal atoms can be fully investigated for a complete dynamical analysis, since the direct exchange interactions are not separable and accordingly they coherently interfere with each other. Studies of electron-impact excitation of alkaline-earth metal atoms have recently been reported by various groups (e.g., Law et al., 1991; Hamdy et al., 1992). One interesting trend in the physical characteristics of alkaline-earth metal atoms is related to the considerable variation of electric dipole polarisibilities; for example, the electric dipole polarisibility increases from Ca to Ba by almost a factor of two (Sadlej et al., 1991). Such and other influences in the excitation of these more complicated atoms will certainly lead to interesting effects of complete dynamical analyses of electron and heavy-particle impact excitation. Although we are far away from systematic studies of impact excitation of all alkaline-earth atoms, we show in Fig. 12 an example for electron impact excitation of the first excited ‘PI state of Sr. The dramatic variations of the photon Stokes parameters extracted from electron-photon coincidence data are noticeable in this figure and related to the effects discussed previously. The theory of such excitation processes will have to take into account all possible electron-electron correlations, relativistic effects and possible couplings between the inner and outer electronic shells.
111. Approaches to “Complete” Experiments in Heavy-Particle Atom Collisions and Photoionisation of Atoms Heavy-particle atom collisions include atoms and ions as projectiles. While approaches to complete experiments by applying particle-photon coincidence measurements have indeed been studied extensively, spin experi-
HOW PERFECT ARE COMPLETE ATOMIC COLLISION EXPERIMENTS?
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ments with regard to collisions between heavy-particles and atoms are still quite rare (Lutz, 1992). In comparison to electron-atom collisions the problems for ion-atom and atom-atom collisions are more complicated due to the coupling between the electronic and nuclear motion (particularly in the quasi-molecular collision regime at lower impact energies). Experimental progress, however, is underway for spin-analysed collisions with one-electron ions and atoms. The relevant spin analysis may be described in analogy to electron-atom collisions described by direct and exchange amplitudes and their interference amplitudes or alternatively by singlet and triplet scattering amplitudes. For example, He’ Na( t) collisional excitation of spinpolarised sodium atoms Na( T) into the first excited zP states may result in a circular polarisation of the emitted NaD radiation. In a direct excitation process of the Na 3s electron to the Na 3p state the spin polarisation remains unchanged while in an electron exchange interaction between the Is electron of He’ and the Na 3s electron the spin polarisation of the sodium atom will be reduced, which can be traced in the variation of circular polarisation of the D lines. The experimental data of such experiments show a remarkable variation as a function of the impact energy; this can be interpreted in a quasimolecular approximation with singlet and triplet wave functions describing the spin exchange contribution of the excitation process (Osimitsch et al., 1989; Jitschin et al., 1986). Clearly, promising developments of intense spinpolarised projectile beams of one-electron ions will be necessary to “complete” the spin analysis with spin-polarised one-electron atoms providing the direct and exchange and their interference terms of the excitation process. Progress has already been made in the production of intense spinpolarised Sr ’ion beams of around 10 pA intensity and 30% spin polarisation in the ’S,,, ground state and the 5D metastable excited state (Reihl, 1992). In thermal atom-atom collisions between spin-polarised and unpolarised alkali atoms spin exchange processes have already been analysed and reported in the 1970s (Pritchard et al., 1970; Pritchard and Chu, 1970). Again a full and complete analysis for alkali-alkali atom collisions requires both spin-polarised projectiles and targets of alkali-atoms. Attention should also be drawn to some interesting and relevant spin effects as theoretically predicted for collisions of fast hydrogen atoms incident on hydrogen or alkali atoms (Swenson et al., 1985) and for collisions between spin-polarised bare atomic nuclei with atomic targets (Khalid and Kleinpoppen, 1982). For example the H(2p) excited state formed by spin-polarised protons in charge exchange collisions with atomic targets such as atomic hydrogen can be a critical test of different theories of electron capture. Electrons moving from the one collision partner to the other move through high electric fields, and the problems of adiabatic spin conservation or nonadiabatic spin flip of the particle involved have neither theoretically nor experimentally been dealt with.
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HOW PERFECT ARE COMPLETE ATOMIC COLLISION EXPERIMENTS?
247
While there is almost an infinite number of choices for heavy-particleatom collisions only some “simple” processes such as p + H -,H(2P) p or H’ + He -+ H(2p) He’(1s) with subsequent emission of Lyman-a radiation are indeed very appropriate candidates for “complete” experiments and test cases for basic theoretical approximations. Measurements of the parameter A = a,/(a, + 2 ~ and ~ the ) orientation parameter 0; of the latter process have been carried out by Hippler et al. (1987). While the experimental data for these quantities appear to be in reasonable agreement to theoretical predictions of Macek and Wang (1986) and Fritsch (private communication), the total polarisation of the Stokes parameters IPI is not quite unity (only between 0.6 and 0.8). For this deviation various reasons such as cascade population of the H(2p) state, neutral hydrogen atom contamination of the incoming proton beam and effects from finite target sizes as reported before have been considered. Interesting alignment and orientation effects and left-right scattering asymmetry effects have indeed been observed in electron-impact excitation of the 3 ’PI state (Silim et al., 1985) and in charge transfer processes between protons and sodium atoms (Royer et al., 1988; Finck et al., 1988; Hippler et a!,, 1985; Dowek et at., 1990; Houver el af., 1992). It has been demonstrated that asymmetry effects of the scattered electrons in coincidence with circularly polarised photons from deexcitation of excited ‘P states are directly related to the circular polarisation of the coincident photons (Silim et a/., 1985). Alternatively such scattering asymmetries in charge transfer processes are related to “circular” atomic states produced by the absorption of circularly polarised laser radiation (Houver et al., 1992). New developments of “complete” photoionisation experiments have been reported that extended the method of “complete” analysis by measuring the spin polarisation of photoelectrons (see, e.g., Heinzmann, 1980; Heckenkamp et al., 1986). It has been suggested by Klar and Kleinpoppen (1982) that angular distributions of photoelectrons from aligned or oriented atoms can be described completely by complex matrix elements and their phase differences. This method has first successfully been applied by Siege1 et a!. (1983) for photoionisation of aligned metastable rare-gas atoms. An important subsequent application was reported by Kerling et al. (1990) of laserexcited aligned ytterbium atoms extracting a full set of relative s- and d-wave amplitudes and their phase differences for photoelectrons from the P, Yb(6s ~ P ) ~state. Another approach to a complete photoionisation has been reported by Hausmann et a/. (1988); in their experiment on the photoionisation of atomic magnesium, the angular distributions of the photoelectron and Auger electrons provide full “complete” description in terms of complex matrix elements for continuum s- and d-electrons and their relative phase (or cosine
+
+
248
H . Kleinpoppen and H . Hamdy
of the phase). Other partial stages of developments of complete photoionisation experiments have been reported in connection with alignment measurements of the produced ions in which separately angular distributions of Auger and photoelectrons or the polarisation of fluorescence radiation is determined (e.g., recent paper on Sr and Ca by Hamdy et al., 1991, and review on present and future experiments by Mehlhorn, 1991). We also refer to measurements of alignment parameters of Ar 2p hole states produced by photoionisation in the soft X-ray region (Becker et al., 1988; Becker, 1989). For the first time these authors measured partial photoionisation cross sections and angular distributions of the Ar 2p photolines and their subsequent LMM Auger lines.
IV. Concluding Remarks The development of “complete” atomic collision experiments represents an ideal situation or aim for experimentalists. The basic description of atomic collision processes in terms of appropriate amplitudes and their phases is known to us since the beginning of modern quantum mechanics. The striking results extracted from the physical data of complete collision experiments are particularly connected to quantum mechanical interference phenomena associated with the amplitudes describing the scattering or collisional process. Alternatively, parameters describing the target state after the collision can be derived from the scattering or collisional amplitudes. Combined knowledge of cross sections and collisional amplitudes is the basis of detailed understanding the atomic collision process, which is required as test ground for theoretical approximations. In this chapter we attempted to draw attention to the problems of “complete” collision experiments, particularly with regard to possible disturbing geometric effects and the still existing considerable error bars in some cases. We are, however, on the right track with the development of perfect or complete collision experiments, and we are entering a stage where highly sophisticated theories of the calibre that Sir David Bates has initiated and advanced can be put to severe tests. REFERENCES Anderson, N., Gallagher, J. W., and Hertel, I. V. (1988). Physics Reps. 165 (1-2), 1. Anderson, N.,Hertel, I. V., and Kleinpoppen, H. (1984). J . Phys. B 17,L901-908. Awe, B., Kemper, F., Rosicky, F., and Feder, R. (1983). J. Phys. B 19,603. Bates, D.R. (1978). Physics Reps. 3 X (4), 305. Becker, K., Crowe, A., and McConkey, J. W. (1992). J. Phys. B, to be published.
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Becker, U., Langer, B., Kohler, E., Heiser, F., Kerkhoff, H. G., and Wehlitz, R. (1988). Book of Abstracts, l l r h ICAP, Paris (C. Fabre and D. Delande, eds.), p. V7. Becker, U. (1989).X V I ICPEAC, A I P Conference Proceedings. (A. Dalgarno, R. S. Freund, M. S. Lube11 and T. B. Lucatorto, eds.), p. 162. Bederson, B. (1969). Comments At. Mol. Phys. 1, 71, and 2, 65. Beijers, J. P. M., van den Brink, van Eck, J., and Heidemann, H. C. M. (1986).J. Phys. B 19, L581. Beijers, J. P. M., Madison, D. H., van Eck, J., and Heidemann, H. G. M. (1987).J. Phys. B 20,167. Berger, O., and Kessler, J. (1986). J. Phys. B 19, 3539. Beyer, H.-J., Kleinpoppen, H., McGregor, I., and McIntyre, L. C. (1982). J. Phys. B: A t . Mol. Phys. 15, L545-548. Beyer, H.-J., Blum, K., and Standage, M. C. (1985). Fundamental Processes in Atomic Collision Physics. (H. Kleinpoppen, J. S. Briggs and H. 0.Lutz, eds.), Plenum Press, New York, 134, pp. 573-577. Beyer, H.-J., Blum, K., Silim, H. A,, Standage, M. C., and Kleinpoppen, H. (1988).J. Phys. B: At. Mol. Phys. 21, 2953-2959. Blum, K., and Kleinpoppen, H. (1979). Phys. Rep. 52, 203-261. Blum, K., and Kleinpoppen, H. (1983). Adv. At. Mol. Phys. 19, 188. Blum, K., da Paixao, F. J., and Csanak, G. (1980). J. Phys. B 13, L257. Bohn, J., and Fano, U. (1990). Phys. Rev. A 41, 5953. Born, M., and Wolf, E. (1976). Principle of Optics, Pergamon Press, Oxford. Bukhari, M. A. (1991). Ph.D. thesis, Stirling University. Bullard, E. C., and Massey, H. S. W. (1931). Proceed. Roy. Soc. A 130, 579. van der Burgt, P. J. M., Corr, J. J., and McConkey, J. W. (1991). J. Phys. B 24, 1049. Corr, J. J., van der Burgt, P. J. M., Plessis, P., Khakoo, M. A., Hammond, P., and McConkey, J. W. (1991). J. Phys. B 24, 1069. Crowe, A., King, T. C. F., and Williams, J. F. (1981). J. Phys. B 14, 2309. Crowe, A,, and Rudge, M. R. H. (1988). Comments At. Mol. Phys. 22, 147. Csanak, G., and Cartwright, D. C. (1988a). In: X V t h I C P E A C , Brighton 1987 (H. B. Gilbody, W. R. Newell, F. H. Read, and A. C. H. Smith, eds.), North-Holland. Csanak, G., and Cartwright, D. C. (1988b).In Proceed. Int. Symp. on Correlation and Polarization in Electronic and Atomic Collisions, Belfast 1987 (A. Crowe and M. R. H. Rudge, eds.), World Scientific, p. 31. Csanak, G., and Cartwright, D. C. (1988~).Physics of Electronic and Atomic Collisions (H. B. Gilbody, W. R. Newell, and F. H. Read (eds.), p. 177; and Correlation and Polarization in Electronic and Atomic Collisions (A. Crowe and M. R. H. Rudge, eds.), p. 31. da Paixao, F. J., Padial, N. T., Csanak, G., and Blum, K. (1980). Phys. Rev. Lett. 45, 1164. da Paixao, F. J., Padial, N. T. and Csanak, G. (1984). Phys. Rev. A 30,1697. Dowek, D., Houver, J. C., Pommier, J., Royer, T., Anderson, N., and Palsdottir, B. (1990).Phys. Rev. Lett. 64, 1713. Eminyan, M., MacAdam, K. B., Slevin, J., and Kleinpoppen, H. (1973). Phys. Rev. Lett. 31, 576579. Eminyan, M., MacAdam, K. B., Slevin, J., and Kleinpoppen, H. (1974). J. Phys. B 7 , 1519. Eminyan, M., MacAdam, K. B., Slevin, J., Standage, M. C., and Kleinpoppen, H. (1975).J . Phys. B 8, 2058. Fano, U. (1957). Revs. Mod. Physics 29, 74. Fano, U., and Macek, J. H. (1973). Rev. Mod. Physics 45, 553. Fano, U., and Pan, X. C. (1991). Comments At. Mol. Phys. 26, 203. Finck, K., Wang, Y.,Roller-Lutz, Z., and Lutz, H. 0. (1988). Phys. Rev. A 38, 6115. Haberland, R., Fritsche, L., and Nome, J. (1986). Phys. Rev. A 33, 2305. Hamdy, H., Beyer, H.-J., West, J. B., and Kleinpoppen, H. (1991). J . Phys. B 24, 4957.
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Hamdy, H., Beyer, H.-J., Mahmoud, K. R., Zohny, E. 1. M.,Hassan, G., and Kleinpoppen, H. (1991). XVIIth ICPEAC. Book of Abstracts (I. E. McCarthy, MacGillivray, ??, and M. C. Standage, eds.), Criffith University, Brisbane, p. 131. Hamdy, H., Beyer, H.-J., Mahmoud, K. R., Zohny, E. 1. M., Hassan, G., and Kleinpoppen, H. (1992). Book of Abstracts, 13th ICAP, Munich. Hanne, G. F. (1992). XVIIth ICPEAC, Brisbane, 1991, Adam Hilger, IOP, Bristol. Hausmann, A., Kammerling, H., Kossmann, H., and Schmidt, V. (1988).Phys. Rev. Lett. 61,2669. Heckenkamp, C. H., Schafers, E., Schonhense, G., and Heinzmann, U. (1986).2. Phys. D2,257. Heinzmann, U. (1980). J. Phys. B 13, 4353. Hertel, I. V., and Stoll, W. (1974).J . Phys. B 7, 570. Hippler, R. (1992). Invited paper, ICPEAC Satellite Meeting, Adelaide. Hippler, R., Faust, M., Wolf, R., Kleinpoppen, H., and Lutz, H. 0.(1985). Phys. Reu. A 31, 1399. Hippler, R., Faust, M., Wolf, R., Kleinpoppen, H., and Lutz, H. 0.(1987). Phys. Rev. A 36,4644. Hippler, R., Madeheim, H., Lutz, H. O., Kimura, M. and Lane, N. F. (1989). Phys. Rev. A 40, 3446. Hollywood, M. T., Crowe, A,, and Williams, J. F. (1979). J . Phys. B 12, 819. Houver, J. C., Dowek, D., Richter, C., and Anderson, N. (1992). Phys. Rev. Lett. 68, 122. Ibraheim, K. S. (1986). Unpublished thesis, University of Stirling. Ibraheim, K. S., Beyer, H.-J., and Kleinpoppen, H. (1985). Proc. 2nd Eur. Conf. on Atomic and Molecular Physics (A. E. de Vries and van der Wiel, eds.), Amsterdam Free University Press, Amsterdam, p. 198. Jitschin, W., Osimitsch, S., Reihl, H., Mueller, D. W., Kleinpoppen, H., and Lutz, H. 0.(1986). Phys. Rev. A 34, 3684. Kerling, C., Bowering, N., and Heinzmann, U. (1990). J . Phys. B 23, L629. Kessler, J. (1991). Aduances At. Mol. and Optic. Phys. 27, 81. Klar, H., and Kleinpoppen, H. (1982). J. Phys. B 15, 933. Kleinpoppen, H. (1971). Phys. Rev. A 3, 2015. Kleinpoppen, H. (1974). Proceedings 4th Atomic Physics Conference, Plenum Press, New York, p. 449. Kohmoto, N., and Fano, U. (1981). J. Phys. B 14, L477. Law, M. R.,Houghton, R. K.,andTeubner, P. J. 0.(1991).X V I l t h ICPEAC(1. E. McCarthy,?? MacGillivray and M. C. Standage, eds.), Griffith University, Brisbane, p. 131. Lee, C. (1986). Phys. Rev. A 33, 921 and 34, 959; (1987),36, 66 and 36, 74; (1989), 39, 554. Lorentz, S. R., Scholten, R. E., McClelland, J. J., Kelley, M. H., and Celotta, R. J. (1991). Phys. Rev. Lett. 67, 3761. Lutz, H. 0. (1992). Proceed. 6th Int. Symposium on Correlations, Polarization in Electronic and Atomic Coltisions and ( e . 2 e } Reactions (P. J. 0. Teubner and E. Weigold eds.), to be published. Macek, J., and Jaecks, D. H. (1971). Phys. Rev. A 4, 2288-2300. Macek, J., and Wang, C. (1986). Phys. Rev. A 34, 1787. MacGillivray, ? ?, and Standage, M. C. (1991). Comments At. Mol. Phys. 26, 179. Madison, D. H. (1979). J . Phys. B 12, 3399. Madison, D. H., Csanak, G., and Cartwright, J. (1986). J . Phys. 19, 3361. Martus, K. E., and Becker, K. (1989). J . Phys. B 22, L437. Martus, K. E., Zheng, S.-H., and Becker, K. (1991). Phys. Rev. A 44, 1682. McClelland, J. J., Kelley, M. H., and Celotta, R. J. (1989). Phys. Rev. 40,2321. McConkey, J. W., van der Burgt, P. J. M., Corr, J. J., and Plessis, P. (1990).Proceed. Int. Symp. on Correlation and Polarization in Electronic and Atomic Collisions ( P . A. Neill, K. H. Becker, and M. H. Kelley, eds.), NIST Special Publication 789, p. 1 1 5. McEachran, R. P., and Stauffer, A. D. (1986). f.Phys. B 19,2523.
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McCregor, I., Hils, D., Hippler, R., Malik, N. A,, Williams, J. F., Zaidi, A. A,, and Kleinpoppen, H. (1982). J . Phys. B 15, L41 I. Mehlhorn, W. (1991). In: Today and Tomorrow in Photoionization, Proceed. of U.K./U.S.S.R. Seminar, St. Petersburg (M. Y . Amusia and J. B. West, eds.), SERC publication DL/Sci/R19, p. 15.
Murray, P. D., Cough, S. F., Beill, P. A,, and Crowe, A. (1990). J . Phys. B 23, 2137. Nickich, V., Hegemann, T., and Hanne, G. F. (1992). Zeitschr. 1: Physik, to be published. Nishirnura, H., Danjo, A,, and Takahashi, A. (1986). J . Phys. B 19, L167. Mo, H. O., and Riera, A. (1989). Phys. Rev. A 40, 2958. Osimitsch, H. 0.. Pan, X. C., and Chakratorty, A. (1990). Phys. Rev. A 41, 5962. Percival, I. C., and Seaton, M. J. (1958). Philos. Trans. Roy. Soc. London, A 251, 113. Pritchard, D. E., Carter, G . M., Chu, F. Y., and Kleppner, D. (1970). Phys. Rev. 2, 1922. Pritchard, D. E., and Chu, F. Y. (1970). Phys. Rev. 2, 1932. Raith, W. (1988). In: Fundamental Processes in Atomic Dynamics, Plenum Press, New York, p. 459. Register, D. F., Trajmar, S., Csanak, G., Jensen, S. W., Fineman, M. A., and Poe, R. T. (1983). Phys. Rev. A 28, 151. Reihl, H. (1992). Ph.D. thesis, Bielefeld, and to be published. Royer, T., et al. (1988).2.Phys. D10, 45. Sadlej, A. J., Urban, M., and Gropen, 0. (1991). Phys. Rev. A 44, 5547. Scholten, R. E., and Teubner, P. J. 0. (1991). VIIth I C P E A C , Griffith University, Brisbane, p. 108. Siegel, A,, Ganz, J., Bussert, B., and Hotop, H. (1983). J. Phys. B 16, 2945. Silim, H. A., Beyer, H.-J., and Kleinpoppen, H. (1985). 2nd European Conference on Atomic and Molecular Physics, Amsterdam, Book of Abstracts, p. 299. Slevin, J. A,, and Chwirot, S. (1990). J. Phys. B 23, 165. Slevin, J., Porter, H. Q., Eminyan, M., Defrance, A,, and Vassilov, G. (1980).J. Phys. B 13, 3009. Standage, M. C., and Kleinpoppen, H. (1976). Phys. Rev. Lett. 36, 577. Steph, N. C., and Golden, D. E. (1980). Phys. Rev. A 21, 759-770. Swenson, D. R., Tupa, D., and Anderson, L. W. (1985). J. Phys. B 18,4433. Zaidi, A. A., McCregor, I., and Kleinpoppen, H. (1980). Phys. Rev. Lett. 45, 1168 and 2078. Zenter, P. W., Trajmar, S., and Casanak, G. (1990). Phys. Rev. A 41, 5980.
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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 32
ADIABATIC EXPANSIONS AND NONADIABATIC EFFECTS R . McCARROLL Laboratoir de Dynamique Moleculaire et Atomique Universite Pierre et Marie Curie, Paris, France
D . S. F. CROTHERS Theoretical and Computational Physics Research Division The Department of Applied Mathematics and Theoretical Physics Queen’s University of Bevast Belfast. Northern Ireland
I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Quantum Mechanical Approach . . . . . . . . . . . . . . . . . . . A. Generalities and Definitions . . . . . . . . . . . . . . . . . . B. Born-Oppenheimer Adiabatic States . . . . . . . . . . . . . . C. Eckart Coordinates . . . . . . . . . . . . . . . . . . . . . . D. Hyperspherical Coordinates . . . . . . . . . . . . . . . . . . E. Concluding Remarks on Reaction Coordinates . . . . . . . . . . 111. Semiclassical Formalism. . . . . . . . . . . . . . . . . . . . . . . A. State-Dependent Translation Factors . . . . . . . . . . . . . . B. Common Translation Factors . . . . . . . . . . . . . . . . . . IV. Some Experimental Evidence. . . . . . . . . . . . . . . . . . . . . V. Relevant Work of D. R. Bates (1962-1983) . . . . . . . . . . . . . . VI. Understanding Nonadiabatic Transitions and Effects (1971-1992). . . . . VII. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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I. Introduction The adiabatic approximation, which permits a separation of the fast and slow variables of a dynamical system is widely used in many fields of physics. In classical mechanics, it leads to the notion of adiabatic invariants, used for example by Bates (1982) and Sakimoto (1984) for taking account of molecular 253 Copyright Q 1994 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003832-3
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rotation in slow ion-molecule reactions. In quantum mechanics, the adiabatic approximation was first developed by Born and Oppenheimer (1927) for the separation of the electronic, vibrational and rotational degrees of freedom in molecules. The adiabaticity being associated with the small ratio of the electron to nuclear masses, it is reasonable to expect the Born-Oppenheimer picture of a molecular complex to be a natural representation for describing low-energy atom and ion-atom collisions (Mott and Massey, 1965). In the adiabatic approximation, the first step is to solve the Schrodinger equation for the electron motion assuming the nuclei to be clamped at some fixed internuclear separation R . In this way a series of adiabatic electronic states can be generated for any given R. The eigenenergies E J R ) depend parametrically on R , while the projection An of the electronic angular momentum in the direction of the internuclear axis is conserved. If, as in a collision, R varies, the system will tend to remain adiabatic at slow velocities provided the energy curves E,(R) stay well separated from each other. If, however, some of the energy curves become quasi-degenerate (avoided crossings between states with same A or real crossings with different A), then the Born-Oppenheimer approximation breaks down and transitions from one adiabatic state to another can occur. For a wide variety of ion-atom and atom-atom systems, these quasi-degeneracies tend to be well isolated, so that only a small number of nonadiabatic transitions are probable. There exist many experimental results obtained using photon emission or ion (atom) translational energy spectroscopy in slow ion-atom and atomatom collisions, which confirm the validity of this simple picture. Of these, perhaps the most spectacular are those concerning processes of state selective electron capture involving multiply charged ions in collision with neutral atoms, where the selection rules obey exactly the predictions of the adiabatic approximation (Wilkie et al., 1986; Ciric et al., 1985; Hvelplund et al., 1987). On the other hand, quantitative agreement between theoretically calculated cross sections and experimentation is much more difficult to achieve. Of course, if an order of magnitude estimation is all that is required, models of the Landau-Zener, Demkov or Nikitin type may be adequate (Landau, 1932; Zener, 1932; Rosen and Zener, 1932; Demkov, 1963; Nikitin, 1962). But apart from being restricted to two-state systems and simple analytic representations of the nonadiabatic couplings, these models tend to circumvent (or assume to be unimportant) certain fundamental difficulties associated with the use of an adiabatic representation to describe a collision complex. Basically, there is a problem in matching the boundary conditions of the collision (with moving nuclei) using a basis of a few adiabatic functions (with clamped nuclei). This becomes particularly severe in the case of rearrange-
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ment processes such as electron capture or electron exchange. It should also be observed that a related problem occurs in molecular spectroscopy concerning rotation-vibration states close to the dissociation threshold energy in the Born-Oppenheimer approximation (Macek and Jerjian, 1986). There have been several reviews in recent years on the use of the adiabatic representation to describe the physics of ion (or atom)-atom collisions (Delos, 1981; Kimura and Lane, 1989; McCarroll, 1988; Solov’ev, 1989; to cite only a few). Inevitably, there will be some overlap with these other works, but our main objective in this work is to concentrate on putting the different theoretical approaches in perspective, rather than an exhaustive survey.
11. Quantum Mechanical Approach A. GENERALITIES AND DEFINITIONS
In view of the fact that there is much confusion in the literature between the difference of terminology in the quantum mechanical and semiclassical formulation of nonadiabatic transitions, both derivations will be presented. The confusion stems in part from the choice of independent variables used to derive the semiclassical approximation. These may or may not be identical to the variables used for the adiabatic separation. The physical results are of course independent (within certain limits) of the particular choice of semiclassical variables, but if these are different from the adiabatic variables, the explicit introduction of appropriate phase (or translation) factors is unavoidable. Let us first consider the general formulation of a typical rearrangement process involving the elementary three-body problem consisting of two ionic cores a, b of masses M , and M , and an electron e. Initially the electron is assumed to be bound to center a in state i. The problem is to determine the cross section for the capture of the electron to center b in a bound state5 Atomic units are adopted, with the electron mass, charge and h taken to be unity. Denoting by R the position vector of b with respect to a, and by r,, rb the position vectors of e with respect to a and b, we may note that the kinetic energy operator T (after separation of the center of mass) is separable in each
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of the three Jacobi coordinate systems (R, r), (Ra,ra), (Rb,rb), where r is the position vector of e relative to the center of mass of a, b:
To define the initial and final channel states, the potential energy may be written in either of the following forms:
v=K+q K=Ke w=cb+be I/ = v/ + Wf v/ = be w/= K b + Ke
(5)
(6)
where I/as denotes the interaction potential between a and B (a, B = a, b, e). The initial and final channel states mi,mf are, respectively, eigenfunctions of (T + K) and (T + Vr) with wave vectors ki, kf
mi= exp(ik,.RJ+(r,)
Qf= exp(ikf.Rb)4,(rb)
(7)
and q5i, Cp/, respectively the bound state wave functions of (ae),and of (be)/.It is clear that no one Jacobi coordinate is adequate to describe the collision process, since we require (R,, r,) to define mi and (Rb,rb) to define Of. B. BORN-OPPENHEIMER ADIABATICSTATES Let us now expand Y the eigenfunction of the total Hamiltonian H on the Born-Oppenheimer (BO) adiabatic basis set
where
ADIABATIC EXPANSIONS AND NONADIABATIC EFFECTS
257
Here, we use the conventional BO coordinates (r, R) with R taken to be the slow adiabatic variable. Substitution of Eq. (8) in the Schrodinger equation yields the following set of coupled differential equations (given in matrix notation):
A,
+ 2iP.V + i V * P+ P 2
where Pmn =
(x~IVRIX,)
(1 1)
Note that the derivative VR is carried out with respect to r fixed. The defects of (10) are immediately evident. For example, if X , and xn dissociate to the same center as R -,co and if these two states are connected by a dipole transition, then the matrix elements P,, have a nonvanishing asymptotic limit. It is then not possible to extract a meaningful S-matrix from the solution of (10).Even in the case of a limited basis set, where there are no nonvanishing matrix elements as R + 00, other hidden defects are associated with a nonphysical dependence on the reduced mass of the nuclei. (See the later discussion on translational invariance in the semiclassical approximation.) To overcome the difficulties with the BO adiabatic states, which first became apparent in the pioneering work of Bates et al. (1953), Bates and McCarroll (1958) showed that a formal solution to the problem could be obtained by introducing appropriate electron translation factors (ETF), which allow for a correct matching of the asymptotic boundary conditions. This way of allowing for the effect of translation appears natural in a semiclassical (or impact parameter) description of the collision, where it is assumed a priori that the internuclear distance is some explicit function of time (usually corresponding to rectilinear trajectories of the nuclei). The ETFs of Bates and McCarroll allow for the translation of the bound electron (either in the initial or final state) with respect to an arbitrary time origin and the resulting dynamical equations possess the interesting property of being Galilean invariant. This is not the case in the unmodified BO states. However, in a quantum mechanical description, the introduction of ETFs does not lend itself to practical calculations (at least under conditions where the semiclassical approximation is invalid). An alternative approach to the problem, first suggested by Mittleman (1969), is to use a suitable set of scattering coordinates that can automatically describe both the excitation and rearrangement channels. This idea was more fully developed by Thorson and Delos (1978), who succeeded in finding a relatively simple way to meet these objectives. Their method is easy to implement since it involves
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R. McCarroll and D. S. F . Crothers
nonadiabatic matrix elements that can be calculated by the conventional techniques of quantum chemistry (Gargaud et al., 1987). Subsequently, Robert and Baudon (1986a-c) drew attention to the suitability of Eckart coordinates (Eckart, 1934) for the three-body problem, and they argue convincingly in favour of these coordinates for rearrangement collisions. It is easy to show, however, that in practice the Eckart coordinates are closely linked to the Thorson-Delos system. It would therefore seem appropriate to develop briefly the formulation of the collision dynamics in terms of the Eckart adiabatic states.
C. ECKARTCOORDINATES Introducing the mass-scaled Jacobi coordinates
we form the 2 x 2 Gram matrix G , whose elements are G,,
(a, p = 1, 2)
= R;R,
(13)
The matrix G has two real positive eigenvalues r: and r: with rl > r2 and is diagonalized by a rotation matrix characterized by an angle q. The three-unit vectors ui (i = 1,2,3) defined by rl = rl u1 = cosqR1 + sinqR2 r2 = r2u2 = -sinqR1
+ cosqR2
(14) (15)
u3 = u1 x u2 (16) correspond to the principal axes of inertia of the instanteous solid. The Eckart coordinates are rl, r2, q and a, p, y the Euler angles of the inertial frame Q with respect to a fixed frame. As shown by Eckart, the Hamiltonian can be expressed in the following form: M: M i p:+p:+1+-2-+r2 rl
2M3p C3
where pl, p 2 , p 3 are the momenta conjugate to rlr r2, q:
Mi
=M
.u~
(i = 1, 2, 3)
M being the total angular momentum of the system.
ADIABATIC EXPANSIONS AND NONADIABATIC EFFECTS
259
Using the fact that ma, mb >> 1, it is easily verified that r I and r2 are related to R and r by the approximate relations r:
+ r2 cos’ 6
= pa,bR2
r: = r2 sin26
(19)
where 6 is the angle between R and r. Except when the two nuclei are close together, R and r are of the same order of magnitude, in which case r: >> rt and
Since ui lies almost in the direction of R and since the kinetic energy terms of (17) are separable, it would appear natural to treat rl as a slow variable and rz as a fast variable. The Eckart adiabatic states are then defined as eigenstates of an electronic Hamiltonian Heckobtained by fixing rl and Q . We thus have
where r I is treated as a parameter just as R is treated as a parameter in the BO Hamiltonian. It is of interest to examine the behavior of r1 and r2 in the dissociation limits (ae, b) and (a, be) as R + 00. Using the relation that
which, for small q, gives
and we then have
r-R rz=r-TR R rb In the limit (a, be) as -,0, R
260
R. McCarroll and D. S. F. Crothers
while in the limit (ae, b) as 5 -+ 0, R
r l = &Ra r2 = ra (27) It is thus clear that the adiabatic Eckart states allow for a correct description of the asymptotic states with no need for the introduction of translation factors. Of course there still remains the problem of determining the adiabatic Eckart states and eigenvalues of HEck. Obviously finding the exact eigenfunctions of H E c k is far from a trivial task, but fortunately, these eigenstates, labelled tn(rz; r l ) ,will not differ appreciably from the BO states, so that to a good approximation we may simply assume w 2 ;
r1) = X n k
(28)
With assumption (28) the only real difference from the BO basis is that the Eckart states are defined with rl fixed whereas the BO states are with respect to R fixed. The total wave function may be expanded in the form y(r, R,
=
FEck.n(rdtn(rZ, n
(29)
4, ?;
which, because of relation (28), may be rewritten as
This is identical to the expansion proposed by Thorson and Delos, who express the adiabatic variable in the form
rl =
A (R +
s)
when their switching function is chosen such that
A straightforward (but lengthy) calculation then yields
-
- [A + 2i(P + A)*V+ iV (P + A) + (P + A)’]F(R)
= EF(R)
(33)
where
In accordance with (24) to (26), it can be easily verified that the matrix
ADIABATIC EXPANSIONS AND NONADIABATIC EFFECTS
26 1
+
elements (P,,,,, A,,,,,) vanish in the asymptotic limit and that the spurious isotope effects are eliminated. In reality, the essential property of Eckart coordinates is the hypothesis of r l as the adiabatic variable. Because of relation (28) there is no need to introduce the coordinates r2 explicitly, and as a consequence, the matrix elements P,, and A,,, involve operators that may be simply expressed in terms of the Jacobi coordinates (R,r). The calculation can therefore be carried out by the conventional techniques of quantum chemistry. A partial wave decomposition (Gargaud et al., 1987) can be made in the usual way (bearing in mind that the adiabatic wave functions are defined with respect to a body fixed axis). We obtain two types of coupling matrix elements: modified radial matrix elements of the form that couples states with An = A,
and modified rotational matrix elements of the form that couples states with A,, = A,,, f 1
where (x, y, z ) are the components of r with the z-axis in the direction of R, and the y-axis perpendicular to the classical collision plane. We should stress, however, that the simplifying assumption (28) that allows for the practical implementation of the Eckart coordinates will not be valid in close collisions when r / R (and the rotation angle q) can become large. As shown by Robert and Baudon, it is only when q is small that condition (28) is valid. No simple way of calculating the Eckart adiabatic states for such a case has yet been proposed. However, in the Thorson-Delos formalism, the introduction of an (empirical) switching function in the definition of the reaction coordinates allows one to treat the case when r / R is large.
D. HYPERSPHERICAL COORDINATES The use of hyperspherical coordinates, used successfully by Macek and Jerjian (1986) to describe vibrational states of HD+ close to the dissociation continuum, has been proposed by Macek et al. (1987) as providing an alternative means of bypassing the necessity of translation factors. However, as noted by Robert and Baudon (1986a-c), the electronic part of the Hamiltonian in hyperspherical coordinates includes all rotations (rotation of
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R. McCarroll and D. S . F. Crothers
the system as a whole as well as rotation of the electron) via the generalized angular momentum operator of Smith (1960). On the other hand, both the Eckart and the BO adiabatic states consider separately the electron angular momentum. The result is that the use of hyperspherical coordinates would not seem to be very practical when rotational coupling is likely to be of importance. But in the case where only radial coupling is of importance, as for example in a typical two-state Landau-Zener crossing, Solov’ev and Vinitsky (1985) have shown the hyperspherical approach to have some attractive features (even if the term hyperspherical is not explicitly used in their work). To illustrate the hyperspherical method, we shall follow the approach of Solov’ev and Vinitsky, who assume at the outset that the angular momentum terms of the total Hamiltonian can be neglected. Making the scale transformation r’ = r/R, the Schrodinger equation for Y becomes
where p=(l
+ ; r1t 2 )
(39)
We observe that the radial matrix elements of the form
(the radial derivative of which must be taken with respect to the r‘ constant) will always vanish in the asymptotic limit, even if the dissociating states are connected by a dipole transition. Of course, the scale transformation is not enough since the existence of cross derivative terms in (38) still gives rise to nonvanishing asymptotic matrix elements. But, instead of R, let us now introduce the hyperspherical radius 93, defined by
and making the transformation
ADIABATIC EXPANSIONS AND NONADIABATIC EFFECTS
263
we obtain (after some tedious algebra) the equation
Equation (43) now has no cross derivatives and contains the specific properties that allow for the adiabatic separation of the slow adiabatic variable 93 and the fast variable r’. We may then define hyperspherical adiabatic states as eigenfunctions of
As in the case of the Eckart adiabatic function, the smallness of the ratio l/p ensures the hyperspherical adiabatic functions will differ little from the BO states. We may remark also that when r a / R -+ 0
In the case when r b / R 0, it suffices to replace a by b. The evaluation of the radial coupling matrix (40) with respect to the r’ constant (and not the r constant) is simple in spheroidal coordinates -+
It suffices to calculate the derivative with respect to the 5, q constant. Calculations carried out by Gargaud et al. (1987) for a number of simple two-state systems using (40) and (36) indicate that when the neglect of Coriolis coupling is justified, both the Eckart and hyperspherical approaches give results in close agreement with each other. But it should be recalled that, if Coriolis couplings are present, the hyperspherical approach of Solov’ev and Vinitsky is not satisfactory.
REMARKS ON REACTIONCOORDINATES E. CONCLUDING
In spite of the elegant features of reaction coordinates, which enable bypassing translation factors without the necessity of ad-hoc assumptions, some intrinsic limitations should not be overlooked. Both the Eckart and hyperspherical coordinate systems depend only on kinematic considerations
264
R. McCarroll and D. S. F. Crothers
(position vectors of the interacting particles and their masses). Their introduction allows for the removal of the most serious defects of the BO basis set, but some (perhaps) minor problems remain. As a simple illustration, let us consider an atom in collision with an inert particle such as a neutron. Physically the probability of excitation in this system must be zero just as for a freely translating atom. But had the system been treated by the use of reaction coordinates, we would obtain some nonzero nonadiabatic matrix elements in contradiction with the case of the freely translating atom. For example, at finite R, the Eckart matrix element (36) is nonzero between states connected by a quadrupole transition. Of course, this matrix element is small and it does vanish at large R, but formally its existence indicates that reaction coordinates are not perfect. On the other hand, the Bates-McCarroll translation factor approach, which, to some extent, makes allowance for the interaction potential, does not encounter problems of this kind. Rather than pursue the detailed analysis of reaction coordinates at this stage, it is preferable to postpone further discussion until Section 111, which is devoted to the semiclassical approximation. This involves no sacrifice of rigor, since numerical calculations reveal that any defects of the Eckart (or hyperspherical) coordinates become of practical importance only at those collision energies for which semiclassical methods are perfectly reliable.
111. Semiclassical Formalism For collision energies greater than a few tens of eV/amu, the relative linear momentum of the nuclei is large and the wave function Y(r, R)presents rapid oscillations. Defining a vector k to be in the direction of ki, the incident wave vector, and k2 = 2pE, most of the oscillatory structure of Y can be removed by writing Y(r, R) = exp(ik * R)I,9sc
(48)
Substituting (48) in the Schrodinger equation, we obtain to lowest order the semiclassical or eikonal (impact parameter) equation for $sc: i-= at
He,$sc(r,t )
R
=b
+ vt
(49)
where b is the impact parameter, v ( = k/p) and t the time. Other variants of the semiclassical approximation are possible with, for example, (r, R) replaced by (ra,RA,(rb, R), or ( P - '%, r d i n (48) and (49).
ADIABATIC EXPANSIONS AND NONADIABATIC EFFECTS
The solution of (49) corresponding to the entry channel (ae)i set equal to T + q) is given by @sc,i
+ b (with H,,
- +p’u’t)exp[( - i E i ( c o ) t ]
= 4i(r,)exp(-ipv-r
while the solution corresponding to a given by (Dsc,, = qjJ(rb)exp(- iqv * r
265
(50)
+ (be)’ (with He, set equal to T + V’) is - $ q 2 v 2 t )exp[( - i E f ( c o ) t ]
(51)
where P=
M,
Ma
q = l - p =
Mb
+ Mb
Ma
+ Mb
We may observe, at this stage, that because of the notion of rectilinear trajectories in solving (49) the total (but not differential) cross sections must scale with the velocity v (or the energy/amu). The total cross sections will therefore be independent of Ma and M , (and of p , 4). This translational invariance with respect to p makes for an added constraint in constructing a formally correct theory. To determine the probability Pi, of the transition, we must solve (49) subject to the condition
subject to the condition that
4- 00)
= 6n.i
(57)
The probability PiJ is then given by Pif = Ia/(m)12
Equation (56) suffers from the same defect as Eq. (lo), since the matrix element -(a/at)lx,) will tend to a nonvanishing asymptotic limit under
266
R . McCarroll and D. S . F. Crothers
the same conditions as (1 1). In addition the matrix element depends on p , so that the probability P , may not have the required invariance properties. To remedy the defects of the expansion (55), the obvious solution is to modify the basis set X,by an electron translation factor of the form (59) l, = x, expCif,(r, R)v * r l where f, is some suitable function of r, R and n. The form off, is to some extent arbitrary but asymptotically it must satisfy the following constraints: f,(r,R)
-
-P
if state n dissociates to (ae), + b
(60)
(1 - p)
if state n dissociates to a + (be),
(61)
R-tm
R-tm
If translational invariance of the basis function (59) is also to be ensured then we must also have fn(r’, R) =f& R) + P - P’
(62)
There are two distinct varieties of ETFs: those that depend on the adiabatic state to which they are assigned and those that do not. A. STATE-DEPENDENT TRANSLATION FACTORS
By state-dependent ETFs, we mean those for which the functionf, depends on n. Of these the simplest are of the plane wave type proposed by Bates and McCarroll, where fn=
-P
f, = (1
-
p)
if n dissociates to (ae), + b
(63)
if n dissociates to a + (be),
(64)
Plane wave translation factors (PWTF) are satisfactory only at large R, where the electron is effectively localized around either center a or center b. In the case of adiabatic states subject to avoided curve crossings, where the localization of the electron may change rapidly with R from one center to the other, the physical notion of a PWTF is not clear. (This problem may be alleviated by the introduction of some type of diabatic representation, but a discussion of this point is beyond the scope of this review as there is no absolute definition of what is meant by a diabatic state. See the review of Kimura and Lane, 1989.) There is also a difficulty with (63), (64) for symmetric systems. In order to achieve the correct boundary conditions in that case, the ETFs must be assigned to symmetric and antisymmetric combinations of the gerade and ungerade states (Bates and McCarroll,
ADIABATIC EXPANSIONS AND NONADIABATIC EFFECTS
267
1958). On the other hand, PWTFs do allow correctly for momentum transfer in electron capture processes and are accordingly simply correlated to the basis functions in the two-center atomic expansion (Bates, 1958; Fritsch and Lin, 1982). To introduce more flexibility in the choice off,, Crothers and Todd (1981) allowed f, to vary with R, but not r. In principle f, can be optimized variationally but in practice this involves extensive calculations. The basic problem is that the presence of factors of exp(iv r) in the nonadiabatic matrix elements makes the calculations rather unwieldy. For that reason, there have been few applications to systems other than the one-electron systems H z + and HeHZ+.
-
TRANSLATION FACTORS B. COMMON By allowing fn to depend on both r and R, it is possible to choose f, independent of state n and yet satisfy the constraints (63), (64). The notion of such common translation factors (CTF) was first introduced by Schneidermann and Russek, who proposed for the case (M, = M b , p = 1/2)
where p is some appropriate constant. We observe, however, that while (65) is consistent with the asymptotic conditions (60) and (61) it does not satisfy the condition of translational invariance (62). However, the practical advantage of CTFs over state-dependent ETFs is that the nonadiabatic matrix elements can be easily expressed in terms of standard molecular integrals. Other variants of CTF have been proposed by Levy and Thorson (1969), Vaaben and Taulbjerg (1981), Errea et al. (1982),which attempt to take some account of the nature of the interaction potentials V,, and Ke. These introduce significant corrections when nonadiabatic effects at small internuclear distances are of importance. Of special interest is the CTF of Errea et al.:
f(r, R) =
R r-R R +p2
which, in the particular case of b = 0, (66) takes the form 1 f(r, R) = - r * R
R
(67)
We observe that (67) satisfies the condition of translational invariance (62).
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R . MeCarroll and D. S. F . Crothers
The radial and rotational coupling matrix elements in this case take the simple form, respectively,
; 1 ;t/
(Xm
+
X”)
which are identical to (36) and (37) obtained using the adiabatic Eckart states. We thus see that the CTF (67) is identical to using the Eckart coordinates in a semiclassical approximation. As mentioned earlier, the use of (36) and (37) is not valid in close collisions. Whereas it would appear difficult to calculate the relevant adiabatic Eckart states for R small, we can see that, in the semiclassical approximation, the problem of close collisions can be taken account of by a suitable modification of the CTF (such as taking B # 0 in (66)). Of course, there is a price to pay since translational invariance is sacrificed in any such modification. On the other hand, provided there are no strong nonadiabatic matrix elements with states not included in the expansion, this defect does not appear to be too serious (Errea et al.). However, a full discussion of this point is beyond the scope of this review. Another interesting C T F is the one proposed by Solov’ev and Vinitsky (1985):
r2d
f=-
2R
Then introducing the new scaled coordinate r’ = r/R as in Section II.C, and writing the function we obtain
4
[-
H - i - $(r’,
T)
=0
where
-
H=
1
--
2
A,, dt’
o2 + R2V + w1, + rI2 2 1
Vt
arctg P
We may expand
o = bv
(73) (74)
$, in terms of eigenfunctions of A. If we neglect the angular
ADIABATIC EXPANSIONS AND NONADIABATIC EFFECTS
269
momentum 1, and the quadratic term in u2, these eigenfunctions of are identical to the BO states. However, in solving the eikonal equation (72) we must remember that the nonadiabatic matrix elements (identical to (40)) involving VR are calculated with respect to r’ fixed and these will always vanish as R + co.These results are identical to those obtained using a semiclassical version of the hyperspherical coordinates of Section 1I.C.
IV. Some Experimental Evidence This review would not be complete without some realistic assessment in quantitative terms of the effectiveness of reaction coordinates (or translation factors) in formulating a consistent theory of nonadiabatic processes. There is no scope for an exhaustive analysis, but an attempt will be made to highlight the main conclusions accumulated over the past few years. Although it is difficult to generalize, an adiabatic representation of ionatom or atom-atom collisions can be expected to be satisfactory only at collision energies not exceeding a few keV/amu. Of course, certain selective features of nonadiabatic transitions can be operative to much higher energies, but then a large basis set is often required to achieve satisfactory convergence. In such circumstances, a simple interpretation of nonadiabatic processes is then no longer possible. For that reason, it is preferable that the theory first be tested on systems for which nonadiabatic effects can be well localized. Other important considerations of relevance in testing the theory concern the accuracy of the molecular structure calculations (adiabatic energies, nonadiabatic coupling matrix elements) on which the dynamics are based. Care must be taken to ensure that errors in these molecular parameters are considerably less than those that originate from the arbitrariness of the reaction coordinates or translation factors. In view of these constraints, it is scarcely surprising that detailed studies of translation effects are limited mainly to effective one-electron systems (electron capture by singly and multiply charged ions in collision with hydrogen or alkali atoms), for which the adiabatic state calculations are known to be reliable. Translation effects can vary greatly from one system to another. For example, they can be quite small in the case of a typical two-state LandauZener avoided crossing at collision energies for which the cross section is maximum. Under these conditions, the probability of a nonadiabatic transition is dominated by the value of the matrix element (36) or (68) at the avoided crossing, for which the dependence on reaction coordinate or translation factor is minimal. As the collision energy increases, the avoided crossing tends to become diabatic, and the transition tends to occur at
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R . McCarroll and D.S. F . Crothers
distances R for which the influence of translation effects is considerable. An interesting test case concerns the electron capture by C3+ from H into the ( 2 ~ , 3 s ) ~state S of C2+,which is controlled by an avoided crossing around 11.2ao. Both the Eckart and hyperspherical coordinates appear to be satisfactory, whereas the error in using BO states increases rapidly for energies above a few tens of eV/amu (Opradolce, et al. 1988). Another interesting situation occurs when electron capture takes place into a state with nonzero angular momentum. One of the interesting test cases is the capture by A13+ from H into (3p)'P state of A12+ (Gargaud et al., 1988). Three adiabatic states are involved, two C states with an avoided curve crossing at 7a0 and one Il state whose energy curve goes through the avoided crossing. Both radial and rotational coupling are operative so that the hyperspherical (or the Solov'ev-Vinitsky) approach is inappropriate. An analogous situation occurs in the B3+/He system, which has been experimentally analyzed by Roncin et al. (1990). The interference of radial and rotational coupling has a profound effect on the alignment and orientation parameters of the emitted radiation by the (2p)'P state of B 2 + . The most exhaustive comparison between the theory and experiments concerns the C4+/H system, where capture takes place via the 3s, 3p and 3d states of C 3 + . Basically seven adiabatic states are involved, but the dominating feature of the charge transfer process concerns the three-state mechanism of radial and rotational coupling. This strong interference of radial and rotational coupling can be strongly influenced by translation effects. For example, the experimental branching ratio for capture into the 3s, 3p and 3d states of C3+ for the C4+/H system can be satisfactorily explained only when translation effects are taken into account. Another example where the introduction of translation effects is effective is the Si'+/H system (Gargaud et al., 1987). However, a more exhaustive study of three-state systems is necessary before a definitive picture emerges. Finally the system ion-alkali atom still presents a number of features that are not fully understood. These systems are quite different from the multiply charged ion-atom systems. Rather than the Landau-Zener type of avoided crossings, which we have just been dealing with, nonadiabatic effects in singly charged ion-atom system tend to be of the Demkov type (nearly parallel adiabatic energy curves). As a result, the nonadiabatic coupling matrix elements are less localized, and we may expect translation effects to be more critical. This is borne out by the calculations of Allan et al. (1983) on charge transfer H + / N a using a BO basis, which showed the dependence on the parameter p (see Section 1I.C) to be so critical for energies above 1 keV/amu as to be unmeaningful. On the other hand, the inclusion of CTFs (of the type (66) or (67)) in his later calculations (Allan, 1986)gives well-converged results
ADIABATIC EXPANSIONS AND NONADIABATIC EFFECTS
27 1
for energies up to 5 keV/amu, which is just about the limit of what could be expected of an adiabatic approach.
V. Relevant Work of D. R. Bates (1962-1983) As discussed in Section II.A, the problem of lack of gauge and Galilean invariance in nonadiabatic expansions was clearly identified by Bates et al. (1953) and then resolved by Bates and McCarroll (1958) through the appropriate use of electron translation factors. With hindsight this use of travelling molecular orbitals with classical ETF also resolved the problem of elastic divergence, a problem since associated with travelling atomic orbitals and indeed with some common translation factors. It may also be remarked that the informal method of Mott and Massey to obtain coupled equations by projection onto each member of the basis set of expansion may be formalised through the use of second-order Euler- Lagrange variation theory, with its emphasis on boundary conditions. Given these favourable circumstances, it appears timely to review the work of D. R. Bates, in the period 1962-1983, in the field of slow heavy-particle ion-atom collisions. Thirty years ago Bates and McCarroll (1962) gave a timely and comprehensive review of charge transfer, including their low-velocity approximation for both resonance and nonresonance, while Bates (1962a) gave a lengthy and detailed account of slow collisions with particular emphasis on the semiclassical impact parameter treatment. Another influential paper of that period (Bates, 1962b)showed that the noncrossing phase integral theory of Stueckelberg (1932), exactly 30 years earlier, was unsatisfactory in the limit of resonance. Papers on pseudo-crossings (Bates et al., 1964) and rotational couplings (Bates and Williams, 1964) followed. Using the impact parameter treatment, both these papers solved coupled equations numerically with unitarity forced by empirically averaging off-diagonal matrix elements, at low velocities, a technique that has since been justified variationally using a Lowdin- Wannier orthonormalisation of the travelling molecular or atomic orbitals. Bates and Holt (1966) investigated both atomic and molecular expansion methods within the three-dimensional JWKB treatment, thus giving a unified treatment of the impact parameter method and its semiclassical corrections and counterparts, essential to a description of differential cross sections. A review followed (Bates, 1970) in which the problem of convergence was highlighted, concerning the truncation of the travelling molecular orbital basis sets, although in the meantime Bates and Reid (1969) had already studied successfully the two-state truncation for symmetric resonance charge
272
R. McCarroll and D. S. F. Crothers
transfer between protons and excited hydrogen atoms at very low energies. Bates and Crothers (1970) presented a semiclassical treatment of atomic collisions, extending the JWKB approximation down to unprecedentedly low energies by the use of a forced common classical turning point for the trajectories of importance. It may be remarked in passing that their method is easily extended to an R-matrix generalised impact parameter treatment of heavy-particle collisions. At the same time Bates (1970) reviewed mutual neutralization at thermal energies, concluding that Landau-Zener pseudo-crossing explanations were often inadequate and ineffective, thus opening the way for the noncrossing theories of Demkov and Nikitin with their perturbed symmetric-resonant pedigree. Contemporaneously, Bates and Sprevak (1970) combined the work of Bates and Williams (1964) and Bates and Crothers (1970) to describe largeangle scattering in slow H+-H(1s) collisions with full allowance for the Coulomb deflection effects on the trajectories and associated phases. Bates (1972) then reviewed the semiclassical treatment of inelastic atomic collisions including both one- and three-dimensional versions of the JWKB approximation. Subsequently Bates (1978) reviewed “other men’s flowers”’ and in particular the role of Massey’s adiabatic criterion in assessing Landau-Zener nonadiabatic effects. Moreover a full list of his publications to 1983 is included in Bates (1983),while much of his more recent work has tended to be in the related area of reactive scattering, notably recombination. As emphasised by Nakamura (1991), in a wide-ranging discussion of electronic transitions in molecular dynamic processes, the dynamics of superexcited states of molecules such as occur in dissociative recombination, a perennial interest of Bates, are more amenable to description by multichannel quantum defect theory than by adiabatic expansions and nonadiabatic theory. Nevertheless the period 1962-1983, as illustrated here, was a highly productive one for David Bates, and his timely and imaginative contributions to the theory and applications of adiabatic expansions and their associated nonadiabatic effects in the field of ion-atom collisions.
VI. Understanding Nonadiabatic Transitions and Effects (1971-1992) Undoubtedly the most convincing and reliable method of calculation, when adiabatic expansions are appropriate, is to solve numerically coupled differential equations that are variationally sound, gauge and Galilean ‘“Comme quelqu’un pourrait dire de moi que j’ai seulement fait ici un amas de fleurs ttrangeres, n’y ayant fourni du mien que le filet a les lier.” Essais de Michel de Montaigne, Vol. 3, Chapter 12 (1588) (A. L‘Angelier, Paris).
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invariant, unitary and free of elastic divergences. Nevertheless, even the most reliable calculations may benefit from an understanding of the principal mechanisms, nonadiabatic transitions, events and effects. From this point of view, an essential tool is the semiclassical phase integral method pioneered by Stueckelberg (1932), reviewed and developed by Crothers (1971), Child (1978), Crothers (1981) and Nakamura (1991), and supplemented by the comparison-equation method. In the JWKB phase integral method, the channel wave functions are developed in terms of the classical adiabatic actions and their first-order quanta1 corrections. Nonadiabatic transitions are associated with transition points, the complex adiabatic degeneracies, and are the physical manifestation of the Stokes phenomenon, in which the coefficient of the purely exponentially subdominant term changes abruptly by an amount proportional to the dominant coefficient and the particular Stokes’s constant. This occurs on the so-called Stokes line, emanating from the transition point. In the case of the one-transition point problem, analyticity determines the Stokes constant and guarantees symmetry and unitarity. The two-transition point problem is less simple. The phase of the Stokes’s constant is determined only by solving a comparison second-order differential equation exactly in terms of parabolic cylinder functions and by developing their asymptotic expansions in the various sectors, using the method of steepest descent on complex contour integral representations. It is particularly important to recognise the simple topological complexities in moving from one to two transition points. First, strong-coupling asymptotics are required to uniformly account for the approach of the two points; and second, for higher values of angular momentum, the bending of the double Stokes’s line (Barany and Crothers, 1981). In the case of the canonical Nikitin model, the adiabatic term difference includes a double pole, in the vicinity of which the JWKB approximation is uniformly exact. In this case the comparison equation is solved exactly in terms of Whittaker functions. Strong-coupling asymptotics were given by Crothers (1978). A full-phase integral treatment, including logarithmically spiralling double Stokes’s lines, was given by Barany and Crothers (1983). This work was generalised to complex energies and interactions by O’Rourke and Crothers (1992) and particularized to the Demkov case by O’Rourke (1992). It may also be mentioned in passing that a considerable number of recent papers have been devoted to making the theory of the Stokes’s constant more rigorous (Berry, 1989, 1990c, 1991; Boyd, 1990; Berry and Howls, 1990; McLeod, 1992; Paris and Wood, 1992; Paris, 1992a-c); however, this rigor has not yet been extended to two or more transition points. Moreover, less formal work has been done on the periodic transition point problem in terms of the Mathieu function (Barrett, 1981) and on the four transition point
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R. McCarroll and D. S. F . Crothers
problem (Lundborg and Froman, 1988; Zhu and Nakamura, 1992a,b) deploying weak-coupling expansions. Using the dynamical adiabatic treatment-representation (Crothers, 1971; Nakamura, 1991), the phaseintegral description may be applied accurately to describe radial and rotational couplings, symmetric resonance and perturbed symmetric resonance in chemically reactive as well as ion-atom collisions, not to mention condensed-matter physics, surface physics and biological physics, thus extending our understanding of nonadiabatic transitions, events and effects. The derived semiclassical (but nevertheless quantal) phase-integral scattering matrix reflects these nonadiabatic effects in terms of interactive competing Stueckelberg nonadiabatic phases and amplitudes nevertheless parametrised in part by the underlying adiabatic potential energies. Perhaps the most famous nonadiabatic effect of recent years is the geometric phase of Berry (1984), further developed by for instance Berry (1988a, b; 1990a, b) anticipated by, for instance, Longuet-Higgins (1979, Stone (1976), Mead and Truhlar (1979) and Mead (1980), and interpreted by Simon (1983) as an anholonomic parallel transport. Following JWKB phaseintegral discussions of this phase (Reinhardt, 1987; Bender and Papanicolaou, 1988), O’Rourke and Crothers (1992) and ORourke (1992) applied phase-integral analysis to the complex Nikitin (Krstic and Janev, 1988; Sidis et al., 1988), Demkov, Rosen-Zener and parabolic models of atomic collisions and showed that, in the case of a complex non-Hermitean nonunitary Hamiltonian matrix (Dattoli et al., 1990; Miniatura et al., 1990), accidental resonance results in both complex (Coveney et al., 1988) and real closed-circuit contributions to the surviving elastic Stueckelberg-Berry phase. It may be noted in passing that the real closed-circuit phase is not an ordinary dynamic phase (cf. Berry, 1990b) and that there is a surviving elastic amplitude, as well as phase, in the accidental resonance limit. Further wideranging discussion of the Berry phase in relation to atomic, molecular and optical physics is contained in Kuratsuji and Iida (1985, 1986), Aitchinson (1988),Samuel and Bhandari (1988), Aharonov and Anandan (1987), Jackiw (1988a,b), Zygelman (1987, 1990), Wang (1990), Tycko (1991). Aldinger et al. (1991) and Moore (1991).
VII. Conclusions We have outlined the problems that arise in applying adiabatic expansion methods to the theory of ion-atom collisions. First, depending on impact energy and the range of information desired, we may choose a fully quantum mechanical approach, a semiclassical impact parameter approach or indeed a
ADIABATIC EXPANSIONS AND NONADIABATIC EFFECTS
215
semiclassical, semiquantal phase integral approach. Second, we may commence with Born-Oppenheimer adiabatic states, although an alternative semimolecular choice of basis set comprises the continuum distorted wave atomic states (Crothers, 1981). Third, we must address Galilean invariance either through reaction coordinates such as Eckhart or hyperspherical coordinates or through electron translation factors (ETF). In turn, the ETF may be classical or nonclassical, state dependent or state independent and electronic-coordinate dependent or independent. Having reviewed the experimental evidence on these issues in actual applications, we have reviewed the adiabatic expansion work of David Bates in the 1962-1983 period and the phase-integral interpretative understanding of nonadiabatic transitions and effects in the 1971-1992 period, including the Berry phase.
Acknowledgments One of us (DSFC) acknowledges support in part by the Science and Engineering Research Council (UK) through grants GR/G06244 and GR/H 59862.
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ADVANCES f NATOMIC. MOLECULAR, A N D OPTICAL PHYSICS, VOL. 32
ELECTRON CAPTURE TO THE CONTINUUM B. L. MOISEIWITSCH Department of Applied Mathematics and Theoretical Physics Queen’s Universily of Belfast Belfast. Northern Ireland
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 219
11. Relativistic Theory of Electron Capture to the Continuum . . . . . . . .
A. The First-Order Relativistic OBK Approximation . . . . . . . . B. The Second-Order Relativistic OBK Approximation . . . . . . . 111. Cusp Asymmetry for Electron Yield . . . . . . . . . . . . . . . . IV. Nonrelativistic Formulas for Electron Capture to the Continuum . . . V. Comparisons with Experimental Data and Other Theories . . . . . . A. Values of B : / Z i for ECC . . . . . . . . . . . . . . . . . . . B. Cusp Asymmetry Parameter fi for ECC . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
280 28 1 283 284 285 286 . . 287 . . 289 . . 292
. . . . .
. . . . .
I. Introduction Pioneering calculations on the ionization of hydrogen atoms by fast incident protons were carried out by Bates and Griffing (1953) using the Born approximation. The related process of electron capture to the continuum (ECC), in which the electron is ejected in the forward direction and has a small velocity with respect to the projectile ion, was first discussed theoretically by Salin (1969) and Macek (1970). The first detailed theoretical examination of ECC was carried out by Dettman et al. (1974) using the Oppenheimer-Brinkman-Kramers (OBK) approximation. They established that the differential cross section daoBK/du,with respect to the speed u, of the ejected electron has a cusp at u, = u, where u is the speed of the incident projectile. The experiments of Meckbach et al. (1977), Vane et al. (1978) and Suter et al. (1978) showed that this cusp is asymmetric about u, = u whereas the OBK approximation results in a symmetric cusp. As a consequence Shakeshaft and Spruch (1978) inferred that this asymmetry is due to secondorder effects and carried out a second-order Born analysis to establish this. Since then ECC has been the subject of considerable experimental and theoretical investigation. 279 Copyright 0 1994 by Academic Press, Inc. All rights of reproduction in any lorn reserved. ISBN 0-12-W3832-3
280
B. L. Moiseiwitsch
The discussion in this review is based on a relativistic formulation of ECC using the second-order OBK approximation evaluated to leading orders in the fine structure constant c1 (Moiseiwitsch, 1991). Nonrelativistic formulas can be readily obtained by letting v/c become small.
11. Relativistic Theory of Electron Capture to the Continuum Let us consider a collision in which a bare projectile ion P is moving with relativistic velocity v referred to the nucleus of a target hydrogenic ion T and the electron of T is captured into a continuum state of P so that it is moving with relativistic velocity v, referred to T and low nonrelativistic velocity v; referred to P. If Iv,-vI and the angle 0 between v, and v are both sufficiently small, it can be shown that 0; N Y2[(U,
- u)2 + v e V e 2 / p p
(1)
and v; will be nonrelativistic if ( y 2 - 1)02<< 1, where y = (1 - v2/c2)-'I2 is the Lorentz factor. If a is the total cross section for capture to the continuum, k' is the momentum vector of the continuum electron referred to P in units of (c1u0)-' = rnc/h, and dR is an element of solid angle, then
s
da/dv, = (v,Zy4/c3) da/dk'dR Integrating over the solid angle of acceptance to the continuum defined by a cone of semiangle do we obtain
Joo0
(2nrnu)- do(O,)/dv, = (vfy4/rnvc3)
da/dk' sin0 d0
(3)
where , ) ~1)-( y ~ da/dk = 2 n ( c 1 ~ ~ ) ~ ( r n / M -
j"
1gM2qdq
(4)
qmin
Here M, is the mass of the projectile ion P, q is the momentumchange vector, qminis the least value of q, and g(q) is the scattering amplitude for capture into the continuum state with momentum vector k', all momenta being expressed in units of (aao)-'.
ELECTRON CAPTURE TO THE CONTINUUM
28 1
A. THEFIRST-ORDER RELATIVISTIC OBK APPROXIMATION The scattering amplitude given by the first-order relativistic OBK approximation is gomi(q) = (2x)-'(Mp/mbZ~
'exp[i(q
' rT
1 1 drb
drT
+ q' ' rb)ls$~(~b)'rTl$T(rT)
(5)
where rT and rb are the position vectors of the electron referred to the target and projectile frames respectively, $ T is the wave function of the electron attached to the target ion in the ground state, $ p is the wave function of the electron in a continuum state of the projectile with momentum k' and S is the operator that Lorentz transforms the wave function t,hP from the frame of P to the frame of 7: Further 4'
+ (a2,)'
= q" - k"
(6)
where 2, is the atomic number of the target hydrogenic ion. The spatial part of the continuum state wave function is given by $&)=
exp(ik'*rb)exp(xq/2)r(l
+ iq),F,[-iq;
1;
-ik'(rk+ k e r b ) ] (7)
where q = aZ,/k' and ,F, is a hypergeometric function. Assuming that k' << q' we have (2n)-3 N
drb exp[i(q'-k)-rb],F,[i~;
6(q' - k )
1; ik'(rb
+ k'-rb)]
+ az,/qf4
(8)
from whch it can be shown that, to leading order in the fine structure constant a, without spin-jip gaiK1
N
+ 1)/2] 'I2( 1-62/2)aa,(aZ,)5/2aZp 1q'-4 iq)[q2 + (uZ,)']
(4/x)2'l'(Mp/rn)[(y x exp(xq/2)r( I -
(9)
and with spin-pip
+ 1)/2] "'(S/2)(q2 - qlfiin)1/2aao(aZT)5/2aZp x exp(nq/2)r(l- iq)[q2 + (az,)'] 1q'-4 (10) where 6 = [(y - l)/(y + l)] ' I 2 , the evaluation of goBKl being analogous to that gadK1N (4/x)21/2(M,/rn)[(y
~
for capture to bound states (Moiseiwitsch and Stockman, 1980).
282
B. L. Moiseiwitsch
Now we may replace q" by q2 + (aZ,)? since qt2 << k'2, and using lexp(nq/2)r(l -iq)12
= 2nq[1 -exp(-2nq)]-'
= 2na~,k'-'
(1 1)
we obtain, on integrating over q, for capture without and with spin-flip to the continuum state having momentum k : daoBKl/dk N ( 3 2 / 5 ) ( c t U 0 ) ~ ( a Z ~ ) ~ ( a Z 1)-'k'-'[4rfiin p)~(y+ ( a z ~ ) -5Fl(y) ~] (12) where F , = 1 - d2 + 5S4/16 s, = (1 - c?z:)1/2. Since Bo is small
and
qmin= (1 - ysT)/(y2 - 1)'I2
with
from which it follows that
As u, -+ u we get (2ZmU)-' dOoBK1(80)/dUe+ y3B&
(15)
where
BO,
= (32/~)(aao)2(a~,)5(~Zp)3~l(y)(mc2)'(7 - 1)- 'CqHin
+ ( ~ z , )- ~ ] (16)
B: was originally defined by Meckbach et al. (1981) and Andersen et al. (1986) at nonrelativistic energies as the first term of the expansion in Legendre polynomials referred to the projectile frame
(d2u/dQdE),
= BO,
+ B& + (By + B:t($,('cosB)'
+ ...
(17)
where vk=v,-v is the velocity of the continuum electron referred to P in units of uo = e2/h and 8' is the angle between v6 and v. Assuming that the acceptance angle Oo is sufficiently small so that ( y 2 - l)eg << 1 it can be seen that the fall off at sufficiently high collision energies E is E - ' , which is the same as for capture to bound states.
283
ELECTRON CAPTURE TO THE CONTINUUM
B. THESECOND-ORDER RELATIVISTIC OBK APPROXIMATION
A relativistic analysis of electron capture to bound states has been carried out by Humphries and Moiseiwitsch (1984) and Moiseiwitsch (1988) using the second-order OBK approximation. This was generalized to ECC by Moiseiwitsch (1991) assuming that k is small. To leading orders in the fine structure constant he showed that the scattering amplitude is given by gOBKZ = gOBKl + g2
(18)
where goBK1 is given by (9) and (lo), and without spin-yip gil) = (4/n)2’/’(Mp/m)[(y
+ 1)/2] ‘/2aa,(aZ,)5/2aZp
x exp(nq/2)r(l -iq)[q2 +(aZ,)’]-’ x (y+6’/2)[qZ +(az,)’-2(y-
(19)
1)-2(yz- l)l”iazr]-l
and with spin-yip 91’’ = (4/n)2”’(MP/m)[(y
+ 1)/2] ‘/’(6/2)(q2 -~~in)’/2aao(aZ,)5~’~Zp
+
x exp(nq/2)r(1 - iq)[q2 (az,)’] x (2y+ 1)[qz+(uz,)’-2(y-
-
’
(20)
1)-2(yz- l)l”iazT]-l
+
+ (d,)’] - we may write Now using (4) and setting x = [q’ (aZ,)’][q$, for electron capture without and with spin-flip into the continuum state with momentum vector k‘ d‘OBKZ /d k’ = 32(a~~)’(ffz,)~(ffz,)~(yl)-’&-’[qiin +(~rz,)’]-~
xP6G(x)dx (21)
Jl=
where, to leading order in a, G(x) N 1 -6’+b2~/4+(C1 + C ~ X + C ~ X ’ ) [ ( X - U ) ’ + U ’ ~ $ ] -(22) ’ with Cl = a[b4b/2
-
2( 1 - S2/2)c], C , = c2+ 2(1- 6’/2)c- h4(b2 + 2b)/4 - as4b/2,
C , = b4(b’+2b)/4, a
= 2(y+
I), b
=
2y+ 1, c
iT= aZ,6-’.
= y+6’/2,
Using the approximate formula (Moiseiwitsch, 1988) an
x“[(x - a)’
+ a”’]
1 N
- (x -a)’
+
4’1;’
n(x - a) a[(x - a)’ a’i’]
+
“ +2z1 1
(n+l-r)d xr (23)
284
B. L. Moiseiwitsch
we find that da,,,K2/dk'
N
(32/5Mau,)2(az~)5(aZp)3(y - l)-'k'-
'[qiin + (az,)']
-
'J'z(7)
(24) where, neglecting quantities of order
c2,
F2(y)=
+ ~ C , U -1/4~ +[ (2/3)u- ' +( 3 / 2 ) ~ +- ~4u-3 + a-4{n/c, -u/b + 5 In b}] + 5 C 2 U - 2 [ 1 / 3 + + 3 U - 2 U - 3 { 7 t / c , U/b4- 4 In b } ] + 5 C , ~ - ~ [ l / 2 + 2 u - '+ ~ - ~ { n / [ ~ - a /3bIn+ b ) ] (25) -
Thus we see that (2nmu)-' dao~~2(8o)/dU, is the same as (14) and the OBK2 formula for B: is the same as (16) with F2(y) substituted for F,(y). S",/Zg does not depend on Z p in both the OBK2 and the OBKl approximations.
111. Cusp Asymmetry for Electron Yield The shape of the cusp for the electron yield given by (2nrno)- da(O0)/du, as a function of the electron energy referred to the target frame, obtained by using the previous theory, possesses only a slight asymmetry and in the wrong sense. It was shown by Shakeshaft and Spruch (1978) in a nonrelativistic theory that this can be corrected by carrying out the OBK2 analysis to a higher order in UZ,. Using an approach analogous to that introduced by Shakeshaft and Spruch (1978) but in a relativistic theory, it is found that the scattering amplitudes given by (19) and (20) are multiplied by the factor exp{2ia~,q(l+P-q)[q2+ ( a ~ , ) ~ - 2 ( y - 1)-2(y2- ~ ) ' / ~ i a ~ , ] - ~ } Now expanding the exponential to the first order in aZ, we obtain G ( X ) z 1 - c ? ~ + C ? ~ X / ~ ++ (CC~ ~X + C J X ' ) [ ( X - U ) ~ + U ~ ~ ~ T ] - '
+
- 8a2Z,ZpC?-2(~ y2 - 1)1/2(x1/2 -cosO')x(C, *
[(x - a)2
+a y + ]
-
+ C2x+ C3x2) (26)
As u, + u we now find that (2nmu)-' da(Oo)/dv, -,y3(B:B0 - B;8;/4)
(27)
ELECTRON CAPTURE TO THE CONTINUUM
F 2 ( y ) = F,(y) F3(y) =
:j
+
- 160[(y
F4(y)= - 160
r
(C,
+ C2x + C,X’)[(X-U)~ + u ~ C ~ , ] - ~ dx X-~
r
+ 1)/2]1’2 (x+Y’-
(C1
285
(31)
+ C ~ +X C,X’)[(X-U)~ + u ~ [ ; ] - ~ x - ~ ~ x (32)
1)1’2(C1+ C 2 ~ + C 3 ~ 2 ) [ ( ~ - ~ ) 2 + ~ 2 C ~ ] - 2 ~ - 5 d ~
(33)
To the zero order in a, F2(y) is given by (25). Analytical forms correct to the zero order in a can also be derived for F 3 ( y ) and F4(y) but they are quite complicated and will not be given here.
IV. Nonrelativistic Formulas for Electron Capture to the Continuum If we allow u/c to become small, the nonrelativistic limit is reached. This can be done by letting y + 1 + a2(u/u0)2/2where uo = e2/h is the atomic unit of velocity. Then
Now expressing velocities in atomic units, it is found that in the nonrelativ-
286
B. L. Moiseiwitsch
istic limit (Moiseiwitsch, 1991): Fl(y)+ 1, F2(y) + f2(u), F3(y) + f3(u) and F41Y) f 4 ( 4 where f2(u) = 1 - 275/384 + (5/512) In 3 + (5/211)(n/ZT)u (35) +
+ 55/81 + 10 In 3 + (i't/z~)U-((71/16z;)U~] (36) = (5/256)[219+ 1/7 + 53/81 11 In 3 +(7n/8ZT)u-(n/1623,)u3]
f3(v) = (5/128)[113
f4(u)
-
(37)
It follows that in the nonrelativistic limit doeoBK2/dk'N (216/5)~iZ&Z:uk- ' ~ - ' ~ f 2 ( ~ )+(Z,/U)~]-'" [1
(38)
and (2nmu)- do,deo)/due N
(2'6/5)uiZ~Z;(mui)- ' u - "f2(u)[ 1+(Z,/U)~] - lo x ( u e / u 2 ) { [ ( u , - u)2
+ Ueue;]
1 / 2 - 10,
-
(39)
01)
so that Bg
N
(216/5)a~Z~Z~(mu~)-'u-'2f2(~)[1 + (zT/u)2]-10
(40)
It can be readily seen that Bg/Zi does not depend on 2, in the nonrelativistic as well as the relativistic OBKl and OBK2 approximations. Further, the cusp asymmetry parameter is given by
fl =
-(zTzP/u2)f4(u)[f2(u)
+ (zTzP/v2)f3(u)]-1
(41)
which should be valid for Zp/uz<< 1, that is for energies k 0.5 MeV amu-'Z;'. Although we have neglected the term ( z T z p / u z ) f 3 ( u ) in the formula (40) for Bg since it is of the order ZTZp/u2,it is necessary to retain this term in the formula (41) for fl since otherwise fl will become singular as u -+ 0.
V. Comparisons with Experimental Data and Other Theories Calculations on ECC have been carried out using the impulse approximation (IA) and using the second-order Born approximation by JakubassaAmundsen (1983,1984,1988), and by Crothers and McCann (1987) using the continuum distorted wave (CDW) approximation. Also calculations have
287
ELECTRON CAPTURE T O THE CONTINUUM
been performed on electron capture to Rydberg states (ECR) using the CDW approximation by Burgdorfer (1986) and by Dube and Salin (1987). Experimental data on ECC from target helium atoms by incident H + and He2+ ions, as well as for other target atoms and projectile ions, have been obtained by Meckbach et al. (1981), Dahl (1985), Andersen et al. (1986), Knudsen et al. (1986) and Gulyas et al. (1986, 1991).
A. VALUESOF Bg/Z;
FOR
ECC
In Table I we compare values of BE/Z; calculated using the OBK2 approximation (Moiseiwitsch, 1991) given by (40) with the CDW calculations of Dube and Salin (1987) and the experimental data obtained by Knudsen et al. (1986) for ECC from target He atoms by incident H + and HeZ+ions. Dube and Salin (1987) carried out CDW calculations of the ECR cross ) capture to excited levels of the projectile ions with principal sections ~ ( nfor
VALUESOF B t / Z :
FOR
TABLE I ECC FROM TARGETHe ATOMS BY H + AND He2+ PROJECTILE IONS
B:/ZZ (cm2eV-'sr-')
H+ Energy E (MeV amu-') 0.1 0.4 0.6 0.8 1.0 1.2 1.6 2.0 2.6
He2+
OBK2"
CDWb
Exp'
CDWb
1.26(-19)' 1.34 - 21) 1.99(- 22) 4.69( - 23) 1.46( - 23) 5.52( -24) 1.15(- 24) 3.32( - 25) 7.58( -26)
1.56(-19) 8.57(-22) 1.38(-22) 3.M-23) 1.20-23) 4.88( - 24) 1.14(-24) 3.57( -25) 9.24( - 26)
1.32(-19) -
1.23(-19) 6.8q- 22) 1.09(- 22) 2.79(-23) 9.41(-24) 3.8q - 24) 8.85(-25) 2.79(-25) 7.03( - 26)
-
1.02(-23) 3.67( - 24) 7.20(-25) 2.12(-25) 4.3q - 26)
Exp' -
4.88( - 22) 9.69( -23) -
9.28(-24) -
6.85(-25) 1.84(-25) -
'Calculated (Moiseiwitsch, 1991) using OBK2 approximation given by (40). bCalculated (Dube and Salin, 1987) using CDW approximation for capture to excited level of projectile ion (ECR) with principal quantum number n = 10. 'Experimental data of Knudsen et al. (1986). dNumbersin parentheses denote powers of 10.
288
B. L. Moiseiwitsch
quantum number n running from 1 to 10. They supposed that the values given by the formula
Bg(n) =
+In3 47127 * 212:
for n = 10 is a satisfactory approximation to Bg(cm2eV- sr- '). It can be seen from Table I that the CDW calculations of Dube and Salin (1987) are in satisfactory agreement with the experimental data of Knudsen et al. (1986) for H + and He2+ ions incident on helium atoms. Further they found that their values are in satisfactory agreement with the experimental data for incident C6+ and O*+ions. Also it can be seen from Table I that the OBK2 and CDW calculations of Bg/Z: are in reasonable agreement for incident H + and He2+ ions. However, the OBK2 values for incident ions with higher charges become somewhat too large since Bg/Z; is independent of the atomic number 2, of the projectile ion for the OBK2 approximation resulting from the assumption that 2, is small. The CDW calculations of ECR by Dube and Salin (1987) were performed using a Hartree-Fock wave function for the target He atom while the OBK2 calculations of ECC (Moiseiwitsch, 1991) were carried out using a simple variational wave function for He with the variationally determined parameter 2, = 27/16. The same variational wave function for He was used by Crothers and McCann (1987) in their CDW calculations of ECC. Dubk and Salin (1987) repeated their CDW calculations of ECR using the same variational wave function and found near perfect agreement with the results of Crothers and McCann (1987). The experimental data obtained by Dahl (1985) for H + + He show that Bg/Z: has a maximum at about 25 keV. Previous OBK2 calculations used asymptotic formulas valid only for Z,/O << 1, which show no maximum. However the formula (40)for B: passes through a maximum at Z , / U 1 because we can assume that k 2 << q" so that q'2 may be replaced by q2 + (Z,/U)~using (6), producing the factor [l + (Z,/v)2] - l o in (40). This improves the agreement with the experimental data in the lower energy range, from 20-100 keV, where the maximum is observed. Andersen et al. (1986) derived a u-11.3*0.2 velocity dependence for Bg from their experimental data in the range of energies 1-2.6 MeV amu-', which corresponds to an energy dependence E - " with n = 5.65 L- 0.1. Values of the index n, calculated using the OBKl and OBK2 approximations given in Sections I1 and IV for ECC from He atoms, are displayed in Table 11. Although there is satisfactory accordance between the experimentally derived index n in the range 1-2.6 MeV amu-' and the OBK2 value n = 5.60 at 2 MeV amu-', it can be seen that the OBK2 value of n increases to 5.81 at 10-20 MeV amu-' and then begins to decrease, reaching the value 5.66 at
-
ELECTRON CAPTURE T O T H E CONTINUUM
RATEOF DECAYE - "
WITH
TABLE I1 ENERGYE OF f3:
FOR
289
ECC FROM He ATOMS
Values of Index n Relativistic Approximation
Nonrelativistic Approximation Energy E (MeV a m - ' )
OBKl
OBK2
OBKl
OBK2
0.I 0.2 0.5 1.o 2 5 10 20 50 100 200 500 lo00 2000
1.84 3.38 4.75 5.33 5.66 5.86 5.93 5.96 5.99 5.99 6.00 6.00 6.00 6.00
1.83 3.35 4.72 5.29 5.60 5.17 5.8 1 5.81 5.78 5.75 5.71 5.66 5.62 5.59
5.85 5.91 5.92 5.88 5.19 5.60 5.13 4.52 3.72
5.71 5.80 5.18 5.69 5.55 5.30 4.69 4.04 3.41
500 MeV amu- ', and approaches the true OBK2 asymptotic limit 5.5 only for energies > 100 MeV amu-'. However for energies > 100 MeV amu-' it can be seen from Table I1 that the effects of relativity start to become apparent and the index n is changed significantly so that at 500 MeV amu- * the relativistic OBK2 approximation leads to n = 4.69. B. CUSPASYMMETRY PARAMETER p FOR ECC In Table I11 we compare the values of p calculated (Moiseiwitsch, 1991) using the OBK2 approximation formula (41) with the experimentally determined values for ECC from He atoms obtained by Knudsen et al. (1986) for incident H + and He2+ ions and obtained by Gulyas et al. (1991) for incident H + ions. It can be seen that the agreement is fairly satisfactory. Also there is fair agreement with the experimental data of Dahl (1985) for energies k 50 keV amu-'Z;'. However the data of Meckbach et al. (1981) and Gulyas et al. (1986) as well as the CDW calculations of Burgdorfer (1986) and Crothers and McCann (1987) give much lower values of p below 0.5 MeV amu-'Z; '.
B. L. Moiseiwitsch
290
TABLE 111
VALLIES
CUSP ASYMMETRY PARAMETER f l = Bf/B: FOR ECC FROM He ATOMS OF THE
Incident H+ Ions Energy E (MeV amu-'Z;')
OBK2"
Exp
0.1
-0.72
0.2 0.25 0.3 1.O 1.2
-0.64
-0.75b -0.65' -0.59' -0.54'
-0.60 -0.57
1.6
-0.32 -0.28 -0.23
2.0 2.6
-0.19 -0.14
-OMb -0.3Sb -0.42b -0.3Sb (-0.58)b
Incident He2+ Ions Energy E (MeV amu-'Z;') 0.2 0.3 0.5
0.8 1.o
OBK2" -0.63 -0.56 -0.46 -0.36 -0.31
Exp -0.54b -0.53b
-0.53b (-0.57)b
-0.44b
"Calculated (Moiseiwitsch, 1991) using OBK2 approximation given by (41). bExperimental data of Knudsen et al. (1986). The values in the parentheses are uncertain. 'Experimental data of Gulyas et al. (1991).
Impulse approximation calculations have been carried out by JakubassaAmundsen (1983, 1984, 1988), but the p values obtained using the post and prior forms are very different, the prior form giving much smaller values of p than the post form below -0.5 MeV. The second-order Born calculations of p performed by Jakubassa-Amundsen (1983, 1984, 1988) also produce small values of /3 below -0.5 MeV. The situation remains uncertain since different experimental groups (Oswald et al., 1989; Gulyas et al., 1991) are in disagreement with each other about the interpretation of the experimental data. Moreover the OBK2 approximation cannot be relied upon necessarily at such low energies. However, it should be noted that the OBK2 formula (41)
29 1
ELECTRON CAPTURE TO THE CONTINUUM
for B is the ratio of two quantities that, although individually they may not be very accurate at low energies, may nevertheless provide a useful estimate of their ratio. In Table IV we make a comparison between the values of B calculated using the nonrelativistic OBK2 approximation formula (411, with and without allowance for the Thomas peak, and using the relativistic OBK2 approximation formula (30) for H + He(ls2) electron capture collisions to the continuum. It is interesting to see that at 8 MeV B changes sign, being negative below this energy in accordance with the experimental data but positive at higher energies where unfortunately there are no experimental data. This change of
+
TABLE IV PARAMETER p = B://B$ FOR ECC FROM He ATOMSBY H + IONS VALUESOF CUSPASYMMETRY CALCULATED USING THE RELATIVISTICAND NONRELATIVISTIC OBK2 APPROXIMATION
Energy (MeV) of Proton 1 2 3 4 5 6 7 8 9 10 20 50 100 200 500
1000 2000 5000 loo00
Nonrelativistic OBK2 Approximation Without Thomas Peak
With Thomas Peak
- 0.342 -0.215 -0.157 -0.124 -0.102 - 0.087 - 0.075 - 0.067 - 0.060 - 0.054 - 0.028 -0.01 1 -0.006 -0.003 -0.001
-0.322 -0.186 -0.120 -0.079 -0.051 - 0.030 -0.014 -0.001 0.010
0.020 0.076 0.145 0.199 0.257 0.340
Relativistic OBK2 Approximation" with Thomas Peak -0.322 -0.186 -0.120 - 0.079 - 0.05 1 - 0.030 -0.014 -0.001 0.01 1 0.020 0.078 0.149 0.206 0.266 0.317 0.289 0.206 0.095 0.043
"Second-order OBK approximation retaining only terms up to the second order in u.
292
B. L. Moiseiwitsch
sign takes place because of the presence of the Thomas peak in the OBK2 differential cross section. If the terms (n/Z+)u3 and ( ~ / Z , ) Uin f2, f3 and f4 in the nonrelativistic formula (41) for /?, which arise from this peak, are neglected, we obtain values for /? that are negative at all energies and then agree with the CDW calculations of Crothers and McCann (1987) at energies 2 5 MeV amu-'Z;', where it has the u-' velocity dependence given by /? + - 13.8ZTZpu-2. As the energy of the proton approaches 469 MeV at which v = c, the nonrelativistic analysis ceases to be valid. Relativistically /? rises to a maximum value of 0.32 at about 500 MeV proton energy for this process and then falls to zero as the energy increases above this point. At high relativistic energies it is found that /? approaches the asymptotic limit /? + 3 ~ z z , z p [ ~ ( ~ z T ) - 3 y - 3 - ( 4 / 3 ) y - ' ] ,and so /? becomes negative again eventually, although only at ultrahigh relativistic energies where the effect of the Thomas peak becomes negligible. This is in accordance with the physical idea, quoted by Crothers and O'Rourke (1992),that convoy electrons should predominate at backward angles. Finally, it should be mentioned that, although we have not discussed them here, the higher order terms in the expansion in Legendre polynomials (17) may not be negligible. For example, the coefficient B; of P,(cosO'), which is independent of vh, has been experimentally determined by Knudsen et al. (1986) for ECC from He atoms by C6+ and 0 8 + incident ions. They find that B:/B: = -0.4 for 0.8 MeV amu-' 08+ ions.
REFERENCES Andersen, L. H., Jensen, K. E., and Knudsen, H. (1986). J . Phys. B 19, L161. Bates, D. R., and Griffing, G. (1953). Proc. Phys. SOC. A 66, 961. Burgdorfer, J. (1986). Phys. Rev. A 33, 1578. Crothers, D. S. F., and McCann, J. F. (1987). J. Phys. B 20, L19. Crothers, D. S. F., and ORourke, S. F. C. (1992). J. Phys. B 25, 2351. Dahl, P. (1985). f. Phys. B 18, 1181. Dettmann, K., Harrison, K. G., and Lucas, M. W. (1974). J . Phys. B 7 269. DUG, L. J., and Salin, A. (1987). J. Phys. B 20, L499. Gulyas, L., Kover, A., Szabo, Gy., Vajnai, T., and Berenyi, D. (1991). 4th Workshop on HighEnergy Ion-Atom Collision Processes, Debrecen 1990, Springer, Heidelberg. Gulyas, L., Szabo, Gy., Berenyi, D., Kover, A., Groenfeld, K. O., Hoffmann, D., and Burkhard, M. (1986). Phys. Rev. A 34,2751. Humphries, W. J., and Moiseiwitsch, B. L. (1984). J . Phys. B 17, 2655. Jakubassa-Amundsen, D. H.(1983). J. Phys. B 16, 1768. Jakubassa-Amundsen, D. H. (1984).Forward Electron Ejection in Ion Collisions, Lecture Notes in Physics (K. 0.Groenveld, W. Meckbach and I. A. Sellin, eds.), Springer-Verlag, Berlin, p. 17.
ELECTRON CAPTURE T O THE CONTINUUM
293
Jakubassa-Amundsen, D. H.(1988). Phys. Rev. A 38, 70. Knudsen, H., Andersen, L. H., and Jensen, K. E. (1986). J. Phys. B 19, 3341. Macek, J. (1970). Phys. Rev. A 1, 235. Meckbach, W., Chiu, K. C. R.,Brongersma, H. H., and McGowan, J. W. (1977). J. Phys. B 10, 3255. Meckbach, W., Nemirowsky, I. B., and Garibotti, C. R. (1981). Phys. Rev. A 24, 1793. Moiseiwitsch, B. L. (1988). J. Phys. B 21, 603. Moiseiwitsch, B. L. (1991). J. Phys. B 24, 983. Moiseiwitsch, B. L., and Stockman, S. G . (1980). J. Phys. B 13, 2975. Oswald, W., Schramm, R., and Betz, H. D. (1989). Phys. Rev. Lett. 62, 1114. Salin, A. (1969). J. Phys. B 2, 631. Shakeshaft, R., and Spruch, L. (1978). Phys. Rev. L e t t . 41, 1037. Suter, M., Vane, C. R., Sellin, I. A,, Elston, S. B., Alton, G. D., Thoe, R.S., and Laubert, R.(1978). Phys. Rev. Lett. 41, 399. Vane, C. R., Sellin, 1. A,, Suter, M., Alton, G. D., Elston, S. B., Griffin, P. M., and Thoe, R. S. (1978). Phys. Rev. L e t t . 40, 1020.
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ADVANCES IN ATOMIC. MOLECULAR, AND OPTICAL PHYSICS, VOL. 32
HOW OPAQUE IS A STAR? M . J . Seaton* Department of Physics and Astronomy University College London London, England
1. What Is Opacity? . . . . . . . . . . . . . . . . . . . . . . . . . . 11. What Is White?. . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Ultraviolet Radiation from Hot Stars . . . . . . . . . . . . . . . . .
296 297 298
IV. Opacities for Stellar Interiors . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
301
305
In 1963 Harrie Massey told me that he had just seen our College Provost, who had said that it was not absolutely necessary for a professor, appointed by confirment of title, to give an inaugural lecture. Did I wish to give one? I replied that I was rather busy, perhaps it could be postponed a little while. That, said Massey, was the answer he had expected from me. Now, 29 years later, it can be put off no longer. I feel it a great honour to be asked to give this lecture today, as the first Massey Memorial Lecture. I first met Massey in 1946, on my second day as an undergraduate student in physics. At that time he was our Goldsmid Professor of Mathematics. I had had a small difference of opinion with our Senior Science Tutor. While in the services 1 had passed some university examinations in mathematics and I wanted to skip the first-year ancilliary mathematics course for physics students, in which case I could complete my degree in two years and catch up on lost time. The Tutor said that could not be done. In despair I went to our Mathematics Department, not knowing what to do when I got there. Massey stopped me in the corridor, “You look worried, can I help?” I explained the problem . He said yes, I could skip the first-year course but I would have to attend the second-year one, given by Mr. Bates. Just two years later I became a research student of David Bates in Massey’s Department. I always remember that first meeting with Massey as an extraordinary act of kindness on his part. In later years I realized that it was typical of his consideration for other people, no matter how junior or how *Based on the first Massey Memorial Lecture given at University College London, 1 May 1992.
295 Copyright Q 1994 by Academic Press, Inc All nghrs of reproduction m any farm reserved. ISBN 0-12-003832-3
296
M . J . Seaton
senior those people were. He was outstanding both for his own contributions to research and as a research leader. He greatly influenced my scientific career, as he did those of many others in this room today. I shall talk about three related topics: first, early work on the opacity of the solar atmosphere to which Massey contributed; second, observations of the ultraviolet spectra of stars in which Massey’s leadership was of great importance; third, a description of work with which I have been involved in recent years on the calculation of opacities for stellar interiors. That work has required the efforts of quite a large international team, and I think that Harrie Massey would have approved. He liked ambitious international collaborations.
I. What IS Opacity? Opacity is the probability per unit distance of radiation being removed from a beam. Let I(& s) be the intensity of radiation (energy per unit area, unit time and unit solid angle) in a direction S: as a function of distance s. The equation of radiative transfer is dl ds
-=
-d+j
where K is the opacity and j the emissivity. Energy can be removed by absorption or by scattering (changing the direction of the radiation). If K is constant and j is zero, the solution of (1) is I(& s) = Z(6, 0) exp( - KS). The mean-free-path of a photon is 1= 1 / ~This . is the quantity given in BBC “Weather Reports from Coastal Stations.” On a clear day, 1 for atmospheric air may be 20 or 30 miles; in a thick fog it may be down to a few feet. In a stellar interior, 1 may be of order 1 mm or less. Let I, be the intensity at frequency v. In a thermal enclosure (a “black body”), I, is independent of position and direction and depends only on temperature IT: I, = B,(T) where B , is the Planck function,
For a black body, dB,/ds = 0, and hence
which is Kirchhoffs law.
297
HOW OPAQUE IS A STAR?
11. What Is White? White is the colour of sunlight! To the human eye, sunlight has no colour. To a first approximation, sunlight corresponds to radiation from a black body at T = 5770K. That is white hot. A body that is hot enough to give visible radiation but less hot than the sun is described as red hot; if hotter, blue hot. The spectral distribution of sunlight depends on processes of absorption and emission in the sun’s atmosphere. Figure 1 shows a smoothed solar spectrum, omitting all detailed structure due to spectrum lines. In the 1930s a major problem was to explain the overall distribution, as shown in the figure. Hydrogen is the most abundant element but photoionisation from groundstate hydrogen occurs only for ,?. < 912 A, way outside of the visible spectrum. A feature is discernible at the H Balmer limit, 3646 A, due to H in n = 2 states but the sun is too cool to have appreciable absorption by hydrogen in higher excited states, giving absorption for 1> 3646 A.One can rule out absorption by other atoms in excited states, as the main continuum opacity source, since that would give absorption edges which are not observed. The dominant source of opacity in the solar atmosphere was first identified by Wildt in 1939 as being due to the bound-free process of photodetachment from the H - negative ion,
H-
+ hv-H + e
(4)
and the corresponding free-free process,
H
+ e + hv-H + e’
(5)
Wildt refers to the first edition of Massey’s book, Negatioe Ions, published in
I
3-
‘
I
1
I
-
m ‘.
N
E V
FIG.1. Flux from the solar surface. Filled circles joined by dotted lines are mean observed values from “Astrophysical Quantities” (Allen, 1973). Full lines indicate black body at the solar effective temperature of 5770 K.
298
M. J. Seaton
1938, and to receipt in March 1939 of a letter from Massey giving preliminary results from a new calculation of the cross section for (4) (subsequently published as Massey and Bates, 1940)which confirmed that the process would be important in the solar atmosphere. The subsequent history of work on the subject is described in a magnificent essay by Bates (1978), “Other Men’s Flowers.” The emergent intensity from the sun, I , ( @ , can be measured as a function of the angle f3 to the normal to the solar surface. Chalonge and Kourganoff (1946) showed that one could invert the formal solutions of the transfer equation and, using results from accurate measurements of lv(8), deduce the frequency dependence of the opacity K , . A long series of papers by Chandrasekhar and various collaborators, begun in 1942, was concerned with making calculations for (4) and ( 5 ) of ever increasing accuracy. Already by 1950, when the second edition of Negative Ions was published, an impressive level of agreement had been obtained between the calculated and that deduced from observations. The work frequency dependence of IC, continued. Laboratory measurements of the cross section for (4) were made by Branscomb and Smith (1955) and Smith and Burch (1959) and further calculations were made using more modern computing techniques. Results are given by Bates (1978). The best modern estimates of the cross sections, which should be correct to within about I%, are given by Wishart (1979) for (4) and by Bell and Berrington (1987) for (5). The most recent comparisons of computed solar continuum opacities with those deduced from observations are given by John (1992).
111. Ultraviolet Radiation from Hot Stars For stars much hotter than the sun, H- is no longer of much importance. The dominant opacity sources are due to H and He in ground and excited states and to scattering by free electrons. During the 1950s and early 1960s there was a lot of interest in making models of hot-star atmospheres assuming local thermodynamic equilibrium (LTE); i.e., using relation (3). The net outward flux of radiant energy is n
J
F,(r) = 27~ I&, U)cos(U)dcos(U)
(6)
where r is the radial distance outward from the stellar centre and 8 is the angle between 5 and f . For stellar atmospheres, in a plane-parallel approximation, it is required that the integrated flux, F(r) =
s
F,(r)dv
(7)
HOW OPAQUE IS A STAR?
299
should be independent of r. One puts
(8) which define the effective temperature T,,,; g B is the Stefan-Boltzmann constant. The stars are arranged in a sequence of spectral types, 0, B, A, F, G, K, M, in order of decreasing T,,, . The sun is of type G. A second required parameter is the surface gravity, F = IsBT$,
g = G M , fR:
(9)
which enters into the equation for hydrostatic equilibrium. One obtains models of temperature and density as functions of r such that, on solving the equation of radiative transfer, one obtains F to be independent of r. One then adjusts T,,, and y to obtain best agreement between fluxes F , for the models and from observed stellar spectra. All of the earlier models could be fitted only to observations made in the visual spectrum. There was great interest in trying to extend the observations to the ultraviolet, which was, of course, possible only by using space vehicles. Massey was a great enthusiast for space research. He was mainly responsible for building up the whole of the U.K. space research program and was also very active in Europe and internationally. It was in September 1964 that Massey first brought together a small group of experts to discuss a proposal for the launching of a satellite for making UV observations of the stars. It was not until 1978 that his idea finally bore fruit in the launch of the International Ultraviolet Explorer (WE) satellite. There was great excitement when UV observations of stellar spectra were first made: in the United States using Aerobee rockets (Stecher and Milligan, 1962; Chubb and Byram, 1963), and in Australia using U.K. Skylark rockets launched from Woomera (Alexander et al., 1964). In those early experiments the observations were made using broadband filters. There were some sizeable discrepancies between the fluxes reported from the rocket observations and those calculated from the atmosphere models, with the observed fluxes being less than those predicted. Those discrepancies led to intensive theoretical studies and also, of course, to attempts to make further observations. Most of the theoretical work was concerned with finding additional opacity sources. Although the stars observed were all nearby, and showed little evidence for interstellar absorption in the optical, Hoyle and Wickramasinghe (1963) suggested that interstellar absorption in the UV might be much larger than had been supposed. Later studies of the interstellar UV absorption showed that there were, indeed, some surprises in store, but that source of absorption was not large enough to explain the discrepancies for the stellar spectra. Another idea, first discussed by Gaustad and Spitzer (1961), was that there
300
M. J . Seaton
could be a lot of absorption by large numbers of UV spectrum lines. Lines due to iron could be particularly important, since iron has a rich line spectrum over many ionisation stages and a cosmic abundance that is high compared with that for other elements in the same region of the periodic table. There are two effects to be considered: first, the direct effect of line absorption in reducing the UV flux; second, the effect of including the UV line opacity in calculating the models. This is known as line blanketing. For a given value of Kff,the effect of blanketing is to reduce the flux in the UV and to increase that in the optical region. A smaller value of T,,, is then required to match the observed optical spectrum. Calculations of line-blanketed models were made by Mihalas and Morton (1965), followed by similar work of a number of other investigators. Allowance for blanketing was found to improve the agreement between theory and observations but not to solve all of the problems. Further observations were made using rockets (references are given by Wilson and Boksenberg, 1969) and, later, from satellites. Bright stars were observed at high resolution with the Copernicus satellite, and extensive sky surveys using filters were made with OAO satellites of NASA and the TD-1 satellite of the European Space Agency. In some cases it was found that there were significant errors in the earlier rocket measurements. By considering pairs of stars of the same spectral type but having different amounts of interstellar absorption, the wavelength dependence of that absorption could be deduced. In all, over 57,000 stars were observed with TD-1 and for 7,697 stars mean spectra from several scans were compiled at University College London (Carnochan, 1982). Figure 2 shows ratios of fluxes at 1445A and 5550 A using mean values for each spectral type, together with aesults from blanketed models by Kurucz et al. (1974). The differences are easily within the
h
0
2 1
0
05
BO
B5
A0
FIG. 2. Ratios of ultraviolet fluxes (1455A) to optical fluxes (5550A): circles, from TD-1 observations analysed by Carnochan (1982); full line, from models of Kurucz et al. (1974). I am indebted to I. D. Howarth for assistance in drawing this figure.
HOW OPAQUE IS A STAR?
30 1
combined errors of UV and optical observations and the uncertainties in determining the atmospheric parameters. The first high-resolution UV spectra for large numbers of stars were obtained using WE. An interesting discussion of results for one star (0Peg, T,,, = 9600 K) is given by Leckrone et al. (1990), who use nine coadded spectra to reduce noise. They compare the complicated observed line spectra with spectra calculated using extensive atomic data computed by Kurucz.
IV. Opacities for Stellar Interiors Conditions inside a star are very close to those for a black body. Equation (3) can be used. But the radiation intensity I , is not completely isotropic because there is a net outward flux of radiation. The transfer equation (1) can be solved in the diffusion approximation (see Schwarzschild, 1958), assuming a small anisotropy in I , . One obtains
F=
471 1 d B d T 3 K R d T dr
where B = (aB/n)T4 is the integrated Planck function and where the Rosseland mean opacity, K ~ is, defined by
In constructing a model of a star it may be assumed that the total luminosity, L = 471rZFlol, is constant (except for regions of deep interiors where nuclear reactions occur). So long as conduction and convention can be neglected, the total energy flux, Flo,,can be equated to the radiative flux F . Under those circumstances the temperature gradient, d T/dr, is proportional to I C ~ Clearly, accurate determinations of K~ are of great importance for the construction of stellar models. Pulsations in variable stars are sensitive to the structures of the stars and, in particular, to the values used for opacities. Studies of such stars therefore provide important checks on various aspects of the theories used. The monochromatic opacity K(v),is given by K(V) =
Ci Niai(v)Ei(v)
(12)
where N iis the number density of particles of type i, ai(v)is a cross section for absorption or scattering and Ei(v) is a correction factor for stimulated emission (Ei(v) = [l - exp( - hv/kT)] for absorption, Ei(v) = 1 for scattering). In calculating Rosseland means we face three problems:
.
302
0 0
M . J . Seaton
Determine the level populations N , ; that is the problem of the equation of state. Determine the cross sections gi(v); that is a problem of atomic physics. Evaluate the Rosseland mean for a wide range of values of temperatures and densities.
The atomic processes to be considered are bound-bound (spectrum lines); bound-free (photoionisation); free-free; scattering (mainly by free electrons). Earlier work on Rosseland-mean calculations, using the hydrogenic Kramers’s approximation, is described in the book by Schwarzschild (1958). Much of the more recent work has been concerned with making detailed calculations for the “metals,” that is to say all elements heavier than H and He. Extensive use has been made of opacities calculated at the Los Alamos National Laboratory in the United States (see Cox, 1965; and Weiss et al., 1990, for references to more recent work). I will refer to that work as LAOL, Los Alamos Opacity Library. Substantial use has also been made of the St. Andrews opacities (Carson et al., 1968). Reasons for some discrepancies between LAOL and the St. Andrews work have been explained (Carson et al., 1984). The bombshell was Simon’s 1982 paper, “A plea for Reexamining Heavy Element Opacities in Stars,’’ in which it was shown that an increase in heavyelement opacities by factors of 2 to 3 would resolve a number outstanding questions in the theory of pulsating stars. Simon argued that changes of that order were not implausible and urged that further work should be done. The challenge was taken up by two groups. One was C. A. Iglesias, J. F. Rogers and B. Wilson at the Lawrence Livermore National Laboratory in the United States. I shall refer to their work as OPAL, the name of their computer code. The other was the international Opacity Project with which I have been involved and which I will refer to as OP. In both projects the approach has been to use improved equations of state and to inctude very large amounts of atomic data. One must consider hundreds of thousands of atomic energy levels and photoionisation cross sections and millions of spectrum lines. For each line one requires information about the line frequency and transition probability, orfvalue, and about the line profile as a function of temperature and density. In their details, the two approaches differ. Elsewhere I have attempted to give a balanced summary of the methods used by both projects (Seaton, 1992). Results obtained are generally in good agreement; differences in values of rcR are, at most, about 20% and usually much smaller. The remarkable achievements of the OPAL team have been described in a number of papers (Iglesias et al., 1987, 1990; Iglesias and Rogers, 1991a, b; Rogers and Iglesias, 1992). I hope I will be forgiven if, in the present lecture, I give more prominence to the OP work.
303
HOW OPAQUE IS A STAR?
That work is concerned with stellar envelopes, that is to say, those outer parts of interiors for which the densities are not very high and plasma perturbations are not of major importance for opacity calculations. The Boltzmann equation is N i = (N,/g,)g,exp( - E , / k T ) , where N i is the population of level i, N , is that of the ground state, the gi are statistical weights and Ei is the excitation energy of level i. That equation gives N i to be divergent! In essence, the problem is that one cannot have excited states with radii larger than the mean interparticle separations in the plasma. A cut-off procedure must be introduced; that used in the OP work is described by Hummer and Mihalas (1988). A main objective has been to make accurate and extensive calculations of atomic data for all cosmically abundant elements in all stages of ionisation. We use techniques (see Berrington et al., 1987) which make extensive use of those developed over many years at The Queen’s University of Belfast and at University College London: the atomic physics groups at both of those institutions were founded by Harrie Massey, in the 1930s and 1940s. The international O P team includes workers in France, Germany, the United Kingdom, the United States and Venezuela. The atomic physics work is described in a series of papers, “Atomic Data for Opacity Calculations” in the Journal of Physics B (to date, 18 papers are published or in press). A summary of the atomic physics work is given by Seaton et al. (1992). For conditions in stellar envelopes, the ionisation equilibrium for each element k depends on temperature T and electron density N , but is insensitive to the mixture of chemical elements. The OP procedure is therefore to calculate and archive the monochromatic opacities rck(v) for each element, on a mesh of values of T and N , . It is then a simple matter to calculate opacities K ( V ) for any mixture and to evaluate the integral (11) to obtain the Rosseland mean, K ~ as, a function of T and p . We have never kept accounts of the total amount of supercomputer time used in the OP work but in all it must be on the order of several thousand hours. In presenting results it is convenient to use the OPAL variable R = p/T& where p is density in gcm-3 and T6 temperature in lo6 K. For a given stellar least, to within an order of model, R is approximately constant-at magnitude or so. Astronomers describe mixtures in terms of values of X , Y and 2,fractions by mass of hydrogen, helium and metals. The composition of the sun is usually taken to be X = 0.70 and Z = 0.02, implying Y = 0.28. It is also necessary to specify the relative abundances of the metal atoms. I will here give a comparison between the new OP results and the older LAOL ones as tabulated by Weiss et al. (1990). Figure 3 shows Rosseland mean opacities plotted against log(T) for X = 0.70, Z = 0.02 and log(R) = - 3 and - 4, corresponding to envelopes of giant stars of interest for
xi
304
M . J . Seaton 2
1
I
h
u“
v
M 0 G l
FIG.3. Rosseland mean opacities for X = 0.70, Z = 0.02, Cox-Tabor abundances. Open circles are from LAOL, full lines are from OP, for two values of log(R) (see text). The opacities given here are per unit mass (units cm2 per 9); the opacity per unit length, used in (10). is then PKn,
studies of pulsations. I use the “Cox-Tabor’’ relative abundances of metal atoms, mixture M4 of Weiss et al., There are four maxima, or “bumps,” in the opacities, labelled X, I: 2 and A. The main contributions are as follows: X, hydrogen in excited states (hydrogen in its ground state gives absorption at too high a frequency to be of importance in the region of the X bump); E: ground-state hydrogen; 2, complex line spectra of metals; A, abundant elements (C,N, 0,.. . ) in H-like and He-like states. It is seen that the O P (and equally the OPAL) calculations give opacities in the region of the Z bump that are much larger than those from LAOL, by as much as a factor of 3 for the case of log(R) = -4. The 2-bump opacity is largely responsible for driving stellar pulsations. It is of interest to consider the complexity of the monochromatic spectra. One uses the variable u = hv/kT. There are negligible contributions to rcR from u > 20 and the dominant contributions are from 0 2 u 2 10. For individual elements it is convenient to use the extinction cross section a(u) such that K(U) = Na(u), where N is the atom number density. Iron is a major contributor to the opacity in the region of the 2 bump. Figure 4 shows values of a(u) for iron, log(T) = 5.30, log(N,) = 18, corresponding to log(R) = -3.604 for the composition of Fig. 3. There are three regions: (1) 0 < u < 10 (so many lines that much of the plot is completely black); (2) 4.9 < u c 5.1 (individual lines just beginning to emerge); (3) 4.99 < u < 5.01 (at last, the lines fully resolved). For Fig. 4, I have used lo6 data points in the range 0 c u < 20. In practice one can use a smaller number (I usually use lo4, which gives K~ correct to within about 1%)-the technique is that of opacity
305
HOW OPAQUE IS A STAR?
N
m
.-c
0
b h
2 b
v
-2
v
tQ
0 -4
2
0
0
6
4
10
U = hu/kT 0
-1 h
b
v
w
3
-2
L
4.9
I 5
, 5.1
4 99
5.00
5.01
U
U
FIG. 4. Extinction cross sections for iron, log(T) = 5.30, log(N,) = 18. Stimulated-emission factors are included.
sampling. There is no shortcut in the need to compute huge amounts of atomic data. The OP atomic data will have many other applications. Selected data will be published in a book. The complete data are being made available through a database, TOPBASE (The Opacity Project), developed by Claudio Mendoza and Walter Cunto of IBM Venezuela. It is very gratifying that the new opacities from the two projects, OPAL and OP, are in substantial agreement. They will be of importance for all future studies of the evolution, structures and properties of the stars. REFERENCES Alexander, J. D. H., Bowen, P. J., and Heddle, D. W. 0. (1964). Proc. Roy. SOC. A 279, 510. Allen, C. W. (1973). Astrophysical Quantities, 3rd ed., Athlone Press, London. Bates, D. R. (1978). Phys. Rep. 35, 305.
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Bell, K. L.,and Berrington, K.A. (1987). J . Phys. B 20,801. Berrington, K. A., Burke, P. G., Butler, K., Seaton, M. J., Storey, P. J., Taylor, K. T., and Yu Yan. (1987). J . Phys. B 20,6379. Branscomb, L. M., and Smith, S. J. (1955). Phys. Rev. 98,1028. Carnochan, D. J. (1982). Mon. N o t . R . Astr. Soc. 201, 1139. Carson, T. R., Mayers, D. F., and Stibbs, D. W. N. (1968). Mon. N o t . R . Astr. SOC.140,483. Carson, T. R., Hiibner, W. F. Magee, N. H., and Merts, A. L. (1984). Astrophys. J . 283,466. Chalonge, D., and Kourganoff, V. (1946). Ann. Astrophys. 9, 69. Chubb, T. A., and Byram, E. T. (1963). Astrophys. J. 138,617. Cox, A. N. (1965). In: Stars and Stellar Systems (L. H. Aller and D. B. McLaughlin, eds.), 8, University of Chicago Press, Chicago. Cunto, W., Mendoza, C., Ochsenbein, F.,and Zeippen, C. J. (1993). Astron. Astrophys., in press. Gaustad, J. E., and Spitzer, L. (1961). Astrophys. J . 134,771. Hoyle, F.,and Wickramasinghe, N. C. (1963). Mon. Not. R . Astr. Soc. 126,401. Hummer, D.G., and Mihalas, D. (1988). Astrophys. J. 331,794. Iglesias, C. A., and Rogers, F. J. (1991a). Astrophys. 1.371,408. Igelsias, C. A., and Rogers, F. J. (1991b).Astrophys. J . 371,L73. Iglesias, C. A,, Rogers, F. J., and Wilson, B. (1987). Astrophys. J . 322,L45. Iglesias, C. A., Rogers, F. J., and Wilson, B. (1990). Astrophys. J. 360,221. John, T.L. (1992). Mon. Not. R . Astr. Soc. 256, 69. Kurucz, R. L., Peytremann, E., and Avrett, E. H. (1974).Blanketed Model Atmospheresfor EarlyType Stars, Smithsonian Institution, Washington, D.C. Leckronne, D. S., Johansson, S., Kurucz, R. L.,and Adelman, S. J. (1990).In: Atomic Spectra and Oscillator Strengthsfor Astrophysics and Fusion Research (J. E. Hansen, ed.), North-Holland, Amsterdam, p. 3. Massey, H. S. W. (1938).Negative Ions, 1st ed., Cambridge University Press, Cambridge; 2nd. ed., 1950; 3rd ed., 1976. Massey, H. S. W., and Bates, D. R. (1940). Astrophys. J . 91,202. Mihalas, D., and Morton, D. (1965). Astrophys. J . 142,253. Rogers, F. J., and Iglesias, C. A. (1992). Astrophys. J . Supp. 79, 507. Schwarzschild, M. (1958). Structure and Evolution of the Stars, Princeton University Press, Princeton, N.J. Seaton, M. J. (1992). Inside the Stars, IAU Colloquium 137, Vienna, April 1992 (W. W. Weiss and A. Baglin, eds.) Astron. Soc. Pacijc 40,222. Seaton, M. J., Yu Yan, Mihalas, D., and Pradhan, A. K. (1993). Mon. N o t . R . Astr. Soc., in press. Seaton, M. J., Zeippen, C. J., Tully, J. A. Pradhan, A. K., Mendoza, C., Hibbert, A,, and Berrington, K. A. (1992). Rev. Mex. Astron. y Astrofs. 23 (Special issue, Workshop on Astrophysical Opacities), 19. Simon, N. (1982). Astrophys. J . 260,L87. Smith, S. J., and Burch, D. S. (1959). Phys. Rev. 116, 1125. Stecher, T.P., and Milligan, J. E. (1962). Astrophys. J . 136, 1. Weiss, A., Keady, J. J., and Magee, N. H. (1990). Atomic data and nuclear data tables, Tables 45, 209. Wildt, R. (1939). Astrophys. J. 89, 295. Wilson, R., and Boksenberg, A. (1969). Ann. Rev. Astron. Astrophys. 7,421. Wishart, A. (1979) Won. Not. R . Astr. SOC.187,59P.
ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 32
STUDIES OF ELECTRON ATTACHMENT A T T H E M A L ENERGIES USING THE FLOWNG AF TERGLO W-LANGMUIR PROBE TECHNIQUE DAVID SMITH and PATRIK S P A N E L Instiiut fur Ionenphysik Universitat Innsbruck Innsbruck. Austriu
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
308
11. Basic Attachment Processes at Low Energies . . . . . . . . . . . . . . 309 111. Theoretical Description of Electron Attachment . . . . . . . . . . . . . 312
IV. Some Experimental Techniques Used to Study Electron Attachment . . . . V. The Flowing Afterglow-Langmuir Probe Technique . . . . . . . . . . . VI. Results from the FALP Experiments; Comparisons with Results from Other Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . A. CCI,, SF, and Some Halomethanes: Activation Energies . . . . . . . B. CH,Br and C,F,: Comparison of FALP Data with Rydberg Data . . C. Dissociative Attachment to Some Haloethanes: Comparison of FALP Data with Data Obtained Using Nonthermal Techniques . . . . . . D. Dissociative Attachment to the Radicals CCI, and CCI,Br . . . . . . E. Br,- Formation in Electron Attachment: The Need for Mass Spectrometry . . . . . . . . . . . . . . . . . . . . . . . . . . F. Dissociative Attachment to Acids and Superacids . . . . . . . . . . G. Dissociative Attachment and Associative Detachment: The Reactions HBr + e S B r - + H; HI + e = I - + H . . . . . . . . . . . . . . VlI. Recent Developments Using the FALP Apparatus. . . . . . . . . . . . VIII. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
313 316 319 319 323 325 327 330 331 332 335 340 34 1 342
*We dedicate this paper to Professor Sir David Bates, F.R.S., a constant supporter and friend, in recognition of his many outstanding contributions to atomic and molecular science, with the hope that the work described in the paper will stimulate him to apply his considerable insight and understanding to the theory of low-energy electron attachment reactions.
307 Copynghl 0 1994 by Academic Press, Inc. All nghts of reproduction in any form reserved. ISBN 0-12-003832-3
308
D. Smith and P . SpanEl
I. Introduction The study of electron interactions with molecules has a long history. Early work was concerned with the determination of ionization cross sections using electron beams into gas targets and ionization coefficients in gas discharges. Later, the phenomenon of electron attachment to molecules, a relatively lower energy phenomenon, came under scrutiny. Much of the early work in these areas is covered in McDaniel’s (1964) book. Electron attachment can occur to any gas atom or molecule that has a stable negative ion state; oxygen is a well-known example. An electrical discharge through air results in the formation of 02-ions and 0-ions (by the processes of “direct” three-body attachment and dissociative attachment; see Section 11), production of the latter ion from an O2 molecule requiring electrons with energies in excess of 4 electron volts. So at thermal energies (i.e., temperatures accessible in laboratory vessels) and at low electron energies, say < 1 eV, 0-cannot be produced from 02.However, electron attachment reactions do occur for a large number of different molecules at thermal energies-SF, is the classic case-and it is thermal electron attachment reactions with which this chapter is largely concerned. Studies of electron attachment are fundamentally very interesting, but are also of practical value in that they are critical to the proper understanding of media in which electrons are converted to negative ions, since attachment alters the physical properties of the media and greatly influences the chemistry also. Thus electron attachment studies are important to the understanding of the terrestrial atmosphere (and other planetary atmospheres, especially the Venusian atmosphere), to the optimization of etchant plasmas for microcircuit production, to the understanding of halogen-containing gas lasers and, not least, to the choice of gases to insulate high voltage devices. The focus of this chapter is the study of electron attachment reactions under thermalized conditions using the flowing afterglow-Langmuir probe (FALP) technique. Several different attachment phenomena have been observed and are described, and comparisons are made between the FALP data and those data obtained using other techniques. To facilitate such comparisons, brief descriptions of the essential features of the other techniques and a more detailed description of the FALP technique are presented. N
309
ELECTRON ATTACHMENT
11. Basic Attachment Processes at Low Energies All electron attachment reactions with molecules are of two types, viz., “dissociative attachment,” exemplified by
CCl,
+ e + C1- + CCl, + AE
(1)
and “direct attachment,” exemplified by SF,
+ e + (SF,-)* + SF,- + AE
(2)
In some dissociative attachment reactions, there are two ionic products, e.g., CCI,Br
+ e + Cl-( + CCl,Br),
Br-(
+ CCl,)
(3)
and in some reactions both direct and dissociative products are evident, as is the case for SF, at electron energies >, 0.1 eV (see later). The potential curves for the dissociative attachment reaction (1) are represented in Fig. l(a), where it can be seen that the attractive CCl, molecular curve (represented by a single C1,C-Cl coordinate) is crossed at its minimum by the repulsive C1,C-CI- curve. The exothermicity of the reaction, AE, is given by the difference of the electron affinity (EA) of the C1 atom, EA(CI), and the CI,C-Cl bond dissociation energy, D(CI,C-Cl); i.e., AE = EA(CI) - D(C1,C-Cl). Reaction (1) is very fast, at 300 K, occurring essentially in every electron-molecule collision (see Section 111). However, the similar dissociative attachment reaction (4) of CCl,F, is relatively slow, at 300K, but becomes faster as the molecule temperature is increased. This is attributed to the influence of an “activation energy barrier,” E,, in this reaction (Wentworth et al., 1967), the origin of which is illustrated in Fig. l(b), where it can be seen that the repulsive ionic curve crosses the neutral molecular curve above its minimum. Thus, excitation energy is required to promote the reaction CCI,F,
+ e -+
C1-
+ CCIF, + AE
(4)
Note, however, that reaction (4) is exothermic to produce C1-, and endothermic to produce F - (see Fig. l(b)), a consequence largely of the strong C-F bonds compared to the weaker C-Cl bonds. Note that the potential curves for CCl,- and CCI,F,- show only very shallow wells, since these negative ions are not stable and at best have very short transient lifetimes (<< 1ps). Other dissociative attachment reactions can proceed by the formation of excited long-lived negative ions that then dissociate into fragments. Such a case is the C1, reaction (Christodoulides et al., 1975), which is represented in Fig. l(c): C1,
+ e + (CI,-)* -,C1- + C1+ AE
(5)
FIG.1. Representative potential curves for some electron attachment reactions. In all four figures the symbols have the following meanings. E.A. is electron affinity; AE is the exothermicity of the reaction; Ea is the activation energy. Note that the internuclear coordinate is in Angstrom units (A) and that the energy coordinate is in electronvolts (eV). (a) The exothermic dissociative attachment reaction of CCI,; the ionic product is CI -. The repulsive ionic curve (no significant well in the CCI,- potential) crosses the attractive curve at its minimum. The reaction is fast at 300 K and does not require activation energy.
EleVl:' 0 -
-1
-
-2
-
-3
-
' * '
I
' ' "'
I
' ' ' "
' ' '
'
' ' '
CCIF,+CI
I I
CCl,F,
'i
.
-4-
: b) -5 -, , , , , , , , , , 0 1 2
,
,
,
,,
3
, ,
, , , , ,:
4
5
R IA1 (b) (b) The exothermic dissociative attachment reaction of CCI,F,; the ionic product is CI-. The repulsive ionic curve (no significant well in the CClzFz- potential) crosses the attractive curve above the minimum, and thus the reaction is relatively slow at 300 K and requires activation energy.
3 10
ELeVl;
-
0 -
' ' '
'
'
I
'
' '
I
'
I
I
'
'
I
c1,
-2 -3 -4 -
'
' "'
a
'
I
'i
:
CI+CI
-1
-5
E,
v
I I I
"
\ \
I *
I
.
_----.-
/
* I k : .
I
I
I
I
n-
cI+cI-:
,OdC
'
cr'
4
-
AE
I
I
I
,
I
I
-
R [A1
(4 (c) The exothermic dissociative attachment reaction of CI,. Note that the potential curve for the stable C1,- crosses the neutral CI, curve above its minimum and thus activation energy is required to promote the reaction. Note also that the reaction is exothermic to produce CI- (and CI), which is the only ionic product observed in the F A L P experiment.
(4 (d) The exothermic direct electron attachment reaction of SF,; the only ionic product at 300 K is SF,-. Note that the SF,- ionic curve crosses the neutral SF, curve at its minimum, so no activation energy is required to produce SF,-. Note also that the SF,- + F asymptote lies above the SF, curve by 0.1 eV, the endothermicity of the dissociative attachment reaction, and that the repulsive curve (dash-dot) crosses the SF, curve above the minimum, indicating that an activation energy is required to promote the dissociative attachment reaction that results in SF,-.
311
D. Smith and P . Span51
312
Note the crossing point of the curves that indicates that reaction (5) requires activation energy, as experiment confirms (Smith et al., 1984). At high pressures in the reaction cell, the (Cl,-)* can be quenched by collisions producing the stable C1,- molecular ion. A similar example is that of SF, (reaction (2)), the potential curves for which are represented in Fig. l(d), but with the very important difference that the SF6- curve crosses the SF6curve at its minimum. Thus the (SF,-)* forms very efficiently at low temperatures, and the loss of some energy results in the stable SF6- ion. At very low pressures, the (SF,-)* can autoionize (lifetime is > 10 ps (Christophorou et al., 1984)) or radiate to stabilize the SF,-, but at high pressures, the usual situation in which SF6 is used as an efficient electron scavenger, collisional stabilization occurs. With increasing temperature and at higher electron energies, SF,- appears as a product of the reaction; i.e., a dissociative channel opens:
+
SF6 e + (SF,-)*
+ SF,-
+ F - AE
(6)
The endothermicity of this reaction is 0.1 eV and it has an activation energy of 0.43 eV (Fehsenfeld, 1970). At even higher electron energies, F- becomes the dominant product; indeed, at high electron energies dissociative electron attachment occurs usually via resonances (Christophorou et al., 1984). We shall not be concerned with this phenomenon in this chapter, but to repeat, we will be concerned only with attachment reactions at and near to thermal energies. These then are the basic, most common attachment processes, but in Section VI of this chapter we present data from the FALP experiment on other kinds of processes, including one for an attachment reaction (C6F6) that slows down with increasing temperature, the formation of Br,- from dibromo organic molecules and the first measurements of dissociative attachment to molecular radicals (CCl, and CC1,Br).
III. Theoretical Description of Electron Attachment No detailed theoretical description of electron attachment that considers energy barriers, resonances, etc., has been constructed. However, of great value is the calculation by Klots (1976) of the capture cross section for slow swave electrons (zero relative angular momentum) by molecules. Thus he obtained the expression nu; q,,,, =(1 -e~p[-4(2crE)''~]) 2E
(7)
ELECTRON ATTACHMENT
313
for the maximum cross section, where a, is the Bohr radius, E is the dimensionless electron energy (in fractions of Rydbergs) and a is the dimensionless polarizability of the molecule (in fractions of a:). In the limit of vanishing electron energy (E << l/a),this reduces to the quantum-mechanical Langevin cross section:
For high electron energies (E >> l/a), omax approaches the de Broglie s-wave cross section, n3tz(= na32E). Although the cross section is the more fundamental quantity, the thermally averaged rate coefficient, p, is more useful to the discussions in this chapter. Thus the maximum value of p, i.e., calculated from n3t2 by averaging over a thermal velocity distribution for the electrons, is 5 x 10-’(300/T,)’’* cm3 s - ’ (Warman and Sauer, 1971). However, using Eq. (7) and again thermally averaging omax, a value of pmaX at room temperature for SF, of 2.64 x lo-’ cm3 s - l is obtained, and similarly for CCl,, p,, at room temperature is 3.29 x lo-’ cm3 s - l (Klots, 1976). Both these values are in good agreement with experiments, indicating that both SF, and CCl, are “perfect” electron scavengers at low electron energies. This p,, concept is useful in that coupled with experimental values of fl, the “efficiency” of electron attachment (/3/jmax) can be obtained. The measured p is very often less than fl,, and this is usually attributed to the influence of activation energy barriers, as described in Section 11.
a,,,,
IV. Some Experimental Techniques Used to Study Electron Attachment The cross sections for electron attachment, a(E),have been measured over wide electron energy ranges for many species using electron beams. The techniques used in that work are fully described in the texts by Massey (1976) and Christophorou et al. (1984) and will not be discussed in this chapter. Rather, here we are concerned with the techniques used to determine electron attachment rate coefficients in the low-energy (thermal or near thermal) regime. The electron swarm technique exploited notably by Christophorou and colleagues (Christophorou et al., 1969, 1984) has resulted in a large body of data on electron attachment. In this technique, electrons are drawn through a buffer gas of number density N (He, Ar and N, are commonly used) under the
314
D. Smith and P . Spang1
influence of an electric field, E. Small admixtures of attaching gases in the buffer gas result in the removal of electrons from the gas and hence in a reduction of the electron current. Thus the attachment rate coefficient is deduced. This is a nonthermal technique; i.e., the “electron temperature” and attaching gas temperature are not equal except at very low &IN. Also the electron energy distribution function, f(E),is usually non-Maxwellian, but f(E)as a function of E / N is known for the buffer gas used and so cross sections, a(E), and rate coefficients, p(E), can be derived by deconvoluting the swarm data. Since the temperature of the attaching gas molecules is usually held constant as the electron energy is varied, this can result in large differences between the derived p ( E ) and the p(T) for the same reactions obtained from truly thermal FALP experiments (see Smith et al., 1989, and Section 1V.C for further discussion of this). In some of these swarm experiments the gas temperature has been increased above room temperature, and from the shape of the electron arrival time spectrum, the autodetachment lifetimes of some transient negative ions have been deduced (see Datskos et al., 1992, and the references therein). In none of these swarm experiments were the product negative ions positively identified by mass analysis. Very worthy of note here is the early “pulse sampling technique” of Wentworth et al. (1967). In this work, gas chromatography was used to prepare samples of attaching gases and electron capture detectors were used to determine electron attachment rates at different temperatures. From these careful experiments, activation energies, E,, were determined for several attachment reactions, some of which have been verified by the much later FALP experiments (see Section V1.A). In the Krypton photoionization method developed by Chutjian and colleagues (Ajello and Chutjian, 1979; Chutjian and Alajajian, 1985), lowenergy free electrons at a given energy are generated in a mixture of Krypton and an attaching gas by photoionizing the Krypton atoms near threshold. By increasing the photon energy, free electrons with well-defined energies can be generated in the gas mixture that react with the attaching gas molecules. The product negative ions are detected and measured using a mass spectrometer. To obtain attachment rate coefficients as a function of electron energy, Maxwellian averaging of the data is performed and the rate coefficients obtained are also normalized to the thermal rate coefficients obtained using other techniques (e.g., the FALP). Most of these experiments have been carried out at a fixed attaching gas temperature ( 300 K), but recently temperature variation of the attaching gas has been introduced (Chutjian et al., 1990). A recent variation of this photoionization technique (Klar et al., 1992) involves the two-laser excitation of Ar atoms to produce high Rydberg states of the atoms (see later) or free electrons and Ar+ ions. In this way, very high energy resolution for the electron has been achieved (-0.2 meV); the N
315
ELECTRON ATTACHMENT
results obtained for attachment to SF, are particularly interesting (Klar et al., 1992). An interesting approach to the study of the electron energy dependence of attachment reactions involves the transfer of the very loosely bound electrons from Rydberg states of rare gas atoms or alkali metal atoms to attaching molecules (Foltz et al., 1977).The assumption is that the distance between the Rydberg electron and the ionic core is so large that the attaching molecule does not interact with both the electron and the core ion. Then the theory predicts (Matsuzawa, 1971) that the rate coefficient for the attachment process should equal that for a free electron having the same velocity as the electron in the Rydberg orbit. The upper limit to the electron energy is about 20 meV, a limit imposed by postattachment interactions (Kalamarides et al., 1989),and the lower limit is about 0.01 meV (Ling et al., 1992), equivalent to Rydberg electrons in orbits for which the principal quantum number n 400 (these are readily field ionized even by weak stray electric fields). For very fast good agreement is obtained attachment reactions (i.e., when /? ?/),, between the /? measured by this method and by “free electron” methods. However, when fl varies with temperature then differences sometimes appear (see Section V1.B). The Cavalleri technique (Cavalleri, 1969) has been used by Crompton and colleagues (Crompton and Haddad, 1983; Petrovik and Crompton, 1987) to obtain accurate data on attachment reactions at thermal energies. In this method, ionization is created in a gas mixture (usually N , with a small admixture of attaching gas) by a burst of X-rays, and the rate of loss of the electrons is determined by observing the light intensity resulting from a timedelayed R F discharge through the N , . The light intensity is proportional to the number of unattached electrons. To date, mass analysis of the product ions has not been carried out in these experiments. The pulsed radiolysis technique developed by Hatano and colleagues (Toriumi and Hatano (1985)) and recently adopted by Shimamori et al. (1992a, b), utilizes a pulsed high-energy electron beam to generate X-rays at a thin tungsten foil. The X-rays then partially ionize a gas mixture in a microwave cavity, and the resonant frequency of the cavity is followed to determine the electron number density; thus the attachment coefficient is deduced. The temperature of the gas can be varied over the range 77-373 K, and the electrons can be heated by microwave energy. Therefore, the separate influences of the electron energy and the attaching molecule temperature on the attachment rates can be investigated, if the electron energy can be determined. Measurements of the rate coefficients for attachment to several molecules within the mean electron energy range 0.031.0eV have recently been reported (Shimamori et al., 1991, 1992a,b). Again, this experiment does not include mass analysis of the product ions of the attachment reactions.
-
-
316
D.Smith and P . Spang1
V. The Flowing Afterglow-Langmuir Probe Technique The flowing afterglow-Langmuir probe (FALP) technique is remarkably versatile in that it can be used to study a variety of plasma reaction processes under truly thermal conditions. It is the result of the successful application of the Langmuir probe diagnostic method to the flowing afterglow plasma that was first developed to study ion-molecule reactions (Ferguson et al., 1969). The probe technique used had been developed previously for the determination of electron and ion number densities and electron temperatures in pulsed (stationary) afterglow plasmas by Smith and colleagues (Smith et al., 1968; Smith and Plumb, 1972, 1973) and used to determine ambipolar diffusion coefficients (Smith et al., 1972), dissociative recombination coefficients (Plumb et al., 1972), and electron temperature relaxation rates (Dean et al., 1974). The application of this probe technique to the more chemically versatile flowing afterglow (Smith et al., 1975) has allowed the determination of the rate coefficients for ion-ion recombination reactions (Smith and Church, 1976), electron-ion dissociative recombination reactions (Alge et al., 1983; Adams et al., 1984), and the electron attachment reactions discussed in Section VI of this chapter. A schematic of the FALP apparatus is given in Fig. 2. The principle of operation is as follows. Flowing afterglow plasmas are created along a flow tube (- 100 cm long and 8 cm in diameter) by a continuous microwave cavity discharge in the upstream region of fast-flowing carrier gas (usually pure Helium or Argon) at a pressure of typically 1 Torr. The flowing carrier gas transports ions (and electrons) from the discharge down the flow tube, and during the transport the initially “hot” electrons cool to the carrier gas temperature and thus a thermalised afterglow plasma is created in the downstream region. Relaxation of the electron temperature (energy) occurs largely in collisions with the carrier gas atoms; this is particularly fast for Helium carrier gas at the pressures used, but much slower for Argon at the same pressure due to the deep Ramsauer minimum in Argon (see Dean et al., 1974, and Section VII). The electron (?I=), positive ion (n,) and negative (n-) ion number densities in the afterglow plasma can be determined by a single, cylindrical Langmuir probe, which can be readily positioned at any point along the axis of the flow tube (coordinate z) to a spatial accuracy of about 1 mm. The plasma flow velocity is readily determined (Adams et al., 1975) and is typically lo4 cm s- l, so the equivalent time resolution is typically 10 ps. A pinhole orifice-differentially pumped mass spectrometer-detection system is located at the end of the flow tube (see Fig. 2), which is used to determine the positive ion and negative ion composition of the afterglow plasma, Temperature variation over the approximate range 80-600 K is achieved by heating h-
317
ELECTRON ATTACHMENT
electron
Roots pump microwave cavlty and discharge
c -;
+tiprobe
mass spec. channeltmn collapse of ambipolar field,
reaction
free diffusion of electrons
m
Z
electron I ion afterglow
pos. ion I n e g . ion afterglow
Y C
1
80
1
1
60
1
1
1
LO d i s t a n c e , 2 I cm 1
1
20
diffusion or recombination controlled
1
0
FIG. 2. A schematic diagram of the flowing afterglow-Langmuir probe (FALP) apparatus, showing the plasma diagnostic instruments (the movable probe and the mass spectrometer), ionization source (microwave discharge), and gas entry ports (inset). Typical variations are indicated of the electron (ne),positive ion ( n , ) and negative ion ( n - ) number densities along the flow tube in the downstream region where electron attachment occurs. In the upstream region electrons and positive ions are lost only by the slow process of ambipolar diffusion. 8 represents the "end correction"; see text.
or cooling the complete flow tube and by preheating-precooling the carrier gas. The procedure used to determine electron attachment coefficients is as follows. In Helium carrier gas the metastable Helium atoms (2 'S and 2 3S) generated in the discharge must be prevented from entering the downstream afterglow plasma, where they can ionize most molecules and thus be an unwanted source of electrons. These metastable atoms (He") are readily removed by the addition of a small amount of Argon upstream; the Penning reaction between the He" and Ar result in the production of Ar+ ions and electrons. The Ar atoms also rapidly charge transfer with He, ions (formed from the He+ + 2He reaction), and thus only the atomic ions He+ and Ar+ together with electrons constitute the downstream afterglow plasma. Since these atomic ions do not recombine with electrons at a significant rate, the only loss process for electrons from the plasma is ambipolar diffusion, +
318
D . Smith and P . Spang1
which is slow at the pressures used (see the graph in Fig. 2). Now electron attaching gas is introduced into the afterglow plasma at a measured flow rate in order to establish a suitable number density of the attaching gas, n,, in the plasma. The attaching gas is introduced via the “ring ports” (shown schematically in Fig. 2) and against the carrier gas flow; this ensures that the gas is rapidly dispersed across the diametric plane and the “end correction” is thus minimised (being typically 2 cm). Electron attachment then occurs and results in an increase of the gradient of n, down the flow tube, as is illustrated in the graph in the lower part of Fig. 2 (see also Figs. 5 and 6). Proper analysis of the n, versus z data, coupled with the known value of n,, provides a value for the attachment coefficient, Thus,
where vd = Dae/A2, v, = nmp, and A is the characteristic diffusion length of the flow tube. (For details of the derivation of this equation, see the original papers by Oskam, 1958, and Biondi, 1958, and the paper by Smith et al., 1984.)The value of v,, and hence p, is obtained by a computer curve-fitting procedure, the value of the diffusion time constant v,, being obtained in the absence of attaching gas. It is most important in these experiments to be aware of the possible loss of electrons by dissociative recombination of the molecular ions that can be formed in the reactions of He’ and Ar’ ions with the attaching gas. Such ion-molecule reactions are usually very efficient, and for slow attachment reactions that necessitate a relatively large n,, the nonrecombining atomic ions He’ and Ar’ will be quickly converted to recombining molecular ions. Then n, would reduce due to recombination as well as by electron attachment. To avoid this disastrous complication, the experiment must be conducted at a low initial n, (i.e., n,(O) in Eq. (9)) to prevent recombination occurring (in practice, n,(O) must be 109cm-3). Also, when studying the most rapid attachment reactions (such as SF, and CCl,), n, must be small enough that n, does not reduce too rapidly along z. However, n, must always be greater than n, to prevent a significant axial gradient in n, that would result in a departure from first-order kinetics (this would complicate the data analysis, but see Section V1.D). In practice, the condition n, >> n, is readily achieved when n,(O) is < 108cm3. When n,(O) is deliberately increased to be close to n,, then it is possible to study the process of “secondary electron attachment”; in this way the p for the dissociative attachment reactions of the radicals CCl, and CC1,Br were determined (see Section V1.D). Using the FALP technique with thought and care, an exceptional range of attachment processes can be studied. Summaries of the results obtained to date are presented in the next section.
-=
ELECTRON ATTACHMENT
319
VI. Results from the FALP Experiments; Comparisons with Results from Other Experiments The variety of the thermal energy attachment processes studied with the FALP apparatus is quite wide, including very fast attachment reactions (p pmaX)that show little variation of p with temperature and slower for which the p usually increase with temperature and for reactions (p << bma3 which activation energies have been derived; the first observations of attachment to unstable molecular radicals (CCl, , CC1,Br); observations of Br, - formation from organic dibromides; attachment to acids and superacids (e.g., H2S04, FS0,H) and dissociative attachment to the hydrogen halides (HBr, HI).We address the interesting results of these studies in this section and compare the FALP results with those obtained using other techniques. The differences in the results obtained using these different techniques are dramatic for some reactions. The accumulated data from the FALP studies are given in Table I, segmented according to the papers in which they were reported. The attachment coefficients, p, are given at 300 K together with the ion products and the E, values where appropriate. (The /3 values at several temperatures are given for some reactions in Fig. 3).
-
A. CCl,, SF,
AND
SOMEHALOMETHANES: ACTIVATIONENERGIES
The first FALP study of electron attachment reactions (Smith et al., 1984) included the CCl, and SF, reactions, both well known to be very rapid (see the data compilation in Smith et al., 1984). To measure the p for these (and other fast reactions), the attaching gas had to be diluted by Helium to facilitate the accurate measurement of its flow rate into the flow tube. Thus a 1% mixture was usually prepared. The p for CCl,, i.e., p(CCl,), was measured at 205, 300,455, and 590 K and was close to p,, (see Section 111) at all four temperatures; clearly, no activation energy barrier exists to this reaction (see Fig. 3). The p(CCl,) was measured to be about 15% greater than &,, calculated from the formula by Klots (1976), which is just within the percentage error figure we placed on the measured values at 300 K. A weak but discernible reduction of p(CCl,) with increasing temperature occurred, which is consistent with theoretical expectations for uninhibited s-wave capture (see Section 111). The value obtained for p(SF,), as expected, was also close to p,,, but again some 15% higher. When the FALP values of p for fast reactions are compared with the theory and values obtained using other techniques (see the table in Smith et al., 1984), it is hard to escape the conclusion that the FALP experiments are indeed some 10% too large,
TABLE I
RATE COEFFIcmm, B, AND THE PERCENTAGE (%) PRODUCT IONSFOR THE ELECTRON ATTACHMENT REACTIONS OF THE M O L E C U INDICATED L~ AT 300 K USINGTHE FALP APPARATWS. THE/3 VALUES ARE GIVENAS, FOR EXAMPLE, 2.q- 9) cm3s-' MEANING 2x cm3s- '. WHEMNo % 1s GIVFN, THENONLYONEPRODUCT IONWASOBSERVED OR THE PRODUCT RATIO COULDNOTBE DETERMINED ACCURATELY. THEE , (GIVENIN meV) ARE m ACTIVATION ENERGIES DETERMINED FROM THE SLOPES OF ARRHENIUS-TYPE PLOTS; - MEANS THATB DOES NOT INCREASE WITH TEMPERATURE; MEANSTHAT AN E, WASNOTDETERMINED 5
Molecule
B
E,
Products (%)
Molecule
(cm3s-')(meV)
W
0
B
E. (cm3s-')(meV)
a
SF, CCI, CHCI, CCI,F CCI,F,
2.q -9) 3.1(-7) 3%-7) 4.q - 9) 2 4 - 7) 3.24 -9)
CH,Br, CH,Br CF,Br CH,I C-CP,,
b 9 4 - 8) 6.q- 12) 16-8) 1.a-7) 6 4 - 8)
50
c1-
-
SF,-
-
c1c1-
-
120 20 150
c1-
f CF,Br, CFBr, CF,BrCF,Br CH,BrCH,Br CH,CICH,Br
3.q--7) 4.8( -9) 1.q - 7) 1.y-8) 1.q-9)
CCI,Br CCI, CC1,Br
8.2( - 8)
-
48
<44 70 156
40
Br-(85), Brz-(15) Br-(90), Br,-(lO) Br-(80), Br,-(20) Br-(97), Br,-(3) Br-(80), C1-(20)
c1g
50 300 80 25
Products (%)
Br BrBrIc-C,F 14
2 4 - 7) 1s-7)
u
h HBr HI
<3( - 12) 3.q - 7)
--
300 -
Cl-(-55), Br-(-45) c1CI-, Br-
BrI-
I
/
dd % *n LL4 u u
0
P ?
321
322
D.Smith and P . spang1
FIG.3. Arrhenius-type plots (i.e., In fl versus l/<) for several electron attachment reactions studied using the FALP apparatus for truly thermal conditions (i.e., = T,).Activation energies, E,, for the reactions are derived from the slopes of the continuous lines. The E , are given for these and some other reactions in Table I. The steep line for the CH,Br reaction is indicative of a large E , of 300meV. The dashed line represents fl,, = 5 x lo-' (300/T,)"' cm3 s - l (see Section 111 of the text). The C - C , F , ~reaction is a direct attachment reaction; the rest are dissociative attachment reactions. The data are obtained from various papers; see Table I for the source references.
implying some unidentified systematic error in the measurements. This is hardly disastrous but quite perplexing. There appeared to be a small increase in P(SF,) with increasing temperature. Concomitant with this increase, SF, appeared as a product (see reaction (6)) together with the major product SF,-. The appearance of SF,- with increasing temperature is consistent with a small activation energy of 0.43 eV for SF,- production (see Fig. l(d)). It is well known (e.g., see Klar et al., 1992) that the electron capture cross section for SF, is very sensitive to the electron energy, E , in the low-energy regime, and thus P(SF,) will also be sensitive to E. We address this interesting phenomenon in Section VII. Several halogenated methanes were included in these FALP studies (see
ELECTRON ATTACHMENT
323
Alge et al., 1984, and Table I). For these reactions there is a wide spread in the values obtained at 300K and a wide variation in the derived E,. Thus, B(CC1,F) and j3(CH31)are both close to but somewhat smaller than fl,,.., and the small increase in the fl with T indicate small E, of 20 meV and 25 meV, respectively for these reactions (the single product ions are Cl- and I-, respectively). The j3(CH,Br2) is also quite large, the E, for attachment to this molecule is 50meV and the only product is Br- (no Br,- is observed; see Section V1.E). However, P(CCI,F,) and P(CHC1,) are only about 1% of B,,, at 300K, but they increase rapidly with 7: The E , were obtained from the slopes of plots of lnj3 versus T - ' , according to the Arrhenius law, /? = A exp - E,/kT, which we assume applies to these reactions. (Note that A in this expression may loosely be equated to p,, for the reactions.) The Arrhenius plots for these reactions and for several other attachment reactions are given in Fig. 3. The E, are relatively large for the CCl,F, and CHCI, reactions, being 120 meV and 150 meV, respectively, implying that at low temperatures these attachment reactions will be very slow. The origin of the E, for the CCI,F, reaction (4) is indicated in Fig. l(b) (after Wentworth et al., 1967), where it is implied that the increasing vibrational excitation of the reactant molecule with increasing molecule temperature, T,, promotes the dissociative attachment reaction. We refer again to this important principle in Section V1.C. The C1, reaction is also slow at 300 K, and an E, of 50 meV is indicated from the temperature studies (Smith et al., 1984). Notice, however, as is indicated in Fig. l(c), that C1,- is a stable, well-known negative ion, but nevertheless the only product observed under the conditions prevailing in the FALP plasma was C1- from the dissociative attachment reaction. B. CH,Br
AND
C6F.5: COMPARISON OF FALP DATAWITH RYDBERGDATA
The most dramatic increase of dissociative attachment reaction CH,Br
p with T is observed for the CH,Br
+ e + Br- + CH,
(10)
cm3 sK1 at 300 K to /?(CH,Br) increases from the low value of 6 x 2.5 x cm3 s - ' at 585 K (see Alge et al., 1984, and Fig. 3), which is equivalent to a large E, of 300meV, the largest so far measured using the FALP technique. (Note that reaction (10) is exothermic by 0.36 eV.) These data (the p and the E,) have been reproduced by Crompton and colleagues using the Cavallieri method (PetroviC and Crompton, 1987). In contrast, the reaction C6F6
+ f2 + C6F6-
(1 1)
324
D. Smith and P . Span61
rapidly decreases with increasing T from a value close to B,,, at 300 K to a value some 50-100 times smaller at -450 K (see Adams et al., 1985). At the time of measurement, such anomalous behaviour was unprecedented, but subsequently this observation was confirmed in swarm experiments (Spyrou and Christophorou, 1985; Knighton et al., 1992) and using the CavaIleri method (Kondo and Crompton, 1986), although it must be said that the decrease in B with observed in these experiments was not so great as the FALP data indicates. While this phenomenon is still not totally understood, it seems most likely that the rapid decrease in B is due to an increase in the autodetachment rate of the (C6F6-)* excited negative ion with increasing temperature (Spyrou and Christophorou, 1985). Recently, similar behaviour has been observed in the attachment of electrons to c-C4F6 (Datskos et al., 1992). It is very interesting here to contrast the FALP results for the CH3Br and C6F6 reactions with the data obtained for these molecules using the very interesting Rydberg atom method for determining values (Hildebrandt et al., 1978; Marawar et al., 1988). It seems very significant that, while there is reasonable agreement between the /? values derived by the two methods for fast reactions that show little or no temperature variation (Smith et al., 1984), the B values for the CH,Br and C6F6 reactions are quite different as determined by the two methods. The B(CH,Br) derived from the Rydberg experiment at 300 K is lo-’ cm3 s - l , i.e., more than lo3 times greater than the FALP result at 300K; and B(C6F6) from the Rydberg experiment it is 3 x lo-* cm3 s - l , i.e., about a factor of 4 smaller than the FALP result at 300 K. The Rydberg experiments ostensibly are equivalent to experiments performed at 300 K, but this appears not to be so, the sense of the differences between the Rydberg and FALP studies implying “equivalent temperatures” of the Rydberg measurements significantly greater than 300 K. It has been suggested (Marawar et al., 1988) that a possible reason for the lower P(C6F6) obtained in the Rydberg studies is due to autodetachment of the (C6F6-)*, a process that is inhibited if the excited ion can be relaxed in collisions with buffer gas atoms or molecules. This can happen in a swarm medium such as the FALP plasma if the lifetime of the excited ion against autodetachment exceeds the collision time. However, such reasoning cannot be applied to the CH,Br reaction (10). It seems that some details of the interaction of the Rydberg atoms with the attaching molecules are not properly understood. (How important are post-attachment interactions?; see Zheng et al., 1990) Comparisons of the B data obtained for several molecules using the pulsed radiolysis technique and others including the Rydberg atom technique and the FALP technique are given by Shimamori et al. (1992b). To determine whether electron attachment rates decreased with the increasing T for other substituted benzenes, electron attachment to C6F,Cl,
ELECTRON ATTACHMENT
325
C6F5Br and C6F,I was studied using the FALP apparatus (Herd et al., 1989a). However, electron attachment to these molecules was in no way similar to that for the symmetrical molecule C&6. The B for all three reactions are significantly smaller at 300 K than B,,, and each increase with T,, thus exhibiting a significant E , (see Table I). Also, the product of the C6F5Clreaction is only the parent negative ion C,F,CI-, the major product of the C,F,Br reaction is C6F,Br- (with 3% of Br- that increases with T) and the major product of the C6F51 reaction is C6F5- (with the parent C6F,I- ion only -5%). Clearly, different mechanisms are operative in electron attachment to these structurally similar molecules.
-
c. DISSOCIATIVE ATTACHMENTTO SOME HALOETHANES: COMPARISON OF FALP DATAWITH DATAOBTAINED USING NONTHERMAL TECHNIQUES The /Ifor the reactions of electrons with CH,CCl,, CH,ClCHCl,, CF,CCl,, and CF,CICFCl, were determined at 298 K, 385 K, and 470 K under thermalized conditions in the FALP plasma (Smith et al., 1989). The range of p values at 300 K for these four reactions is large, ranging from 2.4 x lo-' cm3s-' for CF,CCl, (and a zero E,) to 3.1 x 10-'0cm3s-' for CH,ClCHCI, (and a large E , of 200 meV) (see Table I). These results are not especially noteworthy in themselves, but the value of the data is seen when it is compared to the fl values derived from the nonthermal swarm experiments of McCorkle et al. (1982). The differences between the swarm and FALP data for the slow reactions (B << B,,,) are dramatic, as can be seen by scrutinizing Fig. qa). How can this be explained? Consider first the differences between the conditions of the FALP and swarm experiments. In the FALP plasma, the electron and attaching molecule temperatures are the same and equal to the Helium carrier gas temperature, T, (see Section VII for further discussion); thus the electrons possess a Maxwellian velocity distribution and the mean energy .!? can be equated to 3/2kT,. In the swarm experiment, the buffer and attaching gas temperature is usually held constant and the E of the electron swarm is increased by increasing their drift velocity by applying an electric field. Now the differences between the two sets of data are clear. Note first that the fl values for all four gases are essentially the same at an .!? of 39 meV (i.e., T, = T, = 300K), that is when the electron temperature, T,, and attaching gas temperature, T,, in the swarm experiment also are closely equal. Note also that, for the fast reaction (CF,CCl,), for which p varies only very slowly with and E, the data from the swarm and FALP are the same. But, for the slow reactions, the /? from the FALP increase rapidly with T,, implying relatively large E, values (see Table I for the values), whereas in the swarm experiments the B increase only slightly, if at all, over the same range
D. Smith and P . SpanEl
326 10' 'UI
"
5
------
a*.
I
d
'
i
Qi 10'
loc
10'
10"
40
50
b0
70
eo
90
100
L)
i [rnevl FIG. 4. Comparisons of the attachment rate coefficients, /3, obtained from the FALP experiments under truly thermal conditions with those derived from the nonthermal swarm and Kr photoionization experiments. The FALP data were obtained at three temperatures, 298,385, and 470 K, but they are plotted in terms of the mean electron energy 6 according to E = 3/2kT to facilitate comparisons with the nonthermal data which were obtained at a fixed attaching gas (see temperature of -298 K and at elevated I?. The upper lines on both figures represent pmax Section 111 of the text). These figures were taken from Smith et al. (1989), with permission. (a) FALP data (filled symbols) and swarm data (open symbols) for the four molecules indicated. Note the rapid increase with E(i.e., with T) of the /3 values measured in the FALP experiment for the three relatively slow reactions. (b) FALP data (filled symbols) and Kr photoionization data (open symbols) for the two molecules indicated. Note again, the rapid increase with l? (i.e., with T) of the /3 values measured in the FALP experiment and the decrease in the fl values determined using the nonthermal technique. Note that a decreasing fl with increasing T, is observed for the CH,Br,, CF,CCI,, and CH,CCI, reactions using the pulsed radiolysis method also (Shimamori et al., 1992b).
of E. Clearly this must be a manifestation of the increasing attaching gas temperature, presumably resulting from the enhanced vibrational excitation of the molecules (refer to Fig. 1 for clarification) as was forcibly pointed out by Smith et al. (1989). A similar comparison can be made between the data obtained using the FALP method and the Krypton photoionization method for the attachment reactions of CH,CCl, and CH,Br,. (The FALP data are from Alge et al., 1984; the Kr data are from Alajajian and Chutjian, 1987.) Again, the differences are startling as can be seen in Fig. qb); the fl from the FALP
327
ELECTRON ATTACHMENT
experiment increase with T, (and hence with E ) and those obtained using the Kr method, in which T, is held constant, decrease with increasing E. As before, the reason for the differences is surely that increasing the temperature of the CH,Br, molecule promotes the dissociative attachment reaction. Clearly these differences are real and not “discrepancies,” as they have been described previously (Alajajian et al., 1988). Generally speaking, the /?decrease with increasing electron energy E, and for slow reactions (i.e., for which p < p,,) the fi increase with increasing T,. In these FALP experiments, both T, and T, (i.e., E ) increase with increasing temperature, and so the change of p then reflects these two simultaneously occurring, often opposing, effects. This may explain the apparent maxima in the B for the CH,I and CFCl, reactions (Alge et al., 1984; Smith et al., 1984). Usually, however, the increase in p due to an increase in T, is much greater than decrease in p due to the same increase in T,. This is very well illustrated for the CF,Br, CHCl,, and CH,CCl, reactions, which have been studied recently using the pulsed radiolysis-microwave heating method (Shimamori et al., 1992b).The difference between the p values obtained by varying both and T, together (FALP) and just T, (pulsed radiolysis) are graphically illustrated in the papers by Shimamori et al. (1992a, b). Clearly, it is desirable to study the influence of both T, and T, separately in one experiment. This we are now able to do using the Innsbruck FALP apparatus (see Section VII, especially Fig. 8). D. DISSOCIATIVE ATTACHMENT TO THE RADICALS CCl,
AND
CC1,Br
In the first FALP paper on electron attachment (Smith et al., 1984), we speculated that sequential dissociative attachment reactions might occur in some reactions, such as CCl, s ( c l - ) + c c l ,
s(cl-)+ CCl, s
3..
(12)
The first attachment stage is known to be fast, the second (secondary attachment) stage is exothermic to produce CC1, C1-, but the energetics of the tertiary stage are not known. Clearly, if secondary (and tertiary) attachment does occur for CCl, and other molecules, then each molecule can remove two (or three) free electrons from an ionized gas; and it is obviously important to appreciate this (particularly for electron capture detectors). How can such an effect be investigated? This has been accomplished using the FALP apparatus and reported in detail (Adams et al., 1988). Primary and secondary attachment do indeed occur for CC1, and also for CCl,Br, and the
+
D. Smith and P. Spang1
328
first measurements of j for the dissociative attachment reactions of the two unstable radicals CCl, and CC1,Br have been determined:
+e
+ CCl, CC1,Br + e + C1-( + CClBr), Br-(+CCI,) CCl,
+
C1-
(13) (14)
The basic idea of the measurement is simple but it requires careful experimentation. The primary attachment reaction to CC1, is fast, and so only relatively small number densities of the attaching gas, n,, are required in the afterglow plasma to produce an acceptable decrease of n, along z (see Section V). Also, n, must be much less than n, if first-order kinetics are to properly describe n, vs. z; i.e., if nm is to be invariant with z. However, if n, 7 nm then the attaching gas will be partially consumed, and n, will decrease along z. This decrease in n, at large n, will effectively reduce the gradient of n, along z, and if it is not accounted for, an erroneously small value of B will be deduced from the data. This phenomenon is well illustrated in Fig. 5(a), which shows the experimental data for the rapid dissociative attachment reaction of CH31: CH,I
+ e + I - + CH,
(1 5 )
This reaction was chosen to illustrate the effect because in this case there is no possibility of secondary dissociative attachment to CH, radicals at thermal energies (there are no exothermic channels). It is clear in Fig. 5(a) that, at large initial values of n,, its gradient along z is smaller than at lower n,; this is the result of the removal of CH31from the plasma. However, this loss of reactant gas can be properly accounted for in the kinetic analysis and a consistent value for B obtained at all initial values of n, by computer fitting the data (see Adams et al., 1988, and the caption to Fig. 5(a) for more details). Note that the curve obtained for the lowest initial n, provides a value for b( = 1.2 x lo-' cm3 s - ' at 300 K) directly, because no significant depletion of CH31 occurs (i.e., first-order kinetics apply). When the same procedure is applied to the CCl, reaction, a different result is obtained. The obtained from the low initial n, data is consistent with the known /?(CCl,) but, very significantly, when depletion of CCl, is accounted for at the higher initial n, (and using the known P(CCl,)), then a good fit cannot be obtained between the predicted n, versus z curve and the experimental points (see Fig. 5(b)). The experimental data indicate a faster decrease of n, than predicted, and this is a clear indicator of secondary attachment; in effect, more attaching gas is present in the plasma than is accounted for. The secondary attachment reaction (13) can now be included in the kinetic analysis and the experimental data computer fitted to obtain a secondary attachment coefficient. When this is done, a good fit is obtained to the experimental n, versus z data for a secondary attachment coefficient of
-
lotQ
10'
-
10'
- Port
lo" ne
CCI,,
C)
1olQ
-
10'
-
e L C C i 3 * CI-
C C I ~ e-%C12+
cI-
lo8 - Port
Position
10'
60
50
ko 30 Axiol Position Icm)
20
FIG.5. Plots of the logarithm of electron number density, n, (in cm-'), versus position, z, along the axis of the FALP plasma in the presence of attaching gas at fixed number densities, n,, for several different initial n,. All the data were obtained at 300 K. These figures were taken from A d am et al. (1988), with permission. (a) Attachment to CHJ for n, = 7.5 x lo9 cm-,. Note the faster decay of n, with z at the low initial nerwhere no significant depletion of the CH,I occurs due to the attachment process, and the slow decay at the highest n,, where depletion of CH,I does occur. The continuous curves are computer fits to the data points for a = 1.2 x lo-' cm3s- taking into account the depletion of CH,I. The close fit to the data points indicates that secondary attachment does not occur for the CH, radical, as expected. (b) and (c) Attachment to CCI, for n,,, = 5.0 x lo9cm-'. The data points in (b) and (c) are identical. Note, again, the faster decay of n, along z at the lowest initial ne when no significant depletion of CC14 occurs. The continuous curves in (b) are computer-generated decay curves for a 8, = 4.2 x 10' cm3 s - ' and accounting for the depletion of CCI, but ignoring secondary attachment. The poor correlation of the computer curves and the experimental points indicates that another attachment reaction is occurring. The continuous curves in (c) are the computer-generated curves taking into account both depletion of CCI, and secondary attachment to CCI, radicals with a p2 of 2.4 x lo-' cm' s-'. The good fit of the curves to the data points at the higher initial n, is a clear indication of the occurrence of secondary attachment to CCI, radicals.
',
329
330
D. Smith and P. Spang1
2.4 x lo-’ cm3 s - l at 300 K (see Fig. 5(c)).The same approach can be taken to the analysis of the CCI,Br data, and after this is done a rate coefficient for 1 x lo-’ dissociative attachment to CC1,Br can be estimated to be cm3s-l. To our knowledge, these were the first measurements of p for unstable molecular radical species, but subsequently these observations were confirmed by Mock et al. (1989), using an electron capture detector operating at atmospheric pressure. Presumably secondary reactions occur following other primary attachment reactions. Further FALP investigation of this phenomenon are in progress at Innsbruck.
-
IN ELECTRON ATTACHMENT:THENEEDFOR MASS E. Br,- FORMATION SPECTROMETRY
If the phenomenon of electron attachment is to be properly understood, then it is clear that not only the p are required for particular reactions, but positive identifications of the reaction products are necessary. This is forcibly demonstrated by the FALP results obtained for electron attachment to some bromomethanes and bromoethanes (Smith et al., 1990).The FALP work was prompted by the work of Alajajian et al. (1988),using the Kr photoionization method, who observed that, in the electron attachment reaction of CF,Br,, both Br-(80%) and Br,-(20%) were produced. It is not generally expected that molecular halogen ions can sometimes be produced in these reactions. The FALP study not only confirmed the findings of Alajajian et al. (1988),but also showed that Br,- production is not an uncommon occurrence. Indeed, Br,- was even observed from the reaction BrF,C-CF,Br
+ e + Br- + BrF,C-CF, Br,- + F,C=CF, +
(80%)
(16a)
(20%)
( 16b)
In this reaction, Br atoms from separate C atoms in the molecule bond together in a concerted reaction with the electron to produce Br2-. Indeed, it is possible that (Br, -)* is the most important nascent product, which then partly dissociates to give Br- and Br and is partly stabilized by collisions with the carrier gas atoms (He) (or by radiation emission) to produce stable Br2-. However, for Br- and Br to be products, a double C 4 bond must be formed to satisfy the energy requirements, i.e., F2C=CF, must be the other neutral product (see reaction (16b)). Clearly, without mass spectrometric identification of the product ions, these very unusual reactions would not have been recognized.
33 1
ELECTRON ATTACHMENT
F. DISSOCIATIVE ATTACHMENT TO ACIDS AND SUPERACIDS
A FALP study has been carried out of the attachment reactions of the common acids HNO, and H,SO,, the superacids FSO,H, ClSO,H, and CF,SO,H (“triflic acid”), and the anhydride of triflic acid (CF,SO,),O (Adams et al., 1986). These acids are the so-called Bronsted acids that donate a proton to any base in either the liquid phase or the gas phase. In a plasma, dissociative electron attachment reactions can occur with these acids that is equivalent to Bronsted acid behavior with the free electron acting as the proton acceptor (the base); thus, H 2 S 0 4+ e + HS0,-
+ (H’,
e) + H S 0 4 -
+H
(17)
The study of these reactions using the FALP apparatus was not so easy as usual because of the “sticky” nature of these acids, and thus the accuracy of the derived p values was somewhat diminished (typically SO- 100% error rather than the usual _+ 15%). Nevertheless, it was clear that the reactions were very efficient, since in all cases the b were an appreciable fractions of the p,, estimated using the Klots theory (see Section I11 and Table I). The /? for the triflic acid reaction (18) was 1 x lo-’ cm3 s-’ at 300 K:
-
CF,SO,H
+ e + CF,S03- + H
(18)
In this reaction, the very stable triflate anion, CF,SO,-, is the product, as it is also for the (CF,SO,),O reaction. The rapid reactions of these acids and superacids with electrons demonstrate their extreme acidity. The stability of the product anions has been demonstrated by a study of their reactivity in SIFT experiments (Paulson et af., 1986; Viggiano et al., 1992), the results of which indicate that both the radicals CF,SO, and FSO, have electron affinities greater than any of the halogen atoms. Note that in Table I, only a lower limit is given to the b for the CIS0,H reaction. This is because CIS0,H is partially dissociated in the gas phase to SO, and HC1, neither of which react rapidly with electrons at 300K (Adams et al., 1986). The CF,SO, - anion is also formed in the dissociative attachment reaction of the triflate esters CF,SO,CH, and CF,SO,C,H, (Herd et al., 1989b). However, for these molecules, the FALP study has shown that the p are small at 300K ( - 2 x lo-’’ cm3s-’; see Table I), i.e., but that the p increases with T for both reactions, but especially rapidly for the CF3S0,CH3 reaction: CF,SO,CH,
+ e -+ CF3S0,- + CH,
(19)
Hence an activation energy, E,, of 170meV is obtained from the data for
332
D.Smith and P . Spane'l
reaction (19). This reaction can be viewed as occurring via the donation of a CH,' ion to the electron producing a methyl radical and a CF,S03- ion, analogous to the donation of a proton. In a parallel SIFT study of the reactions of these esters with F-, C1-, Br-, and I-, and using thermochemical cycles (Herd et al., 1989b), it was shown that dissociative electron attachment to both esters is exothermic. So the small /3 for these reactions is indeed due to the existence of activation energy barriers.
G. DISSOCIATIW ATTACHMENT AND ASSOCIATIVE DETACHMENT: THE R E A C T I O N S H B ~ + ~ ~ BH~I-++eHs ;I - + H Using the available bond energies of the hydrogen halides (HF, HCl, HBr, HI) and the electron affinities of the halogen atoms, it can readily be shown (see Section 11) that dissociative attachment to HF and HCl are too endothermic to be observable at thermal energies, and so they cannot be studied using the FALP apparatus in the conventional way. However, the HBr dissociative attachment reaction is endoergic only by about 0.3 eV and the HI dissociative attachment reaction is apparently close to thermoneutral, so these two molecules are candidates for FALP studies. Such studies have been carried out together with a parallel SIFT study of the reverse associative detachment reactions (Smith and Adams, 1987). The measurement of P(HBr) at 300 K provided only an upper limit value of <3 x cm' s-', which is at the lower limit of the measuring capability of the FALP. Therefore, B(HBr) for the reaction HBr + e $ Br-
+H
(20)
was measured at 515 K and a value of 3 x lo-'' cm3 s-' was obtained. The n, versus z data obtained are shown in Fig. 6(a),when it can be seen that n, decreases monotonically with z, as is usual for attachment reactions (see also Fig. 5). The corresponding data for the HI reaction at 300 K are shown in Fig. 6(b),but the profiles of the curves (at two values of the HI number density in the plasma) are very different from that in Fig. 6(a). As can be seen, following a steep decrease in n,, the slope of the curves decrease before falling away again very rapidly due to the free diffusion of electrons (as is ultimately expected; see Smith et al., 1984). Why should the slope of the curve decrease so, a phenomenon not previously observed in the large number of electron attachment reactions studied in the FALP? The answer is simple. It is due to the fact that the reaction HI + e + I -
+H
(21)
is very close to thermoneutral, and so the reverse associative detach-
333
ELECTRON ATTACHMENT
ment reaction can proceed at a significant rate at 300K. Thus, the forward dissociative attachment reaction proceeds rapidly (p = 3 x cm3 s-' N J p,, and then, with the loss of electrons and the build-up of Iions and H atoms, the reverse reaction can proceed at an increasing rate. Hence, an approach to equilibrium occurs in reaction (21). So the distortion of the decay curve is a manifestation of the thermoneutrality of the reaction; this was not observed before because all of the attachment reactions studied previously are appreciably exothermic and hence their reverse reactions are endothermic. In this regard, the question must be asked as to why the curve for the HBr reaction (see Fig. 6(a))was not similarly distorted. The most likely answer is that the reaction of H atoms with HBr (exothermically producing H2 + Br) removes the H atoms from the plasma and thus prevents the back reaction occurring. This is possible since much higher number densities of HBr are used than are HI (see Figs. 6(a) and 6(b)) because of the slower dissociative attachment reaction of HBr. It is worthy of note that a detailed
Diffusion only
attar hment
I
Inlet port position
T=S15 K
. position 1071
60
50 LO Position along flow tube I t m l
b)
60
r=BOK I
I
I
I
LO
30
2
Position along flow tube l c m l
FIG.6. Plots of the logarithm of electron number density, n,, versus position along the axis of the FALP plasma in the presence of(a) HBr at 7 = 515 K and (b) HI at = 300 K. Also shown are the n, versus z plots in the absence of attaching gas when the decay of n, is due only to the slow process of ambipolar diffusion. Note in (a) that the gradient of ne continuously increases due to the attachment loss ofelectrons and that, as expected, n, falls to zero at the transition between the electron-positive ion and the negative ion-positive ion plasma (see also Fig. 2). Note in (b) the unusual decrease in the gradient of n, for a probe position near to 50 cm for both the decay curves obtained at the two number densities of the attaching gas (HI), which is due to the occurrence of the reverse associative detachment reaction of I - ions with H atoms. The simultaneous occurrence of dissociative attachment and associative detachment indicates that these reactions are closely thermoneutral (see the text). (These figures were taken from Smith and Adams, 1987, with permission.)
D.Smith and P . Spang1
3 34
calculation of the reaction rate coefficient of thermal electrons with HI has been made (Clary and Henshaw, 1987) that indicates a value of 4 x lo-' cm3 s - l at 300 K, in acceptable agreement with the FALP result. That the dissociative attachment reactions of HF, HCl, and HBr are endothermic suggests that the reverse associative detachment reactions, e.g., F-,
CI-,
Br- + H + H F ,
HCl, H B r + e
(22)
might be efficient. Thus a parallel study was carried out using the variable temperature SIFT apparatus (Smith and Adams, 1988) to measure the rate coefficients, k, for these reactions and also for the I - + H reaction; the experimental details are given in Smith and Adams (1987) together with references to previous work. The values obtained for these k are given in Table I1 at both 300 K and 515 K together with the corresponding values. Note that the k for the F - , C1-, and Br- reactions are appreciable fractions of the Langevin collisional rate coefficient (which for all four reactions is the same at 1.92 x lo-' cm3 s-'), and that the k for the F- and C1- reactions decrease with increasing temperature. For the Br- reaction, the k is the same at both temperatures 7:and the k for the I- reaction is greater at 515 K than at 300 K. These trends are interesting in themselves (Smith and Adams, 1987) but not directly relevant to this discussion of attachment. However, the k for Br- and I - are valuable in that coupled with the p values, the equilibrium constants for the reactions (20) and (21) can be obtained simply as p/k. Then by calculating the entropy changes in the reactions, the enthalpy changes can be obtained (Smith and Adams, 1987). This procedure shows that the HI reaction (21) is indeed closely thermoneutral (but perhaps slightly endoTABLE I1 FALP VALUESOF THE DISSOCIATIVE ATTACHMENT COEFFICIENTS, p, AND THE SIFT VALUES OF THE (REVERSE) ASSOCIATIVE DETACHMENT COEFFICIENTS, k, FOR THE REACTIONS INDICATED AT THE TEMPERATURES INDICATED. THEB AND k VALUES ARE GIVEN AS, FOR EXAMPLE, 8(- 10) MEANING 8 x lo-'' cm3 S K I . (TAKEN FROM SMITH AND ADAMS,1987.) THE
Rate coefficients (cm' s-')
300 K Reaction
H F + e = F- + H HCI + e e C I - + H HBr+eeBr-+H HI + e e I - + H
515K
B
k
B
k
-
15(-10)
-
8( - 10)
3( - 10) -
q-10) 7( - 10)
-
<3(-12)
3(-7)
9( - 10) 7(-10) 3(-10)
q-w
ELECTRON ATTACHMENT
335
thermic) with a AH at 300K of (23 f 20meV), and that the AH for the endothermic dissociative attachment reaction to HBr is 351 meV at 300 K and (373 k 30) meV at 515 K. The AH for the HI reaction as calculated from the molecular data (i.e., the bond energy of HI and the electron affinity of I atoms) is - 5 meV, indicating that the attachment reaction is very slightly exothermic. This may be why the k for the slightly endothermic reverse associative detachment reaction increases with temperature. Further details of these very interesting measurements and the deductions from them are given in Smith and Adams (1987). To our knowledge, these are the first coordinated studies of such forward and reverse reactions, studies made possible using the powerful SIFT and FALP techniques.
VII. Recent Developments Using the FALP Apparatus In all the FALP programmes outlined in the previous sections, the data were obtained under truly thermalized conditions, i.e., the carrier and attaching gas temperature (T,) and the electron temperature (T,) were the same. To ascribe a temperature to the electron gas implies a Maxwellian velocity distribution for the electrons. However, the question was posed in Section V1.C as to which of T, or T, has the greater influence on the efficiency of attachment reactions at and near to thermal energies. Clearly in the light of the information presented in this chapter this is going to depend on the particular attaching molecule; for the halogenated organic molecules included in the study described in Section VI.C, T has a major influence on the at thermal energies, whereas for SF, the electron energy (T,)has the major influence (Klar et al., 1992). What is obviously needed is an apparatus to study the separate influence on B of both T, and T at thermal energies. This can be achieved to some extent by varying 5 in swarm experiments, Kr photoionization experiments, and pulsed radiolysis-microwave heating experiments, but, as yet, not a great deal has been reported on this. Now, in Innsbruck, we have developed the FALP technique to study electron attachment (and dissociative recombination) reactions at different T (within the range 80-600K, but restricted at the lower temperatures by condensation of the attaching gases) and at different T, (for T, 2 T,). How has this been achieved? First, T, had to be increased above q ;and second, a reliable method had to be developed to determine T, reasonably accurately (see later). The first could not be achieved in Helium carrier gas because of the large momentum transfer cross section between thermal electrons and Helium atoms. Thus, even if T,is momentarily increased in Helium (as it is in the microwave discharge that is the source of the afterglow plasma) it rapidly
D. Smith and P . Span21
336
decreases to q. However, elevated T, can be readily achieved in the FALP plasmas using Argon carrier gas principally because of the deep Ramsauer minimum in the momentum transfer cross section between electrons and Argon atoms (McDaniel, 1964; Dean et al., 1974). Thus, an elevated T, decreases relatively slowly in the upstream region of the flow tube, and so T, > q in the downstream afterglow. The magnitude of T, in the afterglow plasma can be adjusted by varying the position of the microwave cavity (and to some extent the microwave power in the discharge). The presence of metastable Argon atoms in the afterglow plasma effectively heats the electron gas, since the electrons undergo superelastic collisions with the metastables. However, the metastable atoms can be removed, if necessary, by the addition of a gas with low ionization energy that is Penning ionized by the metastables. Further use can now be made of the great versatility of the Langmuir probe technique that, of course, we routinely use to measure n, in the afterglow plasmas. It can also be used to determine T, and electron energy distribution functions f(E)in these afterglow plasmas. The physics behind the measurement of f(E)is described by the following equations (see Swift and Schwar, 1970). Consider a plasma in which f(E)is Maxwellian, when
fMaxw(E) = 27~(7rkT,)-~/~ exp
(
--
Then the electron current, I, to a Langmuir probe biased at a negative voltage, K with respect to the plasma potential will decrease exponentially with increasingly negative K as Z=Zoexp(-g);
then, d2Z - const Sexp dV2
(- Q)
(24)
Thus, in a Maxwellian plasma, a plot of log d2Z/dVZversus V will be linear and the slope of the line provides a value for T,. For a general distribution function f(E),where E = eV: d21 dVZ
- - - Const V-
f(e V)
and thus f(E)can be obtained from a determination of d2Z/dV2as a function of K This approach to the determination of f(E)in plasmas has a long history (see Swift and Schwar, 1970). It has been applied mostly to the study of glow discharge plasmas, but recently it has been applied to the study of flowing afterglow plasmas using frequency modulation of the probe voltage to determine d2Z/dY2 (Glosik et al., 1983). Now, a more accurate method has been developed in Prague (Spanel and Glosik, 1991) and further improved in
337
ELECTRON ATTACHMENT
Innsbruck (Spanel and Smith, 1993a) to process numerically Langmuir probe data acquired from FALP plasmas using on-line microcomputer to determine d21/dV2, and some very interesting results are being obtained. A sample of data obtained from the Innsbruck FALP experiment in Helium carrier gas (where T, = = 300 K) is shown in Fig. 7(a). Note that the plot is linear over some three decades in dZ1/dV2,indicating that f ( E ) is closely Maxwellian in this Helium afterglow. A similar plot is also shown in Fig. 7(a) in Argon carrier gas, but in this case the T,is much greater at 2500 K.
lo-'
I c
0.0
0.5
1.0
v (volts)
v (volts)
FIG.7. Plots of the second derivative of the probe current, I, with respect to the probe voltage, d21/dVZ, against probe voltage, K obtained in the FALP plasmas. The plots have been normalized to d21/dVZ= 1 at the peaks of the curves. Note the rapid reduction of d21/dVz (to the left of each figure) as the probe voltage is swept through space potential (zero on the V scale). (a) Data obtained in pure helium (steep line) and pure argon afterglow plasmas at a T, = 300 K in both cases. The linearity of the plots indicates that f(E)is closely Maxwellian, at a T, = 300 K in the Helium afterglow and T, = 2500 K in the Argon afterglow. (b) Data obtained in an Argon ). (i) was obtained 1 cm afterglow with a small admixture of SF, (n, = 8 x lo9 ~ m - ~ Curve upstream of the inlet port (see Fig. 2); curve (ii) was obtained 1cm into the SF, flow; curve (iii) was obtained 2cm into the SF, flow. Note that the linear portion of each plot (to the right of each plot) is essentially unchanged, the steep parts of which indicate an equivalent T, for the high energy electrons in the distribution of 800K. However, the removal of the lower energy electrons is obvious as the electron attachment process to SF, begins. This is quite consistent with the known increase in the cross section (or equivalently the increase in p ) for attachment to SF, with decreasing electron energy (see Section V1.A).
338
D. Smith and P . Span61
Note that the range of linearity of the plot in the Argon afterglow plasma is not so wide and that the f(E) is perhaps not perfectly MaxwelIian but nevertheless a close approximation to it. So, as mentioned previously, using Argon carrier gas we can now vary T, in the afterglow plasmas, determine f(E)and T, and measure 3/ as a function of T,. Thus the separate influences of T, and 5 on electron attachment (and dissociative recombination) reactions can now be investigated because, of course, the temperature of the carrier gas and the attaching gas can also be varied. Samples of the first data we have obtained on the influence of electron attachment on the f(E)in Argon afterglow plasmas, and the variation of /I with T, are given in Fig. 7(a) and Fig. 8, respectively. However, before commenting on these data, an important point should be made. If departures from an initial Maxwellian f(E)are to be observed due to the occurrence of electron attachment, then the electron number density n, must be lo7 cm-3
-=
1 o-’
CF,Br+e
-
n
I
u)
n
E0
10“
+ Br-
+cFl
W
Q
10-o
1 ~
1 o2
o3
1
1o4
FIG. 8. The variation of the attachment coefficient, /3, for CF,Br with both the gas temperature, (x, obtained in helium carrier gas at a pressure about 1 Torr), and the electron temperature, T, (A, obtained in Argon carrier gas at a pressure of about 0.3 Torr and a T, of 300 K). Both sets of data were obtained using the FALP apparatus. The only ionic product observed in both experiments was Br- (dissociative attachment). The rapid increase of /3 with T, implies an activation energy of 80 meV for the reaction (see Section V1.A). The decrease of /3 with T, implies a reduction in the rate of capture of the electrons by the molecule (see Section 111).
<
339
ELECTRON ATTACHMENT
to ensure that electron-electron collisions d o not rapidly re-randomize the electron energies. Conversely, to ensure that a Maxwellian, or nearMaxwellian, f ( E ) is maintained when processes are occurring that tend to remove low-energy electrons from the afterglow plasma (e.g., by attachment) or introduce higher energy electrons into the afterglow plasma (e.g., by Penning ionization), n, must be >109cm-3. We have determined these limiting n, values from a theoretical study of electron energy relaxation times at low energies in afterglow plasma (Trunec et al., 1993). Now consider Fig. 7(b), which shows how f(E)is modified as the Langmuir probe is moved in the downstream direction from a pure Argon flowing = 300 K) into the same plasma into afterglow plasma (at T, = 800 K, which a trace of SF, is introduced. Note the severe depletion of the lowenergy electrons as the attachment reaction proceeds. The effect is dramatic but consistent with the known increase of fi with increasing electron energy (see Section V1.A). Deconvolution of these data and data like them for other attachment reactions will provide data on the variation of D with electron energy in this low-electron energy regime. For the first study of the influence of T, on the p we chose to study electron attachment to the molecule CF3Br, for which our previous FALP studies (Alge et al., 1984) had shown that p increased rapidly with increasing T, (see Fig. 3). The data in Fig. 8 show the startling difference between the variations of p with T, and T,, the D increase rapidly with T, and decrease with T,! This variation of @ with T, for CF3Br is very similar to that derived from pulsed radiolysis experiments by Shimamori et a/. (1992b). Thus, this FALP experiment demonstrates conclusively the fundamental difference between Tg and T, variations of p, a point we forcibly made in a paper a few years ago (Smith et al., 1989). The new data on the distortion of f ( E ) due to electron attachment, the CF3Br data, and the experimental details will soon be reported (Spanel and Smith, 1993b). Clearly, if the attachment process at thermalenergies is to be properly understood, then it is essential to study the phenomenon as a function of both T, and T, for variety of molecules. Such work is being vigorously pursued in our Innsbruck laboratory. Our most recent FALP experiment has been to investigate electron attachment to the fullerene molecule C6,. Thus we find that for T, = Tg = 300K, the p is relatively small ( d 1 0 - 9 c m 3 ~ - 1 ) , but the fi increases rapidly with increasing T, (at fixed T, = 300 K), and at T, 4500 K the measured p is 3 x 10- cm3s-', which at this temperature is about 3 (see Section III)! A preliminary estimate of the times greater than p,, associated "energy barrier" for this attachment reaction is 0.25 eV (note, with respect to T, and not to T,). These studies of electron attachment to C,, and some other fullerenes are continuing.
-
-
'
-
340
D. Smith and P . Spang1
VIII. Concluding Remarks This review has concentrated on the study of electron-molecu.,: attachment reactions under truly thermalized conditions using the FALP technique and the most recent exciting development of the technique by which we can study electron attachment also at elevated electron temperatures, up to about 3000 K. A substantial body of data is now available from these FALP studies on a variety of different attachment phenomena. The highlights include the determinations of the activation energies for several dissociative attachment reactions, the remarkable-at the time of measurement-unique observation of the inverse temperature variation of the electron attachment coefficient for the C6F6 molecule, and the first measurements of the attachment coefficients to the unstable molecular radicals CCI, and CCI,Br, which has serious implications to the sensitivity of electron capture detectors. The somewhat surprising observation that Br, - molecular ions can be produced in electron attachment to several dibromo organic molecules emphasizes the necessity of mass analysis of the product ions of electron attachment reactions, a point also self-evident from the studies of dissociative attachment to acids and superacids where the identifications of the product ions was crucial to the proper interpretation of the reaction mechanism (i.e., Bronsted acid behavior). The measurements of dissociative attachment and associative detachment reactions involving HBr and HI are also unique and nicely demonstrate the thermoneutrality of the HI dissociative attachment reaction. The realization that the efficiencies of electron attachment reactions at thermal energies are often influenced differentially and dramatically by changing the attaching gas temperature, and the electron energy (or temperature, T,) was forcibly indicated when the FALP data for electron attachment to some halomethanes and haloethanes were compared to the data obtained using nonthermal methods. It is important to be able to quantify these differences if electron attachment reactions are to be understood in detail, which demonstrates a need to investigate the variations of and T,. With such data, attachment rates (and product ions)'with both perhaps, electron attachment rates and product ions may be predicted at higher temperatures, inaccessible to many laboratory experiments but that pertain to real plasmas such as arc discharges, etchant plasmas, rocket plumes, and some planetary ionospheres. However, in high-temperature plasmas dissociation of molecules can occur, and then electron attachment to molecular radicals and (endothermic) associative detachment may occur. Perhaps the FALP (and the SIFT) studies reviewed here can point the way to further studies of these phenomena. Clearly, theoretical work is needed in
<,
ELECTRON ATTACHMENT
34 1
this area, and we hope the work described in this chapter may stimulate such work. Our very new development of the technique for rapidly and accurately determining f(E)and T, in afterglow plasmas offers exciting opportunities for much further work on the variation of attachment and recombination reaction rates at very low-electron energies and on other interesting plasma processes such as T, relaxation rates in the presence of molecular gases, studies we began many years ago using much simpler diagnostics (Smith and Dean, 1975). The study of the variation of the attachment coefficient for CF,Br as a function of both T, and T, is just the beginning of a wider study that will include many other molecular gases. A body of such FALP data is obviously desirable. A final point worth making is that all the FALP data to date have been obtained in carrier gases at pressures of about 1 Torr. Obviously it is important to extend these studies to higher pressures at which the mechanisms, and therefore the rates and the products of some reactions, will change because, for example, the quenching rates of excited, nascent negative ions can increase and so autodetachment can be inhibited at higher pressures (see, for example, Mock et al., 1989). The study of electron attachment to clusters (the condensed phase) of some of the molecules included in this study would also be interesting, especially those molecules for which the attachment coefficients are seen to vary rapidly with T, and T,. For example, how would clusters of (CF,Br),, and (HI),, react to low-energy electrons? We hope that the experiments and the results obtained, which are described in this review, will stimulate these and other experiments in this exciting area of science.
Acknowledgments D.S. wishes to thank all his coworkers in the development of the Langmuir probe technique for use in afterglow plasmas, especially I. C. Plumb, and in the exploitation of the FALP technique to study electron attachment and other processes, especially N. G. Adams, E. Alge, and C. R. Herd. Special thanks are also due to J. F. Paulson for his constant support and encouragement in all aspects of the work and to W. Lindinger and T. D. Mark for providing a beautiful laboratory in which we are able to pursue our FALP and SIFT work. We wish to thank J. Glosik for his help in the development of the numerical on-line probe technique to determine electron energy distribution functions. The financial support of the FALP programmes by the United States Air Force and the Austrian Fonds zur Forderung der wissenschaftlichen Forschung is gratefully acknowledged.
342
D.Smith and P. Spand REFERENCES
Adams, N. G., Church, M. J., and Smith, D. (1975). J. Phys. D 8, 1409. Adams, N. G., Smith, D., and Alge, E. (1984). J . Chem. Phys. 81, 1778. Adams, N. G., Smith, D., Alge, E., and Burdon, J. (1985). Chem. Phys. Letts. 116, 460. Adams, N. G., Smith, D., and Herd, C. R. (1988). Int. J . Mass Spectrom. Ion Processes 84, 243. Adams, N. G., Smith, D., Viggiano, A. A., Paulson, J. F., and Henchman, M. J. (1986). J. Chem. Phys. 84, 6728. Ajello, J. M., and Chutjian, A. (1979). J. Chem. Phys. 71, 1079. Alajajian, S. H., and Chutjian, A. (1987). J. Phys. B 20, 5567. Alajajian, S. H., Bernius, M. T.,and Chutjian, A. (1988). J. Phys. B 21, 4021. Alge, E., Adams, N. G., and Smith, D. (1983). J. Phys. B 16, 1433. Alge, E., Adams, N. G., and Smith, D. (1984). J. Phys. B 17, 3827. Biondi, M. A. (1958). Phys. Rev. 109, 2005. Cavalleri, G. (1969). Phys. Rev. 179, 86. Christodoulides, A. A., Schumacher, R., and Schindler, R. N. (1975). J. Phys. Chem. 79, 1904. Christophorou, L. G., Chaney, E. L.,and Christodoulides, A. A. (1969).Chem. Phys. Letts. 3,363. Christophorou, L. G., McCorkle, D. L., and Christodoulides, A. A. (1984).In: Electron-Molecule Interactions, and Their Applications, 1, Academic Press, New York. Chutjian, A., and Alajajian, S. H. (1985). Phys. Reo. A 31, 2885. Chutjian, A., Alajajian, S. H., and Man, K.-F. (1990).Phys. Rev. A 41, 1311. Clary, D. C., and Henshaw, J . P. (1987). Int. J . Mass Spectrom. Ion Processes 80, 31. Crompton, R. W., and Haddad, G. N. (1983). Aust. J . Phys. 36, 15. Dean, A. G., Smith, D., and Adams, N. G. (1974). J. Phys. B 7, 644. Datskos, P. G., Christophorou, L. G., and Carter, J. G. (1992). Chem. Phys. Letts. 195, 329. Dunning, F. B. (1989). J . Chem. Phys. 91, 4 4 1 1 . Fehsenfeld, F. C. (1970). J . Chem. Phys. 53, 2000. Ferguson, E. E., Fehsenfeld, F. C., and Schmeltekopf, A. L. (1969). Adv. At. Mol. Phys. 5, 1. Foltz, G. W., Lather, C. J., Hildebrandt, G. F., Kellert, F. G., Smith, K. A., West, W. P., Dunning, F. B., and Stebbings, R. F. (1977). J . Chem. Phys. 67, 1352. Glosik, J., Pavlik, J., and Sicha, M. (1983). Czech. J. Phys. B33, 1230. Herd, C. R., Adams, N. G., and Smith, D. (1989a).Int. J . Mass Spectrom. Ion Processes 87, 331. Herd, C. R., Smith, D., and Adams, N. G. (1989b).Int. J . Mass Spectrom. Ion Processes 91, 177. Hildebrandt, G. F., Kelbrt, F. G., Dunning, F. B., Smith, K. A., and Stebbings, R. F. (1978). J. Chem. Phys. 68, 1349. Kalamarides, A., Walter, C. W., Lindsay, B. G., Smith, K. A., and Dunning, F. B. (1989).1.Chem. Phys. 91, 4 4 1 1 . Klar, D., Ruf, M.-W., and Hotop, H. (1992). Chem. Phys. Letts. 189,448. Klots, C. E. (1976). Chem. Phys. Letts. 38, 62. Knighton, W. B., Bognar, J. A,, and Grimsrud, E. P. (1992). Chem. Phys. Letts. 192, 522. Kondo, K., and Crompton, R. W. (1986). Unpublished observations. Ling, X.,Lindsay, B. G., Smith, K. A., and Dunning, F. B. (1992). Phys. Rev. A 45, 242. Marawar, R. W., Walter, C. W., Smith, K. A., and Dunning, F. B. (1988).J . Chem. Phys. 86,2853. Massey, H. S. W. (1976). Negatioe Ions, Cambridge University Press, Cambridge. Matsuzawa, M. (1971). J . Chem. Phys. 55, 2685. McCorkle, D., Szamrej, I., and Christophorou, L. G. (1982).J . Chem. Phys. 77, 5542. McDaniel, E. W. (1964). Collision Phenomena in Ionized Gases, Wiley, New York. Mock, R. S., Zook, D. R., and Grimsrud, E. P. (1989).Int. J. Mass Spectrom. Ion Processes 99, 327.
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Oskam, H. J. (1958). Phillips Research Report 13, 335. Paulson, J. F., Viggiano, A. A., Henchman, M. J., and Dale, F. (1986). Proc. Symposium on Atomic and Surjace Physics, Obertraun, Austria, University of lnnsbruck Press, Innsbruck, p. 7. PetroviC, Z. L., and Crompton, R. W. (1987). J. Phys. B 20, 5557. Plumb, I. C., Smith, D., and Adams, N. G. (1972). J . Phys. B 5, 1762. Shimamori, H., Nakatani, Y., and Ogawa, Y. (1991). Joint Symposium on Electron and Ion Swarms, and Low Energy Electron Scattering, Bond University, Gold Coast, Australia, Abstracts, p. 28. Shimamori, H., Tatsumi, Y., Ogawa, Y., and Sunagawa, T. (1992a). Chem. Phys. Letts. 194,223. Shimamori, H., Tatsumi, Y., Ogawa, Y., and Sunagawa, T. (1992b). J. Chem. Phys. 97, 6335. Smith, D., and Adams, N. G. (1987). J. Phys. B 20,4903. Smith, D., and Adams, N. G. (1988). Adu. Atom. Mol. Phys. 24, 1. Smith, D., Adams, N. G., and Alge, E. (1984). J. Phys. B 17, 461. Smith, D., Adams, N. G., Dean, A. G., and Church, M. J. (1975). J. Phys. D 8, 141. Smith, D., and Church, M. J. (1976). Int. J . Mass Spectrom. Ion Physics 19, 185. Smith, D., and Dean, A. G. (1975). J. Phys. B 8, 997. Smith, D., Dean, A. G., and Adams, N. G. (1972). 2. Physik. 253, 191. Smith, D., Goodall, C. V., and Copsey, M. J. (1968). J. Phys. B 1, 660. Smith, D., Herd, C. R., and Adams, N. G. (1989). Int. J. Mass Spectrom. Ion Processes 93, 15. Smith, D., Herd, C. R., Adams, N. G., and Paulson, J. F. (1990). Int. J . Mass Spectrom. Ion Processes %, 341. Smith, D., and Plumb, I. C. (1972). J. Phys. D 5, 1226. Smith, D., and Plumb, I. C. (1973). J. Phys. D 6, 196. Spanel, P., and Glosik, J. (1991). In: Proceedings of the Pentagonale Workshop on Elementary Processes in Clusters, Lasers, and Plasmas, Kuhtai, Innsbruck, Austria (T. D. Mark and R. W. Schrittwieser, eds.) University of Innsbruck Press, Innsbruck, p. 253. Spanel, P., and Smith, D. (1993a). Manuscript in preparation. Spanel, P., and Smith, D. (1993b). Znt. J. Mass Spectrom. Ion Processes, in press. Spyrou, S. M., and Christophorou, L. G. (1985). J . Chem. Phys. 82, 1048. Swift, J. D., and Schwar, M. J. R. (1970).Electrical Probesfor Plasma Diagnostics, Iliffe, London. Toriumi, M., and Hatano, Y. (1985). J. Chern. Phys. 82, 254. Trunec, D., Spanel, P., and Smith, D. (1993). Contrib. Plasma Phys., submitted. Viggiano, A. A., Henchman, M. J., Dale, F., Deakyne, C. A., and Paulson, J. F. (1992). J . Am. Chem. Soc. 114, 4299. Warman, J. M., and Sauer, M. C., Jr. (1971). Int. J . Radiat. Phys. Chem. 3, 273. Wentworth, W. E., Becker, R., and Tung, R. (1967). J . Chem. Phys. 71, 1652. Wentworth, W. E., George, R., and Keith, H. (1969). J . Chem. Phys. 51, 1791. Zheng, Z., Ling, X.,Smith, K. A., and Dunning, F. B. (1990). J. Chem. Phys. 92, 285.
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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS. VOL. 32
EXACT AND APPROXIMATE RATE EQUATIONS IN ATOM-FIELD INTERACTIONS S. SWAIN Department of Applied Mathematics and Theoretical Physics Queen’s University of Belfast Belfast. Northern Ireland
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Scope of the Review . . . . . . . . . . . . . , . , . . . . . . . B. Previous Work . . . , . . , . . . . . . . . . , . . . . . . . . 11. Basic Features of the REA; the Two-Level Atom . . . . . . . . . . . . A. Rate Equations for the Two-Level Atom . . . . , . . . . . . . . . B. Exact Reduction of the Density Matrix Equations . . . . . . . . . . C. Comparison Between the Solutions for the Exact and Approximate Rate Equations . . . . . . . . . , . . . . . . . . . . . . . . . . . 111. Generalization to N Atomic Levels . . . . . . . . . . . . . . . . . A. The Model . . . , . . . . . . . . . . . . . . . . . . . . . . . B. The Rate Equations . . , . . . . . . . . . . . . . . . . . . . . C. Diagrammatic Evaluation of the Transition Rates . . . . . . . . . . IV. Extensions of the REA . . . . . . . . . . . . . . . . . . . . . . . A. Additional Damping Mechanisms . . . . . . . . . . . . . . . . . B. Interaction with a Squeezed Vacuum . . . . . . . . . . . . . . . . V. Rate Equations in the Dressed-Atom Picture . . . , . . . . . . . . . VI. Summary , . . . . . . . , . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . VII. Appendix: Derivation of the Exact Rate Equations . . . . . . . . . . . References , . . . . . , . . . . . . . . . . . . . . . . . . . . .
345 345 347 348 348 352 354 356 357 358 360 362 362 363 368 371 372 373 376
I. Introduction A.
SCOPE OF THE
REVIEW
Rate equations have a long history in the study of the interaction of atomic systems with the electromagnetic field. One of the first and most famous examples of their use is Einstein’s (1917) derivation of the Planck distribution law, which he obtained by considering a two-level system interacting with an incoherent electromagnetic field. In this review we are concerned in a general way with the theory of the interaction of N-level atomic systems with
345 Copyrighf 0 1994 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003832-3
346
S. Swain
electromagnetic fields. Rate equations, because they can often be written down intuitively, are frequently introduced as a first approximation in the treatment of such systems. A more accurate way of dealing with these problems is to write down the equations of motion for the density matrix. If the fields can be treated classically and the rotating wave approximation can be applied, there are N 2equations. Here we see one difficulty of the density matrix approach: the labor involved in solving such a set increases rapidly with N. If the rotating wave approximation cannot be invoked, the system of equations may not terminate at all (i.e., we have an infinite number of equations). A second difficulty with the density matrix approach involves the physical interpretation. The diagonal elements pii(t) have a direct physical meaning as the probability that the system will be found in the state i at time t , but the off-diagonal elements p i j ( t )or “coherences,” as they are often called, are much more difficult to interpret. This fact can inhibit the construction of suitable physical approximations to simplify the system of density matrix equations. The rate equation approach (REA, for short), on the other hand, leads to a system of only N equations for the occupation probabilities pii(t) only. The coherences have been eliminated; this simplifies the physical interpretation and eases the imposition of physical approximations. However, rate equations are themselves often obtained by making fairly drastic approximations whose physical implications are not immediately clear. It is the purpose of this review to show that exact rate equations can be derived that have the same advantages of physical interpretation as the approximate ones more frequently met with. Our basic assumption is that the intensities of any applied fields are time independent. The rate equations themselves can be written down by inspection once the system is specified. The physics of the problem is transferred to the calculation of the transition rates between the atomic levels. The general expression for these rates is complicated (not surprising, since we have solved a complicated problem exactly). However, we present diagrammatic methods that simplify their evaluation in specific problems. Also, because the rate equations have a simple interpretation, it is often possible to use physical arguments to approximate the rates. For example, if a transition is well off-resonance with an applied field mode, it may be sufficient to calculate the rate for this transition to lowest order. The system we consider is an N-level atomic system interacting with an applied monochromatic (or almost monochromatic field). This field may be treated as either classical or quantized. For convenience, we will consider only the classical case here. (The quantum case is no more difficult to treatit just requires a more complicated notation to specify it.) In this approach, it is easy to take into account such dissipative processes as collisions and the
RATE EQUATIONS IN ATOM-FIELD INTERACTIONS
347
effects of nonmonochromaticity in the driving field; we indicate how this can be done. In addition, the system may interact with a squeezed vacuum. The interaction of atomic systems with nonclassical fields, such as squeezed light, is a topic of considerable current interest. There are two basic approaches: we can describe the atom in the “bare” representation, or we can use the “dressed-atom” picture. In Sections II-IV we use the former method, and in Section V the latter. The relationship between our exact rate equations and the more usual approximate ones is discussed in Section 11, and figures presented that compare their predictions for specific examples. The dressed-atom method (e.g., Cohen-Tannoudji, 1977) has the advantage of conceptual simplicity, but is somewhat more complicated algebraically to set up. It has been shown to be particularly helpful in the discussions of resonance fluorescence and related phenomena (Cohen-Tannoudji and Reynaud, 1977), and the study of collisions in intense laser fields (Reynaud and Cohen-Tannoudji, 1982). Here we adopt a didactic approach. For this reason, the bulk of our discussion is concerned with the two-level atom. In spite of its simplicity, this model is sufficient to illustrate all the main features of the REA. We also show in Section 111 how the generalizations to an N-level system may be affected. Mathematical details are avoided in the main part of the text, but for completeness we present the derivation of the exact rate equations in the Appendix. We show that many of the features of the interaction of atomic systems with applied classical light fields and squeezed vacua can be understood in terms of the generalized transition rates that we introduce. WORK B. PREVIOUS It is impossible to give a complete bibliography of work that involves such a widely used technique as rate equations. Instead we pick out a few papers that have features in common with our approach. As we have mentioned, rate equations have frequently been introduced on an ad hoc basis to describe some physical process without any firm justification for their use. However, a well-defined method for deriving rate equations from the density matrix (or Bloch equations), based on the assumption that the off-diagonal elements of the density matrix decay much more rapidly than the diagonal ones, was outlined by Wilcox and Lamb (1960). We shall take the Wilcox-Lamb procedure to define the “conventional” rate equation approach. There was considerable interest in the 1970s in the use of rate equations to describe multiphoton excitation in multilevel atomic systems (Ackerhalt and Eberly, 1976; Hodgkinson and Briggs, 1979). Ackerhalt and Shore (1977) used
348
S. Swain
the Wilcox and Lamb approach to calculate the rate equations for a simple system and to compare the solutions with those obtained by a numerical solutions of the exact Bloch equations. (We present some similar comparisons in Section 1I.C.) Other examples of the use of rate equations in atom-field interactions are Haken (1970), Stone et al. (1973), CohenTannoudji (1975), Agarwal(l978) and Eberly and O’Neill(1979), Abraham et al. (1985), Cook and Kimble (1985), and Hassan and Bullough (1988a, b). A derivation of exact rate equations for a general multilevel system interacting with quantized field modes, rather than the classical case to be described here, was obtained (Swain, 1980) and applied to a variety of problems (Osman and Swain, 1980, 1982; Jackson et at., 1982; OBrien and Swain, 1983; Swain, 1982, 1984, 1985). Generalizations to deal with atoms interacting with classical driving fields (Smart and Swain, 1992b) and squeezed vacua (Smart and Swain, 1992a, c, d) have recently been presented. The mathematical technique we employ in the Appendix is the continued fraction method (for a review, see Swain, 1986). Other algebraic techniques that are similar in spirit to the continued fraction one have been derived by Ben-Reuven and Rabin (1979) and Rabin and Ben-Reuven (1979), Jackson and Swain (1981) and Dalton (1982).
11. Basic Features of the REA; the Two-Level Atom A. RATE EQUATIONS FOR
THE
TWO-LEVEL ATOM
We can illustrate the basic features of the REA by comparing the rate equations obtained for this system with the corresponding equations obtained in the density matrix approach. In this, we follow the work of Ackerhalt and Shore (1977). With (0) denoting the ground state and 11) the excited state, Einstein’s (1917) treatment is equivalent to assuming rate equations of the form
Po = U1OPI - U , , P , P1= U01P, - U , , P , where P i ( t )is the population of the ith level and U i jis the rate of transitions (assumed to be time independent) between levels li) and Ij) (see Fig. 1). These equations have a simple physical interpretation: Eq. (1) expresses the fact that the rate of change of the population of level (0) is equal to the difference between the rate of transitions into level (0) (the first term on the right) and
349
RATE EQUATIONS IN ATOM-FIELD INTERACTIONS
UOl
u10
10) FIG. 1. The transition rates in a two-level system in the steady-state situation. In the generalized REA, the rates U , are replaced, in Laplace space, by U;, = y i j + G&), where yij is the incoherent and G i j ( z )the coherent generalized transition rates.
the rate of transitions uut of level (0)(the second term). Equation (2) has a similar interpretation. This interpretation generalizes straightforwardly to the N-level system. In Einstein’s treatment, the transition rates were assumed to be of the form uol= Bu,
u,o = A + Bu
(3)
A and B being the Einstein A and B coefficients and u the spectral energy density of the incident incoherent radiation. In these equations, Bu is the rate of stimulated transitions, and A the rate of spontaneous transitions. It was, of course, the introduction of the concept of spontaneous emission that enabled Einstein to derive the black-body distribution law. Nowadays, most experiments in optical interactions are performed with lasers, which usually have to be considered as coherent, rather than incoherent, sources and the question naturally arises as to whether the REA can usefully be employed in this situation. It is the purpose of this review to show that this question can be answered affirmatively and to present examples of the use of rate equations. To compare the REA with that of the standard quantum mechanical method, which involves the use of the density matrix (or more generally, the master equation), we again analyze a two-level system interacting with a monochromatic field, such as may be produced by a laser. We consider the two-level system for mainly didactic purposes; in realistic problems one may deal with tens of interacting atomic levels. A primary motive in developing the REA is that there are only N rate equations to solve in an N-level problem, whereas there are N 2equations in the density matrix approach.
3 50
S . Swain
The density matrix equations for a two-level system interacting with a constant amplitude, monochromatic (classical) electromagnetic field are Po0
= m p 0 1 - PlO) + 4
$11
= - 3 Q ( P 0 1 - PlO) - 4
$01
=
-$Wii
(4)
1 1
-~oo)-f$A
(5)
1 1
+ Q + @poi
(6)
= bio*
where as before A is the rate of spontaneous emission from the excited state to the ground state, 6 is the detuning (the difference between the Bohr frequency of the atomic transition and the frequency of the field), 0 is the Rabifrequency (twice the product of the dipole moment of the transition with the amplitude of the field) and we have defined Q to be the collisional phase destruction rate. As we have introduced dissipation through the parameters A and Q, Eqs. (4) to (6) are not obtained from the Von-Neumann equation for the density matrix, but from the master equation (cf. Agarwal, 1974; Cohen-Tannoudji, 1977; Swain, 1980). In the latter approach, one considers a system interacting with a reservoir (such as the vacuum electromagnetic field), and one obtains an equation for the density matrix of the system alone (the master equation) by eliminating the reservoir variables from the Von-Neumann equation for the density matrix of the system plus reservoir combined. A standard way (Wilcox and Lamb, 1960) of obtaining rate equations from a system such as (4) to (6) is to set the time derivatives of the off-diagonal elements to zero and solve the resulting subset of equations for the diagonal elements in favor of the diagonal elements. With just one off-diagonal element (plus its conjugate), this procedure is trivial:
This expression for pol is then substituted into Eqs. (4) and ( 5 ) to give the following equations for the diagonal elements alone:
Po = Wl0Pl- Wo,P, P, = WOlPO - W,,P,
where Pi3 pii and Wol = w,
Wlo = w + A ,
with w =
+ Q)
$Q2($A ($4 Q)2
+
+ 62
(10)
RATE EQUATIONS IN ATOM-FIELD INTERACTIONS
35 1
These equations have the same structure as Eqs. (1) to (3), bearing in mind that a*is linearly proportional to the intensity of the applied field so that we have w = B’u. The rates W,, and W,, of Eq. (10)therefore have the same form as the rates U,, and Ulo defined in Eq. (3). We shall call such rate equations as these, obtained by means of the Wilcox and Lamb approximation and involving time-independent rates, Conventional rate equations. The natural question is, how well do the solutions to Eqs. (8) to (10) approximate the solutions to Eqs. (4)to (6)? The first point to notice is that the latter set of equations involve no approximation at all, if one is seeking only the steady-state solutions. This is obvious, because to obtain the exact steady-state solutions, one sets all the time derivatives of the density matrix elements on the righthand side of Eqs. (4)to (6) to zero and then solves the resulting set of equations. Since we have established that Eqs. (8) to (10) yield the exact steady-state solution, we consider henceforth the question of time dependence: how well do these equations represent the dynamics of the system? At first sight, the approximation (7) does not appear to be a good one. We have replaced timedependent coherences by their steady-state values, but according to Eqs. (4) and (5) the diagonal elements decay exponentially with weights A , while from Eq. (6) the decay constant for the off-diagonal elements is ($A + Q). Thus if Q = 0 (no collisions) the coherences will decay more slowly than the probabilities, and there is little apparent justification for this procedure. On the other hand, if Q >> A , then the coherences will decay much more rapidly than the probabilities, and we may expect the REA to be a good approximation. However, there is another factor to be taken into account in estimating the quality of the approximation. If we write Eqs. (8) and (9) in matrix form: P = KP the matrix K has real eigenvalues, so that the probabilities will consist of a sum of exponentially decaying terms. On the other hand, if we write Eqs. (4) to (6) in matrix form, the corresponding matrix is not Hermitean, and the solutions will consist of sums of exponentially decaying and oscillating terms. The oscillations are in fact the Rabi oscillations: it is well known that the populations of a two-level atom interacting with a sufficiently intense monochromatic field undergo oscillations at the Rabi frequency R. The conventional rate equation approach neglects these completely. In the next subsection, we show that we can solve the system of Eqs. (4)to (6) exactly, but in such a way that the equations for the populations possess the gain-loss structure characteristic of rate equations.
352
S. Swain
B. EXACTREDUCTION OF THE DENSITY MATRIXEQUATIONS The fact that the coefficients in the equations for the density matrix elements are time independent suggests that we should solve the system (4) to (6) by introducing the Laplace transform P = P(z) of p(t):
Noting that Y [ Q= z?(z) - f(O), the Laplace transform of Eqs. (4) to (6) is
''a' ' a= -31 (Po1 - P 1 0 ) - 4 h 1
ZPOO
- P O O ( 0 ) = 9 (Po1 - P10) + A P l l
(12)
ZPll
-P1,(0)
(13)
zBio -
o(0) = N V P i 1 - doo) - (& +IQ - i 4 P i o
(14)
We again solve for the coherences in terms of the populations:
In the great majority of applications, the initial coherences will be zero. For the sake of simplicity, we shall consider this to be the case here. It is not difficult to carry out the analysis without making this assumption, but it does add extra terms (which we write down later). Substituting (15) into Eqs. (12) and (13) we obtain a pair of equations for the populations that have the form of rate equations in Laplace space. They correspond very closely to Eqs. (8) and (9): zFo - Po(0) =
El0F, - EOlFO
(16)
zp", - P,(O) =
J%olp", - lop",
(17)
but with w of Eq. (10) replaced by %(Z)=
ii,
= ii,(z),
+
&2*(z + A + Q ) ( z +&I Q)'+62
+
Note that w = iG(O), showing once again but in a different way that the steady-state solution of Eqs. (8) and (9) is equal to the steady-state solution of the exact system (16) to (18). By taking the Laplace transform, one can obtain the exact generalized rate equations (16) and (17) for the two-level problem with no more labor than that needed to derive the approximate rate equations (8) and (9). We shall call rate equations of the form (16) and (17), in
RATE EQUATIONS IN ATOM-FIELD INTERACTIONS
353
Laplace space, exact or generalized rate equations, to contrast them with the approximate forms (8) and (9). The simplest way to obtain the exact time-dependent occupation probabilities is to solve Eqs. (16) and (17) and then perform the inverse Laplace transform. However, if we take the inverse of (16) and (17) first, we obtain
These equations show that the population of any level la) at time t depends upon the populations of the other levels at all previous times 7.For the latter reason, it may be termed a non-Markofian rate equation, in contrast with equations of the type (8) and (9), which may be termed Markofian. Although Eqs. (19) and (20) are more complicated than Eqs. (16) and (17), they still have a simple physical interpretation. For example, the first term on the righthand side of (19) is the probability that the system is in the state 11) at time z multiplied by the probability W,,(t-~)dz that it makes a transition from 11) to (0) in the remaining interval t - z. This has to be integrated over all times 7 from 0 to t to obtain the increase in the occupation probability of level 10) at time t due to transition into it from other levels. The final term on the righthand side is the net rate of transitions out of this level to all other levels. Before considering the generalization of this treatment to the multilevel atom, we describe the corrections to be made if we had not taken the initial value of the coherences to be zero. It is easy to show that the extra terms AFo and Apt where
would have to be added to the righthand side of Eqs. (16) and (17), respectively. They represent transients that contribute nothing to the steadystate solutions. Finally, we remark that to calculate some properties of atomic systems, we need the off-diagonal, rather than the diagonal, density matrix elements. This is true of the resonance fluorescence spectrum for example. There is no problem in finding such quantities: once the diagonal elements have been determined explicitly, the off-diagonal ones may be calculated by expressions such as Eq. (15).
354
S. Swain
C. COMPARISON BETWEEN THE SOLUTIONS FOR APPROXIMATE RATEEQUATIONS
THE
EXACTAND
Here we compare the time-dependent occupation probabilities calculated from the approximate rate equations (8) and (9) and the exact form (16) and (17). The solution of the latter is
+
+
Fo(z) = zPo(0) Wol(z) - zPl(0)+ i t ( z ) A z[Wo,(z)+ W1O(z)] z C z + 2 W + A l
+
(22)
+
Fl(Z) = ZP,(O)+ Wl,(Z) - zP,(O) G(z) A Z[WO1(Z)+ WIO(Z)] z C z + 2 W ) + A l
From the Laplace identity
we obtain the steady-state solution
This is easily seen to be equal to the steady-state solutions of Eqs. (8) and (9), with the correspondence w = it(O), if we impose the normalization conditions Pl(t)+P,(t) = 1. In Figs. 2 , 3 and 4 we compare the time-dependent solutions for the exact 0.8,
TIME
FIG.2. The ground-state occupation probability Po(t) under the initial condition Po(0) = 0 with y = 6 = 1, Q = 5 and Q = 0. The solid line is the solution of the exact rate equations while the dashed line is the solution of the approximate rate equations (8) and (9).
RATE EQUATIONS IN ATOM-FIELD INTERACTIONS
355
TIME
FIG.3. The parameters are the same as in Fig. 2 but the initial condition is P,(O) = 0.5.
and conventional rate equations, corresponding to the initial conditions P,(O) = 0, 0.5 and 1, respectively. We take 6 = y = 1, R = 5 and Q = 0. The latter choice corresponds to assuming pure radiative damping only. The dashed line is the solution obtained from the conventional rate equations (8) and (9) and is identical to expression (22) with k(z) replaced by iir(0) = w. The solutions are obtained by Laplace inversion. It can be seen that the agreement is good only close to the steady state. The approximate solution
TIME
FIG.4. The parameters are the same as in Fig. 2 but the initial condition is Po@) = I
356
S. Swain
shows no Rabi oscillations and tends to follow the average of the exact solution over the Rabi oscillations, although the area under this curve is greater than that under the exact solution. However, as we have discussed in Section II.A, the conventional rate equation approach is a good approximation only when the incoherent rates of the system dominate the coherent ones-that is, when the damping mechanisms present are so large as to wash out the Rabi oscillations. This feature is demonstrated in Fig. 5, where we have the same parameter values as in Fig. 3, but introduce an additional damping mechanism (collisions) by setting Q = 5. The Rabi oscillations are greatly reduced in amplitude, and the agreement between the approximate and exact expressions is greatly improved. The trend continues in Fig. 6, where we increase Q to the value 12. Here the Rabi oscillations have died out completely, and the agreement between the predictions of the approximate and exact rate equations is good.
111. Generalization to N Atomic Levels In this section we generalize the treatment for the two-level atom to the Nlevel atom case. As in the previous sections, we restrict our attention to the case where the applied monochromatic field can be treated classically. The extension to the situation where the applied field is quantized is straightforward (Swain, 1980). We also specify precisely the model to be studied here.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
TlME
FIG.5. The parameters are the same as in Fig. 2, the initial condition is P,(O) have introduced a finite collision rate Q = 5.
= 0.5, but
we
RATE EQUATIONS IN ATOM-FIELD INTERACTIONS
357
1
0.68,
TIME
FIG.6. The parameters are the same as in Fig. 5 but with Q increased to Q = 12.
Because the equations are exact, the general expressions are complicated, and we relegate the details of the calculation to the Appendix. It is based on derivations presented in Smart and Swain (1992a, c). In this section we quote only the results and give a physical discussion. A. THEMODEL
For simplicity, we consider an atomic system with nondegenerate energy levels Ei,which are labelled in such a way that Ei > E j if i >j . We use a system of units in which h = 1. The Hamiltonian for the interacting system of atom and fields may be written
H
=
1 Eili)(il + V i
(26)
where pij,l = e r i j g l is a coupling constant that is equal to the product of the dipole matrix element erij of the atom with the field strength of the electromagnetic field mode characterized by the index 1,whose frequency is ol. In general, the mode index 1 is a four-dimensional vector consisting of the wave vector and polarization index. As usual, we make the rotating wave approximation. This involves neglecting the antirotating terms in the interaction Hamiltonian V, which then becomes
358
S. Swain
Next we assume that a given mode A can cause transitions between only one pair of energy levels, i andj. We can then uniquely identify a frequency w1 with this pair of levels, so that a convenient change of notation is o*ZE w i j (29) where, to be definite, we here assume i > j . This definition implies that oijis positive. (Note that we are not defining w i j as the Bohr frequency (Ei-Ej)the transition is not necessarily resonant.) We may also write pij,*= pij, as the I label is now redundant. We may allow some of the oiJto be equal if necessary. By these means we cover the great majority of the situations that arise in practice. The interaction then takes the form I/ =
1 (pijli)(jle-iwrJ'+ p$lj)(ileiwJ)
(30)
i>j
The final equation may be written in a more compact form if we fix our convention so that w . . = -a.. ,r
(with w i j > 0 for i > j , and wii
= 0)
(31)
and interchange the subscripts i and j in the final term of Eq. (30). Then we may write V(t) = C pijli)(jle-imij'
(32)
iJ
where p i j is assumed to be a Hermitean matrix, with pii = 0.
B. THERATEEQUATIONS We now present our main results. The density matrix for the N-level system is (e.g., Agarwal (1974)): bob
= -iEo.bPab
-
1
[%c(t)Pcb
- P a ~ K b ( ~+ )l
1
YcoPcc - l-abPab
(33)
C
C
where Yba is the spontaneous decay rate for the transition b + a if b > a and is zero if b a, and r o b = Cc(yoc+ Ybc). As in the two-level case, we obtain generalized rate equations for the populations by taking the Laplace transform of this set of equations:
-=
( z + iEa,b
+ rob)?ob(Z)
= Pab(O) -
1
[pac?cb(z
C
+
YC~?CC(~) C
+ ioac)
- pcb?ac(z -!-iwcb)l
(34)
RATE EQUATIONS IN ATOM-FIELD INTERACTIONS
359
and formally eliminating the off-diagonal matrix elements from the equations for the diagonal elements (populations). The N equations for the Laplace transforms of the populations can be written in the rate equation form: ZFJZ)
- f‘,(O)
=
1 CRAzPc(z)-
c+a
RC(z)F,(z)]
(a = 1,. .
. ,N )
(35)
where P,(O) is the initial occupation probability of level la) (for simplicity we assume all initial coherences to be zero) and
Ra(z> = Yca + G C J 4
(36)
is the total transition rate between levels Ic) and la). The first term, yca, is the rate of incoherent transitions between these levels and the second term, Gc,(z) is the rate of coherent transitions induced by the applied field. Since the only decay mechanism considered here is radiative decay, the incoherent decay rate yCa is equal to is the Einstein A coefficient for spontaneous decay in the Ic) + la) transition. Equation (35) is clearly a straightforward generalization of Eqs. (16) and (17). The set of equations (35) have the same form for all N-level systems. All the physics of a specific problem is contained in the generalized transition rates and in particular the coherent rates, which are defined by Eqs. (90) and (86) of the Appendix. The general expression for these quantities is of course complicated, but for the moment we do not need their explicit form. In subsection C we describe a diagrammatic method for their evaluation, which is more convenient for application to specific problems than substituting into the algebraic formula. Taking only the first term in Eq. (90) and evaluating the quantity (ca) of Eq. (86) to zeroth order in p, we obtain the lowest order expression for GJJz) as
R,(z),
which is a Lorentzian function of the detuning, Sc,, = E , , - coca.In the steady state (z = 0) and in the limit of zero spontaneous decay, the Lorentzian becomes a delta function, and we have
This shows that the lowest order approximation to the steady-state transition rates is just the Fermi “golden rule” transition rate of standard perturbation theory, representing a single-photon transition between states Ic) and la). This justifies our calling the Gcageneralized transition rates.
S.Swain
360
If we go to a higher approximation, Eqs. (90) and (86) give Gca(z)= 2 Re
-
Z+rcaAifi
PacPca pcdpda
I
...
+**]
(39)
This shows that in general the denominator has a continued fraction structure, which implies that the “bare” widths TCaand level shifts Sc,a are modified by the field-dependent dipole interaction p. There are also additional terms in the series, of order p 3 , p4, etc. These represent multiphoton transitions from state a to state c. In the time domain, the N-level generalization of Eq. (35) is
and the N-level generalization of Eqs. (8) and (9) is Pa(t) =
1 ~@tc.(o)pc(t)- % c ( o ) p a ( f ) ~
c#a
+
where Ra(0)= yca Gca(0).This is the set of conventional rate equations which would be obtained by a generalization of the Wilcox-Lamb procedure. Similar conclusions apply here as in the two-level case. Thus we may summarize the position for the N-level system as follows: the conventional rate equation approach will give the exact solution for the steady-state populations, whether the fields are incoherent or coherent, provided the correct expressions for the transition rates, obtained from Eq. (90) with z = 0, are used. We cannot, however, expect it to correctly describe the time evolution unless the damping constants are so large that the Rabi oscillations are completely damped out. C. DIAGRAMMATIC EVALUATION OF THE TRANSITION RATES
Expression (90)is undeniably complex, and it is simpler in practice to employ a diagrammatic method for evaluating the transition rates. The diagrams that contribute to Gcaare illustrated in Fig. 7. Since we are dealing with transitions between density matrix elements there are two lines at the beginning of each diagram, each labelled c to represent the initial “state” pcc,and two lines at the end, labelled a, to represent the final state pa.. We examine Eq. (33) to determine the interactions that cause a transition from state c to a. In the bottom line of Fig. 7(a) we insert the interaction ipca,and in the upper line we insert the complex conjugate of this quantity (ipca)= - ipac.We draw dotted lines vertically downward from these points of interaction, and associate the
RATE EQUATIONS IN ATOM-FIELD INTERACTIONS
--lpac
C
361
a
A
1pco V
C
(0c)-’
a
(a)
C
a
\/
A
(4 FIG. 7. The diagrams that contribute to G&). The contribution of each diagram is obtained by taking the product of the matrix elements and dividing by the (ij)functions. A sum is performed over all intermediate states. Figure 7(a) shows the lowest (second-order) diagrams. The contribution is ( - ip,,J(ipcO)/(ac).Figure 7(b) gives the third-order diagram, whose con~ ~ Finally, ) / ( c a in , Fig. 7(c) we show the effect of including the tribution is ( i ~ ~ ~ ~ - i p ~ ~ - i pda). squeezing interaction. The contribution of this diagram is (- ip,,,)( - ip@J(-tc,,)/(ac, ca).
factor (ac)-’ with the intermediate “states” paC(but not the initial or final “state”). The product of all these factors gives the contribution of the diagram. This diagram represents the process
--i P a c
Pcc
iUca
Pac
Paa
It is important to note that corresponding to this diagram is a mirrorimage diagram that contributes the complex conjugate. This is a general rule, and accounts for the 2Re factor in front of Eq. (90). Thus to find the
362
S . Swain
contribution to a given order, one need list explicitly only half the total number of diagrams (omitting any diagram for which the mirror image is already present), calculate the contribution of each and then take twice the real part of the sum. In Fig. 7(b) we show the diagram corresponding to the second term in Eq. (90).The same procedure is used for calculating the contribution except that, where there are two or more intermediate “states” as here (pca and pdc), the contributing factor is (ca, da)-’. The total contribution to a given order is obtained by summing the contributions of all diagrams up to and including those of that order. A similar diagrammatic approach can be used to calculate the (ca) functions. In this case, the start and end states are both equal to pea, and each contribution must be multiplied by the additional factor (- 1). The zeroth order contribution is z + rca+ ic5c,a,and the second-order process is
iPRC
Pea
ipda
Pcd
Pca
(cf. the denominators of expressions (37) and (39), which give the zeroth and second-order contributions to (ca)). The physical interpretation of (ca)-’ is that it is the Laplace transform of the probability of the coherence pca persisting for a time t. For brevity, we do not give the diagrams for (ca) here, but they may be found in Smart and Swain (1992c), where further details of the diagrammatic approach are presented.
IV. Extensions of the REA A. ADDITIONALDAMPING MECHANISMS In this section we generalize our model to take into account some additional effects. We begin by looking at additional damping mechanisms. The rate equation approach, being founded on the master equation method, can easily accommodate many of these. First we consider the situation where the intense driving field is not monochromatic, but possesses a finite linewidth due to phase diffusion (Haken, 1970). It is easiest to deal with this in the situation where the driving field is quantized, rather than classical. Then the states of the system are of the form la, n), where a labels the atomic state as
RATE EQUATIONS IN ATOM-FIELD INTERACTIONS
363
before and n is the number of photons in the driving field. Corresponding to Eq. (33) we have the following equation of motion for the density matrix: ban,bm
=
-I"H,
Plam.bm
+ b'absnm
YcaPcn,cn - r a b P a n , b m
-
qn- m)2Pan,bm
(42)
C
where the final term on the righthand side accounts for the driving laser having a finite linewidth L due to phase diffusion (Osman and Swain, 1980; Swain, 1980). Note that the linewidth does not appear in the equations of motion for the diagonal elements. When the driving field is treated classically, it is usually straightforward to write down the effect of a finite linewidth by analogy with the quantum case-that is, it is usually clear by inspection what the values of n and m would be if the system was being treated quantum mechanically. We can also take into account elastic collisions in a phenomenological way by adding to the term r o b the term QZb where Q:b is the rate of elastic collisions between states la) and Ib). If one is dealing with collisions in intense laser fields in any detail, a better procedure is to introduce the damping parameters in the density matrix equations written in the dressed picture (Reynaud and Cohen-Tannoudji, 1982). B. INTERACTIONWITH A SQUEEZED VACUUM A topic of considerable current interest is the interaction of atomic systems with nonclassical light-particularly the squeezed vacuum. (For reviews of squeezedstates, see Walls (1983), Loudon and Knight (1987), Teich and Saleh (1989) and Zaheer and Zubairy (1990).)The important feature of such states, as far as this review is concerned, is the correlation between the annihilation and creation operators 6 and b' of the squeezed vacuum, which are
(b'(t)b(t'))= XNd(t - t')
(43)
(b(t)b'(t'))= x(N + l)d(t - t')
(44)
(b(t)b(t'))= ~ M d ( t t')e-i"('+") (bt(t)b+(t'))= ~ M * d (t t')e"'('+'')
(45)
(46)
where v is the center frequency of the squeezed field, x is a damping parameter, N is real and M is in general complex: M = 1MI eid.We note the inequality lMlZ
< N ( N + 1)
(47)
364
S. Swain
If M = 0, then the set of correlation functions (43) to (46) are those that would be obeyed by a single mode of the black-body radiation field-that is, they represent a completely incoherent excitation. On the other hand, if M # 0, then the correlations (45) and (46) are nonzero, and we have a quantum field without a classical analog. Clearly, we expect the maximum nonclassical effects when IMl is maximal-that is, when the equality holds in Eq. (47). The master equation for a multilevel system interacting with an applied field and a squeezed vacuum has been derived by Gardiner and Collett (1985) by making use of the correlations (43) to (46).A separate broadband squeezed source (characterized analogously to Eqs. (43) to (46) by the parameters Nab and M a b ) is assumed to interact with each atomic transition a -+ b. Assuming the center frequency of the squeezed mode corresponding to the laser frequency wab to be equal' to that frequency, the master equation is
where the first contribution represents the first two terms on the righthand side of Eq. (33), we have assumed that the squeezed light field provides the only source of damping and that tab
= XabMab
(49)
(Recollect that a > b implies E , > Eb, where E , is the energy of the atomic state la).) In the previous, r . 6 = &(yaC + &), and Xab ( a > b) is the spontaneous decay rate in the absence of a squeezed field. Note that the values of y and r have been redefined in Eq. (48) as compared with Eq. (33). When Nab = 0, the two definitions become identical. Equation (48) is formally identical to Eq. (33) apart from the last term, which connects an off-diagonal element of the density matrix with its transpose. It can be solved in a similar way to Eqs. (33), and a diagrammatic interpretation provided. The modification to the diagram rules is that an additional interaction due to the squeezing that causes the two lines representing the state of the density matrix to cross over. The second-order contribution in the two-level case is shown in Fig. 7(c). With each such crossover, we associate the quantity - t a b where b denotes the upper line in the diagram and a the lower line; the other rules are unaltered. (For more
RATE EQUATIONS IN ATOM-FIELD INTERACTIONS
365
details, see Smart and Swain, 1992~).The diagram in Fig. 7(c) corresponds to the process
- ipac
Pcc
-L a
Pac
-
Pca
ipac Paa
+
We have used this approach to discuss the excitation of a two-level atom interacting with a classical applied field and a squeezed vacuum taking into account the effect of finite laser linewidths and collisions (Smart and Swain, 1992d).We find that the atomic steady-state occupation probabilities depend strongly on the degree of squeezing N and the squeezing phase 0,which is defined as follows. If plo = IpLlol exp($J, then @ = 2&-4
(51)
(where 4 is the phase of M). For the probability of the atom being found in the excited state when the steady state has been attained, we find P,
w + Yo1
=
2w
(52)
+ Yo1 + Y l O
where w is the coherent transition rate W =
in2(r+ X M
C O S ~
(53)
T2 + 6’ - x 2 M 2
+
+
+ +
1) and l- = %yol ylo) Q L. The term 6 is the detuning that we henceforth in this section take to be zero. Equation (52) shows that the dependence on @ will be greatest when the coherent rate w is not dominated by the incoherent rates yol and ylo. Equation (53) reduces to Eq. (37) for wca(z)in the limit z -P 0, if we set N = M = 0. For N large and IMI maximal, it takes on a range of values, which depend strongly on the value of 0.Taking Q = L = 6 = 10 for simplicity, we have 0 = 21plo1,yol = x N , yl0 = x ( N
lim w =
NDI
-
4R2NIX R2/(4N~f
{
@=O
(54)
@=x
A plot of w as a function of N for @ = 0, x/4,x/2,3x/4 and x, is shown in Fig. 8 (upper graph). For @ 0 the contributions of the two diagrams in Figs. 7(b) and 7(c) interfere constructively, whereas for @ x they interfere
-
destructively. By examination of Eq. (52), it is clear that Pl(oo)will saturate to the valuef for N >> 1 (strong incoherent excitation). It will also saturate if w is the
366
S . Swain
-
dominant quantity, which is the case if the intensity of the driving laser is sufficiently large and 0 0. These features are shown in the lower graph of Fig. 8, where we plot the steady-state probability for the atom to be found in its excited state as a function of N and @ in the absence of collisions under excitation by a monochromatic driving field. Here we have taken the Rabi
_______----:
_...-
10'-
0
'
05
1
1s
2
25
3
35
4
45
5
3
3.5 35
4
4.5 45
5
N
0.36
0
0.5 05
1
1.5 15
2
2.5
N
FIG. 8. The coherent transition rate k(0) (upper graph) and the steady-state occupation probability for the excited state (lower graph) of a two-level system interacting with a resonant (6 = 0) classical field and a squeezed vacuum, as a function of the degree of squeezing N for various values of the squeezing phase: rP= 0, 4 4 , nf2, 3n/4 and x. We have assumed (MI= JN(N+I), and 7 = @ = 1, with zero linewidth of the classical field ( L = 0) and no elastic scattering (Q = Q. = 0). In each figure, @ vanes from 0 = 0 for the top plot to @ = x for the bottom plot.
RATE EQUATIONS IN ATOM-FIELD INTERACTIONS
367
frequency to have the value f l z = 10, which corresponds to a strong but not saturating intensity in the absence of squeezing. (Note y = 1.) We observe that the greatest variability with @ occurs for an optimum value of N-in this n case, N 0.5. This is the value of N for which the coherent rate w with approximately equals the incoherent rate yol. For these values of the parameters a considerable degree of control can be exercised over the excited state population, as compared with its value when the squeezed vacuum is not applied, by varying @. In Figs. 9 and 10 we plot w and P1(co)respectively for some nonzero values of L and Q. Somewhat surprisingly, the introduction of finite values for these damping parameters increases the dependence of P , on the squeezing phase. This effect arises because these nonzero values of L and Q greatly reduce the value of w,which has the consequence that the optimum value of N occurs for smaller, and therefore less saturating, values of w. We have also shown that, in the absence of external damping, the squeezing induces marked, phase-dependent asymmetries in the resonance fluorescence spectra when the detuning 6 is nonzero (Smart and Swain, 1992d).
-
-
( b ) : L = 1 , Q, = 0
(a): L = 0 , Qe = 0 I
600 I
. I
O=D
/I
I
/
_ _ - -.___..__._...._...-0 00
1 30
2 60
(c): L = 0 , Oe = 1
J 90
N
520
(d): L = Qe = 1
12
FIG.9. Here we show the effects of allowing nonzero values for Q, and L. Here, and in Fig. 10. we take y = 1, R = 1.5 and 6 = 0. (From Smart and Swain, 1992d.)
368
S. Swain (a):
_,_
a
a =L=0
(b): L = 1 , Qe = 0
lr72-- -------
____----___----
\
\. 0'"
0.45
0.0
1 .J
1.6
M Y
0 / /
5.2
3.9
N
0
FIG.10. The steady-state population of the excited state for the same parameters as in Fig. 9. (From Smart and Swain, 1992d.)
V. Rate Equations in the Dressed-Atom Picture In this section we adopt a new approach: we transform the equations of motion to the dressed-state picture before eliminating the off-diagonal elements. In this way we obtain dressed-state rate equations. The principle of the dressed-state approach is to consider first, in isolation, the Hamiltonian for the interacting atom-applied monochromatic field system, in the rotating wave approximation. We neglect all damping mechanisms for the moment. Then if the atom is modeled as an N-level system, the N eigenvectors of the system can in principle be found. These are the dressed states. The next step is to consider the density matrix for the full system, with all the damping mechanisms included, and to transform it to the dressed-state picture. CohenTannoudji and Reynaud (1977) have shown that this approach is very powerful in the discussion of resonance fluorescence (see also Courty and Reynaud, 1989.)
RATE EQUATIONS IN ATOM-FIELD INTERACTIONS
369
We illustrate the approach by considering a two-level atom interacting with a classical, monochromatic field and a squeezed vacuum. The equations for the density matrix in the bare picture are
boo = -YolPoo+YloPll -i(g*P,o-gPo,)
(55)
+ i(g*p,o - WOl)
(56)
b11
= YOlPOO - YlOPll
b0l
= -(To1
-
+ ig*(poo - P11) - (Po1 Tol = gyol+ y l o ) = T. For
i6)POl
where g = Iplol exp(i0,) and henceforth consider the case of exact resonance, 6 states, which are eigenfunctions of the Hamiltonian
= 0.
(57)
simplicity we Then the dressed
are
Transforming the density matrix equations to the dressed state representation, they become
where I'= y l o
+ yol
= (2N
+ l)y, and
2yM cos0 = R,
2yM sin0 = 1
(62)
First we note that if 1 = 0 ( M = 0 or 0 = n), then the equations for the diagonal elements are uncoupled from the off-diagonal elements, and the equations for the populations are already in exact rate equation form. The equations for the off-diagonal elements however, are always coupled to the populations. In certain circumstances, the secular approximation may be invoked to simplify this latter set. This approximation asserts that those variables on the righthand side of such equations as (60) that are oscillating at a vastly different rate from the variable on the lefthand side be ignored. Thus, close to the steady state, pas has the approximate time whereas paa(t)and pss(t) are approxdependence pas@) exp( - iRt)pas(0),
-
370
S. Swain
-
imately time independent and psu(t) exp(iRt)p,(O). It follows that if R >> r,R we may approximate Eqs. (60) and (61), even if I # 0, as
If the secular approximation is not valid, we may use the same procedure as in Section I1 to eliminate the off-diagonal elements and obtain the rate equations Zijuu(Z) -Pu,(O) = &p(Z)Wsu(4 - F , , ( Z ) K p ( Z )
where
= - CzPsp(4-
Pps(0)l
(65)
wj = yij + wij(z), Yus = Ysu
=
(r+ 4114
(66)
and in the steady state,
Explicitly, these have the following solution for the dressed state populations
There are some interesting differences between the rate equations in the bare and dressed pictures. In the former case, the incoherent rates are different but the coherent rates are equal; in the latter case, the incoherent rates are equal but the coherent ones differ. The secular approximation is consistent here with setting wUs= wsu = 0 in Eq. (69). (It is equivalent to letting R -+ co.)Then it is clear that the dressed-states populations are equal. (We are assuming exact resonance.) However, if we do not make the secular approximation, this is no longer the case, and large population differences between the dressed states may be obtained providing @ n/2 and R is not too large. This is shown in Fig. 11. We also find qualitative differences between predictions made with and without the use of the secular approximation for the spectrum in resonance fluorescence. In particular, the secular approximation always leads to a symmetric spectrum at exact resonance, whereas the exact calculation produces an asymmetric one. This is illustrated in Fig. 12.
-
RATE EQUATIONS IN ATOM-FIELD INTERACTIONS
0
0.5
I
1.5
2
2.5
3
3.5
4
37 1
4.5
RABI FREQUENCY
FIG. 11. The population of the dressed state lor) for the two-level atom as a function of the Rabi frequency of the applied, resonant, monochromatic classical field for various values of the squeezing parameter N. We have assumed IMI = J”+lj. In the secular approximation, the dressed states have equal populations.
VI. Summary We have derived exact rate equations for an atomic system interacting with classical electromagnetic fields and squeezed vacua. It is most convenient to represent them in Laplace space, where they have a simple algebraic form and a simple physical interpretation. For the two-level atom, we have compared the time-dependent solutions of the exact and “conventional” rate equations, obtained using the Wilcox and Lamb technique. For the N-level case, we have given exact expressions for the transition rates i?l,.(z), which appear in the rate equations, and presented diagrammatic methods for their evaluation. Provided the rates have been correctly calculated, the conventional rate equations will give correct results for the steady-state solutions, but will in general describe the time-dependent behavior incorrectly. They will give a good approximation to the dynamics when the dissipative mechanisms are sufficiently large for the Rabi oscillations not to appear in the motion. We have also derived rate equations when the dressed atom representation is used and used these to obtain the resonance fluorescence spectrum for a twolevel atom interacting with a squeezed vacuum without making the secular approximation. We have shown that many features of the interaction between atoms and
S . Swain
372 0.8
Fluom m a Frcsucncy
FIG.12. The resonance fluorescence spectrum for a two-level atom in a squeezed vacuum as calculated exactly (solid line) and using the secular approximation (dashed line). Here only radiative damping is operating, and we take y = 1 , b = 0. In the upper figure R = 4 and in the lower R = 8. We have assumed (MI = Note that the asymmetry, which is not predicted by the secular approximation, is significant.
Jm.
electromagnetic fields can be qualitatively (and quantitatively) understood by considering the incoherent and generalized coherent transition rates.
Acknowledgments This work was supported by the United Kingdom Science and Engineering Research Council, and by a N.A.T.O. Collaborative Research Award.
RATE EQUATIONS IN ATOM-FIELD INTERACTIONS
373
VII. Appendix: Derivation of the Exact Rate Equations This mathematical derivation of the exact rate equations may be omitted by readers uninterested in the details, although such readers may wish to refer to Eq. (90) et seq. for the transition rates. We begin with the master equation for the density matrix: bob
= -iE~,bPab-i
c
1
[ ~ ~ ( ~ ) p c b - p a c ~ b ( ~ ) l
C
YcaPcc-rabPob
(70)
C
where = E i - E j , r . 6 = 1/2(ra+rb) with ra= Ccyac being the total spontaneous decay rate out of level la). The term yac is the spontaneous decay rate from level la) to level Ic) (with a > c). Defining the Laplace transform of f ( t ) by T(z) we find the Laplace transform of the master equation to be
By putting a = b in this equation, we see that the diagonal elements paa(z)are connected to off-diagonal elements only of the form + iwbo),where the argument z + imba is determined by the matrix labels, a and b. Defining gab
?ab(Z
+ iuba)
(72)
the equation for this quantity becomes
where
We thus have a set of linear inhomogeneous equations for the quantities 6. We have shown (Swain, 1986) how the set of equations a i j x j= bi
( i = 1, 2 , . . .,n)
(76)
i= 1
may be solved in terms of continued fractions as
The starred summation sign means that none of the variables summed over
374
S. Swain
may be equal to each other or to the solution index i. Thus, in the second term, it is understood that j # i, and in the third term that j # i and k # i,j. Only off-diagonal elements a i j therefore appear in the numerators of the sums. The solution (77) is exact if the system of equations is finite. Turning now to the (i), (i,j), . .. functions, we note that the order of the terms in the parentheses is unimportant: e.g., ( i , j , k) = ( j , i, k ) = (k,j , i), etc. The first member is defined as
the second as
(k.17 = (i)(j)i= ( i ) j ( j )
(79)
and higher order members similarly. For example, (i, j , k )
(i)(j)i(k)ij= (i)i(j)(k)ij= (i)jk(jh(kL
etc
(80)
The quantities ( i , j , . . .)abc.. are defined in a way similar to Eqs. (78) to (80), except that, in addition to the restrictions implied by the starred summation sign, the variables summed over in the definition may not be equal to any of the subscripts a, b, c, . . . . Thus the subscripts in the quantity of interest (i,j,. . define the additional restrictions on the sums. Equations (78) to (80) provide a recursive definition for the ( i , j , . . .) functions. For a finite system of equations, the restrictions on the sums cause the series to terminate, and the exact solution is obtained. Equation (71) may be written in the form of Eq. (76) if we adopt tetradic notation, replacing each label i in Eq. (76) by an ordered pair of labels, ab:
1'
&ab,cd%d
= Bab
c.d
where the prime on the sum means c # d (i.e., only off-diagonal X ' s appear on the lefthand side), B o b is defined in Eq. (75), and &ab.cd
The solution is
where
=
{
ipacdbd
&ab
- ipdbdca
(ab # cd) (ab = cd)
RATE EQUATIONS IN ATOM-FIELD INTERACTIONS
Explicitly, the first few terms of the solution for
ab, cb)
*;
-t
[
pacpcdBdb - (ab, cb, db)
- pcbpdcBad
+
(ah, ac, ad)
-1
(ab, ac) +
(Tab
375
are
1
pacpdbBcd (ab, cb, cd)
+
pcbPadBdc (ab, ac, dc)
(85)
(a z b)
The values of (a, b) are given by
Equation (85) gives the solution for the off-diagonal elements in terms of the diagonal elements and the initial coherences Pab(0) through the factors Bob. We substitute these solutions into the equation for naa= paa(z)= Pa(z): (Z
+ raa)Pa = Paa(0) - i cC+ a pacoca + i c1 pca(Tac + C Y c a b c c #a c#a
(87)
For simplicity, we assume Pab(0)= 0 for all a, b with a # b. That is, there are no initial coherences. This assumption is not essential; it is straightforward to rework the steps that follow without making it. Substituting into Eq. (71) with b = a, we obtain
This equation may be rearranged into rate equation form (in Laplace space): cz
+ ra + %a(z)lFa(z)= Pa(0)+
Crca cfa
+ %ca(z)~Fc(z)
(89)
376
S. Swain
where, to the third order in p of the denominators: Gca(z) = 2 Re
{-
PaePca
-i
c*[ (da, dc)
pcapadpdc
d
&#adpdc + -
+- (ca, da)
pcapadpdc]
+
.}
..
(90)
and
where the (ca), (ca,da), etc., are defined in Eq. (86). We have used the properties (ij, kl, .. .) = ( j i , Ik, . . .I*,
(ij,
kl)ab.cd ,... =
(kl, @b*o.dc ,...
(92)
which are easily inferred from the definitions. It can be shown (Swain, 1980) that
Using this property, we may write Eq. (89) in a form having a gain-loss structure:
(where we have made use of the definition
raa= Zc+.yac).
REFERENCES Abraham, N. B., Lugiato, L. A,, and Narducci, L. M. (1985). J . Opt. SOC.Am. B2, 7. Ackerhalt, J. R., and Eberly, J. H. (1976). Phys. Rev. A 14, 1705. Ackerhalt, J. R., and Shore, B. W. (1977). Phys. Rev. A 16, 277. Agarwal, G. S. (1974). Springer Tracts in Modern Physics, 70, 1. Agarwal, G. S. (1978). Phys. Rev. A 18, 1618. Ben-Reuven, Y., and Rabin, A. (1979). Phys. Rev. A 19, 2056. Cohen-Tannoudji, C. (1977). In: Frontiers in Laser Spectroscopy ( R . Balian, S . Haroche and S. Liberman, eds.) North-Holland, Amsterdam. Cohen-Tannoudji, C. (1975). Atomic Physics, Vol. 4. (G. Zu Putlitz et a/., eds.) Plenum Press, New York. Cohen-Tannoudji, C., and Reynaud, S. (1977). J. Phys. B 10, 345. Cook, R. J., and Kimble, H. J. (1985). Phys. Reo. Lett. 54, 1023. Courty, J.-M., and Reynaud, S. (1989) Europhys. Lett. 10, 273. Dalton, B. J. (1982). J . Phys. A 15, 2157. Eberly, J. H., and ONeil, S. V. (1979). Phys. Rev. A 19, 1161. Einstein, A. (1917). Phys. Z . 18, 121. Gardiner, C. W., and Collett, M. J. (1985). Phys. Rev. A 31, 3761. Haken, H. (1970). In: Handbuch der Physik ( S . Flugge, ed.), 25, Part IIc. Springer-Verlag, Berlin and New York.
RATE EQUATIONS IN ATOM-FIELD INTERACTIONS
377
Hassan, S. S., and Bullough, R. K. (1988a). Physica A 151, 397. Hassan, S. S., and Bullough, R. K. (1988b). Physica A 151, 425. Hodgkinson, D. P., and Briggs, J. S. (1979). Chem. Phys. Lett. MA, 511. Jackson, R. I., and Swain, S. (1981). J. Phys. A 14, 3169. Loudon, R., and Knight, P. L. (1987). J. Mod. Opt. 34, 709. OBrien, D. P., and Swain, S. (1983). J . Phys. B 16, 2499. Osman, K. I., and Swain, S. (1980). J . Phys. B 13, 2397. Osman, K. I., and Swain, S. (1982). Phys. Rev. A 25, 3187. Rabin, Y., and Ben-Reuven, A. (1979). Phys. Rev. A 19, 1697. Reynaud, S., and Cohen-Tannoudji, C. (1982). J . Physique 43, 1021. Smart, S., and Swain, S. (1992a). In: Quantum Measurements in Optics (P. Tombesi and D. F. Walls eds.), Plenum Press, New York, p. 265. Smart, S., and Swain, S. (1992b). Opt. Commun. 88, 218. Smart, S., and Swain, S. (1992~).Phys. Reu. A 45, 6857. Smart, S., and Swain, S. (1992d). Phys. Rev. A 45, 6863. Stone, J., Thiele E., and Goodman, M. F. (1973). J. Chem. Phys. 59, 2909. Swain, S. (1980). Adv. At. Mol. Phys. 16, 159. Swain, S. (1982). J. Phys. B 15, 3405. Swain, S. (1984). J . Phys. B 17, 3873. Swain, S. (1985). J. Opt. SOC. America B2, 1666. Swain, S. (1986). Adu. At. Mol. Phys. 22, 387. Teich, M., and Saleh, B. E. A. (1989). Quantum Opt. 1, 153. Walls, D. F. (1983). Nature, Lond. 306,141. Wilcox, L. R., and Lamb, W. E., Jr. (1960). Phys. Rev. 119, 1915. Zaheer, K., and Zubairy, M. S. (1990). Adu. At. M o l . Phys. 28, 143.
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ADVANCES IN ATOMIC. MOLECULAR, AND OPTICAL PHYSICS. VOL. 32
ATOMS IN CAVITIES AND TRAPS H . WALTHER Sekrion Physik der Universirat Miinchen and Max-Planck-lnsli!ul fur Quantenoptik Munich, Germany
1. Introduction . . . . . . . . . 11. Review of the One-Atom Maser.
. . . . . . . . . . . . . . . . . . . Ill. Dynamics of a Single Atom. . . . . . . . . . . . . IV. A Source of Nonclassical Light . . . . . . . . . . .
. . . .
. . . .
. . . .
. . . .
. . . . . . . . . . . . . . . . . . . . .
319 380 383 385
V. A New Probe of Complementarity-The One-Atom Maser and Atomic Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Experiments with Ion Traps . . . . . . . . . . . . . . . . . . . . . VII. Order versus Chaos: Crystal versus Cloud . . . . . . . . . . . . . . . VIII. The Ion Storage Ring . . . . . . . . . . . . . . . . . . . . . . . . 1X. Ordered Structures in the Storage Ring and Comparison with the Theory . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
389 39 1 393 391 399 404
I. Introduction The modern techniques of laser spectroscopy allow one today to investigate single-atom phenomena. A prominent example is the work on the photon correlation in resonance fluorescence (see Cresser et al., 1982 for a review). In the present chapter two groups of more recent experiments of this type will be reviewed with special emphasis on applications to study quantum phenomena. The first group deals with the one-atom maser and the second with the application of trapped ions (e.g., Wineland et al., 1984). In recent years there was also considerable progress in trapping neutral atoms (Chu et al., 1986; Pritchard et al., 1988). These techniques are very promising and are undergoing a rapid development; due to lack of space they will not be discussed in detail here. 379 Copyright 0 1994 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003832-3
380
H . Walther
11. Review of the One-Atom Maser The most fundamental system to study the generation process of radiation in lasers and masers is to drive a single-mode cavity by single atoms. This system, at first glance, seems to be another example of a “Gedanken” experiment, but such an one-atom maser (Meschede et al., 1985)really exists and in addition can be used to study the basic principles of radiation-atom interaction. The advantages of the system are (1) It is the first maser that sustains oscillations with less than one atom on the average in the cavity.
(2) The system allows one to study the dynamics of the energy exchange between an atom and the radiation field (Jaynes and Cummings, 1963). (3) The setup allows one to study in detail the conditions necessary to obtain nonclassical radiation, especially radiation with sub-Poissonian photon statistics in a maser system directly.
(4)It is possible to study a variety of phenomena of a quantum field including the quantum measurement process. What are the tools that make this device work? The enormous progress in constructing superconducting cavities together with the laser preparation of highly excited atoms-Rydberg atoms-have made such a one-atom maser possible. Rydberg atoms have quite remarkable properties (Haroche and Raimond, 1985; Gallas et al., 1985) that make them ideal for such experiments. The probability of induced transitions between neighboring states of a Rydberg atom scales as n4, where n denotes the principle quantum number. Consequently, a single photon may be enough to saturate the microwave transition between adjacent levels; moreover, the spontaneous lifetime of a highly excited state is very large. We obtain a maser by injecting these Rydberg atoms into a superconducting cavity with a high quality factor. The injection rate is such that on the average there is less than one atom present inside the resonator at any time. The experimental setup of the one-atom maser is shown in Fig. 1. A highly collimated beam of Rubidium atoms passes through a Fizeau velocity selector. Before entering the superconducting cavity, the atoms are excited into the upper maser level 63 p3/2by the frequency-doubled light of a cw ring dye laser. The laser frequency is stabilized onto the atomic transition 5 s1/2+ 63 p3/2,which has a width determined by the laser linewidth and the transit time broadening corresponding to a total of a few MHz. In this way, it is possible to prepare a very stable beam of excited atoms. The ultraviolet light is linearly polarized parallel to the electric field of the cavity. Therefore,
ATOMS IN CAVITIES AND TRAPS
38 1
Field ionization
Laser excitation
"CS
mic beam
FIG. 1. Scheme of the one-atom maser. To suppress black-body-induced transitions to neighboring states, the Rydberg atoms are excited inside the liquid-Helium-cooled environment.
only Am = 0 transitions are excited by both the laser beam and the microwave field. The superconducting Niobium maser cavity is cooled down to a temperature of 0.5 K by means of a 3He cryostat. At this temperature the number of thermal photons in the cavity is about 0.15 at a frequency of 21.5 GHz. The cryostat is carefully designed to prevent room temperature microwave photons from leaking into the cavity. This would considerably increase the temperature of the radiation field above the temperature of the cavity walls. The quality factor of the cavity is 3 x 10" corresponding to a photon storage time of about 0.2 s. Two maser transitions from the 63 p3/2 level to the 61 d,/, and to the 61 d5/2level are studied. The Rydberg atoms in the upper and lower maser levels are detected in two separate field-ionization detectors. The field strength is adjusted to ensure that in the first detector the atoms in the upper level are ionized, but not those in the lower level. The lower-level atoms are then ionized in the second field. To demonstrate maser operation, the cavity is tuned over the 6 3 ~ 3 ~ 2 61d3,, transition and the flux of atoms in the excited state is recorded simultaneously. Transitions from the initially prepared 63 p3/2 state to the 61 d3/2 level (21.50658 GHz) are detected by a reduction of the electron count rate. In the case of measurements at a cavity temperature of 0.5 K, shown in Fig. 2, a reduction of the 63 p3/2 signal can be clearly seen for atomic fluxes as small as 1750 atom+. An increase in flux causes power broadening and a
H . Walther
382
N = 17501s
N = 5000h 0,
e
Y
4-
C
3
0 L)
I
-YX)KHz
I
I
*28M(ttz
21.506 577 400 GHz
Cavity
frequency
FIG.2. A maser transition of the one-atom maser manifests itself in a decrease of atoms in the excited state. The flux of excited atoms N governs the pump intensity. Power broadening of the resonance line demonstrates the multiple exchange of a photon between the cavity field and the atom passing through the resonator.
small shift. This shift is attributed to the ac Stark effect, caused predominantly by virtual transitions to neighboring Rydberg levels. Over the range from 1,750 to 28,000 atoms/s the field ionization signal at resonance is independent of the particle flux, which indicates that the transition is saturated. This and the observed power broadening show that there is a multiple exchange of photons between Rydberg atoms and the cavity field. For an average transit time of the Rydberg atoms through the cavity of 50ps and a flux of 1,750 atoms/s we estimate that approximately 0.09 Rydberg atoms are in the cavity on the average. According to Poisson statistics this implies that more than 90% of the events are due to single atoms. This clearly demonstrates that single atoms are able to maintain a continuous oscillation of the cavity with a field corresponding to a mean number of photons between unity and several hundred. It was also demonstrated by Brune et al. (1987) that a two-photon maser with Rydberg atoms can be realized. The Rubidium Rydberg series offers suitable states with a near resonant intermediate level, which enhances the
ATOMS IN CAVITIES AND TRAPS
383
transition probability, so that maser operation with a few atoms at a time in the cavity is possible.
111. Dynamics of a Single Atom These experiments have been performed with a cavity temperature of 2.5 K. When low atomic-beam fluxes are used, then the cavity of the single-atom maser contains essentially thermal photons only. At high fluxes, the atoms deposit more energy in the cavity, and the number of photons stored in the cavity increases and their statistics change. This change in photon statistics leads to a change of the dynamics of the atom-radiation interaction; i.e., the Rabi nutation. For a coherent field with a Poissonian probability distribution, a dephasing of the Rabi oscillations is observed, causing P,(t), the probability of finding the atom in the upper maser level, to collapse (Eberly et al., 1980). After the collapse P,(t) starts oscillating again in a very complex way. Such collapses and revivals recur periodically, the time interval being proportional to the square root of the photon number stored in the cavity. Both collapse and revival in the coherent state are pure quantum features without any classical counterpart (Eberly et al., 1980). Collapse and revival also occur in the case of a thermal field, where the spread in the photon number is far larger than for a coherent state and the collapse time is much shorter. In addition, revivals overlap completely and interfere, producing a very irregular time evolution. On the other hand, a classical thermal field represented by an exponential distribution of the intensity shows collapse, but no revival at all. From this it follows that revivals are pure quantum features of the thermal radiation field, whereas the collapse is less clear-cut as a quantum effect (Knight and Radmore, 1982; Yo0 and Eberly, 1985). The above-mentioned effects have been demonstrated experimentally by Rempe et al. (1987), using the Fizeau velocity selector to vary the interaction time of the atoms with the cavity (Fig. 1). Figure 3 shows a series of measurements obtained with the single-atom maser, where P,(t) is plotted against interaction time for increasing atomic flux N . The strong variation of P , ( t ) for interaction times between 50 and 80 p s disappears for larger N and a revival shows up for N = 3000s-' for interaction times larger than 140ps. The average photon number in the cavity varies between 2.5 and 5, about two photons being due to the black-body field in the cavity corresponding to a temperature of 2.5 K.
T=25K
--n.
N.3000
5.'
OS7i 0.6
c
2 0.5-
fiP
0.L-
-
05Rb 63p,,,
0.3-
L
1
I
I
I
1
61d,,
1
I
I
I
I
1
1
1
I
FIG.3. Quantum collapse and revival in the one-atom maser. Plotted the probability P,(t) of finding the atom in the upper maser level for different fluxes N of the atomic beam. The measurements were made with a cavity of 2.5 K (Rempe et al., 1987).The solid curves follow from theory (Jaynes).
384
ATOMS IN CAVITIES AND TRAPS
385
IV. A Source of Nonclassical Light One of the most interesting questions in connection with the one-atom maser is the photon statistics of the electromagnetic field generated in the superconducting cavity. This problem will be discussed in the following. Electromagnetic radiation can show nonclassical properties (Walls, 1979, 1983, 1986), that is, properties that cannot be explained by classical probability theory. Loosely speaking we need to invoke “negative probabilities” to get deeper insight into these features. We know of essentially three phenomena that demonstrate the nonclassical character of light: photon antibunching (Kimble et al., 1977; Cresser et al., 1982), sub-Poissonian photon statistics (Short and Mandel, 1983) and squeezing (Slusher et al., 1985; Loudon and Knight, 1987). Mostly methods of nonlinear optics are employed to generate nonclassical radiation. However, the fluorescence light from a single atom caught in a trap also exhibits nonclassical features (Carmichael and Walls, 1976; Diedrich and Walther, 1987). Another nonclassical light generator is the one-atom maser. We recall that the Fizeau velocity selector preselects the velocity of the atoms; hence, the interaction time is well-defined that leads to conditions usually not achievable in standard masers (Filipowicz et al., 1986a-c; Lugiato et al., 1987; Krause et al., 1987, 1989; Meystre et al., 1988; Slosser et al., 1990).This has a very important consequence when the intensity of the maser field grows as more and more atoms give their excitation energy to the field. Even in the absence of dissipation this increase in photon number is stopped when the increasing Rabi frequency leads to a situation where the atoms reabsorb the photon and leave the cavity in the upper state. For any photon number, this can be achieved by appropriately adjusting the velocity of the atoms. In this case the maser field is not changed any more, and the number distribution of the photons in the cavity is sub-Poissonian (Filipowicz et al., 1986a-c; Lugiato et al., 1987); that is, narrower than a Poisson distribution. Even a number state that is the state of a well-defined photon number can be generated (Krause et al., 1987, 1989; Meystre, 1987; Brune et al., 1990) using a cavity with a high enough quality factor. If there are no thermal photons in the cavity-a condition achievable by cooling the resonator to temperatures below 100 mK-very interesting features such as trapping states show up (Meystre et al., 1988). In addition, steady-state macroscopic quantum superpositions can be generated in the field of the one-atom maser pumped by two-level atoms injected in a coherent superposition of their upper and lower states (Slosser et al., 1990). Unfortunately, the measurement of nonclassical photon statistics in the cavity is not that straightforward. The measurement process of the field
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invokes coupling to a measuring device, with losses leading inevitably to a destruction of the nonclassical properties. The ultimate technique to obtain information about the field employs the Rydberg atoms themselves. For this purpose the population of the atoms in the upper and lower maser levels is probed after they left the cavity. This is done using state selective field ionization. The technique maps the photon statistics of the field inside the cavity via the atomic statistics; the number of maser photons can be inferred from the number of atoms detected in the lower level. In addition, the variance of the photon number distribution can be deduced from the number fluctuations of the lower-level atoms (Rempe and Walther, 1990). In the experiment, we are therefore interested mainly in the atoms in the lower maser level. Under steady-state conditions, the photon statistics of the field is essentially determined by the dimensionless parameter 0 = ( N e X+ l)l’ZRtin,, which can be understood as a pump parameter for the one-atom maser (Filipowicz et al., 1986). Here, N , , is the average number of atoms that enter the cavity during the lifetime of the field T,, tint is the time of flight of the atoms through the cavity and R is the atom-field coupling constant (onephoton Rabi frequency). The one-atom maser threshold is reached for @ = 1. At this value and also at 0 = 271 and integer multiples thereof, the photon statistics is super-Poissonian. At these points, the maser field undergoes firstorder phase transitions (Filipowicz et al., 1986). In the regions between these points sub-Poissonian statistics are expected. The experimental investigation of the photon number fluctuation is the subject of the following discussion. In the experiments (Rempe et al., 1990), the number N of atoms in the lower maser level is counted for a fixed time interval T roughly equal to the storage time T, of the photons. By repeating this measurement many times the probability distribution P ( N ) of finding N atoms in the lower level is obtained. The normalized variance (Mandel, 1979) Q, = [ ( N 2 ) - ( N ) ’ ( N ) ] / ( N ) is evaluated and used to characterize the deviation from Poissonian statistics. A negative (positive) Q, value indicates sub-Poissonian (super-Poissonian) statistics, while Q, = 0 corresponds to a Poisson distribution with ( N 2 ) - ( N ) 2 = ( N ) . The atomic Q, is related to the normalized variance Q f of the photon number by the formula
Q, = &PQr(2+ QJ)
(1)
which was derived by Rempe and Walther (1990). It follows from Eq. (1) that the nonclassical photon statistics can be observed via sub-Poissonian atomic statistics. The detection efficiency E for the Rydberg atoms reduces the subPoissonian character of the experimental result. The detection efficiency was 10%in our experiment; this includes the natural decay of the Rydberg states between the cavity and field ionization.
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As an example we discuss the result for the 63p3,,-61 d5,, maser transition with 0 = 44 kHz shown in Fig. 4. Fast atoms with an atom-cavity interaction time of ti,, = 35 ps were used. A very low flux of atoms of N , , > 1 is already sufficient to generate a nonclassical maser field. This is the case since the vacuum field initiates a transition of the atom to the lower maser level, thus driving the maser above threshold. The sub-Poissonian statistics of atoms near N,, = 30, Q , = -4% and P , = 0.45 (see Fig. 4) is generated by a photon field with a variance ( n 2 ) ( n ) , = 0.3 ( n ) , which is 70% below the shot noise level. Again, this result agrees with the prediction of the theory (Filipowicz et al., 1986a-c; Lugiato et al., 1987). The mean number of photons in the cavity is about 2 and 13 in the regions N , , x 3 and N,, = 30, respectively. Near N,, = 15, the photon number changes abruptly between these two values. The next maser phase transition with a super-Poissonian photon number distribution occurs above N,, z 50. We emphasize that the reason for the sub-Poissonian atomic statistics is the following: a changing flux of atoms changes the Rabi frequency via the
d
i
"'5
I0
'
io'j,'iO'So'€i Nex
FIG.4. Variance Q,, of the atoms in the lower maser level as a function of N e x .Shown in the region above threshold for the 63 p3,2 -, 61 d,,, transition. The solid line represents the result of the one-atom maser theory using formula ( 1 ) to calculate Q.. The agreement with the experiment is very good.
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stored photon number in the cavity. By adjusting the interaction time, the phase of the Rabi-nutation cycle can be chosen so that the probability for the atoms leaving the cavity in the upper maser level increases when the flux and therefore the photon number is raised or vice versa. We observe subPoissonian atomic statistics in the case where the number of atoms in the lower state is decreasing with increasing flux and photon number in the cavity. This feedback mechanism is also demonstrated when the anticorrelation of atoms leaving the cavity in the lower state is investigated. Measurements of this “antibunching” phenomena for atoms are described in the following. In Fig. 5 we plot the probability gt2)(t) of finding an atom in the lower maser level 61 d5,2 at time t after a first one has been detected at t = 0. The probability g(’)(t) was determined from the actual count rate by normalizing with the average number of atoms determined in a large time interval. Time is given in units of the photon storage time. The error bar of each data point is determined by the number of about 7,000 lower-level atoms counted. This corresponds to 2 x lo6 atoms that have crossed the cavity during the total measurement time. The detection efficiency is near E = 10%. The measurements were taken for N,, = 30. The time of flight of the atoms through the cavity was tint= 37 ps, leading to 0 = 9. The fact that anticorrelation is observed shows that the atoms in the lower state are more equally spaced than expected for a Poissonian distribution. It
g‘”(t 1 100
--- - - -
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means when two atoms enter the cavity close to each other the second one performs a transition to the lower state with a reduced probability. The experimental results presented here clearly show the sub-Poissonian photon statistics of the one-atom maser field. In addition, the maser experiment leads to an atomic beam with atoms in the lower maser level showing number fluctuations which are up to 40% below those of a Poissonian distribution found in usual atomic beams. This is interesting, because atoms in the lower level have emitted a photon to compensate for cavity losses inevitably present under steady-state conditions. Although this is a purely dissipative process giving rise to fluctuations, nevertheless the atoms still obey sub-Poissonian statistics.
I. A New Probe of Complementarity-The One-Atom Maser and Atomic Interferometry The preceding section discussed how to generate a nonclassical field inside the maser cavity. But this field is extremely fragile because any attenuation causes a considerable broadening of the photon number distribution. Therefore it is difficult to couple the field out of the cavity while preserving its nonclassical character. But what is the use of such a field? In the present section we want to propose a new series of experiments performed inside the maser cavity to test the “wave-particle” duality of nature, or better said “complementarity” in quantum mechanics. Complementariy (Bohm, 1951; Jammer, 1974)lies at the heart of quantum mechanics: matter sometimes displays wavelike properties manifesting themselves in interference phenomena, and at other times it displays particlelike behavior thus providing “which-path’’ information. No other experiment illustrates this wave-particle duality in a more striking way than the classic Young double-slit experiment (Wootters and Zurek, 1979; Wheeler and Zurek, 1983). Here we find it impossible to tell which slit light went through while observing an interference pattern. In other words, any attempt to gain “which-path” information disturbs the light so as to wash out the interference fringes. This point has been emphasized by Bohr in his rebuttal to Einstein’s ingenious proposal of using recoiling slits (Wheeler and Zurek, 1983) to obtain “which-path” information while still observing interference. The physical positions of the recoiling slits, Bohr argued, are known only to within the uncertainty principle. This error contributes a random phase shift to the light beam which destroys the interference pattern. Such random-phase arguments illustrating in a vivid way how the “whichp a t h information destroys the coherent wavelike interference aspects of a
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given experimental setup are appealing. Unfortunately, they are incomplete: in principle, and in practice, it is possible to design experiments that provide “which-path” information via detectors that do not disturb the system in any noticeable way. Such “Welcher Weg” (German for “which path”) detectors have been recently considered within the context of studies involving spin coherence (Englert et al., 1988). In the present section we describe a quantum optical experiment (Scully and Walther, 1989) that shows that the loss of coherence occasioned by “Welcher Weg” information, that is, by the presence of a “Welcher Weg” detector, is due to the establishment of quantum correlations. It is in no way associated with large random-phase factors as in Einstein’s recoiling slits. The details of this application of the micromaser are discussed by Scully et al. (1991). Here only the essential features are given. We consider an atomic interferometer where the two particle beams pass through two maser cavities before they reach the two slits of the Youngs interferometer. The interference pattern observed is then also determined by the state of the maser cavity. The interference term is given by
(@if), @y) I @y,@y’) where I@?))and denote the initial and final states of the maser cavityj. Let us prepare, for example, both one-atom masers in coherent states I@:)) = laj) of large average photon number (m) = lajlz > 1. The Poissonian photon number distribution of such a coherent state is very broad, Am x a >> 1. Hence the two fields are not changed much by the addition of a single photon associated with the two corresponding transitions. We may therefore write
I@.”)
x
ICYj)
which to a very good approximation yields
(@if),
I a(/),my)) x (al,a2 1 a l , a 2 ) = 1
Thus there is an interference cross term different from zero. When, however, we prepare both maser fields in number states Inj) the situation is quite different. Since the number states are orthogonal, the interference term disappears whenever a passing atom deposits a photon in one of the cavities, therefore no interference is observed in such a case. At first sight this result might seem a bit surprising when we recall that in the case of a coherent state the transitions did not destroy the coherent cross term, i.e., did not affect the temporal interference fringes. However, in the example of number states we can, by simply “looking” at the one-atom maser state, tell which “path” the atom took. The atomic interference experiment in connection with one-atom maser
ATOMS IN CAVITIES AND TRAPS
39 1
cavities is a rather complicated scheme for a “Welcher Weg” detector. There is a much simpler possibility, which we will discuss briefly in the following. This is based on the logic of the famous “Ramsey fringe” experiment. In this experiment two microwave fields are applied to the atoms one after the other. The interference occurs since the transition from an upper state to a lower state may occur either in the first or in the second interaction region. In order to calculate the transition probability, we must sum the two amplitudes and then square them, thus leading to an interference term. (For details see Englert et a[., 1992). We conclude this section by emphasizing again that this new and potentially experimental example of wave-particle duality and observation in quantum mechanics displays a feature that makes it distinctly different from the Bohr-Einstein recoiling-slit experiment. In the latter the coherence, that is, the interference, is lost due to a phase disturbance of the light beams. In the present case, however, the loss of coherence is due to the correlation established between the system and the one-atom maser. Random-phase arguments never enter the discussion. We emphasize that the argument of the number state not having a well-defined phase is not relevant here; the important dynamics are due to the atomic transition. The fact that whichpath information is made available washes out the interference cross terms (Scully et al., 1991).
VI. Experiments with Ion Traps In contrast to neutral atoms, ions can easily be influenced by electromagnetic fields because of their charge. In most of the experiments the Paul trap is used. It consists of a ring electrode and two end caps as shown in Fig. 6. Trapping can be achieved if time-varying electric fields (Paul et al., 1958; Fischer, 1959) are applied between ring and caps (the two caps are electrically connected). A dc voltage in addition changes the relation of the potential depth along the symmetry axis (vertical direction in Fig. 6 ) to that in a perpendicular direction. The equation of motion of an ion in such a situation is the Mathieu differential equation, well known in classical mechanics, which-depending on the voltages applied to the trap (dc and radiofrequency voltages)-allows stable and unstable solutions. Another way to achieve trapping is the use of a constant magnetic field aligned along symmetry axis leading to the Penning trap (Wineland et a]., 1984; Dehmelt, 1967). In this case only a dc voltage has to be applied between ring and cap electrodes. To produce the ions in the Paul trap, a neutral atomic beam is directed
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Laser-Beam
Electrons
Atomic-Beam
FIG.6. Sketch of the Paul trap. The fluorescent light is observed through a hole in the upper electrode.
through the trap center and ionized by electrons. The resulting trapped ions have a lot of kinetic energy rendering them useless for most applications, such as spectroscopy; therefore the ions have to be cooled. This is done by laser light. The laser frequency v is tuned below the resonance frequency, so that the energy of the photon is not sufficient to excite the atom. Crudely, the ion can extract the missing energy from its motion and thus reduce its kinetic energy. In other words, the atomic velocity Doppler shifts the atom into resonance to bridge the detuning gap A between laser and resonance frequency, and the atom absorbs the photon of momentum hk = hv/c. After the absorption process the momentum of the atom is reduced, lowering its kinetic energy. The lowest temperature achievable is determined by the Doppler limit (Dalibard et al., 1988), which is in the millikelvin region. The low temperatures can be obtained within a fraction of a second. The results discussed in this chapter were obtained using a Paul trap with a ring diameter of 5 mm and an end-cap separation of 3.54 mm (Diedrich and Walther, 1987; Diedrich et al., 1987).This trap is larger than most of the other ion traps used in laser experiments (Wineland et al., 1984; Neuhauser et al.,
ATOMS IN CAVITIES AND TRAPS
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1978, 1980). The radio frequency of the field used for dynamic trapping is 11 MHz. The trap is mounted inside a stainless steel, ultrahigh vacuum chamber. The ions are loaded into the trap by means of a thermal beam of neutral atoms (Magnesium atoms in our case), which are then ionized close to the center of the trap through a small hole in the lower end-cap (see Fig. 6). In order not to distort the trap potential, the hole is covered by a fine Molybdenum mesh. The neutral Mg beam and the laser beam pass through the gaps between the end-cap and the ring electrodes. The laser frequency is shifted by an amount A below the 3 S1,, + 3 P,,, resonance-transition of 24Mg+ at 280nm to extract kinetic energy from the ions by radiation pressure as discussed previously. In this way, a single ion can be cooled to a temperature below 10mK. The fluorescence from the ions is observed through a hole in the upper end-cap (see Fig. 6), again covered by a Molybdenum mesh. The large size of the trap allows a large solid angle for detecting the fluorescence radiation, either with a photomultiplier, or by means of a photon-counting imaging system. To observe the ions, the cathode of the imaging system is placed in the image plane of the microscope objective attached to the trap; in this way images of the ions could be obtained (Blumel et al., 1988).
VII. Order versus Chaos: Crystal versus Cloud The existence of phase transitions in a Paul trap manifests itself by significant jumps in the fluorescence intensity of the ions as a function of the detuning A between the laser frequency and the atomic transition frequency. These discontinuities are indicated in Fig. 7 by vertical arrows and occur between two types of spectra: a broad and a narrow one, analogous to the fluorescence spectrum of a single, cooled ion. We interpret the broad spectrum as a fingerprint of an ion cloud and the narrow spectrum as being characteristic for an ordered many-ion situation with a “single-ion signature.” Thus the jumps clearly indicate a transition from a state of erratic motion within a cloud to a situation where the ions arrange themselves in regular structures. In such a crystalline ion structure the mutual Coulomb repulsion is compensated by the external, dynamic trap potential. The regime of detunings in which such crystals exist is depicted in Fig. 7 by the horizontal arrow. The existence of the two phases-crystal and cloud-can be verified experimentally by direct observation with the help of a highly sensitive imaging system and theoretically by analyzing ion trajectories via Monte Carlo computer simulations (Bliimel et al., 1988a,b). The excitation spectrum of Fig. 7 and the jumps in it were recorded by
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I
- 800
I
-LOO Laser Detuning t M H z l
1
I
)
0
FIG.7. Fluorescence intensity, that is, photon counts per second, from five ions as a function of the laser detuning A. The vertical arrows indicate the detunings where phase transitions occur. The horizontal arrow shows the range of detunings in which a stable five-ion crystal is observed. The spectrum was scanned from left to right.
altering the detuning A from large negative values to zero. When we scan the spectrum in the opposite direction the jumps occur at different values of A, which means hysteresis is associated with these phase transitions (Blumel et al., 1988a,b).Such hysteresis behavior can be expected with laser-cooled ions because the cooling power of the laser strongly depends upon the details of the velocity distribution of the ions. The behavior of the ions in the trap is governed by the trap voltage, the laser detuning and the laser power. Hysteresis loops appear whenever one of these parameters is changed up or down while the others are kept constant. Figure 8 shows the ion structures as measured with the imaging system. For the measurements only a radio-frequency voltage was applied to the trap electrodes: in this case the potential is a factor of two deeper along the symmetry axis than perpendicular to it, and therefore plane ion structures are observed being perpendicular to the symmetry axis (for details see Blumel et al., 1988a,b). After we have accepted the existence of the ion crystals and the corresponding phase transitions, how do they actually occur? Would the cooling laser not force any cloud immediately to crystallize? A heating mechanism balancing the cooling effect of the laser must be the answer to this puzzle, but what heating mechanism? Since the early days of Paul traps this so-called radio-frequency heating has repeatedly been cited (Wineland et al., 1984). A deeper understanding, however, was missing and was provided only recently
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FIG.8. Two, three, four and seven ions confined by the dynamical potential of a Paul trap and crystallized into an ordered structure in a plane perpendicular to the symmetry axis of the trap. The average separation of the ions is 20 pm.
by a detailed study of the dependence of the cloud +crystal and crystal + cloud phase transitions on the relevant parameters (Bliimel et al., 1988a,b; 1989). The ions are subjected to essentially four different forces: the first one arising from the trapping field, then the Coulomb interaction between the ions, the laser cooling force, and finally a random force arising from the spontaneously emitted photons. Using these forces computer simulations of the motion of the ions can be performed (Bliimel et al., 1988a). Depending on the external parameters such as the laser power, the laser detuning, and the radio-frequency voltage, the experimentally observed phenomena could be reproduced (Bliimel el al., 1988a). Some of the results of the simulations are summarized in Fig. 9. Plotted is the radio-frequency heating parameter K of five ions versus their mean separation (Bliimel et al., 1989). For zero laser power and large r, we did not observe any net heating of the ions. This is confirmed by our experiments in which, even in the absence of a cooling laser, large clouds of ions can be stored in a Paul trap over several hours without being heated out of the trap. When the ions are far apart, the Coulomb force is small, and on short time scales the ions behave essentially like independent
396
H . Walther Chaotic Regime
- I c r
Molhleu Regime
-
quasi periodic
60.-
i
2
6
6
1’0 1’2 IT2“pml
14
16
-100
0
*
18
0 100 x
10 x
FIG.9. Average heating rate
K of five ions in a Paul trap versus mean ion separation. The insets show the power spectrum and the corresponding stroboscopic Poincark sections (plane perpendicular to the symmetry axis) of relative separation of the two ions in three characteristic domains: the crystal state, the chaotic regime, and the Mathieu regime. The units on the axes are in pm. In order to calculate the power spectrum of the “crystal” shown on the lefthand side, the distance of the two ions was displaced by 1 pm from the equilibrium position. The Mathieu regime shown on the right is dominated by the secular motion.
single stored ions. For this reason, we call this part of the heating diagram the Mathieu regime (Bliimel et al., 1988a, 1989). Turning on a small laser, the rms radius r reduces drastically, but comes to a halt at about 14pm. At this distance the nonlinear Coulomb force between the ions plays an important role and the motion of the ions gets chaotic. In this situation the power spectrum of the ions becomes a continuum leading to a radio-frequency heating process. Increasing the laser power further results in an even smaller cloud. The smaller cloud produces more chaotic radio-frequency heating, as seen clearly by the negative slope of the heating curve in the range 8 < r < 14 pm. Finally, in the range 4 < r < 8 pm there is still chaotic heating but the slope of the heating curve is positive. As a consequence of the resulting triangular shape of the heating curve at a laser power of about P = 150 pW,corresponding to r z 8 pm, the chaotic heating power can no longer balance the cooling power
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of the laser light and the cloud collapses into the crystalline state located at r x 3.8 pm. At this point the amplitude of the ion motion is so small that the nonlinear part of the repulsive Coulomb force is negligible again, so that chaotic heating disappears and the phase transition occurs. Due to this collapse of the cloud state, the behavior of the heating rate in the range 3.8 < r < 8 pm cannot be studied by balancing laser cooling and radio-frequency heating. In this case we start out from the crystal state and slightly displace the ions to explore the vicinity of the crystal. We observe no heating for 3.8 c r < 4 pm, but quasiperiodic motion, and thus dub this the quasiperiodic regime. We call the upper edge of the quasiperiodic regime ( r x 4 pm) the chaos threshold. An initial condition beyond the chaos threshold, i.e., satisfying r x 4 pm, leads to heating, and expansion of the ion configuration. For the quasiperiodic, the chaotic, and the Mathieu regimes, respectively, we display the corresponding type of power spectrum as the insets above the abscissa of Fig. 9. The data were actually taken for the case of two ions, but would not look much different in the five-ion case. We obtain a discrete spectrum in the quasiperiodic regime and a complicated noisy spectrum in the chaotic regime. The spectrum in the Mathieu regime is again quite simple and dominated by the secular motion frequency. We also show stroboscopic pictures of the locations of the ions in the plane perpendicular to the symmetry axis of the trap characterizing the three regions (inserts below the abscissa in Fig. 9).
VIII. The Ion Storage Ring A completely new era of accelerator physics could begin if it were possible to produce, store and accelerate Coulomb crystals in particle accelerators and storage rings. To work with crystals instead of the usual dilute, weakly coupled particle clouds has at least one advantage: the luminosity of accelerators (storage rings) could be greatly enhanced and (nuclear) reactions whose cross sections are too small to be investigated in currently existing accelerators would become amenable to experimentation. In the following, we would like to discuss very briefly our recent experiments using a miniature quadrupole storage ring. The storage ring is similar to the one described by Drees and Paul (1964) or by Church (1969). We can observe phase transitions of the stored ions and the observed ordered-ion structures are quite similar to the ones expected in relativistic storage rings, however, much easier to achieve. The motivation for building this small storage ring came from the fact that micromotion perturbs the ion
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H. Walther
structures in a Paul trap and only a single trapped ion is free of micromotion (Wineland et al., 1984). The ring trap used consists of a quadrupole field, leading to harmonic binding of the ions in a plane transverse to the electrodes of the quadrupole and no confinement along the axis (see Fig. 10). Confinement along the axis is achieved, however, by the Coulomb interaction between the ions when the ring is filled; then the total number of ions in the ring determines the average distance between them. A scheme of the ring trap used for our experiment is shown in Fig. 10 (Walther, 1991; Waki et al., 1992). It consists of four electrode rings shown in the insert on the right. The hyperbolic cross section of the electrodes required for an ideal quadrupole field was approximated by a circular one. The experiments were also performed with Mg' ions. The ions were produced between the ring electrodes by ionizing the atoms of a weak atomic beam produced in an oven that injected the atoms tangentially into the trap region. The electrons used for the ionization came from an electron gun, the electron beam of which was perpendicular to the direction of the atomic beam. A shutter in front of the atomic beam oven allowed the interruption of the atomic flux. The ultrahigh vacuum chamber was pumped by an ion getter pump. After baking the chamber, a vacuum of lo-'' mbar could be reached.
FIG.10. Quadrupolestorage ring. The cross section of the electrode configurationis shown on the insert on the righthand side of the figure.
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Under these conditions the number of ions stored in the trap stayed practically constant for several hours. When laser cooling of the ions is started a sudden change in the fluorescence intensity is observed, resembling very much that seen with stored ions in a Paul trap (Fig. 8), which indicates a phase transition and the formation of an ordered structure of ions. The ion structure can also be observed using an ultrasensitive imaging system. Pictures of typical ion structures are shown in Fig. 11. The ions are excited by a frequency-tunable laser beam that enters the storage ring tangentially. In the linear configuration the ions are all sitting in the center of the quadrupole field, therefore they do not show micromotion and it is possible to cool them further to temperatures in the microkelvin region. The new cooling methods proposed by Dalibard et al. (1988) can be applied to the Mg’ ions so that the single-photon recoil limit can be achieved for the cooling process. At this limit the kinetic energy of the ions is smaller than the energy resulting from the “zero-energy’’ motion of the harmonically bound ions; the ion structure then reaches its vibrational ground state; i.e., a Mossbauer situation is generated. That means that the recoil limit for laser cooling of the ions does not exist anymore, the ion structure then corresponds to a Wigner crystal. The observed configurations (Figs. 11 and 12) in the quadrupole ring trap are described in a recent paper by Birkl et al. (1992). We will review the major results reported in that paper in the next section as well as the comparison to the theory (Rahman and Schiffer, 1986; Hasse and Schiffer, 1990).
IX. Ordered Structures in the Storage Ring and Comparison with the Theory In the molecular dynamics calculations (see, e.g., Hasse and Schiffer, 1990),a cylindrically symmetric, static harmonic potential is assumed to describe the confining field. Each particle is subjected to the Coulomb forces of all other particles and to the confining field. The classical equations of motion are integrated for a system of several thousand particles, starting with random positions and velocities, to give the time evolution of the system. Cooling the stored particles is simulated by scaling down the resulting velocities of the stored particles at defined instants of the integration process. After sufficient cooling, ordered structures such as strings, zigzags, shells and multiple shells should arise, owing to the confining field’s harmonic potential. Our experiments are well suited to checking these predictions. To compare the experimental results with theory, we can use the normalized “linear particle density” A, which is given by the ion density multiplied by the ratio of
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H . Walther
FIG.11. Crystalline structures of laser-cooled 24Mg+ions in the quadrupole storage ring. (a) At a low ion density (A = 0.29) the ions form a string along the field axis. (b) Increasing the ion density transforms the configuration to a zigzag (A = 0.92). (c) At still higher ion densities, the ions form ordered helical structures on the surface of a cylinder, e.g., three interwoven helices at A = 2.6. As the fluorescent light is projected onto the plane of observation in this case the inner spots are each created by two ions seated on opposite sides of the cylindrical surface, resulting in a single bright spot.
Coulomb repulsion and confining force of the trap (Hasse and Schiffer, 1990). Low I values correspond to a deep potential well or a small number of ions, resulting in an equilibrium structure closely confined to the field axis making up a string of ions (Fig. Il(a)). This is the micromotion-free configuration discussed previously, and the analog of the single stored ion in a Paul trap, as
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FIG.12. Images and intensity profiles of (a) one shell plus string (A x 4.3), (b) two shells plus string ( A x 12.2),(c) three shells plus string (A x 26), and (d) four shells (A x 34). There are up to x8 x lo5 ions stored in the ring for the four-shell structure.
in both cases the ions sit in the potential minimum and show no micromotion. For higher values of A, the structures extend more and more into the off-axis region, giving rise to (in the order of increasing A) a plane zigzag structure (Fig. 1l(b)) and cylindrical structures with the ions forming helices on cylindrical surfaces. The structure in Fig. 1l(c) consists of three interwoven helices with six ions per pitch. The string and the zigzag have also been observed with laser-cooled Hg' ions in a linear trap (Raizen et al., 1992). Increasing further the number of ions leads to structures with smaller spacings between the ions where we cannot optically resolve individual ions any more. Images of these structures are presented in Fig. 12. The radial intensity profiles displayed on the right-hand side of the figure provide information about the structures as they give a measure of the radial distribution of the ions. For increasing A it becomes energetically more favorable to create a string inside the first ion shell (Fig. 12(a)) to provide space for more particles. This results in a structure that is a three-dimensional analog of the plane seven-ion crystal for a Paul trap (Fig. 8). The inner string turns into a second shell at higher densities: a string then develops inside this
402
H . Walther
second shell (Fig. 12(b)),and so on. Figures 12(c) and 12(d) show structures consisting of three shells plus string, and four shells, respectively. We have been able to observe all possible structures, from the string up to four shells plus string. The formation of multiple-shell crystalline structures in the quadrupole storage ring contrasts with the observation of shell structures in Penning traps, where the ions do not occupy fixed positions inside the shells (Gilbert et al., 1988). Figure 13(a) gives a summary of experimental data for all recorded images in which the ions were individually resolved. The depth of $, of the potential well and the ion density per unit length are the experimental parameters. The theoretical boundaries between the different shell structures, predicted by Hasse and Schiffer (1990) in terms of the functional dependence of 1 on $,, and the ion density are given by the straight lines with constant 1. String structures are expected for 1 < 0.709, zigzag structures in the range 0.709 < A < 0.964 and single shells in the range 0.964 < 1 < 3.10. Many different structures that are degenerate in energy are expected within the single-shell regime. We obtained stable configurations near 1 = 1.3 and 1= 2.0 (two interwoven helices) and near 1 = 3.0 (three interwoven helicesFig. ll(c)). The observed structures agree with the predicted scheme for a large range of experimental parameters, thus confirming the theoretical results. A summary of the experimental observations for ordered-shell structures with up to four shells plus string and without resolution of individual ions is presented in Fig. 13(b). The depth of the confining potential well is again one of the experimental parameters. As the ion density cannot be determined directly from the images, the radius p of the structures is used instead as the second parameter. The theoretically predicted boundaries between the different shell structures are again given as straight lines of constant 1 following Hasse and Schiffer (1990).The observed ion configurations are seen once more to be determined by 1for a wide range of potential depths and ion densities. Our results have important implications for two very different fields. Consider first the physics of low-energy particles: an ion string in a quadrupole ring, being free of micromotion, can be cooled to its vibrational ground state in the confining potential using recently proposed laser cooling techniques (Dalibard et al., 1983). This would place the string in the LambDicke regime with a vanishing first-order Doppler effect because the spatial amplitude of the motion is smaller than the wavelength of the atomic transition. Furthermore, the second-order Doppler effect, which can be reduced only by further cooling, also disappears making the stored ions very interesting for frequency standards. The large number of ultracold ions available in the ring will lead to a high signal-to-noise ratio. Finally, cooled
+,,
-
-' E
0.2
-
0.1
-
4
a
Y
3 h
0.05-
.L(
10
5
f l 0.03.
-
0.02.
L-(
0
0
0
P.
0.2 0.3
0.5
I
I
2
1
5
3
Potential Depth
10 $o
20 30
[eV]
(a)
I
I
1
2
3
Potential Depth
5
7
*, [eV]
1
(b) FIG.13. Summary of the experimental results. (a) Individual ions resolved, where the observed structures are characterized by the ion density per unit length and the depth of the potential well $o. These two parameters can be combined to give the normalized linear particle density I that fully determines the ion configuration. The straight lines show critical 1values separating the regions corresponding to the various theoretically expected structures. The observed configurations are labelled with different symbols for each structure. (b) Individual ions unresolved, where the observed shell structures with up to four shells plus string are characterized by their radius p and the potential depth +o. The various observed structures are again separated by lines of theoretically determined critical 1(for details see Birkl et al., 1992).
403
404
H . Walther
ions in the ring represent a quantum object of macroscopic dimensions: a Wigner crystal. Second, with the experimental confirmation that the ordered structures expected in high-energy ion storage rings can indeed be formed, a completely new era of accelerator physics will emerge, if it is possible to reproduce such Coulomb crystals in these rings. The enhanced luminosity of the corresponding beams would allow studies of ionic reactions whose cross sections are too small for investigations in existing accelerators.
REFERENCES Birkl, G., Kassner, S., and Walther, H. (1992). Nature 357, 310. Bliimel, R., Chen, J. M., Quint, W., Schleich, W., Shen, Y. R., and Walther, H. (1988a). Nature 334,309. Bliimel, R., Chen, J. M., Diedrich, F., Peik, E., Quint, W., Schleich, W., Shen, Y. R., and Walther, H. (1988b).Atomic Physics, 11 ( S . Haroche, J. C. Gay and G. Grynberg, eds.), World Scientific Publishing, Singapore. Bliimel, R., Kappler, C., Quint, W., and Walther, H. (1989). Phys. Rev. A 40,808. Bohm, D. (1951). Quantum Theory, Prentice-Hall, Englewood Cliffs, N.J. Brune, M., Haroche, S., Lefevre, V., Raimond, J. M., and Zagury, N. (1990). Phys. Rev. Lett. 65, 976. Brune, M., Raimond, J. M., Gay, P., Davidovich, L., and Haroche, S. (1987). Phys. Rev. Lett. 59, 1899. Carmichael, H. J., and Walls, D. F. (1976). J . Phys. B 9, L43, 1199. Chu, S., Bjorkholm, J. E., Ashkin, A., and Cable, A. (1986).Atomic Physics, I0 (H. Narumi and S. Shimamura, eds.), North-Holland, Amsterdam. Church, D. A. (1969). J. Appl. Phys. 40, 3127. Cresser, J. D., Hager, J. Leuchs, G., Rateike, M., and Walther, H. (1982). Dissipative Systems in Quantum Optics (R. Bonifacio, ed.), Topics in Current Physics 21, Springer, Berlin, p. 21. Dalibard, J., Salomon, C., Aspect, A., Arimondo, E., Kaiser, R., Vansteenkiste, N., and CohenTannoudji, C. (1988).Atomic Physics, 11 ( S . Haroche, J. C. Gay and G. Grynberg, eds), World Scientific Publishing, Singapore). Dehmelt, H. G. (1967).Adu. Atom. Molec. Phys., 3 (D. R. Bates and I. Estermann, eds.), Academic Press, New York, pp. 53-72. Diedrich, F., and Walther, H. (1987). Phys. Reo. Lett. 58, 203. Diedrich, F., Peik, E., Chen, J. M., Quint, W., and Walther, H. (1987). Phys. Rev. Lett. 59, 2931. Drees, J., and Paul, W. (1964). Z. Phys. 180, 340. Eberly, J. H., Narozhny, N. B., and Sanchez-Mondragon, J. J. (1980). Phys. Rev. Lett. 44,1323. Englert, B.-G., Schwinger, J., and Scully, M. 0.(1988). Found. Phys. 18, 1045. Englert, B-G., Walther, H., and Scully, M. 0. (1992). Appt. Phys. B 54, 366. Filipowin, P., Javanainen, J., and Meystre, P. (1986a). Opt. Comm. 58, 327. Filipowicz, P., Javanainen, J., and Meystre, P. (1986b). Phys. Rev. A 34, 3077. Filipowicz, P., Javanainen, J., and Meystre, P. (1986~).J. Opt. Soc. Am. 83, 906. Fischer, E. (1959). Z . Phys. 156, 1. Gallas, J. A., Leuchs, G., Walther, H., and Figger, H. (1985). Advances in Atomic and Molecular Physics, 20, Academic Press, New York, p. 413. Gilbert, S. L., Bollinger, J. J., Wineland, D. J. (1988). Phvs. Rev. Lett. 60,2022.
ATOMS IN CAVITIES AND TRAPS
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Haroche, S., and Raimond, J. M. (1985).Advances in Atomic and Molecular Physics, 20, Academic Press, New York, p. 350. Hasse, R. W., and Schiffer, J. P. (1990). Ann. Phys. 203, 419. Jaynes, E. T., and Cummings, F. W. (1963).Proc. I E E E 51, 89. Jammer, M. (1974). The Philosophy of Quantum Mechanics, Wiley, New York. Kimble, H. J., Dagenais, M., and Mandel, L. (1977). Phys. Rev. Lett. 39, 691. Kimble, H. J., Dagenais, M., and Mandel, L. (1978). Phys. Rev. A 18, 201. Knight, P. L., and Radmore, D. M. (1982). Phys. Rev. Lett. !MA, 342. Krause, J., Scully, M. O., and Walther, H. (1986). Phys. Rev. A 34, 2032. Krause, J., Scully, M. O., and Walther, H. (1987). Phys. Rev. A 36, 4547. Krause, J., Scully, M. O., Walther, T., and Walther, H. (1989). Phys. Rev. A 39, 1915. Loudon, R., and Knight, P. L. (1987). J. Mod. Opt. 34, 707. Lugiato, L., Scully, M. O., and Walther, H. (1987). Phys. Rev. A 36, 740. Mandel, L. (1979). Opt. Lett. 4, 205. Meschede, D., Walther, H.,and Muller, G. (1985). Phys. Rev. Lett. 54, 551. Meystre, P. (1987). Opt. Lett. 12, 669. Meystre, P., Rempe, G., and Walther, H. (1988). Opt. Lett. 13, 1078. Neuhauser, W., Hohenstatt, M., Toschek, P., and Dehmelt, H. (1978). Phys. Rev. Lett. 41, 233236. Neuhauser, W., Hohenstatt, M., Toschek, P.,and Dehmelt, H. (1980). Phys. Rev. A 22, 1137. Paul, W., Osberghaus, O., and Fischer, E. (1958). Ein Ionenkujig, 415 Forschungsberichte des Wirtschafts- und Verkehrsministeriums Nordrhein-Westfalen, Diisseldorf. Pritchard, D. E., Helmerson, K., and Martin, A. G. (1988). Atomic Physics, 11 ( S . Haroche, J. C. Gay and G . Grynberg, eds.), World Scientific Publishing, Singapore. Rahman, A., and Schiffer, J. P. (1986). Phys. Rev. Lett. 57, 1133. Raizen, M. G., Gilligan, J. M., Berquist, J. C., Itano, W. M., Wineland, D. J. (1992).J . M o d . Opt. 39, 233. Rempe, G., Schmidt-Kaler, F., and Walther, H. (1990). Phys. Rev. Lett. 64, 2783. Rempe, G., and Walther, H. (1990). Phys. Rev. A 42, 1650. Rempe, G., Walther, H., and Klein, N. (1987). Phys. Rev. Lett. 58, 353. Scully, M. O., and Walther, H. (1989). Phys. Rev. A 39, 5229. Scully, M. O., Englert, B.-G., and Walther, H. (1991). Nature 351, 11 1. Short, R., and Mandel, L. (1983). Phys. Rev. Lett. 51, 384. Slosser, J. J., Meystre, P., and Wright, E. M. (1990). Opt. Lett. 15, 233. Slusher, R. E., Hollberg, L. W., Yurke, B., Mertz, J. C., and Valley, J. F. (1985).Phys. Rev. Lett. 55, 2409. Waki, I., Kassner, S., Birkl, G., and Walther, H. (1992). Phys. Rev. Lett. 68, 2007. Walls, D. F. (1979). Nature 280, 451. Walls, D. F. (1983). Nature 306,141. Walls, D. F. (1986). Nature 324, 210. Walther, H. (1991). Proc. of the Workshop on Light Induced Kinetic Effects o n A t o m , Ions and Molecules (L. Moi, S . Gozini, C. Gabbanini, E. Arimondo and F. Strumia, eds.), ETS Editrice, Pisa. Wheeler, J. A., and Zurek, W. H. (1983). Quantum Theory and Measurement, Princeton University Press, Princeton, N.J. Wineland, D. J., Itano, W. M., Bergquist, J. C., Bollinger, J. J., and Prestage, J. D. (1984). Atomic Physics, 9 (R. S . van Dyck, Jr. and E. N. Forston, eds.), World Scientific Publishing, Singapore. Wootters, W., and Zurek, W. (1979). Phys. Rev. D 19, 473. Yoo, H. I., and Eberly, J. H. (1985). Phys. Rep. 118, 239.
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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS,VOL. 32
SOME RECENT ADVANCES IN ELECTRON-IMPACT EXCITATION OF n = 3 STATES OF ATOMIC HYDROGEN AND HELIUM J . F. WILLIAMS and J . B. WANG Centre for Atomic, Molecular and Surface Physics Department of Physics University of Western Australia Perth. Australia
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. The Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . .
Ill. Coincidence Measurements . . . . . . . . . . . IV. Radiation Trapping . . . . . . . . . . . . . . Acknowledgments. . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
. . . .
. . . .
. . . .
407 41 1 415 418 424 42 5
I. Introduction This chapter concerns measurements of the wave function Y of an excited atom, or more correctly the squared modulus of the wave function IYI’. Generally the atomic wave function may be described by a set of eigenfunctions $, with amplitudes f,, and the measurement will average over some quantity determined by the apparatus so that (I’YI’)
=
1 (f,*fm*>$z$m,
mm‘
=
1 pmm,$z$m,
mm‘
(1)
which defines the density matrix p,,,,,,.. When m and m’ are the magnetic quantum numbers, combinations of the pmmrgive rise to simple rotational properties upon coordinate frame transformation. These combinations are defined as state multipoles qq
Gq=
1 ppmm.(-l)k-j-m’(j’m‘j- mlkq)
(2)
mm’
where (j’m’j - mlkq) are Clebsch-Gordon coefficients and they form the basis for formulating the theory of angular correlations of particles emitted 407 Copyright Q 1994 by Academic Press, Inc. All nghts of reproduction in any form reserved. ISBN 0-12-003832-3
408
J . F. Williams and J . B. Wang
from a collision complex. The state multipoles describe the angular distribution of the electron charge cloud of the excited state; i.e., the height, width, length and the angle of the axis of the charge cloud with respect to the quantisation axis (the alignment), and the internal motion (orientation) of the charge cloud. This description of the excited state is similar to that obtained using expectation values of components of angular momentum but it provides a more transparent picture for contemporary measurements of an excited atom. The concept of a coincidence experiment, in which two or more of the scattered or emitted particles from a collision complex are detected in time coincidence, to determine the elements of a density matrix or the state multipoles of an excited atomic state was discussed explicitly by Fano (1957). Using density matrix theory and Racah algebra, the polarisation or directional correlation of light emitted from a collision fragment was related to the multipole moments of the fragment excited state. However, the experimental implementation had to await the technology and the development of the relevant techniques in atomic physics. The theory of perturbed angular correlations of nuclei (Fano and Racah, 1959) contained the necessary theoretical methods. The work of Fano and Macek (1973) provided a unified framework for disentangling dynamical from geometric information in particle-impact atomic excitation processes, such as described by Macek and Jaecks (1971). The way in which these ideas have developed has been reviewed, for example, by Andersen et al. (1988) and many of the earlier reviews quoted therein, is particularly instructive. The topical review by Hippler (1992) gives an excellent overview of basic ideas with particular application to fewelectron atom-atom collisions. Subsequent discussion here concerns electron impact excitation. Scattering theory and models are not discussed here; for recent reviews, see Madison (1990), Scholz (1992), Stelbovics (1992) and Walters et al. (1992), for example. The large body of knowledge generally concerns excited P states of atoms for which both theoretical and experimental considerations are relatively not too difficult. In general the methods for the theory of the measurements are well known, and they have been implemented to keep pace with experimental techniques. However, the implementation was laborious, particularly prior to the introduction of computer algebra software such as Maple or Mathematica. For D state excitation the difficulty of the experiments increases considerably. Recent reviews by Chwirot and Slevin (1991) and Stelbovics (1992) address progress in studying some aspects of the excitation of n = 3 P and D states of atomic Hydrogen, which continues to be an interesting atom for the development of both the theory and measurement. The work of Mahan et al. (1976), using a noncoincidence but time-modulated beam method, stands out from earlier
ELECTRON-IMPACT EXCITATION OF ATOMIC HYDROGEN AND HELIUM
409
work because it determined the differential cross sections for the separated 3S, 3P and 3D states over a wide energy range and the coincidence methods have not yet yielded the same information. The density matrix formalism has been applied to the scattered electron, radiated photon, angular and polarisation correlation measurements for the n = 3 states of atomic Hydrogen by Blum et al. (1978), and for the D states of Helium by Nienhuis (1980) and Van den Heuvell et al. (1981, 1983), for Sodium by Herman and Hertel (1982) and for Lithium by Andersen et al. (1983). Further developments for Hydrogen were given by Heck and Gauntlett (1986) in the authors’ laboratory and also by Chwirot and Slevin (1985,1987) to interpret measurements for the 32Pj and 3*Dj states. Figure 1 shows the energy level diagram, state lifetimes and decay schemes that
j=1/2
j=1/2.3/2
j=3/2.5/1.
AlD
+Pl/23/2
32s/2S/z
e= O J w O (158) nsec
e =O.m=O
e=1
(5.3)
e =2 (15.5)
+2 +1
m= 0 -I -2
F64.5 M H Z
-4 -1 1 cm = 3.10 M H z = 1.24 x 10 eV y = decay constant
FIG.1. The energy level diagram for atomic hydrogen showing the n lifetimes and fine-structure and decay schemes.
=
1,2 and 3 states, their
410
J . F . Williams and J . B. Wang
influence the measurements. The fine and hyperfine interaction effects are usually time averaged over nanoseconds in present coincidence measurements and appear as a partial depolarisation of detected particles. Such a system is well suited to the mixed quantum state description offered by the density matrix formalism. The same information can be obtained about the state multipole moments for the D state using either the Balmer alpha or Lyman alpha radiation in coincidence with the scattered n = 3 energy loss electron. From this basis Chwirot and Slevin (1985, 1987) and Farrell et al. (1990) made a series of measurements of Lyman alpha radiation in coincidence with the scattered electron at 20" and 25" electron scattering angles for 54.4 and 100eV to characterise the 3P and 3D states excitation process. For the 32Pjstates the ii and R values were determined from angular correlations. For the 32Djstates angular (in-plane) and polarisation (normal-to-the-plane) correlations permitted some normalised rank 2 state multipoles to be deduced. The measurements of the circular polarisation parameter P, (normal to the scattering plane) have uncertainties so large as to limit their usefulness; for 0.22 whereas the limiting example, P, at lOOeV and 25" equals -0.02 values are f0.14 because of the depolarisation of the cascade radiation due to the fine structure interaction. In addition to these large uncertainties arising only from the statistics of the coincidence signals, there are presumably unknown systematic uncertainties. Generally all the data did not agree with first Born approximation values or with distorted wave Born approximation values (Katiya and Srivastava, 1987).The values of iiand R for the 3P state did not have close agreement with those for the 2P state at the same angles and energies. This result is contrary to the expectation derived from the observation for Helium (Csanak and Cartwright, 1988)that similar values of angular correlation parameters have been measured for 2lP and 3'P state excitation. These measurements lead to the following comments. The time-coincident detection of the scattered energy loss electron and the radiated photon selects ideally a given ensemble of atoms specifying a unique quantum state of the excited atom, that is, with specific amplitudes and phases. The experimental progress towards the measurement of these quantities is still limited by technology in some way. In principle there are two essential components of an apparatus, namely, sources and detectors of incident and scattered (radiated) particles, which are essential to specify the (pure or mixed) quantum states involved. For several decades, sources and detectors of spin-polarised electrons and hydrogen atoms as well as detectors of radiation from transitions into the n = 2 and 1 states have existed. However, the absolute efficiencies of these devices still limit the type and accuracy of measurement. Progress has also been limited of course by the available funding and personnel for the significant tasks of apparatus
ELECTRON-IMPACT EXCITATION OF ATOMIC HYDROGEN AND HELIUM
41 1
construction and development of the theory of the measurement. Experimentally the basic difficulty is demonstrating that the measured quantity is indeed that specified mathematically. When work in our laboratory started on the problems of n = 3 excitation, about 1985, a new approach to the determination of rank 3 and 4 multipoles was attempted. Some recent results of that work are addressed here, including the introductory theory, sequential cascading photon correlations and multipole-rank-dependent radiation trapping. A series of papers on these topics is in press.
11. The Basic Theory Some aspects of the description of the coherently excited states pertinent to the ideas developed here are considered. The intensity of the emitted photons from the excited-state quantum ensemble is derived in terms of the density matrix p and the corresponding multipole moments. Unpolarized target and incident electrons are used, and an average over the total spins S = 0 and 1 is made. The density matrix before the collision is written in the usual notation (Blum, 1981) as pi =
C
llOOkSMs)
SMS
~
2s+ 1
(lOOkSMsI
(3)
After the collision, at t = 0, the density matrix has the form p(0) = TpiTt, where T is the T-matrix operator describing the excitation process. The principle quantum numbers are suppressed, and a summation over all quantum numbers for the coherently excited sublevels has been made. The matrix elements of the T-matrix operator are the scattering amplitudes fi, written as
where
The excited states relax into total angular momentum J states since the lifetime of the excited states is considerably longer than the LS precession time of about 10- s. The time development is given by P(t) =
w)P(o)ut(t)
(6)
412
J . F . Williams and J . B. Wang
where U ( t ) l n J M ) = exp[-t(io,,
- ynJ/2]lnJM)
and E,j
= hOnJ
The density matrix describing the emission of a dipole photon at time t given by
(7) = to is
If we take the trace over the intermediate state angular momentum, the resulting matrix element
gives a reduced density matrix over the helicity states (A = _+ 1) of the photon. The preceding density matrix forms can be applied recursively to describe any electron impact excitation that leaves the atom in an excited state that then decays in a cascade of dipole photons (Stelbovics, 1993). By taking combinations of the elements, all the Stokes parameters describing the polarisation states of the emitted photon can be determined. The density matrix can be expanded in tensor operators such that the state multipole expansion of the coherent states ( J M ) is defined by p =
1
IJ’M’)(J’M’lplJM)(JMI =
JMJ‘M’
1
(T(J‘J)HQ)T(J’J)KQ(10)
J’J‘KQ
where the coefficients ( T ( J ’ J ) i Q )=
+
(- 1)J’-M’(2K l)’.’ M’M
(:I
-M
K, Q (J’M‘JpJJM)
(1 1)
are the state multipoles of rank K and component Q, whose determination gives a complete description of the coherent scattering. Their definition limits the rank of the state multipoles to 0 < K G J’ + J , where J and J’ refer to the angular momentum of the coherently excited states. Of importance to the way in which the measurements have evolved is that the state multipoles also display any symmetry in the scattering event in a transparent manner. The state multipoles required to describe the excited n = 3 states are, for the 3s state,
ELECTRON-IMPACT EXCITATION OF ATOMIC HYDROGEN AND HELIUM
413
Here aim= (Ifim12) and a1= Zmolm. The state multipoles describing the 3S3P-3D coherence are not given since their measurement requires an external field, and while this has not yet been attempted for an (e y1 y z ) triple coincidence experiment, the method has been indicated by Heck and Williams (1987) for the 2 s and 2 P state mixing. The triple coincidence measurement (e y1 y2) has access to all ranks of state multipoles and thus provides a complete description of the excited 3D state. It is noted that for ion-atom collisions in the presence of an electric field, Havener et al. (1986) determined the density matrix for n = 3 state excitation. However, for electron impact, the double coincidence measurements (e y l ) and (e y2) can determine only up to rank K = 2 state multipoles because of detection of a single-dipole photon while a measurement of the two photons (yl y2) in
J . F. Williams and J . B. W a n g
414
coincidence sheds some light on the higher rank multipoles averaged over the electron scattering angles. Heck and Gauntlett (1986) derived the general formulae for the density matrix corresponding to the triple (e y1 yz) coincidence measurement where the sequential cascading Balmer-alpha photon and Lyman-alpha photon are detected in coincidence with the scattered n = 3 energy loss electron. A schematic representation of the experiment and reference frame is shown in Fig. 2. They showed that the density matrix describing the rate of emission of the photons from the n = 3 to the n = 2 to the n = 1 levels has the form
)(n2, t ; to, AlnlyAZA2 = C(WJ
1
(-1~L;+L2-S3-J2+k+L1+L2
L, J2L;J"LikQq
e 1 ec tron detector
T i m e Coincidence
A
A '
photon detector 2 photon detector I
FIG.2. A schematic representation of a triple (e y , y 2 ) coincidence experiment and reference frame. The incident electron beam defines the z axis, the scattered energy-loss electron defines the scattering xy plane and the y axis is chosen to complete the normal righthanded Cartesian coordinate system. For normal (e y) coincidence measurements, the photon direction is defined by the spherical polar coordinates B and 4. For sequential cascading photons the Balmer-alpha photon direction is and a and for the Lyman-alpha 0 and 4. The electron detector includes a 180" energy analyser with the normal input optics and the photon detectors includes wavelength filters.
ELECTRON-IMPACT EXCITATION OF ATOMIC HYDROGEN AND HELIUM
41 5
Because the count rates for a triple (e y1 y 2 ) coincidence measurement are extremely low, it will be some time before this type of measurement can be expected to yield useful information about the higher rank multipoles. Further discussion concerns measurements that have not observed one of the three outgoing particles.
111. Coincidence Measurements In the ( e , y 2 ) measurements the first photon is not detected and the density matrix elements in Eq. (15) must be summed over both helicities I , = f 1 and integrated over all directions nl. This approach was used by Chwirot and Slevin and is the topic of further comment in papers in publication by Stelbovics, Kumar and Williams and are not discussed further here. However, the other possible detection scheme of not detecting the scattered electron has recently been the subject of measurement by Williams et al. (1992b). The detection of sequential cascading photon-photon, angular or polarisation correlations has not been used previously to determine state multipoles in atomic scattering studies. However, they have been observed previously in a wide variety of measurements to determine lifetimes, g-values and branching ratios (Imhof and Read, 1977), absolute quantum efficiencies of photon detectors (Christofori et al., 1963), the angular distribution of the two-photon decay of the 2s state of He+ (Lipeles et al., 1965) and of atomic Hydrogen (O'Connell et al., 1975), directional correlations of cascading photons (Fry, 1973), polarisation correlations (Freedman and Clauser, 1972), the perturbation of a polarisation correlation due to an external magnetic field (Dumont et al., 1970) and polarisation correlations using time-varying polarisers (Aspect et al., 1982)in experimental test of hidden-variable theories of quantum mechanics. For (y,, y z ) measurements, Eq. (15) is integrated over all scattered electron (undetected) directions, and summed over the helicities of both photons since polarisations were not measured. The schematic representation of the experiment shown in Fig. 2 was modified for this work as follows. The incident beam direction provides a quantisation axis and the photon detectors for the n = 3 -,2 (656.2 nm) and n = 2 -+ 1 (121.6 nm) photons were located in the scattering plane defined by the incident electron beam and the
J. F. Williams and J. B. Wang
416
first of the detected cascading photons. It was shown (Williams et al., 1992a) that the radiated photon-photon intensity is given by (T(OO);O)P
+ iP,(CO@
- PI)]
r
1
where
A is a constant whose detail depends on the photon-detection solid angles and energies; P and 6 are the angles of the first and second photons with respect to the Z axis. The A function of angles can be reduced to a sum of associated Legendre functions, which do not admit a simple analytic reduction. For the coplanar experiment in which the atomic source possesses axial symmetry, all the state multipoles vanish except those with Q = 0. The additional attraction for such measurement is indicated by rewriting the relevant multipoles in the more explicit notation:
(WxO)
= c3s
(T(22);O)
= (1/$)[2c3d2
(T(22):O)
= ($/$)C2d3d2
( T(22)&)= ($/@)cc3d,
+ 263d1 + d3dol
117)
- ‘3dl - O3dol -4c3dl
+ 3a3dol
where 63d, is the cross section for exciting the 3d magnetic sublevel m. There is no other way at present of determining these integrated cross sections for electron-impact excitation. The measurement of the sequential cascading photons and their angular correlations are difficult because of the near degeneracy of the principal 3’P, and 3’D, decay radiation to the quantum number levels. The 32S1,2, n = 2 levels cannot be separated by convenient optical filters, but their
ELECTRON-IMPACT EXCITATION OF ATOMIC HYDROGEN AND HELIUM
417
lifetimes [t(3'S) = 158, t(3'P) = 5.3 and t(3'D) = 15.5 ns] are sufficiently different and longer than instrumental timing resolutions of the order of 1 ns that they can in principle be separated under appropriate conditions. The apparatus (Williams, 1981) was a typical crossed electron and atom beams geometry. The electron beam current was of the order of several uA with an energy spread of about 0.75eV. The 656.2nm photons were sampled in a solid angle of 5 x lo-' sr (i.e., an in-plane angle of 30")and detected by a photomultiplier type EM1 9883. Similarly, the 121.6nm photons were sampled in a solid angle of 5 x lO-'sr (i.e., an in-plane angle of 20") and detected by a channel electron multiplier type Philips B418 BL. The wide angular range of the measurements was essentially 270" subject to the angular sizes of the incident electron source and photon detectors. Thirteen data points collected for two positions of the first photon detector, i.e., angles fl = 40" and 90" and then several values of 8,' = (p-0). The data are normalised to the B = 40" and = 180"datum point and are shown in Fig. 3. It is seen that the angular correlations for both fl = 40" and 90" clearly follow the P,(cosO,,) relationship. For a fixed fl the earlier equation is of the form I@, 8) = a + b cos0,,. The effect of a nonzero value of fl is to reduce, through P2(cosfl),the amplitude of the sinusoid. The accuracy of determining the state multipoles is best when the first photon detector is placed as close as possible to either fl = 0 or n. A least squares regression of the data to the above equation determined the
I
0
.
1
,
.
I
,
90 180 270 Angle between photon detectors (deg)
,
360
FIG 3 The photon (656.2nm)-photon (121.5nm) coincidence count rate IS shown as a function of the angle between the photon detectors for /I= 40" (open circles) and 90" (closed circles) The lines are a fit of Eq (16) in the form (a + b cos0) to the data.
J. F. Williams and J. B. W a n y
418
four state multipoles, normalised by setting T(OO),+,equal to the Born value at 290eV to give the values of (T(OO),,)
= 0.128[ - 1](26),
(T(22)Zo)
= 0.32[ - 2](6),
{ T(22)oo) = 0.37[ - 2]( 15),
(T(22)40)
=
-
0.05[ - 2](29)
and a,,=0.128[-1](26), 63d,
= 0.10[ - 2](20)
ajdo =
and
-0.05[-2](20), 63d2
= 0.33[ - 2](5) a.U.,
where the numbers in brackets indicate powers of ten and parentheses contain the statistical uncertainties of plus or minus one standard deviation. The first Born values for these quantities, in the same order, for the (T(LL'),,) are 0.128[ - 1],0.430[ - 2],0.160[ -2],0.428[ - 31 and for the a3,,, are 0.128[- 11, 0.138[-21, 0.129[-21, 0.283[-21, respectively. It is clear that, while the experimental feasibility of the measurement is good, an improvement in the accuracy of measurement is required before accurate quantities can be deduced. The study has determined for the first time a rank 4 multipole and the magnetic sublevel cross sections for a D-state of Hydrogen. The measurements show the feasibility of extending the method of sequential (or nonsequential) cascading photon-photon angular correlations to higher levels. Second, the way is open for the use of an external field to mix even and odd parity states as in the work of Heck and Williams (1987) for the 2 s and 2P states. For photon-photon coincidences there is the considerable simplification that the angle of scattering of the energy loss electron does not have to be determined in the presence of the field. The interferences of the even and odd parity states can then be explored for n 2 3 states. The next step towards a determination of the n = 3 density matrix is to measure triple coincidences between the two cascading photons and the scattered energy loss electron, which will establish a lower symmetry and determine, with polarisation analysis, the odd-rank tensors and the nonspherical components.
IV. Radiation Trapping The preceding theoretical description of electron impact excitation and the subsequent radiative decay concerns single collisions. In the usual experimental conditions single collisions would be attained as the target atom density approaches zero, but then so does the scattered electron and radiated photon signals, and the statistical accuracy or the time required to ac-
ELECTRON-IMPACT EXCITATION OF ATOMIC HYDROGEN AND HELIUM
419
cumulate data becomes unacceptable. In practice the target density is increased usually until reasonable signal rates are obtained and the singles rates are still linearly dependent upon the density. This means that there is always a finite probability of the absorption and reemission of radiation, particularly between the resonant excited and ground states, which prolongs the time taken for the detection of the radiation and so is called radiation trapping. This process can also decrease the polarisation of the detected radiation field and hence its angular intensity distribution. The basic problem is how to quantify the effects and to know when they have affected coincidence measurements. The phenomenon of radiation trapping is well known, see Mitchell and Zemansky ( 1 934), for example, but its effects on coincidence measurements have been not well explored, mainly because of the long measurement times. The effect on the Stokes parameters, for example, had been thought to be small because the photon intensities detected at angles of zero and 90" were affected by radiation trapping in a similar manner (Murray et al., 1991) or the effect was mainly on alignment rather than orientation (Eminyan et al., 1974). However, a comprehensive literature describes the effect of radiation trapping on the diagonal elements and lifetimes, see, for example, the theories of Holstein (1947) and Holt (1976). The development of optical pumping, magnetic depolarisation, double resonance, coherence narrowing and similar phenomena (see Kastler, 1957; Series, 1959; and Happer, 1972, for reviews) focused attention on individual elements of the density matrix of the excited state. A general theory was developed by Barratt (1959) and DYakonov and Perel (1965) to describe coherence narrowing in terms of the density matrix. The recent work of Williams et al. (1992b) showed how these earlier studies could be applied to angular and polarisation correlations coincidence measurements. It was shown (DYakonov and Perel, 1965) that the time evolution of the state multipoles can be simply described by ( T ( J ;t)ktq) = ( T ( J; t = O)kt,) exPC - Y k tl
(18)
where yk
= y(l -
and A, = 0.3(2J
{; ; ;}l
+ 1)[6 +(- l)k]
k ZO;A ,
=1
and J , J i are the angular momentum quantum number for the upper and lower states, respectively, and q is the probability of trapping of the radiation between source and detector. Here yk is independent of the multipole
420
J . F . Williams and J . B. Wang
component q, because the atomic vapour density distribution has been assumed to be uniform and the radiation trapping relaxation isotropic, but this may not always be realised in experimental conditions. The result that the state multipoles decay exponentially with different decay rates for different rank multipoles has the important consequence that radiation trapping effects for each rank multipole cause an increase in the lifetime and a decrease in the polarisation of the detected radiation. The effects will be observable when the experimental method permits separation of quantities containing the appropriate rank multipole, such as in coincidence measurements. The consequences of this theory was confirmed for 3 'P excitation in Helium but has not yet been explored for atomic Hydrogen. The preceding equation for Yk applies when there is only one radiative path. For the 3 'P decay Yk = yk(2 'S)+y(l -Ak$, however, yk(2 9)is much smaller than y( 1 'S) and the effect is negligible. For this excitation the totaI intensity I, the product of the intensity and the Stokes parameters, I P i , as well as the total right- and left-handed circularly polarised intensities I + and I - respectively, are related to the state multipoles, their relaxation rates and the photon detector position relative to the scattering plane described by the spherical angles (8, cp), as follows:
( T(L);,)
+ (T(L):,)2
sin28 cos2cp - ( T(L):,) sin28 coscp
sin8 sincp
(19)
ELECTRON-IMPACT EXCITATION OF ATOMIC HYDROGEN AND HELIUM
421
1
- ( T ( L ) l , ) sin28 cosq + - (T(L)&)(3 coszO- 1
$
The measured coincident intensity ](a, 8,s) is given in terms of the Stokes parameters Pi by
I(a, fi, 6) = +(I+ I P1[cos2(a -B) cos2B - sin2(a-
B) sin2B cosd]
+ I Pz[cos2(a -p) sin2P + sin2(a B) cos2B cosdJ + 1P3[sin2(a-B) sins]} -
(24)
where a is the angle of the transmission axis of the linear polariser, B the angle of the fast axis of the retarder relative to the incident electron beam direction and 6 is the retardance of the phase retarder. These expressions indicate that the measured intensities I and the Stokes parameters Pi will be pressure dependent, because the decay rates yo, y1 and y z differ from the natural decay rate y (i.e., without radiation trapping) by different factors related to the trapping probability r] and also that the observed lifetime depends not only on gas cell pressure, but also on the position and polarimetry of the photon detector. The consequences of these relations were determined in scattered electronradiated photon angular and polarisation correlation measurements for 3 P excitation at 81.2 eV with an electron scattering angle of 20". As the pressure increases, the most evident change in the radiation field is that the dumbbell pattern becomes broadened. This is implied by Eq. (19), in which the first term represents the spherically symmetrical component of the radiation and the second term the alignment. Because the decay rate yo for the monopole is smaller than the decay rate y z for the rank K = 2 multipole, the spherical component decays more slowly than the alignment and this results in a broader radiation pattern as shown in Fig. 4. At low pressure, the trapping probability is small and the ratio of yo and y 2 is close to unity. An increase of the target pressure increases the probability of trapping and produces an increasing difference between y o and y2, and thus the measured radiation field is broadened. The consequences for PI and P, are readily deduced, and the
'
422
J. F. Williams and J. B. Wang t = O
t=t
n +
+
FIG.4. The depolarisation of radiation field by radiation trapping, as obtained from the expression for total intensity, indicating schematically the effects of the different decay rates of the rank K = 0 and 2 multipoles upon the radiation field.
depolarisation is readily seen. Measurements of the Stokes parameters for 3 'P Helium at 81.2eV also confirmed the ratio P,/P, is independent of pressure, shown in Fig. 5, as predicted by the previous equations. Measurements were made of the intensities I(a,/?,6 ) normal to the scattering plane in the manner detailed by Wedding et al. (1991), as well as I ( & C#I = 0) in-plane angular correlations, to determine the Piat various gas jet driving pressures. Typical results are shown in Fig. 6 for a range of angles a, /? and 8. The variations in the observed lifetimes are revealed in the figure. A detailed analysis of the decay curves deduced the decay rates yo, y1 and y z of 0.42,0.50and 0.47 ns- at q = 0.26 (P = 0.525 Torr); and 0.1 1,0.34 and 0.25 at q = 0.81 (P = 16.5 Torr) and 0.04,0.33 and 0.22 at q = 0.94 (P = 32 Torr). The lifetime for zero pressure thus determined was 1.7 f0.1 ns, which is in agreement within experimental uncertainty with earlier determinations of Wedding et al. (1990) and theoretical predictions of Gabriel and Heddle (1960).This consequently increases confidence in the observations. These measurements have confirmed the following features that are evident from the previous equations. The total intensity I decays proportionately to combinations of y o and y2, and the ratio of those two components depends on the values of state multipoles for the excited state and on the position of the photon detector (0, cp). The angular momentum of the excited atom, which is
-0.2
a" P
B
-0.3
&-
t? -0.4 9
s&
-0.5
3
2
-0.6 -0.7 0
20 30 40 Pressure (Tor)
10
50
60
FIG.5. Pressure dependence of P, (open triangles) and Pz(solid triangles) for the 3 l P state of Helium. The incident electron energy was 81.2 eV, and the electron scattering angle was 20".
e=-ii o e=-i30 e=-45 400
500
450
550
a=45 a=60 a=l35 350
400
500
450
p=0 p=135 p45 400
420
440
480
480
500
Time (x0.51 ns) FIG.6. Measurements of the intensities 1(u, 8,s):(1) in-plane angular correlation without polarisation analysis (P= 16.5 Torr); (2) linear polarisation analysis normal to the scattering plane with varying angle a (P = 16.5 Torr); (3) circular polarisation analysis normal to the scattering plane with varying angle 8 with retardance angle d = -63" (P = 4.7 Torr).
423
424
J . F . Williams and J . B. Wang
determined by the intensity difference (I+ - I -) = I P3,decays as yl, while the circularly polarised components of the light I, and I - decay as a combination of yo, y1 and y z but differently from each other. The products IP, and IP, decay as y2 and Pl/Pz is pressure independent within experimental uncertainty. The time evolution of the excited state has not been considered previously in determining the state multipoles except in the presence of external fields or separable fine-structure effects. The total time-integrated intensity is usually the prime recorded quantity. The present work indicates the time evolution of the excited state enables the extent of radiation trapping, the separation of true and random signals and the measurement of the state multipoles to be quantified more accurately. The benefits of observing the time dependence are most apparent when subtracting I, and I - to determine the circular polarisation Stokes parameter P,, because the pure P3 signal must be contained within the rank 1 decay time. Clearly any signal at other times is not part of P,. This can be used as an important check on both the magnitude and the sign of P, measurements which are not always straightforward (Wedding et al., 1991, for example). Decay rate measurements provide an efficient way to obtain natural lifetimes and at the same time quantifies the effect of radiation trapping, which they do much more quickly and accurately than the dependence of the orientation and alignment parameters on pressure, as used in previous experiments. The method can be extended to higher rank multipoles of higher angular momentum states as well as to the components q of the multipoles l& for the appropriate geometry; however, there will be considerable difficulty in obtaining time coincidence data with sufficient statistical accuracy to separate the smaller higher rank multipoles. In conclusion it has been indicated that the method of photon-photon angular correlations can give information on the spherical components of the higher rank multipoles and the magnetic sublevel excitation cross sections. Also the detailed analysis of the time decay of the coincident signals quantifies the effects of radiation trapping in a way dependent upon the rank of the state multipoles observed in the measurements. The usefulness of the state multipoles description of the excited state is shown.
Acknowledgments The chapter has been written in acknowledgment of the influence of Professor Sir David Bates, as well as of the many staff and students at Queens University, Belfast, on the development of the first author’s appreciation of
ELECTRON-IMPACT EXCITATION OF ATOMIC HYDROGEN AND HELIUM
425
physics, particularly during his stay from 1970 to 1980. The authors acknowledge the contributions of their coauthors, A. T. Stelbovics, M. Kumar and A. G. Mikosza, in the referenced papers and the assistance of R. Hippler with helpful comments on the manuscript. The work was supported by the Australian Research Council and the University of Western Australia.
REFERENCES Andersen, N., Andersen, T., Dahler, J. S., Nielsen, S. E., Nienhuis, G., and Refsgaard, K. (1983). J. Phys. B: At. Mol. Opt. Phys. 16, 817. Andersen, N., Gallagher, J. W., and Hertel, I. (1988). Physics Reports 165, 1. Aspect, A,, Dalibard, J., and Roger, G. (1982a). Phys. Rev. Lett. 49, 180. Aspect, A., Grangier, P., and Roger, G. (1982b). Phys. Rev. Lett. 49 91. Barratt, 3. P. (1959). J. Phys. Radium 20, 541, 633, 657. Blum, K. (1981). Density Matrix Theory and Its Applications, Plenum Press, New York. Blum, K., Fitchard, E. E., and Kleinpoppen, H. (1978). Z. Physik A 287, 137. Christofori, F., Fenici, P., Frigerio, G., Molho, N., and Sona, P. G. (1963). Phys. Lett. 6, 171. Chwirot, S., and Slevin, J. (1985). J. Phys. B: At. Mol. Phys. 18, L881. Chwirot, S., and Slevin, J. (1987). J. Phys. B: At. Mol. Phys. 20, 3885. Chwirot, S., and Slevin, J. (1991). Comments At. Mol. Phys. 26, 11. Csanak, G., and Cartwright, D. C. (1988). Phys. Rev. A 38, 2740. Dumont, A. M., Camy-Val, C., Dreux, N., and Wry, N. (1970). Compt. Rend. B 271, 1021. DYakonov, M. I., and Perel, V. I. (1965). Soviet Physics JETP 20, 997. Eminyan, M., MacAdam, K. B., Slevin, J., Standage, M., and Kleinpoppen, H. (1974). J. Phys. B 7, 1519. Fano, U. (1957). Rev. Mod. Phys. 29, 74. Fano, U., and Macek, 3. (1973). Rev. Mod. Phys. 45, 553. Fano, U., and Racah, G. (1959). Irreducible Tensorial Sets, Academic Press, New York. Farrell, D., Chwirot, S., Srivastava, R., and Slevin, J. (1990). J. Phys. B: At. Mol. Phys. 23, 315. Frauenfelder, H., and Steffen, R. M. (1965). Alpha, Beta and Gamma R a y Spectroscopy (K. Siegbahn, ed.), North-Holland, Amsterdam. Freedman, S. J., and Clauser, J. F. (1972). Phys. Rev. Lett. 28, 938. Fry, E. S. (1973). Phys. Rev. A 8, 1219. Gabriel, A. H., and Heddle, D. W. 0.(1960). Proc. R o y . SOC.A 258, 124. Happer, W. (1972). Rev. Mod. Phys. 44, 170. Havener, C. C., Rouze, N., Westerveld, W. B., and Risley, J. S. (1986). Phys. Rev. A 33, 276. Heck, E. L., and Gauntlett, J. P. (1986). J. Phys. B: At. Mol. Opt. Phys. 19, 3633. Heck, E. L., and Williams, J. F. (1987). J . Phys. B: At. Mol. Phys. 20, 2871. Herman, H. W., and Hertel, I. V. (1982). Comments At. Mol. Phys. 12, 61, 127. Hippler, R. (1992). J . Phys. B: At. Mol. Opt. Phys., Topical Review, in press. Holstein, T. (1947). Phys. Rev. 72, 1212. Holt, H. K. (1976). Phys. Rev. A 13, 1442. Imhof, R. E., and Read, F. H. (1977). Rep. f r o g . Phys. 40, 1. Kastler, A. (1957). J . Opt. SOC.47, 460. Katiya, A. K., and Srivastava, R. (1987). J . Phys. B: At. Mol. Phys. 20, L821. Lipeles, N., Novick, R.,and Tolk, N. (1965). Phys. Rev. Lett. 15, 690. Macek, J., and Jaecks, D. H. (1971). Phys. Rev. A 4, 1288.
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Madison, D. H. (1990).In: Proc. Sixteenth Int. ConJ Elect. A t . Coll (A. Dalgarno, R. S. Freund, P. M. Koch, M. S. Lubell and T. B. Lucatorto, eds.), A I P , New York, p. 149. Mahan, A. H., Gallagher, A,, and Smith, S . J. (1976). Phys. Reo. A 13, 156. Mitchell, A. C. E., and Zemansky, M. W. (1934). Resonance Radiation and Excited Atoms, Cambridge University Press, Cambridge. Murray, A. J., MacGillivray, W. R.,and Standage, M. C. (1991). Phys. Rev. A 44,3162. Nienhuis, G. (1980). In: Coherence and Correlation in Atomic Collisions (H. Kleinpoppen and J. F. Williams, eds.), Plenum Press, New York. OConnell, D., Kollath, K. J., Duncan, A. J., and Kleinpoppen, H. (1975). J . Phys. B 8, L214. Scholz, T. T. (1992). In: Proc. Seventeenth Int. ConJ Elect. A t . Coll. (W. R . MacGillivray, I. E. McCarthy and M. C. Standage, eds.), Adam Hilger, New York, p. 181. Series, G. W. (1959). Rep. Prog. Phys. 22, 280. Stelbovics, A. T. (1993). In Press. Van den Heuvell, H. B. v. L., Nienhuis, G., van Eck, J., and Heideman, H. G. M. (1981).J. Phys. B: At. Mol. Opt. Phys. 14, 2667. Van den Heuvell, H. B. v. L., van Gasteren, E. M., van Eck, J., and Heideman, H. G. M. (1983). J. Phys. B: A t . Mol. Opt. Phys. 16, 1619. Walters, H. R. J., Scholz, T. T., Scott, M. P., and Burke, P. G. (1992). In: Correlations and Polarization in Electronic and Atomic Collisions and (e, 2e) Reactions (P. J. 0.Teubner and E. Weigold, eds.). Wedding, A. B., Mikosza, A. G., and Williams, J. F. (1990). J . Elect. Spect. Rel. Phen. 52, 689. Wedding, A. B., Mikosza, A. G., and Williams, J. F. (1991). J . Opt. Soc. Am. A 8, 1729. Williams, J. F. (1981). J. Phys. B 14, 1197. Williams, J. F., and Heck, E. L. (1988). J. Phys. B: At. Mol. Opt. Phys. 21, 1627. Williams, J. F., Kumar, M., and Stelbovics, A. T. (1992a). In Press. Williams, J. F., Mikosza, A. G., Wang, J. B., and Wedding, A. B. (1992b).Phys. Reo. Lett. 69,757.
Index
A Adiabatic approximation, 253-254 avoided crossings, 254, 266,270, 273 Bom-Oppenheimer states, 256 Eckart states, 258-261 nonadiabatic couplings, 261,263,265,268,271 Aligned atoms, 247 Alignment, 224,247-248,408,421422 Alignment angle, 243 Alignment tensor, 229 Alkali atoms, 245 Alkali metal atoms, 244 Alkaline-earth metal atoms, 244 Amplitude(s), 226, 241-242, 248 for elastic scattering, 242 AnguIar correlation, 226228. 243 formula, 226 function, 227 Angular correlation parameters A. and x. 227, 229-230.233 Angular distributions, 247-248 Auger and photoelectrons, 248 Auger electrons, 247 photoelectrons. 247 Angular part, 226 Anisotropic population, 229 magnetic substates. 229 Antibunching, 285, 385
427
Anticorrelation of atoms, 288, 388 Antihydrogen formation, 212, 2 15-2 17 Approximation methods close coupling, 208,210 complex coordinate rotation, 207-208, 214-215 coupled static, 210 distorted wave, 210 Kohn variational, 210. 215,218 R-matrix, 209,220 stabilization, 208, 215 Associative detachment, 332,334 Astrophysical plasmas, 73-74 Asymmetry effects, 247 of scattered electrons, 247 Asymptotic expansions, 104 quantum electrodynamic corrections, I04 relativistic corrections, 104 Atom-atom collisions, 245 between spin-polarised and unpolarised alkali atoms, 245 Atom-field interactions, see Rate equation approach Atomic collision physics, 223-224 Atomic collision process, 224, 248 Atoms, see also Relativistic electronic structure in cavities and traps, 379-404 dressed-atom picture, rate equations, 368-37 I dynamics, 383-384
Index
428 heavy, relativistic effects, 44 many-electron, relativistic electronic structure, 172- 175 N-level, rate equation approach, 356-362 open-shell, relativistic electronic structure, 175-179 two-level, rate equation approach, 348-356 Attachment coefficient influence of electron temperature on, 339 obtained using the FALP apparatus, 320 theoretical description, 3 12 Attractive polarisation forces, 235 Average level scheme, 177
B Basis sets, 179-183 BENA module, 177 Bethe logarithm, 107 Bootstrap method, 105 Born approximation.236 for positronium formation, 20.30 Born series methods, 4 7 4 9 Breit interaction. 172, 183 Breit-Pauli Hamiltonian, 44 C
Casimir-Polder effect, 107 Cavalleri technique, 315 Channel functions, 42 Chaotic regime, 396 Charge cloud, 243 distributions, 243 Charge exchange collisions, 245 Charge transfer processes, 247 Chemistry circumstellar carbon chain molecules, 194 CO formation, 188, 199,202 fullerenes, 195 H2 formation, 188, 199 polycyclic aromatic hydrocarbons (PAHs), 195 interstellar, 187 solid state circumstellar, 194 interstellar, 203 Circular atomic states, 247 Circular polarisation, 245 of D lines, 245 Circumstellar envelopes (CSEs), 191
carbon-rich, 193 oxygen-rich, 193 masers, 193 Close coupling approximation, 84 Cloud configuration, 393 Clouds interstellar, 5 8 ,6 2 4 5 ion, 393 Coherence, 227-228 complete, 228 data, 243 of heavy rare-gas, 243 degree of, 228 of orthogonal light vectors, 228 of photon radiation, 228 Coherence correlation factor, 228 Coherence parameters, 224,227, 234,243 Coherence properties, 227 of photon radiation, 227 Coherent excitation, 227 Coherent oscillators, 227 Coherent superpositions,241 of Coulomb direct amplitude, 241 and exchange interaction amplitude, 241 Coincidence, 225 double, 413 measurement, 408,410,414415,420 photon-photon, 41 1,414,416,418 triple, 413414,418 Collisional amplitudes, 248 Collisional-radiativerecombination, 61, 120. 133-134 Collision dynamics boundary conditions, 257 impact parameter, 264,271 JWKB methods, 272-274 quantum mechanical, 256-263 rearrangement collisions, 256-257 semiclassical, 264-269, 273-274 Collision process, 225 Collision strength, 43 Complementarity, 289,389-391 Complete, 224 Complete analysis. 228, 241, 245 for alkali-alkali atom collisions, 245 of electron-atomic hydrogen scattering, 241 ofelectron scattering, 241 of orbital angular momentum transfer, 228 of scattering amplitudes, 241 Complete atomic collision experiments, 248 Complete collision experiments, 236,248
INDEX Complete dynamical analyses, 244 of electron and heavy-panicle impact excitation, 244 Complete electron impact analysis, 225 Complete experiments, 244, 247 in heavy-particle atom collisions, 244 in photoionisation of atoms, 244 Complete photoionisation experiments. 247 Complete spin analysis, 245 Configurational state, 176 Continuum distorted wave (CDW) approximation for ECC, 286289,292 Continuum electrons, 247 Convoy electrons, 292 Correlation angular, 407,409-410,415,417,423 polarisation, 40'3410,415,423 sequential cascading photons, 41 I. 414 Correlation functions, 42 Cosmic ray, 63 Coulomb direct amplitudes. 225 Coulomb exchange amplitudes, 225 Coulomb Schrodinger equation, 97 Coupled channel equations, 21-23 in momentum space, 23 and two-centered basis, 22 Coupled-channel optical model, 47 Coupled integrodifferential equations, 4 2 4 4 Coup1ing between electronic and nuclear motion, 245 between inner and outer electronic shells, 244 Cross sections, 226227.248 differential, 227 partial, 226 total, 227 Cusp asymmetry for electron yield, 284-285 cusp asymmetry parameter p for ECC, 285286,289-292 high energy asymptotic behaviour of p, 292
D Damping mechanisms, rate equation approach extension, 362-363 Decay process, 227 Decay rate, 407.41 1 4 1 2 Deexcitation, 225 Degree of coherence, 224 Dielectronic recombination, 1 19, 124-127 radiative stabilization, 124-125 Density matrix, 407.41 1-412 N-level system, 358
Density matrix element, 229 describing excitation process, 229 Density matrix equations, exact reduction, 352-354 Dielectronic recombination, 7 1 Differential cross section, 41,236 Differential excitation cross section, 226 Dipole polarisibilities, 244 electric, 244 Dirac-Coulomb Hamiltonian, 173 Dirac Hamiltonian, 44 Dirac-Hartree calculation, 173-174 Dirac-Hartree-Fock method, 173-1 74 Dirac radial spinor components, 174 Dirac spectrum, basis sets, 181-183 Dirac's wave equation, 170 Direct amplitude, 241, 245 Direct excitation, 245 Direct scattering amplitude, 243 Dissociative recombination, 119, 127-133 characteristics, 128 configuration mixing, 131 H3+ interstellar space, 63-64 Jovian planets, 60-61 starburst galaxies, 65 indirect mechanism, 130 multichannel quanta1 defect theory, 131 Distorted wave approximation, 76,79,84 for positronium formation, 20 CDW, 33-35 distorted wave Born, 30-32.35 DWPO, 31.35 Distortion, 235-236 of electron waves, 236 of incident electron partial waves, 235 Doppler limit, 392 Double-slit experiment, 389 Dressed-state picture, 368 Dressed state populations, 370 Dust grains, in space ice mantles, 203 Nova ejecta, nucleation, 199 Red Giant winds, 192 surface catalysis, 203 Dynamical analysis, 243 of orbital angular momenta, 243 Dynamical pararneter(s), 237-238, 240 Dynamics atoms, 383-384 photon exchange, 383
429
430
Index E
ECC, see Electron capture to the continuum ECR, see Electron capture to Rydberg states Effective electron-atom interaction potential, 235 Eigenfunctions spin, 41 target, 41 Eikonal approximation for e+ scattering, 33-35 Eikonal-Born series, 48 Elastic electron-xenon scattering, 243 Electric dipole polarisibilities, 244 Electron angular distribution, 224 Electrowatom collisions, 225,245 Electron atom scattering, 39-53 coupled integrodifferential equations, 42-44 e--F e+ scattering, 50 e--H scattering. 47.50-51 at intermediate and high energies, 44-9 Born series methods, 47-49 optical potential methods, 4 6 4 7 at low energies, 40-44 scattering equations. 40-42 Electron attachment, 308-341 activation energy, 309,3 19,322.33 1 Br2- formation, 330 to c60, 339 to CC14, SF6 and some halomethanes, 319 to CH3BR and c@6. 323 direct, 309, 322 dissociative, 309 to acids and superacids, 331 to some haloethanes. 325 to the radicals CC13and CC12Br, 327 experimental techniques, 3 13-3 15 potential curves for, 3 10 thermal energies, 308 Electron capture detector, 314, 327 Electron capture ion-atom collisions, 120-123. 269-270 comparison with experiment, 269-27 1 multicharged ions, 270,273 radial coupling, 270 rotational coupling, 270 singly charged ions, 270-271 translation effects, 270 Electron capture to bound states, 281-283 Electron capture to Rydberg states, 287-288 Electron capture to the continuum, 279-293 B:/Z: values for ECC, 287-288 cusp asymmetry parameter p values for ECC, 289-292
rate of decay with energy, 282,288-289 Electron charge distribution, 224 Electron-Aectron correlation(s), 235, 244 Electron energy distribution functions, 336 Electron exchange, 48 forces, 241 Electron impact excitation, 227 of atoms, 227 Electron impact ionization highly charged ions, 74-75 isoelectronic series, 73.77 isonuclear series, 73, 79 Electron impact process, 224 Electron-ion recombination, 124-1 34 collisional-dissociative, 134 collisional-radiative model, 120, 133-134 dielectronic radiative, 124-127 direct radiative, 1 2 4 1 2 6 dissociative, 127-1 33 superdissociative, 132-133 temolecular, 123 unified treatment of, 125-126 Electron-ion scattering, 39-53 coupled integrodifferential equations, 4 2 4 at intermediate and high energies, 4 4 4 9 Born series methods, 47-49 optical potential methods, 46-47 at low energies, 4 0 4 4 scattering equations, 4 0 4 2 Electron partial waves, 235 Electron photon angular correlations, 229 Electron-photon coincidence(s), 227,243-244 experiments, 243 technique, 234,241,244 Electrons, 241 self-energy. contribution to hydrogenic energy, 171 spin polarised, 241 Electron scattering, 227 angle, 226 Electron spin effect, 225 Electron spin technique, 241 Electron swarm technique, 313,325 Energy, levels, 57-58 Energy-resolved electron beams compared with space charge neutralization, 74 experiments, 73-74.84 space charge neutralization method, 73 Envelope chemistry, Supernova 1987A, 6 5 4 6 Equilibrium constant, 59 Exchange amplitude, 241,245 Exchange effects, 99
INDEX Exchange excitation, 24 1 Exchange interaction, 245 Excitation, e'-He 1 1S-23S, 49-50 Excitation amplitudes, 226 and phase differences, 226 Excitation autoionization, 7&7 1,75-76,84 Excitation-deexcitation, 228, 230 Excitation energy, average, 48 Expectation values, 229 for orbital angular momenta, 229
F Fadeev-Watson expansion, and positronium formation, 32-33 Fano-Macek alignment tensors, 230,243 FaneMacek orientation vector, 230,243 Feshbach projection operators, 46 Fine linewidth, 362 Fine structure formula, 170 Fine-structure levels, 44 Finite size effect, 234 First-order many-body theory (FOMBT), 235 Flowing afterglow-Langmuir probe (FALP) plasma, elevated electron temperature in, 336 technique, 308,316318,335 Forward scattering, 240
G Galaxies, starburst, 65 Galilean invariance, 257, 265-266 Gaussian basis functions, 23 Geometric effects, 248 Glauber amplitude, 48 Glauber approximation(s), 236,239-241 GRASP, 183-184 GSI Darmstadt, 73
H H3+. 5 7 4 6 dissociative recombination interstellar space, 63-64 Jovian planets, 6 0 4 1 starburst galaxies, 65 extraterrestrial. 6 M 6 interstellar space, 62-65 Jovian planets, 6 0 4 2 Jupiter magnetosphere, 62 starburst galaxies, 65 Supernova 1987A. 65-66 infrared absorption spectrum, 59
43 1
infrared emission, 57 interstellar, 57 terrestrial, 58-59 Hamiltonian effective, two-electron atom, 173 nonrelativistic, 174 Hartree-Fock approximations, 88-89 Hartree's proof, 95, 102 H2D+, 63-65 Heavy particle-atomic collisions, 244, 247 Heavy rare-gas atoms, 244 Heisenberg's method, 99 Helicity, 412, 415 Helium, 225,243-244,422 Helium spectrum comparison with experiment, 110 singly excited states, 95 total energies, 107 variational calculations, 103 Hertzsprung-Russell diagram, 189 Hot stars, 298 UV observations, 299 Hydrogen, 244,245,409 Hydrogenic atoms, relativistic electronic structure. I7&172
I Ideal atomic collision experiments, 224 Impact polarisation, 240 theories, 240 Impulse approximation (1A) for electron capture to the continuum, 290 Inelastically scattered electron, 226 Infrared absorption spectrum H3'. 59 starburst galaxies, 65 Interference amplitudes, 245 Interference effects, 224,227 of partial waves, 224 Interference phenomena, 224,248,236 quantum mechanical, 248 in scattering processes, 224 Interstellar space, H3+. 6 2 4 5 Ion-atom collisions, 245 Ion-ion recombination, 135-144 mutual neutralization termolecular, 122-1 23, 1 3 6 1 4 3 tidal, 137, 143-144 Ionization energy, helium, I 1 3 Ions cooled, 392
432
Index
heavy, relativistic effects, 44 storage ring, 397-399 Ion traps, 391-392
J Jovian planets, 57 H3+, 60-62 ionospheres, 58 Jupiter, magnetosphere,62
K Kinetic balance, strict, 180 K-matrix, 43 Krypton photoionisationmethod, 314,326
L h and x data, 228 h and x parameters, 227,229-230.233 h and R parameters, 1s-2p excitation, 5 1 Lamb-Dicke regime, 402 Lamb shift, 107, 110, 113 Langmuir probe, 316,336 Laplace space, rate equations, 352 Laser-assisted superelastic electron scattering, 241,244 Laser cooling, 392 Left-right scattering asymmetries, 243 Light, nonclassical, source, 385-389 Linear algebraic equations method, 43 Linearity, 224 Linear particle density, 399 Lines, absorption, 59 Line strengths, 59 L-Spinors, 181-182 Lyman- radiation, 247
M Magnetosphere,Jupiter, 62 Maser one-atom, 380-383 two-photon, 382 Mass polarization, 107 Master equation, 350 squeezed vacuum, 364 Mathieu differential equation, 391 Mathieu regime, 396 Matrix variational method, 43 Maximum information, 224 on excitation process, 224
Metastable ions in beams, 73,75, 80 Micromotion, 398 Model of coherent impact excitation, 227 Model of completely coherent excitation, 228 Molecules, see Relativistic electronic structure Multichannel Landau-Zener model for electron capture, 158 calculations for Ar4+, A& + H, 160 calculations for 03+ + H, 161 calculations for S3+ + H, 166 Multiconfiguration Dirac-(Hartree)-Fock method, 176-178 Multiple ionization, 71-72, 80-84.
N NaJ3 radiation, 245 Nebulae, planetary formation, 195 molecules, 196 Negative ions, 298 H-, 297 Negative ions, formation, 308-341 Negative probabilities, 385 I/n expansions, 95, 101 Nonadiabatic transitions, 272-274, Nonclassical light, 385 Nonrelativistic formulas for electron capture to the continuum, 285-286 Number state, 385 Nuclear attraction, 241 Nuclear size correction, 107 Number state, 385
0 OBKl approximation for ECC, 281-282 OBK2 approximation for ECC, 283-292 Occupation probabilities, time-dependent, 354 One-atom maser, 380-383 One-electron atoms, 241,244 Opacity, 296 line blanketting, 300 Rosseland mean, 301 Oppheimer-Brinkman-Kramers (OBK) approximation, 279 Optical potential methods, 4&47 Orbital angular momentum transfer, 236 Ordered configuration, 393 Ordered many-ion situation, 393 Ore gap, 20 Orientation, 224,247,408
433
INDEX Orientation parameter, 247 Orientation vector, 229 Oriented atoms, 247
P Parameter, 227 h, 51,227,229-230.233.247 Partial electron waves, 236 Partial-wave expansion, 241 Particle-photon coincidence, 244 Partition function, 59 Pauli's relation, 180 Paul trap, 391-392 phase transitions, 393 Penning trap, 391 Perfect, 223 Perfect collision experiments, 224, 248 Perfect complete atomic collision experiments, 223 Perfect electron-atom collision experiments, 225 Perturbation expansion, 47 Phase difference, 228 of atomic excitation process, 228 of light radiation, 228 Phase integral methods, 272-274 Phases, 224 Phase transitions, 393, 399 Paul trap, 393 Photoionisation, 247 of aligned metastable rare-gas atoms, 247 of atomic magnesium, 247 atomic nitrogen excited states, 15 ground state, 10 atomic oxygen excited states, 7 ground state, 2 Photoionization of ions, 72,74, 87 . Photon density matrix, 228 Polarisation, 248 of fluorescence radiation, 248 Polarisation correlation, 231,233, 237, 243 Polarisation correlation parameters, 227 Polarised electrons, 242-243 Polarizability, 104 dipole, 104 nonadiabatic correction, 96, 104 quadruple, 104 Polarization potential, 98, 235 Positron annihilation, 219 Positron beams, 205
Positronium beams, 205,213 Positronium formation, 19,205-210,212,217 coupled model for, 21-23 and double scattering, 31 from e+-H, 19-20,23-25,32-34 from e+-He, 19-20,28-29.3435 from e+-Li, 19, 25-26 from e+-Na, 1 9 , 2 6 2 8 and impulse approximation, 33 perturbation models for, 29-35 Positronium hydride, 214 Positronium molecule, 214 Positronium negative ion, 214 Positronium scattering, 205-206 by atoms, 213-214 by H, 214 by He, 214 by Ps, 214 by charged particles, 215 Positron scattering, 205-206 by alkali atoms, 209 by H, 207 by Li, 210 by H, 23-25,32,34 by He, 28-29,3435 by Li, 25-26 by molecules, 2 17-220 by H<->2<->, 2 17-220 by Na, 26-28 resonances in, 23 Potential energy surfaces, 57 Probability amplitudes, 236-237.239 Projectile beams, 245 spin polarised, 245 Pseudostates, 45, 5 I in coupled channel models, 24, 28 IP state excitation, 225 Pulsed radiolysis technique, 315, 339 microwave heating, 327 Pure exchange excitation, 236
Q Quantum defect, 93, 113-1 14 fits to norrelativistic energies, 105-106 fits to total energies, 107, 109-1 12 Quantum dynamically complete experiments, 224 Quantum electrodynamics, 171 atoms and molecules, 179-183 Quantum mechanical interference, 236 of partial waves, 236 Quantum mechanically pure state, 224
Index
434 Quasi-hydrogenic ion, 93 Quasi-molecular approximation, 245 Quasi-molecular collision, 245
R Rabi frequency, 350-351.387-388 Rabi oscillations, 351, 356 Radiation absorption, 419 depolarisation, 422 re-emission, 419 trapping, 41 1,418,420,422,424 probability, 419,421 Radiation-atom interaction, 380 Radio-frequency heating, 394 Ramsey fringe experiment, 391 Rare gas atoms, 241 Rate equation approach, 345-76 derivation of exact rate equations, 373-376 dressed-atom picture, 368-37 1 extensions damping mechanisms, 362-363 fflteraction with squeezed vacuum, 363-368 generalization to N atomic levels, 356-362 diagrammatic evaluation of transition rates, 36&362 model, 357-358 rate equations, 358-360 previous work, 347-348 two-level atom, 348-356 density matrix equation reduction, 352-353 solutions for exact and approximate rate equations, 354-356 Rate equations conventional, 351,355 exact or generalized, 353 non-Markoffian, 353 Reaction coordinates for 3-body problem Eckart, 258-261 hyperspherical, 26 1-263 Jacobi, 256 Recombination processes capture-stabilized theory, 12G123 collisional-radiative model, 120, 133-1 34 characteristics of, 118 electron-ion, 124-134 history of theory, 118-120 ion-ion, 135-144 macroscropic view of, 120-121 nomenclature, 117-1 18 transport-influenced reactions, 123, 142-143
Reflection invariance, 226 Relativistic contributions, 244 Relativistic corrections, I07 recoil, 107 reduced mass, 107 Relativistic effects, 244 heavy atoms and ions, 44 Relativistic electronic structure, 169-1 84 basis sets and quantum electrodynamics of atoms and molecules, 179-183 beginnings of theory, 170-175 hydrogenic atoms, 170-1 72 many-electron atoms, 172-175 open-shell atoms, 175-179 Relativistic theory of electron capture to the continuum, 280-292 continuum distorted wave (CDW) approximation, 286-289, 292 fine structure constant a,280,283-285 first-order relativistic OBK approximation, 28 1-282 second-order relativistic OBK approximation, 283-292 Repulsive forces, 235 Resonance fluorescence, 370 Resonances, 209-210.214-216 Feshbach, 207,214-215 in positron scattering, 23 shape, 209,215 Resonant excitation autoionization READI. 7 1.77-78 REDA, 71.75.77-78.80.82-83 REPA, 72,82 REQA, 72.82-83 RETA, 72.82-83 Reviews of ionization, 72,74,77 Ritz defect, 94-96, 100, 102, 107 Ritz expansion, 94-96, 105 adjustments to, 107 comparison with perturbation expansions, 97 and mass polarization, 114 and relativistic corrections, 114 Ritz variational method, 172 R-matrix, 50 calculations, 85.87 intermediate-energy, 45, 51 R-matrix method, 43 Rydberg atoms, 380 Rydberg series, 93 Rydberg state, 62 rare gas atoms, 315,324
INDEX S Scattering, 248 amplitudes, 41, 224,241 asymmetry effects, 247 in charge transfer processes, 247 in electron-impact excitation, 247 Scattering parameters k and x, 227, 229-230, 233 Schrodinger equation, 4 M 1 Second Born term, 48 Secular approximations, 369 Selected ion flow tube ( S I R ) apparatus, 334 Self-consistent field method, 172 Semiclassical model, 235 Semiempirical theory, 75 Singlet scattering amplitudes, 245 Singlet-triplet mixing, I07 Singlet wave functions, 245 Slater integrals, 176 S-matrix, 43 Sodium atom(s), 245 spin polarisation of, 245 spin polarised, 245 Spin-analysed collisions, 245 with one-electron ions and atoms, 245 Spin analysis, 243 Spin conservation. 245 adiabatic, 245 Spin effects, 244, 245 Spin exchange contribution, 245 of excitation process, 245 Spin exchange effects, 243 Spin experiments, 244 Spin flip, 245 amplitude, 241, 243 nonadiabdtic, 245 Spin-orbit coupled states, 243 Spin orbit interaction, 241 Spin polarisation. 245, 247 of photoelectrons, 247 Spin-polarised one-electron atoms, 245 Spin-polarised Srf ion beams, 245 Squeezing, 265,385 Stars binary, 189, 193 evolution, 189 main sequence, 189, 191 novae, 197 ejecta dust formation, 199 molecules, 200
435
nucleosynthesis, 199 red giant, winds, 191 supergiant, 189 core, 200 supernovae, 199 ejecta, molecules, 200 SN1987A, 200 white dwarf, 189, 193 young stellar objects, winds, 188 State multipoles, 224, 229 State of the photon, 227 State-selective electron capture in H, 149 by photon emission spectroscopy, 150 by translational energy spectroscopy, 151 Ar6+ + H, 160 results for Ar4+, A?, results for C2+ + H, 157 results for C3' + H, 154 results for 02++ H, 163 results for 03++ H, 161 resuits for s3++ H, 164 State vector(s), 225,227 Statistical tensors, 229 Stokes parameters, 227-232.243-244, 246-247, 407,412,417,420 Storage ring, 397 ions, 397-399 ordered structures in, 3 9 9 4 0 4 Sturmian expansion, 52 Sturmian functions, 45 Sub-Poissonian photon statistics, 385, 387-388, 385 Superdissociative recombination. 132-1 33 Superelastic electron-impact deexcitation, 234 of atoms, 234 Superelastic scattering, 234 of electrons, 234 Supernova 1987A, 6 5 4 6 S-wave capture, 312,319 Symmetry, 240 cylindrical, 240
T Temperature rotational, 60 translational, 60 vibrational, 60 Termolecular recombination, 136-143 Bates's universal curve, 143 at higher gas densities, 139-142 at lower gas densities, 123, 137-142 Theories of electron capture, 245
436
Index
Theory of excitation processes, 244 Thermonuclear plasmas, 73.79 Thomas double scattering, 31 Thomas peak, 291-292 Tidal ion-ion recombination, 119, 137, 143-144 Time-inverseprocess, 234 excitation of atoms, 234 to inelastic electron impact, 234 Transition probabilities, 59 Transition rate diagrammaticevaluation, 360-362 generalized, 359 total, 359 Transitions, coherent and incoherent, 359 Translational energy spectroscopy in H, 149 principles underlying electron capture studies, 151 apparatus, 152 Translation factors common, 267-269 plane wave, 257,266267,271 switching factor, 266-268
Target parameters, 224 Triplet scattering amplitudes, 245 Triplet wave functions, 245 Two-electron atoms, 241,244 heavy, 241 Two-photon maser, 382
V Vacuum, squeezed, interaction of atomic systems with, 363-368 Vector polarisation, 228
W Wave-particle duality, 389 Which-path information, 389-390 Wigner crystal, 404
Y Ytterbium atoms, 247 laser-excited aligned, 247n
Contents of Volumes in thk Serial
The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A. R. Samson
Volume 1
Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. Hall and A. 'I: Amos
The Theory of Electron-Atom Collisions, R. Peterkop and V Veldre
Electron Affinities of Atoms and Molecules, B. L. Moiseiwitsch Atomic Rearrangement Collisions, B. H. Bransden The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K . Takayanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H. Pauly and J . P. Toennies High-Intensity and High-Energy Molecular Beams, J . B. Anderson, R. P. Andres, and J . B. Fen
Volume 2
The Calculation of van der Waals Interactions, A. Dalgarno and W D. Davison Thermal Diffusion in Gases, E. A. Mason, R. J . Munn, and Francis J . Smith Spectroscopy in the Vacuum Ultraviolet, W R. S. Garton
Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J . de Heer Mass Spectrometry of Free Radicals, S . N . Foner Volume 3
The Quanta1 Calculation of Photoionization Cross Sections, A. L. Stewart Radiofrequency Spectroscopy of Stored Ions I : Storage, H. G. Dehmelt Optical Pumping Methods in Atomic Spectroscopy, B. Budick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H . C. Wolf Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum Mechanics in Gas Crystal-Surface van der Waals Scattering, E. Chanoch Beder Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J . Wood
431
438
Contents of Volumes in this Serial
Volume 4
H. S. W. Massey-A Sixtieth Birthday Tribute, E. H. S. Burhop Electronic Eigenenergies of the Hydrogen Molecular Ion, D. R. Bates and R. H . G. Reid Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A. Buckingham and E. Gal Positrons and Positronium in Gases, P. A. Fraser Classical Theory of Atomic Scattering, A. Burgess and I . C. Percioal Born Expansions, A. R. Holt and B. L. Moiseiwitsch Resonances in Electron Scattering by Atoms and Molecules, P. G. Burke Relativistic Inner Shell Ionizations, C. B. 0. Mohr Recent Measurements on Charge Transfer, J . B. Hasted Measurements of Electron Excitation Functions, D. W 0. Heddle and R. G. W Keesing Some New Experimental Methods in Collision Physics, R. F. Stebbings Atomic Collision Processes in Gaseous Nebulae, M . J . Seaton Collisions in the Ionosphere, A. Dalgarno The Direct Study of Ionization in Space, R. L F . Boyd Volume 5
Flowing Afterglow Measurements of IonNeutral Reactions, E. E. Ferguson, F. C . Fehsenfeld, and A. L. Schmeltekopf Experiments with Merging Beams, Roy H. Negnaber Radiofrequency Spectroscopy of Stored Ions 11: Spectroscopy, H. G. Dehmelt
The Calculation of Atomic Transition Probabilities, R. J . S. Crossley Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations sns’”pq,C. D. H. Chisholm, A. Dalgarno, and F. R. lnnes Relativistic 2-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle Volume 6
Dissociative Recombination, J . N . Bardsley and M . A. Biondi Analysis of the Velocity Field in Plasmas from the Doppler Broadening of Spectral Emission Lines, A. S. Kaufman The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazu Itikawa The Diffusion of Atoms and Molecules, E. A. Mason and 7: R. Marrero Theory and Application of Sturmian Functions, Manuel Rotenberg Use of Classical Mechanics in the Treatment
of Collisions between Massive Systems, D. R. Bates and A. E. Kingston
Volume 7 Physics of the Hydrogen Master, C. Audoin, J . P. Schermann, and P. Grivet Molecular Wave Functions: Calculation and Use in Atomic and Molecular Processes, J . C. Browne Localized Molecular Orbitals, Hare1 Weinstein, Ruben Pauncz, and Maurice Cohen General Theory o f Spin-Coupled Wave Functions for Atoms and Molecules, J . Gerratt Diabatic States of Molecules-Quasi-Stationary Electronic States, Thomas F. O’Mafley
The Spectra of Molecular Solids, 0. Schnepp
Selection Rules within Atomic Shells, B. R. Judd
The Meaning o f Collision Broadening of Spectral Lines: The Classical Oscillator Analog, A. Ben-Reuven
Green’s Function Technique in Atomic and Molecular Physics, Gy. Csanak, H . S. Taylor, and Robert Yaris
CONTENTS OF VOLUMES IN THIS SERIAL
A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Nathan Wiser and A. J. Greenfield Volume 8
439
Recent Progress in the Classification of the Spectra of Highly ionized Atoms, B. C. Fawcett A Review of Jovian Ionospheric Chemistry, Wesley 7: Huntress, J r .
Interstellar Molecules: Their Formation and Destruction, D. McNally
Volume 11
Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C. Keck
The Theory of Collisions between Charbd Particles and Highly Excited Atoms, I. C. Percival and D. Richards
Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C. Y Chen and Augustine C. Chen Photoionization with Molecular Beams, R. B. Cairns, Halstead Harrison, and R. I . Schoen The Auger Effect, E. H . S . Burhop and W N. Asaad
Electron Impact Excitation of Positive Ions, M . J . Seaton The R-Matrix Theory of Atomic Process, P. G. Burke and W D. Robb Role of Energy in Reactive Molecular Scattering: An Information-Theoretic Approach, R. B. Bernstein and R. D. Levine Inner Shell Ionization by incident Nuclei, Johannes M . Hansteen
Volume 9
Stark Broadening, Hans R. Griem
Correlation in Excited States of Atoms, A. W Weiss
Chemiluminescence in Gases, M . F . Golde and B. A. Thrush
The Calculation of Electron-Atom Excitation Cross Sections, M . R. H . Rudge
Volume 12
Collision-Induced Transitions between Rotational Levels, Takeshi Oka
Nonadiabatic Transitions between Ionic and Covalent States, R. K . Janev
The Differential Cross Section of Low-Energy Electron-Atom Collisions, D.Andrick
Recent Progress in the Theory of Atomic Isotope Shift, J . Bauche and R.-J. Champeau
Molecular Beam Electric Resonance Spectroscopy, Jens C. Zorn and Thomas C. English
Topics on Multiphoton Processes in Atoms, P. Lambropoulos
Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McElroy
Optical Pumping of Molecules, M . Brayer, G. Goudedard, J . C. Lehmann, and J . UguL
Volume 10
Time-of-Flight Scattering Spectroscopy, Wilhelm Raith Ion Chemistry in the D Region, George C. Reid
Highly Ionized Ions, Ivan A. Sellin Relativistic Effects in the Many-Electron Atom, Lloyd Armstrong, Jr. and Serge Feneuille The First Born Approximation, K . L. Bell and A. E. Kingston Photoelectron Spectroscopy, W C. Price Dye Lasers in Atomic Spectroscopy, W Lange, J . Luther, and A. Steudel
Volume 13
Atomic and Molecular Polarizabilities-A Review of Recent Advances, Thomas M . Miller and Benjamin Bederson
440
Contents of Volumes in this Serial
Study of Collisions by Laser Spectroscopy, Paul R. Berman Collision Experiments with Laser-Excited Atoms in Crossed Beams, I. !I Hertel and w Stoll Scattering Studies of Rotational and Vibrational Excitation of Molecules, Manfred Faubel and J . Peter Toennies Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R. K . Nesbet Microwave Transitions of Interstellar Atoms and Molecules. W B. SomeroiIle Volume 14 Resonances in Electron Atom and Molecule Scattering, D. E. Golden The Accurate Calculation of Atomic Properties by Numerical Methods, Brian C. Webster, Michael J . Jamieson, and Ronald F . Stewart (e, 2e) Collisions, Erich Weigold and Ian E. McCarthy Forbidden Transitions in One- and TwoElectron Atoms, Richard Marrus and Peter J . Mohr Semiclassical Effects in Heavy-Particle Collisions, M. S. Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francis M . Pipkin Quasi-Molecular Interference Effects in IonAtom Collisions, s. V. Bobasheo Rydberg Atoms, S. A. Edelstein and Z F. Gallagher UV and X-Ray Spectroscopy in Astrophysics, A. K. Dupree Volume 15 Negative Ions, H. S. W Massey Atomic Physics from Atmospheric and Astro. physical Studies, A. Dalgarno Collisions of Highly Excited Atoms, R. F . Stebbings
Theoretical Aspects of Positron Collisions in Gases, J. W ; Humberston Experimental Aspects of Positron Collisions in Gases, 'I: C. Griflth Reactive Scattering: Recent Advances in Theory and Experiment, Richard B. Bernstein Ion- Atom Charge Transfer Collisions at Low Energies, J. 3. Hasted Aspects of Recombination, D. R. Bates The Theory of Fast Heavy Particle Collisions, B. H . Bransden Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H. B. Gilbody Inner-Shell Ionization, E. H. S. Burhop Excitation of Atoms by Electron Impact, D. W 0.Heddle Coherence and Correlation in Atomic Collisions, H. Kleinpoppen Theory of Low Energy Electron-Molecule Collisions, P. G. Burke Volume 16 Atomic Hartree-Fock Theory, M . Cohen and R. P. McEachran Experiments and Model Calculations to Determine Interatomic Potentials, R. DCren Sources of Polarized Electrons, R. J. Celotta and D. Z Pierce Theory of Atomic Processes in Strong Resonant Electromagnetic Fields, S. Swain Spectroscopy of Laser-Produced Plasmas, M. H.Key and R. J . Hutcheon Relativistic Effects in Atomic Collisions Theory, B. L. Moiseiwitsch Parity Nonconservation in Atoms: Status of Theory and Experiment, E. N . Fortson and L. Wilets Volume 17
Collective Effects in Photoionization of Atoms, M. Ya. Amusia
CONTENTS OF VOLUMES IN THIS SERIAL
44 1
Nonadiabatic Charge Transfer, D. S. F. Crothers
Atoms with Fully Stripped Ions, B. H. Bransden and R. K . Janev
Atomic Rydberg States, Serge Feneuille and Pierre Jacquinot
Interactions of Simple Ion-Atom Systems, J . I: Park
Superfluorescence, M . F. H . Schuurmans, Q. H . F. Mehen, D. Polder, and H . M . Gibbs f .pplications o f Resonance Ionization Spectroscopy in Atomic and Molecular Physics, M . G. Payne, C. H . Chen, G. S. Hurst, and G. W Foltz
High-Resolution Spectroscopy of Stored Ions, D. J . Wineland, Wayne M . Itano, and R. S . Van Dyck, J r .
Inner-Shell Vacancy Production in lonAtom Collisions, C. D. Lin and Patrick Richard Atomic Processes in the Sun, P. L. DuBon and A. E. Kingston
The Reduced Potential Curve Method for Diatomic Molecules and Its Applications, F. JenE
Volume 18 Theory of Electron-Atom Scattering in a Radiation Field, Leonard Rosenberg Positron-Gas Scattering Experiments, Talbert S. Stein and Walter E. Kauppila Nonresonant Multiphoton Ionization of Atoms, J . Morellec, D. Normand, and G. Petite Classical and Semiclassical Methods in Inelastic Heavy-Particle Collisions, A. S . Dickinson and D. Richards Recent Computational Developments in the Use of Complex Scaling in Resonance Phenomena, B. R. Junker Direct Excitation in Atomic Collisions: Studies of Quasi-One-Electron Systems, N. Anderson and S. E. Nielsen Model Potentials in Atomic Structure, A. Hibbert Recent Developments in the Theory of Electron Scattering by Highly Polar Molecules, D . W Norcross and L. A. Collins Quantum Electrodynamic Eflects in FewElectron Atomic Systems, G. W F. Drake
Volume 19 Electron Capture in Collisions of Hydrogen
Spin-Dependent Phenomena in Inelastic Electron-Atom Collisions, K . Blum and H. K leinpoppen
The Vibrational Excitation of Molecules by Electron Impact, D. G. Thompson Vibrational and Rotational Excitation in Molecular Collisions, Manfred Faubel Spin Polarization o f Atomic and Molecular Photoelectrons, N. A. Cherepkov
Volume 20 Ion-Ion Recombination in an Ambient Gas, D. R. Bates Atomic Charges within Molecules, G. G. Hall Experimental Studies on Cluster Ions, 'I: D. Mark and A. W Castleman, Jr. Nuclear Reaction Effects on Atomic InnerShell Ionization, W E. Meyerhofand J.-F. Chemin Numerical Calculations on Electron-Impact Ionization, Christopher Bottcher Electron and Ion Mobilities, Gordon R. Freeman and David A. Armstrong On the Problem of Extreme UV and X-Ray Lasers, I . I. Sobel'man and A. K Vinogradou Radiative Properties of Rydberg States in Resonant Cavities, S. Haroche and J . M . Raimond Rydberg Atoms: High-Resolution Spectroscopy and Radiation Interaction-Rydberg Molecules, J . A. C. Gallas, G. Leuchs, H . Walther, and H . Figger
442
Contents of Volumes in this Serial
Volume 21
Subnatural Linewidths in Atomic Spectroscopy, Dennis P. O’Brien, Pierre Meystre, and Herbert Watther Molecular Applications of Quantum Defect Theory, Chris H. Greene and Ch. Jungen Theory of Dielectronic Recombination, Yukap Hahn Recent Developments in Semiclassical Floquet Theories for Intense-Field Multiphoton Processes, Shih-I Chu Scattering in Strong Magnetic Fields, M . R. C. McDowell and M . Zarcone Pressure Ionization, Resonances, and the Continuity of Bound and Free States, R. M . More Volume 22
Positronium-Its Formation and Interaction with Simple Systems, J . W Humberston Experimental Aspects of Positron and Positronium Physics, T C.GrrfFth Doubly Excited States, Including New Classification Schemes, C . D. Lin Measurements of Charge Transfer and Ionization in Collisions Involving Hydrogen Atoms, H. B. Gilbody Electron-Ion and Ion-Ion Collisions with Intersecting Beams, K . Dolder and B. Peart Electron Capture by Simple Ions, Edward Pollack and Yukap Hahn Relativistic Heavy-Ion-Atom Collisions, R . Anholt and Haruey Gould Continued-Fraction Methods in Atomic Physics, S. Swain
Point-Charge Models for Molecules Derived from Least-Squares Fitting of the Electric Potential, D. E. Wiliams and Ji-Min Yan Transition Arrays in the Spectra of Ionized Atoms, J . Bauche, C. Bauche-Arnoult, and M . Klapisch Photoionization and Collisional Ionization of Excited Atoms Using Synchrotron and Laser Radiation, F. J . Wuilleumier, D. L. Ederer, and J . L. PicquC Volume 24
The Selected Ion Flow Tube (SIFT): Studies of Ion-Neutral Reactions, D. Smith and N . G. A d a m Near-Threshold Electron-Molecule Scattering, Michael A. Morrison Angular Correlation in Multiphoton Ionization of Atoms, S. J. Smith and G. Leuchs Optical Pumping and Spin Exchange in Gas Cells, R. J . Knize, Z. W, and W Happer Correlations in Electron-Atom Scattering, A . Crowe Volume 25
Alexander Dalgarno: Life and Personality, David R. Bates and George A. Victor Alexander Dalgarno: Contributions to Atomic and Molecular Physics, Neal Lane Alexander Dalgarno: Contributions to Aeronomy, Michael B. McEIroy Alexander Dalgarno: Contributions to Astrophysics, Dauid A. Williams Dipole Polarizability Measurements, Thomas M . Miller and Benjamin Bederson
Volume 23
Flow Tube Studies of Ion-Molecule Reactions, Eldon Ferguson
Vacuum Ultraviolet Laser Spectroscopy of Small Molecules, C . R. Vidal
Differential Scattering in He-He and He+He Collisions at KeV Energies, R. F . Stebbings Atomic Excitation in Dense Plasmas, Jon C . Weisheit
Foundations of the Relativistic Theory of Atomic and Molecular Structure, Ian P. Grant and Harry M . Quiney
CONTENTS OF VOLUMES IN THIS SERIAL
443
Pressure Broadening and Laser-Induced Spectral Line Shapes, Kenneth M . Sando and Shih-I Chu
Associative Ionization: Experiments, Potentials, and Dynamics, John Weiner, Franpoise Masnou-Sweeuws, and Annick Giusti-Suzor
Model-Potential Methods, G . Laughlin and G. A. Kctor
On the p Decay of '"Re: An Interface of Atomic and Nuclear Physics and Cosmochronology, Zonghau Chen, Leonard Rosenherg, and Larry Spruch Progress in Low Pressure Mercury-Rare Gas Discharge Research, J . Maya and R. Lagushenko
Z-Expansion Methods, M . Cohen Schwinger Variational Methods, Deborah K a y Watson Fine-Structure Transitions in Proton-Ion Collisions, R. H . G. Reid Electron Impact Excitation, R. J . W Henry and A. E. Kingston Recent Advances in the Numerical Calculation of Ionization Amplitudes, Christopher Bottcher The Numerical Solution of the Equations of Molecular Scattering, A. C. Allison High Energy Charge Transfer, B. H . Bransden and D. P. Dewangan Relativistic Random-Phase Approximation, W R. Johnson Relativistic Sturmian and Finite Basis Set Methods in Atomic Physics, G . W F. Drake and S. P. Goldman Dissociation Dynamics o f Polyatomic Molecules, 7: Uzer Photodissociation Processes in Diatomic Molecules of Astrophysical Interest, Kate P. Kirby and Ewine F. van Dishoeck The Abundances and Excitation of Interstellar Molecules, John H . Black Volume 26 Comparisons of Positrons and Electron Scattering by Gases, Walter E. Kauppila and Talbert S. Stein Electron Capture at Relativistic Energies, B. L. Moiseiwifsch The Low-Energy, Heavy Particle CollisionsA Close-Coupling Treatment, Mineo Kimura and Neal F. Lane Vibronic Phenomena in Collisions of Atomic and Molecular Species, V. Sidis
Volume 27 Negative Ions: Structure and Spectra, David R. Bates Electron Polarization Phenomena in Electron-Atom Collisions, Joachim Kessler Electron-Atom Scattering, I . E. McCarthy and E. Weigold Electron-Atom Ionization, I . E. McCarthy and E. Weigold Role of Autoionizing States in Multiphoton Ionization of Complex Atoms, I/: I. Lengyel and M . 1. Haysak
Multiphoton Ionization of Atomic Hydrogen Using Perturbation Theory, E. Karule Volume 28 The Theory of Fast Ion-Atom Collisions, J . S. Briggs and J . H . Macek Some Recent Developments in the Fundamental Theory of Light, Peter W Milonni and Surendra Singh Squeezed States of the Radiation Field, Khalid Zaheer and M . Suhail Zuhairy Cavity Quantum Electrodynamics, E . A . Hinds Volume 29 Studies of Electron Excitation of Rare-Gas Atoms into and out of Metastable Levels Using Optical and Laser Techniques, Chun C. Lin and L. W Anderson
444
Contents of Volumes in this Serial
Cross Sections for Direct Multiphoton Ionization of Atoms, M . !J Ammosov, N. B. Delone, M . Yu. Ivanov, I . I. Bondar, and A. K Masalov Collision-Induced Coherences in Optical Physics, G. S. Agarwal Muon-Catalyzed Fusion, Johann Rafelski and Helga E . Rafelski Cooperative Effects in Atomic Physics, J. P. Connerade Multiple Electron Excitation, Ionization, and Transfer in High-Velocity Atomic and Molecular Collisions, J . H. McGuire
Volume 30 Differential Cross Sections for Excitation of Helium Atoms and Helium like Ions by Electron Impact, Shinobu Nakazaki Cross-Section Measurements for Electron Impact on Excited Atomic Species, S. Trajmar and J . C. Nickel The Dissociative Ionization of Simple Molecules by Fast Ions, Colin J . Latimer Theory of Collisions Between Laser Cooled Atoms, P. S. Julienne, A . M . Smith, and K . Burnett Light-Induced Drift, E. R. Eliel Continuum Distorted Wave Methods in lonAtom Collisions, Derrick S. F. Crothers and Louis J . DubP
Volume 31 Energies and Asymptotic Analysis for Helium Rydberg States, G. W F. Drake Spectroscopy o f Trapped Ions, R. C. Thompson Phase Transitions of Stored Laser-Cooled Ions, H . Walther Selection of Electronic States in Atomic Beams with Lasers, Jacques Baudon, Rudolf Duren and Jacques Robert Atomic Physics and Non-Maxwellian Plasmas, MichPie Lamoureux
Volume 32 Photoionisation of Atomic Oxygen and
Atomic Nitrogen, K . L. Bell and A. E. Kingston Positronium Formation by Positron Impact on Atoms at Intermediate Energies, B. H. Bransden and C. J . Noble Electron-Atom Scattering Theory and Calculations, P. G. Burke Terrestrial and Extraterrestrial H, +,Alexander Dalgarno Indirect Ionization of Positive Atomic Ions, K . Dolder Quantum Defect Theory and Analysis of High-Precision Helium Term Energies, G. W F. Drake Electron-Ion and Ion-Ion Recombination Processes, M . R. Flannery Studies of State-Selective Electron Capture in Atomic Hydrogen by Translational Energy Spectroscopy, H. B. Gilbody Relativistic Electronic Structure of Atoms and Molecules, I . P. Grant The Chemistry of Stellar Environments, D. A. Howe, J . M . C. Rawlings, and D. A. Williams Positron and Positronium Scattering at Low Energies, J . W Humberston How Perfect are Complete Atomic Collision Experiments?, H. Kleinpoppen and H . Hamdv Adiabatic Expansions and Nonadiabatic Effects, R. McCarroll and D. S. F. Crothers Electron Capture to the Continuum, B. L. Moiseiwitsch How Opaque I s a Star?, M . J . Seaton Studies of Electron Attachment at Thermal Energies Using the Flowing AfterglowLangmuir Technique, Smith and Patrik SoanZ.1
Exact and Approximate Rate Equations in Atom-Field Interactions,
s, Swain
Atoms in Cavities and Traps, H. Walther Some Recent Advances in Electron-Impact Excitation of n = 3 States of Atomic Hydrogen and Helium, J . F. Williams and J . B. Wang
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