Advances in ATOMIC AND MOLECULAR PHYSICS VOLUME 4
H. S. W . Massey
ADVANCES IN
ATOMIC AND MOLECULAR PHYSICS Edited by
D. R. Bates DEPARTMENT OF APPLIED MATHEMATICS THE QUEEN’S UNIVERSITY OF BELFAST BELFAST, NORTHERN IRELAND
Immanuel Estermann DEPARTMENT OF PHYSICS THE TECHNION ISRAEL INSTITUTE OF TECHNOLOGY HAIFA, ISRAEL
VOLUME 4 In honor of H . S. W . Massey on the occasion of his sixtieth birthday
@) 1968 ACADEMIC PRESS New York
London
0
COPYRIGHT 1968, BY ACADEMIC PRESS INC. ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED I N ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.
ACADEMIC PRESS INC. 111 Fifth Avenue, New York, New York 10003
United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London W.l
LIBRARY OF CONGRESS CATALW CARDNUMBER : 65-18423
PRINTED I N THE UNITED STATES OF AMERICA.
List of Contributors Numbers in parentheses indicate the pages on which the authors’ contributions begin.
D. R. BATES, School of Physics and Applied Mathematics, The Queen’s University of Belfast, Belfast, Northern Ireland (13) R. L. F. BOYD, Mullard Space Science Laboratory, Department of Physics, University College, London, England (41 1) R. A. BUCKINGHAM, Institute of Computer Science, University of London, London, England (37) A. BURGESS, Department of Applied Mathematics and Theoretical Physics, Cambridge University, Cambridge, England (109) E. H. S. BURHOP, Department of Physics, University College, London, England (1) P. G. BURKE,’ Theoretical Physics Division, AERE Harwell, Didcot, Berkshire, England (1 73) A. DALGARNO, Harvard College Observatory and Smithsonian Institution Astrophysical Observatory, Cambridge, Massachusetts (38 1) P. A. FRASER, Department of Physics, University of Western Ontario, London, Ontario, Canada (63) E. GAL, Institute of Computer Science, University of London, London, England (37)
J. B. HASTED, Department of Physics, University College, London, England (237) D. W. 0. HEDDLE, Physics Department, University of York, Heslington, York, England (267) A. R. HOLT,’ School of Physics and Applied Mathematics, The Queen’s University of Belfast, Belfast, Northern Ireland (143) Present address: School of Physics and Applied Mathematics, The Queen’s University of Belfast, Belfast, Northern Ireland. Present address: Department of Mathematics, University of Essex, Colchester, England. V
vi
LIST OF CONTRIBUTORS
R. G. W. KEESING, Physics Department, University of York, Heslington, York, England (267) C. B. 0. MOHR, Department of Theoretical Physics, University of Melbourne, Victoria, Australia (221) B. L. MOISEIWITSCH, School of Physics and Applied Mathematics, The Queen’s University of Belfast, Belfast, Northern Ireland (143) I. C. PERCIVAL, Department of Physics, Stirling University, Stirling, Scotland ( 109) R. H. G. REID, School of Physics and Applied Mathematics, The Queen’s University of Belfast, Belfast, Northern Ireland (13)
M. J. SEATON, Department of Physics, University College, London, England (331) R. F. STEBBINGS, Department of Physics, University College, London, England (299)
Foreword This serial publication is intended to occupy an intermediate position between a scientific journal and a monograph. Its main object is to provide survey articles in fields such as the following: atomic and molecular structure and spectra, masers and optical pumping, mass spectroscopy, collisions, transport phenomena, physical and chemical interactions with surfaces, and gas kinetic theory. The present volume is offered as a tribute to Professor Sir Harrie Massey whose sixtieth birthday is in 1968. All the contributors have been directly or indirectly greatly influenced by him; most are former students of his. The articles naturally have had to be restricted to the field normally covered by “Advances in Atomic and Molecular Physics.” Because of this it has regrettably not been possible to include articles by some of Sir Harrie’s closest friends.
Belfast, Northern Ireland Haifa, Israel February, 1968
D. R. BATES I. ESTERMANN
vii
This Page Intentionally Left Blank
Contents LIST OF CONTRIBUTORS
V
vii xiii
FOREWORD CONTENTS OF PREVIOUS VOLUMES
H. S. W. Massey-A Sixtieth Birthday Tribute E. H . S . Burhop 1
Text
Electronic Eigenenergies of the Hydrogen Molecular Ion D . R. Bates and R. H. G . Reid 13 14 17 21 23 25 35
I. Introduction Quantum Numbers Calculation of Exact Eigenenergies Expansions JWKB Approximation Appendix References
11. 111. IV. V.
Applications of Quantum Theory to the Viscosity of Dilute Gases R . A . Buckingham and E. Gal I. Introduction 11. The Transition from Classical to Quantal Mechanics 111. Reduced Variables and Law of Corresponding States
IV. General Quantal Effects at Low Temperatures V. Special Cases References
31 38 39 43 47 60
Positrons and Positronium in Gases P . A . Fraser I. Introduction 11. The Fate of Positrons in Gases 111. Experimental Results IV. Theoretical Results V. Other Areas of Positron Atomic Physics VI. Basic Questions Review Works References
ix
63 65 71 87 103 103 104 105
CONTENTS
X
Classical Theory of Atomic Scattering A . Burgess and I. C. Percival I. Introduction 11. Classical Cross Sections
111. IV. V. VI.
Binary Encounters Perturbation Theories and Threshold Laws Orbit Integration and Monte Carlo Methods Correspondence Principle and Conclusions References
109 111 117 126 128 137 139
Born Expansions A . R . Holt and B. L. Moiseiwitsch I. Introduction 11. Born Expansion for the Scattering Amplitude 111. Convergence of Born Expansions
IV. Time-Dependent Collision Theory V. Rearrangement Collisions References
143 144 156 162 169 171
Resonances in Electron Scattering by Atoms and Molecules P . G . Burke I. Introduction 11. Experimental Observations 111. Resonance Scattering Theory IV. Further Results and Conclusions References
173 175 186 208 214
Relativistic Inner Shell Ionization
C. B. 0 . Mohr I. Introduction 11. Relativistic Wave Functions
111. IV. V. VI.
Inner Shell Energies K Ionization by Electrons Ionization by Protons Ionization by Photons References
22 1 22 1 224 226 23 1 233 235
Recent Measurements on Charge Transfer J. B. Hasted I. Introduction 11. Total Cross Sections for the Symmetrical Resonance Process
237 237
CONTENTS
111. IV. V. VI. VII. VIII. IX.
Total Charge Transfer Cross Sections for Unlike Ions and Atoms Differential Scattering with Capture Pseudocrossing of Potential Energy Curves Molecular Charge Transfer Processes at Low Energies Experimental Techniques Role of Excited Species Miscellaneous Topics References
xi 242 243 246 248 249 254 259 263
Measurements of Electron Excitation Functions D. W. 0 . Heddle and R . G . W . Keesing I. Introduction 11. The Excitation Equilibrium
111. IV. V. VI. VII. VIII.
The Angular Distribution of the Light Simultaneous Ionization and Excitation High Resolution Measurements Time-Resolved Measurements Related Measurements Comparison of Observations References
261 267 278 28 1 284 289 292 294 296
Some New Experimental Methods in Collision Physics R. F. Stebbings I. Introduction 11. Flowing Afterglows 111. Merged Beams IV. Ion Beam Measurements V. Electron Beam Measurements VI. Photoelectron Spectroscopy VII. Metastable Atom Measurements References
299 300 304 308 318 324 321 329
Atomic Collision Processes in Gaseous Nebulae M . J . Seaton I. Introduction 11. Recombination Spectra 111, The Forbidden Lines
References
331 332 356 378
Collisions in the Ionosphere A . Dalgarno I. Introduction 11. The Slowing Down of Fast Electrons
381 382
xii
CONTENTS
111. Electron Cooling Processes
IV. Ion Cooling Processes V. Ion-Molecule Reactions VI. The Slowing Down of Fast Protons References
390 394 399 405 405
The Direct Study of Ionization in Space
R.L. F. Boyd 1. Introduction 11. The Space Situation
111. IV. V. V1. VII.
Theory of Electron and Ion Probes Ungridded Probe Systems Gridded Probe Systems Transverse Field Analyzers Ion Mass Spectrometers References
AUTHORINDEX SUBJECT INDEX
41 1 412 417 423 428 433 437 441 443 458
Contents of Previous Volumes Volume 1 Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. Hall and A. T. Amos 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 . PauIy and J . P. Toennies High Intensity and High Energy Molecular Beams, J. B. Anderson, R. P. Andres, and J . B. Fenn AUTHORINDEX-SUBJECT INDEX Volume 2 TheCalculation 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. Carton The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A. R. Samson The Theory of Electron-Atom Collisions, R. Peterkop and V. Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J. de Heer Mass Spectrometry of Free Radicals, S. N. Foner AUTHORINDEX-SUBJECTINDEX Volume 3 The Quanta1 Calculation of Photoionization Cross Sections, A. L. Stewart Radiofrequency Spectroscopy of Stores Ions. I: Storage, H. G. Dehmelt Optical Pumping Methods in Atomic Spectroscopy, B. Budick Energy Transfer in Organic Molecular Crystals: A Survey ot Experiments, H. C. Worf 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 AUTHORINDEX-SUBJECT INDEX
...
XI11
This Page Intentionally Left Blank
H. S. W. MASSEYA SIXTIETH BIRTHDA Y TRIBUTE E. H . S. BURHOP Department of Physics, University College London, England
It is hard to think that H. S. W. Massey is approaching his 60th birthday. His tremendous mental activity and enthusiasm for physics is at as high a level as ever and would put most of his colleagues, even those half his age, to shame. Although his great wisdom and judgment on matters of policy have been recognized by his appointment to one of the highest government advisory scientific posts, and mountains of documents come for him to peruse and assess, his personal output of significant scientific work and the leadership he gives in his department and in many scientific fields outside of it, are undiminished. Physically, also, he remains remarkably active. My first contact with Massey was in Melbourne in 1928 when one afternoon he came in to demonstrate to a group of first-year physics students in the absence of our regular demonstrator. I can recall very few of the other thousand or so classes I attended as an undergraduate but I remember this one quite vividly, even the experiment I was doing at the time. I suppose this must be attributed to his remarkable personality and enthusiasm and to the fact that even though he was then only in his first year as a postgraduate student, he was already something of a legend among the undergraduates of his own university, and some of his more pungent comments had passed round a wide circle. Massey was born in Victoria and received his early education in the tiny rural school of a little settlement called Hoddle’s Creek, in the heart of the bush, some thirty or forty miles east of Melbourne. In those days the primary and central school courses in Victoria normally took eight years and ended with the award of the Merit Certificate at about the age of 13. The rural schools of the Victorian out-back were presided over by a single teacher, and in order to ease his task the children did their work in four groups, each group covering two years’ work. Young Harrie Massey romped through each of the successive two-year courses in one year, with the unique result that he obtained his Merit Certificate at the age of nine, far younger than anybody else, before or since. 1
2
E. H. S. Burhop
From Hoddle’s Creek his family moved to Melbourne, and he attended University High School, one of the best known State secondary schools in Victoria, and entered Melbourne University in 1925. At that time it was still possible, in principle, to do a science degree with full honors courses in both physics and chemistry. It was a difficult option, but naturally Massey went through the course with ease, finishing up with a first class B.Sc. degree and prizes in both physics and chemistry, while in the following year he sat for final honors in mathematics, easily obtaining a first class B.A. degree and a prize. The many brilliant successes that marked his progress through high school and university were by no means confined to the academic sphere, however. He was a very successful sportsman, competent at cricket, tennis, and baseball (for which he represented Melbourne University and won a half-blue). He also showed great prowess at billiards, winning a tournament while at University. Perhaps there may have been some connection between his interest in the collision of billiard balls and his subsequent emergence as the leading authority on atomic impact phenomena! He has in fact remained a keen cricketer. He played club cricket regularly in Belfast, and for several years after coming to London, he would turn out each weekend in the season for Chislehurst. The award to him of the Hobbs bat, for his success in club cricket, presented by the great J.B.H. himself, is one of his proud recollections. For many years after coming to University College London, he would turn out each year for the annual match between staff and students of the Physics Department, which he dominated. I am not sure whether he ever actually carried his bat through the staff innings and then went on to capture all the students’ wickets, but the reputation of his prowess was such that everybody expected him to do so. He can still be seen occasionally at Lords, especially when the Australians are playing. I n his first postgraduate year at Melbourne, in addition to taking final honors in mathematics, Massey wrote a 400-page dissertation consisting of a critical survey of the then rapidly developing subject of quantum mechanics. The extensive and encyclopedic knowledge of the early literature he acquired in the process undoubtedly provided a very sound basis for his later work. In addition, however, he joined with C. B. 0. Mohr in an experimental investigation of the reflection of soft x rays, thus beginning a notable partnership in scientific research which continued during Massey’s four years at Cambridge and long afterwards, and was very fruitful in producing some of the basic papers on the application of collision theory, mainly to atomic and electronic collisions, but also to nuclear collisions. In August 1929 Massey left Melbourne for Cambridge as the holder of an Aitchison Travelling Scholarship. He was fortunate in that his period at the Cavendish coincided with the Golden Age of Rutherford’s postwar period. Apart from Rutherford there were: Chadwick, engaged at the time in the
H. S. W. MASSEY-A
SIXTIETH BIRTHDAY TRIBUTE
3
preliminary experiments leading up to his discovery of the neutron in 1932; Cockcroft and Walton, building up the high tension apparatus which culminated in the first artificial disintegration of the lithium nucleus, also in 1932; Blackett and Occhialini, constructing the first counter-controlled cloud chamber, which enabled them to observe the production of positron-electron pairs in 1933. It was a time of great ferment in theoretical physics too. Dirac was at the summit of his creative period. Mott was writing his classical papers on electron scattering, and R. H. Fowler, to whom fell most of the burden of supervising theoretical physics research students, was striving to ensure that on the theoretical, not less than on the experimental side, the Cavendish Laboratory should be a recognized center of excellence. Massey fitted easily into this atmosphere of intensive research, and very soon his views came to be regarded seriously. He set to work with E. C. Bullard in the old “ garage at the Cavendish on an experimental study of the scattering of slow electrons in argon, and this work soon established the appearance of maxima and minima in the elastic scattering angular distribution due to the diffraction of the electron waves by the spherically symmetrical scatterers, the argon atoms. Massey continued similar work with E. C. Childs when Bullard turned to geophysics, and in 1930, when C. B. 0. Mohr also came to the Cavendish from Melbourne, Massey encouraged him to embark upon a parallel experimental program of inelastic scattering. At the same time, Massey and Mohr embarked on their extensive series of papers on the scattering of electrons and atoms by atoms and molecules. Massey, together with N. F. Mott, was invited by the Oxford University Press to prepare their volume on the “Theory of Atomic Collisions.” This soon became the classic treatise on the subject and remains so today, having been brought up to date in the second and third editions successively by Massey almost alone, since Mott’s interests went off into other directions. Massey’s four years at the Cavendish were probably the most enjoyable of his life. Never since has he had similar freedom from teaching, administrative, and policy-forming activities. It was a period of remarkable achievement, during which he and his collaborators produced no fewer than 25 substantial original papers in scientific journals. In addition to the experimental and theoretical study of electron scattering, these papers covered a wide field, including gas kinetics, scattering of positive and negative ions, surface phenomena, and theory of X rays and of nuclear collisions. Some of the papers were of fundamental importance, opening up new fields, not only in atomic physics, but also in theoretical chemistry. In 1933 he took up the post of Independent Lecturer in Mathematical Physics at Queen’s University, Belfast. For the next six years, together with one assistant (R. A . Buckingham), he was responsible for initiating and ”
4
E. H. S. Burhop
giving all the courses in applied mathematics of what was already quite a large university, and was then badly understaffed. The amount of lecturing involved was prodigious. Anybody who has worked with Massey knows the care and thought he gives to the choice and presentation of material in lectures. For a lesser man the tasks faced at Belfast would have led to a drying up of creative ideas and research output. But no such thing happened. Massey maintained contacts with many of his former collaborators and continued to direct their work by correspondence. Several of us recall with pleasure the working vacations we spent during this period in the Massey household in Belfast. Later, when we became even more dispersed, Mohr in Capetown, R. A. Smith in St. Andrews, and myself back in Melbourne, the intensive collaboration continued across the world. I think that during his Belfast period Massey missed most the experimental physics that had been part of his life. Massey is a very talented mathematician and a great calculator, never daunted by the magnitude of the sheer effort involved. Nevertheless, he is first and foremost a physicist. The basis of the study of nature lies in experiment. His mathematics is not an end in itself but a technical aid for the interpretation and coordination of the results of experiment. Both in Melbourne and at Cambridge he had greatly enjoyed the actual carrying out of experiments. In Belfast that was no longer possible. Every summer, however, as soon as the examinations were over, he would move to Cambridge with his family for the long vacation, and, for the seven or eight weeks until the people in the Cavendish disappeared for their own vacations, usually toward the end of August, he would enjoy once again the contact with, if not the close participation in, experiment that was the mainspring of his ideas. In the course of time Massey built up his own school at Belfast. With the help of John Wiley he built a differential analyzer, forerunner of the modern computer, and something of a rarity at that time. In spite of his burdensome teaching commitments, the Belfast period was only slightly less prolific than the Cambridge period. When he left Queen’s in 1939 to take up the Goldsmid Chair of Applied Mathematics at University College London, he had published approximately 20 additional original works. He had become involved in at least two new fields: the study of negative ions, on which he had written a monograph which was to be the standard work on the subject for many years; and the study of the basic collision processes in the ionosphere, his interest in which had been kindled particularly by the influence of T. H. Laby, his old professor of Melbourne days. Some of Massey’s most gifted pupils (D. R. Bates and J. Hamilton are among those who have established their own reputation in theoretical physics) date from the Belfast period. Soon after coming to London
H. S. W. MASSEY-A
SIXTIETH BIRTHDAY TRIBUTE
5
Massey was elected to Fellowship of the Royal Society, at the very early age of 31. Massey had scarcely settled into his new chair at UCL when war broke out, and before long he found himself called upon to apply himself to problems related to the national effort. For a time he was attached to the Admiralty Research Laboratory at Teddington. But even during the worst period of the blitz and the blackout he never lost interest in his basic research problems. He could no longer devote his working days to those problems but, on the long train journeys between his home in Chislehurst and Teddington, he completed some most intricate and significant calculations with the help of a cylindrical slide rule. The paper with R. A. Buckingham on neutrondeuteron scattering, forerunner of a long series on the few-nucleon problem, dates from this period. In 1941 Massey was asked to move to Havant to take charge of the Mine Design Department of HMS Vernon. The Mine Design Department was organized as a typical Services laboratory able to cope with its normal peacetime tasks, but quite unable to meet the challenging technical problems posed by the blockade of a powerful and determined enemy. Massey approached his new responsibilities with characteristic energy and resourcefulness. He has never had difficulty in gathering around him capable and dedicated young people who respect his leadership and scientific integrity. Many physicists who have made their mark in the postwar scientific world came to work under him at Havant : D. R. Bates, R. A. Buckingham, Francis H. C. Crick, J. C. Gunn, C. H. Mortimer, and H. L. Penman. Together they brought imaginative new ideas and organizational concepts, which transformed the laboratory and enabled it to make a significant contribution to the naval war effort. In the meantime, interest was building up in the development of a nuclear weapon. At the Quebec Conference of 1943 arrangements had been made for the cooperation of British scientific personnel in the Manhattan project, and Massey was asked to join M. L. Oliphant with a group of British physicists going to the Radiation Laboratory of the University of California, to assist in the development of the electromagnetic process for the separation of the isotopes of uranium. Massey was put in charge of the theoretical physics group which included such well-known U. S. theoretical physicists as E. U. Condon and David Bohm. When Oliphant returned to England early in 1945, Massey took over the leadership of the British personnel in Berkeley. Although his main responsibility was concerned with the theoretical side of the project, Massey soon noted that the magnet and vacuum tank of the 37-inch cyclotron was not being used, and he arranged with E. 0. Lawrence, the director of the Radiation Laboratory, for the theoretical group to run it in order to study basic processes related to the operation of the ion sources
6
E. H. S. Burhop
being used in the isotope separation process and to the mechanism of space charge neutralization that made possible the production of sharply focused ion beams. A number of important reports, which appeared after the war in the official publication of the work of the Manhattan project, described in detail the various regimes encountered in a low voltage arc operated in a strong magnetic field and measurements of ionization cross sections important for its operation. These investigations came too late to influence the design of the isotope separators used in the production process. They did, however, have the effect of further stimulating Massey’s own interest in a whole range of collision phenomena. Returning to University College London after the war, in October 1945, Massey found the state of disorganization consequent upon the destruction of a large part of the College in the bombing and its evacuation to various partsof the country far worse than he had imagined. I had gone from Australia to join Massey’s group in Berkeley and had been invited to come to England to work with him at University College. In the 35 years I have known Massey well, this is the only time I have seen him really despondent and temporarily lacking in enthusiasm. The mood lasted only a week or so. Soon he was busily tmgaged in rebuilding his department, assisted not only by the old UCL staff, but also by a number of colleagues, including Bates, Buckingham, and Gunn, who had worked with him during the war. Massey was still very interested in carrying out experiments on cross sections of collision processes of importance in the interpretation of ionospheric phenomena. It was typical of the enlightened attitude of University College and of the Provost at the time, Sir David Pye, that nobody seemed to think it strange that experiments on atomic (and nuclear) physics should be carried out in the Mathematics Department. The College was chronically short of space, so that no additional room could be provided for the purpose, but experiments were set up in two small rooms and in a landing on a stairway, and the strength of the Mathematics Department was increased by the appointment of two “physics” research assistants: R. L. F. Boyd, who had been trained as an engineer at Imperial College End had actually spent some time at the Mine Design Laboratory, and J. B. Hasted, an Oxford chemist. Such were the humble origins of the large experimental research groups in atomic and high energy physics that have now been built up at UCL. When E. N. da C. Andrade resigned as Quain Professor of Physics to take the post of Director of the Royal Institution, it was obvious that his successor should be H. S . W. Massey. Massey spent a weekend of agonizing indecision before bowing to the inevitable and accepting the Quain professorship, involving much larger administrative responsibilities than had fallen to him in the Mathematics Department.
H. S. W. MASSEY-A
SIXTIETH BIRTHDAY TRIBUTE
7
During his five years in the Mathematics Department he had been able to return to many of the theoretical problems which he had had to lay aside during the war, relating to general questions of nonrelativistic collision theory, the theory of collision processes in the upper atmosphere and in astrophysics, and the theory of nuclear forces. It was during this time that he also took particular interest in the problem of teaching mathematics to physics and chemistry students, an interest which culminated in the appearance of a comprehensive text for the purpose (with H. Kestleman) a few years later. During this period also, I assisted him in the preparation of a work on the physics (as distinct from the theory) of electronic and ionic collisions, which played a role in rekindling interest in these topics in the postwar period, leading to the present enormous volume of work in the field. With the move across to the Physics Department in October 1950, Massey’s horizons broadened markedly. At last he had a department in which he could build up experimental groups of a viable size. Most of the staff who had come to work with Massey at UCL since the war moved to the Physics Department with him, as did the theoretical physics research students, so that he was able to continue building up his school of research in the theory of collisions, which was now beginning to achieve wide recognition. In the course of a few years the Physics Department was transformed into one of the largest physics research departments in the country, with nearly 100 postgraduate students and 30 or more postdoctoral fellows. I t is a tribute to Massey’s tact and understanding, no less than to that of the older members of the Physics Department, that the rapid expansion involved was achieved almost without friction and without prejudicing the good internal relations which have always been a feature of the departments over which Massey has presided. For the first time it became possible to dispose experimental resources of appropriate size for carrying out the research program on electronic and ionic collisions that had been close to his heart for so long. In addition to groups led by J . B. Hasted studying slow collisions of atoms and ions and by R. L. F. Boyd studying collision processes in gas discharges, a group under D. W. 0. Heddle started working on basic optical excitation processes. But Massey’s interests have always been very broad, extending over the whole range of physics, so that it is not surprising that more than half the effort of the department went into nuclear and elementary particle physics. Exploiting a development that had taken place first in Canada, C. Henderson, F. F. Heymann, and R. E. Jennings constructed a small microtron for the purpose of measuring radiative reaction effects in fast electron scattering. This work was later extended by R. E. Jennings, who built a 28-MeV microtron for electron scattering investigations. The arrival on the staff of T. C. Grifith, who had worked at Aberystwyth with E. J. Williams, and later G . R. Evans, on the operation of a high pressure
8
E. H. S. Burhop
cloud chamber, aroused interest in the possible use of such an instrument in the study of unstable elementary particles in the cosmic radiation. In collaboration with the University of Padua, such an apparatus was installed in Mount Marmolada in the Dolomites and continued to collect data there for several years. Another chamber was used by a UCL group at AERE Harwell for studying nucleon scattering in hydrogen and helium. The Marmolada project, involving UCL research personnel at a distant station, set the pattern of things to come. Earlier than most, Massey had realized that the increasing cost and size of equipment for research in high energy physics implied that future work by university groups in this field would need to be carried out by visiting teams at just a few central accelerators. One of the first British groups to carry out an experiment at CERN, Geneva was from UCL, under F. F. Heymann, while an emulsion group under myself was set up to examine elementary particle decays and interactions in photographic emulsions exposed to radiations from large accelerators. At a later stage a group, under C. Henderson, was set up to study similar processes in bubble chambers. Massey has always made sure that the department had on its staff people of great technical competence and experience. He was able, therefore, to undertake responsibility for the design and construction of the 160-cm British national heavy liquid bubble chamber, a task that was carried out with signal success by H. S. Tomlinson, directing a joint group drawn from UCL and the Rutherford Laboratory. The third major field of Massey’s experimental interests in recent years has been related to studies of space physics. This began almost casually, and quite unexpectedly, with a telephone call from the Royal Aircraft Establishment in 1953, asking whether he would be interested in using rocket vehicles for studies of the upper atmosphere. Despite commitments in other directions that would have already seemed more than sufficient for most physicists, Massey immedia.tely saw the immense opportunities this offer opened up and threw himself into this new field of work with characteristic energy. With the help of R. L. F. Boyd, and later of G. V. Groves and A. P. Willmore, he drew up a program of research, realistic and interesting, and the department was soon heavily involved in a long series of experiments using Skylark rockets launched at the Woomera rocket range in Australia. Massey’s interest in problems of the ionosphere dated back to the thirties. He had been chairman of the Royal Society Gassiot Committee which provided support for such research in Britain. He was now able to engage in experimental research on the ionosphere, not only by studying collision cross sections in the laboratory, but also by rocket studies in the field. Right from the beginning, Massey has guided Britain’s scientific effort in space, as Chairman of the Royal Society’s British Committee for Space Research, and
H. S . W. MASSEY-A
SIXTIETH BIRTHDAY TRIBUTE
9
later as representative of the United Kingdom on COSPAR, the UN committee on space research, and on the Governing Board of ESRO, the European Space Research Organization which he served as Chairman. He has been responsible for negotiating UK participation in joint space projects with NASA, the US space organization, resulting in the launching of US satellites containing experiments prepared by laboratories in the UK. It was a great satisfaction to him that in the first of these satellites, UK 1, six of the nine experiments came from his own laboratory at UCL. The great contribution made by Massey’s group at UCL to the UK space program has been recognized by the establishment (following a generous gift from Mullard Ltd. of the Mullard Space Laboratory at Holmsbury, under R. L. F. Boyd, as a separate section of the Physics Department of UCL. Characteristically, Massey’s influence in guiding U K policies in Space Research has always been used to ensure that the comparatively meager resources this country can provide for the scientific program are used to carry out timely and meaningful experiments. With all his other preoccupations, Massey has managed to continue his own personal research at all times, either alone, or in collaboration with other members of staff or the many research students he still manages to supervise in problems of electronic and atomic collisions, and in relation to few-nucleon collisions. Massey has had the satisfaction of seeing many of his former students come to occupy senior positions in the scientific life of the UK, the Commonwealth, and the USA. Three have been elected to Fellowship of the Royal Society, and many more to University chairs in physics. Several have returned to occupy senior posts in his own department: J. Hamilton and L. Castillejo in high energy theoretical physics, M. J. Seaton in theoretical atomic physics, and R. L. Stebbings, A. P. Willmore, and F. R. Stannard on the experimental side. Massey’s contributions to the forming of national policy on science have not been confined to the field of space physics. In nuclear and high energy physics also, he has occupied an influential post as Chairman of the Nuclear Physics Committee of the old DSIR (Department of Scientific and Industrial Research) and as a member of the Visiting Committee of the Rutherford High Energy Laboratory. His facility for seeing the essential content of a problem or of a scientific theory through a bewildering mass of detail has enabled him to continue to make significant contributions in many different branches of physics, despite the daunting growth of the volume of relevant scientific literature. I t is also, however, the key to the noted contributions he has been able to make on many committees for which the paper work can be even more daunting and fully explains why, following the drastic reorganization of governmental scientific administration in 1965, he was invited to become
10
E. H . S . Burhop
Chairman of the Council on Scientific Policy (CSP), the key scientific advisory committee to the Ministry of Science and Education. Successful as he is in such work, however, he does it more from a sense of duty than from any love of it. He always begrudges the time it takes away from his scientific work. His period as Chairman of the CSP has been of great importance, and he has been responsible for initiating subcommittees to report on all aspects of scientific life in the UK, including manpower and financing problems, and the proper utilization of computing techniques. The results of these enquiries will determine the pattern of British scientific life for a long time to come. Even during his busiest period as Chairman of the CSP, however, he has continued to supervise effectively the running of his department, to produce a steady stream of significant research papers, and to continue to work on several scientific books. This same aptitude for distinguishing the essential physical content of a problem from a mass of unessential detail has also been the basis of his success as a teacher, where he is able to make the most difficult and erudite ideas of modern physical theory comprehensible. It has enabled him to build up a growing reputation as a popular expositor of the ideas of modern physics in public lectures and in a number of popular books, such as the “New Age in Physics,” which convey to the reader some of his own enthusiasm for physics and its modern developments. Massey’s unique contributions to physics in the United Kingdom have been recognized by many universities, which have awarded him honorary degrees, and by the Royal Society, through the award of its Hughes Medal in 1955 and its Royal Medal in 1958. With all his great gifts, Massey remains the friendliest and most unassuming of men, interested in humble everyday affairs no less than in the finest achievements of the human mind. He is equally at home discussing the most difficult points of a new theory with a physicist, or details of the family, children’s ailments or achievements, housekeeping, or the cost of living with the physicist’s wife. Many have occasion to recall, like myself, the encouragement and wise advice he has been able to give them, as young physicists at the outset of their careers. Massey shows infinite patience in listening to the problems of others, but is the last person to inflict his own worries and doubts in return. As a collaborator, one can be sure that Massey has contributed far more than his share of the work before he would allow his name to go forward as a joint author. He will not easily commit himself to a particular course of action, but once he has done so, one can be sure that he will adhere absolutely to the assurance he has given. He runs his large department effectively, but unobtrusively, without ever needing to exert his authority. “Never tell anyone to do anything, but see that it gets done ” has been his guiding principle, and it has worked because
H. S. W. MASSEY-A
SIXTIETH BIRTHDAY TRIBUTE
11
his reliability and friendliness have inspired in his staff a confidence and affection equaled only by their respect for him as a scientist. His knighthood in 1960 was regarded by all who knew him as a just recognition of the great services he had rendered. His relations with his colleagues in other departments are also very close. Traditionally, in universities there is rivalry, or worse, between the Physics and Chemistry Departments. University College London must be almost unique in the very close and cordial relations that are maintained between Massey and R. S . Nyholm, the Head of the Chemistry Department. A large joint research project on the study of reactions between molecules in molecular beams has just been initiated by the two departments. The fact that both Nyholm and Massey are Australian is scarcely sufficient explanation of these unusually good relations. Throughout his whole career Massey has enjoyed a very happy family life with the devotion and support of his wife, Jessica, whom he married while still an undergraduate in Melbourne. The lot of the wives of hard-working physicists, with their husbands poring over their papers night after night and throughout the weekend, is a hard one. Many of the relaxations open to other men’s wives are denied them. More recognition should be given to the contributions made by long-suffering physicists’ wives in easing the tasks of their husbands during their difficult creative periods. The people who most influenced Massey at the outset of his career were his professors, J. H. Michell and T. H. Laby in Melbourne, and Rutherford and J. Chadwick in Cambridge. Nobody could work under Rutherford without being infected with the sheer joy and satisfaction of discovery. Laby, in particular, inculcated in his students a love and respect for physics as a discipline worthy of every effort and sacrifice. He wore himself out in the attempt to instill, into a mostly unresponsive public, the great importance of scientific research to a young country like Australia. It is interesting to reflect that many of the things for which Laby strove so assiduously, but failed to accomplish, such as an adequate recognition by governments of the importance of scientific research, has seen fulfillment partly as a result of the achievements and career of his most talented pupil, H. S . W. Massey.
This Page Intentionally Left Blank
ELECTRONIC EIGENENERGIES OF THE HYDROGEN MOLECULAR ION D.R. BATES and R . H . G. REID School of Physics and Applied Mathematics The Queen's University of Belfast, Belfast, Northern Ireland I. Introduction ..................................................... 11. Quantum Numbers ................................................ 111. Calculation of Exact Eigenenergies ................................. IV. Expansions ....................................................... A. Large Internuclear Distance ..................................... B. Small Internuclear Distance ..................................... V. JWKB Approximation ............................................. Appendix ........................................................ References .......................................................
.13 14 .17 21 .21 .22 23 25 .35
I. Introduction The electronic eigenfunctions
of H 2 + satisfy the equation
(V + E + - r1+ - r22 , Y = O in which r , and r2 are the distances from the two protons, A and B, in atomic units and E is the electronic eigenenergy in Rydberg units. Burrau (1927) pointed out that Eq. (1) is separable in elliptic coordinates (A, p, 4) where
+
A = (ri r2)/R, cc = ( r i - r 2 ) / R (2) R being the distance between the protons, and where 4 is the azimuthal angle. He also obtained eigenenergies for the ground state by numerical integration. Wilson (1928), Teller (1930), Hylleraas (1931), Jaff6 (1934), and Baber and HassC (1935) developed a much more powerful general procedure based on representing the eigenfunctions by suitable expansions. During the past fifteen years several sets of systematic computations have been carried out by this procedure (cf. Table I). We extend these here (Section 3) giving the values of the eigenenergies of 70 of the lower states over a wide R-range which may readily be increased indefinitely by use of the relevant asymptotic series (Section 4). In addition we present a simple formula derived from the 13
D. R . Bates and R . H . G . Reid
14
TABLE I SYSTEMATIC COMPUTATIONS OF THE ELECTRONIC EIGENENERGIES OF THE HYDROGEN MOLECULAR ION
Reference
States treated
Bates et al. (1953)” Wallis and Hulbert (1954)b
Wind (1965) Peek (1965)b a
2s0, 3su, 3pu, 4pu 3du, 4fo, 2prr, 3drr Isu, 2s0, 3su 2pu, 3po 4pu 3do, 4du 4fu Zprr, 3prr 3drr I su Isu, 2pu
}
Internuclear distances (a.u.)
Number of significant figures given
0(0.2)5(0.5)9
5
0(0.2)5(0.5)10
5
O(5) IS( 10)45
8
0(0.05)20 1 (0.5)30
I 13
Give also separation constant and eigenfunction expansion coefficients. Gives also separation constant.
JWKB approximation (Section 5). With its aid other eigenenergies may be found quickly. Before proceeding with our main task we shall recall the relations between the quantum numbers used to describe the electronic states of H,’ in the general case (with R finite and nonzero) and in the united atom and separated atoms limits (Section 2). We shall follow the treatment of Morse and Stueckelberg (1929) who deduced these relations from the conservation of the number of nodal surfaces of the eigenfunction of a state as R is varied.
II. Quantum Numbers Writing = Q,(4>M(P)W
(3)
and substituting in Eq. (1) it is found that where rn = 0, * I , + 2 , . . . , and that (5)
d
- ((A’ - 1) dA
$)+ (-C + 2RA
- p2Az - A2m-2 1 ] A = O
(6)
ELECTRONIC EIGENENERGIES OF THE HYDROGEN MOLECULAR ION
15
where p2= - ~ R ~ E
(7) and where C is a separation constant. We shall denote a coordinate quantum number, that is, the number of nodal surfaces in the coordinate, by N with the symbol for the coordinate affixed as a subscript. The quantum numbers ( N + , N , , N,) specify a particular state. It is apparent from (4) that = m.
N,
(8)
Because of the axial symmetry of the potential rn is a good quantum number at all separations. In the united atom limit the (unnormalized) eigenfunction expressed in spherical polar coordinates (r, 0, 6)is Y = exp(imc$)Pj"l(cos 8)(2r/n,)'~:,'y+_:- 1(2r/nu) exp( - r/n,)
(9)
in which P and L are the associated Legendre and Laguerre polynomials indicated, the notation of Morse and Feshbach (1953) being followed, and in which nu and 1 are the principal and azimuthal quantum numbers.' Noting that
it may be seen that nu - 1 - 1, N , = N o = 1 - [mi.
N,
=N,=
Consider next the separated atoms limit. Let (tl, v ] , , 4) and parabolic coordinates centered on A and B, respectively, with
tl = r , ( l
+ cos el),
q 1 = r l ( l - cos 0,)
(13)
+ cos 0,)
(14)
and
t2 = r2(1 - cos 02),
(tZ, q 2 , 6)be
q2 = r2(l
in which the polar angles 0, and 0, are measured in the same sense from the A B axis. They are related to the elliptic coordinates in that
A+ 1 + h-1 -= R
1
+-?ZR as R+co.
p + -1
+ 51 -= R
(15)
52
1--
R
' The subscript u is affixed to n as a reminder that we are here concerned with the united atom limit. The subscript s will be used similarly for the separate atoms limit.
D. R. Bates and R . H. G . Reid
16
The electron may be attached to either proton. Hence the (unnormalized) eigenfunction is given by
where
is that for an isolated hydrogen atom, n, being the principal quantum number and k the parabolic quantum number. The i,nodal surfaces of the general case are ellipsoids with A and B as foci. They must have a one-one correspondence with the 11 nodal surfaces so that N , = N , = n, - k - I m I - 1.
(18)
The position regarding the p nodal surfaces is somewhat less simple. They are hyperboloids (again with A and B as foci). If N , is even each focus has +N, of them concave toward it: hence fN, =N,
= k.
(19)
However if N , is odd the corresponding number is f ( N , - 1) and the remaining nodal surface is the plane normal to and bisecting A B (which we shall call the midplane) : hence f ( N , - 1) = Ng
= k.
(20)
Even values of N , are associated with the positive sign in combination (16); odd values are associated with the negative sign. In the former case Y is symmetrical with respect to reflection in the midplane; in the latter case it is antisymmetrical. By combining relations (1 1) and (12) with relations (18)-(20) the quantum numbers of a state in the separated atoms limit may be expressed in terms of those of the same state in the united atom limit: thus
while
n, = nu - f { l - Iml k =f{Z Iml - l}
-
+ 11
if
{ I - [mi} is odd.
(22)
The sign of m is significant only in 0 and for convenience we shall henceforth take it to be positive.
ELECTRONIC EIGENENERGIES OF THE HYDROGEN MOLECULAR ION
17
111. Calculation of Exact Eigenenergies The object is to find the electronic eigenenergy E as a function of the internuclear distance R for a state specified by the quantum numbers (nu,1, m) in the united atom limit. This may be achieved by first using the eigenvalue equation (5) to calculate the separation constant C as a function of the parameter p defined in (7); then using the eigenvalue equation (6) to calculate R as a function of p and thence of E ; and finally carrying out the necessary inverse interpolation. The eigenfunctions of (5) may conveniently be represented by M(P) =
if S C +
(23)
S(P)
the summation being over even or odd values of s according to whether (1 - m) is even or odd (Wilson, 1928). Substitution in the eigenequation and use of the properties of the associated Legendre functions yields the three term recurrence relation as f s + 2
where as =
(s
(2s
+ bs f s +- cs f s - 2
=0
+ 2m + l)(s + 2m + 2) + 2m + 3)(2s + 2m + 5 )
bs =
2(s + m)(s + m + 1) - 2m2 - t - (s (2s + 2m + 3)(2s + 2m - 1)
cs =
s(s - 1) (2s + 2m - 1)(2s 2m - 3)
+ m)(s + m + I) - C P2
(26)
+
(Morse and Feshbach, 1953). The boundary conditions in (24) are fs=O for s < O ; fs/fs-2-0 as s + c o . (28) Stratton et al. (1956) have described an iterative scheme for finding the eigenvalues C(1,m Ip) from the recurrence relation and have compiled an extensive table of them. We judged it better to program the scheme rather than read from the table. Direct and inverse power series expansions in p are known (Flammer, 1957), but their useful ranges are too limited. Following Jaffk (1 934) we took m
in which CT
= ( R / p )-
m-1
and
5 = (1 - l)/(A
+ 1).
18
D. R. Bates and R. H. G. Reid
For this to be a solution of ( 6 ) the expansion coefficients must satisfy the three-term recurrence relation %Sf+l - A S ,
+ YrSt-1 = 0
(32)
where
+ m + t>(l + t ) p, = oA, + B, a, = (1
(33) (34)
and y, = o2
+ o(2 + m - 21) + (1 + m + t ( t - m - 2))
(35)
in which A , = -(2p
+ 1 + m + 2t)
(36)
B,
I p ) + p 2 + 4pt - m ( m + 1) + 2tZ.
(37)
and = C(1, m
If R is one of the eigenvalues then Sf -, $ - ( + + R I P )
exp( -4p1/2t112)
as t -, 00
(38)
(Baber and Hasse, 1935). Recurrence relation (32) leads to a continued fraction equation in o the roots of which determine these eigenvalues. Thus writing
6, = %-1Yt and
and noting that
=o
(40)
+ Bo - PI= 0.
(41)
9-1
it may be seen that oAO
We found the lower roots from (41) by the following method. Proceeding in convenient increments of p from the origin where o = nu - m - 1
(42)
a zero order approximation, oo , to the root at the next p is obtained by extrapolation. Using this in (39) the continued fraction Fl(o0)is computed. Its
ELECTRONIC EIGENENERGIESOF THE HYDROGEN MOLECULAR ION
19
value is substituted in (41) which is now regarded as a linear equation in 0 the solution being the first order approximation =
01
{F"ll(.o)
- Bo)/Ao *
(43)
{91(0l) - Bo)/Ao.
(44)
A second order approximation is =
02
Combination of the information in (43) and (44) gives as a third order approximation 03
+
=02
(01
- ff2Y
2tT1 - 0 0
(45)
-02
The sequence is continued until self-consistency is attained. Convergence is rapid. The method just described is not suitable for the determination of the higher roots because the proximity of these to poles in 9 , ( c ) causes difficulties. However the difficulties associated with a particular pole may be avoided by inverting the continued fraction one or more times. It is seen from (39) that
- t(t -
+ m){02 + (2 + m - 2t)a + 1 + m + t ( t - m - 2)) 0 4 + Bt -
*
(46)
9 t + l
Using this once in (41) yields an equation which may be regarded as a quadratic in 0 in the same sense as the original equation is linear. The equation resulting from - 1 inversions is clearly of the form
c 'X,CU+F,1 t
1-
1
tYuOU=
u=o
u=o
0
(47)
in which 'Xuand Yu are coefficients. Consideration of the equation obtained if (46) is substituted in (47) leads to the recurrence relations t+
1
xu= - B, 'Xu- A, [Xu-
1
- t(t
+ rn) + + m - 2t)'Y,-, + tYu-2]
x [ ( t - I)(? - 1 - m)'YU (2
(48)
With the aid of (48) and (49) the coefficients may be built up on a computer from 'XI = - A o ,
'x, = - B , ,
'Yo = 1.
50)
D. R . Bates and R. H . G . Reid
20
After these have been found the procedure is much as before. For each of the successive approximations to a root an equation of degree t in CT is solved (in the case o f t - 1 inversions). This is done by Newton’s method. The convergence of the successive approximations is again rapid. We treated all states for which
n,
<5
where n, is the principal quantum number in the separated atoms limit. In presenting the results it is convenient to take as independent variable, not the internuclear distance R (atomic units), but rather a reduced internuclear distance
S = R/n,. A common interval and common range for the various states may then be adopted. It is also convenient to replace the electronic energy E (Rydberg units) by a reduced electronic energy of opposite sign
F = -En:.
(53) This is positive; has the value 4n,2/nu2when S is zero, nu being the principal quantum number in the united atom limit; and tends to unity as S tends to infinity. The results are listed in the appendix. They should be correct to within one unit in the last decimal place retained. States which, in the united atom limit, are specified by the same azimuthal quantum number I and magnetic quantum number m are grouped together. They are distinguished by the number I
= n,
-E
=N,
+ 1.
(54) Consider a pair of states which have the same Z and the same m and which have values of I given by 1,=m+2r,
12=m+2r+1
( r = 0 , 1 , 2 ,...).
(55)
It is clear from (21) and (22) that they are described by the same set of quantum numbers n, ,k( = r), and (of course) m in the separated atoms limit. These states are distinguished by their electronic parity which is even (gerade, 9) or odd (ungerade, u) according to whether 1 is even or odd. As R becomes large the difference between the electronic energies of such a g-u pair decreases exponentially. In contrast to this parity splitting the Stark splitting between states having the same n, and the same m but different k decreases as some inverse power of R in the asymptotic region (see Section 1V.A). The tabulation of the appendix extends into the region of large S where parity splitting vanishes to the number of decimal places carried. Furthermore
ELECTRONIC EIGENENERGIES OF THE HYDROGEN MOLECULAR ION
21
expansion (57) of Section 1V.A (up to and including R - 4 terms) allows the tabulation to be extended to still larger S with an accuracy of at least four decimal places. Thus it reproduces to this accuracy the exact values of F for the largest S for which results are presented except in the cases of the three g-u pairs of states indicated in Table 11. The difference AF between the exact values and the values obtained from the expansion may readily be extrapolated. TABLE I1 DIFFERENCE BETWEEN
THE
EXACTVALUES OF F A N D THE VALUES EXPANSION (57)
OBTAINED FROM
S (a.u.) 44
40
rn 0 1 1
I 8 or 9 7 or 8 1 or 2
48
52
lo5 x AF (Rydberg units)"
I 1 1 4
70 32 -22
41 19 - I4
26 12 -9
16 7 -6
AF = /=(exact)- /=(expansion 57).
IV. Expansions A.
LARGEINTERNUCLEAR DISTANCE
Coulson and Gillam (1947) have used pertur ,ation theory to a-tain an asymptotic expansion for the electronic energy of a hydrogen atom in the field of a proton. They expressed the unperturbed eigenfunction of the atom in parabolic coordinates with quantum numbers (nl, n, ,m) where m is as always and n , and n, are related to n, and k of Section I1 by2
n, = k,
n2 = n, - k - m - 1.
(56)
Expanding the energy in the form
with C, = l/n:
* There is no standard usage of the quantum numbers n, and n z in the literature, the definitions given in Eq.(56) being sometimes interchanged.
D.R. Bates and R. H . G. Reid
22
they found that
c, = 2
(59)
C2 = 3n, A
(60)
C,
= nS2{1 -
n:
n3 C , = 8 {n,[19
+ A[59
+ 6A2} + 17nS2- 36’
- 9m2]
- 39nS2+ 109A2 - 9m2]}
(62)
in which A=n,-n,.
(63)
Further terms of the series are known for the Iso, 2po, 2pn, and 3dn states (Coulson, 1941 ; Dalgarno and Stewart, 1957; Robinson, 1958). By consideringp to be large and solving Eqs. (5) and (6) iteratively, Ovchinnikov and Sukhanov (1965) have obtained expansions in powers of 1/R and exp( - R) for the wave functions and the electronic energies of the l s o and 2po states. They derived an expression for the g-u splitting: E(2pa) - E( Isrr) = 8R
(64)
Their method is applicable to higher states. For these, however, the convergence is poor and the coefficients of the expansions are cumbersome. Thus, if the numeric value of the electronic energy is required, it seems doubtful if the expansions have any practical advantage. Smirnov (1964) has obtained a simple general expression for the g-u splitting by adopting a large -R approximation to the wave functions based on work by Gershtein and Krivchenkov (1961) and substituting into the surface integral g-u splitting formula of Firsov (1951) and Gor’kov and Pitaevskii (1964). The accuracy is low in the region where the g-u splittingis appreciable. B.
SMALLINTERNUCLEARDISTANCE
By treating the change in the potential due to the separation of the nuclei as a perturbation on the united atom Bethe (1933) has shown that if R is small then 4{1(1+ 1) - 3m2}R2 nuI ( I 1)(21- 1)(21+ 1)(21+ 3)
+
+
...I.
(65)
It is not easy to extend the perturbation treatment. Brown and Steiner (1966) point out that the attempts which have been made to do so contain errors.
23
ELECTRONIC EIGENENERGIESOF THE HYDROGEN MOLECULAR ION
Turning to the separated equations (5) and (6), and using expansions (23) and (29) they showed that for the lsa state
4R2
8R3
176R4
+-32R5 log 2R + O(Rs)] . 9
(66)
The presence of a logarithmic term is interesting.
V. JWKB Approximation Gershtein et af (1965) have demonstrated that the JWKB approximation is quite accurate even for terms with low quantum numbers. Equation (5) is rather awkward to treat because the effective potential involved does not in general have the form of an ordinary well. Gershtein et a1 showed how to overcome the difficulty using the complex turning point method proposed by Pokrovskii and Khalatnikov (1961j. However the JWKB approximation does not here seem to have any advantage, at least as far as the eigenvalues are concerned, since the program for finding them by the scheme of Stratton et a1 (1956) is simple to write and is very effective for all m even when 1 is large. The position with regard to Eq. (6) is different. Thus considerable labor is entailed in determining further eigenvalues of R by the method described in Section 3; and results that are sufficiently precise for most purposes may easily be obtained using the JWKB approximation. There are a number of possible transformations. The one which we tested is given below. It differs from the one suggested by Gershtein et a1 which is not directly suited for combination with the exact solution of Eq. (5) as it involves a different separation constant (cf. Arnold and Bates, 1968). In order to make the range of the independent variable infinite instead of semi-infinite put (67)
A-l=eZ
(Langer, 1937); and in order to eliminate a term involving the first derivative Put A
= Z(l
+
(68)
Equation (6) then becomes d 2 Z / d z 2+ K(A)Z = 0
with
K(A) =
1
4(A
+ 1>* {4(A2 - 1 ) ( - p 2 L 2 + 2RA-
C) - (A - l)(A
+ 3) - 4mZ}. (70)
D . R. Bates and R. H . G . Reid
24
According to the JWKB approximation, the eigenvalues R satisfy the integral equation Y ( R )=
-
l:
[K(1
+ exp z ) ] ~ ”
dz
=(N,
+ +)n,
where N Ais the integer specified in (1 1) and where
1, and 1, being roots of K(1) = 0
(73)
1 < 11 < 1, < 00.
(74)
such that Given m,p, C, and R it is a straightforward task to find these roots and then to evaluate Y ( R ) of (71) by quadrature. The computer program which arranges this may readily be combined with the program which yields C for given 1, m, and p so that, having chosen n u , I, m, and R, the corresponding p , and hence energy E, may be determined by iteration. Comparison with the exact values shows that even when n, is as low as 5 the JWKB approximation to E is in error by less than 1 %.
ACKNOWLEDGMENTS This work has been supported by the Culham Laboratory of the United Kingdom Atomic Energy Authority, to whom thanks are due for permission to publish. We gratefully acknowledge grants of computer time by the Atlas Computing Laboratory of the United Kingdom Science Research Council. Most of the programming was carried out by Mrs. Norah Scott. One of us (R.H.G.R.) is indebted to the Ministry of Education, Northern Ireland, for financial support.
APPENDIX. REDUCEDELECTRONIC EIGENENERGJES F (RYDBERG U N I ) FOR SPECIFIED REDUCED INTERNUCLEAR DISTANCE S (a.u.)
1= 0
=. S 0 .0 0 0.25 0.50 0.7s
1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.50
4.00 4.50 5.00
5.50 6.00 6.50 7.00 8.00 9.00
10.00 12.00 14.00 16.00 ia.00 20.00 24.00 2a.uo 32.00 36.00 L0.00
44.00 L8.00 52.00
1 .
I= 1 4.00000 3.79711 3.469Y8 3.16477 2.90357 2.68359 2.49798 2.34029 2.20527 z.oaa79 1.90765 1.89931 1 .a2179 1.69311 1.59217 1.51232 1.440b4 1. w a i o 1.35727 1.32412 1.29690 1.25514 1.22461 1.20116 1.16700 1.14300 1.12507 1.11116 1.10003 1 .oa335 1.07164 1 .06250 1.05556 1.05000 1.01546 1.04167 1 .03846
1
m= 0
0
I. 2 1.00000 3.72059 3.38340 3. 1oaw 2.88692 2.70427 2.55110 2.42063 2.30812 2.21 01 1 2.12405 2.04799 I .98044 1.a6621 1.77422 1.69930 1 .63?69 1 .58645 1.54334 1.50661 1 . 47491 1 .62203 1.3a161 1 .-34799 1.29615 1.257a7 1.22039 1 .20497 1.18591 1.15677 1.13553 1.11935 1 .10663 1 .09616 1 .oa7a9 1 .oaa79 1.07475
I. 3 4.00000 3.60732 3;isaaa 3.12276 2.92672 2.76554 2.m80
I= 4 4.00000 3.67219 3.3741s 3.14967 2.97122 2.82431 2.70000
2i32339 2.24470 2.17451 2.11174 2.00411 ii91595 1 .a4275 Ii 7 a 1 2 2 1.72892 1 .6a38a 1.64462 1.b1003 1.55164 1.5Qk01 1 .L6423 1 .40132 1.35362 1.31615 1 .2as90 1.26096 1.22223 1 ;i9353 1.17110 1.15382 1.13951 1.I2766 1.11763 1.10907
2.51620 2.34237 2.27610 2.21651 2.11344 2;027U2 1 .955a0 1 .a9445 1 .BLISS I .79532 1 .?5452 1.nai4 1.65500 1 .6 0 6 0 7 1.56020 1 .4a990 1.43559 1.39231 I .35695 1.32751 1.20121 1.2464I 1.21935 1.19763 1 .179a4 1.I6499 1.15241 1.14161
I. 5 4.00000 5.66581 3.38592 3.17772 3.01262 z.am5 2.76018 2.65970 2.57142 2.49303 2.42206 2.35964 2.30239 2.20202 2 . 11932
2.0ca~a I .9a734 1.9341 5 1 .a8728 1 .a4553 1 .a0801 1 .74309 1 .baa60 1.64205 1.56635 1.50718 1 .45951 1 .42021 I ,38721 1.33iaz 1.29503 1.26373 1.25849 1.21767 1.20020 1.18533 1.17253
I= 1 i.00000 1.ooa3a 1.03377 1.07547 1.12963 1.18915 1.22636 1 29576 1 .3350? 1.36416 1 .3a414 1.39653 1 .402a4 1.40242 1.39114 1.37410 1. 3 w a 1 .33440 1.31462 1 .295a1 1.27826 1.24721 1.22131 1 .199a0 1.rb678 1.14296 1.12S07 1.1111s 1.10003 1. 0a33s 1.07144 1 .O6250 1 .05556 1.05000 1 .OL546 1 ,04147
1 .0JBi6
1. 2 1.7777a 1.81682 1.PI452 2.00327 2.Q4351 2.04496 2.02542 1.99546 1 .960U8 1 .92471 1. a a m 1 .a5335 1.81916 1.75641
I= 3 2.25000 2.3279a 2.46654 2.47037 2.43660 2.3aza1 2.32391 2.26545 2.20954 2.15689 2.10766 2.06175 2.01897 1.94189
9.69987
1.87466
1.64949 1 .bob69 1. 5 6 4 0 2 1.52928 1.49753 1 .L6909 1.42041 1.38067 1.34762 1.29610 1 .2578? 1 .22a3v 1.20497 1 .1a591 1.15677 1.13553 1.1193s 1.I0663 1.09636 1 .oa?a9 1.08079 1.07475
1.81573 I .76379 1.71778 (.ma2 1.64017 1.60724
1;5505b 1 .503S9 1. 46407 1.40129 1.35362 1.31615 1.2as90 1.26096 1.22223 1.19353 1.17140 1.153a2 1.13951 1.12764 1.11763 1.10907
I. 4 2.56000 2.67098 2.75005 2 . 7 1 197 2.64082 2.56~a7 2.49179 2.42361 2.36063 2.30~59 2.2490a 2.19965 2.15390 ~.07196 2.00079 1.93847 1 .caw 1.83467 1.79106 1.751aa 1 .?I 652 1.65519 1 .6030L 1.56020 1.40988 1.43S59 1.39231 I .35695 1.32751 1.20121 1.24644 1.21935 1.19763 1.179a4 1.16499 1.15241 1.14161
I. 5 2.7777a 2.90009 2.93223 2.057va 2.76955 2.60435 2.40576 2.53401 2.46855 2.40061 2.35312 2.3031 1 2.25634 2.17266 2.09994 2.0361 5 1.97913 1.92917 1. a a 4 ~ 2 1.84379 1 .a0696 1.74270 1 .baa16 1.64200 1 .56634 1.5ona 1.45951 1 .&PO21 1 ,18721 1.33402 1.29593 1.26375 1 .23a49 1.21767 1.20020 1 .?a533 1. 17253
h,
m
APPENDIX (continued)
15 3 m= 0
1= 2 m= 0 S
0.00 0.25 0.50 0.75 1 .00 1 .L5 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.50 4.00 4.50
5.00 5.50
I.
1
1.777711 1 .?a349 1 .0014a 1.83433
1.00622 1.96130 2.06003 2.17363 2.20579 2.30030 L.44010 2.48704
2.49996 2.4b737 2.38a09 2.28aw z.iwo4
2.00390
7.00 0.00 9.00 10.00 12.00 14.00 16.00 10.00 20.00 24.00 28.00 32.00 36.90
1.90930 1.90252 1.a2388 1.69002 1 .>a423 1.50213 1.39070 1.32252 1.27644 1.24251 1 zi6ia 1 17770 1 15088 1 13110 1 1 1 590
40.00
1 1 03a6
6.00 6.50
44.00 48.00
52.00
1 09409 1 00600 1 079 19
I. 2 2.25000 2.26243 2.50429 2.38704 2.51 199 2.64299 2.73632 2.70346 2.79520 2.70315 2.75503 2 .?I 61 7 2.6701 0 2.56574 2.45529 2.34630 2.24282 2.14674 2.05a74 1 .970111 1.90664 I. ?a369 1 .6a5a1 1.60065 1 .49a99
1 .4256S 1.37225 1.33109 I .29a22 1 .24a85 1.21351 1 .?a695 1.16627
1.14970 1.13613 1.12402 1 .?I524
.I 3 2.56000 2.58065 2.65510 2.79419 2.92002 2.99194 3.00059 2.97052 2.93991 2.09213 2.03921 2.7~34~ 2.72631 2.61156 2.50001 2.39145 2.29629 2.20605 2.12371 2.04095 1 .9a134 1.06566
.I
4
2.77770 2.80790 2.92064 3,O 7659 3.14702 3.1461 1 3.11 077 3.06056 3.00370 2.94407 2.88361 2.02335 2.76391 2.64006 2.54029 2.43910 .3m9 .26040 .I8242 ,11154 .O4731
.93690
1 .?7269 1.69026 1 .5a869
.04739 .77475
1.51146 1.b5301
.50407
1 .40683
1.36929 1.31101 1 .26904 1 .2~7a3 1.21260 1.19221 1.17539 1.16127 1.14925
.66497
,52205 .4?302
.43199 .36027 .32100 .20451 .25540 .ma3 .21210 .19560 .10143
.I 1 1 .OOOOO 1.00000
1.00320 1 .00720 1.01315 1.02099
1.03107 1 .OL377
1.05962 1.07927 1. ( O M 1.13295 1.16797 1.25200 1.34194 1 .42065 1 .47909 1.51667 1 .53670 1.56336 1.540CI
.I 2 1 .44000
I. 3 1.77770
1 .45945 1 .4?610
1 .a1507
1 .44207 1.64040
1.49994 1.53331 1 .5?062 1.63500 .69947 .75966
.00912 .04503
.88686
.av~a .09136
.07336 .04040
.01953 .?0042 .75647
1.51695 1 .b0226
.69330
1.37608 1.3’1070
1 .2756b
,49200 .42411 .37194
1.24235
.33103
1 .44468
1.21615
1.17770
1.15000 1.13110 1 .I1590 1.10306 I.09409 1.00600 1.07919
.63410 .50070
.29021 .24005 1i21351 1.10695
1.16627
1.14970
1.13613
1i12402 1.11S24
1.7a158
1.79344
1.05022 I .90449 1.97884 2.057116 2 .1 201 6 2.15945 2 .I0000 2.1 6765 2. 10626 2.16666 2.13111 2.09499 2.05275 2.00946 1.96639 1.92134 1 .a.3381 1 .sow 1.74113 1.68191 1.50490 1.51066
1.45286 1 .LO600 1 .36920 1.31101 1 .269a1 1 .23?03 1.21260 1.19221 1.17539 1.16127 1 .I4925
I.
4
2.04002 2.04669 2i06552 2.1021 2 2.16590 2.25676 2.34055 2.39053 2 . 41161 2.41451 2.40632 2.391 20 2.371 6 4 2.32492 2.27337 2.22046 2.16auz 2.11705 2.0601 3 2.02155 1.97747 1.0V6U0 1 .a2603
I ,76408
1 .66265
1.50442 1.52277 1. b7301 1.43199 1.36027 1.32100 1.20b5I 1 .25560 1.23103 1.2121a 1 .I9560 1.10143
P b a
31
A,
APPENDIX (continued)
. 1
=. S
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.50 1.00 4.50 5.00 5.50 6.00 6.50
7.00
0.00
9.00 10.00 12.00 11.00 16.00 10.0 0 20.00 24.00 28.00 32.00 36.00 10.0 0 41.00 48.00 52.00
4
1- 5
ma 0
0
I= 1 1.44000 1 .44094
1.46376 1.46053 1.45534 1.46432 1.47569 1.48972 1.50603 1 .5275? 1.55275
1.50340
1.62081 1.72055 1.85144 1 .99004 2.10704 2.10007 2.23230 2.24484 2.23240 2.15039 2.05250 1 .V4O50 1 .?3V06 1.58569 1 .4?066 1.40631 1.35529 1.28610 1 .LCOI3
1.206V3 1.10101 1.16215 1.11632 1.13331 1.12243
1. 2 1. 77178 1 779 19 1.?a470
1 .TY350 1 .a0650 1 .a2405 1 .017ZI
1 .a7715
1.91146
1 .96990 2.03650 2.11420 2.1931 4 2.31970 2.39332 2.42775 2.43551 2.42421 2,39067 2.36226 2.31 765 2.21204 2.10003 1.99020 1 .?9V62
1.65595
1 .55419 1. 10209 1 .42951 1.35316 1.30010 1.26107 1.23098 1.20711
1.18771
1.17163 1.15009
I. 3 2.04002 2.04346 2.05153 2;06554 2.00649 2.1 1632 2.1 5053 2.21 029 2.29032 2.30051 2 46042 2.52704 2.56566 2.60127 2.60303 2.50496 2.55423 2.51 490 2. 1 6 9 4 4 2.11949 2,36631 2.25464 2.1 4267 2.03673 1 .a5532 1 .?I073 1.62079 1 .34921 1.49591 1.01220 1.15395 1.31010 1.27604 1.24060 3.22624 1 .20751 1 ,19165
I= 1 1.00000 1.00029 1.00117 1.00265 1 .OOb71 1 00740 1.01071 1.01475 1 .0194? 1.02494 1.031Z2 1 .03038 1 .Oh651 1 .06615 1.09144 1.I2140 1.16020 1.22520 1.29129 1 .36a39 1. 4 3 m 1 .5C352 1.59014 1.61L05 1.50228 1 .51944 1.15521 1.39931 1 .35347 1.206U9 1 .24013 1 .20693
:
1.10101
1.16215 1,14632 1.13331 1.12243
I= 2 1.30612 1.30670 1.30015 1.311LO 1.31559 1.321 00 1.32797 1.3363V 1 .34651 1. E m 0 1.37294 1.39007 1.11065 1 .L659S 1 .564?1 1.63630 1 .?l 556 1 .??23S 1 .a0907 1 .a32811 1 .0L406 1.01401 1 .a2300 1 .?a046 1 .7030L 1.63715 1.51101 1.17910 1 . 12865 1.35312 1.30010 1.26107 1.23098 1 .2071 1
1.10771
1.17163
1.1saov
1. 3 1.56250 1.56345 1.56632 1.57119
1.57017 1.50745
1 .59933 1.61 126 1 .63200 1.65630 1.60616 1 .?2441 1 .7?220
1.08179
1.96759 2.01705 2.04393 2.05141 2.05422 2.04635 2.03200 1 .994t1
1 .94600 1 .a9376
1 .?a056 1.69363 1.61390 1.54722 1.49345 1.11210 1.35395 1.31010 1.27604 1.21060 1.22621 1.20751 1.19165
N 00
APPENDIX (continued)
S
0.00
0.25 0.50
I. 1 1.30612 1 .30647 1.30752
2 1.56250 1.
1.56307 1 .56470 1.56766 1.5717L 1.57707 1 583i1 1.59175 1.60130 1.61253 1.62561 1.64003 1.65053
i
2.50 2.75 3.00 3.50 4.00 4.50 5.00
5.50 6.00 6.50 7.00 0.00 9.00 10.00 12.00 14.00 16.00 18.00 20.00 24.00 24.00 32.00 36.00
40.00 44.00 48.00 52.00
1.34300 1.35146 1.36OYI 1.30367 1.41194 1.44020 1.49534 1.55784 I .63927 1 .73641 1 .a3713 2.00362 2.09756 2.12739 2.06140 1.91029 1 .7607V 1.64096
1.54170 1.41573 1 .34209 1.29165 1.25435 1.22557 1 .LO266 1.10399 1.16817
1 . 0 m= 0
1 . 7 m- 0
11 6
m= 0
1 .70369 1.76795 1 .05PC3
1.96991 2.0699 4 2.14400 2.19716 2.2329 1 2.26360 2.25789 2.22539 2.10761 1.95751 1.01287 1.69155 1 .59773 1 .47494 1.39793 1.34311 1.35169 1 .26V23 1.243O0 1.221 5 6 1.20355
1- 1 1.OOOOO 1.00015 1.00060 1.00136 1.00243 1.00300 1.00569 1.00751 1.00985 1.01254 1.01558 1.01099 1.02270 1.03157 1.06214 1.05473 1.06967 1.08742 1.10066 1.13441 1 .16619 1.25516 1.37550 1.49244 1.62951 1.65501 1.62270 1.56863 1.51102 1.41239 1.34105 1.29164 1.25435 1.22557 1 .20266 1.18399 1. 16 8 4 7
1. 2 1.23657 1 .23603 1 .23561 1.23691 1 23874 1.24112 I .24605 1 .24756 1.25170 1.25645 1.261U6 1.26000 1.27490 1;2912a 1 .31175 1.33743 1.37039 1.11b23 1.47256 1.54339 1.61127 1 . I 2 1 05 1 .703V3
:
1 .a1665 1.02176 1 ;77046 1 .71 361 1.6460U I .5780? 1.47314 1.39782 1.34311 1.30169 1.26923 1.24300 1.22151 1.20355
I. 1 1 .23457 1 .23674 1. 23520 1.23616 1 23741 1.23902 1.24101 1 .24337 1.24611 1.24926 1.25201 1 .25670 1.26120 1.27144 1.20370 1.29825
:
1.31540
1 ,33563 1.35955 1 .38815 1 .62287 1 .51V90 1 .6632V 1.01042 2.01555 2.05821 2.0061 7 1.90420 1 .70025 1.50953 1.46621 1.30807 1 .33506 1. 2 W 2 1.26378 1 .23851 1.21767
1 .
m.
9
0
1 1 .ooooo 1 .00009 1.00037 1.00003 1. 0 1140 1. 0 1231 1. 0 1334 1. a 1656 1 . 0 1597 1. 0 I750 1 . 0 1940 1. 0 1 4 1 1. 0 364 0 875 0 !481 a I1 07 .. .03994 .04921 .05902 .07192 .00552 .11951 .16511 .23014 .43036 .59926 .67320 .68039 .65040 -55221 .45078 .38759 .33503 -29512 1 .26378 1.23051 1.21767
I.
b
# bY
f? 2 a
2.
APPENDIX (continued)
1.
1
1. 2 m= 1
m= 1 S 0.00 0.25 0.50 0.75 1.0 0 1.25 1.50 1 .75 2.00 2.25 2.50 2.75 3;OO 3.50
4.00 4.50 5.00
5.50 6.00
6;50 7.00 0.00 9.00 1u.00 12.00 14.00 16.00 18.00 20.00 24.00 28.00 32.00 36.00 40.00 44.00 48.00 52.00
I= 1 4.00000 3.93875 3.79206 3.61381 3.43017 3.25453 3.09155 2.94237 2.00660 2.68322 2.57108 2.46901 2.37593 2.21305 2.07609 1.96020 1.86173 1 .???a4 1 .?0627 1.64519 1.59301 1.51002 1.44~15 1.40067 1,33227 1 .2a463 1.24914 1.22156 1 .IV949 1.I6635 1.14265 1.12485
1.11101 1.09992 1.09085 1 .oa329 1 .or689
I. 2 4.00000 3.91536 3.74166 3.55317 3.37421 3.21001 3.06333 2.93040 2 . a1 059 2.70208 2.60352 2.51367 2.43152 2.28691 2.1641s 2.05922 1 .V6912 1 .a9152 1 .a2417 1 .?6660 1 .?I649 1 .63449 1 .5?067 1.51949 1 .44175 1 .3a4a3 1.34105 1.30626 1. 27792 1.23454 1.20287 1.17873 1 .I5912 1.14437 1.13170 1.12108 1.11201
I- 3 r.OOOOO 3.89655 3.70929 3;52214 3.35199 3.20020 3.06490 2.94377 2.83475 2.73607 2.64633 2.56435 2 . 46917 2.35617 2.26243 2.14446 2.05967 1.98605 1.92197 1 .a6602 1.81696 1 .73529 1 .66999 1.61631 1.53242 1 .46925 1 ,61970 1 .37971 1 .34673 1 ,29546 1.25746 1 .22ao9 1.20476 1.la575 1.I6990 1.I5667 1 .14530
1. 4 4.00000 3.80152 3.68843 3.50619 3.34107 3.20283 3.07100 2.96466 2.86359 2.77205 2.68866 2.61232 2.54214 2.41748 2.31025 2.21733 2.13641 2.06570 2.00372 1.94919 1.90095 1 .a1956 1 .7532a 1.69791 1 .609?9 1.54219 1 .baa43 1 .44454 1 .4079a 1.35050 1.38730 1.27361 1.24660 1 .22446 1.20596 1.t9029 1 .1 7683
I= 1 1 . 77778 1 .7ao56 I.?a845 1.80012 1.81360 1 .a2669 1 .a3740 1 .0446a 1 .a4763 1 .a4627 1 .a4091 1 .a3206 1 .a2031 1 .?9039 1.75513 1 .?I746 1.67933 1.64200 1.60620 1 .ST237 1 .540?0 1.ha404 1.13501 1.39501 1.33116 1 .2a443 1.2491 0 1.22156 1.19949 1.16635 1 .1 4265 1.124a5 1.11101 1.09992 1 .090as 1.00329 1. 0 7 m
I= 2 2. 25000 2.25577 2.27009 2.28554 2.29461 2.29348 2.2021 5 2.26250 2.23710 2.20774 2.17605 2. 1 4315 2. I0982 2.04393 1.911092 1.92188 I .a671 7 I .a1680 1.77059 1 .7202a 1 .6a957 1.62177 1.56491 1.51696 1.44129 I .3a475 1.34104 1.30626 1.27792 1.23454 1.20287 I .17a73 1.15972 1.15437 1.13170 1.12108 1.11204
I= 3 2.5600~
1. 4 2.77778
2.4b501 2.40060 2.35639 2,31309 2.27110 2.19176 2.11890 2.05263 1 .99225 1.93735 1 .a8739 1 .a41a9 1 .a0039 1.72778 1.66672 1.61692 1.53210 1.46921 1.41969 1.37971 1.34673 9.29546 1.25744 1 .22ao9 1.20476 1 .la575 1.16998 1.15661 t.ir530
2.57599 2.52428 2.47435 2.42650 2.38083 2.29601 2.219~0 2.15020 2.08761 2.03009 1 .9793a 1 .93247 1 .a8967 1.81463 1.75120 1 .69705 1.60965 1 .5&21? 1.4~8~2 1.44454
1 .LO798 1.35050 1.30730 1 .27361 1 .26660 I .22446 9.20596 1.19029 1.17683
5-
P
APPENDIX (continued)
S
r 0.25 0.50 0.75 1 .uo 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.50 4.00 1.50 5.00 5.50 6.00 6.50 7.00 8.00 9;OO 10.00 12.00 14.00 16.00
10.00 20.00 24.00 28.00 32.00 36.00 40.00 44.00 48.00 52.00
I= 1 2.25000 2.25302 2.26211 2.27734 2.29069 2.32376 2.35762 2.39259 2.42839 2.46249 2.49251 2.51664
2.53371 2.5C509 2.527b9 2;4a645 2.42762 2.35744 2.28131 2.203L4 2.12671 1.9a321 1 .US761 1.75099 1.59005 1 .4a430 1.41330 1.36223 1.32293 1 .2656a 1.22574 1 .I9624 1.17357 1.15550 i.ic09a 1.12080 1.11869
I* 2 2.56000 2.56caa 2.57956 2.60376 2.63593 2.67211 2.707711 2.73670 2.75622 2.76523 2.76452 2.75540 2.73959 2.69233 2.63074 2;56007 2.4a400 2.40543 2.32612 2.24975 2.17504 2.04010 1.92232 1 .a2240 1.67047 1 .56753 1.49473 1 .b3969 1 .39606 1.33070 1.28391 1 .24071 1.22128 1.19929 1.ta127 1.16623 1.15350
I= 3 2.77770 2.7a467 2.00525 2.a3777 2.07616 2.91057 2.93286 2.94041 2.93475 2.91 a70 2.09 527 2.86640 2.833i-a 2.76146 2.6a395 2.6041a 2.52392 2.44451 2.3671 1 2.2926a 2.22191 2.092a2 1 ,pa106 1 .a8608 1. 7 4 0 4 4 1.63902 1 .36463 1 . SO675 1 .15997 1.30053 1.33639 1.29660 1.26523 1. m a 5 1 .2’1a91 1.20132 1 .1a635
.
-1 9
i
1 44000 1 .44oao 1 .44319 1 .44721 1 .45291 1 46053 1 .16955 1 .baot.s 1 -49369 1 .sua70 1 .52565 1 .54141 1 .56473 1 .6oa34 1 .6520? 1 .69107 1 .7217a 1 .?4253 1 .75335 1. m 2 a 1 .74979 1 .72275 1 .60290 1 -63800 1 .54aao 1 47190 1 .41024 1 .36151 1 .32277 1 .26567 1 .22574 1 .19624 1 ,17357 1 .15550 1 1 409a 1 1 2aaa 1 1 1 a69
.
.
.. .
I= 2 1 .????a 1 .??923 1 .?a363 1 .?PI05 1.80161 1.81541 1 .a3264 1 .05309 I.a7634 1.90141 1iV2716 i .951a3 1.97407 2.0077a 2.02565 2.02950 2.02237 2.00707 I .9a5a8 1.96057 1.93252 1 .a7220 1.01078 I .?516b 1 .6463a 1.56090 1.49315 1 .43935 1 .39599 1.33OTO 1.28391 1.21a71 1.22128 1.19929 1.10127 1.16623 1.15350
19
3
2.01002 2.04306 2.049a4 2.06131 2.07778 2.09920 2.1 2506 2.1 5375 2. 18260 2.20~163 2.22956 2.24442 2.25323 2.25543 2.24237 2.21930 2.109a1 2.15633 2. 12069 2.00313 2.0~59a 1.97210 1 .90192 I.a3690 1.72476 1.63501 1 .563?3 1.50657 1 .45994 1.38813 1.33639 1.29660 1.26523 1.23905 1.21091 1.20132 1.1a635
b
P
B
2 8
APPENDIX (continued)
1. 5 m= 1 S 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
1 . m 7 n .nab1 . ? 8 11 0 ,78530 .79122 1-
.?9896
.a0857 .a2017 .n3~ta .84984 .86818
4;oo 4.50 5.00 5.50 6.00 6.50 7 .0 0
a.oo
9.00 1 0 .0 0 12.00 14.00 16.00 18.00 20.00 24.00 28.00 32.00 36.00
40.00 44.00 48.00 52.00
2.02943 2 . 0 9 496 2.156e7 2.20853 2.24696 2.27084 2.28067 2.26377 2 . 2 0 9 23 2 . 1 3047 1 .V4765 1 ,778no I .66404 1.54469 1.47367 1.38025 1.31914 1.27517 1.z4in8
1.21579 1.I9479 1.I7751 1.16305
1. PI=
1. 2 2.04oa2 2.04209 2.04594 2.05243 2.061 65 2.07378 2.08899 2.10751 2.12956 2.15517 2.18425 2.21 620 2.24993 2.31655 2.37229 2 . 4 1 110 2.~32a8 2.44007 2.43542 2.42127 2.3994a 2.33842 2.26051 2.17229
I. 1 .30612 .30645 .30742 .30904 .31133 .31430 .31796 .32233 .32745 .33334 .34005 .34761 ,35607 .37593
1.99005
.72523 .66337 .59223 .52537 .46764 1.37984 1.31912 1.27517 1. 2 4 1 8 a 1.21579 1.I9679 1.17751 1. 1630 5
I.a2864
1 ,700VO 1.60617 1.53654 1.4L007 1.37410 1.32518 I. 2 8 a 0 ~ 'I . 2 5 a 3 9 1.23625 1 .21422 1.19735
1 .
6 1
.40011 .42907 .46303 .50154 .54324 ,58579 .626ca .69343 .73426
.74957
I= 2 1.56250 1.56303 1.56462 1.56729 1.57106 1.57597 I. 5 a 2 0 6 1.5av3v 1.59104 1 .6uao9 1.61964 1. m a 0 1.64767 I. 6 a 2 a 3 I.7218a 1.??I35 1.8174a 1.a5838 1. a 9 1 17 1.91519 1.93102 1.94211 1 .V3232 1 .9oao5 I. a w v 1.74935 1.66703 1.59642 1.53311 1 .43986 1.37LOP 1. 3 2 5 4 a 1.20aoa 1.25a39 1.23C25 1.21 L22 1 .19735
7 1
r.
' 1 .5 6 2 5 0 .5 6 2 0 5
. 5 ,6390 . 5 '6565 .5 ,6612 .5 ,7131 . 5,7525 .5'7995 .5 , 0 5 4 1 . 519176 .5 '9094 .6 , 0 7 0 2 .681606 .6 b3723 - 616301 .6b9407 . I ' 3 1 11 . l'7462 .B I2433 . 8 17869 1,93471 2.03790 2.11223 2.15172 2.1 4275 2.05450 1 .V2775 1.79743 1 .&a301 1.52119 1 .L2656 1i 3 6 3 5 0 1.31712 1.2a133 1,25281 1.22960 1.21028
1-
m.
a 1
I. 1 1.23657 1.23474 1.23525 1.23610 1.23729 1.23aa4 1.2C073 1.26299 1.24561 1 .21a61 1.25200 1.2557a I .25vva 1.26967 1.20123 1 . 2 ~ ~ ~ 4 1.31074 1.32922 1.35061 1.31529 1 .C0350 1.4710L 1.54619
1. m a t . 1.72493 I,75189 1. 7 3 ( 6 6 1- 6 8 2 4 7 1.62349 1.51152 1.42569 1 .363C4 1.31711 1.za133 1.25204 1.22960 1.21028
APPENDlX (continued) 1. 2
1. 3 2
=.
m= 2
S 0.00
0;25 0.50 0.75 1 .oo 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.50 4.00 4.50 5.00 5.50
6.00 6.50 7.00 8.00 9.00 10.00 12.00 14.00
16.00
10.00 20.00 24.00 2a.00 32.00 36.00 40.00
44.00
*a. 00 52.00
I. 1 4.00000 3.97227 3.89716 3.79154 3.66994 3.54259 3.41 563 3.29248 3.17494 3.06376 2 .V5918 2.861 09 2.76924 2.60278 2.45681 2.32847 2.21530 2.11 524 2.02660 1 ,94800 1 .a7828 1.76173 1.67039 1.59864 1.49523 1.42413 1.37151 1.33067 1.29795 1 .24a72 1.21344 1.I8691 1 .1 6624 1.14960 1.13612 1.12481 1.11523
I. 2 4.00000 3.96391 3.87262 3.75376 3.62565 3.49717 3.37455 3.25784 3.14~24 3.04574 2.95003 2.86067 2.77721 2.62613 2.49344 2.37635 2.27260 2.1~031 2,091321 2.02493 1.95956 1 .a4928 1.76113 1.69116 1.58594 1.51021 1 .45235 1 ,40643 1 .36902 1.31160 1.26977 I .2377a 1.21257 1.I9219 1 .17537 1.16126 1.I4924
I8 3 4.00000 3i95613 3.85242 3.72619 3.59651 3. 17114 3.35286 3.24240 3.13961 3.04405 2.95514 2 .a7230 2.79499 2.65502 2.531 a2 2. 12275 2.32572 2.23908 2.161 56 2.09211 2.02987 1.92407 1 ;a3883 1 .76935 1 .66280 1. 5 m 3 1.52225 1 47264 1.43173 1.36814 1.32093 1 .2a447 1.25545 1.23181 1.21211 1.I9559 1.18142
1: 1 2.25000 2.24999 2.24978 2.24894 2.24686 2.24292 2.23663 2.22770 2.21604 2.20176 2. 1a51 o 2 . 1 6636 2.14590 2.10118 2.05341 2.00454 1.95594 I. 9 o a 5 5 I.a6297 1,81956 1 .77852 1 .?03?6 1.63853 1. 5 m 5 1.49126 I .4232a 1.37134 1 33064 1.29794 1. 24872 1.I1344 1.10691 1.16624 1.14968 1.13612 1.12401 1.11523
I= 2 2.sbooo 2.55995 2.5591 7 2.55620 2.SLQ42 2.53780 2.521 10 2.19974 2.47651 2.44628 2.11591 2.3841 o 2.35146 2.28542 2.22033 2.15164 2.09807 2 . 0 4 1 96 1 .PI)PLO 1 .Pb034
1.89466 1 .a1270 1 .?6225 1.68153 1 .5a3a1 1.5097a I .45227 1 .40641 1.36901 1.31168 1.26977 1 .2377a 1.21257 1.19219 1.17537 1.16126 1.14924
I. 3 2.77778 2.77764 2.77580 2.7691 a 2.75542 2.73406 2.70613 2.67324 2.63699 2.59069 2.55936 2.51971 2.48027 2.40331 2.33013 2.26140 2.19726 2.13764 2.08232 2 . 0 3 1 05 1 .9a354 1 .a9876 1.82590 1 .76312
1.66151
1 -58359 1 .s2221 1.57263 1 .43173 1.36816 1 .32093 1 .2a447 1.25545 1 .231a1 1.21217 1.19559 1.1~142
P ?cl
Brl
APPENDIX (continued) 1; 4 m= 2
S 0.00
0.25 0.50
0.75 1 .uo 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.50 4.00
4.50 5.00 5.50
6.00
6.50 7.00 0.00 9.00
10.00 12.00 14.00 16.00
18.00
20.00 24.00 28.00 32.00 36. aa 10.00 64.00 40.00
52.00
1 2.56000 2.56118 2.56467 2.57035 2.51795 2.58712 2.59732 2.60791 2.61017 2.62735 2.63476 2.63984 2,64210 2.63776 2.62107 2.59307 2.55530 2.50939 2.45693 2.399L0 2.33061 2.21273 2.009 01 1 .??632 I. 78693 1 .64792 1.55049 1.40100 1.62052 1.35277 1. 29999 1.26096 1.23092 1.20707 1.10760 1.17161 1.15007 1:
1. 5 m= 2
2 2.77770 2.77944 2.70420 2.79187 2.0U162 2.01 174 2.021 39 2.02094 2.03319 2.03341 2.02931 2.82090 2.00077 2.77420 2.73022 2.67901 2.62300 2.56391 2.50255 2.4390? 2.37656 2.25130 2.13275 2.02461 1.064Y4 1.71250 1.61773 1.
1.54769
1.49306 1.41103 1.35376 1.31006 3.27597 1.24063 1.22621 1.20749 1.191b3
19
1
1.77770 1.77033 1.77999 1 .?a274 1 .?a660 1.791 53 1.79752 1 .a0453 1.81252 1.82130 1.03102 1.04129
1 .a5199
1 .a7382
1 .a9456 1 .91220 1.92546 1 .93321 1 .9352C 1.93177 1.92329 1 .a9412 1 .a5350 I.a0655 1 ,70020 1.61767 1.54099 1 .&?a45 1 .42790
1 ,35274
1.29999 1.26096 1.23092 1 .20707 1.1876a 1.17161 1 .1 5007
1- 6 m= 2
2 2. 04002 2.04167 2.0~~21 2.04042 2.054~7 2.0616~1 2.07051 2.00058 2.091 59 2.10315 2.1 1 679 2.12600 2.13628 2.15243 2.361 05 2.16169 2 .I5499 2.14210 2.12621 2.1 0 2 4 4 2.07772 2.02240 1 .96335 1 .90366 I.
1.79074
1.69292 1.61 201 1.54625 1 .49273 1.41181 1 .35376 1.31 0 0 6 1 .27597 1.24863 1.22621 1 .2a?iv 1.19163
I= 1
2.04002 2.041 42 2.04325 2.04629 2.05056 2.05607 2.06205 2.07005 2.0001 3 2.09067 2.10245 2.11543 2 .1 2954 2 . 1 6060 2.19440 2.22003 2.261 26 2.28939 2.31144 2.32637 2.33306 2.32763 2.29703 2.24719 2.10060 1.95041 I ,00400 1.60537 1.59394 1 .47300 1.39747 1.34287 1 .30155 1.26913 1 .26302 1.22152 1.20352
1 s
.=
7 2
I= 1 1 .56250 1.56279 1 56366 1.56591 1.56715 1.56970
1.57300
1 .5760b
1.50129 1.50637 1 .59209 1.59846 1.60549 1.62156 1 .64032 1.661 69
1 .68537 1.71a02 1.73717 1 .76335 1 .70017 1 .a2966 1 .a5607 1 .a6614 1 .a4496
1 .7a026 1.71 601
1.64436 1.57813 1 .47236 1.39738 1.34206 1.30155
1.26913 1.24302 1.22152 1 .20352
8 z W W
APPENDIX (continued)
1s 3 m= 3 S
0.ou 0.25 0.50 0;75 1 .vo 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.50 4.00 4;50 5.00 5.50 6.00 6.50
7.00
8.00
9.00 1v.00 12.00 14.00 16.00
18.00
20.00 24.00 28.00 32.00 36.00 40.00 44.00 48.00 52.00
I= 1 4.VV000 3.98436 3.94002 3.~17314 3.79078 3.69907 3.60270 3.50493 3.40797 3.31 322 3 . 2 2 1 52 3.13335 3.04896 2 . n v i 6s 2.74914 2.62030 2.50378 2.39830 2.30268 2 . 2 1 588 2.13702 2.00017 1.a8718 1 .79456 1.65649 1.56146 1.491n9 1.43814 1.39510 1.33025 1 .2n367 1.24857 1.22119 1.19923 1.18123 1.16620 1.15348
I= 4
1 .
3
m-
I=
I= 2 4.00000 3.98064 3.92147 3.85095 3.76099 3.66475 3.56681 3.46990 3.37561
J.ZB~TB
3.19784 3.11492 3.03601 2 .a8976 2.75778 2.63855 2.53063 2.43273 2.34372 2.26266 2.18876 2.05985 1.95276 1 ,86407 1.72975 1.63429 1 .56226 1.5053? 1 .C5900 1.38809 1.33616 1.29645 1.26513 1 .239?9 1.21886 1.20129 1 . 1 ~ 3 3
I= 1 I. 2 2.56000 2.77778 2.55896 2.77631 2.55581 2.77177 2.76390 2.55046 2.54219 2.75241 2.53271 2.73717 2,71829 2.52019 2.50526 2.69607 2 . 4 ~ ~ 0 6 2.67098 2.64353 2.56878 2.44764 2.61 423 2.58358 2.42193 2.40090 2.55201 2.48746 2.34995 2,42279 2.29669 2.24264 2.35940 2 . 1 ~ 3 2.29815 2 . 1 3634 2.23953 2 . I 8378 2.08512 2.1 3099 2.03652 2.08117 1.98983 1.90346 1.9901 3 1.82641 1 .90983 1 .a3921 1 .?582L 1.64553 1 .?22?6 1.55869 1.63263 1.56190 1.49126 1.50530 1.43801 1.45906 1.39508 1.33025 I .3aa09 1.33614 1.28367 I.24a57 1.29645 1.22119 1.26513 1.23979 1 .I9923 1 .1a123 1.21aa6 1.20129 1 .I6620 1.15348 1 .jab33
5 3
I. 1 2.71778 2,77800 2.77864 2.77960 2.78075 2.78180 2.78274 2.78308 2.78260 2.78103 2.77814 2.77374 2.76768 2.75037 2.72624 2.69589 2.66021 2.6201 4 2.57652 2.53002 2 . 4 8 1 1a 2.37828 2 . 2 7 1 51 2.16527 1.96999 1,81039 I .6a883 1.59970 1.53321 I, 4 3 8 8 ~ 1,37352 1.32516 1 .2a?aa I. 2 5 a 2 6 1.23616 1.21417 1.19731
1= 6 m= 3 I. 1 2.04082 2.041 1 2 2.04202 2i04352 2.04559 2.04820 2;05131 2.05487 2 . 0 ~ 8 ~ 2 2.06309 2.06'157 2.0721 a 2.07679 2.oa5s5 2.09287 2;097ac 2.09979 2.09828 2,09317 2.08453 2.0ma 2.0401 6 1.9989a 1.95209 1.a5139 1 -75349 1.66632 1.592L1 1.53118 1. i 3 8 ? 6 1 * 37351 1.32516 1.28788 I .25a26 1.23616 1.21416 1.19731
1 .
.m
4 4
I. 1 4.00000 3.98999 3.96099 3.91569 3.85759 3.19025 3.71675 3.63962 3.56078 3.48164 3. 4 0 3 2 2 3.32623 3.2511 3 3.10773 2.9741 8 ziasorb 2.73649 2.631 34 2.53440 2.44499 2.36248 2 . 2 1 5 ~ 2.09060 1.98376 1.81637 1.69725
1.61040
1,5131)~ 1.490a0 1.41080 1.35321 1.30974 1 .2?5?7 1.26850 1.22612 1 .20?43 1.19158
1- 5 4
m:
I= 1 2.77778 2.77643 2.77239 2.76568 2.75633 2.74443 2.71 3 0345 69
2 . 69 471 2.67408 2.65179 2.62809 2.60320 2.55077 2.49611 2i44054 2.38505 2.33039 2.27708 2.22546 2.0 18 72 56 74 7
1.99806 1.92185 1 .?9254 1.68985 1 .boa45 1.54312 1.49069 1.41079 1.35321 9.30974 1.27577 1.2b850 1.22612 1.20743 1.19158
P ?J
B2
ELECTRONIC EIGENENERGIES OF THE HYDROGEN MOLECULAR ION
35
REFERENCES Arnold, J. and Bates, D. R. (1968). In preparation. Baber, W. G., and Hasse, H. R. (1935). Proc. Cambridge Phil. SOC.31, 564. Bates, D. R., Ledsham, K., and Stewart, A. L. (1953). Phil. Trans. Roy. SOC.London A246,215. Bethe, H. (1933). In “Handbuch der Physik” (S. Fliigge, ed.), Vol. 24, part 1, p. 527. Springer, Berlin. Brown, W. Byers, and Steiner, E. (1966). J. Chem. Phys. 44, 3934. Burrau, 0. (1927). Kgl. Danske. Videnskab. Selskab 7, No. 14. Coulson, C. A. (1941). Proc. Roy. SOC.Edinburgh A61, 20. Coulson, C. A., and Gillam, C. M. (1947). Proc. Roy. SOC.Edinburgh A62, 362. Dalgarno, A., and Stewart, A. L. (1957). Proc. Roy. SOC.(London) A238, 276; 240, 274. Firsov, 0. B. (1951). Zh. Experim. i Teor. Fiz. 21, 1001. Flammer, C. (1957). “Spheroidal Wave Functions.” Stanford Univ. Press, Stanford, California. Gershtein, S. S., and Krivchenkov, V. D. (1961). Soviet Phys. JETP 13, 1044. Gershtein, S. S., Ponornarev, L. I., and Puzynina, T. P. (1965). Soviet Phys. JETP 21, 418. Gor’kov, L. P., and Pitaeyskii, L. P. (1964). Soviet Phys. “ Dokludy” 8, 788. Hylleraas, E. A. (1931). Z. Physik. 71, 739. Jaffe, G. (1934). Z. Physik. 87, 535. Langer, R. (1937). Phys. Rev. 51, 669. Morse, P. M., and Feshbach, H. (1953). “Methods of Mathematical Physics,” p. 1503. McGraw-Hill, New York. Morse, P. M., and Stueckelberg, E. C. G. (1929). Phys. Rev. 33,932. Ovchinnikov, A. A., and Sukhanov, A. D. (1965). Soviet Phys. “ Doklady” 9, 685. Peek, J. M. (1965). J. Chem. Phys. 43, 3004. Pokrovskii, V. L., and Khalatnikov, I. M. (1961). Soviet Phys. JETP 13, 1207. Robinson, P. D. (1958). Proc. Phys. SOC.71, 828. Srnirnov, B. M. (1964). Soviet Phys. JETP 19, 692. Stratton, J. A., Morse, P. M., Chu, L. J., Little, J. D. C., and Corbat6, F. (1956). “Spheroidal Wave Functions.” Wiley, New York. Teller, E. (1930). Z. Physik. 61, 458. Wallis, R. F., and Hulbert, H. M. (1954). J. Chem. Phys. 22, 774. Wilson, A. H. (1928). Proc. Roy. SOC.(London) A118, 517 and 635. Wind, H. (1965). J. Chem. Phys. 42, 2371.
This Page Intentionally Left Blank
APPLICATIONS OF QUANTUM THEORY TO THE vrscosm OF DILUTE GASES R. A . BUCKINGHAM and E. GAL Institute of Computer Science, University of London London, England
I. Introduction
......................................................
37
11. The Transition from Classical to Quantal Mechanics. ...................38
111. Reduced Variables and Law of Corresponding States ..................39 .43 IV. General Quantal Effects at Low Temperatures ....................... V. Special Cases. .................................................... .47 A. The Helium Isotopes ........................................... .47 B. Ortho- and Para-Hydrogen ..................................... .50 C. Atomic Hydrogen .............................................. 54 References ....................................................... .60
I. Introduction The practical evaluation of transport properties for dilute gases at temperatures such that quantal effects become significant owes its origin to a happy combination of the statistical theory of transport properties developed by Chapman and Enskog and the quantal theory of atomic collisions developed by Mott and Massey. In 1933 two papers appeared which effectively brought these theories together. The first, by Uehling and Uhlenbeck (1933), introduced a quantum mechanical form for the Boltzmann equations for dilute gases; the other, by Massey and Mohr (1933), evaluated the quantal collision cross sections and collision integrals for a gas of rigid spheres from suitably symmetrized wave functions. From these results they were able to derive the temperature variation of the viscosity of hydrogen and helium. Since 1933 an extensive series of calculations has been carried out for the transport coefficients of dilute gases in which the molecular interaction can be adequately represented by a spherically symmetrical potential. The theoretical basis has been presented by Hirschfelder et al. (1954), together with results of the earlier calculations. Some more recent results are summarized in this article. 37
38
R. A . Buckingham and E. Gal
II. The Transition from Classical to Quanta1 Mechanics In classical mechanics the derivation of transport properties which depend only on binary collisions between the molecules requires the evaluation of collision cross sections : Q(")(g)= 271 lom(l- cos" x)b d b
where ~ ( bg), is the angular deflection which results from the collision of two molecules with relative velocity g and impact parameter b. If V ( r )is the potential energy for the internuclear separation r, and p the reduced mass of the system, then dr -
.'
f m
where a is the nearest distance of approach according to classical mechanics. A relatively simple change is required to make the same formulation acceptable in quantum mechanics. This is to write (1 - cos" x)cr(g, x) sin x dx
(3)
where a(g, 1)is now the quanta1 probability of the deflection x occurring in a binary collision. The cross sections Q'")can now be expressed in terms of the usual asymptotic phase shifts 6,(k) associated with the solution of the equation
where k = pg/h. The application of standard scattering theory with properly symmetrized wave functions leads to the following particular results : Q'"(k) = Q'"(k)
=
c wL')(L + 1) sin2(6,+ - 6,) 471 c wL" ( L +2L1)(L+ 3+ 2) sin2(6,+, - 6,) k~ 4rr k~
+
+
(4)
+
( L 1)(L 2)(L 3) . s1n2(6,+ 3 - 6,) (2L 3)(2L 5 )
+
+
+ 2L + 1) sin2@,+ + 3(L(2L+ 1)(L2 - 1)(2L + 5 )
1
-63)
(6)
QUANTUM THEORY AND VISCOSITY OF DILUTE GASES
39
where wz'), wL2), and wL3) are weighting coefficients depending on fundamental properties of the colliding particles. The last result is due to Munn el af.(1965). Q(')(k)is the cross section which is associated with coefficients of diffusion, Q'*'(k)with viscosity, and Q(3'(k)with the thermal diffusion factor. When the gas consists of indistinguishable particles, the wave functions must be correctly symmetrized which determines the coefficients wL. The values of w(Lz)areshown in Table I, for particles with spin s subject to Bose-Einstein or Fermi-Dirac statistics. TABLE I
B-E s=o
L even L odd
S Z O
s=o
2s O
Z
T
l
SZO
2s
2(s+ 1) 2s+ 1
2
Boltmann
F-D
ZTi 2
+
2(s 1) 2s+ 1
All s 1
1
When the particles are distinguishable it is appropriate to use Boltzmann statistics, and therefore wi2) = 1 or all L. For observable diffusion effects it is axiomatic that the particles are distinguishable and in such cases w f ) = 1 also. The formulas which have been given so far allow for diffraction effects arising in quantum mechanics, and these are particularly significant for low collision energies. In addition to the symmetry effects mentioned above, there are other consequences of symmetrization which may appear in gases when multiple collisions are significant. These are, of course, density-dependent and will not be considered in the context of this paper. We should note here how the transition from quanta1 to classical mechanics takes place for high energy collisions. First, and also Consequently the summation over L in, for example, Eq. (9,can be replaced by integration over b.
111. Reduced Variables and Law of Corresponding States In order to demonstrate the effects of diffraction and statistics introduced by quantum mechanics, it is convenient to use standard units for distance and
40
R.A . Buckingham and E. Gal
energy associated with the intermolecular potential. Let this be expressed in the form
where I,, and E represent the position and depth, respectively, of the minimum of V(r) (see Fig. lb), and f ( p ) defines the shape of the potential. Then the radial wave equation becomes
' = p r m 2 / 2 h 2 Both . q and I are nondimensional, and where q2 = k2rm2and 1 for a given function f ( p ) the phase shift 6 , depends only on L, q, and I. We also introduce the following reduced variables : reduced specific volume
u* = u/rm3;
reduced kinetic energy
K
reduced temperature
(9)
= pg2/2&= q2/12;
T* = uT/E.
FIG.1. A binary atomic collision. (a) Collision parameters. (b) Interatomic potential parameters.
QUANTUM THEORY AND VlSCOSITY OF DILUTE GASES
41
The cross sections Q(")are conveniently reduced by dividing by the corresponding quanta1 cross sections for a gas of rigid spheres of diameter r m . Thus
a=-
n even
n + 1'
[see Hirschfelder, et al., (1954, p. 525)]. In particular 4 Q*("(q, A ) = 7 wZ"(L q r .
+ 1) sin2(6,+
6 (L Q*"'(q, A ) = 7 C wL') 4
L
- 6,)
+ 1)(L + 2) sin2(6L+2- 6,). 2L + 3
(14)
The first approximations to the coefficients of diffusion and viscosity are given, according to the Chapman-Enskog formulation, by integrated averages of these cross sections with respect to a Maxwellian distribution. We denote these' by Q*( 1, 1) and R*(2, 2), respectively, where
R*(n, s) = -Jm (s l)! 0
+
and x
= KIT* = q2/A2T*.The
e-xxs+
1
Q*'"'(q,
A) dx
Q*'s are functions of T * and 1. TABLE I1
The notation of Halpern and Buckingham is also used in later sections. Various notations have been used for these cross sections and we take note of those given in Table 11. It should be noticed that the quantities in columns (1) and (4) are (r,,,/u)' times the quantities in the other columns, u being the position of the zero in the intermolecular potential V(r), and r,,, the position of the minimum.
42
R . A . Buckingham and E. Gal
Reduced coefficients, D" and fil*, of diffusion and viscosity are defined in terms of the usual dimensioned coefficients D and fil, by the relations
'I*=- r m
2
(2/.l&)'/2file
(17)
Finally,
The factorsf, and& which have been introduced into (1 8) and (19) correspond to higher approximations. In general, these factors differ from unity by a few per cent at most. If we consider only the second approximation of Chapman and Enskog, then the factors are expressible in terms of the integrals R*(n, s). For example, f:"= 1 bT2 bl lb22 - bT2 where bll = 412*(2, 2)
+
b1,
= 7Q*(2,2)
- 8Q*(2, 3)
301 b22 = -R*(2,2) - 28R*(2, 3) + 2OR*(2,4). 12 Higher order approximations have been derived, and also alternative forms, depending upon the method of expansion used in the transport theory; a discussion of these is given by Hirschfelder et al. (1954, pp. 604-610). In concluding this section we must refer to the significance of the parameter A, or of the quantity A, = 2x12, which is more commonly used. We see that A,,, = h/(2p~)'/~r,
(21)
and is the ratio of the de Broglie wavelength for collision energy E to the atomic diameter r,; it is therefore a critical parameter in determining the importance of diffraction effects in the collisions occurring in the gas. The more this ratio departs from zero and approaches or exceeds unity, the more important these effects become. For very small A, , which is generally true for the heavier molecules, and temperatures such that T* > 1 , the departures from classical mechanics are also slight.
43
QUANTUM THEORY A N D VISCOSITY OF DILUTE GASES
To indicate the species for which quantal effects can be appreciable at low temperatures, we have included a table of approximate values of Amand 1' (Table 111). The values of E / I C correspond to the absolute temperature at which T* = 1. TABLE 111
H3(C) A, A2 E/K
3.1 4.2 161"
He3 2.68 5.5
lo"
He4
H2
2.33 7.3 10"
1.57 16.7 34"
HD
D2
1.36 25 34"
1.11 33 34"
Ne 0.50 160 36"
Ar 0.16 1150 119"
The special case of collisions between H atoms in the lowest 'X state has been included in the table in view of an application which is discussed later. It is clear that for atoms as heavy as neon, quantal effects are becoming negligible at temperatures of general interest.
IV. General Quanta1 Effects at Low Temperatures Before considering particular applications of the theory so far presented, it is desirable to have a general appreciation of the quantal effects involved and of their dependence on the parameter 1 and the method of symmetrization. We shall, therefore, illustrate these effects using the results of calculations with a typical interaction potential and the following range of values of 1 ' : A2
= -, loo
n
n = 5(1)10(2)20(2$)25.
The corresponding range of A, is from 1.41 to 3.14. The potential selected for the purpose has a form introduced by Buckingham and Corner (1948), in which the shape function in (7) f(p) = f , ~ - ~+( lbp-') - f 2 e - a ( P - 1 )
with
f, = a/{a(l
+ b ) - 6 - 8b},
=-
f2 =
-1
+ (1 + b)f1.
This form is used when p 1 ; when p < I , fl is multiplied by a factor exp{ -4(1 - p ) ' / p 3 } which rapidly removes the p - 6 and p - * terms but maintains continuity of f ( p ) and several derivatives at the minimum. Numerical values of 13.5 and 0.2 are given to a and 6, respectively. In order to exhibit the quantal effects it is unnecessary at this stage to specify numerical values for the potential parameters E and rm.
R.A . Buckingham and E. Gal
44
Some of the computed results, derived by the authors using the Atlas computer in the University of London, are presented in Figs. 2-4. Figure 2 shows the variation of Q * ( 2 ) as a function of K (or q2/,12)and of Q*(2, 2) as a function of T * , for ,I2 = 5 and A’ = 8.33. The K and T * scales are chosen so that for equal lengths, K = 3T*; this corresponds to the fact that the weighting function x3e-x in Eq. (15) has its maximum when x = K/T* = 3. The results of summation for even and odd values of L , corresponding to Bose and Fermi statistics, are shown separately in each diagram, as well as the result of including all values of L, equally weighted, corresponding to Boltzmann statistics.
t
-0 w
la)
2. II I
I
0
1
03
06
I
K
09
12
15
04
05
I6
K
_ _ _---------_-_____________ 0
01
0.2
0.3 T”
04
05
0
I
0
01
02‘
03
06
To
FIG.2. Variation of Q*(’) with K and n*(2,2)with T* for (a) xz = 5 and (b) xz = 8.33. Broken lines correspond to Boltzmann statistics.
The dominant features of the curves of Q*(’) are the rapid rise at small collision energies; and the considerable oscillations of the curves which involve symmetrization about those for classical statistics. For larger values of K these oscillations soon become small, and as one would expect they are much less marked in the variation of R*(2, 2), which represents the result of averaging Q*(” with respect to the Maxwell velocity distribution. For larger values of ,I than those illustrated, the oscillations persist but decrease in amplitude.
QUANTUM THEORY AND VISCOSITY OF DILUTE GASES
45
Figure 3 shows only the variation of R*(2, 2) for values of T* up to about 0.15 but for a greater variety of values of 1’. Again the results for R E and F-D statistics are shown separately; and the complexity of the behavior at these small T* values is evident. The overall behavior is a combination of (a) symmetrization or statistics, (b) the quantal mass effect, and in Fig. 3 these effects are superimposed. Figure 4 achieves some separation of the symmetrization effect by depicting the difference of the Bose and Boltzmann cross sections relative to the former. This is shown for various I’ for values fo T* up to 0.25. The effect is strongly dependent both on L2 and T*.
T’
T‘
FIG.3. Variation of n*(2,2) with T* for various h2 for (a) B-E statistics and (b) F-D statistics. Circles indicate points of intersection of corresponding curves for B-E and F-D statistics.
The quantal mass effect also depends strongly on both parameters. It is characterized by the departure of the quantized Boltzmann cross section from the corresponding classical cross section associated with a zero value of A, (or infinite value of A). The path of the Boltzmann cross sections is indicated by the circles in Fig. 3 which denote crossing points of the Bose and Fermi curves. For a certain range of values of I , the Boltzmann cross sections are substantially greater than the classical for small T*; this is most marked for A’= 12.5 (A,= 1.8). As 1’ drops below 10 (A, > 2), there is a swing over so
R. A . Buckingham and E. Gal
46
04
02
q.Oo -02
-04
-06
0025
005
0075
01
0125 0150 T"
0175
0200
0225 0250
FIG.4. The effect of symmetrization on averaged viscosity cross sections. Variation of
that the quantal cross sections become much smaller than the classical. A reference to Table I11 in the previous section shows that the quantal mass effect for the He isotopes should be different from that for H, at comparable values of T*. The calculations reported here' are an extension of earlier calculations by Buckingham and Gilles (1957). The present range of T* extends from 0.00625 to 2.3. The results illustrated in this section have not been taken beyond 0.25 as the behavior for higher values of T* does not show any features of special interest. Other computations have been carried out recently using the Lennard-Jones interaction, notably those by Imam-Rahajoe er al. (1965) for A*= 1, 2, 3 and T* from 0.3 to 28, and the most extensive work of Munn et al. (1965) which covers A*=O(O.5) 3.5 and T* from 0.01 to 100. (Values of A* are approximately 15 % greater than those of A, .) In concluding this section we should again observe that as T* becomes larger or as A, becomes small (< I), the transport cross sections should tend to the classical results given by (3). This has led to a considerable literature on " semiclassical approximations " to the quantal cross sections, based either on the use of approximate phase shifts, e.g. the JWKB approximation, or The computations are continuing and in due course tables of cross sections will be deposited with the US Library of Congress.
QUANTUM THEORY AND VISCOSITY OF DILUTE GASES
47
attempts to expand the cross sections in powers of Planck's constant. The first approach, though much earlier in origin, was further encouraged by the analysis of potential barrier penetration and rainbow scattering by Ford et a/. (1959); the second is exemplified by the work of Curtiss and Powers (1964) which led to an expansion of the phase shifts S, in powers of h. A related paper by Wood and Curtiss (1964) applied this expansion to the transport cross sections ; however, satisfactory numerical results using this expansion have not yet been achieved. Considerable insight into the range of applicability of JWKB and other approximations in simplifying the calculation of quantal cross sections is provided by Munn et a/. (1964) and Bernstein et a/. (1966).
V. Special Cases We shall now illustrate some of the quantal effects in the theory of viscosity by reviewing in greater detail three particular cases. These are: A. the isotopes of helium, He3 and He4; B. ortho- and para-forms of molecular hydrogen, H,; C. atomic hydrogen. Each case exemplifies different effects which can arise. A. THEHELIUM ISOTOPES
Helium has always been an obvious choice by which to test the use of quantal cross sections in the Chapman-Enskog theory. Not only does helium persist as a monatomic gas to very low temperatures, but the intermolecular potential is reasonably well-known from other sources, and the existence of two isotopes satisfying different quantal statistics makes it an interesting test case. Apart from the early rigid sphere calculations already mentioned, the first quantal results were those of Massey and Mohr (1934) for He4, based on the theoretical Slater interaction for two He atoms. The viscosity was evaluated at temperatures of 15" and 20°K. Their results were extended for a similar potential over the range 2"-30"K by Massey and Buckingham (1938), and still further by Buckingham ef al. (1941) for six different potentials, particularly for the range 2"-5"K. By 1938, experimental values of He4 viscosity were available at temperatures down to 1.64"K, and all these results were useful in confirming current knowledge about the helium interaction, on the assumption that He4 atoms satisfied Bose-Einstein statistics. (Only even-order phases were included in Eq. ( 5 ) for Q'2'.)
48
R. A . Buckingham and E. Gal
Independent investigations were carried out over a number of years by de Boer and his collaborators, consistently using a Lennard-Jones (12, 6) potential for the atomic interaction. These are represented at this stage by a comparison of theoretical and experimental viscosity for He4 in the range up to 2.74"K(de Boer, 1943). By 1954 reliable viscosity measurements had been made by Becker et al. (1954) for He3 gas as well as various mixtures of He3 and He4 at temperatures between 1" and 5°K. This made a detailed study of these isotopes worthwhile, in line with arguments presented by Halpern (1951). Preliminary results for He3 were already published by Buckingham and Temperley (1950), de Boer and Cohen (1951), and Buckingham and Scriven (1952), the last paper including also the thermal diffusion ratio for He3-He4 mixtures. All these showed that on the hypothesis that He3 atoms followed Fermi-Dirac statistics, very significant differences between the viscsosity variation for the two isotopes were to be expected. With the help of the experimental results, more detailed analyses by Cohen et al. (1954) and Halpern and Buckingham (1955) established beyond doubt that the quantal formulation was adequate to explain these results on the basis of a common interatomic potential for the two isotopes, with the assumption that He4 and He3 atoms behaved as B-E and F-D particles, respectively. Both groups of workers also studied the behavior of He3-He4 mixtures up to 5"K, adapting for this purpose the mixture formula of Curtiss and Hirschfelder (1949). Here the comparison with experiment was not entirely satisfactory. The essential feature of a gas of He3 atoms is that it contains two species with unlike spins. One-half of all collisions are therefore between dissimilar particles and should not be subject to symmetrization; the remainder are between like F-D particles and should be symmetrized accordingly. On the other hand, He4 atoms are without spin and satisfy B-E statistics. In consequence, the values of w L in Fig. (5) are as shown in the following tabulation.
Leven L odd
He4
He3
wL=2 WL = 0
9
4
The interatomic potential is effectively the same for both isotopes, but because of the mass difference there is a difference in the quantal parameter 1 (or A,,,). The second virial coefficient of helium gas at low temperatures is well represented by a B-C type of potential with parameters such that wmZ= 122 x erg cm2; the corresponding values of 2' for He4 and He3 are 7.27 and 5.48 (A,,, = 2.33, 2.68, respectively).
QUANTUM THEORY AND VISCOSITY OF DILUTE GASES
49
The effect of the quantal mass factor associated with these two values of A2 is shown in Fig. 5, taken from Halpern and Buckingham (1955). The full-line curves correspond to the quantal statistics expected to hold, using the above values of w,; the broken curves to the classical Boltzmann statistics. It should be noted that the classical mass factor m'" does not enter at this stage, so that the departure of each full-line curve from its accompanying broken curve shows the cross section is affected by using the correct statistics, that is, the symmetrization effect. The same relation between K and T* scales has been used as in the earlier Fig. 2. In Fig. 5 it appears that when T* = 0.5 the curves are all fairly close to unity, which is the classical value for rigid spheres. However, for large T* the value of S , < 1 because the effective collision radius is then less than rm. The other important point is that when T* > 0.1 the symmetrization effects, being in opposition for the two isotopes, enhance the mass effect, and, indeed, over much of the temperature range the difference in viscosity of the isotopes arises more from the different statistics than from the different masses. In plotting S, against K, and S, against T*, it is not necessary to specify the energy parameter E . This can, in practice, be chosen to give the best overall
0.3 0.6 0.9
1.2
3,
1.5 1.8 K=E/e
2.1 2.4
FIG.5. Quanta1 viscosity cross sections of pure He3 and He4. (a) S,, before averaging, as function of reduced collision energy K . (b) after averaging, as function of reduced temperature T * . Full-line curves correspondto actual statistics, dashed curves to Boltzmann statistics. [From Halpern and Buckingham (1955).]
s,,,
R . A . Buckingham and E. Gal
50
agreement with observed viscosity values, as shown in Fig. 6. Here the theoretical values of q*(T*)-'I2 for He3 and He4 are shown against the experimental values of Becker et al. (1954), the temperatures being reduced by assuming E / K = 10.2"K.The classical mass factor is now included, as shown in formula (17), for q*. The excellent agreement for both isotopes is a clear vindication of the theory in general, of the method of symmetrization, and to a limited extent, of the choice of interaction. In this respect it should be noted that the corresponding calculations of Cohen et al. (1954), using a Lennard-Jones interaction, also lead to good agreement. A more recent and extended analysis by Monchick et al. (1965), which considers also thermal conductivity, diffusion effects, and the equation of state, shows that this wide range of properties of the He isotopes can be well correlated using the L-J interaction. There are, however, some residual discrepancies between results at low and high temperatures, which indicate the need for a more flexible potentialin which the repulsive and attractive parts can be independently specified. 0.3
-
0.2 '
.-/--He
0.1'
I
0
,'
0. I
0.2
0.3
0.4
0.5
FIG.6. Comparison of theoretical and observed viscosities of pure He3 and He4. Theoretical full-line curves correspond to actual statistics, dashed curves to Boltzmann statistics. Experimentalpoints, due to Becker et al. (1954) have been reduced byassuming E / K = 10.2", r,. = 2.94 A. [From Halpern and Buckingham (1955).]
B. ORTHOAND PARA-HYDROGEN A somewhat different and more difficult mixture problem arises in molecular hydrogen. At low temperatures hydrogen gas may be considered as a mixture of para- and ortho-molecules, each in their lowest rotational states, and we must distinguish three cross sections S,, S,,, So corresponding to binary collisions between para-para, para-ortho, and ortho-ortho molecules,
QUANTUM THEORY AND VISCOSITY OF DILUTE GASES
51
respectively (the subscript q is here omitted for convenience, otherwise the notation is still that of Halpern and Buckingham). If the same centrally symmetrical potential is assumed for all types of collision, and ortho-molecules are assumed to have nine independent and equally abundant subspecies, then the appropriate values of wi') are as shown in the accompanying tabulation. From the corresponding averaged cross
L even
LO
wiz)= 2
1
9
0
1
3
L odd
8
sections S,, Spo,and So, it is convenient to define the viscosity difference ratios
So) and from the above tabulation it then follows that
A,, = ( 8 / W P o f
(22)
With the help of a suitable mixture formula it is possible to derive values of A,, and A,, from differential viscosity measurements of ortho-para mixtures. Thus, if Amix= ( S , - Smix)/S,, , the appropriate mixture formula can be written
*
,
mix
+ - Apo)l/(l + P ) + x,zAoo 1 + 2x,x,{(2Apo - Aoo)/(1 - A,O)l/(l + PI
=2 x p x o ~ p o ~ P 1/(1
(23)
where xp , x, ( = 1 - x p ) are the fractions of para- and ortho-molecules present in the mixture, and p is here the ratio of two collision integrals related to their mutual diffusion. This involves no algebraic approximation, but if Ap0 and A,, are small and quadratic terms can be neglected, then
Amix N xo(2xpApo
+ xoAool*
(24)
This first order approximation depends linearly on A,, and A,, and quadratically on the concentration x p . For normal-H, , with x,, = t,
A(n - HZ) 1;A,, provided (22) is valid. Experimental work in this field has also been carried out by Becker and co-workers, who used (24) to derive estimates of A,, and A,, at several temperatures for ortho- and parahydrogen mixtures (Becker and Stehl, 1952) and
52
R.A . Buckingham and E. Gal
ortho- and paradeuterium mixtures (Becker et al., 1953). We quote their results for molecular hydrogen in Table IV, denoting these results by Abo and Aio. Thus we see first that these viscosity difference ratios are very small, and secondly that the relation (22) is not satisfied. The values of A& are relatively large, showing that the simple assumptions made at the beginning are not valid, at least for ortho-ortho collisions. In making comparison with the present theory, therefore, it is advisable to start with Apo only. TABLE IV
15.0 20.3 63.2 77.3 90.1
-0.0042 -0.00253 -0.00055 -0.00037 -0.00028
-0.0099 -0.00835 -0.00275 -0.00223 -0.00187
Two theoretical investigations were made at the time when these results became available, by Cohen ef al. (1956) and Buckingham ef al. (1958a,b). These used the Lennard-Jones potential and a Buckingham-Corner type potential, respectively. The results are illustrated in Figs. 7 and 8. The first of these figures shows how the difference in the viscosity cross sections S p and Spo varies with collision energy. The difference exhibits the characteristic oscillations, and there is reasonable agreement between the two sets of calculations, at least for larger values of q2. Figure 8 makes the comparison of Apowith the values estimated by Becker and Stehl. At 15" the agreement is reasonably good, but at higher temperatures the viscosity difference is considerably underestimated by the theory. It must be remembered, however, that the calculations are difficult and involve substantial cancellations of positive and negative contributions beyond T* = 0.5. The calculations reported in this paper differ slightly from those of Buckingham et al. (1958b), but do not lessen the discrepancy shown in the inset figure. To understand the discrepancy it is necessary to recall the main approximations which have been made in the theory: (1) any angular dependence of the potential has been ignored, in particular the long range quadrupole-quadrupole interaction for which the spherical average is zero (2) the same potential has been assumed for p-p, p-0, and 0-0 collisions; (3) inelastic collisions, which could arise when an ortho-molecule is involved, have been ignored, together with excitation of higher rotational or vibrational states.
QUANTUM THEORY AND VISCOSITY OF DILUTE GASES
53
FIG. 7. Ortho- and para-hydrogen. Variation of viscosity cross section, AS = S, - S,, , with collision energy. Note the wide difference in scales between (a), (b), and (c). [From, Buckingham et al. (3958b,).] 0.03-
APO
0.02
-
0.01
-
-Oooo5 -0001
t/ II
Of:
. I0
1.5
T*
2
0
0
-
-0.01
\ \
'..*.'
,
-0.02L
FIG. 8. Ortho- and para-hydrogen. Comparison of theoretical and experimental values - &,)/s,. The inset diagram shows tail of curve considerably magnified. The broken curve is a rough construction through the points of Cohen et nl. (1956). [From Buckingharn et a/. (1958b), with corrected ordinate scale.!
of viscosity ratio, Ap0 =
(s,
R. A . Buckingham and E. Gal
54
Some progress in investigating these problems has been made in a series of papers by Takayanagi. A study of molecular collisions allowing for a nonspherical interaction was begun by Takayanagi and Ohno (1955) but led to inconclusive results. Subsequently, Niblett and Takayanagi (1 959) examined the nonspherical potential effective in p-o collisions in more detail, and concluded that the difference between viscosity cross sections for p - p and p-o collisions could be explained by the nonspherical nature of the potential superimposed on the effects of the statistics. They also gave arguments based on differences of interaction for expecting Aoo to be at least twice as large as Apo.
C . ATOMIC HYDROGEN The third case for detailed consideration is that of atomic hydrogen. From the theoretical side this is interesting because the spin interaction of the hydrogen atoms leads to alternative interaction potentials for atoms in their ground states, the 'Z, potential corresponding to the normal H, molecule, and the 3Zupotential corresponding to the lowest repulsive state of H, . There is one chance in four that atoms will interact according to the singlet state, and the effective cross section is therefore a mean of the cross sections for the singlet and triplet states weighted in the ratio 1 : 3. Scattering in the singlet state also has intrinsic interest because of the great depth of the potential well compared with the B-C and L-J type potentials so far considered, and the consequent large number of bound states associated with it. The effect of this on the phase-shift calculations will be seen later. It is preferable to reformulate the collision of hydrogen atoms in atomic units. Thus the radical wave equation may be written
+
L(La2 1)) J p " = o
where a is in units of a , , and the potential energy of interaction VB or V, is in units of e2/a,;k is the wave number of relative motion in units of l/a,, and M the ratio of the proton and electron masses. The viscosity cross section corresponding to (5) in units of nao2,with the above weighting of states and correct symmetrization of wave functions, becomes 4
S(2)(k)= k2
c ( L +2 L1)(L+ 3+ 2)
x { f w L , @sin2(&+, - 6'9
+ 2wL," sin2(6",+2- S,")}
(26)
QUANTUM THEORY AND VISCOSITY OF DILUTE GASES
55
where WL,g
= $12 - ( -
WL,u
= t{2
WL,g
+ W L , u = 2.
11,)
+ (-- 1)")
If the identity of protons could be ignored, as in the hypothetical collision of two particles of the same mass as the proton and interacting according to the 'Zg+ and 'Xu+ potentials, then WL,g
= WL," = 1
and there is no discrimination between even and odd values of L. When it is desirable to distinguish between these two cases, we shall refer to the former cross sections as S$,', ,corresponding to protons satisfying Fermi statistics, and to the latter as S ~ ~ ~corresponding y m , to hypothetical dissimilar particles satisfying Boltzmann statistics. It is convenient to define a reduced temperature T ' b y
(27) and the averaged viscosity cross section
:j
S ( 2 ) ( T '= )
C y y 3 S 2 ) ( kd) y
(28)
where y = k2/T'. This differs by a numerical factor from the definition of R*(2, 2). It represents the first approximation to the viscosity cross section; a second approximation can be derived as in (20). It is apparent from (26) that two sets of phase shifts SLg and 6," must be calculated using the singlet and triplet interactions, respectively. Some preliminary calculations were made in 1961 by Buckingham and Fox, using for the singlet case a square well to represent the major interaction, together with attractive tail varying as o - ~ and , for the triplet the B-C type potential with suitably chosen parameters. The viscosity was then derived for the temperature range from 25" to 300°K (Buckingham and Fox, 1962). Prior to this, Kudriavtsev (1 958) had applied classical theory to derive viscosity cross sections at higher temperatures, and Dalgarno and Smith (1962) also used classical theory to calculate viscosity and thermal conductivity up to lo5 OK. These authors based their calculations on careful analytic fits to the accurate interaction potentials derived earlier by Dalgarno and Lynn (1956). At the same time, they used the approximate potentials adopted by Buckingham and Fox to enable comparisons to be made, and concluded that the classical theory was likely to be adequate above 100°K.
56
R.A . Buckingham and E. Gal
In order to make this comparison more directly useful, Buckingham, et al. (1965) repeated the quantal calculations with the accurate potentials of Dalgarno and Lynn, and we shall summarize their results here. First, in Fig. 9 are shown the phase shifts derived from the singlet potential. These are particularly interesting in exhibiting many sharp resonances which are associated with the quasi-bound states of the effective potential
L(L
+ l)/a2+ MVg(a).
They are characterized in most cases by a sudden change of 71 in the phase shift at some value of k . For L > 9 nearly every curve shows at least one such resonance, and a few curves, e.g. for L = 22, 24, and 26, show three. It should be noted that dogtends to 1% for small k , corresponding to the 15 vibrational levels known to exist in the ground state of the H, molecule. These phenomena are discussed in detail by Buckingham and Fox (1962). Figures 10 and 11 show the contributions to the total viscosity cross sections (26) arising from the singlet and triplet states, respectively, together with some of the values derived by Dalgarno and Smith (1962). (The upper curve in each case is the unsymmetrized cross section.) There is good agreement in the general trend of the cross sections for k > 1, but deviations due to the quantal effects are important and, for the singlet case, complex. The complicated behavior is explicable in the light of the phase shifts in Fig. 9; for example, the high peak in Sun&,,, near k = 0.45 arises from two partial cross sections involving 6;. The lower curve in Fig. 10 shows that proper symmetrization is significant over the whole energy range covered. This curve has a structure resembling that of the upper curve. From Fig. 11 it is clear that for the triplet case symmetrization is significant for k less than about 0.7. It should be noted that the lower curve has the opposite sign to that in Fig. 10. Averaged cross sections S',' were derived by Buckingham et al. (1965) for temperatures between 1" and 400°K. In these, the sharp variations apparent in Fig. 10 are largely smoothed out. The singlet and triplet contributions to correctly symmetrized, are comparable in magnitude although the triplet is the larger. Table V gives an idea of the variation of the two contributions, and of the resultant first approximation to the viscosity. The last column in Table V indicates the extent to which the classical calculations of Dalgarno and Smith differ from the quantal results. The small differences at the higher temperatures should perhaps be taken as evidence that there is no significant error in either calculation. Comparison of these theoretical results with experiment is necessarily somewhat indirect because measurements must be made on mixtures of atomic and molecular hydrogen. The derivation of viscosity coefficients for atomic
s(,),
QUANTUM THEORY AND VISCOSITY OF DILUTE GASES
57
FIG.9. Atomic hydrogen. Variation of phase shifts with wave number for scattering of two H atoms in singlet ground state; even values of L, full curves; odd values, broken curves. [From Buckingham er al. (1965).]
R. A . Buckingham and E. Gal
58
FIG.10. Atomic hydrogen. Variation of viscosity cross sections with wave number for lower to the singlet potential; upper curve refers to S,,!!Fm, = S.6’.:%; 0,Dalgarno and Smith (1962) classical values. [From Buckingham er nl. (1965).1
I
Wave number # (in units of ‘/ao)
FIG,1I. Atomic hydrogen. Variation of viscosity cross sections with wave number for the triplet potential; upper curve refers to lower to AS(’)’= S(’)’ sym, - Sunsym (’)’ .0 Dalgarno and Smith (1962) classical values, [From Buckingham er al. (1965).]
&’Arm,
7
QUANTUM THEORY AND VISCOSITY OF DILUTE GASES
TABLE
I 5 10
25 50 75 100 200 300
400
52.09 80.45 102.43 84.20 57.62 51.48 48.53 41.93 38.19 36.30
62.14 153.98 128.27 105.97 96.13 90.43 86.35 16.38 10.41 66.25
59
V
3.35 3.65 5.25 10.06 17.61 23.36 28.38 45.16 60.68 74.65
4.1 4.0 2.4 1.7 0.4 -0.4 4 . 4
hydrogen, therefore, involves some assumptions about the atom-molecule interaction and the appropriate mixture formula, which also involves the mutual diffusion coefficient. The most recent investigation of this problem is by Browning and Fox (1964), who present measurements of viscosity and mutual diffusion at three temperatures, -SO", O", and 100°C, and also reanalyze earlier experimental results. Their final comparison with theory is shown in Fig. 12. Two sets of points are given which depend on whether the
- -1.0 O'I
FIG.12. Atomic hydrogen. Variation of viscosity coefficients with temperature; upper curve refers to yvnsym, lower to A 7 = ~ . . . y m - rlaYm,the ordinate scale for the lower being 50 times that for the upper and labeled for negative values only; 0, Browning and Fox (1964) values derived on the basis of (12,6) and (exp, 6) fits, respectively,to H-H2 interaction potentials. [From Buckingham et ul. (1965).]
+,
60
R. A . Buckingham and E. Gal
H-H, potential is assumed to have the Lennard-Jones (12, 6) form or an (exp, 6 ) form in which the repulsion is represented by an exponential. This choice is not too critical and there is a good quantitative agreement between theory and experiment. Figure 12 also shows that the symmetry effect from the identity of the protons persists over the whole temperature range covered, although it falls to less than 1 % for T > 70°K.
REFERENCES Becker, E. W., and Stehl, 0. (1952). Z. Physik 133, 615. Becker, E. W., Misenta, R., and Stehl, 0. (1953). Z. Physik 136,457. Becker, E. W., Misenta, R., and Schmeissner, F. (1954). Z. Physik 137, 126. Bernstein, R. B., Curtiss, C. F., Imam-Rahajoe, S., and Wood, H. T. (1966). J. Chem. Phys. 44,4072. Browning, R., and Fox, J. W. (1964). Proc. Roy. SOC.A278, 274. Buckingham, R. A., and Corner, J. (1948). Proc. Roy. SOC.A189, 118. Buckingham, R. A., and Fox, J. W. (1962). Proc. Roy. SOC.A267, 102. Buckingham, R. A., and Gilles, D. C. (1957). Unpublished work. Buckingham, R. A., and Scriven, R. A. (1952). Proc. Phys. SOC.(London)65, 376. Buckingham, R. A., and Temperley, H. N. V. (1950). Phys. Rev. 78,482. Buckingham, R. A., Hamilton, J., and Massey, H. S. W. (1941). Proc. Roy. SOC.A179, 103. Buckingham, R. A., Davies, A. E., and Davies, A. R. (1958a). Proc. Conf Thermodyn. Transport Properties Fluids, London, 1957. Inst. Mech. Engrs., London. Buckingham, R. A., Davies, A. R., and Gilles, D. C. (1958b). Proc. Phys. SOC.(London) 71,457. Buckingham, R. A., Fox, J. W., and Gal, E. (1965). Proc. Roy. SOC.A284,237. Chapman, S. (1917). Phil. Trans. Roy. SOC.London A217, 115. Chapman, S., and Cowling, T. G. (1960). Mathematical Theory of Non-Uniform Gases,” 2nd ed. Cambridge Univ. Press, London and New York. Cohen, E. G. D., Offerhaus, M. J., and de Boer, J. (1954). Physica 20, 501. Cohen, E. G. D., Offerhaus, M. J., van Leeuwen, J. M. J., Roos, B. W., and de Boer, J. (1956). Physica 22, 191. Curtiss, C. F., and Hirschfelder, J. 0. (1949). J. Chem. Phys. 17, 550. Curtiss, C. F., and Powers, Jr., R. S. (1964). J. Chem. Phys. 40, 2145. Dalgarno, A., and Lynn, N. (1956). Proc. Phys. SOC.(London) A69, 821. Dalgarno, A., and Smith, F. J. (1962). Proc. Roy. SOC.A267, 417. de Boer, J. (1943). Physica 10, 348. de Boer, J., and Cohen, E. G. D. (1951). Physica 17,993. Enskog, D. (1922). Arch. J. Marhmatik, Asrronomi och Fysik 16, 516. Ford, K. W., Hill, H. D., Wakano, M., and Wheeler, J. A. (1959). Ann. Phys. (N.Y.)7, 239. Halpern, 0. (1951). Phys. Rev. 82, 561. Halpern, O., and Buckingham, R. A. (1955). Phys. Rev. 98, 1626. Halpern, O., and Gwathmey, E. (1937). Phys. Rev. 52, 944. Hirschfelder, J. O., Curtiss, C. F., and Bird, R. B. (1954). “Molecular Theory of Gases and Liquids,” Chapter 10. Wiley, New York. Imam-Rahajoe, S., Curtiss, C. F., and Bernstein, R. B. (1965). J. Chem. Phys. 42, 530. ‘I
QUANTUM THEORY AND VISCOSITY OF DILUTE GASES
61
Kudriavtsev, V. S. (1958). Soviet Phys. JETP (English Transl.) 6, 188. Mason, E. A. (1954). J. Chem. Phys. 22, 169. Massey, H. S. W., and Buckingham, R. A. (1938). Proc. Roy. SOC.A168, 378; A169, 205. Massey, H. S. W., and Mohr, C. B. 0. (1933). Proc. Roy. SOC.A141,434. Massey, H. S. W., and Mohr, C. B. 0. (1934). Proc. Roy. SOC.A144, 188. Monchick, L., Mason, E. A., Munn, R. J., and Smith, F. J. (1965). Phys. Rev. 139, A1076. Mott, N. F., and Massey, H. S. W. (1933). “Theory of Atomic Collisions,” 1st ed. Clarendon Press, Oxford. Munn, R. J., Mason, E. A., and Smith, F. J. (1964). J . Chem. Phys. 41, 3978. Munn, R. J., Smith, F. J., Mason, E. A., and Monchick, L. (1965) J. Chem. Phys. 42, 531. Niblett, P. D., and Takayanagi, K. (1959). Proc. Roy. SOC.A250,222. Takayanagi, K., and Ohno, K. (1955). Progr. Theorer. Phys. (Kyoto) 13, 243. Uehling, E. A., and Uhlenbeck, G. E. (1933). Phys. Rev. 43, 552. Wood, H. T., and Curtiss, C. F. (1964). J. Chem. Phys. 41, 1167.
This Page Intentionally Left Blank
POSITRONS AND POSITRONIUM IN GASES P . A . FRASER Department of Physics, University of Western Ontario London, Ontario, Canada
I. Introduction
..........
......................................... .......................
63 65 A. General Description ........................................... .65 .............................. 67 B. Experimental Methods . . . . . . . ....................... 68 tronium.. ..............69 ....................... 71 ....................... 71 A. Positronsin Gases .............................................. 71 B. Positronium in Gases. . .......................... 78 IV. Theoretical Results ............................................... .87 A Positron-Atom Collisions B. Orthopositronium-Atom .................103 V. Other Areas of Positron VI. Basic Questions ...... .................103 Review Works ................................................... 104 References ...................................................... ,105 11. The Fate of Positrons in
I. Introduction In some ways this does not seem the time to review research into positron and positronium physics in gases. Behind one there are available many excellent review works, addressed to varying audiences; in particular we have the recent comprehensive book by Green and Lee (1964).' Ahead one sees the prospect of new types of experiment which will completely alter the nature of the work in this field. Indeed, one cannot help but feel that within a year or two there may well be many new experimental and related theoretical results. On the other hand, this may be a good time to take stock of what has been done, particularly of late, and to draw attention to theoretical work that to some extent may anticipate the forthcoming experiments. We shall attempt to give a rather different emphasis from that of recent reviews, guided both by the subject matter of the series of which this volume forms a part and by the interests of the man to whom this volume is respectfully dedicated.
' A separate list of review works is included at the end of the article. 63
64
P.A . Fraser
Thus this will not contain a further account of the modes of annihilation of positrons, nor of the fine details of the structure of positronium and their impressive relation to the predictions of quantum electrodynamics. Further, we shall ignore aspects that could perhaps be more properly termed “chemistry,” and except insofar as they appear relevant to experiments in gases, we shall ignore the large amount of work in liquids and solids. There are other places where such work could more properly be described. In short, we shall look particularly at the phenomena arising from the interaction of low kinetic energy positrons (less than a few tens of electron volts) with isolated atoms and some of the simpler molecules. It is clear that there are significant physical differences between positron and electron interactions with atomic systems. Effects of the identity of the projectile with components of the target, exchange effects, do not arise in positron problems; while this would apparently lead to simplification, in fact the concomitant absence of symmetries can make calculation more difficult. While an electron is attracted by the mean field of an atom, a positron is repelled. Yet, the force arising from distortion of the target is attractive in both cases. If its kinetic energy is high enough, a positron may pick up an electron from the target, forming the very light and large positronium atom; or it may, as do electrons, excite or ionize the target. At sufficiently low energies a positron will scatter only elastically. The presence of positrons and positronium may be detected by the characteristic radiation of their eventual annihilation, which is sufficiently improbable compared with atomic processes to make the notion of collision meaningful. The exciting prospect in view is that of beams of low energy positrons with small energy width. Herring and his collaborators (1967) are working on the extraction of such a positron beam from a tungsten target bombarded by the beam from a 30-MeV high current electron linear accelerator; the positrons arise from pair creation in the target by sufficientlyhigh energy bremsstrahlung. Their preliminary studies indicate that their first planned experiment, a measurement of the total cross section for the scattering of positrons by helium in the energy range 1-1000 eV by the attenuation of a positron beam, will produce measurable counting rates of annihilation prays. To the present, the available positrons have come from nuclear beta decays, and the experiments have of necessity dealt with moderating swarms of positrons. This prospect is mentioned at the beginning, as it is well to keep it in mind; the lack of such beams has decreed the type of experimental work done so far. In the following we shall note some recent researches that have come about as if in response to the remarks of Green and Lee (1964): “A determined experimental and theoretical attack on positrons in helium, instead of the usual argon, would be most valuable”; and of Osmon (1965b); “There is at
POSITRONS AND POSlTRONlUM IN GASES
65
present no overlap of experimental and theoretical studies of slow positronatom collision processes. This is unfortunate when progress has been accelerating from both directions.”
LI. The Fate of Positrons in Gases A. GENERAL DESCRIPTION Full accounts of the fate of positrons in gases have been given previously (e.g., Green and Lee, 1964), and it will be sufficient here to describe the phenomena but briefly. The positron sources have been radioactive nuclei, usually Na22. The great majority of positrons from this nucleus have a maximum energy of 0.54 MeV and a most probable energy of 0.12 MeV; the decay has a half-life of 2.6 years. The birth of a positron in this case is signaled by an almost simultaneous 1.3-MeV pray (within lo-” sec) emitted by the excited state of NeZ2to which the NaZZp-decays, and this fact is of course made use of in delayed coincidence measurements of annihilation events. In argon, for example, such a positron will slow down to about 10 eV in a time roughly given by (lOO/p) nsec, with p the pressure in atmospheres (Tao et al., 1963). The slowing down time in argon, and the number of electrons available in each atom, are such that a substantial fraction, some 30%, of the positrons are estimated to annihilate before slowing down to atomic excitation energies. The fraction is subtantially less in helium, for example; indeed it appears to be almost negligible. The formation of positronium (which, following the example of Green and Lee (1964), from now we shall term Ps) in its ground state can take place if the positron energy is greater than (E, - EB),where Er is the atomic ionization energy and E , is the binding energy of Ps (6.8 eV). It is possible to argue that such formation will take place principally in what is known as the “ Ore gap,” following the original considerations of Ore (1949a), where it exists. The Ore gap exists if the atomic excitation energy E,, is greater than (EI - EB),the Ps formation threshold, and is the name given to the range of positron kinetic energies lying between (E, - EB)and E,, . For positron energies greater than E,, , atomic excitation competes with Ps formation; and for positron energies greater than E,, Ps would be formed with kinetic energy greater than 6.8 eV and would be likely to break up in a further collision. Formation of Ps in excited states is relatively improbable, as the necessary minimum positron kinetic energy is great enough that excitation of the atom will compete. On the basis of such considerations, and making the assumption of a uniform distribution of positron kinetic energies between zero and Er following the last
P. A . Fraser
66
ionization collision, it is possible to estimate broad limits on the fractionf of positrons that form Ps : Eex
- @ I - EB) E,X
EI
- (EI EI
--
EI'
The lower limit neglects the competition of elastic collision within the Ore gap, and the upper, in addition, those of atomic excitation collisions and the competing formation of excited Ps. For argon, for example, the inequality gives 0.22
Ap = 0.80 x 10"
sec-',
which gives a mean life very short compared with others that occur, and thus one quarter of the Ps that is formed will disappear very fast. The natural annihilation rate (into three photons) of o-Ps is, on the other hand,
A. = 0.71 x lo7 sec-'. We shall see that this is generally slow compared with observed annihilation rates of slow positrons in a gas, and that the latter rate in turn is generally slower than the p-Ps annihilation rate.
POSITRONS AND POSITRONIUM IN GASES
67
The long life of 0-Ps may be shortened by various processes, which bring about a two-photon decay, and the 0-Ps is then said to be quenched. These processes are discussed in Section I1,D. To return to a positron that has not formed Ps: with a kinetic energy less than ( E , - EB), the threshold for Ps formation, it may take part only in elastic collisions in a monatomic gas. Some will be moderated to thermal energy before annihilation ; some will annihilate before being thermalized. Annihilation rates are discussed more quantitatively in Section I1,C. It is possible that some become bound to the atoms or molecules before annihilation.
B. EXPERIMENTAL METHODS The experimental techniques used have been admirably discussed in detail by Green and Lee (1964), and there is no need here for more than a brief summary, particularly of methods used in work to be discussed. As suggested above, the 1.3 y-ray accompanying the positron from Na22 may be counted in delayed coincidence with an annihilation quantum, thus giving a time spectrum of the number of annihilations. Time to pulse-height converters, whose outputs are fed to multichannel pulse-height analyzers, are used in such work. A typical delayed coincidence time spectrum of annihilation counts in a gas contains three distinct features (over the background) : (i) the prompt peak ; attributed to annihilations in the source, annihilations before slowing down to atomic energies, annihilation of p-Ps, and possibly annihilation of bound systems of positrons or Ps with atoms or molecules. (ii) the intermediate region ; attributed to the annihilation of positrons with kinetic energies generally below that of the Ps formation threshold. (iii) the long-lived tail; attributed to decay of 0-Ps. It is possible to suppress either (ii) or (iii) by the admission of appropriate other gases, and by suitable source construction (e.g., Falk and Jones, 1964) to reduce the number of counts in (i). The annihilation quanta from two-photon annihilation each has energy 0.51 MeV (for slow positrons), and are emitted in essentially opposite directions. The quanta from three-photon annihilation are emitted essentially in a plane and are distributed in energy in a rising spectrum from zero to 0.51 MeV. I n studies of the formation and quenching of 0-Ps, two other methods have also been used; those involving a measurement of changes in the y-ray spectrum (e.g., Heymann et al., 1961), and those that measure, under varying conditions, the steady three-photon coincidence rate (with the counters set 120" apart) (e.g., Celitans and Green, 1964), or the two-photon coincidence rate (e.g., Pond, 1952).
68
P.A . Fraser
Further, angular correlation measurements of two-photon events tell something of the momentum of the annihilating pair; a narrowing of the angular distribution, for example, indicates an increase in the two-photon decays from thermalized p-Ps. Related to such work is the recent study by Brimhall and Page (1966) of the Doppler width of the 0.51-MeV annihilation line in 0 2 .
c. RATES OF ANNIHILATION OF SLOW POSITRONS We shall lay out here some useful relations that help in comparing the various experiments and calculated quantities. These relate annihilation rates to density or pressure and to effective numbers of electrons per atom or molecule. A slow positron in a medium containing electrons will annihilate with an essentially coincident electron into two photons at a rate, the singlet rate, given by A, = 4 7 ~ r ~ ~ c ( ~ n , ) , (1) where lne is the average number density of electrons in the medium at the position of the positron and in a singlet state relative to the positron (e.g., Ferrell, 1956; Wallace, 1960; Deutsch and Berko, 1965). ro is the “classical cm), and c is the speed of light. electron radius” (2.8 x Now 1 ne = bne where n, is the average number density of electrons in the medium at the position of the positron in any spin state. Thus A, is usually written
A, = nro2c(n,).
(2) The contribution of three-photon annihilation to the total decay rate is 11370 of A,, and is here ignored. We may write, in the case of free positrons in a gas, and in the absence of collective effects (e.g., Section III,A,2),
-
(3) ne = n~ Zeff where nA is the number density of atoms or molecules, and Zeffis an effective number of electrons per atom (or molecule) for the annihilation process. Z,,, is a number calculable from a knowledge of the wave function of the positronatom system (see Section IV,A,l). Noting the proportionality of nA to p, the mass density of the medium, or to p , the pressure, under ideal gas conditions, the rate A, may be written alternatively as 1 I , = (4.51 x lo9) -pZef, (sec-’), (4) M.4
POSITRONS AND POSITRONIUM IN GASES
69
where p is in grams per cubic centimeter, and M A is the atomic or molecular mass in atomic mass units, or
As(,) = (2.01 x i O 5 ) p z e f f
(appropriate for O°C)
(54
A s ( 2 5 ) = (1.84 x 105)pZeff
(appropriate for 25°C)
(5b)
with p in atmospheres. As an estimate, it is not unreasonable to take Zeffto be the atomic number (which gives the so-called “ Dirac rate ”). However it is found experimentally and it is to be expected theoretically that Zeffdepends on positron energy. Thus the relations (4) - (5b) apply strictly only to positrons of a single energy, or with energies spread over a short range. It is worth noting that an annihilation cross section per atom may be defined, and is related to As by
where c is the positron speed. Recalling that r,
= (1/137)2a,,
where a, is the Bohr radius (0.53 x lo-’ cm), it is seen that, even for thermal positrons in equilibrium with helium at 4.2”K,this annihilation cross section per atom is very small compared with atomic areas. Thus it is meaningful to consider positron-atom collisions ignoring the possibility of annihilation during the collision process.
D. MODESAND RATESOF QUENCHING OF ORTHOPOSITRONIUM It has already been remarked (Section II,A) that when the normal threephoton decay of 0-Ps is changed to a two-photon decay, thus shortening its life, the 0-Ps is said to be quenched. Such quenching processes, other than that which one may call the “deliberate quenching” by the application of an external magnetic field which mixes the m =O state of 0-Ps with the singlet p-Ps (e.g. Deutsch, 1953), have been listed and discussed by Heymann et a/. (1961), whose scheme we closely follow. (i) Concersion quenching. I n a collision with an atom or molecule with one or more unpaired electrons (e.g., H, NO, 0 2 )it, is possible for 0-Ps to be converted to p-Ps through exchange scattering and hence be quenched. This process was analyzed by Ferrell (1958), who pointed out that it is misleading to regard the quenching as arising necessarily from “ spin-flip” by exchange. Wallace (1960) has given very full details of the case typified by 0-Ps collisions
70
P. A . Fraser
with O,, and Fraser (1961) has considered the case of 0-Ps collisions with H, which is identical in the exchange properties with NO. The cross sections for these conversion processes are expected to be of the order of atomic areas, and hence the quenching rates would be large. Experiments are described in Section III,B,l. (ii) Chemical quenching. The 0-Ps may become bound to an atom or molecule, with a consequent great increase in the annihilation rate. This appears to occur in Cl,, for example (Gittelman and Deutsch, 1956, 1958). (iii) Pickof quenching. The positron of the 0-Ps may, during a collision, find itself at the location of an atomic electron with which it forms a singlet spin state. It may then promptly annihilate. This effect is energy dependent, as it depends on the degree of penetration of the 0-Ps and the atom (or molecule). The pickoff rate will be proportional to the number density of atoms. Experiments are described in Sections III,B,2 and 3. (iv) Spin reversal. It is conceivable that the magnetic forces experienced in a collision between 0-Ps and an atom or molecule could convert 0-Ps to p-Ps and hence produce quenching. Consideration of the most favorable case, paramagnetic 02,by Ore (1949b) showed the cross section for the process to or lo-’ of collision cross sections, an estimate conbe of the order of firmed by Massey and Mohr (1954). We shall not consider this case further. While measurements are made of quenching rates, it seems better in the case of conversion quenching to think in terms of a cross section 6, related to the quenching rate 1, via 6,
= Iq/unA
(7)
where v is the 0-Ps speed and nAthe number density of atoms or molecules. It is recalled that 6, is expected to be of the same order as a collision cross section. If it is felt that an 0-Ps speed is ill-defined (i.e., perhaps the 0-Ps is not thermalized) then a “volume rate” R , = ‘‘ 6,v”
= l,/nA
(8)
is appropriate (Paul, 1958). On the other hand, in the case of pickoff quenching it seems desirable to think in terms of the number of electrons per atom or molecule in a singlet state relative to the positron of the 0-Ps, which we shall call ‘Zeff.The “ triplet” electron already in the 0-Ps is thus excluded from consideration. Following Eq. (l), we may thus write
I , = (47trO2c)n, lzerf
(9)
where nA is again the number density of atoms or molecules. In terms of the mass density p (gm/cm3), or the pressure p (atm), we thus have
POSITRONS A N D POSITRONIUM IN GASES
Aq = (1.804). 10'o(l/M,)p 'Zeff (sec-')
A,,,
= (8.05 x
jLq(zs) = (7.37
105)p 'Zerf (sec-')
(for OOC)
71 (10) (IW
x 105)p 'Zeff (sec-')
(for 25°C) (1lb) M Ais the atomic or molecular mass in atomic mass units. One could expect, in the noble gases for example because of the repulsive exchange force, 'Zef,to be somewhat less than 2/4, where Z is the atomic number, and thus the related cross sections would be very small. The pickoff process will always occur, but when conversion quenching is possible, it will be swamped. The number of 0-Ps atoms would decay with a rate of (A, + A,), and the fraction of 0-Ps annihilating by three-photon decay would be A,/(A, + A,), where A, = 0.71 x lo7 sec-', the natural 0-Ps three-photon decay rate.
E. UNITS In the discussions of experimental and theoretical work to follow, it will often be convenient to work in atomic units. The unit of energy will be taken to be the Rydberg, 13.6 eV; thus the binding energy of Ps is 4 in these units. The kinetic energy of a positron will be written as k z in Rydberg units. k is proportional to the positron speed, and the relation is u
(in cmjsec) = (2.18 x 10')k.
Cross sections will generally be given in units of nu,', where a, is the atomic unit of length (Bohr radius) 0.53 x lo-' cm. The positron de Broglie wavelength in a, is given by AdeB = 2 4 k . k is sometimes referred to as the positron wave number in a;'. Scattering lengths will be given in a, length units, and phase shifts in radians.
III. Experimental Results A.
POSITRONS IN
GASES
I. Mainly Argon Among the most significant of recent experiments have been those that have shown clearly that the annihilation rate of positrons in gases depends on their energy. Tao et al. (1963) drew attention to the fact that, at least in argon and other atomic gases, the slowing down time to thermal energies was comparable to the annihilation mean life, and hence that many positrons would annihilate before being thermalized. Their estimate that up to 30 % of the positrons from
72
P. A. Fraser
a Naz2 source would annihilate before being slowed down to atomic excitation energies accords with the observation of Deutsch and his collaborators (Gittelman and Deutsch, 1956) that about one third of such positrons in argon are annihilated without either forming Ps or falling below the Ps formation threshold. While it now appears that Tao et al. (1963) overestimated the slowing down time from the Ps formation threshold to thermal energies through using in their calculation a momentum transfer cross section for positrons in argon that was too small at low energies, their suggestion that a proportion of the positrons would annihilate before being thermalized from energies of order 10 eV has been confirmed. With high resolution delayed coincidence counting apparatus, Tao et al. (1964) and, independently, Paul (1964) detected a flat shoulder in the plot of the logarithm of the counting rate versus time following the prompt peak (see Section 11,B). Following this shoulder two exponential decays are seen: the shorter time one being attributed to the annihilation of thermalized positrons, the longer one to 0-Ps decays. The width (in time) of the shoulder was found to be inversely proportional to the pressure, and hence the feature could be connected with the slowing down process. The presence of the shoulder was taken to indicate an energy dependent annihilation rate for positrons with energies below the Ps formation threshold. The shoulder was suppressed in the presence of the diatomic gases 0, and N, , which would be better thermalizers of positrons than argon. For “pure” argon Tao et al. (1964) obtained a “ shoulder-width ” pressure product of about 90 nsec-atm, while Paul (1964) obtained between 200 and 300 nsec-atm. The shoulder was also independently found in argon by Falk and Jones (1964), who obtained a shoulder-width pressure product of about 270 nsec-atm, and was confirmed by Osmon (1965a). In addition, Falk and Jones (1964) observed a shoulder in krypton, and Osmon (1965a) in both krypton and xenon, and not so clearly in helium and neon. Tao and Bell (1967) have now attributed the discrepancy in the shoulder-width pressure product results to the presence of air impurity in the argon of Tao et a/.(1964), as suggested by Paul (1964), by observing the effect of deliberately admitting impurity. A typical argon shoulder appears in Fig. 1 [from Falk et a/.1965)], curve (a), the case of electric field zero. Paul (1964) derived an annihilation rate as a function of time from his data, and it shows a rise from an average Zeffof about 15just after the prompt peak, to a plateau value of about 30 over a time of 50 nsec in about 5 atm of argon. As there is at any time a distribution of positron velocities, the annihilation rate, except at the longest times when equilibrium has been reached, cannot be easily converted to Z,,, as a function of positron energy. Paul (1964) made an exploratory calculation to derive Z,,, from his data, assuming a certain initial positron velocity distribution, and using a momentum transfer cross section
POSITRONS AND POSITRONIUM IN GASES
1 0 :
- 10'
73
R
Id
1 0 :
-
- 10: 105
1 0 :
19
0
-
-Id
10 20
40
€0 80 100 Time (nsec)
120 140
160
FIG. I . Positron annihilation spectra in argon for three values of applied electric field (the background and 0-Ps component have been subtracted). The argon pressure was 10.5 atm at 25°C. Lifetimes of the exponential decays are (a) 17.9 rt 0.7 nsec, (b) 26.9 f 1.0 nsec, and (c) 34.8 f 1.5 nsec. The solid lines represent the fit to the data using the momentum transfer cross section of Eq. (12), and the Z,,, of Eq. (13b). (From Falk ei a]., 1965.)
available at that time in a diffusion equation. Clearly reliable collision cross sections are needed to analyze such data. Jones and his collaborators (Falk et al., 1965) have performed similar experiments, with, in addition, the annihilation region subjected to an electric field of several hundred volts per centimeter. The annihilation rate as a function of time depends on the electric field as the equilibrium energy of the positrons will now be relatively considerably higher, and presuming an energy dependence o f Z e f f .Figure 1, taken from their paper, shows that the shoulder indeed narrows with increasing electric field, and almost disappears at the highest field shown. We see also from Fig. 1, curve (c), that a field of 682 V/cm in 10.5 atm of argon at 25°C halves the final (equilibrium) annihilation rate, compared with the zero field case, showing a considerable dependence of Z,,, on energy. The electric field gives an extra parameter for the analysis. Assuming certain functional forms in zi, the positron speed, for the momentum transfer cross section and for Zeff,which are used in a diffusion equation for the positron
P . A . Fraser
74
velocity distribution, Falk et al. (1965)find that the annihilation rate data as a function of electric field and of time may be fitted quite well by the choice of two parameters: the magnitudes of the momentum transfer cross section and of Zeff.A uniform initial positron velocity distribution in the region below the Ps formation threshold was assumed. The solid lines in Fig. 1 are those so calculated. In their analysis, Ps formation is neglected; while Ps formation is enhanced by an electric field (Marder et al., 1956), Falk et al. (1965) kept the field sufficiently low to keep the loss of positrons via Ps formation hopefully negligible. Their curves fit the experimental data quite well, except at the shortest times, and assuming a validity for the functional forms assumed, their derived results should hence be quite good up to a few electron volts, if not to the Ps formation threshold. They deduce the following for the momentum transfer cross section ( b d ) and for the annihilation cross section (0,): bd(u)
= 1.32(vt,/o)na02
o,(u)= 3.80 x 1 0 - 6 ( ~ t h / ~ ) 1 . 5 n a o 2
(12) (134
where uth is the positron velocity at the threshold of Ps formation (8.9 eV; kth = 0.81). The a,(u) may be converted, using Eq. (6), to
Zeff= 7.1/Jk.
(13b)
The equilibrium Zefis for the nonzero electric fields shown in Fig. 1 (curves (b) and (c)) correspond to energies of about 0.25 and 0.7 eV, clearly small enough to be consistent with their assumption of negligible Ps formation. The value of b d derived in this way by Falk et al. (1965)accords well in the threshold region with the measurement of Marder er al. (1956; Teutsch and Hughes, 1956). The latter authors obtained for the case of argon b d = 1.5nao2 (f25%). Marder et al. (1956) studied the enhancement of Ps formation by electric fields, and worked at very much higher fields (or more precisely, much higher field/pressure ratios) than Falk et al. (1965).A parameter in the theory of the enhancement (Teutsch and Hughes, 1956) was the momentum transfer cross section below the Ps formation threshold, which was taken to be constant. This latter assumption is probably better than it would at first seem, as the high fields used would give equilibrium positron energies greater than a few electron volts. Further, their assumption that Zeffw 2 is probably fairly good at these energies; their results were not sensitive to this parameter in any case. Marder et al. (1956) have also measured in this way b d for helium and for neon. Together with the work of Falk et al. (1965)on argon, these represent apparently the only positron-atom collision cross section measurements.
POSITRONS AND POSITRONIUM IN GASES
75
Jones and his collaborators are continuing their work on argon (Jones and Orth, 1967). In this later work they have used three sets of theoretical cdand Zeffin their diffusion equation, and compared the results with observation. The models are discussed in more detail in Section IV,A,4. It sufficesto say here that one model adapted from that which gives good results for efectronargon collisions gives a final Zeffabout a factor of 10 down from experiment. The other models, which contained an adjustable parameter which was chosen so that the final equilibrium annihilation rate was close to the zero electric field observation, gave a dependence of the equilibrium annihilation rate on (E/p) quite different from experiment. They conclude that the short-range effects of the positron-atom interaction need more careful consideration in calculations. Tao and Bell (1967; Tao 1965) have measurements of the decay rate as a function of time in pure argon that indicate some structure to Z,,, , which they term " annihilation resonances." This is an interesting recent development. A summary of the various measured equilibrium decay rates in argon is included in Table I, and it seems that the thermal energy value ofZeffis well established to be about 29 or 30. TABLE I MEASURED EFFECTIVE NUMBERS OF ANNIHILATION ELECTRONS PER ATOMOR MOLECULE, Zerr,FOR THERMALIZED POSITRONS Remarks (gas, room temperature, unless otherwise stated)
Atom or molecule
References
He
Daniel and Stump (1959)
3.2
Duff and Heymann (1962) Falk e t a / . (1965)
3.25 f 0.22 3.92 5 0.04
Osmon (1 965a)
2.46
Roellig and Kelly (1965)
3.43 f 0.19
Falk and Jones (1966)
3.80 i0.04
Roellig and Kelly (1967b)
3.96
zerr
Gas, wide temp. (7°K to room temp.) and density range; one measurement in liquid at 5.1"K.
0.04
Caption to their Fig. 3 should give lifetimes (a) 23.2 & 0.27 nsec, (b) 24.9 f 0.24 nsec. (Falk and Jones, 1966) From Osmon's A (Osmon, 196%). Gas, 4.2"K; average result. Error estimated by author. Gas, 77°K mainly; wide density range.
P.A . Fraser
76
TABLE I (conrinued)
Atom or molecule
References
Zerr
* 0.32 * 0.34
Paul and Graham (1957) Wackerle and Stump (1957) Liu and Roberts (1963a)
3.92 3.12 3.67
Ne
Osmon (1965a)
3.6
Ar
Paul and Saint-Pierre (1963) Tao et a/. (1964) Paul (1964) Falk and Jones (1964)
25.6 & 0.4
Falk el al. (1965) Osmon (1965a)
28.8 & 1.1
Tao and Bell (1967)
25.7 i 0.9
Jones and Orth (1967) Liu and Roberts (1963b) Falk and Jones (1964)
28.0 i0.4 12.6 + 0.5 60.5
Osmon (1965a)
31.4
Xe
Osmon (1965a)
143
HZ
Osmon (1965c) Liu and Roberts (1963a)
11.7i0.8 6.9 f0.3
Dz
Liu and Roberts (1963b)
6.1 i0.2
N2
Tao et al. (1964) Osmon (I 965c) Liu and Roberts (1963a)
27.2 21.4 5.5 13.8 i 0.5
Osmon (196%) Liu and Roberts (1963a) Osmon (1965~)
19.5 & 1.6 13.7 It 0.6 24.4 f 8.1
He
Kr
0 2
COz
0.12
*
30.0 1.0 30.7 k 1.0 26.5 i2.0
15.1
*
Remarks (gas, room temperature, unless otherwise stated) Liquid, 4.2"K. Liquid, 4.2"K. Liquid, 1.5-4.2"K, superfluid and normal; rate proportional to density. From Osmon's A (Osrnon, 196%). Average of three results. Pressures converted to 0°C equivalent; averageof four results, From Osmon's A (Osmon, 1965~). Estimated by author from their figures. Liquid, 86°K. Pressures converted to 0°C equivalent. From Osmon's A (Osmon, 1965~) From Osmon's A (Osmon, 196%). Liquid and solid lO-20"K; rate proportional to density. Liquid, 20.4"K; compare with their liquid and solid H z result. Liquid (and solid! see text), 55-77°K. 1Liquid, 90°K.
POSITRONS AND POSlTRONlUM IN GASES
77
2. Helium The case of positrons in helium has not received the detailed treatment given to argon, as outlined previously. However there have been many measurements of the equilibrium decay rate, presumably belonging to thermalized positrons. The measurements, with references and remarks, are included in Table I. Falk et al. (1965) detected a shoulder in helium at a pressure of 59.7 atm, and this has been sharpened in later experiments with higher purity helium (Falk and Jones, 1966). They are also performing electric field experiments in helium, which will provide a good test of theoretical momentum transfer cross sections and Z,,, for a system more amenable to theoretical treatment than argon. Marder et a/. (1956; Teutsch and Hughes, 1956) also determined c d for positrons in helium, again presumably valid for energies close to the Ps formation threshold. The very low value they obtained, od = 0.023naO2 ( + 2 5
%)
has been hard to understand theoretically (see Section IV,A,3). Yet an approximate calculation of the slowing down time of positrons from Ps formation threshold to thermal energies based on this cross section value at threshold, and assuming l/tl dependence, gives a result in good accord with the shoulder width in helium observed by Falk et al. (1965). [We recall the accord between the Marder et al. (1956) and Falk et al. (1965) measurements in argon.] This approximate argument tends to support their low value, but we must await the results of at least electric field experiments in helium for confirmation or otherwise. Experiments on the annihilation rate of thermalized positrons in helium have been carried out under various conditions of temperature and state, as listed in the remarks of Table I, and there is interesting agreement in general between the results in both gas and liquid. However, we mention particularly the work of Roellig and Kelly (1965) on positron lifetimes in helium gas at 4.2"K and pressures of the order of an atmosphere. From the observed time spectrum of annihilation counts they subtracted the background and the long lifetime contributions. The remaining spectrum showed, following the prompt peak, an exponential decay lasting some 20 nsec, then a narrow hump followed by a second faster exponential decay. The first lifetime they later attributed (Roellig and Kelly, 1967b) to the decay of free positrons (see Table I for the derived Zeff);the hump and subsequent more rapid decay (from two to five times faster than the first) they attribute to the formation of clusters of helium atoms attracted to the thermalized positron, with the electron density at the positron thus enhanced (Roellig and Kelly, 1967b). Certainly the changes in the spectrum, which are not seen in their experiments at 77°K over a wide density range, suggest a change in the mode of positron annihilation under these conditions, not describable in terms of Z,,, , Eq. (3).
78
P . A . Fruser
3. General Remarks Table I also shows practically all the available results for the annihilation rates of thermalized positrons in atomic media, and a few molecular results. These are all reduced to the unifying notion of Z,,,; the original data in the literature are given in many forms ranging from psec-' atm-', through sec-' amagat-' to volume rates of cm3 sec-' ! Except where otherwise stated by the authors, the gaseous data have been presumed to be taken at a room temperature of 25°C. One hesitates to make generalizations from such a table, yet the well studied cases of helium and argon suggest that the Z,,, for very slow positrons is about twice Z. For other than helium, the Zeffin condensed media appears to be about half the gaseous value. Liu and Roberts (1962, 1963a, 1963b) found that the annihilation rate was proportional to density through the liquid-solid phase transition in a number of condensed gases, and indeed used this observation to make the first determination of the density of solid nitrogen by means of positron annihilation. Paul and Saint-Pierre (1963) and Osmon (1965~)have measured annihilation rates in polyatomic gases, and in some diatomics not shown in Table I. The observed rates in some cases, e.g., C1, (see also Tao, 1965), are so very large that interpretation in terms of the annihilation of free positrons with molecular electrons seems untenable (one obtains Zeff'sin the thousands, for example). The suggestions have been made that in such cases the positron becomes bound to the molecule (see Section V) before annihilation or that long-lived collision complexes are formed (Gol'danskii and Sayasov, 1964), each circumstance giving a large average electron density at the positron position. While it is outside the scope of this article, it is interesting to note that it has also recently been observed in metals that positrons are not always thermalized before annihilation (Kim et a/., 1967).
B. POSITRON~UM IN GASES There is little one can add to the recent thorough description of Green and Lee (1964) concerning the fraction of positrons from a radioactive source that forms Ps. The observed fractions are often within the wide limits given by the Ore gap considerations (Section II,A), but in some cases they are outside. The recent measurement in argon of Falkand Jones (1964), which confirms that the fraction of the Ps formed as 0-Ps is 3, as expected from statistical arguments, should be noted. Their measurements give the fraction as 69 f 7 (%); it is possible that their low central value reflects the suggestion of Mohr (1955) that overall perhaps a little more than 25 % of the Ps formed will be p-Ps, as 0-Ps
POSITRONS AND POSITRONIUM IN GASES
79
formed with higher kinetic energies may break up before decay, while the p-Ps decays promptly upon formation. The area of research into Ps in gases we wish to concentrate on is that of the quenching of 0-Ps by the conversion and pickoff processes (Section I1,D).
I.
Conversion Quenching
The most clearcut observed case of conversion quenching of 0-Ps is that by NO. The most recent measurement is that of Heymann et al. (1961), who measured, via the y-ray spectrum method, the decrease of the 3 y counting rate with increasing pressure of the quenching gas. They found a quenching cross section per molecule of 0.14 x cm2 or O.l6naO2,assuming thermalized 0-Ps. This result is clearly consistent with the interpretation that it is a conversion process by which NO quenches 0-Ps, in that the cross section is of the order of atomic areas. The quenching by NO is so efficient, that this gas is introduced into other gases in small quantities to suppress the long-lived tail of the annihilation time spectrum (Section 11,B). The case of O2 is not quite so clear. The quenching rate per atmosphere of Heymann et al. (1961) gives a cross section, using Eq. (7) and assuming thermalizedo-Ps,of(0.10 x x 10-'6cm20r(0.11 x 10-2)7ra,2,which is in good agreement with the more accurate value of (0.114 & 0.015) x lo-' x cm2 or (0.130 kO.017 x 10-2)nao2 from lifetime measurements of Celitans et al. (1964). I f the quenching process is interpreted as due to pickoff, the result of Celitans et al. (1964) leads to a 'Z,,, of 31 [Eq. (llb)], which being about twice Z is rather large. The observed cross section is 1/100 of that in NO, and is thus considerably smaller than atomic areas. The analysis of Ferrell (1958) of conversion processes cannot suggest any reason for a substantial difference between the two molecules. While Celitans et al. (1964) conclude that the cross section is too small for the process to be considered as conversion quenching, it would seem on balance that conversion is the preferable interpretation and indeed it is so considered in a later unpublished report by Tao and Green (Tao, 1967). Further support comes from the work of Gittelman and Deutsch (1956; Gittelman et a[., 1956), which showed that in the presence of oxygen, in argon for example, a magnetic field quenches more than 3 of the three-quantum decay, showing that collisions with oxygen molecules could reorient the 0-Ps spin (but the effect was smaller than in the case of NO). I t is worth remarking that at low partial pressures of oxygen in argon, less than atm, Gittelman and Deutsch (1956) found that the quenching cross section of O2 was at least as large as that of NO, but at higher pressures was at least 100 times smaller in accord with the results quoted above. The work of Celitans et al. (1964) was done at pressures greater than 2 atm, and that of Heymann et al. (1961) at pressures greater than 3 atm.
80
P . A . Fraser
Gittelman and Deutsch (1956; Deutsch and Berko, 1965) suggest that the large quenching in low partial pressure of 0, comes about through unthermalized o-Ps exciting the singlet state of O2 at 1.6 eV; this process would not be available to thermalized o-Ps, which is consistent with the observed lower quenching rates at higher 0, partial pressures. Fraser (1961) has calculated the conversion quenching cross section of o-Ps by atomic hydrogen, which is analogous to NO in having a single unpaired electron. He found a very large cross section, at low energies many times larger than atomic areas (Section IV,B,2). It would be an interesting model calculation to consider the conversion quenching of o-Ps by orthohelium; the results may shed some light on the smallness of the 0, cross section, and such a calculation is planned. 2. Pickoff Quenching
We shall leave the case of quenching by helium until the following section as recent results show this case to be more complicated than earlier believed (the complication could apply to other gases also). We shall consider, other than He, the cases of Ne, Ar, H, ,N, and CO, . There appears to be not much else in the simpler substances. The results for lZeff,Eq. (9), the number of annihilation electrons per atom or molecule in a singlet state relative to the positron, are summarized in Table 11, and have been obtained by essentially two methods : (a) the decrease in the three-photon counts with pressure by observation of the pray spectrum (Heymann er al., 1961) or by observation of three-photon coincidences (Celitans and Green, 1964); (b) the change with pressure in the slow decay measured by delayed coincidence techniques (e.g., Duff and Heymann, 1962; Celitans et al. 1964). In argon the expected linear relation between the inverse of the three-photon counting rate and pressure (Section II,D) is observed by Heymann etal. (1961) for pressures greater than 15 atm, and up to 32 atm, at room temperature; the departure from linearity at lower pressures is well accounted for by noting that at the lower pressures not all the positrons are stopped in the observation area (Celitans and Green, 1963). Correcting the observed low counting rate for this effect, Celitans and Green (1963) show that the data of Heymann et al. (1961) give linearity with pressure from 4 atm (the lowest pressure used) to the 32 atm limit of their experiment. and For argon-oxygen mixtures, with the 0, fraction between 5 x 5 x and working in the pressure range 2 or 3 atm to 20 or 25 atm, Celitans and Green (1964) and Celitans et al. (1964) obtain the same result for quenching in argon as Heymann et al. (1961). The quenching rate was independent of the O2 fraction in the range mentioned. For 0, fractions greater
81
POSITRONS AND POSITRONIUM IN GASES
TABLE I1 MEASURED EFFECTIVE NUMBERS OF SINGLETANNIHILATION ELECTRONS PER ATOM OR MOLECULE, Iz,rr,FOR PICK OFF QUENCHING OF ORTHOPOSITRONIUM
Atom or molecule Ar
Hz
Nz
CO,
References Heyniann et al. (1961) Celitans and Green (1964) Celitans ef al. (1964) Jones and Orth (1967) Paul (1958)
'Zdf
0.339
I=
o.340 0.36 0.22
Remarks (gas, room temperature, unless otherwise stated)
* 0.007 ~
o,oo7 0.05
* 0.02
Osmon (I 965c) Liu and Roberts (1962) Liu and Roberts (1963a) Liu and Roberts (1962) Liu and Roberts (1963a)
0.108 i0.041 0.039 & 0.002 0.043 & 0.004 0.070 k 0.004 0.079 & 0.007
Tao et a/. (I 964) Celitans and Green (1964) Celitans et al. (1964) Osmon (I 965c) Paul (1958) Liu and Roberts (1963a) Liu and Roberts (1963a)
0.24 0.28 i0.02 0.30 7 0.02 0.28 i0.14 0.156 t 0.008 0.164 i 0.01 5 0.31 L 0.03
Heymann et al. (1961) Osmon (1 96%)
0.77 0.37 f 0.20
Liquid, 875°K Liquid, 15-20°K. Liquid, 15-20°K. Solid, 9-14°K. Solid, 9-14°K.
Liquid, 77.5"K. Liquid, 63-77°K. Solid, 62°K; density as measured by positron annihilation rate.
than the conversion quenching by O2 took over. However, for lower 0, fractions (1 and 2 x in argon both Celitans and Green (1964) using the three-photon coincidence rate technique, and Celitans el al. (1964) using the delayed coincidence lifetime technique, obtained a quenching rate (essentially the same in the two experiments) as a function of pressure which is better fitted by assuming it proportional to (pressure)2. This behavior is indeed difficult to understand, as not only did Celitans and Green (1964) apply the " stopping fraction " correction (Celitans and Green, 1963), but also the lifetime measurement gave the same result. At the higher pressures, above 15 atm, at these low 0, fractions the observations are in agreement with those at the higher 0, fractions, which displayed the linear pressure dependence. In view of the effect of small air impurity in argon on the width of the shoulder in the positron annihilation spectrum (Tao and Bell, 1967), though with larger
P . A . Fraser
82
impurity fractions, it is possible that at the low O2 fractions the o-Ps has not been moderated to thermal energy and that Celitans and Green (1964) and Celitans et al. (1964) have observed the effect of a strongly velocity dependent pickoff rate. There would seem to be no connection between this observation and that of the large quenching cross section of 0, at low partial pressures (Section III,B,l), as firstly the latter would give an enhancement not observed and secondly the partial pressure of O 2in the work of Celitans and Green (1964) and Celitans et al. (1964) is far less than that in the 0, quenching experiments. As Celitans et al. (1964) suggest, further experiments seem called for. In nitrogen-argon mixtures, with the N, fraction between l o w 4and Celitans and Green (1964) and Celitans et al. (1964) observed almost the same quenching rate as in the oxygen-argon mixtures in the 0, fraction range of 5 x to 5 x They point out that this is evidence that in both cases the quenching mechanism is that of pickoff in argon. In pure nitrogen the rate is only slightly less than in argon at the same pressure. The results summarized in Table I1 support the qualitative notion that 'Zefffor pickoff would be a small fraction of Z. The pickoff rate in liquids is smaller than in the corresponding gas. This may be due to the appearance of cavities or bubbles in the liquid, which are discussed in greater detail in connection with helium in the following section. Hence the 'Zeff'sfor liquids in Table I1 should perhaps be considered as apparent values. For example, while Liu and Roberts (1963a) found the positron annihilation rate proportional to density through the liquid-solid phase change, as noted in Section 111,A,3,they noted adoubling of the o-Ps decay rate in H, in going from liquid to solid (the density change is about 20%). Liu and Roberts (1963a) remark that the rate in the gas at comparable densities differs little from that in solid H, . In the case of N 2 , assuming a density of solid N, as measured by the rate of positron annihilation by Liu and Roberts (1963a), their measured o-Ps lifetime leads to a lZeffin good agreement with that for gaseous N, . 3. Quenching in Helium
Until fairly recently it seemed that we had a good quantitative knowledge of the pickoff process in gaseous helium. The room temperature experiments of Heymann et al. (1961) (y-ray spectrum method) and of Duff and Heymann (1962) (lifetime method) gave, respectively, 'Z,,, = 0.135 k 0.068 and 0.118 0.011. The helium pressures at which they worked were 8 to 11 atm in the first experiment, and 34 and 55 atm in the second; these correspond' to helium densities of less than lo-, gm/cm3. In the Duff and Heymann (1962) experiment, freon was introduced at a partial pressure of 600 mm Hg to remove promptly the free positron annihilation component. The work of
+
83
POSITRONS AND POSITRONIUM IN GASES
Daniel and Stump (1959) over a wide temperature range (9.2"K to room temperature) and over a wide density range (from 50 to 300 times the STP density) suggests a 'Z,,, of about 0.1 with a large uncertainty due to a wide scatter of points. At high densities of helium, such as in the liquid and in the low temperature gas under pressure (the density of liquid helium at 4.2"K is 0.124 gm/cm3), the observed annihilation rate of 0-Ps is only a little greater than that of the isolated 0-Ps atom, very much less than could reasonably have been expected if the pickoff process continued simply to these densities. The numerous observations, listed in Table 111, made under different conditions of temperature and state, give an 0-Ps decay rate of close to 1.1 x lo7 sec-'; the natural rate is 0.7 x lo7 sec-'. To explain the early observations of this low rate Ferrell(l957) suggested that the repulsive exchange force between 0-Ps and He could lead to the rapid formation of bubbles in the liquid, and by implication in the high density gas, thus reducing the annihilation rate toward that of isolated 0-Ps. Daniel and Stump (1959) observed the long life in gaseous helium at temperatures of 5.3 to 7°K at a density of about 0.06 gm/cm3. A variation with temperature and pressure of the decay rate in the bubble predicted by Ferrell (1957) through the details of his model, which included the bubble being filled with saturated helium vapor, has not been observed. Tndeed, the bubble seems rather empty (see below); the constancy of the observed decay rates suggests a constant low helium density in at least the very low temperature bubbles. TABLE I11 MEASURED ANNIHILATION RATESOF ORTHOPOSITRONIUM IN HIGHDENSITY HELIUM References Paul and Graham (1957) Wackerle and Stump (1957) Paul (1958) Daniel and Stump (1959)
Rate (in lo' sec-') 1.10 0.06 0.83 + 0.14 1.18 & 0.06 1.06 & 0.06
0.89 & 0.06 0.95 0.78 1.15
Liu and Roberts (1963a)
1.14 & 0.10
Conditions Liquid, 4.2"K. Liquid, 4.2"K. Liquid He 11, 1.4"K. Liquid, 5.loK,just below critical point. Liquid, 4.2"K. Gas, 5.3"K Density about Gas, 53°K 6 x lo-* gm/cm3 Gas, 7°K (rates estimated by author from their figure). Liquid, 4.2K".
I
P.A . Fraser
84
Roellig and Kelly (1967a) have observed the onset of the bubbles at suitably high density in helium gas. They measured the decay rate of 0-Ps at 77°K and at 4.2"K over a range of densities, and the rates showed a significant downward break at close to the predicted densities for bubble formation. Their figures are reproduced in Fig. 2. We give a sketch of the method of prediction. The exchange force repulsion between very slow 0-Ps and a helium atom, reflected in a positive scattering length, presents a potential barrier to an 0-Ps atom found in the medium and attempting to move around in a random but dense distribution of helium atoms. The height of the barrier is proportional both to the scattering length and the number density of helium atoms, and inversely proportional to the 0-Ps mass. The theory of the effect, which was first discussed by Fermi, has been recently summarized by Levine and Sanders (1967), to whose paper on a related topic we shall return shortly. At a high enough density of helium atoms, at a given temperature, it becomes energetically favorable for the 0-Ps to exist in a bubble (or cavity) in the medium, which requires work to create, rather than to move through the medium. By consideration at a fixed temperature of the energy of the system of an 0-Ps atom trapped in a spherical cavity, which consists of the zero point energy of the 0-Ps bound in the cavity and the pressure-volume work required to create the cavity (the surface energy is neglected), Roellig and Kelly (1967a)
I
0
I
0.01
I
1
002 003 Density (g/cm3)
I
004
005
006
007
008 QO9
Density (glcm')
FIG. 2. Annihilation rate of 0-Ps in helium gas at 77°K and 4.2"K (the dashed curves, labeled by cavity radii, are theoretical results for the cavity rate taking into account barrier penetration by the 0-Ps). The ordinate scale begins at the natural 0-Ps rate, 0.71 x lo7 sec-'. (From Roellig and Kelly, 1967a.)
POSITRONS AND POSITRONIUM IN GASES
85
find an equilibrium cavity radius possible for densities greater than a certain minimum. Using a scattering length for o-Ps and He calculated by Fraser (1962), Roellig and Kelly (1967a) find that for 77°K and for 4.2"K, the minimum densities for cavity formation are, respectively, 6.4 x and 7.3 x gm/cm3. It is reasonable to expect that the bubbles form at a lower density at a lower temperature. The corresponding equilibrium cavity radii were about 6 and 18 A. The experimentally observed annihilation rates gm/cm3 for 77°K and at just show a downward break at densities of 7 x less than lo-' gm/cm3 at 4.2"K; Roellig and Kelly (1967a) consider that this good agreement between observation and the calculated estimates represents a direct verification of the exchange force induced cavity model. The results of Daniel and Stump (1959) at 77°K also show what must now be considered a significant break in the right direction at a density of between 7 and 9 x gm/cm3. At least for low temperatures, less than 7"K, it would appear from the work of Daniel and Stump (1959) that for densities somewhat greater than the minimum for bubble formation, the rate falls to the value of about lo7 sec-'. For example at 5.3", 5.8", and 7°K Daniel and Stump (1959) observe this rate at a density of close to 6 x gm/cm3 which is surely greater than the minimum density for bubbles at these temperatures, according to the calculations and observations of Roellig and Kelly (1967a). On the other hand, at 9.2"K they observe what appears to be a "no bubble" rate at a density of about 3 x gm/cm3, which is between the minimum densities calculated by Roellig and Kelly (1967a) for 4.2"K and 77°K. A difficulty of interpretation shows up particularly in the 77°K decay rate results of Roellig and Kelly (1967a) shown in Fig. 2. While they reported no decay rates at this temperature for densities less than 1.4 x lo-' gm/cm3, their results suggest a sharp, perhaps linear, rise in the decay rate from 1, , the natural rate for o-Ps, at zero density, to about 2.51, at a density of lo-' gm/cm3. This rise is followed by a linear region of smaller slope, which continues to the bubble onset density of 7 x lo-' gm/cm3. This second linear portion extrapolates to give at zero density a rate about twice the natural o-Ps rate 1,.It would thus seem inconsistent to interpret this region as simple pickoff with a rate (A, A,), A, being proportional to density. Roellig and Kelly (1967a) give two results at 4.2"K at densities less than the minimum for bubble formation which are seen in Fig. 2 (0.66 and 0.82 gm/cm3), and these rates fall on the sharply rising first region of the 77°K data. Interpreting these rates each as natural rate plus a density proportional pickoff rate, they correspond to lZerfE 0.25 k 25 %. It is possible that the second linear region in the 77°K data corresponds to a transition region (in density) between ordinary density proportional pickoff for gm/cm3 and the complete onset of bubbles at 7 x densities less than gm/cm3. While this region seems wide, the corresponding change in average
+
86
P. A . Fraser
interatomic spacing is not relatively large, and is certainly much less than the change in spacing for densities zero to lo-' gm/cm3. A possible difficulty with this interpretation is that a similar transition region at 4.2"K is not clearly observed. A wide transition region is observed in the mobility of electrons in high density and low temperature helium gas, the problem considered by Levine and Sanders (1967) in the paper referred to earlier in connection with the theory of bubble formation. Low energy electrons too are repelled by helium atoms, and the spectacular decrease of mobility with increasing density of helium is attributed to the formation of similar bubbles. Roellig and Kelly (1967a) remark that the ratio (scattering length)/mass is approximately the same for both the o-Ps -helium and the electron-helium systems. Thus for a given density of helium, the potential barrier is very nearly the same height for both 0-Ps and electrons, and the cavities will be much the same size. The theory considered by Levine and Sanders (1967) is analogous to that of Roellig and Kelly (1967a) and predicts a density for complete bubble onset, but does not treat the transition region adequately. [This is not the place to say more on this fascinating question of the mobility of electrons in high density helium; we refer the reader to the paper of Levine and Sanders (1967) for complete references.] The value of 'Z,,, of about 0.25 one can obtain from the results at 4.2"K of Roellig and Kelly (1967a) is about double the apparently reliable value of Duff and Heymann (1962) of 0.1 18 f 0.01 1, measured at room temperature but at comparable densities. However, we recall that the Duff and Heymann (1962) helium was deliberately contaminated with 600 mm of Hg of freon to remove the free positron component of the annihilation time spectrum promptly. Roellig (1967) has suggested that bubbles form in freon containing o-Ps at freon pressures at room temperature greater than about 8 atm, and that these may have affected the Duff and Heymann (1962) result. Roellig bases his suggestion on the results of the measurements of Deutsch (1951) on decay rates in freon; Deutsch's figure shows decay rates equal to the natural o-Ps rate of 0.7 x lo7 sec-' for freon pressures greater than about $ atm, but also shows one point at & atm with a decay rate of about lo7 sec-' with small uncertainty. Roellig suggests that there is pickoff in freon at low pressures, but that somewhere between atm and $ atm bubbles set in, thus affecting the Duff and Heymann (1962) experiment. We should point out that Deutsch (1951) attributed this anomalous point to the effect of free positron annihilation at low freon pressures. It is perhaps interesting to note that the observed decay rate of 1.1 x lo7 sec-' of o-Ps in high density helium at low temperature corresponds to a helium density of about 0.3 x lo-' gm/cm3, if we assume 'Z,,, = 0.25. The bubbles are thus not completely devoid of atoms on this interpretation, and a bubble of radius 12 A [a radius suggested by the work of Roellig and Kelly
POSITRONS A N D POSITRONIUM IN GASES
87
(1967a) at 4.2”KI would contain about 4 helium atoms. It is possible that barrier penetration by the o-Ps contributes to the excess of the cavity rate over the natural (Roellig and Kelly, 1967a).
IV. Theoretical Results A. POSITRON-ATOM COLLISIONS The differences from electron-atom collisions, mentioned in the Introduction, make the study of positron-atom collisions of particular theoretical interest. Further, Z,,, may be calculated from a knowledge of the collision wave function, and since the annihilation process requires the essential coincidence in space of the positron and an electron, this number “probes” the internal region of the wave function. Comparison with experiment thus provides a severe test of an approximate wave function in a very different region from that which gives the approximate result of a collision experiment. The at present impractical case of positron collisions with hydrogen atoms has, for obvious reasons, received the most attention. There is a growing volume of work on the practical case of helium. Work on other atoms is sparse, and on molecules almost negligible.
I . Calculation of the Annihilation Rate We consider the wave function, Y, ignoring spin variables, of the system of a low energy positron in the field of an atom to have asymptotic form
-
Wl,rz, r 3 , ... ;rp)r p + m +Ar1, rz, r 3 , . . .)Wp),
(14)
where ri are the electron coordinates, rp those of the positron, $ A the atomic wave function normalized to unity, and F(rp)is normalized to correspond to a density of one positron per unit volume asymptotically. Then Z,,, , Eq. (3), the average number of electrons per atom at the position of the positron is given by Zeff =
cSd.,
dr, dr,
. . . I y ( r l , r 2 , r 3 , . . . ;ri>12.
(15)
This represents a generalization of the quantum electrodynamic result for the case of very slow positrons of Ferrell (1956), as given by Wallace (1960). The expression may also be obtained by a “naive” calculation in terms of the probability density and assuming the locality of the annihilation process.
P . A . Fraser
88
For the case that the positron motion is undisturbed by the atom, and the atom in turn undistorted by the positron, we have Y = $k1, r 2 , r 3 , . ..) expCikz,l
(16)
everywhere, and clearly Zeff = 2,
the number of atomic electrons. This approximation gives the " Dirac rate " (Section I1,C); one can see that the interaction of the positron and the atom could produce rates different from the Dirac rate-for example a mutual attraction could give an enhancement. For the case of an approximate wave function of the form = $ A 0 1 9 rz 3 r3 9
* *
mr,)
(18)
everywhere, with F(r,) in general different from the plane wave of Eq. (16), Zeff
s s
=
dr p(r)IF(r)I
'9
(19)
where p(r) is the electron number density of the atom with
dr p(r) = 2.
As it is not yet a practical case, Z,,, for the positron-hydrogen atom system has apparently not been calculated from the various approximate collision wave functions. It would be of considerable theoretical interest to compare the predictions, some of which would presumably be quite accurate. For the case of helium, with a spatial wave function Y(rl, r , ; r,), symmetric in the electron co-ordinates, Zeff = 2
s
dr1 dr2 Iy(r1, r2 ;r1)12.
(20)
2. Positron-Hydrogen Collisions Up to close to the Ps formation threshold in hydrogen, 6.8 eV or kZ = +, we can consider that the elastic collision problem is solved. We have the definitive variational I = 0 and 1 phase shifts of Schwartz and Armstead (Schwartz, 1961; Armstead, 1964) and the variational lower bound calculations for 1 = 0, 1 and 2 (with extrapolations) of Spruch and his collaborators (Hahn and Spruch, 1965; Kleinman et al., 1965). These results form a target (and a criterion) for those of other methods more applicable to other problems. The 1 = Ophase shift of Schwartz (1961) is shown in Fig. 3, curve(a), and the 1 = 1 result of Armstead and Schwartz (Armstead, 1964) is shown in Fig. 4, curve (a). We note here that the discrepancy between the I = O results of
FIG.3. The 1 = 0 phase shift, 70 in radians, for positron-hydrogen atom elastic scattering. The positron kinetic energy is k 2 in units of 13.6 eV, and the Ps formation threshold is at k2 = 1/2.Curve (a): Schwartz (1961). Squares: rigorous lower bounds of Hahn and Spruch (1965). Vertical bars: extrapolated estimates of Hahn and Spruch (1965). Curve (b): mean static atomic field approximation (MSF). Curve (c) : from expansion in hydrogen states ls-2s-2p-3s-3p-3d (McEachran and Fraser, 1965). Curve (d): MSF and virtual Ps (Cody e f a/., 1964). Curve (e): MSF and Temkin-Lamkin polarization potential (T-L) (Cody e f al., 1964). Triangles: MSF, virtual Ps, and T-L (Cody e t a / . , 1964).
k
FIG. 4. The I = 1 phase shift, rll in radians, for positron-hydrogen atom elastic scattering. Curve (a): Armstead (1964). Curve (b): rigorous lower bounds of Kleinman ef a/. (1965). Vertical bars: extrapolated estimates of Kleinman et a/. (1965);not shown for k = 0.1 and 0.2,for which cases they fall on curve (a) with uncertainties less than &3 %. Curve (c): from expansion in hydrogen states ls-2s-2p-3s-3p-3d(McEachran and Fraser, 1965).
P.A . Fraser
90
Rotenberg (1962), based on an expansion of the wave function in Sturmian functions, and those of Schwartz (1961), has now been removed following the discovery of a numerical error in Rotenberg’s work (Drachman, 1965, footnote 2). Figure 3 also shows the rigorous lower bounds on the 1 = 0 phase shift of Hahn and Spruch (1965), the small squares at k = 0.2, 0.4, and 0.6, and their extrapolated estimates, based on the bound calculations, of the correct phase shift, the vertical bars (whose extent indicates the uncertainty of the estimate). The rigorous lower bounds for I = 1 and 2 of Kleinman et al.(1965) are shown, respectively, in Fig. 4, curve (b) and Fig. 5, curve (a), together with their extrapolated estimates as vertical bars.
0.10
k
FIG.5. The / = 2 phase shift, T~ in radians, for positron-hydrogen atom elastic scattering. Curve (a): rigorous lower bounds of Kleinman et a / . (1965). Vertical bars: extrapolated estimates of Kleinman ef a/. (1965); not shown for k = 0.1. Curve (b): from expansion in hydrogen states ls-2s-Zp-3s-3p-3d(McEachran and Fraser, 1965).
Much of the work on this problem has been summarized by Mott and Massey (1965), and from the earlier work we shall select only a few items. Spruch and Rosenberg (1960) showed by a variational calculation with a trial function that included virtual Ps formation, that the scattering length for this problem had a negative upper bound: a
< - 1.397a0.
This was the first indication in hydrogen that, at very low positron energies at least, the distortion effects could produce a net attraction, overcoming the repulsive mean static field of the atom. They also gave a “reliable, but not
POSITRONS AND POSITRONIUM IN GASES
91
rigorous ” (Temkin, 1962)lower bound to the I = 0 phase shift, qo ,at k = 0.2: qo 2 0.156 rad.
The elaborate variational calculation of Schwartz (1 961) established that a
< -2.10ao
;
further, Schwartz judged from the nature of the convergence of his calculations that indeed a 2 -2.11ao.
For illustrative purposes the 1 = 0 phase shift for the mean static (repulsive) atomic field is shown in Fig. 3, curve (b). The vast difference between this result and that of Schwartz (1961)displays the difficulty of accurate calculation. I n contrast to the corresponding approximation, with exchange, in the electrun-hydrogen problem, this first approximation bears no resemblance to the truth. McEachran and Fraser (1965) have calculated the I = 0, 1 and 2 phase shifts below the Ps formation threshold in a “ close-coupling ” approximation using only hydrogen states, up to the 3d in the expansion (the mean static field result above corresponds to taking only the first 1s term in such an expansion). According to the results of Hahn el al. (1964), McKinley and Macek (1964),and of Gailitis (1964) [reviewed in this series by Peterkop and Veldre (1966)],such calculations provide a lower bound to the correct phase shift (strictly to tan q l ) . In Figs. 3 (curve (c)), 4 (curve (c)), and 5 (curve (b)) are shown only the “best,” according to the bound theorems, results of McEachran and Fraser (1965), the ls-2s-2p-3s-3p-3d approximation. These are immediately seen to be completely inadequate, in contrast to corresponding approximations in the electrun-hydrogen problem with exchange included (see the article by Burke in this volume). The work of Spruch and Rosenberg (1960)suggests that the inclusion of Ps states in the eigenfunction expansion, that is inclusion of virtual Ps formation, would be important. Smith (1961)has formulated the positron-hydrogen collision problem in the form of an expansion of the wave function in a series of eigenfunctions of both H and Ps. Cody et al. (1964)have treated the 1 = 0 case for such an expansion, truncated at the 1s H and Ps states (the virtual Ps approximation). Their results, which give a lower bound, are shown in Fig. 3,curve (d), and we note that the scattering length is positive, and the phase shift negative. This is perhaps surprising in the light of the work of Spruch and Rosenberg (1960), and of Bransden (1962)who included virtual Ps formation by a perturbation procedure and obtained a positive phase shift for the smaller k. The corresponding approximation in the case of He (Section IV,A,3) gives a negative scattering length and a positive phase shift for the smaller k.
92
P . A . Fraser
Cody et al. (1964) have also calculated the 1 = 0 phase shift for the case of adding to the static atomic field a polarization potential as given by the Temkin-Lamkin approximation (e.g., Mott and Massey, 1965). The results, Fig. 3 (curve (e)) are closer to those of Schwartz (1961), but do not provide a rigorous bound. Further Cody et al. (1964) have considered the model of adding the Temkin-Lamkin polarization potential to the virtual Ps approximation, and their results, shown as triangles in Fig. 3, considerably overshoot those of Schwartz (1961), except at k = 0.6. Drachman (1965) has considered the effect of the exact (to second order) adiabatic polarization potential of Dalgarno and Lynn added to the repulsive mean static field. For I = 0 the phase shift was rather larger (by some 30 % at the maximum) than that of Schwartz (1961), but he found that with the introduction of a single energy-independent parameter the result of Schwartz could be accurately reproduced. The parameter tl measures the amount of the monopole (spherically symmetric and here short range) part of the adiabatic potential retained. The choice of c1 = 0.1 gave the best results, but the case of complete monopole suppression tl = 0 gave results almost negligibly different. The effect of monopole suppression on the 1 = 1 and 2 cases is naturally less, and Drachman’s (1965) results (not shown in Fig. 4 and 5 for reasons of clarity) are not significantly different from the rigorous lower bounds of Kleinman et al. (1965). They are thus considerably less, at the higher k, than the extrapolated estimates of Kleinman et al. (1965), and for I = 1, the result of Armstead and Schwartz (Armstead, 1964). This calculation is of particular interest, as Drachman goes on to apply the derived “ rule” of complete monopole suppression to the case of positron-helium collisions (Section IV,A,3). A related calculation has been reported by Bransden and Jundi (1966), who considered the monopole, dipole and quadrupole parts of the DalgarnoLynn potential used by Drachman (1965). For 1 = 0, the three contributions give a result larger than that of Schwartz (1961) (some 13% at the maximum), while the dipole and quadrupole parts together give results below those of Schwartz (some 10 % at the maximum). Stone (1966) has introduced an adiabatic polarization potential by calculating variationally the coefficients (dependent upon the positron distance) of a limited expansion in hydrogen states of the distorted atomic wave function. His 1 = 0 phase shift is slightly larger than, but not very different from, that of Cody et al. (1964) using the Temkin-Lamkin potential (Fig. 3, curve (e)). Recently Drachman (1967) has calculated a lower bound on the I = 0 phase shift qo (strictly on tan so) using a trial function that included the first order adiabatic change in the atomic wave function that gives the second order Dalgarno-Lynn polarization potential used in his earlier work (Drachman, 1965). The assumed trial function is of the form to which the bound theorem of
POSITRONS AND POSITRONIUM IN GASES
93
Gailitis (1964) applies, and the results are somewhat lower than those of the more elaborate calculation of Hahn and Spruch (1965). Reported calculations on the formation of Ps by positron impact on H are few. They have been described by Bransden (1965) in this series, and are the Born approximation calculation of Massey and Mohr (1954) (with a distorted wave improved result at k = l), and the impulse approximation calculation of Cheshire (1964). Other calculations on this reaction are in progress (e.g., Bransden and Jundi, 1967; Fraser, 1967). Reliable results are of theoretical interest,as this reaction, with its product of small mass,constitutes a different type of rearrangement collision from those usually considered in atomic physics. Bransden and Jundi (1967) have extended the two state (H and Ps ground states) calculation of Cody et al. (1964) to above the Ps formation threshold for both 1 = 0 and 1. Further they have considered the effect of the adiabatic distortions of both the hydrogen and Ps wave functions (see Bransden and Jundi, 1966), and find that these polarization effects have a profound influence on the cross sections, particularly on the 1 = 0 part of the Ps formation cross section which is increased (to a magnitude of order mo2)by several orders of magnitude. The Ps formation reaction in hydrogen is an ideal case, as exact bound state functions are known, to which to apply the methods of Gailitis (1964) that give bounds. These methods, which add to a truncated eigenfunction expansion " correlation terms " which may be treated variationally, have been used with considerable success by Gailitis (1965) and by Burke and Taylor (1966) in the electron-hydrogen problem. The lack of orthogonality of the H and Ps states complicates the projection operator formalism, but various schemes are available (Hahn, 1966; Chen and Mittleman, 1966; Chen, 1966; Fraser, 1967). Bransden and Jundi (1967) plan a calculation that gives bounds. Mittleman (1966) has shown explicitly that, in analogy with the electronhydrogen case, positron-hydrogen scattering exhibits resonances just below the thresholds for excited states of both H and Ps. These resonances arise from the degeneracy of the excited states, and thus Mittleman makes no prediction about the existence of a resonance below the Ps formation threshold. Mittleman (1966) remarks that Temkin and Bhatia are carrying out a detailed calculation in this energy region. It may be relevant to point out that Russian authors (Shmelev, 1959; Din-Van-Hoang, 1964) have reported a quasi-bound state of the positron-hydrogen system about 0.9 eV below the Ps formation threshold. Yet accepting the correctness of the Schwartz and Armstead results (Schwartz, 1961; Armstead, 1964) any such resonance must be within 0.136 eV of the threshold, for 1 = 0 and 1. For completeness we note the classical calculations of Percival and Valentine on Ps formation in H. These are described elsewhere in this volume in the article by Burgess and Percival.
94
P . A . Fraser
3. Positron-Helium Collisions and Annihilation Rates
Perhaps the earliest calculation on the positron-helium collision problem, or indeed on any positron-atom collision problem, was that at zero energy of Ore (1949a), the pioneer of so much of the atomic physics of positrons and Ps. Ore took into account the polarization of the atom by describing the positronatom interaction by a potential that was of a Morse form found suitable for the HeH' molecule in the inner region, smoothly joined to the long-range l/r4 attractive polarization potential whose coefficient was given by the empirical He polarizability. He obtained a scattering length of
thus bringing out at an early date the importance of the attractive forces. Ore used his wave function to calculate Z,,, at zero energy, using Eq. (19), obtaining the value 2.8. Calculations ignoring polarization effects (e.g., Massey and Moussa, 1958; Malik, 1961) gave positive scattering lengths and negative phase shifts. A typical 1 = 0 phase shift given by a simple mean static repulsive field for this problem is shown in Fig. 6, curve (a) (from Kraidy, 1967). Allison et al. (1961)
FIG.6. The I = 0 phase shift, 710 in radians, for positron-helium atom elastic scattering. The positron kinetic energy is k2 in units of 13.6 eV, and the Ps formation threshold is at k2 = 1.31. Curve (a): typical result from mean static atomic field (MSF) (Kraidy, 1967). Curve (b): MSF and adiabatic polarization potential with complete monopole suppression (Drachman, 1965). Curve (c): MSF and virtual Ps (Kraidy and Fraser, 1967). Curve (d): MSF, virtual Ps, and Temkin-Lamkin polarization potential (Kraidy and Fraser, 1967).
POSITRONS AND POSITRONIUM IN GASES
95
obtained a negative scattering length (-0.16 ao) by a Kohn variational calculation allowing for distortion of the atom. Kestner et al. (1965) considered the elastic scattering of both electrons and positrons by helium. For the positron case they adapted a local “pseudopotential,” which incorporating both exchange and polarization effects, gave electron results in good agreement with experiment. Dropping, of course, the exchange contribution in the positron case, they obtained a negative scattering length ( -0.575a0) and an 1 = 0 phase shift that is positive for k < 0.6 and which rises to a maximum of about 0.066 rad at k = 0.3. Kestner et al. (1965) also calculated an 1 = 1 phase shift by a variational method, and this is presumably positive. Their elastic cross section exhibits a Ramsauer dip (the zero is removed by the significant p-wave contribution) and levels off at energies near the Ps formation threshold to 0.1 lna,’. Drachman (1966a) has extended the “ rule ” arising from his positronhydrogen calculation (Section IV,A,2 ; Drachman, 1965) to the helium case. For the helium wave function he used the uncorrelated, shielded hydrogenic form to apply the method of Dalgarno and Lynn for the polarization potential. Drachman makes two choices of the shielding factor and the amplitude of the atomic distortion, both choices being required to give an empirically correct asymptotic polarization potential. The choices give very similar results (the 1 = 0 phase shifts differing by at most 12 %), and since Drachman uses only one case in his calculation of Z,., (Drachman, 1966b), we consider only that case for convenience. In this choice the atomic distortion was taken to be as given by a direct calculation (i.e., unit amplitude) and the atomic shielding factor then turns out to be 1.5992, about 5 % less than the familiar 27/16 given by the simple variational calculation for the He atom. With complete monopole suppression, which his hydrogen calculations suggested would give good results for at least 1 = 0, Drachman (1966a) obtains a scattering length of -0.66a0, and an I = 0 phase shift as shown in Fig. 6, curve (b). The results are similar to those of Kestner et al. (1965), though slightly larger in magnitude. Drachman also calculated the 1 = 1 and 2 phase shifts, which were positive, and which for smaller kZ are proportional to k2 as they must be when the long range polarization force is present (e.g., Mott and Massey, 1965). Estimating the 1 = 3 phase shift, Drachman (1966a) calculates the elastic and the momentum transfer cross sections ; the latter is shown in Fig. 7, curve (b), and is seen to approach a value of 0.19nao2 at the Ps formation threshold. This is an order of magnitude greater than the measured value of Marder et a/. (1956), shown as a hatched block on the right edge of Fig. 7. From his hydrogen results (Drachman, 1965), Drachman judges that his 1 = 1 and 2 phase shifts for helium are underestimates, and shows that an increase of these phases increases the disagreement with experiment.
96
P. A . Fraser
FIG.7. The momentum transfer cross section for positron-helium atom collisions, in mO2. The hatched block represents the experimental result of Marder et al. (1956). Curve (a): mean static atomic field (MSF) and Temkin-Lamkin polarization potential (T-L) (Massey et al., 1966). Curve (b): MSF and adiabatic polarization potential with complete monopole suppression (Drachman, 1966a). Curve (c): MSF and virtual Ps (Kraidy and Fraser, 1967). Curve (d): MSF, virtual Ps, and T-L (Kraidy and Fraser, 1967).
Drachman (1966b) has calculated Z,,, as a function of positron energy from his collision wave function, through Eq. (20), using the distorted atomic wave function. His result is shown in Fig. (8), curve (c), plotted against k,and at the lowest energies is about 50% greater than the observations (Table I). In this paper Drachman also considers the state of excitation of the residual He+ ion following an annihilation. Massey et al. (1966) have surveyed the effect of simple polarization potentials on positron-noble gas collision calculations. They considered that such potentials would underestimate the effective attraction, virtual Ps formation not being taken into account, but that as such procedures gave adequate low energy results in electron-noble gas atom calculations, it would be worthwhile to treat the positron case. In the case of He, Massey et al. (1966) used the Ternkin-Lamkin adiabatic approximation (e.g., Mott and Massey, 1965).
97
POSITRONS AND POSITRONIUM IN GASES
J
l-
----
0
I
(a)
I
I
I
I
I
I
I
1
FIG. 8. Z,rr for positron annihilation in helium. Curve (a): mean static atomic field (MSF) (Kraidy, 1967). Curve (b): MSF and Temkin-Lamkin polarization potential (T-L) (Massey er a/., 1966). Curve (c): MSF and adiabatic polarization potential with complete monopole suppression (Drachrnan, 1966b). Curve (d): MSF and virtual Ps (Kraidy and Fraser, 1967). Curve (el: MSF, virtual Ps, and T-L (Kraidy and Fraser, 1967). The square at k = 0 in the result of Ore (1949a). The bars represent two experimental results (Table I): KR (Roellig and Kelly, 1967b) at 77°K; FJ (Falk and Jones, 1966) at 25°C.
Their momentum transfer cross section is shown in Fig. 7, curve (a), and again is much greater near the Ps formation threshold than the observed value. Through Eq. (19), Massey et a/. (1966) calculated Zeff,shown in Fig. 8, curve (b). At low energies it is about 40 % of the observed values (Table I). Hashino (1966) has performed a variational calculation at zero energy, including the dipole and quadrupole distortions of the atom, and his best value of the scattering length is Q
=
-0.5462~0.
Kraidy and Fraser (1967; Kraidy, 1967) have included virtual Ps formation in their calculation. Their result for the / = 0 phase shift is shown in Fig. 6, curve (c), and it is seen that the effect is much larger than is the case for hydrogen (Cody et a/., 1964), giving a scattering length of - 0 . 9 2 ~ and ~ a positive phase shift for k < 0.44. The He wave function used by Kraidy and Fraser (1967) was of necessity an approximation, indeed the simplest variational form, so the bound theorems (e.g., Peterkop and Veldre, 1966) do not
98
P.A , Fraser
strictly apply. Yet in Drachman’s phrase, the results do give a “semirigorous ’’ lower bound. Kraidy and Fraser (1967) have added to the above virtual Ps approximation an atomic polarization potential via the Temkin-Lamkin procedure, and the resulting I = 0 phase shift is shown in Fig. 6, curve (d). The scattering length is increased in magnitude to -2a,. This procedure destroys any possible application of the bound theorems, and their results may well overestimate the correct phase shift, judging from the results of Cody et al. (1964) for the corresponding approximation in hydrogen. The effect of the polarizability of the Ps, which is eight times that of hydrogen, in the region of the Ps formation threshold will further raise the I = 0 phase shift and decrease its magnitude, and this is being investigated. Kraidy and Fraser (1967) have also computed higher order phase shifts in the above approximations, and their positive phase shifts are larger than those of Drachman in the case of atomic polarization. Their momentum transfer cross sections for virtual Ps formation without and with atomic polarization are shown in Fig. 7, curves (c) and (d), respectively, and are again far from the experimental value. Their Z,,, results for the two approximations are shown in Fig. 8, curves (d) and (e), respectively. The virtual Ps plus atomic polarization result shows a strong k dependence and is in fair agreement with the most recent experimental results, which are also shown in Fig. 8. Drachman (1967) has extended his work on bounds on tan q,, in hydrogen to the case of helium, obtaining semi-rigorous bounds lying considerably below the results of his earlier (Drachman, 1966a) no-bound calculation and the semi-rigorous bounds of Kraidy and Fraser (1967). Wardle and Smith (1966; Moussa et al., 1967) are considering the elastic scattering using a trial wave function with a more accurate He wave function, virtual Ps formation, and correlation terms. Massey and Moussa (1966; Moussa, 1967) are carrying out a variational calculation on electron and positron scattering in helium, along the lines of that of Schwartz (1961) for hydrogen. This is an extremely elaborate calculation that should produce reliable results. On Ps formation in helium there are just two calculations: the approximate Born calculations of Massey and Moussa (1961) and the extension above threshold of the two state approximation by Kraidy and Fraser (1967; Kraidy, 1967). The latter are considering in particular the effect of Ps polarization on the cross section, but while the calculations are not complete at the time of writing, it appears that the effect is large, as found by Bransden and Jundi (1967) in the hydrogen case (Section IV,A,2). Without the inclusion of excited states of He or Ps in the expansion, these results have but a limited range of possible validity in the threshold region (some two electron volts).
99
POSITRONS AND POSITRONIUM IN GASES
4. Positron-Argon Collisions and Annihilation Rates
In the case of argon we have the experimentally derived momentum transfer cross section and Zeff,Eqs. (12) and (13b) of Falk et al. (1965). These are shown respectively in Fig. 9, curve (a) and Fig. 10, curve (a). The hatched block at the right edge of Fig. 9 represents the derived cross section of Marder et al. (1956). The vertical bar at k z 0.05 on Fig. 10 represents most of the thermal positron annihilation rate results for Z,,, . Massey e: 01. (1966) have considered the polarization effect by using the semiempirical polarization potential found by Holtsmark to be quite successful for the low energy electron-argon collision problem. Their result for the momentum transfer cross section is shown in Fig. 9, curve (b), and for Z,,, using Eq. (19), in Fig. 10, curve (b). The Zeffresult [also obtained by Jones and Orth (1967)l is seen to be a factor of ten low at small energies, which reflects the importance of the short range distortion effects in the calculation of Z,,, .
"
O
C
N
5
c
P
U
v) v) v)
0
W
!03
0
01
02
03
04
05
0.6
07
08
k
FIG.9. The momentum transfer cross section for positron-argon atom collisions, in mO2. The positron kinetic energy is k2 in units of 13.6 eV, and the Ps formation threshold
is at kZ = 0.66. Curve (a): derived from experiment, Eq. (12) (Falk e t a / . , 1965). Curve (b): mean static atomic field (MSF) and Holtsmark polarization potential (Massey et al., 1966). Curve (c): MSF and modified Holtsmark polarization potential (Jones and Orth, 1967). The hatched block represents the experimental result of Marder et a/. (1956).
P. A. Fraser
100
t
l 0
l 01
l
02
l
03
l
04 k
l
05
l
06
l
l 07
l
00
FIG. 10. Zerrfor positron annihilation in argon. The curve labeling corresponds to that in Fig. 9 (with curve (a) from Eq. (13b)). The bar at k z 0.05 is representative of most of the experimental results (Table I).
As discussed in Section III,A,l, Jones and Orth (1967) carried out two additional related calculations. In these the polarization potentials were adjusted in their internal “cutoff” parameter to give Zeffat zero energy as observed for thermalized positrons. In the one case they modified the Holtsmark form by decreasing the cutoff radius and thus increasing the internal magnitude, and in the other they used a polarization potential of the l/r4 form modified by a factor that was asymptotically unity and which near the origin behaved as r8. The Zeffresults were not significantly different, and only the first case is shown in Fig. 10,curve (c). The momentum transfer cross sections were similar up to k z 0.3, but that given by the second case levels off to 1.75rcao2 as the Ps formation threshold is approached. This is higher than the result of the first case which alone is shown in Fig. 9 curve (c). As discussed in Section III,A, 1, Jones and Orth (1967) have applied a critical test to the three calculated momentum transfer cross sections and Zeff’s. Through a diffusion equation, they have calculated the final equilibrium annihilation rate as a function of (electric field)/(pressure). None of the three models gives a result resembling experiment. However the cross section results in the threshold region are closer to the observations of Marder et al. (1956) than is the case in helium. It seems clear that in order to obtain agreement with experiment over the below threshold energy range, good account needs to be taken of the distortion of the atom during the collision.
POSITRONS AND POSITRONIUM IN GASES
101
5. Other Cases
Massey et al. (1966) have also considered the cases of neon and krypton in their survey of the effect of polarization forces on the positron motion. Near the Ps formation thresholds they find odz Trao2
for
Ne,
o,,% 2.5naO2
for
Kr.
and
The measured value for Ne (Marder et al., 1956) is 0.12naO2(&25 %). For Zeff, the results of Massey et al. (1966) extrapolate to low energies to give about 1.3 for Ne and 5.5 for Kr. For Ne, Osrnon's (1965a) measurements lead t o Z,,, = 3.6, and for Kr, those of Falk and Jones (1964) give Z,,, = 60.5. While molecules do not strictly fall in the category of this section, there has been so little theoretical work on positron-molecule collisions that a separate section is not needed. The experimental study of positrons in H, is of course possible (e.g., Osmon, 1965a), and a theoretical study not intractable. Massey (1967) has commenced calculations on positron-molecule collisions. A very preliminary calculation on positron-H, collisions has been reported by Massey and Moussa (1958).
B. ORTHOPOSITRONIUM-ATOM COLLISIONS 1. Calculation of the Pickoff Quenching Rate In analogy with the calculation of Z,,, for positron annihilation (Section it is possible to calculate 'Zeff,Section Il,D, Eq. (9), for pickoff quenching of 0-Ps from a knowledge of the wave function of the o-Ps-atom system. We consider the wave function here to include dependence on spin variables, and to leave out of consideration the triplet electron in the 0-Ps, one must project the wave function on the singlet spin function, xoo(sp,s,), of the positron and the ith electron, obtaining IV,A,l)?
ai(r1, s,; r 2 , s2 ; . . . ; ri ; . . . ; rp) = (xoo(s,,
si), Y),
(21)
where s p and si are the positron and electron spin variables. Placing rp = T i , integrating (including summing over remaining spins), and summing over i, one obtains n
102
P. A . Fruser
2. Collisions of Orthopositronium with Atoms Massey and Mohr (1954) treated, in Born approximation, the collisions of slow 0-Ps and H, and their results suggested that the conversion cross section (Section II,D) could be quite large and could depend strongly on energy. In this approximation, the conversion cross section is 4 of the elastic cross section. Fraser (1961) considered the problem, for the case I = 0 only, using appropriately symmetrized approximate wave functions involving (as a product) the ground state eigenfunctions of Ps and of H. This procedure could be considered as the retention of only the first terms of symmetrized eigenfunction expansions of the wave functions, and thus in the light of the later work on bounds (e.g., Peterkop and Veldre, 1966), his results give a lower bound on the tangents of the 1 = 0 phase shifts for energies less than the first excitation threshold at 5.1 eV. With only the I = 0 contribution considered, Fraser (1961) found the elastic cross section to vary strongly with energy, being 192nuO2at zero energy and falling to 2.9nuO2at 6.8 eV. The conversion cross section was a fraction of this, the fraction ranging from 0.176 at zero energy to 0.070 at 6.8 eV. The fraction is not necessarily 4 when the exchange symmetries are allowed for. The calculations on 0-Ps collisions with hydrogen have been discussed as an example of a rearrangement collision (the conversion process) by Bransden (1965) in this series. The earlier work of Fraser (1962) on slow 0-Ps collisions with He has been repeated with greater accuracy and extended to the p - and d-waves by Fraser and Kraidy (1966). The simplest He function was used in this work. Their elastic cross section varies from 14.2naoz at zero energy to 7.5nuo2 at 6.8 eV, and the I = 2 contribution is not negligible at the higher energies. Fraser and Kraidy (1966) have also calculated 'Zeff,Eq. (22), from this model and obtain the value of 0.033 at very low (thermal) kinetic energies. This compares with the result of Ferrell (1958) of 'Zeffz 0.02 for very slow 0-Ps in low density helium, and both values are very far from that derived from the work of Roellig and Kelly (1967a) of 0.25 and the earlier value of Duff and Heymann (1962) of 0.1 18 (Section 111,B,3). Both calculations took into account only the exchange repulsion between 0-Ps and He, and there must be compensating attractive effects such as the Van der Waals force between the two atoms as pointed out by Ferrell (1956). Bransden and Barker (1967) are considering the collision problem taking into account the Van der Waals force, and Fraser (1 967) is formulating a zero energy variational calculation. While the calculated 'Zeffof Fraser and Kraidy (1966) is far from adequate, the scattering length given by the model, 1.88~0,is perhaps much less bad. Roellig and Kelly (1967a) used an earlier value of 2 . 1 ~(Fraser, ~ 1962) in their bubble calculations (Section III,B,3), and obtained fair agreement with experiment. Again, then, it appears that while a certain approximate wave
POSITRONS AND POSITRONIUM IN GASES
103
function may well be adequate for collision results, better account needs to be taken of short range effects to obtain good theoretical Z,,, and ‘Z,,, .
V. Other Areas of Positron Atomic Physics We merely mention two other areas of atomic physics involving positrons. First there are the possible bound states of positrons or positronium with atoms or molecules, formation of which can greatly increase annihilation rates. Much of such work has been discussed by Green and Lee (1964). Among the recent calculations have been that of Khare et al. (1964) on a positronhelium bound state and that of Gol’danskii et af. (1964) on positron-negative hydrogen ion bound states. The possibility of using positrons as probes of plasma has been explored by Toptygin (1962, 1964).
VI. Basic Questions Though much of the following is strictly outside the scope of these volumes, it is well to remind ourselves in closing of the role that positrons and positronium continue to play in very basic questions. Longe (1965) has calculated the annihilation rate into two photons of o-Ps in the excited n = 2, I = 1, state, and finds rates of the order of lo4 sec-’ for the J = 0 and 2 states (the decay into two photons from the J = 1 state is forbidden). Theriot et af. (1967) have reported the first direct measurement of the p-Ps decay rate, and the measurement agrees with theory. In this work they used theoretical values of the o-Ps life and the fine structure splitting of o-Ps and p-Ps, but these have been confirmed by earlier measurements (see e.g., Deutsch, 1953). Sadeh (1963) has reported under the title “Experimental evidence for the constancy of the velocity of prays using annihilation in flight.” Rich and Crane (1966) have measured the g-factor of the positron by a measurement of the sinusoidal fluctuations in the two-photon coincidence rate of Ps decay in a magnetic field, the Ps being formed from the polarized positrons of P-decay. Theg-factor anomaly is found to be in agreement with that of the electron and that calculated by quantum electrodynamics. Mills and Berko (1967) have sought evidence for C nonconservation in electron-positron annihilation. They find the singlet 37/27 branching ratio to be less than about 3 x the singlet 37 decay being C forbidden. The Klein-AlfvCn (AlfvCn, 1965) cosmology, that involves overall equality between matter and antimatter, has stimulated, for example, the suggestion of
104
P.A . Fraser
Vlasov (1964) that the emission spectrum of Ps (amongothers) be searched for astronomically. It is thus interesting to recall that the original suggestion of the bound state of a positron and an electron by Mohorovicic in 1934, was in connection with the spectra of some nebulae (e.g., Green and Lee, 1964).
ACKNOWLEDGMENTS This article was written while the author was on leave-of-absencein 1966-1967 from the University of Western Ontario, supported in part by a Senior Research Fellowship from the National Research Council of Canada. Both the leave and the support are gratefully acknowledged. The author would like to thank Dr. J. H. Tait and Dr. P. G. Burke (AERE, Harwell) and Professor K. Smith (Royal Holloway College) for their kind hospitality during this leave. Correspondence from Drs. Bransden, Drachman, Herring, Jones, Paul, Roellig, and Tao to M. Kraidy and the author is gratefully acknowledged. The author would like to take this opportunity of thanking Professor Massey for steering him into positron and positronium research some years ago.
REVIEW WORKS For convenience, those of the listed review works specifically cited in the body of the text are also given below in the References. DeBenedetti, S.,and Corben, H. C. (1954). Positronium in Ann. Rev. Nucl. Sci. 4, 191-218. Deutsch, M. (1953). Annihilation of Positrons in Progr. Nucl. Phys. 3, 131-158. Deutsch, M., and Berko, S. (1965). Positron Annihilation and Positronium in “Alpha-, Beta-, and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), Vol. 2, pp. 1583-1598. North-Holland Publ., Amsterdam. Ferrell, R. A. (1956). Theory of Positron Annihilation in Solids in Rev. Mod. Phys. 28, 308. Green, J. H. (1966). Positronium Formation and Reactions in Endeavour 25, 16. Green, J., and Lee, J. (1964). “Positronium Chemistry.” Academic Press, New York. Heymann, F. F. (1961). Positronium in Endenuour 20, 225. Simons, L. (1958). Positronium in “Handbuch der Physik” (S. Fliigge, ed.), Vol. 34, pp. 139-165. Springer, Berlin. Wallace, P. R. (1960). Positron Annihilation in Solids and Liquids in Solid State Phys. 10, 1-69. The following recent book, published while this article was in proof, contains several review papers : Stewart, A. T., and Roellig, L. 0. (eds.) (1967). “Positron Annihilation” (Proc. Conf. Positron Annihilation). Academic Press, New York.
POSITRONS AND POSITRONIUM IN GASES
I05
REFERENCES Alfven, H. (1965). Rev. Mod. Phys. 37, 652. Allison, D. C. S., McIntyre, H. A. J., and Moiseiwitsch, B. L. (1961). Proc. Phys. SOC. 78, 1169. Armstead, R. L. (1964). Ph.D. Thesis, Lawrence Radiation Lab., Univ. of California, Berkeley, California. Bransden, B. H. (1962). Proc. Phys. SOC.79, 190. Bransden, B. H. (1965). Advan. Atomic Mol. Phys. 1, 85-148. Bransden, B. H., and Barker, M. 1. (1967). Private communication. Bransden, B. H., and Jundi, 2. (1966). Proc. Phys. SOC.89, 7. Bransden, B. H., and Jundi, Z. (1967). Proc. Phys. SOC.92,880. Brimhall, J. E., and Page, L. A. (1966). Nuovo Cimenro 43B, 119. Burke, P. G., and Taylor, A. J. (1966). Proc. Phys. SOC.88, 549. Celitans, G. J., and Green, J. H. (1963). Proc. Phys. SOC.82, 1002. Celitans, G. J., and Green, J. H. (1964). Proc. Phys. SOC.83, 823. Celitans, G. J., Tao, S. J., and Green, J. H. (1964). Proc. Phys. SOC.83, 833. Chen, J. C. Y. (1966). Phys. Rev. 152, 1454. Chen, J. C. Y., and Mittleman, M. H. (1966). Ann. Phys. ( N . Y.) 37, 264. Cheshire, I. M. (1964). Proc. Phys. SOC.83, 227. Cody, W. J., Lawson, J., Massey, H. S . W., and Smith, K. (1964). Proc. Roy. SOC.(London) A278,479. Daniel, T. B., and Stump, R. (1959). Phys. Rev. 115, 1599. Deutsch, M. (1951). Phys. Rev. 83, 866. Deutsch, M. (1953). In “Progress in Nuclear Physics” (0. R. Frisch, ed.), Vol. 3, pp. 131-158. Pergamon Press, Oxford. Deutsch, M., and Berko, S. (1965). In “Alpha-, Beta-, and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), Vol. 2, pp. 1583-1598. North-Holland Publ., Amsterdam. Din-Van-Hoang (1964). Dokl. Akad. Nauk Belorussk. SSR 8, 711. Drachman, R. J. (1965). Phys. Rev. 138, A1582. Drachrnan, R. J. (1966a). Phys. Rev. 144, 25. Drachman, R. J. (1966b). Phys. Rev. 150, 10. Drachman, R. J. (1967). In “ V International Conference on the Physics of Electronic and Atomic Collisions, Leningrad, U.S.S.R., July 17-23, 1967: Abstracts of Papers” (I. P. Flaks and E. S. Solovyov, eds.), pp. 106109. Nauka, Leningrad. Duff, B. G., and Heymann, F. F. (1962). Proc. Roy. SOC.(London) A270, 517. Duff, B. G., and Heyrnann, F. F. (1963). Proc. Roy. SOC.(London) A272, 363. Falk, W. R., and Jones, G . (1964). Can. J . Phys. 42, 1751. Falk, W. R., and Jones, G. (1966). Private communication. Falk, W. R., Orth, P. H. R., and Jones, G. (1965). Phys. Rev. Letters 14, 447. Ferrell, R. A. (1956). Rev. Mod. Phys. 28, 308. Ferrell, R. A. (1957). Phys. Rev. 108, 167. Ferrell, R. A. (1958). Phys. Rev. 110, 1355. Fraser, P. A. (1961). Proc. Phys. SOC.78, 329. Fraser, P. A. (1962). Proc. Phys. SOC.79, 721. [Erratum submitted.] Fraser, P. A. (1967). Work in progress. Fraser, P. A,, and Kraidy, M. (1966). Proc. Phys. SOC.89, 533. [Erratum submitted.]
106
P.A . Fraser
Gailitis, M. (1964). Zh. Eksperim. i Teor. Fiz. 47,160 [Sovief Phys.-JETP (English Transl.) 20, 107 (1965)l. Gailitis, M. (1965). In “Cross Sections of Electron-Atom Collisions” (V. Veldre, ed.), pp. 155-177. Latvian Acad. Sci., Riga. Gittelman, B., and Deutsch, M. (1956). Mass. Inst. Technol. Lab. Nucl. Sci. Ann. Progr. Repf., 152. Gittelman, B., and Deutsch, M. (1958). Mass. Inst. Technol. Lab. Nucl. Sci. Ann. Progr. Rept., 139. Gittelman, B., Dulit, E. P., and Deutsch, M. (1956). Bull. Am. Phys. SOC.1, 69. Gol’danskii, V. I., and Sayasov, Yu. S . (1964). Phys. Leffers13,300. Gol’danskii, V. I., Ivanova, A. V., and Prokop’ev, E. P. (1964). Zh. Eksperim. i Teor. Fir. 47,659 [Soviet Phys.-JETP (English Transl.) 20,440 (1965)l. Green, J., and Lee, J. (1964). “Positronium Chemistry.” Academic Press, New York. Hahn, Y. (1966). Phys. Rev. 142,603. Hahn, Y., and Spruch, L. (1965). Phys. Rev. 140,A18. Hahn, Y., O’Malley, T. F., and Spruch, L. (1964). Phys. Rev. 134,B397. Hashino, T. (1966). Progr. Theoref.Phys. (Kyofo)36,671. Heinberg, M., and Page, L. A. (1957). Phys. Rev. 107, 1589. Heymann, F. F., Osmon, P. E., Veit, J. J., and Williams, W. F. (1961). Proc. Phys. SOC. 78,1038. Herring, D. F. (1967). Private communication. Jones, G., and Orth, P. H. R. (1967). Private communication. Kestner, N. R., Jortner, J., Cohen, M. H., and Rice, S. A. (1965). Phys. Rev. 140,A56. Khare, H. C., Wallace, P. R., Bach, G. G., and Chodos, A. (1964). Can. J. Phys. 42, 1522. Kim, S. M., Stewart, A. T., and Carbotte, J. P. (1967). Phys. Rev. Letters 18, 385. Kleinman, C. J., Hahn, Y., and Spruch, L. (1965). Phys. Rev. 140,A413. Kraidy, M. (1967). Ph.D. Thesis, Univ. of Western Ontario, London, Ontario, Canada. Kraidy, M., and Fraser, P. A. (1967). In “ V International Conference on the Physics of Electronic and Atomic Collisions, Leningrad, U.S.S.R., July 17-23, 1967: Abstracts of Papers” (I. P. Flaks and E. S. Solovyov, eds.), pp. 110-1 13. Nauka, Leningrad. Levine, J. L., and Sanders, T. M. (1967). Phys. Rev. 154, 138. Liu, D. C., and Roberts, W. K. (1962). Phys. Chem. Solids 23, 1337. Liu, D. C., and Roberts, W. K. (1963a). Phys. Rev. 130,2322. Liu, D. C., and Roberts, W. K. (1963b). Phys. Rev. 132, 1633. Longe, P. (1965). Phys. Lefters 19, 381. McEachran, R. P., and Fraser, P. A. (1965). Proc. Phys. SOC. 86, 369. McKinley, W. A., and Macek, J. H. (1964). Phys. Leffers10, 210. Malik, F. B. (1961). Z. Nafurforsch. 16a,500. Marder, S., Hughes, V. W., Wu, C. S., and Bennett, W. (1956). Phys. Rev. 103,1258. Massey, H.S. W. (1967). Private communication. Massey, H. S. W., and Mohr, C. B. 0. (1954). Proc. Phys. SOC.A67, 695. Massey, H. S. W., and Moussa, A. H. A. (1958). Proc. Phys. SOC. 71, 38. Massey, H.S. W., and Moussa, A. H. A. (1961). Proc. Phys. SOC. 77,81 1. Massey, H. S. W., and Moussa, A. H. A. (1966). Private communication. Massey, H. S. W., Lawson, J., and Thompson, D. G. (1966). In “Quantum Theory of Atoms, Molecules and the Solid State: A Tribute to John C. Slater” (P.-0. Lowdin, ed.), pp. 203-215. Academic Press, New York. Mills, A. P., and Berko, S. (1967). Phys. Rev. Letters 18,420. Mittleman, M. H. (1966). Phys. Rev. 152,76. Mohr, C. B. 0. (1955). Proc. Phys. SOC.A68, 342.
POSITRONS AND POSITRONIUM IN GASES
107
Mott, N. F., and Massey, H. S. W. (1965). “The Theory of Atomic Collisions” (3rd ed.) Oxford Univ. Press, London and New York. Moussa, A. H. A. (1967). In “ V International Conference on the Physics of Electronic and Atomic Collisions, Leningrad, U.S.S.R., July 17-23, 1967: Abstracts of Papers” (1. P. Flaks and E. S. Solovyov, eds.), pp. 116-1 17, Nauka, Leningrad. Moussa, A. H. A., Massey, H. S. W., Smith, K., and Wardle, C. (1967). In “ V International Conference on the Physics of Electronic and Atomic Collisions, Leningrad, U.S.S.R., July 17-23, 1967: Abstracts of Papers” (I. P. Flaks and E. S. Solovyov, eds.), pp. 113-1 16. Nauka, Leningrad. Ore, A. (1949a). Uniu. Bergen Arbok Naturvitenskap. Rekke No. 9. Ore, A. (194913). Univ. Bergen Arbok Naturvitenskap. Rekke No. 12. Osmon, P. E. (1965a). Phys. Rev. 138, B216. Osmon, P. E. (1965b). Phys. Letters 16, 271. Osmon, P. E. (1965~).Phys. Rev. 140, A8. Paul, D. A. L. (1958). Can. J . Phys. 36, 640. Paul, D. A. L. (1964). Proc. Phys. SOC.84, 563. Paul, D. A. L., and Graham, R. L. (1957). Phys. Rev. 106, 16. Paul, D. A. L., and Saint-Pierre, L. (1963). Phys. Rev. Letters 11, 493. Peterkop, R., and Veldre, V. (1966). Advan. Atomic Mol. Phys. 2, 263-326. Pond, T. A. (1952). Phys. Rev. 85, 489. Rich, A., and Crane, H. R. (1966). Phys. Rev. Letters 17, 271. Roellig, L. 0. ( I 967). Private communication. Roellig, L. O., and Kelly, T. M. (1965). Phys. Rev. Letters 15, 746. Roellig, L. O., and Kelly, T. M. (1967a). Phys. Rev. Letters 18, 387. Roellig, L. O., and Kelly, T. M. (1967b). Private communication. Rotenberg, M. (1962). Ann. Phys. ( N . Y . ) 19, 262. Sadeh, D. (1963). Phys. Rev. Letters 10, 271. Schwartz, C. (1961). Pbys. Rev. 124, 1468. Shmelev, V. P. (1959). Zh. Eksperim. i Teor. Fiz. 37, 458 [Soviet Phys.-JETP (English Transl.) 10, 325 (1960)]. Smith, K. (1961). Proc. Phys. SOC.78, 549. Spruch, L., and Rosenberg, L. (1960). Phys. Rev. 117, 143. Stone, P. M. (1966). Phys. Rev. 141, 137. Tao, S. J. (1965). Phys. Rev. Letters 14, 935. Tao, S. J., (1967). Private communication. Tao, S. J., and Bell, J. (1967). In “Positron Annihilation” (Proc. ConJ Positron Annihilation) (A. T. Stewart and L. 0. Roellig, eds.), pp. 393-399. Academic Press, New York. Tao, S. J., Green, J. H., and Celitans, G. J. (1963). Proc. Phys. SOC.81, 1091. Tao, S. J., Bell, J., and Green, J. H. (1964). Proc. Phys. SOC.83, 453. Temkin, A. (1962). Proc. Phys. SOC.80, 1277. Teutsch, W. B., and Hughes, V. W. (1956). Phys. Rev. 103, 1266. Theriot, E. D., Beers, R. H., and Hughes, V. W. (1967). Bu//.Am. Phys. SOC.12, 74. Toptygin, I. N. (1962). Zh. Eksperim. i Teor. Fiz. 43, 1031 [Soviet Phys.-JETP (English Transl.) 16, 728 (1963)l. Toptygin, I. N. (1964). Zh. Tekhn. Fir. 34, 645 [Soviet Phys.-Tech. Phys. (English Transl.) 9, 499 (1964)l. Vlasov, N. A. (1964). Astron. Zh. 41, 893. [Soviet Asrron.-AJ (English Transl.) 8, 715 (1965)l. Wackerle, J., and Stump, R. (1957). Phys. Rev. 106, 18. Wallace, P. R. (1960). Solid State Phys. 10, 1-69. Wardle, C., and Smith, K. (1966). Private communication.
This Page Intentionally Left Blank
CLASSICAL THEORY OF ATOMIC SCATTERING A . BURGESS Department of Applied Mathematics and Theoretical Physics Cambridge University, Cambridge, England
and C. PERCIVAL
I.
Department of Physics, Stirling University Stirling, Scotland
I. Introduction
....................................................
11. Classical Cross Sections.. . .
109
................................ 1 1 1 ........................ . . . . . . . . . . . . . . 1.1 1
A. Total Cross Sections B. Differential Cross Sections .................................... .113 C. Formal Theory: Individual Collisions .......................... .114 D. Ensembles and Liouville Expansions ............................ 11 5 111. Binary Encounters . ........................................... 117 A. Theory before 194 ............................... 117 B. Theory since 1958 ........................................... .119 C. Symmetrized Binary Encounter Theory ......................... .120 D. Excitation and Ionization by Heavy Particles . . . . . . . . . . . . . . . . . E. Applications . . ...................................... 122 F. Charge Transfer ....................... IV. Perturbation Theories and Threshold Laws . . . . . . . . . . . A. Perturbation Theories . ......................... 126 B. Threshold Laws . . . . . . ......................... 127 V. Orbit Integration and Monte Carlo Methods ....................... .128 A. Theory ...................................................... 128 B. Microcanonical Distribution ........................... C. Numerical Integration .................................. 131 D. Application to Chemical ses .............................. 132 E. Application to p-H and e-H Collisions .......................... .134 F. Wigner-Keck Variational Method ...... . . . . . . . . . . . . . . . . . ..136 VI. Correspondence Principle and Conclusions . . . . . . . . . . . . . . . . . . . . .137 References .............................. 139
I. Introduction The original classical theories of atomic scattering due to Thomson (1912) and Rutherford (19 1 1) were proposed before quantum mechanics was known. Despite the discovery of quantum mechanics and its wide application to atomic scattering, classical methods continue to be used for their comparative simplicity. 109
110
A . Burgess and I. C. Percival
There are two distinct reasons why classical mechanics can be applied to atomic scattering: (1) The Correspondence Principle. When the action integrals associated with the interactions are large compared with h, classical mechanics applies to a good approximation. Collisions between atoms, or between atoms and simple molecules, are of this type, provided there are no resultant transitions between their electronic states. Classical theories have been widely used in this field of chemical kinetics since the thirties, especially since digital computers have enabled Monte Carlo methods to be used, as described in Section V. Collisions involving high states of excitation of atoms and ions are also of this type, and have recently become important in the theory of plasmas (Bates et al., 1962). When classical methods are applied to such collisions they complement close coupling and variational methods, which are more appropriate when the number of quantum states involved in a collision is relatively small. (2) The Rutherjord Scattering Identity. The classical and quanta1 differential cross sections are the same for the scattering of two nonidentical particles which interact through an inverse square law of force (Mott and Massey, 1965). Classical theory can then be used where the uncertainty principle (see Williams, 1945) would lead us to suppose that it could not. Although this identity does not apply directly to many-particle collisions, it is probably behind the success of the applications of classical theory to electronatom and proton-atom collisions. However, there is still no solid theoretical foundation for most of the applications of classical mechanics to these collisions. Early developments in this theory were partially summarized in the review articles of Williams (1945) and Bohr (1948). Much of the early work was repeated when the paper of Gryzinski (1959) started a new wave of interest in classical scattering. Atomic processes are often easier to understand and cross sections easier to calculate using classical instead of quantum mechanics. Once the approximation of replacing quantum by classical mechanics has been made, little further approximation may be necessary to obtain a cross section. However, specific quantum phenomena, such as interference, resonances, and the tunnel effect cannot be understood in terms of classical theory alone, although the modification required to incorporate such effects approximately may be small. Because of the bias in our interests we concentrate on binary encounter and Monte Carlo and related methods. We consider quantum mechanics only where it enables us to understand or improve the applicability of the classical methods. We do not deal with the classical path methods for heavy particle collisions whereby the theoretical problem is reduced to the solution of a time-dependent
111
CLASSICAL THEORY OF ATOMIC SCATTERING
Schrodinger equation. Nor do we deal with the semiclassical (JWKB) methods and related classical methods for two-particle scattering, such as are used for atom-atom scattering when there is no change in the internal state of either atom. Bernstein (1966) has recently written a useful review article on the latter subject, and the papers of Massey and Mohr (1933, 1934), Ford and Wheeler (1959a,b), Smith (1965), and Smith et al. (1965) are also helpful. Ion-ion recombination has also been excluded from this review; we merely note the remarkable agreement between the detailed quanta1 calculations of Bates and Moffet (1965, 1966) and the simple classical theory of Thomson (I 924) corrected by Loeb (1 939).
11. Classical Cross Sections A. TOTALCROSS SECTIONS A classical theory of Atomic scattering requires : (1) The choice and interpretation of a classical model. (2) The solution of the classical model collision problem.
In practice, the choice must be made first, but for convenience we describe the solution first. Consider a system of N particles i of masses m iand positions ri(t) at time I, which interact classically through potentials V i j ( r i j )where r i j = I ri - rjl. Neglecting relativistic and magnetic effects, and noncentral forces, the particles then obey Newton’s laws of motion mi
d2r. dt2
1 =-
2 v iKj(rij).
all j # i
Let X(to) represent all the positions and velocities (ri(to),vi(to))at time t o . Then the classical state X ( t l ) of the system at any other time t, is determined uniquely by X ( t o ) and the laws of motion (1). X defines a point in the 6Ndimensional position-velocity space of the system, and the motion X ( t ) defines a trajectory in this space for all 1. The most elementary type of scattering problem for N particles concerns an indiaidual collision or scattering event. The N particles are initially bound into two separate groups, bound systems or bodies, A and B, containing N , and N , particles. At some time t - in the very distant past the system is in the classical state X- . The centers of mass of A and B move toward one another initially in approximately straight lines whose distance apart is the impact parameter b. When the systems are close to one another, all N = N A N , particles interact. With the exception of some special singular cases, at some
+
112
A . Burgess and I. C. Perciual
time t + in the very distant future the particles are bound into two or more separate systems C, D,E, . . . containing N,, N , , N E , . . . particles. This final state depends on the initial state through the laws of motion. We shall refer to the grouping of the particles into bound systems as the channel of the whole system, by analogy with the channels of quantum mechanics. For example, in a classical model collision between He+ and H, some of the final channels are
+ + +e +
He+ H He+ H + He H +
(direct scattering channel) (ionization channel) (charge transfer channel).
No distinction is made between different states of excitation in defining the channel, as the classical energy has a continuous range of values. In practice, the initial states of atomic systems cannot be controlled sufficiently to study individual collisions precisely. Observed physical phenomena depend on statistical properties of ensembles of collisions, that is to say, on very large numbers of collisions, with variable initial states whose probability distribution is determined by the macroscopic conditions, e.g. the apparatus. This statistical aspect of scattering theory is always present, even in the quantum theory, but is often obscured by the formalism. A cross section is a statistical property of collisions. The initial statistical distributions depend on the experiment. Classical cross sections are defined in terms of the following standard distribution. The internal states X , and X , of A and B, if there are any, have distributions pA(XA)and p,(X,) which are always stationary and usually isotropic. The center of mass of B with position and velocity (r, ,v,) lies at rest at the origin of a coordinate system, whereas a uniform beam of the bound systems A impinges upon it with velocity u = V2. The initial unperturbed distribution p r ( X ) for the whole system is then
In the absence of interaction B would strike the xy plane at the point with polar coordinates (b, cp), where b is the impact parameter. For a given (b, cp), if pb+,(c)is the probability of the final channel being c, then the cross section for that final channel is
Q(c) = S’;b 0 d b 1;’dp
Pb+,(c).
(3)
For isotropic pA(XA)and p,(X,), the value of pbb(C) = Pb(c) is independent of cp, so that
Q(c) = 2n: Joab d b Pb(c).
(4)
CLASSICAL THEORY OF ATOMIC SCATTERING
113
This expression corresponds to quanta1 partial wave summation, and will be used in the rest of the article. If c is not the direct scattering channel, Pb(c)is zero beyond some maximum impact parameter b,,,Jc). The total cross section integral diverges for direct scattering except for cutoff potentials which are identically zero beyond some fixed distance. In quantum mechanics the direct scattering is made up of elastic and inelastic scattering, and for short range potentials which decrease as fast as r - 2 or faster the total direct cross section is finite. This fundamental distinction between classical and quantum mechanics has been discussed by Massey and Mohr (1 933) for two-particle collisions. In classical mechanics every dynamical variable has a continuous range of possible values, including binding energies, angular momenta, and other normally quantized variables. Classical cross sections for these variables are differential cross sections. The connection between these differential cross sections and the total quantum mechanical cross sections will be considered in Sections IIIE and VI.
B. DIFFERENTIAL CROSSSECTIONS For the initial distribution, Eq. (2), we define the differential cross section for the continuous variable s in terms of the probability pb(s, As). This is the probability that for impact parameter 6, the system reaches a final state for which the variable represented by s lies within a small region As of the value s. The differential probability density with respect to s is then
and the differential cross section is
The variable s might be an angle of scattering, or the internal enegy E transferred to the system B. We use the same definition when s represents more than one continuous variable, for example the solid angle w defined by the scattering angles (0, cp). The As then represents an area or volume including the point s in the space of these continuous variables, e.g. for the solid angle, the space is the unit sphere. For collisions of electrons with classical hydrogen atoms the channels correspond to different ranges of the energy transfer E to the internal energy of
A . Burgess and I. C. Percival
114
motion of the original H atom. Supposing the proton has infinite mass, U is the binding energy of the hydrogen atom, and El, (El > U ) the initial energy of the incident electron. Then for ionization to take place, the bound electron must gain at least the energy U to leave the proton, but the incident particle must not lose more energy than E, or it will become bound to the proton, and result in classical charge transfer. For E > 0 the situation may be summarized as follows: &
>El
direct scattering (excitation)
c=o
ionization
c=1
classical rearrangement (exchange)
c = 2.
(7)
Notice that exchange is due to more violent collisions than ionization. The ionization Q , and rearrangement QR cross sections can be expressed as integrals over dQ/& : aQ(Ei, 6 ) QdEA = dc: (W
The analytic behavior of dQ/& as a function of El and E has a crucial effect on the classical threshold laws, as discussed in Section N , B . If the incident energy El is less than the ionization energy U, then only direct and rearrangement channels are available : E E
< El >U
direct scattering rearrangement.
(9)
The gap
(10) contains collisions in which both electrons are bound to the nucleus for a period of time, until their interaction leads to ejection of one or other of the electrons. These classical collisions correspond to resonance collisions of quantum scattering. E,<&
C. FORMAL THEORY: INDIVIDUAL COLLISIONS The evolution operator U(t, to) for individual classical systems has been briefly discussed in connection with the Monte Carlo method (Abrines and Percival, 1966a). In the notation of Section A it is defined by the relation w t , t o ) X(t0) = X(r)
(1 1)
where the classical states X ( t ) and X(to) lie on the same trajectory defined by
CLASSICAL THEORY OF ATOMIC SCATTERING
115
the classical laws with Hamiltonian H. It is evidently a nonlinear operator, but in other respects it is simpler than the quantum evolution operator, since the classical states are defined by the values of the coordinates and velocities, whereas quantum states require functions of position to define them. Let Ho be the Hamiltonian and Uo the classical evolution operator for two or more noninteracting systems and let these become H a n d U when an interaction potential is introduced which results in scattering. The classical scattering operator S may then be defined as S = lim U o ( t o ,t+)U(t+ , t - ) U o ( t - , to) t*+fa,
(12)
where the arbitrary time to corresponds to the arbitrary choice of phases in the interaction representation in quantum mechanics. The scattering operator S is also nonlinear, but effectively describes the net effect of classical scattering. It reduces to the unit operator in the absence of interaction. It defines the simultaneous effect on both positions and velocities of the particles, unlike the quantum scattering operator, for which this is prevented by the uncertainty principle. The above operator formalism has proved useful in writing computer programs for orbit integration, in which each operator corresponds to a particular part of the program. Because of the relative simplicity of classical states, X , the effect of a classical evolution operator on a particular classical state, can be obtained directly by numerical integration of the equations of motion, but because of nonlinearity it is more difficult to handle analytically than a quantum evolution operator.
D. ENSEMBLES AND LIOWILLE EXPANSIONS RCsibois (1959), following Prigogine and HCnin (1957), has developed a linear formal theory of scattering of statistical ensembles of classical systems with cutoff potentials which follows closely the formal theory of scattering (Lippmann and Schwinger, 1950) and is based on the Liouville equation (Landau and Lifshitz, 1958). We describe the theory as applied to scattering by a central potential. The density P Y X , 0 = PQ, v, t ) (13) for ensembles of classical particles defined the relative probability of finding a particle within a small region AX of the classical state X . The Liouville equation was then W ( r , v, at
t)
a p’(r, v, t ) = 1 - V(r) - - p’(r, v, t ) . +v ar r:[ ] v: *
(14)
116
A . Burgess and I . C. Percival
From our point of view, the important property of this equation is that it is linear, and may be treated in a similar fashion to the Schrodinger equation. Rksibois defined a Hermitian unperturbed Liouville operator L‘ and a perturbation 6L‘ as
The Liouville equation then became i aPf - = (L’ at
+ I6c)p’.
To remove the effect of the free motion, he used the interaction representation p(r, v, t ) for the density: p(r, v,
t ) = exp(itl’)p’(r,
v, t).
(17)
Note that the unitary evolution operator for free motion was real. By transforming to a wave number representation and back he obtained an integral equation for the stationary distribution functionf(r, v) of the scattering process : av(rf> a f(r, v) = poo(v) I d3r‘ G(r - r’, v) -* -f(r’, v) (18) art av
+
s
where poo(v) = 6(v - voa)/(Rvo)represented the initial unperturbed beam, supposed to be contained in a volume R and switched on for a period 2T. The kernel G was the Green’s function for the unperturbed Liouville operator G(r - r’, v) =
c
d7 6(r - r f - vz).
(19)
Equation (1 8) may be expanded in powers of 1 giving a Liouville-Born series. For long times T and large volumes R the asymptotic velocity distribution was obtained in the form PO(V, T ) = P o 0 W
+R 2T [;b
db J?v
Cf@, v) - PO0(V)l
(20)
where r = ( z , b, 4), in cylindrical coordinates. The second term was the scattering term, giving a differential cross section for scattering into solid angle w with unit vector 0 # of
Light used the Liouville theory for practical calculations of reaction rates, as in Section IV,A.
117
CLASSICAL THEORY OF ATOMIC SCATTERING
111. Binary Encounters A. THEORY BEFORE 1949 In the application of this theory to collisions of electrons with atoms, the transfer of energy E from the incident electron 1 to a bound electron 2 is equated to the energy transfer between two free electrons. It is assumed that during the period of significant interaction between these electrons, the other electrons and the nucleus play no role. A necessary condition for this to hold is that this period should be short compared with the orbiting time of 2. This condition is easier to satisfy for collisions with large energy transfers such as occur in ionizing collisions and pure exchange collisions, rather than in direct excitation. For large energy transfers the binary encounter takes place in a region which is small relative to the size of the atom. Thomson (1912) first applied the method, with the assumption that the atomic electron is at rest. The energy transfer E in a Coulomb collision between a particle of mass m, and charge zle with initial kinetic energy E l , and a particle of mass m, and charge z,e with initial kinetic energy E, = 0, is related to the impact parameter b’ (we use b’ to distinguish from the incident particle-nucleus impact parameter) by
Thus, the cross section for an energy transfer in the range
For an energy transfer in the range given by
E
to
(E
E~
< E < e2 is
+ LIE) the cross section ~ Q ( E )is
,
in e4 d-Q(4 - 7Tz12z22 dE in2 El&’ ~
For ionization by electrons we have z , = z 2 = - 1, m, = m2 = m, and we may consider either E 2 U in Eq. (21) or U < E < El in Eq. (22) or (23) to obtain the usual Thomson ionization cross section
where N is the effective number of electrons in the atom, and U is the ionization energy of the atom.
118
A . Burgess and I. C. Percival
Thomson (1906) also calculated the stopping power using the same method. For a review of this and later treatments of stopping power see the work by Bohr (1948); for a more concise summary see also the work of Jackson (1962). The method was generalized to the case of nonzero E, by Thomas (1927a,b) and Williams (1927) who considered a spherically symmetric distribution of velocities of the atomic electron. Denoting the momentum of particles 1 and 2 before the collision byp, (= m l v l ) andp, (= rn2v2),respectively, and after the collision by pl’ and p,’, respectively, Thomas obtained the important intermediate result that the cross section dQ(q; E ) for an energy transfer between E and simultaneous momentum transfer between q and (q + dq) is given by
with
[For more detail on the derivation of this far from obvious result, other, later papers e.g. Gryzinski (1965a,b) should be consulted.] Thomas (1927b) goes on to consider the case of m l= m 2 , z1 = z2 = - 1, E > 0, and vl‘ > v2 (i.e. v1 > v,’ from energy conservation), when it is easy to show that the simple energy transfer cross section dQ(e) is given by
Thomas (1927a) also gives dQ(E)/dt:for the case of z1 # z2 , m, % m, , which is relevant to, e.g., proton-atom collisions. Williams (1927) gives, for the general case of z1 # z2 , ml # m, , dQ(E) - - - 7ce4z12z22m1 dE
m2
4E
El
(this can also be obtained easily from Thomas’s results), but with no statement of the velocity regions for which it is valid. It is in fact true for
(Gerjuoy, 1966, see Section 111, D), which is the case of most usual interest.
CLASSICAL THEORY OF ATOMIC SCATTERING
119
Thomas (1927b) also introduced the idea that since the binary encounter takes place effectively in a small region, the incoming particle should gain kinetic energy beforehand by interacting with the nucleus and electrons in the atom other than the one with which it has the binary encounter. He chose this gain in kinetic energy to be equal to the magnitude of the initial potential energy of the struck electron. Thus in Eqs. (29) and (30), E, is replaced by El E2 U . Webster er al., (1933) modified the Thomas (1927b) result by introducing an outside factor in the cross section to account for the focusing of the incident beam by the nuclear field before the binary encounter (they had particularly in mind the case of inner shell ionization where this effect may be of importance). From a conservation of angular momentum argument they deduced this factor to be (El + E2 U ) / E l ,thus converting the final Thomas result back to (29).
+ +
+
B. THEORY SINCE 1958 Interest in the method then appeared to lapse for many years but was rather dramatically revived by the publication of the work of Gryzinski (1959). This was essentially an independent reworking of the nonzero E, ThomasWilliams calculation. In Gryzinski’s calculation, which was based on stellar collision work of Chandrasekhar (1941) and Chandrasekhar and Williamson (1941), scattering in the center of mass frame of the incident and struck electrons is considered and followed by transformation to coordinates at rest relative to the nucleus. Gryzinski compared his calculations with a wide range of experiments and obtained startlingly good agreement. This had the very good effect of drawing attention to this and other classical methods useful for estimating collision cross sections, but the degree of agreement with experiment was misleadingly exaggerated, partly due to selection of experimental data which was not always the best available and mostly due to an inessential approximation in the integrations. This was pointed out by Ochkur and Petrun’kin (1963) and by Stabler (1964) whom we quote: “Gryzinski’s cross sections are in remarkably good agreement with experimental data for a wide variety of inelastic processes. The results appear to indicate that the properly calculated impulse approximation is superior not only to the earlier classical theory, but also to many first- and second-order perturbation theories of inelastic electron-atom collisions. . . . A subsidiary approximation made by Gryzinski in averaging over the initial angular distribution is responsible for the fact that his cross-sections are in any better agreement with experiment than the Born approximation. This second approximation, rather than simplifying the forms of the cross-sections, actually complicates them; and while it does improve the results, it enters in an arbitrary fashion which removes much
A . Burgess and I. C. Percival
120
of the selfconsistency of the calculation (e.g. the cross-sections do not behave properly under time reversal).” As Ochkur and Petrun’kin rightly point out, this makes the Gryzinski (1959) theory of only semiempirical nature, and simpler (and perhaps more effective) semiempirical formulas can be proposed. Both sets of authors carried through the calculation without this approximation (Stabler, in a particularly elegant direct way, avoiding the use of center of mass coordinates) and thus rederived the Thomas-Williams formula (29) but generalized to include the case of 0, > v l ’ (but with m, = m,), viz
with E the smallest of El, E , , El’, and E,’, where El‘ =
and E,’
=
In a later paper, Gryzinski (1965a) rederived the Thomas differential momentum and energy transfer cross section (25). C. SYMMETRIZED BINARY ENCOUNTER THEORY
As was discussed by Burgess (1964a, b), in the case of electron impact, the binary encounter may be treated as a quantum mechanical collision between identical particles with proper symmetrization and treatment of interference between direct and exchange scattering. As before, the initial and final atomic states are treated classically in terms of an orbiting electron with definite initial and final kinetic energy, the change in kinetic energy being related to the angle of scattering. The idea of calculating exchange cross sections classically was also discussed by Ochkur (1963). Burgess also allowed for the acceleration of the incident electron in the same way as Thomas (1927b), the kinetic energy of the incident electron before the binary encounter being set equal to El E, U(contrary to the statement by Vriens (1966a) that it was set equal to El U). In this way, Burgess showed that cross sections for direct and exchange excitation and ionization could be calculated, together with the effect (if it can occur) of interference between direct and exchange scattering. This latter effect could not be calculated analytically without further approximation, and an assumption was made analagous to the case of maximum interference discussed by Rudge and Seaton (1965) in the proper quanta1 treatment of ionization by electron impact. However, the calculations were carried out by transforming between nuclear and center of mass coordinates, and in transforming an invariant cross section was assumed, which is incorrect. If instead, one assumes an invariant reaction rate, as is correct, one obtains, cf. Vriens (1966a, b),’
+ + +
N.B. In Vriens (1966a), Qlotis incorrect.
CLASSICAL THEORY OF ATOMIC SCATTERING
121
where, for El > E , 1 1 ---+~i
E,
ne4
Qex
=E,
_ ~f’)]
2E2 _ 1 _ 3 (c12
1
+ E, + U
El
+ U - E,
1
(El
+U-
1
-
El
~2)’
(El
(33)
+U-
+U-
~1)’
(34)
Qd , Q,, , and Pin,being the direct, exchange, and interference cross sections, respectively. Vriens (1966a,b) shows that for E , > E l , in (33)-(35), E, must be replaced by El. These formulas are algebraically simpler than the invariant cross-section ones given in Burgess (1964b). The invariant cross-section results and the invariant rate results (33)-(35) are identical if we set E , = 0; they are also identical to first order when El 9 E , (i.e., in the high energy limit) but for smaller incident energies the invariant rate results give smaller cross sections than the invariant cross-section results, and are usually in rather better agreement with experiment in this region. The differential form of (32)-(35), obtained by setting = E , E, = E + dE, is sometimes useful, viz.,
nQ(4 -- dE
ne4 El + E, + U
[ (f
+
4E
7 $)
+
((El
+
1 U - E)’
+ 4-3 ( E , + U -
E ) ~
In applying (32)-(35) to ionization, care must be taken to avoid counting events twice, i.e., we must set el = U , E, = (El U ) / 2 ,when we obtain
+
To account for electron-positive ion collisions, Burgess (1964) introduced an outside factor
F =1
+ (ze2/ElF)
(38)
where F is the mean radius and z is the charge number of the ion, to account
122
A . Burgess and I. C. Percival
for the focusing effect of the long range Coulomb field of the ion. In fairness, it must be said that this should only be viewed as a semiempirical factor; if focusing is included, and in strictly working out the consequences of the model (rather than obtaining semiempirical formulas) we feel that it should, then the focusing factor obtained by Webster et al., (1933) should be used. Thus, in Eqs. (33)-(37), one should strictly replace the factor (El E2 V ) by El to obtain cross sections more truly representative of the model, even though this destroys some of the low energy agreement with experiment. Equations (32)-(37) may be usefully employed as they stand but are then of a semiempirical nature.
+ +
D. EXCITATION AND IONIZATION BY HEAVY PARTICLES McDowell(l966) and Vriens (1967) have worked out the theory for a binary encounter between an incident proton and an initially bound electron with E, # 0, to first order in m,/m,, using the correct invariant rate transformation as discussed in Section II1,C. McDowell gives only the ionization cross section, but Vriens gives the more general quantity dQ(e)/de which is then easily integrated to give excitation or ionization cross sections. The results of Vriens for dQ(e)/deagree exactly with those of Thomas (1927a). Percival and Valentine (1966) performed the proton impact calculation but used the invariant cross section transformation. Gerjuoy (1966) has treated the general case of arbitrary m,, m, , zl, and z , , giving energy transfer cross sections for all the possible ranges of energies of the particles. In the relevant energy range his results agree with those of Williams (1927) as noted in Section III,A, while for m, = m,, z1 = z2 they check with those of Ochkur and Petrun’kin (1963) and Stabler (1964).
E. APPLICATIONS Several of the papers mentioned above which deal with the various binary encounter theories include results of applying those theories to the calculation of cross sections for specific atomic systems. Some additional results of this kind will now be mentioned, but without attempting to be exhaustive since this type of material is sometimes published as only a small part of a calculation or experimental report. Prasad and Prasad (1963) have used the Gryzinski (1959) theory with E, = U, to calculate ionization cross sections for several atoms and diatomic molecules by electron impact and by proton impact. Kingston (1964a,b, 1966a,b) has applied the Gryzinski (1959) theory and the corrected form of this theory (see Section III,B) to electron impact ionization from the ground and several excited states of hydrogen. He gives cross sections obtained from putting E2 = U and from averaging over the proper quantum mechanical velocity
CLASSICAL THEORY OF ATOMIC SCATTERING
123
distribution, and compares them with Born approximation results. Sheldon and Dugan (1965) have applied Gryzinski (1959) to electron impact excitation of Cs. Saraph (1 964) has criticized a further approximation sometimes made in using Gryzinski's (1 959) theory to calculate excitation cross sections. Robinson (1965) has modified the Gryzinski (1959) treatment by introducing a rough cutoff in the theory, so that impact parameters for which the collision time 2 the period of the bound electron are ignored. By setting E, equal to the mean kinetic energy calculated by using Slater's rules, rather than setting E, = U , he claims to get improved results for ionization cross sections. Particularly interesting is the good agreement with experiment that he obtains for ionization of neon, since the observed cross section is abnormally small. McDowell ( 1966) has integrated the electron impact ionization cross section obtainable e.g. from Eq. (31), over the correct distribution function for u2 (see Section V,E) and obtained an analytic result, but this does not agree with the numerically integrated result of Abrines, et al. (1966) for H(1s). McDowell (1966) also gives results for proton impact ionization cross sections numerically integrated over u, . Vriens (1967) has given an analytic expression for this in the case of H( I's) ionization. Considering their simplicity, the binary encounter theories have in general yielded surprisingly good results. For ionization, to which the method is particularly well suited since rather large energy transfers are involved and there is no ambiguity in choosing the classical band of energies to represent the final state, the results are often superior to the Born approximation except at high incident energies. The method can also yield very useful estimates of cross sections for change of spin multiplicity under electron impact; again this is partly due to the large energy transfer required, although in this case there is difficulty in representing the final state. Another reason for the success of the theory in these two cases is that the electron-electron interaction is not treated as a small perturbation. For direct excitation of low lying states the method is usually less satisfactory since the energy transfer is small and there is a large amount of ambiguity in choosing the final state energy band. It is also difficult to see how to split up this classical energy band so as to represent the various possible quantized angular momentum final states separately. Burgess (1964a) has suggested that this separation may perhaps be approximately achieved by using as final angular momentum state weights, the quantities w/
j
Ill/i(r) +I(')I~ d3r9
where ll/i and are initial and final state wave functions and wf is the final state statistical weight, but this suggestion has not been tested widely enough to judge its usefulness.
124
A . Burgess and I. C. Percival
For large incident energy the binary encounter excitation and ionization cross sections (in common with the exact classical cross sections) have the wrong functional form (of order l/El instead of (log El)/El), for any finite choice of E, or when averaged over any quantal or reasonable classical distribution of E, (see Section V,E). Gryzinski (1965b), in order to force the correct high energy behavior onto the binary encounter cross sections, has introduced the following distribution function for v2 : f(Ud = au; exp( - P/u,>, (39) where a and P are constants. This is completely at variance with any quantal velocity distribution: the fact that it yields an infinite mean kinetic energy also leads to difficulties. We feel that it could not be accepted by any who interpret atomic structure according to quantum mechanics in the final analysis. However, as semiempirical formulas the Gryzinski (1965b) cross, section results may be of interest and Gryzinski (1965c), Bauer and Bartky (1965), and McFarland (1965) have applied them to excitation and ionization for several atoms and diatomic molecules. It should be noted that although one can in this way force a (log E J E , behavior of the cross section, there is no guarantee that the coefficient of this term is predicted at all accurately since the coefficient in fact depends on the oscillator strength of the struck system. The (log EJE, high energy behavior arises quantum mechanically because of the effect of distant encounters, which may be thought of in terms of virtual photon swapping. Hence, the dependence of the coefficient of this term on the oscillator strength, and the logarithmic behavior, which is connected with the infrared divergence. An interesting light was thrown on the classical situation by Williams (1931) who pointed out that quantal and classical calculations of stopping power give similar results; at high energies they both have a dominant logarithmic term, but, classically this term arises from small energy transfers, quantally from finite energy transfers (excitations and ionizations). In other words, the log term is not absent in classical theories, it has just moved to a different region of energy transfer; and of course the classical small energy transfers occur for distant encounters. Burgess (1964a,b) has obtained the correct (log E,)/El high energy behavior in a semi classical theory by combining a symmetrized binary encounter treatment of close collisions with an impact parameter (classical orbit, quantal perturbation theory) treatment of the distant collisions. The binary encounter contributions should be modified as discussed in Section II1,C.
F. CHARGE TRANSFER
A detailed classical binary encounter theory of charge transfer due to Thomas (1927~)has been revived largely through the work of Bates and Mapleton (1965, 1966, 1967), who have also introduced a modification for
CLASSICAL THEORY OF ATOMIC SCATTERING
125
transfer from a stationary heavy atom to a light fast incident nucleus which agrees well with experiment. Bohr (1948) and Gryzinski (1965~)contented themselves with less elaborate treatments than Thomas. Thomas' basic assumption was that the electron was transferred as a result of two binary encounters, first with the light incident nucleus and then with the heavy nucleus, so that it finished with a kinetic energy relative to the incident nucleus less than the potential energy, and thus became bound to it. The analysis of the double binary collision is rather complicated. Thomas assumed that the atomic potential could be approximated in the region of interest by me I l r 2 where I is a constant and r is relative to the heavy nucleus, and that for the atomic distribution all accessible elements of phase space were occupied, so that
was the distribution function for the magnitude of the positions r and velocities u of the bound electrons (mass me) in energetically accessible regions of phase space. This gave 2me 2 z , e 2 712 ( 2 4 3 / 4 v ; 1~ QCT. = 3fi3
(7)
where z,e and V , are the charge and velocity of the incident nucleus. Thomas approximated an integral over a finite range of r by an infinite range, and thereby obtained c % 3.5, (42) which disagrees with experiment. Bates and Mapleton obtained better estimates for the limits and thus for the integral. C was no longer a constant, but was obtained as a function of x = me E , / ( M U ) where M is the mass and El the initial energy of the incident nucleus and U the ionization potential of the heavy nucleus. The normalized transfer cross section would then also be a function of x.This relationship agreed well with experiment for protons in noble gases, nitrogen, and oxygen, which all followed approximately the same curve. The function obtained by Bates and Mapleton (1966) agreed well with experiment for 1 < x < 3, but elsewhere disagreed by factors of up to 3. The agreement at higher incident energies was improved by using Hartree-Fock-Slater distributions (Bates and Mapleton, 1967) which showed the effect of the shell structure.
\
126
A . Burgess and I. C. Percival
IV. Perturbation Theories and Threshold Laws A. PERTURBATION THEORIES With the aid of classical perturbation theory of the interaction potential, and using action and angle variables, as in celestial machanics, Fowler (1925) obtained formulas for mean energy transfer from a passing charged particle to a classical hydrogen atom. Unfortunately, he did not take into account the detailed balance relation between energy loss and energy gain cross sections, and as a result he obtained zero energy loss in first order perturbation theory. Recently (Percival and Richards, 1967), a further adiabatic approximation has been used to obtain dQ/dc in the limit of very small E . In this limit the cross section had a singularity of the form
The theory was used to obtain an approximate inelastic cross section for transitions between certain highly excited states of hydrogen. For incident proton or electron 1 of kinetic energy E, and mass m,,and bound electron of mass m 2 and binding energy u=--1 e2 (45) 2a0 n2 ’ in state with principal quantum number n, the resulting symmetrized inelastic cross section was
The primed quantities are final values. The rather restricted conditions for the approximate validity of this expression are 1 4 In - n’l
+ n,
m1 E U <m2
-= U2/)&I
(47)
Light (1962) has obtained an adiabatic theory of the three-particle inelastic scattering problem, in which two of the particles are bound by a cutoff harmonic oscillator potential, using the Liouville equation theory of Prigogine and Henin (1957) and Resibois (1959) (Section 11,2). He treated the exact solution as a perturbation about the solution of the soluble two body problem for the bound particle. The expansion parameter was the inverse of the oscillator frequency, and not the interaction potential, so he was able to apply the theory to the dissociation of a diatomic molecule in a single collision with a passing particle even when the interaction between them was strong.
CLASSICAL THEORY OF ATOMIC SCATTERING
127
Table I compares experiment and theory for the dissociation of bromine in argon. Agreement is reasonably good. TABLE I THEDISSOCIATION OF BROMINE IN ARGON' T (OK) 298 400 1200 2000
Theory 10.2 7.6 3.1 2.2
7.1 4.8 1.3 0.8
Experiment 13.0 8.4 2.4 1.5
12.4 8.1 1.6 0.8
Normalized reaction rate kDexp(ED/kT) x cc/rnole-sec for bromine in argon.' The dissociation rate is kD and E D is dissociation energy. The first three columns are from the Light theory with different interaction potentials. The fourth colunin is derived from experiment.
B. THRESHOLD LAWS There is not yet any general theory of threshold laws for classical scattering. For electron collisions with classical hydrogen atoms, the threshold laws are closely related to the behavior of the differential cross section dQ/dc as a function of El and E . For El > U and E > c1 > 0 binary encounter theory and Monte Carlo calculations indicate that dQ/dc is a nonsingular nonzero function of these variables for fixed cl. The Monte Carlo orbits also indicate that a binary encounter is not a bad qualitative description of what occurs in an ionization even near the threshold for ionization, although the numbers are wrong. No proof is available that this is the correct behavior of dQ/dE, although it seems very reasonable. If dQ/de is neither singular nor zero, then by Eq. @a), Q,(E,)
N
k(E, - U)
( k a constant)
(48)
near threshold. Also, the threshold cross section is linear for excitation into any energy band between fixed positive E , and c 2 . The singular behavior for very small energy transfers, which is valid for any incident charged particle, is given by the classical adiabatic perturbation theory of Section IV,A. Wannier (1953) considered the quantum threshold law for ionization of H by electrons. On the very reasonable assumption that only quantum states of zero angular momentum need be considered, the problem was reduced to motion in two dimensions, and since the dynamics of the motion with both electrons distant from the proton was crucial, he considered the two-dimensional classical motion of the electrons.
A . Burgess and I. C. Percival
128
On this basis, and on a quasiergodic hypothesis that the orbits responsible for ionization occupy the available phase space with a nonsingular probability density, Wannier obtained a quantum threshold law : Q,(E1)
N
k ( E , - U)"'''
(49)
This result does not conflict with the linear law for classical ionization, as it refers to classical collisions in which the electrons are restricted to orbits with total angular momentum zero, which for three-dimensional classical scattering have zero probability. The quantization of angular momentum is crucial to Wannier's theory. But Wannier's result does conflict with the more recent theory of Rudge and Seaton (1965) which predicts a linear quantum threshold law for e-H ionization. Wannier has generalized his result to multiple ionization (1956). Light (1964, Appendix B) has derived a linear threshold law for endothermic ion-molecule exchange reactions with classical distributions in energy and angular momentum. He considered the effect of quantization and thus the range of energies over which the classical law was expected to hold.
V. Orbit Integration and Monte Carlo Methods A. THEORY These methods consist of computer scattering experiments, in which collisions are simulated on a digital computer. Consider the application of the orbit integration method to an individual collision. The state of the system at a time t is represented in the computer by the values X of the positions and momenta of the particles at that time, or by generalized coordinates and momenta. Initially these are set to their values X - at the time t - . No dynamical approximation is made. The equations of motion for X are solved by stepwise numerical integration, with the physical time t as the independent variable, until the values X , at the final time t , are obtained. The values of any final parameters such as the channel, angles of scattering, and binding energies are then printed out. In practice, only a finite number of steps can be integrated, giving rise to the following errors : (El) Truncation errors in the difference formula for the stepwise numerical integration of the equations of motion. (E2) Time cut oflerrors: The initial and final states Xi are those at finite times ti , or else the computation time would be infinite, whereas the scattering parameters are defined in terms of limits as t -+ k co. The Monte Carlo method proper applies to general collisions, involving a defined statistical ensemble of initial conditions. The initial conditions for
CLASSICAL THEORY OF ATOMIC SCATTERING
129
each orbit are sampled from the initial statistical distribution p(X, r - ) , using random numbers which are fed into the computer, or a pseudorandom sequence which is generated within the computer (Hammersley and Handscomb, 1964, p. 27). The method of production of the initial sample of the distribution p(X, t - ) from the random numbers varies from problem to problem, and will be discussed in the next section. Each set of initial conditions defines an individual collision which is solved by orbit integration. The final states X , form a sample from which approximate total and differential cross sections can be obtained as in a laboratory collision experiment. Only a finite number of orbits can be integrated, so that there are: (E3) Sratisricalerrors, due to the sample being finite. These can be expressed as error bars on the estimated cross sections. The orbit integration method is described by Wall et al. (1958), the statistics of the Monte Carlo method by Blais and Bunker (1962), and by Karplus and Raff (1964), and in simple terms by Bunker (1964b). Applications to ionization and charge transfer are discussed by Abrines and Percival(1966a,b). If T, is the total time of computation, then for the usual Monte Carlo methods the statistical errors in cross sections are proportional to TC-"with v = 4. For fourth order stepwise integration, the truncation errors for the whole orbit are proportional to T,-" with v E 4. The time cutoff error depends on the potential, but even for the worst case decreases much more rapidly than T,-'". Thus with relatively little additional computation the systematic errors ( E l ) and (E2) can be reduced to negligible size compared with the statistical error (E3). Relative statistical errors are generally of the order of 5 % or more. We have described a direct simulation hit-or-miss method, which is close in conception to a laboratory experiment, but which is the crudest and least efficient Monte Carlo procedure. At the price of sacrificing the close similarity to the actual collision process, the method may be improved by application of several techniques which are described by Hammersley and Handscomb (1964) for general Monte Carlo methods. Some of these methods have been applied to collisions. Abrines and Percival ( I 966a) used importance sampling, biasing the sample towards important collisions and correcting the bias analytically. Stratified sampling was used by Abrines et a / .(1966). They divided the initial range of X - into partial ranges or strata, and ensured the number of sample orbits corresponding to each stratum was exactly proportional to its statistical probability. Without stratified sampling the number was proportional in the mean, and there was additional statistical fluctuation about this mean which increased the statistical errors in the cross section. Percival and Valentine (1966) used a modified method of control variates, whereby a large sample of orbits was integrated quickly using few steps of integration and with relatively large errors (El)
130
A . Burgess and I. C. Percival
and (E2), and then orbits which were close to the borderline between final channels were repeated using more steps to see if the channel changed. Each of these methods was helpful, and each reduced the time of computation by a factor of rather less than 2 in the cases in which it was applied. The exact improvement could not be obtained for the more sophisticated methods, as for them there was no analytic expression for the statistical error. B. MICROCANONICAL DISTRIBUTION
In order to use the Monte Carlo method we must obtain a sequence of initial states drawn from the initial distribution p ( X , t - ) . The factors for the internal motion are the most difficult to obtain. For beam experiments, we usually require a microcanonical distribution in the phase space of each of the colliding systems; for simplicity we choose the system A , and drop the suffix A in this section. Usually only one system is compound. If we divide the sequence of random numbers uniformly distributed between 0 and 1 into groups of n members each, then each group defines a point in an n-dimensional cube, and these points are uniformly distributed within the cube. By linear transformation we obtain a uniform distribution in any n-dimensional cuboid. We require a sample from a microcanonical distribution in phase space of a bound system, which we suppose has three degrees of freedom:
p E . ( X )= K 6 ( H ( X ) - E’)
( K a constant)
(50)
where X here represents the generalized coordinates and momenta ( q , , q 2 ,q 3 , pl, p 2 , p 3 ) of the bound system and H ( X ) is its Hamiltonian function. The distribution is confined to the energy shell for E ’ and has the property that when integrated over E‘ from E’ to ( E ’ A E ) it gives a distribution which is uniformly distributed in phase space between these limits. Blais and Bunker (1962) used a projection method for the microcanonical ensemble as follows. Let Y = (q,, q 2 , q 3 , p 2 , p 3 ) represent all coordinates except p , . We can obtain a distribution which is uniform in Y within a cuboid which contains the E ‘ shell for all values of p,. We then project this distribution onto the E ‘ shell by solving
+
N P , , y >= E‘
(51)
for p,. Suppose there were only one solution for p , , say p l ’ . The projected distribution has the property that when integrated overp,’, fromp,’ to (pl’ A p , ’ ) , it is uniform in phase space. Thus, although it occupies the energy shell, it has
+
CLASSICAL THEORY OF ATOMIC SCATTERING
131
the wrong density on that shell. To correct, we multiply the density by
This is just the generalized velocity corresponding to p 1 and for the case of interest has a maximum value lql lmax. Bunker and Blais therefore selected a fraction lcjl I/ Id1lmax of their sample for a given Y, again using independent random numbers to decide which values of Y should or should not be included. The value of lql lmaxcould be slightly overestimated without additional error or serious loss of computing time. Usually Eq. (51) has two solutions which are chosen with equal probability. In the work of Abrines and Percival (1966a) on collisions with hydrogen atoms, the microcanonical distribution was factorized into one-dimensional distributions, each of which were uniform and so could be obtained directly from the random number sequence. Many cases are intermediate between these two extremes, and can be partially solved by factorization. The statistical distribution for the relative motion is comparatively easy to obtain. As it is an infinite distribution, some artificial upper limit b,,, must be put on the impact parameter b, and this is generally chosen to be slightly greater than the estimated b,,,(c) for the final channels of interest. In the case of excitation, in which the final and initial channels are the same, b,,, must be limited to keep the time of computation finite, and this puts a practical lower limit on the magnitude E of the energy transfer that can be investigated by the method. Smaller values of E must be treated by an alternative method, such as the adiabatic method of Section IV,A. In practice, the initial time t - is chosen separately for each orbit to ensure that the colliding systems are sufficiently far away initially to interact very little, and thus minimize the time cutoff error. This is simple for chemical problems with short range forces, but for charged particles with asymptotic dipole forces and particularly for low incident velocities, the time t - has to be chosen carefully to ensure that time cutoff errors are small when compared with the statistical errors. C. NUMERICAL INTEGRATION
Either Cartesian or generalized coordinates have been used for the numerical integration. It is not clear which is quicker numerically, as no direct comparisons are available, but Cartesian coordinates are simpler. Runge-Kutta or Runge-Kutta-Gill methods were used for stepwise integration ; for the chemical processes a constant interval of integration was used. For ionization and charge transfer the attractive singularity of the
132
A . Burgess and I. C. Perciual
coulomb force and its long range effect made variation of the step length essential. It was changed at every step according to a fairly complicated formula which depends on the scaling properties of the coulomb forces (Abrines and Percival, 1966a). The termination of the integration at time t+ was also more complicated for Coulomb forces.
D. APPLICATION TO CHEMICAL PROCESSES The early history of the orbit integration and Monte Carlo methods is summarized in Table 11. Evidently, the success of the method depended crucially on the development of computing technology. Hirschfelder et al. (1936) had no automatic computer, and their orbit integration was incomplete. The fact that this particular orbit was complicated and spent a long time in the reaction region encouraged collision complex theories of reaction. More recent calculations with better statistics demonstrate that the orbit was not typical, and illustrate the danger of inadequate statistics. Wall et al. (1958, 1961) pioneered the modern orbit integration methods, and were able to discuss some characteristics of the orbits, but under their conditions the number of orbits was insufficient to predict reaction rates. In any case, no attempt was made to obtain a correct statistical sample for the initial conditions. The first successful application of Monte Carlo methods to reaction rates was that of Blais and Bunker (1962), using methods which were also used by Bunker (1962) on triatomic dissociation problems. The application of Monte Carlo methods to molecular beam studies is reviewed by Herschbach (1966). Best comparisons are the for K + CH, I reaction, treated theoretically with CH, as a single particle. The experiments of Herschbach (see Karplus and Raff, 1964) have been used as a basis for comparison. The original calculations of Bunker and Blais were for two-dimensional reactions, and Karplus and Raff (1964) were able to show that the offplane reactions were important, and hence that full three-dimensional studies must be carried out. However, their own calculations gave total cross section about 60 times larger than experiment. They were able to trace this discrepancy to a spurious attraction between the reactants occurring in the trial Morse-type potential used by Bunker and themselves. Consequently other conclusions of the calculations are suspect. More recent calculations of Raff and Karplus (1966) on the same reaction used four different functional forms for the potential energy, and three give reasonable agreement with measured values of cross sections. In particular, 80 % or more of the reaction energy remained in
TABLE I1 ORBIT INTEGRATION AND MONTECARL0 METHODS FOR SIMPLE CHEMICAL REACTIONS Publ. date
Authors
Machine
Number of orbits
0 r P rn
Problem
zn
1936
Hirschfelder et a/.
Desk
Part of 1
Colinear H
+ H z reaction
1958
Wall et a/.
Illiac
Several hundred
Colinear H
+ Hz reaction
1961
Wall et a/.
Illiac
700
Three-dimensional H
1962
Bunker
St retch
90,026
Planar dissociation of nonrotating triatomic molecules (6 cases),models of N,O and O3
P r
-1
+ H 2 reaction
;cr
<
2 3
i5 v1
1962
Blais and Bunker
Stretch
?
+
+
Planar reaction M CH31 + CH3 MI (M an alkali atom); CH3 regarded as single particle
1964a
Bunker
Stretch
235,000
RRK M vs. Slater theory for dissociation
n
>
4 -1
m
z!
8
CL
w w
I34
A . Burgess and I. C. Percival
the internal degrees of freedom in both theory and experiment, and the laboratory angular distributions of the products were comparable. They also obtained many other results which are not yet accessible to direct experiment. Thus, the Monte Carlo method is being used to help determine potentials from beam experiments, as much as to obtain qualitative features of the reaction mechanisms. Benson and de More (1965) have evaluated other theories in the light of Monte Carlo methods, and Cross and Herschbach (1965) have made an orbit integration study of collisions between an atom and a rigid rotator, and have thereby tested various approximate theories. Wolf (1965) has carried out computer experiments on ion-molecule interactions, treating the molecule as a rigid rotator, whereas Alterman and Wilson (1965) and Kelley and Wolfsberg (1966) have used orbit integration to evaluate approximate translationalvibrational energy transfer formulas. Polanyi and his collaborators (Polanyi and Rosner 1963; Kuntz et a/., 1966) have made orbit integration studies of a two-dimensional potential energy surface. Finally, Raff (1966) has treated the K + C2 H, I reaction as a four body interaction using Monte Carlo methods.
E. APPLICATION TO p-H
AND
e-H COLLISIONS
The detailed theory required for the Monte Carlo calculation of p-H cross sections was applied by Abrines and Percival(l964, 1966a,b) to charge transfer and ionization. Because of the special properties of the coulomb potential the same theory applies to collisions with atoms in any energy level, provided the states of that level are initially equally populated. For very high n it follows from the correspondence principle that classical theory is valid. For low n the validity does not follow from the correspondence principle but the results agree remarkably well with experiment when the hydrogen atom is initially in the ground state. The reason for this close agreement is not fully understood. Since the Coulomb potential is an inverse power potential, one can use dynamical similarity (Landau and Lifshitz, 1960) to relate orbits of different sizes, provided that if for distances r = Or' we also put for times t = 0 3 / 2 t 'and , scale all other variables according to their dimensions. Masses, charges, and dimensionless ratios remain unchanged. Thus if Q is a cross section for a collision of an incident particle with energy E with a hydrogen atom of ionization energy U and semimajor axis a, then Q/(na2)is a fixed function of E / U , since each is dimensionless. Thus the classical cross section need only be calculated for one U . This theory only applies strictly for classical scattering and thus for high quantum numbers, but it is also a reasonable empirical approximation (Elwert, 1952) for low quantum numbers. For all cases we have to choose an initial distribution for the orbits of the electron. In the classical limit the highly excited levels with equal population
CLASSICAL THEORY OF ATOMIC SCATTERING
135
in each state correspond to a uniform distribution in phase space on the corresponding energy shell. This is a microcanonical distribution. The resultant distribution p(p) for the momentum is (Pitaevskii, 1962; Mapleton, 1966) P(P> =
8Pc5 n2(p2 pc2)4’
+
(53)
where the classical momentum p c is given by pc2 = 2m, U .
(54)
For low initial principal quantum number n there is an ambiguity in the choice of the initial conditions for the classical model of the hydrogen atom. However, there are strong reasons for choosing the same distribution. First, the hydrogen atom initially has a stationary, spherically symmetric distribution when all states of a given level are equally populated. As shown in Appendix I of the paper by Abrines and Percival (1966a), the probability distribution for a classical model hydrogen atom which is stationary and spherically symmetric is completely determined except for the distribution in angular momentum. The choice of this distribution is assisted by the fact shown by Fock (1935) that the form of the momentum distribution for the level n of the quanta1 H atom is independent of n, and has the same form Eq. (53), as required by the correspondence principle. The only stationary spherically symmetric distribution with this property is the same microcanonical distribution. Thus Abrines and Percival were able to obtain cross sections for any n by a single classical Monte Carlo calculation. The results for n = 1 ionization and charge transfer are both within the experimental errors. The results are particularly close to the experimental curve for charge transfer (as given by McClure (1966). As expected, the quantum In E/E dependence for ionization is not obtained for incident protons at the higher energies. To obtain the classical cross sections for any n, the units for the energy must be changed to n-’ keV and for the cross section to rr(n’~,,)~.For high n the curves thus represent a range of very much lower incident proton energies. Monte Carlo calculations were also carried out for ionization of classical hydrogen atoms byelectrons(Abrines eta/., 1966; Brattsev and Ochkur, 1967). As expected, the agreement with experiment for n = 1 is not nearly so good as for protons at low incident electron velocities; this disagreement is due at least in part to the quantum mechanical interference between direct and exchange scattering, which is shown to be important by the binary encounter theories. One striking feature of the results is that all purely classical theories have a maximum at 34 eV, well below that of the experiment.
136
A . Burgess and I. C. Percival
The quantum effects become smaller as n increases. Using the same transformations as for incident protons, we can obtain classical Monte Carlo ionization cross sections for arbitrary n, which should be reliable in the corresponding energy range for n 2 3. The Monte Carlo calculations were also compared in detail with the classical binary encounter theory of Vriens (1966a), and comparisons between the dQ/dE were made for various incident electron energies. It showed that the classical ionization cross section was slightly overestimated by the binary encounter theory, owing particularly to a gross overestimate of dQ/dE for low ejected electron energies. Reasonable extrapolation showed that a pure classical binary encounter theory grossly overestimates excitation cross sections, even for an incident electron energy as high as 218 eV. Calculations have also been carried out for positronium formation, in e+-H collisions. These results are given in the article by Fraser in this volume.
F. WIGNER-KECK VARIATIONAL METHOD Wigner (1937, 1939) obtained an upper bound on reaction rates by considering the flux across a surface in phase space between regions which contain the reactants and the regions which contain the products. The method has only been applied to equilibrium reaction rates, but there appears to be no reason why it should not be extended to rates and cross sections at a given energy. The theory was slightly generalized by Horiuti (1938), described in detail and extended to become a variational method by Keck (1960). Let S be a surface which divides the initial channel c- from the final channel c, . In regions where there is negligible interaction S must be a surface with zero flux across it, but where there is significant interaction it can be chosen to be explicitly dependent on arbitrary parameters c i j . These c i j may be varied, so that the surface is varied, and the least Wigner bound obtained. This upper bound to the reaction rate is given by the rate of flow across S(aj) in phase space in the direction from c- to c + , R(aj)=
pv
*
A
do,
(55)
where p is the density in phase space, A is the unit normal and d o is an element of the surface S. This would be the exact rate were it not for the fact that trajectories can cross the surface S more than once, and thus be included in the integral ( 5 5 ) more than once. The calculations have been carried out for the case when p is an equilibrium distribution, P = Po exp( - H / W (56)
CLASSICAL THEORY OF ATOMIC SCATTERING
137
where H is the Hamiltonian function of the point in phase space. In practice, the minima of the trial reaction rates were obtained by numerical trial and error. The method has been extended by Woznik (1965). Keck and Makin (1963) have applied the method to three-body electron-ion recombination in plasmas. They get reasonable agreement with. Bates et a/. (1962) for the case in which collisional deexcitation is the dominant mechanism. Keck (1962) has used a particularly promising combination of the Monte Carlo method and the Wigner-Keck variational method. A rough estimate of the reaction rate was made by the variational method, and then a sample of orbits integrated from the trial reaction surface both ways in time. In this way statistical estimate was made of the number of intersections of these orbits with the trial surface, and thus of the correction to the Wigner-Keck variational bound. Since only the correction is to be estimated by Monte Carlo methods, the overall error for the reaction rate is much smaller than for direct sampling. The method automatically selects important orbits.
VI. Correspondence Principle and Conclusions There are difficulties in applying classical methods to quantum mechanical problems, both when the number of quanta1 states is relatively large, as in most chemical processes, or when the number is small, as in electron-atom excitation. However, in the case of atom-molecule scattering, these difficulties are swamped by the much greater difficulty of finding a suitable potential energy for the interaction between the atoms. m most cases any quantum mechanical effects are likely to be small compared with the errors due to uncertainties in this potential energy, which now present the main obstacle to the application of classical methods to chemical reactions. For applications to atomic excitation, ionization, and charge transfer, it is otherwise. The potentials are known. According to the correspondence principle, for sufficiently highly excited states, classical mechanics should be valid for all processes. If the changes in the quantum numbers are large in an individual collision, this can be proved (Abrines and Percival, 1966b). For recombination of electrons and ions in plasmas, n + n + 1 collisions are of great importance for high principal quantum number n, and the classical theory fails to provide the necessary Q = KE;' In El dependence of the cross sections Q on the incident electron energy El. However, the work of Williams (I93 1) discussed in Section III,E suggests that for high n the same overall effect would be obtained by using classical theory with a classical continuous range of highly excited states as by using quantum
138
A . Burgess and I. C. Percival
theory with quantized states; but wrong results could result from the use of classical cross sections with quantized states. By contrast with the limit of high n, there is no known principle by which one can obtain a unique correspondence between classical theory and quanta1 cross sections when n is small. Indeed, there is some ambiguity in relating the two, and, at present, only plausibility arguments or a posteriori comparison with experiment can be used in choosing a relation. There is ambiguity in the choice of initial conditions. Quantum mechanical states must be represented by probability distributions over classical states. These distributions cannot be uniquely defined in both position and velocity because of the correspondence principle. However, the velocity distribution is much more important than the position distribution for scattering; in fact, the latter usually has no effect in the binary encounter approximation. So it is plausible to use the same initial velocity distributions. The classical distributions in position are still not uniquely defined; for the H atom, the arguments in favor of a microcanonical distribution are strong. There is ambiguity in the final conditions. There is little doubt that total cross sections for ionization and charge transfer channels in classical and quantum theory should correspond. Difficulties arise in the discrete spectrum of quantum dynamical operators. Thus there is difficulty if the final state of interest is an individual bound state or level. The classical probability of finding an atom in an orbit of a given energy is zero. Only finite energy ranges have finite probability and finite cross sections. The usual choice of an energy band between the final level and the next highest level (Gryzinski, 1959; Stabler, 1964) is unsatisfactory since it gives an infinite elastic cross section and does not satisfy the reciprocity relations when applied to deexcitation (Benson et al., 1963). We conclude that for application to excitation of low levels of atoms, classical methods still need a solid theoretical foundation. The classical theory of collisions still consists of a number of parts which are inadequately related to one another. The application of classical methods to atomic collision problems has had a number of successes, particularly for atom-molecule scattering but a lot remains to be done. Clearly, the quoted authors who have worked on the classical binary encounter theory since 1958 would have been helped if the earlier work of Thomas and Williams had been more widely known, although some might have been discouraged by the lack of detail in Thomas’ and Williams’ articles. ACKNOWLEDGMENTS We should like to thank R. Abrines, D. Banks, M. Gryzinski, R. W. P. McWhirter, M. J. Seaton, F. T. Smith, N. A. Valentine, and L. Vriens for helpful discussions.
CLASSICAL THEORY OF ATOMIC SCATTERING
139
REFERENCES Abrines, R., and Percival, I. C. (1964). Phys. Letters 13, 216. Abrines, R., and Percival, I. C. (1966a). Proc. Phys. SOC.(London) 88, 861. Abrines, R., and Percival, I. C. (1966b). Proc. Phys. SOC.(London) 88, 873. Abrines, R., Percival, 1. C., and Valentine, N. A. (1966). Proc. Phys. SOC.(London)88, 885. Alterman, E. B., and Wilson, D. J. (1965). J . Chem. Phys. 42, 1957. Bates, D. R., and Mapleton, R. A. (1965). Proc. Phys. SOC.(London) 85, 605. Bates, D. R., and Mapleton, R. A. (1966). Proc. Phys. SOC.(London) 87, 657. Bates, D. R., and Mapleton, R. A. (1967). Proc. Phys. SOC.(London) 90,909. Bates, D. R., and Moffet, R. J. (1965). Nafure 205, 272. Bates, D. R., and Moffet, R. J. (1966). Proc. Roy. SOC.A291, 1. Bates, D. R., Kingston, A. E., and McWhirter, R. W. P. (1962). Proc. Roy. SOC. A267.297. Bauer, E., and Bartky, C. D. (1965). J. Chem. Phys. 43, 2466. Benson, S. W., and de More, W. B. (1965). Ann. Rev. Phys. Chem. 16, 397. Benson, S. W., Berend, G . C., and Wu, J. C. (1963). J. Chem. Phys. 38, 25. Bernstein, R. B. (1966). Advan. Chem. Phys. 10, 75. Blais, N. C., and Bunker, D. L. (1962). J. Chem. Phys. 37, 2713. Bohr, N. (1948). Kgl. Danske Videnskab. Selskab Mat. Fys. Medd. 18, No. 8. Brattsev, V. F.,and Ochkur, V. I. (1967). Zh. Eksperim. iTeor. Fiz. 52, 955 [Soviet Phys. JETP (English Trans/.) 25, 631 (1967)l. Bunker, D. L. (1962). Nature 194, 1277. Bunker, D. L. (1964a). J. Chem. Phys. 40, 1946. Bunker, D. L. (1964b). Sci. Am. 211, 100. Burgess, A. (1964a). Proc. Intern. Con$ Electron. At. Collisions, 3rd, London, 1963, p. 237. North-Holland Publ., Amsterdam. Burgess, A. (1964b). Proc. Symp. At. CollisionProcesses in Plasmas, Culham, 1964, A.E.R.E. Rept. 4818, p. 63. Chandrasekhar, S. (1941). Ap. J. 93, 285. Chandrasekhar, S., and Williamson, R. E. (1941). Ap. J. 93, 308. Cross, R. J., and Herschbach, D. R. (1965). J. Chem. Phys. 43, 3530. Elwert, G. (1952). Z. Naturforsch. 7a, 432. Fock, V. (1935). Z. Physik 98, 145. Ford, K. W., and Wheeler, J. A. (1959a). Ann. Phys. (N.Y.) 7, 259. Ford, K. W., and Wheeler, J. A. (1959b). Ann. Phys. (N.Y.) 7,287. Fowler, R. H. (1925). Proc. Cambridge Phil. SOC.22, 793. Gerjuoy, E. (1966). Phys. Rev. 148, 54. Gryzinski, M. (1959). Phys. Rev. 115, 374. Gryzinski, M. (1965a). Phys. Rev. 138, A305. Gryzinski, M. (1965b). Phys. Rev. 138, A322. Gryzinski, M. (1965~).Phys. Rev. 138, A336. Hammersley, I. M., and Handscomb, D. C. (1964). “Monte Carlo Methods.” Methuen, London. Herschbach, D. R. (1966). Advan. Chem. Phys. 10, 319. Hirschfelder, J., Eyring, H., and Topley, B. (1936). J. Chem. Phys. 4, 170. Horiuti, J. (1938). Bull. Chem. SOC.Japan 13, 210. Jackson, J. D. (I 962). “Classical Electrodynamics.” Wiley, New York. Karplus, M., and Raff, M. (1964). J . Chem. Phys. 41, 1267.
A . Burgess and I. C. Percival
140
Keck, J. C. (1960). J. Chem. Phys. 32, 1035. Keck, J. C. (1962). Discussions Faraday SOC.33, 173. Keck, J. C., and Makin, B. (1963). Phys. Rev. Letters 11, 281. Kelley, J. D., and Wolfsberg, M. (1966). J. Chem. Phys. 44, 324. Kingston, A. E. (1964a). Phys. Rev. 135, A1529. Kingston, A. E. (1964b). Phys. Rev. 135, A1537. Kingston, A. E. (1966a). Proc. Phys. SOC.(London) 87, 193. Kingston, A. E. (1966b). Proc. Phys. SOC.(London) 89, 177. Kuntz, R. J., Nemeth, E. M., Polanyi, J. C., Rosner, S. D., and Young, C. E. (1966). J. Chem. Phys. 44, 1168. Landau, L. D., and Lifshitz, E. M. (1958). “Statistical Physics.” Pergamon Press, Oxford. Landau, L. D., and Lifshitz, E. M. (1960). “ Mechanics.” Pergamon Press, Oxford. Light, J. C. (1962). J. Chem. Phys. 36, 1016. Light, J. C. (1964). J. Chem. Phys. 40,3221. Lippmann, B. A., and Schwinger, J. (1950). Phys. Rev. 79, 469. Loeb, L. B. (1939). “Fundamental Processes of Electrical Discharges in Gases.” Wiley, New York. Mapleton, R. A. (1966). Proc. Phys. SOC.(London) 87,219. Massey, H. S. W., and Mohr, C. B. 0. (1933). Proc. Roy. SOC.A141,434. Massey, H. S. W., and Mohr, C. B. 0. (1934). Proc. Roy. SOC.A144, 188. McClure, G. W. (1966). Phys. Rev. 148,47. McDowell, M. R. C. (1966). Proc. Phys. SOC.(London) 89, 23. McFarland, R. H. (1965). Phys. Rev. 139, AN. Mott, N. F., and Massey, H. S. W. (1965). “Theory of Atomic Collisions.” Oxford Univ. Press (Clarendon), London and New York. Ochkur, V. I. (1963). Zh. Eksperim. i Teor. Fiz. 45,734 [Soviet Phys. JETP (English Transl.) 18,503 (196411.
Ochkur, V. I., and Petrun’kin, A. M. (1963). Opt. Spectry. (USSR) (English Truns.) 14,457. Percival, I. C., and Richards, D. (1967). Proc. Phys. SOC.(London) 92, 311. Percival, I. C., and Valentine, N. A. (1966). Proc. Phys. SOC.(London) 88, 885. Pitaevskii, L. P. (1962). Zh. Eksperim. i Teor. Fiz. 42, 1326 [Soviet Phys. JETP (English Transl.) 15, 919 (1962)l. Polanyi, J. C., and Rosner, S. D. (1963). J. Chem. Phys. 38, 1028. Prasad, S. S., and Prasad, K. (1963). Proc. Phys. SOC.(London) 82, 655. Prigogine, I. and Henin, F. (1957). Bull. Acad. Roy. Belg. Cl. Sci. 43, 814. Raff, M. (1966). J. Chem. Phys. 44, 1202. Raff, M., and Karplus, M. (1966). J. Chem. Phys. 44, 1212. Rbibois, P. (1959). Physica 25, 725. Robinson, B. B. (1965). Phys. Rev. 140, A764. Rudge, M. R. H., and Seaton, M. J. (1965). Proc. Roy. SOC.A283, 262. Rutherford, E. (191 1). Phil. Mug. 21,669. Saraph, H. (1964). Symp. At. Collision Processes in Plasmas, Culham, 1964, A.E.R.E. Rept. 4818, p. 74. Sheldon, J. W., and Dugan, J. V. (1965). J. Appl. Phys. 36, 650. Smith, F. T. (1965). J. Chem. Phys. 42,2419. Smith, F. J., Mason, E. A., and Vanderslice, J. T. (1965). J. Chem. Phys. 42, 3257. Stabler, R. C. (1964). Phys. Rev. 133, A1268. Thomas, L. H. (1927a). Proc. CambridgePhil. SOC.23, 714. Thomas, L. H. (1927b). Proc. Cambridge Phil. SOC.23, 829. Thomas, L. H. (1927~).Proc. Roy. SOC.A114, 561.
CLASSICAL THEORY OF ATOMIC SCATTERING
141
Thomson, J. J. (1906). “Conduction of Electricity through Gases,” Cambridge Univ. Press, London and New York. Thomson, J. J. (1912). Phil. Mug. 23,449. Thomson, J. J. (1924). Phil. Mag. 47, 337. Vriens, L. (1966a). Phys. Rev. 141,88. Vriens, L.(1966b). Proc. Phys. SOC.(London) 89, 13. Vriens, L.(1967). Proc. Phys. SOC.(London) 90,935. Wall, F. T., Hiller, L. A., and Mazur, J. (1958). J. Chem. Phys. 29, 255. Wall, F. T., Hiller, L. A., and Mazur, J. (1961). J. Chem. Phys. 35, 1284. Wannier, G. H. (1953). Phys. Rev. 90,817. Wannier, G. H. (1956). Phys. Reu. 100,1180. Webster, D.L., Hansen, W. W., and Duveneck, F. B. (1933). Phys. Reu. 43,839. Wigner, E. P. (1937). J . Chem. Phys. 5, 720. Wigner, E. P. (1939). J. Chem. Phys. 7 , 646. Williams, E. J. (1927). Nature 119,489. Williams, E. J. (1931). Proc. Roy. SOC.A130, 328. Williams, E.J. (1945). Rev. Mod. Phys. 17,217. Wolf, F.A. (1965). Abstracts. Proc. Intern. Con$ Phys. Electron. A t . Collisions, 4th, Quebec, 1965. Science Bookcrafters, Hastings-on-Hudson, New York. Womik, B. J. (1965). J . Chem. Phys. 42, 1151.
This Page Intentionally Left Blank
BORN EXPANSIONS A.
R. HOLT and B. L. MOISEIWITSCH
Departmenf of Applied Mathematics School of Physics and Applied Mathematics The Queen’s University of Berfast Belfast, Northern Ireland
I. Introduction ................................................... 11. Born Expansion for the Scattering Amplitude ........................
.143 .144 A. Variational Principles for the Scattering Amplitude ...............,146 .148 B. Screened Coulomb Potential ................................... C. Coulomb Potential .... ......... .................... 150 D. Scattering of Electrons by Hydrogen Atoms ..................... .151 111. Convergence of Born Expansions .... IV. Time-Dependent Collision Theory . . . . Scattering of Protons by Hydrogen Ato ........................ 164 ....................... .169 V. Rearrangement Collisions . . . . References ...................................................... 1 7 1
I. Introduction If the potential 1V(r) of a center of force is regarded as a small perturbation disturbing the motion of an incident particle, a series expansion in powers of the strength 1of the potential may be obtained for the wave function of the incident particle and for the scattering amplitude. Such an expansion is called a Born series. The leading term of the Born series for the scattering amplitude is called the first Born approximation to the scattering amplitude and has been used extensively to evaluate cross sections. In a number of cases the first two terms of the Born expansion for the scattering amplitude have been evaluated, but owing to the complexity of the analysis involved hardly any investigations have been made employing still higher order terms of the expansion. In the present article we shall be concerned with developing the basic theory of Born expansions, including the important problem of obtaining conditions for their convergence, and with the practical application of the second and higher Born approximations to the calculation of scattering amplitudes and cross sections. 143
A . R.Holt and B. L. Moiseiwitsch
144
II. Born Expansion for the Scattering Amplitude The wave function $(r) describing the scattering of particles of mass m having wave number k by a potential l.V(r) satisfies the Schrodinger equation V2
1
+ k2 - 2m IV(r) $(r) = 0 h
where for large radial distances r $(r)
-
exp(ikj r) + r - l exp(ikr)f(B, +),
(2) ki being a vector of magnitude k having the direction of the incident beam of particles and f(8, 4 ) being the scattering amplitude for particles deflected through polar angles 8, 4. The differential cross section is given in terms of the scattering amplitude by the formula
I ( 4 4) = If(&4)12 while the total cross section is given by
Q=
s’
12’1(0, 4) sin 0 d8 d4.
0
0
(3)
(4)
The total cross section may also be expressed in terms of the imaginary part of the forward scattering amplitude Imf(0) by the optical theorem : 471
Q = - Imf(0). k
(5)
We may write the solution of the wave equation (1) with asymptotic behavior (2) in the integral equation form
$(r) = exp(ik, r)
+I
where
s
G(r, r’)U(r’)$(r’) dr’
1 exp(ik Ir - r’l) C(r, r’) = - 471 Ir - r’l
is the Green’s function for a free particle and’ V(r) =
2m V(r). A
Hereafter we shall refer to XV(r) as the potential.
(6)
(7)
145
BORN EXPANSIONS
Now for large values of r
k Ir - r’l
-
kr - k, * r’,
(9) where k, is a vector of magnitude k having the direction of the scattered particles given by r, and hence we obtain the integral equation for the scattering amplitude exp( - ik, * r‘)U(r’)$(r‘) dr’.
4n
To derive the Born expansion for the scattering amplitude we express the wave function and the scattering amplitude as power series in the strength A of the potential according to the formulas m
and
Now substituting these expansions into Eqs. (6) and (10) and equating the coefficients of A” we obtain
$,,(r)
= JG(r,
r’)U(r’)$,,-l(r’) dr’
( n # 0)
(13)
and
f,(O,
4) = -
where
‘s
exp( - ik, * r’)U(r’)$,,- ,(r’) dr’,
(14)
t,b0(r) = exp(ik, r) is the unperturbed plane wave function representing the incident beam of particles. The ( p 1)th Born approximation to the scattering amplitude is defined by
+
and may be expressed in terms of the pth order wave function
according to the formula
”s
f ( p +‘ ) ( O , 4) = - - exp( - ik, * r‘)U(r’)$(P)(r’) dr’.
471
A . R.Holt and B. L. Moiseiwitsch
146
In particular the first and second Born approximations to the scattering amplitude are given by
f ( 1 ) = Afl
(19)
and f(2)
+ A2f2
= Afl
respectively, where
f,
=
1
- -jexp(i(ki 47c
- k,)
- r’}U(r’) dr’
and
x exp(iki * r2) dr, dr, VARIATIONAL PRINCIPLES FOR THE SCATTERING
A.
.
(22)
AMPLITUDE
The Kohn variational principle is based upon the functional
s
I(k,, ki) = $,(r)[V2 where for large r
$i(r)
-
exp(iki * r)
+ kZ - AU(r)]$i(r)
+ r-’
dr
exp(ikr)fi(k, ki)
(23)
(24)
and
$,(r)
N
- + r-’ exp(ikr)f,(k,
exp( - ik, r)
- k,),
(25)
k being a vector of magnitude k in the direction of r. If we take t,hi and $, to be the exact solutions of the wave equation (1) having the asymptotic forms (24) and (25), respectively, and if 61(k,,ki) is the change in l(ks, ki) arising from infinitesimal variations S$i and a$, such that for large r
S$i
-
r-l exp(ikr) 6fi(k, ki),
(26)
then it has been established by Kohn (1948) that to the first order of small quantities 6I(k,, ki) =
- 4n 6fi(k,, kJ.
This is Kohn’s variational principle for the scattering amplitude.
(27)
147
BORN EXPANSIONS
Let us now introduce trial functions
-
$iT
and
$,T
where for large r
t+hiT(r) exp(ik, * r) + r - l exp(ikr)f,,(k, ki),
(28) and denote the expression obtained by substituting these trial functions into (23) by IT(ks,ki). Then it follows from the Kohn variational principle (27) that to the first order of small quantities 1 f(ks 9 ki) =fiAks 9 ki) + 471 I (ks' k.1 '
since I(k,, ki) vanishes for the exact solution $i. If we take
c An$, P
$iT
=
= $(P)
n=O
and
$,
=
An$,
= $(q)
n=O
where $. is the adjoint function of $,,, we see that
7
ji&, ki) = - - exp( - ik, r)U(r)$(P- ')(r) dr 4rr
= f'P'(k,,
and
s
ki)
IT(ks, ki) = $("(r)[V2 =
-I
s
Hence we obtain from Eq. (29)
+ k 2 - AU(r)]$(p)(r) dr
$(q)(r)U(r)$p(r) dr.
7
f(k,, ki) =f(P)(k,, ki) - - $(q)(r)U(r)$p(r) dr. 4n Noting that the neglected quantity
/d$,[V'
+ k 2 - AU(r)] S$i dr
is of order l p + q + we2 deduce , that expression (34) for the scattering amplitude is correct to order Ap+q+' so that
"s
f ( P + q +')(k,, k,) = f(P)(k,, k,) - - $(q)(r)U(r)$p(r) dr. 4n
(35)
Hence if the solution of the Schrodinger equation is known to order AP, the scattering amplitude can be evaluated to order AZp+'
A . R.Holt and B. L. Moiseiwitsch
148
Another important variational principle, established by Schwinger, is based upon the following expression for the scattering amplitude :
s
f(ks, ki) =
exp(ik, r) dr exp(- ik, * r’)U(r’)$(r’) dr’
--”s$(r)U(r) 47C I-
J $(r)U(r)$(r) dr - I
I-I-
JJ
.
(36)
$(r)U(r)G(r, r’)U(r’)$(r’) dr dr’
This expression for f(k, , k i ) is stationary with respect to infinitesimal variations of the exact solutions $ and $ of the Schrodinger equation and has the additional valuable property of being homogeneous in $ and $. If we take the plane wave trial functions $T(r) = exp(iki * r),
JT(r) = exp( - ik, r)
(37)
and substitute them into the right-hand side of Eq. (36) we obtain the approximate expression for the scattering amplitude :
which, on retaining only terms of order I and approximation formula, Eq. (20).
A’, yields the second Born
B. SCREENED COULOMB POTENTIAL An interesting example of the application of the second Born approximation is provided by the screened Coulomb potential
for which a closed analytical formula results for the scattering amplitude. The first Born approximation to the scattering amplitude is given by (19) with
where K = ki - k, , Choosing the polar axis in the direction of the momentum change vector hK and performing the integrations over the polar angles we obtain
f~ =
1
so “
1
sin Kr exp(-jlr) dr = -
K 2 + jl*
where K = 2k sin(8/2) and 8 is the angle of scattering.
149
BORN EXPANSIONS
To evaluatef, we use the Fourier integral formula
which enables us to separate the variables rl and r2 in (22) and thus to write
‘S
f2 = lim -
3 2 1 ~ 4’ ~ - dq k2 - is
f-10
= Ilm e-10
‘S
1
-
2n2 q 2 - dq k2 - is {(q - k,)’
+ p2}{(q - ki)’ + p 2 }
’
(43)
A closed analytical expression for f 2 can now be obtained by employing the formula derived by Dalitz (1951): 1
1 - is [(q - P)’ + A’]’ - A(P2 + A’ - p’ - 2pAi)
lim
(44)
and the Feynman identity 1
1
-1
dZ [ + ~ ( l Z ) +b(l - Z)]’
+ +
with u = (q
- k,)’
+ p’,
b
= (q
+ p’
- ki)’
and 1 2
- a(l
+ Z) + -21 b(l - Z) = (4 - P)2 + A’.
Then carrying out the integrations over Z we find that
f’
= 2kA
sin(B/2)
[tan-
+
pk sin(8/2) i A 2k’ sin(8/2) +-In A 2 A - 2k2 sin(B/2)
where A’ = p4
+ 4p’k2 + 4k4 sin2(B/2).
(45)
I50
A . R. Holt and B. L. Moiseiwitsch
If we denote the total cross section evaluated to order 1" by Q'"),the first Born approximation to the total cross section is given by fn f2n
Since the imaginary part of the forward scattering amplitude is given by the second Born approximation to be Imf(2)(0) =
kA2
+
p2(4k2 p 2 )
we see that
This is a specizl case of a general result derivable directly from the optical theorem (5) that the total cross section correct to order I" is given by
Q'") = (44k)Imf(")(O).
(49)
C. COULOMB POTENTIAL The case of the Coulomb potential U(r)= - l/r (50) may be obtained by allowing p to approach zero in the example of the screened Coulomb potential treated in the preceding section. We then find that
fi = 1/K2 (51) so that the first Born approximation yields the following differential cross section for scattering by a Coulomb potential : =
If'"(e)12
I2
= 16k4 sin4(e/2) *
(52)
This is identical with the classical Rutherford scattering formula and also with the differential cross section formula obtained by using an exact quantum
151
BORN EXPANSIONS
mechanical treatment of the problem. Because of this latter occurrence it seems rather plausible that the terms of order Lz and higher in the Born expansion of the scattering amplitude should only contribute a phase factor to the formula forf(0). Referring to (46) we see that as p 0 the real part of f z ( 0 ) vanishes while -+
rmfz(e)
1 2k sin(8/2) 4k3 sinz(0/2) In p
)’
(
(53)
which is logarithmically divergent for small p. In addition Dalitz (1951) has determined the most divergent term off3(8), while Kacser (1959) has succeeded in evaluatingf3(0) exactly in the p 0 limit. They find that for small p -+
1
- 8k4 sin2(0/2)
f3(e)
(In(
2k sin(0/2) p
))
(54)
On the basis of (53) and (54) it has been suggested by Dalitz (1951) and emphasized by Kacser (1959) that the Born expansion of the scattering amplitude can be expressed in the closed form
L
2k sin(8/2)
4kZsin2(0/2)
(55)
for small p. The phase of the exponential factor is divergent here as a consequence of the very long range character of the Coulomb potential which perturbs the plane wave form of the incident wave even at great distances from the scattering center. D. SCATTERING OF ELECTRONS BY HYDROGEN ATOMS The second Born approximation to the scattering amplitude f(’) has been used extensively to investigate the scattering of electrons by hydrogen atoms. This is a much more complex problem than scattering by a potential field because of the coupling between the different states of the target atom. It can be readily shown that the scattering amplitude for the excitation of the nth state of a hydrogen atom from its ground state ( n = 1) is given by the second Born approximation to be
where
A = me2/h2= a; fi
=
’,
- 4n jexp{i(ko - k,) r}Vn1(r)dr,
(57)
152
A . R . Holt and B. L. Moiseiwitsch
and
4"being the wave function of the nth state of a hydrogen atom, k, and knbeing the initial and final wave vectors of the incident electron, and the summation in ( 5 8 ) being over all states m of the atom. Using the Fourier transform (42) and Bethe's integral
s
-
4R exp(iK rr) dr' = K 7exp(iK r), Ir - r'l
we find that m
where
with
s
I(n, m ;t) = &,(r)4,(r)(exp(it
r)
- 1} dr.
(63)
Whereas in the case of the screened Coulomb potential we have just a single term (43) to evaluate, we see that an infinite number of terms contribute to fi in the present example so that an exact closed analytical expression for f 2 cannot be obtained. The usual approach is to approximate by truncating the series, all the terms up to a particular level M of the atom being retained and all terms corresponding to higher levels and the continuum being dropped : M f2
2
Cf2,. m=l
Another approach to the problem, known as the Massey and Mohr approximation (Massey and Mohr, 1934; Rothenstein, 1954), is to replace k, by ko in (61) and then to employ the closure theorem
S +m(r')4m(r) = d(r - rr) m
(65)
153
BORN EXPANSIONS
which yields f2 = lim
dq (q2
- ko2 - k)(q - kJ2(q - kJ2
This integral expression can be readily evaluated in closed analytical form by using the Dalitz formula (44), and the Feynman identity (45). A useful modification of this procedure leading to more reliable results is obtained by evaluating exactly all the terms of the series (61) for f2 up to a certain level m = M and setting k, = kM+,in all the remaining terms (m > M l), evaluating them using the closure formula (65) :
+
where,fF+l*"'is the value of f2"' when k, is replaced by kM+l. The differential cross section may be calculated to order A3 using the second Born approximation formula (56) :
where
N O ) = ReCf2(e)/fi(e)I*
(69)
A term of order A4 is also provided by A2f2 but this should be neglected since we have not included the term of this same order arising from the AY3term of the Born expansion for the scattering amplitude. The total cross section can now be calculated using Eq. (68) by performing a numerical integration over the scattering angle according to the formula (4). Detailed calculations have been carried out on the elastic scattering of electrons by hydrogen atoms and on the 1s + 2s and 1s + 2p excitations of atomic hydrogen by electron impact. Let us consider the elastic scattering case first. In Table 1 we display values of the elastic scattering amplitude Ref2(0) for the forward direction 8 = 0. Values of Re,f2(0) obtained by Holt and Moiseiwitsch (1968) using the approximate formula (64) are compared with those obtained by employing Eq. (67) with M = 0, 1,2, 3,4, 5. The M = 0 approximation is just the Massey and Mohr approximation (66) and is seen to be rather unreliable. However, as M is increasedf2(0) converges rapidly and thus confidence in the approximation (67) for M 2 2 is strengthened.
154
A . R. Holt and B. L. Moiseiwitsch
n IW v
.-8 cr
cd
id
W
E
v
3 cd
id
.-
w"
155
BORN EXPANSIONS
Total elastic cross sections calculated to order A2 (i.e. using the first Born approximation) and to order A3 are given in Table 11. Here the calculations of Kingston and Skinner (1961) using formula (64) forf2 are compared with the calculations of Holt and Moiseiwitsch (1968) using the formula (67), both with M = 2. Similar comparisons are made in Tables 111 and IV where total cross sections for the excitation of the 2s and 2p states of hydrogen calculated by Kingston et a/. (1960b), Moiseiwitsch and Perrin (1965), and by Holt and Moiseiwitsch (1968) to orders ,I2 and A3 are displayed. It can be seen from Tables I1 and I11 that for k 5 2a;' the A3 contribution to the cross section is not small in comparison with the A2 term and thus that the convergence of the Born series for the 1s + 1s and 1s + 2s transitions is, at best, slow. However, for the Is + 2p excitation one sees from Table IV that the convergence of the Born series appears satisfactory even down to k = 1.5~~;'. TABLE I1 TOTALCROSSSECTIONS FOR H(ls)
+ e + H(ls) + e
Wave Number
Second Born First Born
ko (in a , ')
Equation (64),M
2.0 3.0 4.0 5.0 a
0.523 0.247 0.142 0.092
=2
Equation (67), M
0.701 0.285 0.154 0.097
=2
0.882 0.321 0.166 0.101
Units of mo2.
TABLE 111 TOTALCROSS SECTIONS FOR H(ls)
Wave Number ko (in a; ')
First Born
2.0 3.0 4.0 5.0
I .019 0.476 0.272 0.175
+ e + H(2s) + e Second Born
Units of 10-'mo2.
Equation (64),M 0.841 0.422 0.252 0.166
=2
Equation (67), M 0.873 0.452 0.266 0.173
=2
A . R. Holt and B. L. Moiseiwitsch
156
TABLE IV TOTALCROSS SECTIONS FOR H(1s) Wave Number
+ e -+ H(2p) + e
(I
Second Born First Born
ko
(in U& 1)
Equation (64). M = 2
1.5 2.0 3.0 4.0 5.0
Equation (67), M
1.427 1.022 0.634 0.437 0.321
1.281 1.041 0.663 0.453 0.329
=2
1.153 0.987 0.648 0.447 0.327
Units of mo2.
Other calculations on the scattering of electrons by hydrogen atoms using the second Born approximation have been carried out by Pomilla and Shapiro (1964), who investigated the 2 s - 3s excitation, and by Taylor and Burke (1964), who investigated the 1s + 2s excitation making allowance for the important effect of exchange which they found reduced the cross section for k 2 2a;’.
III. Convergence of Born Expansions We now turn our attention to the determination of sufficient conditions for the convergence of Born expansions. From Eq. (13) it follows that
and so
where
J
a=max, lU(r’)l dr’ 47c Ir-rI and max denotes the maximum value for all r. Hence r
157
BORN EXPANSIONS
since I $o(r) 1 = 1, and so the Born series converges if Ia < 1. Now a = max r
=
[;1SoU(r’)r’ dr’ + :S r-
V(rf)rrdr‘]
Som
D(r’)r’ dr‘
(73)
where
W ) =max I U ( r , &4)l, e9$
(74)
and so the Born series converges if the potential IU(r) satisfies the condition
I
JOm
D(r‘)r’ dr’
-= 1.
(75)
The integral on the left-hand side of this inequality exists if O(r) has at most an r singularity at the origin and falls off more rapidly than r-’ for large r. The error made in truncating the Born series can also be estimated with ease. We see ihat
and so from Eqs. (10) and (18) it follows that the truncation error in the Born expansion of the scattering amplitude satisfies the inequality
where
‘S
/?= - lU(r’)l dr‘ 471
For p to exist U(r) must fall off more rapidly than r - 3 for large r. The sufficient condition (75) for the convergence of the Born expansion for the total wave function $(r) was established by Manning (1965) using a generalization of the approach described above. This condition is somewhat less stringent than that derived previously by Zemach and Klein (1958) who showed that the Born series converges if Ia < 4. At the present stage it is convenient to investigate the convergence of the Born series for the individual partial waves obtained by expanding the total wave function $ in Legendre polynomials: $(r)
1 “
=-
r
C A , u,(r)P,(cos 6).
I=O
(79)
A . R.Holt and B. L. Moiseiwitsch
158
If the potential 1U is spherically symmetric it can be readily verified that the partial wave function u,(r) satisfies the equation
2 (dg + kZ - 1 U ( r ) - -u l ( r ) = 0 where the asymptotic behavior of ul(r)for large r is given by
u,(r)
-
kr{j,(kr) - tan
?I
N41,
(81)
j , and n, being spherical Bessel and Neumann functions, respectively, and
q I being the phase shift for the Ith order partial wave. The solution of the radial equation (80) having the asymptotic behavior (81) may be expressed in the form of the integral equation
ul(r) = krj,(kr)
+1
i:
Gl(r, r’)U(r‘)ul(r’)dr‘
(82)
where the Green’s function is given by the formula
W r , r‘) = k r , r>.h(kr<)ndkr,),
(83)
r c and r , being the lesser and greater of r and r’, and the phase shift q1is given by the integral equation tan qI = -A
:j
r71(kr‘)U(r’)u,(r’) dr‘.
(84)
The Born expansion of ul(r) may be written
where
Since for every I, (kr)1/21jl(kr)land (rr’)-’l2 lGl(r, r’)l are both bounded for all r and r‘, we see from Eq. (86) that max{(kr)-”2 Iul,,,(r)I}< uI max{(kr)-’/’
Iu,,”- l(r)I}
(88)
where QI =
max{(rr’)-’/2 lGl(r, r’)l} j y r ‘ IKJ(r’)l dr‘, r,r’
(89)
159
BORN EXPANSIONS
from which it follows that
where
rn, = rnax{(kr)’/’ ljf(kr)l}. I
Hence the Born series (85) converges if la,< 1, that is, if s m r ‘lU(r’)l dr‘ < 1 0
tl
where 1
- = max{(rr’>-’/’ t,
lG,(r, r’)l>.
(93)
r,r’
Now to = 1, and thus for the zero order partial wave the condition (92) for the convergence of the Born series becomes fm
which is the same as the condition (75) for the convergence of the Born series for the total wave function when the potential 1U is spherically symmetric. The sufficiency of the condition (94) for the convergence of the Born expansion for the zero order partial wave was first obtained by Jost and Pais (1951). The generalization to all values of I was obtained by Kohn (1954). The values of t , calculated by Kohn for 0 < I < 3 are given in Table V. For large values of I Kohn shows that t,
-
1.036(21 + 1 y 3 .
(95)
The truncation error in the Born expansion of tan can be determined in the same fashion as for the scattering amplitude. It can be readily seen that
TABLE V VALUES OF It
I:
0
1
2
3
t,:
1.000
2.344
3.339
4.198
160
A . R. Holt and B. L. Moiseiwitsch
and so the truncation error satisfies
The condition (92) for the convergence of the Born expansion for the fth order partial wave applies to all values of the wave number k . A less severe condition which is sufficient for convergence can be established for the zero energy scattering case k = 0. If we set
then Eq. (82) becomes ulo(r)= rl+'
+L
Glo(r, r')U(r')u:(r') dr'
(99)
where 0 1 rT1 GI (r, r') = - -21 + 1 r,'
The Born expansion of uIo may be written in the form
where
and Since
for all r and r', we see that 0 maxlr- ( I + ' ) u,,,(r)l
where
(1+1) 0 < CQ" maxlrul,,-l(r)I r
(105)
161
BORN EXPANSIONS
Hence
and so a sufficient condition for the convergence of the Born expansion for the Ith order partial wave when k = 0 is Actlo < 1, that is
1
lom.'
lU(r')l dr' < 21
+ 1.
If we take U ( r ) = -6(r - a ) we find that u?(a) = a'+' n=O
(q 21+ 1
(110)
+
which converges if l a < 21 1. Since the left hand side of the inequality (108) is l a for this particular integral, it follows that if we express the convergence condition in the form l!:r'
IU(r')l dr' < A
lom
and require it to be valid for all potentials l V ( r ) such that r'IU(r')l dr' is finite, the optimum value of A is 21 1. For the zero order partial wave corresponding to Z = 0, condition (108) for the convergence of the Born expansion when k = 0 is the same as the condition (94) for convergence at any energy. Bargmann (1952) has shown that if condition (108) is satisfied there does not exist a bound state of angular momentum h[Z(Z + 1)]'12 for a particle in the field of the potential l U ( r ) . Further, he has shown that if there exists a bound state of this angular momentum then
+
1
IU(r')l dr' > 21 + 1.
This leads naturally to the work of Davies (1960) who established a less restrictive condition than (94) for the convergence of the Born series for the total wave function $(r) and the scattering amplitude f. Davies has shown that a sufficient condition for the convergence of the Born expansion (12) for the scattering amplitude arising from a spherically symmetric potential 1U(r) is that the potential -A1 U(r)I is unable to support a bound state. Moreover, Huby (1963) has proved that this condition is also sufficient for the
162
A . R.Holt and B. L. Moiseiwitsch
convergence of the Born expansion of $ [ ( r )and tan yll for all values of 1. If the potential AU(r) has the same sign for all values of r the above suficient condition for the convergence of the Born expansion is also a necessary condition. Thus if - 11 U(r)I supports at least one bound state, then the Born expansion for $(r) does not converge for all values of r. However, if the sign of the potential AU(r) changes and if the potential is such that both AU(r) and -ALl(r) do not possess bound states but -A1 U(r)I possesses at least one bound state, we do not know whether the Born expansion for $(r) converges for all r or not. In this case the condition that -AlU(r)( cannot support a bound state is a sufficient but not a necessary condition for convergence. An instructive example of the use of the convergence condition established by Davies (1960) is provided by the scattering of electrons by the static potential of a hydrogen atom. Although
for this potential, it does not possess any bound states. Hence the Born series converges according to the theorem derived by Davies even though the sufficient condition (75) is not satisfied. Other interesting investigations of the convergence of Born expansions have been made by Kikuta (1954a,b), Meetz(1962), and Huby and Mines (1964).
N.Time-Dependent Collision Theory The total Hamiltonian operator H of a pair of interacting atomic systems may be expressed in the form
H=H,+V
(112)
where H , is the unperturbed Hamiltonian operator and V is the interaction potential between the two systems which vanishes in the limit when they are at an infinite separation. The collision is then described by the time-dependent Schrodinger equation
a
ih - Y(t)= ( H , at
+ V)Y(t)
where Y ( t ) is the state vector of the whole system. We now exclude the explicit time dependence arising from the unperturbed Hamiltonian H , by making the unitary transformation @(t) = exp( - iHo t/h)Y(t)
(114)
163
BORN EXPANSIONS
to the interaction representation characterized by the new state vector @(t).We then obtain
where H,(t)
= exp(iH,
t/h)V exp( - iH, t/h).
(116)
Let us now regard the time development of the state vector @ ( t )as an unfolding of the unitary transformation @(t) =
U(t)@(- co)
(117)
where U*(t)U(t) = 1. Then the final state vector @(co) may be written in terms of the initial state vector
(119)
where
s = U(co) is called the collision operator. Substituting the state vector formula (177) into Eq. (115) we see that the unitary operator U ( t ) satisfies
aU(0 = H,(t)U(t) ih at
where U( - 03) = 1. Hence we may express the solution of Eq. (121) in the integral equation form
is’ + (3’s’
U ( t )= 1 - h
(122)
H,(t’)U(t’) dt’
-m
which may be solved by an iterative procedure to yield
u(t)= 1 - 1
s’
-a
H,(t,) dt,
H,(t,) dt,
-m
f’
H1(i2)d t ,
-m
164
A . R.Holt and B. L. Moiseiwitsch
If the values of the operator function H,(t) commute with each other for all t, this expansion may be expressed in the exponential form : U ( t ) = exp[ - j:wHl(f') dt']
The first and second Born approximations to U ( t ) are given by
and
+
(3 Z
f
[' Hl(t2) dt, (126) h -m -m respectively. They may be derived either directly from (123) or by using variational principles (Lippmann and Schwinger, 1950). U(,)(t)= 1 -
H,(t,) d t ,
J-mHl(tl) d t ,
SCATTERING OF PROTONS BY HYDROGEN ATOMS As an example of a time dependent scattering problem we investigate the collision between a proton and a hydrogen atom using the impact parameter method, Because of the large mass of the incident proton we may suppose that it is moving with constant speed u parallel to the 2 axis of a rectangular Cartesian frame of reference with the nucleus of the target hydrogen atom located at the origin. We then expand the electronic wave function Y(r, t ) in terms of the wave functions q5,,(r) of the hydrogen atom associated with eigenenergies En according to the formula
r being the position vector of the atomic electron and t being the time. Substituting (127) into the time-dependent Schrodinger equation for the atomic electron i/j dy(r' ') = [Ho(r) at
+ V(r, R(t))]Y(r, t ) ,
where F12
ez
2m
r
Ho(r) = - - v2 + is the Hamiltonian operator of the hydrogen atom, R(t) is the position vector of the incident proton and
BORN EXPANSIONS
165
is the interaction potential between the incident proton and the target atom, we obtain
where
and
the zero of time being chosen so that Z = vt. If the initial conditions are taken to be a1(--a3) = 1
urn(- a)= 0
( m > l),
(134)
the cross section for the excitation of the nth state from the ground state m = 1 of the target atom is Q ( 1 + n> = 27t
Iom
lan(a)12P dp
(135)
where p is the impact parameter, that is, the perpendicular distance from the atomic nucleus to the straight line path of the incident proton. If we first consider the elastic scattering of the incident proton by the target hydrogen atom in the ground state m = 1 and neglect all couplings to higher states rn > 1 , Eq. (131) reduces to
a 1 i - u l ( Z ) = - V,,(Z)u,(Z). az
hV
This equation may be solved by inspection and gives
the relationship to the formula (124) being evident. Although we have been able to find an exact solution of Eq. (136) it is rather instructive to solve it by performing a Born expansion. Expressing Eq. (1 36) in the form of the integral equation a l(Z) =
1-
hv
J -m
V, l(Z’)u l(Z’) dZ’
A . R. Holt and B. L. Moiseiwitsch
166
and carrying out an iterative procedure we obtain 2 z aim = 1 - hu - m Vll(Z1) dZ1 + Vll(Z1) dZ, -m
‘s’
+
a
s
.
+
(ji)
($),r
V l l ( Z l ) d Z l r l V1,(Z2)dZ2
-m
-m
Vll(Z2) dZ2
-.‘r-’ s’l
-m
Vl l(Z,) dZ,
+-
* *
-m
(139) It can be readily verified by integrating by parts that
-m
=
[j
1 n!
-m
-m
Z
V,,(Z’) dZ’1
-m
from which it follows at once that the sum of the series on the right-hand side of Eq. (139) is just the exponential in formula (137). Let us now consider the excitation of the nth state of hydrogen. Rewriting Eq. (131) in the integral equation form
s
a,(Z) = 6,, - -
u,(Z’)V,,(R’)
exp( - ia,,,, Z’) dZ‘
(141)
and setting al(Z) = 1 and arn(Z)= 0 for m > 1 on the right-hand side we get a,(Z) = 6,, - -
Vn1(R’)exp( - ialnZ’) dZ‘,
which is the impact parameter form of the first Born approximation to the amplitude a,. To obtain the second Born approximation we substitute (142) into the righthand side of (141) which yields (Bates, 1958)
x
r1 -m
~ r n 1 ( ~ exp(-iu,mZ2) 2) dZ2
(143)
The calculation of the total cross section Q(1 + n) requires only a knowledge of u,(.o) as a function of the impact parameter p. If we neglect coupling to all
167
BORN EXPANSIONS
states m other than the initial state 1 and the final state n we find that m
a,(oo) =
--
remembering that Vll(Z) is an even function of 2. It is interesting to relate this second Born approximation formula to the distortion approximation which may be derived by substituting the solution (137) of the reduced elastic scattering equation (136) into the right-hand side of (131) and setting a,,, = 0 for all m # 1, n . We obtain
a az
"
i - a,(Z) = - Vnn(R)an(Z)
hu
-icclnZ -
Lj hu
Z
Vll(Z') dZ']], -m
whose solution is
(n # 1). Thus to the distortion approximation we see that
(145)
168
A . R. Holt and B. L. Moiseiwitsch
which, on expanding to the second order in the interaction energy V, yields the second Born approximation formula (144). We now see that the second term within the square bracket on the right-hand side of (144) arises from expanding the phase factor
in the distortion approximation. In the case of the 1s -+ 2s excitation of atomic hydrogen, Vll, V,, , and Vln are all even functions of 2 and then to the second Born approximation a,(co) =
-5 h V [[;vn,(zl)
cos(a,,Z,) d Z ,
The first two terms are pure imaginary while the last term is pure real and thus the term arising from the expansion of the phase factor in (147), to which this real term corresponds, leads only to a contribution to the total cross section which is of the fourth order in the interaction potential and therefore does not contribute to the total cross section when calculated using the third order approximation. Total cross sections for the 1s + 2s excitation of atomic hydrogen by proton impact have been calculated using a wave treatment by Kingston et al. (1960a) employing the third order approximation allowing for: (i) coupling to the 1s and 2s states and (ii) coupling to the Is, 2s, and 2p states. These are presented in Fig. 1 where they are compared with the first Born approximation and the distortion approximation calculations of Bates (1959) using an impact parameter treatment. This comparison is legitimate since it is well known that the wave and impact parameter treatments are equivalent at high energies (Bates and Holt, 1966; Moiseiwitsch, 1966). Also displayed in Fig. 1 is the 1s- 2s excitation cross section calculated by Holt and Moiseiwitsch (1968) using the third order approximation with the coupling terms m = 1, 2 in formula (143) being evaluated exactly and the coupling terms m 2 3 being evaluated by setting Em = E3 and using the closure formula (65).
I69
BORN EXPANSIONS
15
I
I
2.0
2.5
log,,(protcm energy in kev)
FIG. 1. Cross sections for ls+2s excitation of atomic hydrogen by proton impact: curve A, first Born approximation; curve B, distortion approximation (Bates, 1959); curves C and D, third-order approximations, (i) and (ii), respectively (Kingston ef a/., 1960a); curve E, third-order approximation using Eq. (67) with M = 2 (Holt and Moiseiwitsch, 1968).
V. Rearrangement Collisions We conclude this article by directing our attention towards collision processes in which a particle 1 is incident initially upon particles 2 and 3 in a bound state, and particle 2 emerges finally as a free particle leaving particles 1 and 3 in a bound state. The convergence of the Born expansion for such rearrangement collisions is in doubt. Aaron et al. (1961) have attempted to establish that the Born expansion of the transition amplitude for rearrangement processes diverges for all energies of the incident particle in the case of an attraction potential whose Fourier transform is always negative. However, Dettmann and Liebfried ( 1966) have produced a one-dimensional counter example of a rearrangement collision which does not lead to divergence of the Born expansion for all energies of the incident particle and yet for which the arguments of Aaron et al. (1961) should apply. Let us now consider a particular example of a rearrangement collision, namely the capture of an electron by a proton from a hydrogen atom: H+
+ H(ls)+ H(ls) + H+.
( 149)
170
A . R. Holt and B. L. Moiseiwitsch
Investigations of this process using the first Born approximation have been carried out by several authors. The earliest analysis along these lines was performed by Brinkman and Kramers (1930) who supposed that the contribution to the scattering amplitude arising from the internuclear potential could be neglected. Denoting their capture cross section by QBK it is found that at high impact energies QBK
N
26 1 - nao2, 5 E6
where the energy E of the incident proton is in units of 100 keV. Calculations retaining the internuclear potential have been carried out by Jackson and Schiff (1953). Neglecting a very small range of angles about the backward scattering direction 8 = 7c they found that their capture cross section has the asymptotic behavior at high energies Qa 0 . 6 6 1 Q ~ ~ . N
(151)
Mapleton (1964) has shown that at extremely high incident energies the backward scattering contribution to the total cross section arising from the internuclear interaction dominates the capture cross section which then has the form
where M / m = 1837 is the ratio of the proton and electron masses. Since (1 52) is unimportant below 40 MeV it may be disregarded for most practical purposes and the Jackson and Schiff cross section having the E - 6 asymptotic form (1 5 1) may be taken as the first Born approximation cross section. We now come to the application of higher Born approximations to the evaluation of the cross section for the charge transfer process (149). So far no exact evaluation of higher order corrections to the transition amplitude has been performed. However Drisko (1955) has determined the high energy behavior of the second order correction to the transition amplitude by using a peaking approximation to evaluate the relevant integrals. The resulting high energy cross section was found to be
a result which has been verified by Mapleton (1966) using a different Green’s function. We see that the correction term obtained by Drisko (1955) decays as E - 1 ’ / 2 at high impact energies which is a less rapid fall off than the E - 6 decay of the first Born approximation cross section derived by Jackson and Schiff (1953). Because of this surprising feature Drisko (1955) also
BORN EXPANSIONS
171
estimated the high energy behavior of the third order correction to the transition amplitude but he found that the energy decay of the capture cross section was unaltered although the constant 0.295 within the brackets on the right-hand side of (153) was changed to 0.319. Thus the Born expansion does not seem to converge to its leading term. Evidently the situation regarding the high energy behavior of the cross section for the charge transfer process (149) requires further elucidation. A more detailed discussion of atomic rearrangement collisions can be found in the review article by Bransden (1965).
ACKNOWLEDGMENT The work has been sponsored in part by the U.S. Office of Naval Research for the Advanced Research Projects Agency Department of Defense under Contract N62559-4279.
REFERENCES Aaron, R., Amado, R. D., and Lee, B. W. (1961). Phys. Rev. 121, 319. Bargmann, V. (1952). Proc. Natl. Acad. Sci. U. S. 38, 961. Bates, D. R. (1958). Proc. Roy. SOC. A245, 299. Bates, D. R. (1959). Proc. Phys. SOC.(London) 73, 227. Bates, D. R., and Holt, A. R. (1966). Proc. Roy. SOC. A292, 168. Bransden, B. H. (1965). Aduan. At. Mol. Phys. 1, 85. Brinkman, H. C., and Kramers, H. A. (1930). Koninkl. Ned. Akad. Wetenschap. Proc. Ser. 33, 973. Dalitz, R. H. (1951). Proc. Roy. SOC. A206, 509. Davies, H. (1960). Nucl. Phys. 14,465. Dettmann, K., and Liebfried, G. (1966). Phys. Rev. 148, 1271. Drisko, R. M. (1955). Ph.D. Thesis, Carnegie Inst. of Technol., Pittsburgh. Holt, A. R., and Moiseiwitsch, B. L. (1968). Proc. Phys. SOC.(London) B1, 36. Huby, R. (1963). Nucl. Phys. 45, 473. Huby, R., and Mines, J. R., (1964). Nucl. Phys. 54,28. Jackson, J. D., and Schiff, H. (1953). Phys. Rev. 89, 359. Jost, R., and Pais, A. (1951). Phys. Rev. 82, 840. Kacser, C. (1959). Nuovo Cimenro 13, 303. Kikuta, T. (1954a). Progr. Theoret. Phys. 12, 225. Kikuta, T. (1954b). Progr. Theoret. Phys. 12, 234. Kingston, A. E., Moiseiwitsch, B. L., and Skinner, B. G. (1960a). Proc. Roy. SOC.A258, 237. Kingston, A. E., Moiseiwitsch, B. L., and Skinner, B. G. (1960b) Proc. Roy. SOC.A258, 245. Kingston, A. E., and Skinner, B. G. (1961). Proc. Phys. SOC. (London) 77, 724. Kohn, W. (1948). Phys. Rev. 14, 1763.
172
A . R.Holt and B. L.Moiseiwitsch
Kohn, W. (1954).Rev. Mod. Phys. 26, 292. Lippmann, B. A., and Schwinger, J. (1950).Phys. Rev. 79,469. Manning, I. (1965).Phys. Rev. 139,B495. Mapleton, R. A. (1964).Proc. Phys. SOC.(London)83,895. Mapleton, R. A. (1966). Ph.D. Thesis, Queen’s Univ. of Belfast, Belfast. Massey, H. S. W., and Mohr, C. B. 0. (1934). Proc. Roy. SOC.A146, 880. Meetz, K. (1962).J. Mufh.Phys. 3,690. Moiseiwitsch, B. L.(1966).Proc. Phys. SOC.(London)87,885. Moiseiwitsch, B. L.,and Perrin, R. (1965).Proc. Phys. SOC.(London) 85, 51. Pomilla, F. R., and Shapiro, J. (1964). In “Atomic Collision Processes” (M. R. C. McDowell, ed.), p. 330.North-Holland Publ., Amsterdam. Rothenstein, W. (1954).Proc. Phys. SOC.(London)A67, 673. Taylor, A. J., and Burke, P. G. (1964). In “Atomic Collision Processes” (M. R. C. McDowell, ed.), p. 324. North-Holland Publ., Amsterdam. Zemach, G., and Klein, A. (1958). Nuovo Cimenro 10, 1078.
RESONANCES IN ELECTRON SCATTERING BY ATOMS AND MOLECULES P . G . BURKE Theoretical Physics Division, AERE Harwell Didcot, Berkshire, England*
I. Introduction ......................................................
....................................... A. Basic Experiments ............................................ B. Resonances in e--H Scattering ................................. C. Resonances in e- he scattering ................................. D. Resonance States of He ....................................... E. Resonances in Other Systems ................................... Resonance Scattering Theory ..................................... A. Introduction .................................................. B. S-Matrix Theory ..............................................
11. Experimental Observations
111.
C. Multichannel Resonance Theory ............................... D. Close Coupling Theory ....................................... E. Molecular Resonance Theory ................................... F. Effective Range Theory ....................................... IV. Further Results and Conclusions. .................................. References .....................................................
173 .175 175 .176 .177 .178 .181 .186 186 186 .193 .196 .201 .206 .208 . 214
I. Introduction In this review we will be mainly concerned with resonances produced in electron scattering by atoms and molecules. The foundations of this subject were laid by H. S. W. Massey and co-workers in the 1930's and we will see that methods developed then are among the most powerful in use today. We will also discuss resonance states produced in photon absorption and heavy particle collisions and their relationship to resonances in electron scattering. Resonance states of atoms and molecules have been studied for many years. The Auger effect (Auger, 1925) is one of the earliest examples of atomic states found lying above the first ionization threshold. In molecular spectroscopy, the phenomenon of predissociation, reviewed by Herzberg (1950), is another example arising from the overlapping of discrete and continuous states. In
* Present address: School of Physics and Applied Mathematics, Queen's University of Belfast, Belfast, Northern Ireland. 173
174
P. G . Burke
inelastic electron scattering, early observations of doubly excited, or resonance states of He were made by Whiddington and Priestley (1934) and Priestley and Whiddington (1935). Also, observations in the vacuum ultraviolet made by Compton and Boyce (1928) and Kruger (1930) showed emission lines in He which they attributed to transitions between continuum states and normal bound states. Finally, in atomic absorption spectroscopy, Shenstone and Russell (1932) observed autoionizing, or resonance states, of Ca, while Beutler (1935) saw evidence for autoionizing states in Ar, Kr, and Xe. Early theoretical work on the analysis of the interaction between discrete and continuum states was carried out by Rice (1933), while Fano (1935) was able to interpret Beutler’s observations in the rare gases as due to Rydberg series of levels convergent on the 2P1/2 first excited state of the ion and which autoionize into the ion in its 2P3/2ground state plus a free electron. Massey and Mohr (1935) were the first to carry out detailed calculations of the excitation of resonance states of He by inelastic electron scattering, and thus explained the results of Whiddington and Priestley. Early attempts to estimate the position and life times of the resonance states of He were made by Fender and Vinti (1934), Wu (1934), Wilson (1935), and Ma and Wu (1936). We will be concerned with two distinct types of resonances, which we will call “ closed channel resonances ” and “ shape resonances.” Closed channel resonances arise when the interaction potential between the incident particle and an excited state of the target is strong enough to support a bound state. If this state lies energetically above the ground state of the target with the incident particle at infinity, and is coupled to this continuum state by terms in the Hamiltonian, it will decay with the emission of the incident particle. The presence of this intermediate “ bound ” state will cause a resonance in the scattering cross section. The excitation of the target can take several forms. In an atomic system it can be an electronic excitation, or it can involve a change in the quantum state of the target without electronic excitation as, for example, occurred in Beutler’s observations referred to above. For a molecular target, there is also the possibility of rotational and vibrational excitation. Shape resonances, on the other hand, can occur even if only one channel is involved in the scattering. The interaction potential between the incident particle and the target, which may be in an excited state, must now be such that an intermediate state can be formed, without changing the quantum state of the target, where the lifetime is appreciably greater than the collision time. Potentials with this property usually have repulsive barrier tails, which are often caused by the centrifugal force. Because of the important role played by the potential in producing these resonances, they are often referred to as “ potential resonances.” We can now summarize the essential distinction between the two types of resonances. Closed channel resonances lie energetically below the channel or
RESONANCES IN ELECTRON SCATTERING
175
channels to which they are most strongly coupled, while shape resonances lie energetically above. For this reason, closed channel resonances are typically much narrower than shape resonances, because their most important decay modes are energetically forbidden. In atomic systems, most interest has centred around closed channel resonances, although there are several important examples where shape resonances have been found. In molecular systems, however, shape resonances are very important in describing processes such as associative detachment and low energy electron molecule scattering. The rest of this review is arranged as follows. In Section I1 we summarize recent experimental work. In Section 111 we discuss some theoretical interpretations of, and calculation procedures for, resonance reactions. Finally, in Section 1V we describe some recent theoretical calculations in greater detail.
II. Experimental Observations A. BASICEXPERIMENTS Resonances can be observed in basically two types of experiment. Resonance states of the target atom or molecule, denoted in the following by B, may be excited in inelastic collisions according to A
+ B+ A + B*
--f
A
+ (C+ D),
(1)
where C and D are the decay products of B. Depending on the experiment, beams of electrons, protons, and atomic and molecular ions have been used for A . Alternatively, an intermediate compound state M may be formed in the collision A+B+M+C+D,
(2)
where C and D are now the decay products of M . If C = A and D = B, then Eq. (2) corresponds to elastic scattering. In the class of experiments denoted by Eq. (I), the cross section for the reaction is determined basically by the production amplitude for the compound state B*, and not by the decay width of B*. It follows that the probability of observing the resonance state does not depend critically on the energy resolution of the incident beam. In experiments denoted by Eq. (2), a very narrow resonance may easily be missed if the energy resolution of the incident beam is poor. It was for this reason that, apart from work done on photon absorption, most early observations of resonances in atomic scattering corresponded to Eq. (1).
176
P . G . Burke
Recently, with the advent of electrostatic analyzers and monochromators, the picture has changed markedly. It is now possible to produce low energy electron beams with energy resolution in the millivolt range (Simpson, 1964a; Boersch et al., 1964; Lassettre, 1965). As a result, in the last few years many narrow resonances have been identified in electron atom and electron molecule collisions corresponding to the formation of intermediate compound negative ion states. A further important technical advance is the availability of continuum photon sources in the vacuum ultraviolet, using synchrotron light from electron accelerators (Madden and Codling, 1965). This has considerably extended the energy range available for experimental investigation, and allowed states of double electron excitation to be analyzed in the lightest atomic systems. B.
RESONANCES IN
e--H SCATTERING
The first indication that there might be a resonance in the scattering of electrons by atomic hydrogen came from close coupling calculations (Burke and Schey, 1962). It was found that at about 0.6 eV below the n = 2 excitation threshold, the scattering phase shift in the ' S state increased rapidly through R radians. Massey (1962) explained this phenomenon as a closed channel resonance, and recent work has shown that its energy is 9.56 eV and that its width is 0.0475 eV (Burke and Taylor, 1966). The mechanism that produces the resonance has been shown by Gailitis and Damburg (1963a,b) to be the degenerate dipole coupling between the 2s and 2p states of the hydrogen atom. At large distances this gives an r - 2 potential which is sufficiently attractive to support an infinite number of bound states for the total angular momentum equal to 0, 1, and 2 in the absence of coupling to the 1s state of hydrogen. We will discuss this further in Section II1,F. The resonance was first observed by Schulz (1964a), who passed an electron beam through atomic hydrogen and observed an anomaly in the transmitted electron current as the energy varied close to and just below the n = 2 threshold. The anomaly corresponded to a change in the number of electrons scattered out of the incident beam near the resonance. Schulz's result has now been confirmed with better electron energy resolution by Kleinpoppen and Raible (1965) and by McGowan et al. (1965). In the experiment of McGowan et al. (1965), there were indications of further resonances lying closer to the n = 2 threshold. Recent results, discussed in Section IV, have shown that the most important of these is a 3P resonance at 9.74 eV and a ' D resonance at 10.125 eV. It is easy to show that the degenerate dipole interaction will produce closed channel resonances below all the higher thresholds in e--H scattering. These
RESONANCES IN ELECTRON SCATTERING
177
resonances will effect inelastic scattering processes such as the 1s-2s and 1s-2p excitation cross sections. So far there has been no deficitive experiment reporting these resonances, although recent close coupling calculations predict that their effect will be important (Burke et al., 1966b, 1967a,b,d; see also Section IV). C. RESONANCES IN e--He SCATTERING Massey and Moiseiwitsch (1954) first calculated a 2S resonance just above the 23S threshold, which accounted qualitatively for the l'S-23S observations of Maier-Leibnitz (1935). More detailed experiments by Schulz and Fox (1957), which were analyzed by Baranger and Gerjuoy (1957, 1958), indicated that the 2 3 S excitation cross section close to threshold was purely resonant. Below the 2 3S threshold, Schulz (1963) observed a narrow resonance effect at 19.31 eV in the elastic scattering of electrons in helium at 72". The resonance has been confirmed by many other groups (e.g. Fleming and Higginson, 1963; Simpson, 1964b; Kuyatt et al., 1965; Golden and Bandel, 1965a; Ehrhardt and Meister, 1965; Andrick and Ehrhardt, 1966). Ehrhardt and collaborators carried out angular distribution measurements which showed that the resonance occurs in the ' S state, which confirms the conclusion of Simpson and Fano (1963) based on the interference of the resonance with the background. We assign it to the mixed configuration ( 1 ~ 2 3 ~and ) ~ (ls2p')'S. s The resonance is certainly narrower than the corresponding H - resonance, since the coupling with the open channel is weaker. Recent work by Ehrhardt and collaborators has shown that the width is -0.015 eV. Recent measurements above the 2 3 S threshold have shown several resonances which are strongly coupled to the n = 2 levels of He (Schulz and Philbrick, 1964; Chamberlain and Heideman, 1965; Holt and Krotkov, 1966) and several which are strongly coupled to the n = 3 levels of He (Chamberlain, 1965, 1967; Chamberlain and Heideman, 1965). We show Chamberlain and Heideman's result for the 2 3S, 2 ' S , 2 3P, and 2 'P excitation cross sections in the forward direction in Fig. 1. The three peaks at 19.95 eV, 20.45 eV [which is the peak discussed by Baranger and Gerjuoy (1957)], and 21.0 eV in the 2 3 S excitation have recently been shown, by angular distribution measurements, to consist of nearly pure S-wave, pure P-wave, and pure D-wave scattering, respectively (Ehrhardt and Willmann, 1967a). The P-wave and D-wave peaks have been shown to be resonances (Burke et al., 1966a), while the S-wave peak may be the tail of the 19.31-eV resonance below the 2 3 S threshold. Further resonances can be seen, in Fig. 1, below the n = 3 threshold. They give windows in the spin allowed transitions and peaks in the spin changing transitions from the ground state.
P. C. Burke
178 I
I
-
I
I
I
I
I
I
7
3 I16'5
-
I
I
H
1
I_n_, 23~1
I
I 20
I
I 21
I
I
I
I
22 23 Incident energy --eV
I
I 24
FIG. 1. Electron induced excitation of the 23S, 2'S, 23P, and 2'P states of helium measured in the forward direction by Chamberlain and Heideman (1965).
A further resonance at an incident electron energy of 0.45 eV has been reported by Schulz (1965). Schulz's results were similar to early experiments by Ramsauer and Kollath (1929) and Normand (1930). However, Golden and Nakano (1966) have recently failed to find this resonance and have ascribed the earlier observations to an instrumental effect. It would indeed be difficult to explain a narrow resonance at this energy unless the angular momentum barrier were large. However, the interference with the background of this " resonance " seems to require a (ls22s)'S configuration. Recently, Herzenberg and Lau (1 967) have suggested that a narrow S-wave resonance could result from the fact that the He ground state is not a pure closed shell. Finally, we note that states which autoionize only through spin-spin and other fine structure interactions, e.g. the 4P,!,states of Li and He- (Pietenpol, 1961), will not be considered in this review.
D. RESONANCE STATES OF He We give the energy level diagram of He in Fig. 2. Below the ionization threshold at 24.58 eV, we have bound states which decay by photon emission
179
RESONANCES IN ELECTRON SCATTERING
u u u t u u u ~e+(25,2p)+e-
oL
-I 9
Autoiai-iuing states
gmund states
FIG.2. Bound states and autoionizing states of helium.
with lifetimes 5 lo-* sec. Below the higher thresholds we have autoionizing states, bound by the Coulomb potential to excited states of He+, which decay with the emission of an electron into the adjacent continuum with lifetimes S6.1O-l5 sec (see Section IV). Madden and Codling (1963, 1965), using the electron synchrotron at the National Bureau of Standards, Washington, D.C. as a continuum source of radiation, have been able to observe anomalous photon absorption in He gas corresponding to excitation of the 'P doubly excited states of He converging on the n = 2, 3, and 4 states of He+. We reproduce in Fig. 3 their absorption coefficient in the 175-245 A range, where one Rydberg series of resonances is seen converging onto the n = 2 state of He+. The strong mixing between the 2s and 2p states of He+ implies that a correct zero approximation for the configuration of the resonances is (2snp + 2pns)'P and (2snp - 2pns)'P, however only the former " + " series has appreciable autoionization probability and excitation probability from the ground state (Cooper et al., 1963). The " -" series have now been seen interspaced between the " series shown in Fig. 3 in higher sensitivity results obtained by Madden and Codling, but the (2pnd)'P series predicted by Burke et al. (1964a) have not been
+"
180
P. G . Burke Electron volts
Wavelength (angstroms)"
FIG.3. Photon absorption cross section in helium near the second ionization continuum measured by Madden and Codling (1965).
observed yet, owing both to their narrow width and to the small oscillator strength of the transition from the ground state of He. Only 'P states of He are excited in photon absorption experiments. In order to see other states it is necessary to use an alternative method of excitation. Atomic and molecular ion beams have been used by Berry (1961, 1962), Rudd (1964, 1965),and Barker and Berry (1966). Using 75-keV beams of H + and H 2 + , Rudd observed electrons emitted at 160" to the incident ion direction arising from the decay of doubly excited states of He. In this way he has been able to identify the states: (2sns)'S, n =2, 3, 4, 5 ; ( 2 ~ n p ) ~ P , n = 2, 3, 4, 5 ; (2pnp)'D, n = 2, 3, as well as many of the optically allowed states already seen by Madden and Codling. Rudd has also recently extended his measurements to Ne and Ar (Rudd and Lang, 1965; Rudd el al., 1966; Edwards and Rudd, 1967). Electron beams have been used by Silverman and Lassettre (1964) and by Simpson et al. (1964,1965), who have extended the early work of Whiddington and Priestley mentioned in Section I. With low energy, 60 to 400-eV electron beams, Simpson et al. have obtained information on several optically forbidden autoionizing states of He. Also, Kuyatt et al. (1965) and Simpson et al. (1966) have observed resonances at 57.1 and 58.2 eV incident electron energy, which have been interpreted by Fano and Cooper (1965b) as corresponding to the configurations ( 2 ~ ~ 2 p and ) ~ P( 2 ~ 2 p ~ )of~ HeD
RESONANCES IN ELECTRON SCATTERING
E.
RESONANCES IN
181
OTHERSYSTEMS
I . Photon Absorption
Experimental work on absorption lines in the ionization continuum of atomic and molecular spectra has been recently reviewed by Garton (1966) and by Samson (1966). We will therefore confine our discussion here to a few general remarks. It has been shown by Fano (1961) and by Fano and Cooper (1965a) that the absorption line profiles can all be represented by
where E = ( E - E,)/$T, E, is the resonance energy, r is the resonance width, q is called the line profile index and defines the line shape, and oo and ob are slowly varying with energy. A correlation coefficient p can be defined by pz = o,/(o, + a,), which is essentially the overlap between the continuum states produced by nonresonant photon absorption and the continuum state produced by decay of the resonance state. We will see in Section TIT, C that the resonance form given by Eq. (3) is not limited to photon absorption cross sections, but describes the cross sections in the neighborhood of any isolated resonance state. As an example of the application of Eq. (3), we see that :he ' P absorption lines in He shown in Fig. 3 can all be fitted with bb = 0 and q negative. The classification of autoionizing lines in a complex atom or molecule will obviously be an immense task. The complexity of the problem is well illustrated by the work of Madden and Codling (1963-1966), Codling et al. (1967), Codling and Madden (1964), and Samson (1963) in the rare gases. The ground state configuration is (np)6(nd)'o(n 1 s)2(n 1 p)6 'Sin general. Structure in the continuum has been seen corresponding to the following proposed electron excitations: (a) one s shell electron excited; (b) one s and one outer p electron excited; (c) two outer p electrons excited; (d) one d electron excited; (e) one inner p electron excited; (f) one d electron and one outer p electron excited. Recently Codling et al. (1967) have analyzed in detail the autoionizing ' P states in Ne which illustrates some of these features.
+
+
2. Electron Scattering by Atomic Systems
Resonances have been observed in the scattering of electrons by many other atomic systems. In e--Ne scattering, Schulz (1964b) and Simpson (1964b) observed a resonance occurring about 0.5 eV below the first excited state of Ne. Kuyatt et al. (1965) showed that this resonance was a doublet
182
P . G . Burke
with the probable configurations ls22s22p53s22P3,2,1,2,and gave the positions as 16.04 and 16.135 eV. The levels of Ne associated with the 2p53s configuration lie between 16.6 and 16.8 eV, while those associated with the 2p53p configuration lie between 18.4 and 19.0 eV. It is probable that the dipole coupling between these two excited states of Ne provides some of the necessary attraction for the resonances. This resonance doublet has been shown by Franzen and Gupta (1965) to give rise to appreciable electron spin polarization. Kuyatt et al. observed further resonances at higher energies in Ne, and also resonances in electron scattering by Ar, Kr, and Xe. Schulz (1964b) also reported resonances in e--Kr and e--Xe scattering. All the above resonances are of the closed channel type. For electrons scattered from Ar, Kr, and Xe just above the Ramsauer minimum, on the other hand, the D-wave phase shift rises rapidly (Massey and Burhop, 1952) and we obtain a shape resonance. This rapid rise was confirmed by a phase shift analysis of the differential cross section data in Ar by Westin (1946).
3. Electron Scattering by Molecular Systems Massey (1950) and Craggs and Massey (1959) have empahsized the importance of negative ion states in the collision of electrons with molecules. We will see that temporary negative molecular ion states or electron molecule resonances are important in many situations. Closed channel resonances have been observed in the scattering of electrons by H,, HD, and D, by Kuyatt et al. (1964, 1966), and in H, and D, by Golden and Bandel (1965b), and in H, by Menendez and Holt (1966) and Heideman et al. (1966b). The compound H,- states exhibit a vibrational structure which can be interpreted as an electron moving in the field of H, in the (lso2prc) 'IT", 'IT,, excited states (Taylor and Williams, 1965), with a binding energy of some 0.9 eV. The first of these H,- states occurs at an incident electron energy of 11.28 eV (Kuyatt et al., 1966), and is probably a 2Zg+state. A sharp isolated window type resonance (q small in Eq. (3)) has also been observed in electron transmission measurements in N, at 11.48 eV by Heideman et al. (1966a). This lies about 0.4 eV below the E 'C,' excited state of N, . Olmsted et al. (1965) have also observed resonances in the rare gases and in diatomic molecules. Resonances have been observed in the low energy scattering of electrons by N, between 1.8 and 3.5 eV by Schulz (1962, 1964c), by Heideman et al. (1966a), by Golden and Nakano (1 966), and by Andrick and Ehrhardt (1966). Ehrhardt and Willmann (1967b) also carried out angular distribution measurements which indicate that the resonance may be D-wave. These resonances are found to strongly enhance the vibrational excitation cross section, and
RESONANCES IN ELECTRON SCATTERING
183
we summarize the experimental situation in Fig. 4. By considering the isoelec, and Nz-, and using Schulz's experimental work tronic sequence 0 2 +NO, as a guide, Gilmore (1965) predicted that the X 'Ilg potential energy curve of Nz-lay some 2 eV above the 'C,+ ground state curve of N,, as shown in the figure. To understand these peaks in terms of closed channel resonances (Chen, 1964a,b) requires a binding energy of several eV to the nearest excited level of Nz, which seems rather unlikely (Chen, 1966). On the other hand, the resonances appear to have a width not much different from the vibrational spacing of ~ 0 . eV, 3 and thus, if they are shape resonances (Herzenberg and Mandl, 1962) moving in the potential of N, in the ground state, some mechanism must be present to account for the narrowness of the width. It is possible that if the resonance does correspond to a D-wave then the centrifugal barrier may be sufficient to cause this narrow width. Alternatively the peaks may be produced by interference effects between adjacent shape resonances, whose natural width is larger than the spacing. This would fit in with the calculation carried out by Fisk (1936), who found a broad peak at 2 eV in the total e-N, scattering cross section. e-+ N, ( V = 0 )
- N;(
e-+ N, [ V")
V')
excitation
Fmrn G.J. Schulz P.R. 135 A988 (1964)
kvu:1; k72' l 2.0
3.0
eV-
FIG. 4. Resonances in the electron scattering by N 2 . The experimental vibrational excitation cross section measured by Schulz (1964~)is shown together with potential energy curves for the ground state of N2 and for the proposed N z - resonance state (Gilmore, 1965).
P . G . Burke
184
Reconance effects have also been observed in the vibrational excitation cross sections of H, and CO (Schulz, 1964~). Resonance states are also important in the ionization of molecules close to threshold both in photon absorption (e.g. Beutler et al., 1936; Cooke and Metzger, 1964; Dibeler et al., 1965; Doolittle and Schoen, 1965) and in electron impact (e.g. Briglia and Rapp, 1965; McGowan and Fineman, 1965; McGowan et al., 1967). In H, , the ionization via an intermediate resonance state of H, proceeds as follows: e-
+ Hz('&+, v = 0,K )-+e- + H2*(n, v',
K'),
(4)
where H,* is in an excited Rydberg state n with vibrational excitation v' and rotational excitation K'.This state then autoionizes as follows : H2*(n, v', K')-Hz+(2&,+, v", K")+ e - .
(5)
Berry (1966) estimates that strong coupling exists between electronic and vibrational motion for n of about 5 to 8, and between electronic and rotational motion for n of about 15 to 20 or more. It is probably the former coupling that causes the structure in H, ionization. In the examples of resonance states that we have considered up to now, where the autoionization has been caused by the electron-electron interaction, the lifetimes are typically 5 1 0 - l 5 sec. Reaction (5), on the other hand, proceeds through a breakdown of the Born-Oppenheimer approximation, and the autoionization life times may be as long as sec (McGowan et al., 1967). Finally, we consider processes involving the collision of an electron with a molecule, giving a negative ion through the reaction e-
+AB+A- +B
(6)
known as dissociative attachment. Experimental work by Khvostenko and Dukel'skii (1957), Schulz (1959), and Rapp et al. (1965) revealed peaks in the cross section for dissociative attachment in H,. In Fig. 5 we give data obtained by Rapp et al., which also show structure for H D and D , . The peaks are due to the formation of intermediate compound negative molecular ion states. The lower peak in Fig. 5 corresponds to the 'Zg+ repulsive state of H,-, which dissociates into H and H - in their ground states (Bardsley et al., 1966a,b). The attractive ,Xu+ state of H2- calculated by Dalgarno and McDowell (1956), and more recently by Bardsley et al. (1964, 1966a,b) gives rise mostly to enhanced vibrational excitation of H, (Schulz, 1964c), although there is evidence of a low energy dissociative attachment peak (Schulz and Asundi, 1965). The isotope effect apparent in the results shown in Fig. 5 arises since the time of electron capture into the resonant state remains essentially
RESONANCES IN ELECTRON SCATTERING
185
Electron energy (eV)
FIG.5. Dissociative attachment in H2 HD, and Dz. measured by Rapp er al. (1965).
unchanged in the isotopically substituted molecule, but the time of molecular dissociation is drastically altered (Demkov, 1965). For further discussion of some of these points see Section III,E. 4 . Ion-Atom and Ion-Molecule Scattering
We have space here to no more than mention the role of intermediate compound molecular states in heavy particle collisions. Recent experiments (e.g. Barker and Berry, 1966; Rudd et al., 1966; Everhart and Kessel, 1965; Kessel et al., 1965; Lipeles et al., 1965; Afrosimov et al., 1965) have shown that autoionizing states are often produced in ion-atom collisions. In many cases the velocity of the ions is so low that the Massey adiabatic criterion (Massey, 1949) would predict very small ionization cross sections. The observed ionization during the collision has been explained (e.g. Bates and Massey, 1954; Fano and Lichten, 1965) by the formation of an intermediate compound molecular state. The corresponding potential energy curve is assumed to exhibit a pseudocrossing with a potential curve which, in the separated atom limit, leaves one or both atoms in an autoionizing state. The Landau-Zener theory (Bates, 1962) would then predict large ionization cross sections at certain energies and impact parameters.
P. G. Burke
186
III. Resonance Scattering Theory A. INTRODUCTION We now turn to a discussion of the theory of resonance reactions, with particular emphasis on those methods that have found application in atomic and molecular resonance scattering. Perhaps the most fundamental approach is through the analytic properties of the S-matrix first introduced by Wheeler (1937). It has been shown (e.g. Heisenberg, 1943; M~ller,1945) that a knowledge of the S-matrix is sufficient to predict all observable quantities if its analytic and unitary properties are known and the completeness condition is assumed. We will see that resonances in scattering processes are closely associated with poles in the S-matrix in the complex energy plane, and, from general arguments, we are able to discuss the distribution of these poles, and consequently the resonances that can occur. In order to develop the theory of resonance reactions further, we adopt an approach similar to that used by Brenig and Haag (1959) and, in a different context, by Fano (1961). An approximate Hamiltonian is introduced which can be solved exactly in terms of discrete states and continuum states. The neglected part of the interaction causes mixing between the discrete states and the continuum states, giving rise to resonances. We then discuss two models that have proved successful in predicting atomic and molecular resonance reactions. For electron atom scattering, we discuss the close coupling method introduced by Massey and Mohr (1932) and which has been shown by Feshbach (1958, 1962, 1964) to give rise naturally to resonances. Feshbach’s approach, using projection operator methods, also suggests further approximations that have proved useful in understanding resonance scattering. For molecular processes involving the interaction of slow nuclear motion with relatively rapid electronic motion, we discuss recent theories developed by Bardsley et al. (1964, 1966a,b) and by O’Malley (1966). Finally, we consider in detail the analytic properties of the S-matrix in the neighborhood of thresholds for new reactions. We will see that it is possible to extrapolate the S-matrix through thresholds to predict resonances using recent developments in multichannel effective range theory (Bely et al., 1964; Seaton, 1966a,b; Ross and Shaw, 1961 ; Gailitis and Damburg, 1963a,b; Gailitis, 1963).
B. S-MATRIX THEORY 1. Single Channel S-Matrix We consider first the single channel Schrodinger equation. This problem has been studied extensively by many people, including Hu (1948), Humblet (1952), Humblet and Rosenfeld (1961) and Martin (1964).
RESONANCES IN ELECTRON SCATTERING
187
The radial equation for the fth partial wave is d2
1(1
+ 1) + k 2 - V ( r ) u,(k, r ) = 0,
1
(7)
where k 2 is the energy in appropriate units, and we assume that the potential has finite range so that
V ( r )= 0
( r > a).
(8)
The S-matrix, which is here just a number S,(k), is defined by the solution of Eq. (7) satisfying the boundary conditions
u,(k,O)
=
0
The following symmetry relations follow immediately from the properties of Eqs. (7) and (9):
S,(k) = s; , ( k ) ,
S,(k) = S,*( - k*),
(10)
where the asterisk denotes the complex conjugate quantity. It follows from Eq. (10) that S,(k) is unitary, and the phase shift 6,(k), defined by exp[2idl(k)] = S,(k), is real when k is real. Further, if S,(k) has a pole at the point k, then it also has a pole at - k*, and zeros at k* and - k. At a pole of S,(k), the wave function (9) can be written asymptotically u,(k, r)
E
eikr,
r-rm
apart from a normalization constant. This is called the Siegert definition of a resonance state (Siegert, 1939). For negative k2 (k = iK, with K real and positive), Eq. (1 1) becomes u e - K r which , corresponds to theusualdefinition of a bound state. For Im k > 0 the wave function (ll), corresponding to a pole in S,(k), is normalizable. It follows from the Hermiticity of the operator in Eq. (7) that the corresponding value of k2 must be real, and thus the poles must occur on the imaginary axis. In the lower halfk plane,thewave functionisnot normalizable, but it follows from the symmetry relations (10) that the poles occur in pairs symmetric with respect to reflection in the imaginary k axis. We show in Fig. 6 a possible example of the distribution of poles in the k plane. B,, B,, and B, correspond to bound states, while we shall see that R, and R, are resonance poles. The poles S, and S, are required by the reflection principle. In practice, for the square well potential discussed by Nussenzweig (1959) and Wiedenmiiller (1964), for example, there are an infinite number of poles in the lower half k plane, but usually only a few lie
-
P. G. Burke
188
k plane
63 82
FIG.6. Possible configuration of poles in the single channel S-matrix.
close enough to the physical region (real positive k axis) to give observable resonance effects. As the potential strength is increased the complex poles will, one by one, move onto the positive imaginary axis to give bound states. The poles are usually isolated, but for certain potentials there may be double poles. The possibility of double poles in atomic scattering has been discussed by Demkov and Drukarev (1965), while Goldberger and Watson (1964), and Eden and Landshoff (1964) have shown that a resonance corresponding to a double pole does not have a simple exponential decay. Consider the effect of an isolated pole in the S-matrix lying close to the real k axis. We define its position by
kr2 = E, - +ir,
(12)
where we shall see that E, is the resonance position, and I‘is the resonance width. From Eq. (lo), it follows that there is a zero in the S-matrix at E, + +ir. If all other singularities of S are far away we can write, for E close to E., S,(k) = exp(2i6,0)
E - E, - +ir E - E, 3ir’
+
where 6: is a slowly varying “ potential” phase shift. It follows from Eq. (13) that the resonant contribution to the total phase is given by 6,’
= tan-’
HE,-E‘
RESONANCES IN ELECTRON SCATTERING
189
For n overlapping resonances we obtain, in a similar manner, n
6, = 6,'
3ri + 1tan- E,'-E' i= 1
The partial wave cross section can be expressed in terms of the phase shift by 01
=
. p sin' 47c
6, ,
and in the neighborhood of an isolated resonance this gives Q,
=
47c sin' 6,' ( E + 4)' k 1 + & ',
where E = ( E - E,)/+r and q = -cot 6,'. This result has been considered before, see Eq. (3), in connection with photon absorption. Here we see that (Tb is zero. If we have other nonresonant partial waves contributing to the total cross section, then bb will be nonzero. The Breit-Wigner (1936) one level resonance formula follows from Eq. (17) by putting 6,' = 0. We illustrate some of the effects in Fig. 7, where we have assumed that three S-matrix poles lie close to the physical region. The phase shift is a continuous function of the energy and increases through n radians in the neighborhood of each resonance. Since we have assumed a constant potential phase shift equal to 7116 radians, the shape of each resonance given by q in Eq. (17) is the same. 2. Coupled Channel S-Matrix
We will now study the properties of the coupled channel S-matrix. We will restrict our considerations to a finite number of channels with two particles asymptotically in each. We do not include, therefore, ionization processes. After separation of the angular coordinates we obtain coupled equations, describing the radial motion of the scattering particles, of the type
i=l,
..., N ,
j = 1 , ..., N .
The potential matrix is symmetric, from time reversal invariance, and we will assume initially that it has finite range. The channel energies kiz are related to the total energy E of the system by
E = ki2
+E ~ ,
i = 1,
.. ., N ,
(19)
P. G. Burke
190
Energy plane 3
4
Re(energy)
0
-
0
0.1
-0.2
-
Energy
FIG. 7. Effect of resonances on the phase shift and cross section for single channel scattering. The circles correspond to resonances with tr = 0.001,0.01, and 0.1.
where the E ~ which , are ordered in increasing value, are the sum of the internal energies of the two colliding particles in channel i. Finally, the li are the channel angular momenta. Equations of the type (18) describe a large number of collision processes from nucleon-nuclei collisions (Lane and Thomas, 1958) and electron-atom collisions (Massey and Mohr, 1932; Seaton, 1953; Peterkop and Veldre, 1966), to electron scattering by rigid rotators (Arthurs and Dalgarno, 1960). Some of these processes have been recently described in detail by Mott and Massey (1965).
191
RESONANCES IN ELECTRON SCATTERING
We can find, in general, N independent solutions of Eq. (18) regular at the origin. The S-matrix is defined in terms of these by U i j ( 0 ) = 0,
uij(r)
-
r-t m
k; '/'(exp[ - i(ki r - f l i n)] d i j - exp[i(k, r - 41in)]Sij), i = 1, ..., N,
(20)
j = 1, ..., N.
Sij is an N x N matrix, each element of which is a function of the N channel momenta k , , . . . , k, . When the total energy is insufficient to excite all the channels included in Eq. (18), then Eq. (20) contains exponentially increasing waves in the closed channels. These are not physically allowed, and we therefore choose only those solution of Eq. (18) which have exponentially decaying waves in the closed channels. We define an open channel S-matrix by the boundary conditions
Uij(0)
=
uij(r)
N
i = l , ..., N , j=1,..., N a y
0,
k,: '/'(exp[ - i(ki r - f l i n)] S i j - exp[i(ki r - +li n)]Sij),
r-tco
uij(r)
-
r-tm
i = 1, ..., N , , j = 1, ..., Nay Nij exp(-Icir)
(ici
(21)
= -iki),
i = N a + l,..., N , j=1,..., Nay
for N, open channels, where N i j are normalization constants. From the symmetry and reality properties of Eq. (18), we can derive the following relations satisfied by the open channel S-matrix, So,, defined by Eq. (211,
s,.,so,= 1,
so, = so',
(22)
9
where " + " means Hermitian conjugate, and T means transposed. Equations (22) can be continued into the complex energy plane and impose important restrictions on the analytic properties of S (Eden and Taylor, 1964). The relations (19) introduce a complication that did not exist for single channel scattering. Since k i occur in the boundary conditions (20) and (21) defining S, the value of S is only given uniquely by k , if the sign ambiguities in the relations k , = f(k,'
+
El
- &')l/',
. . . ,kN = f ( k I 2 + E l
- EN)"'
(23)
are resolved. These signs can be chosen in 2N-' different ways, and consequently the S-matrix can only be made single valued, or uniformized, by introducing 2 N - ' sheets in the k, plane. There is now no reason why the k,
P. G.Burke
192
plane should be preferred to the E plane, and for physical reasons the uniformization of S [or more correctly of S,!j= ki-1/2Sijkj'2to remove nonessential double valuedness in the off-diagonal elements (Peierls, 1959)] is usually carried out by introducing 2N sheets in the E plane. E plane O
"
,
El
€2
,
€3
FIG.8. Branch cuts in the many channel S-matrix. The circles are bound states lying below the lowest threshold.
We show in Fig. 8 an example of the cut E plane, where all the branch cuts are chosen, conventionally, to run from ki= 0 along the real axis to E = + co. The physical scattering region is along the upper rim of all the cuts. On the physical sheet, obtained by continuing from the physical region without crossing any branch cuts, all the ki have positive imaginary parts. It follows that the wave function at a pole of S on the physical sheet is normalizable, and thus, from the Hermiticity of the Hamiltonian, the poles on the physical sheet must be at a real energy lying below the lowest threshold. Consider now the example of two coupled channels illustrated in Fig. 9. In the absence of coupling between the channels, the S-matrix is diagonal with poles both in S,,(k,) and in S22(k2),as shown in Figs. 9a and 9b, respectively. When coupling potential VI2 is switched on, two things happen. First, except in exceptional circumstances, all poles now occur in all elements of the S-matrix. Secondly, the pole positions depend on the strength of the coupling. Unitarity, given by Eq. (22), forces poles, which in the absence of
E plane
FIG.9. Poles in the single channel and coupled channel S-matrix.
RESONANCES IN ELECTRON SCATTERING
193
coupling would lie on the real E axis with E > E ~ to , move onto a nonphysical sheet of E. Fonda and Newton (1960) also discussed another possibility in which resonances appear only in the strong coupling limit. The situation for weak coupling is summarized in Fig. 9c. We find two types of resonance. Resonance R,,which occurred in S , , in the zero coupling limit, is now strongly coupled to, and decays mainly into, channel one. We have called this resonance a shape resonance in Section I. Resonance R , , similarly, is a shape resonance in channel two. Resonance R, , on the other hand, is a bound state in channel two in the absence of coupling. It corresponds to the closed channel resonance discussed in Section I, and its width, which is caused by the coupling, is typically rather small. The width of the shape resonances is essentially given by the dynamics of the uncoupled channel situation, and is usually rather broad, unless there is some barrier mechanism to strongly inhibit the decay (Bernstein et al., 1966).
C . MULTICHANNEL RESONANCE THEORY We develop in this section an explicit expression for the multichannel S-matrix in the neighborhood of an isolated resonance. Following Brenig and Haag (1959) and Fano (1961), we consider the situation where one discrete state interacts with several continua. We denote our zero approximation discrete state by I$), and the continuum states by l$j(E)), where j denotes the N linearly independent solutions corresponding to N open channels. We assume the normalization (49
4)
= 1,
The Hamiltonian matrix can be written
where the inessential assumption has been made that the Hamiltonian is prediagonalized in the subspace of the continuum states. This restriction has been relaxed by Fano and Prats (1963) and by Altick and Moore (1966). We now diagonalize H in the space defined by 14) and l$j(E)), by introducing r
P. G. Burke
194
It is possible to choose N linearly independent combinations of the continuum states It,hi(E)), so that only one interacts through H with the discrete state. These are given by Xi(E)) =
1i vji(E)$j(E),
(27)
where V j i ( E ) is a unitary matrix whose first column is Vj(E) defined by Eq. (25), but normalized to unit length. The other columns of Vji(E)are orthogonal to each other and to the first column but are otherwise arbitrary. Equation (25) can then be replaced by
which defines the real quantity V(E). In this new representation, the eigenstate which we wish to determine is YI’(E)) = ~
4 +)
s
dE’ b(E’)XI(E’))
Now the asymptotic form of the untransformed continuum states can be written generally as Y(E))
- ($)
r+m
1/ 2
(sin O
+ cos OK,)
x (1
+ K:)-’/’,
(30)
where we have adopted a matrix notation for Y ( E ) ) where the columns represent the independent continuum states Y j ( E ) ) ,and the rows represent the open channels, Also, quantities in roman type are diagonal matrices, e.g. k = ki dii, while quantities in boldface type are not necessarily diagonal. We have written O = kr - $In, while KO is related to the S-matrix, defined by Eqs. (20) and (21), by So = (1 iKo)/(l - iK,). Other quantities are defined by Eq. (1 8). Finally, for convenience of notation, we have suppressed the coordinates of the target and the angular coordinates of the scattered particle, both of which are assumed normalized to unity. Following the method developed by Fano (1961) for a single continuum, and remembering that only the first column of “’(E)) interacts with the
+
195
RESONANCES IN ELECTRON SCATTERING
discrete state, we derive, from the condition that "'(I?)) Hamiltonian, that
K,(1
diagonalizes the
]I'
+ KO2)-'I2 - (1 + KO2)-l/' f Er y- Er ~
(31)
apart from a normalization factor. In Eq. (31), we have defined
Er = E,
+ A,
A(E) = 9
V(E')' dE' E-E'
where 9 denotes the principal value integral. Equation (32) defines the resonance width r and shift A. Rearranging Eq. (31) in the form (30), we define a matrix K, which describes the complete scattering, by
and the S-matrix is
+
where S = (1 iK)/(1 - iK). The partial width amplitudes yi , defined by Eq. (32), are real since we have chosen a real representation (30) for our continuum states. Because of the square root ambiguity in Eqs. (33) and (34), we find it convenient to define a complex width amplitude vector g by y1lzg =
~ 1 / 2 s ~; / 2.
=
(4' H I + ) ,
(35)
where Y') is that combination of the zero approximation continuum states with the asymptotic form
This is proportional to a plane wave state in the diagonal channels with a modified outgoing wave in all channels.
P. G . Burke
196
Equations (33) and (34) are our basic equations describing the behavior of the scattering near an isolated resonance. The background S-matrix S o , which is assumed to be slowly varying with energy, allows for far away resonances and potential scattering. The diagonalization procedure discussed in this section is, in principle, exact, provided that a complete set of states is chosen in the zero approximation. It has been used with success as a basis for a calculational procedure for closed channel resonances in e--He+ single channel scattering by Altick and Moore (1966). For the treatment of shape resonances the choice of basis functions is more difficult, since the widths of such resonances are typically rather broad, and thus the final resonance state may be very different from 14). In the next section, we discuss the close coupling method where shape resonances occur naturally in the continuum states. We conclude by remarking that the cross section elements o j j , obtained from the S-matrix (34) in the usual way (Burke and Smith, 1962), all have the form given by Eq. (3), with the same r but different oa, ob, and q. We have seen in Section III,B,l that ob is zero if there is only one open channel available for the decay of the resonance. Recently, Macek and Burke (1967) and Mies (1967), have discussed resonances decaying into two open channels. It follows immediately from Eq. (33) that, unless KOis changing rapidly with energy, each element of K will always vanish in the neighborhood of the resonance, and consequently S12,which for a 2 x 2 S-matrix has the multiplying factor K,,, will also vanish at the same point. Thus, in this case (Ob)12
= O*
D. CLOSECOUPLING THEORY We will consider in this section a procedure which has proved successful in predicting the properties of both closed channel and shape resonances in atomic scattering processes. The method, called the close coupling approximation, is based on an expansion of the total wave function in eigenstates of the target Hamiltonian. It was first introduced by Massey and Mohr (1932), and has since been shown by Feshbach (1958, 1962, 1964) to give rise naturally to resonances of the closed channel type. The Schrodinger equation describing the scattering of an electron by an atomic system with N electrons can be written W N + l
-JW(X1,X2, * . . , X N + d = O ,
+
(37)
is the (N 1)-electron Hamiltonian, and X i is a collective where H N + l variable describing the space and spin coordinates of the ith electron. If the target is a molecular system, both H N + l and II/ depend on the nuclear coordinates. In this section we will be mainly concerned with atomic systems,
RESONANCES IN ELECTRON SCATTERING
197
and we defer, therefore, a discussion of the modifications necessitated by the nuclear motion in molecular systems to Section II1,E. In the close coupling method the total wave function is expanded in the form
where d antisymmetrizes the complete expression with respect to interchange of any two electrons. The summation in Eq. (38) is over the complete set of eigenstates $ i including , the continuum, of the target Hamiltonian and, except in the case of the scattering of electrons by hydrogen like ions where exact wave functions are available, the $ iare usually approximated by HartreeFock wave functions. The Fiin Eq. (38) describe the motion of the scattered electron in the ith channel. In order to make the method numerically tractable, the infinite summation in Eq. (38) is truncated to a small number of terms. Apart from approximations inherent in the choice of $ i for complex atoms, this is the only approximation made in the method. Coupled second-order integro-differential equations are then derived for the radial parts of the functions Fi by projecting Eq. (37) onto the atomic states $ i , together with the angular and spin coordinates of the scattered electron. If spin-orbit, spin-spin, and other relativistic terms are neglected in the Hamiltonian, which is usually valid for light atoms, the resultant equations are diagonal in the total angular momentum L and total spin S of the system. The coupled equations have been discussed by many authors (e.g. Seaton, 1953; Burke and Smith, 1962; Peterkop and Veldre, 1966) and have the form given by Eq. (18), with the additional complication that the potential matrix is, in general, nonlocal with long range inverse r behavior asymptotically. In Section IV we discuss some examples which have involved the solution of between three and ten coupled equations for each LS combination. The formal properties of the solution of the equations can be conveniently discussed using the projection operator formalism introduced by Feshbach (1962). We partition the ( N + 1) electron Hilbert space into two orthogonal parts by means of projection operators defined by P+Q=R, P 2 = P,
Q2 = Q ,
(39)
PQ = Q P = 0,
where R is a subspace of the full Hilbert space. In the close coupling approximation, R is the space spanned by the terms retained in the expansion (38),
198
P . G . Burke
and P and Q represent some subdivision of these terms. Equation (37) can be formally rewritten as
where $" is the approximate solution in space R. In the limit R = 1, the solution of Eq. (40) equals the exact solution of Eq. (37). We eliminate Q$" from the first equation in (40) by substituting for it from the second equation in (40) to give
The second term in Eq. (41) describes scattering out of the P space into the Q space, propagation in the Q space, and then scattering back into the P space. It has the form of an optical potential. We now consider an explicit realization of the operators P and Q. We take P to project onto the open channels in R , and Q to project onto the closed channels. For the two electron system, for an energy below the n = 2 thresholds, the relevant projection operators satisfying the Pauli principle are (Hahn et al., 1962) P = R - Q, Q = QiQz 9
It is seen that Q$ = 0 when either of the two electrons is in the hydrogenic ground state. Introducing the eigenfunctions of the operator QHN+I Q by
Q H N+ I Q 5 n = En t n
5
(43)
we find that this operator has, in general, a discrete spectrum plus a continuum starting from the lowest threshold in the closed channel subspace. It is the discrete spectrum, which corresponds physically to an electron bound in the field of an excited atom or ion, that gives rise to closed channel resonance solutions of Eq. (41). If the energy E is in the neighborhood of one of the discrete eigenvalues E, of QHN+lQ, then Eq. (41) can be written
199
RESONANCES IN ELECTRON SCATTERING
where we have separated, on the right-hand side of Eq. (44),the rapidly varying part of the optical potential. We introduce solutions I)+ and $-, of the operator Hi+ton the lefthand side of Eq. (44), with outgoing wave and ingoing wave boundary conditions respectively. Equation (44) can then be formally solved to give
P*R)
+ E + i q - 1H ; + 1
= *+)
PHN+tQ~s>(rsQHN+lP*+)
X
E - ES -
(
5 s Q H N+ t P
+
1 iv - H;, + P H N+ t
(45)
Y
QL)
where q is a positive infinitesimal quantity. The T-matrix, which equals S can be easily evaluated from Eq. (45) and is
Tif
= (T&
-i
( * f - P H N + 1 Q 6 > ( r s QHN + t P$i+ )
(
E - ES - L Q H N +t p
9
1
~
iq - H ; , + l
PHN+
- 1, (46)
t~L)
where we have adopted the normalization given by Eq. (36). Expanding the denominator and comparing with Eq. (34), we find that the resonance width is rs = I ( ~ , Q H N ip1c/f+)12, + (47)
1 1
where the summation is taken over all final states. This definition of the width is equivalent to Eq. (35). The bound states in the closed channels thus give rise to resonances in the solution of the close coupling equations. Shape resonances, on the other hand, appear in the operator PHN+lP, and it is not easy to derive an explicit expression for their position and width. Provided, however, that the shape resonance is only closely coupled to a few open channels, there is no difficulty in solving the relevant open channel close coupling equations numerically to obtain the resonance parameters. Closed channel resonances have been found below the n = 2 threshold in e--H scattering, both in the 1s-2s close coupling approximation (Smith et al., 1962) and in the 1s-2s-2p approximation (Burke and Schey, 1962; McEachran and Fraser, 1963). Resonances have also been found below the n = 3 threshold in the ls-2s-2p-3s-3p-3d approximation (Burke et al., 1966b, 1967a,d). Close coupling solutions of several other systems have revealed closed channel resonances. These include e--He+ scattering (Burke e l al., 1963, 1964a,b; Burke and McVicar, 1965), e--Be+ scattering (Burke, 1965; Moores, 1967), e--He scattering (Burke et al., 1966a), e--O+ scattering (Smith et al., 1967), and e--Mg+ and e--Ca+ scattering (Burke and Moores, 1968).
P.G . Burke
200
Shape resonances occur less frequently in electron atom scattering, but recently a ‘P shape resonance has been found in the 1s-2s-2p close coupling approximation just above the n = 2 threshold in e--H scattering (Burke et al., 1967b). Shape resonances have also been identified close to and just above the n = 2 thresholds in e--He scattering (Burke et af.,1966a, 1967~).Finally, it is probable that the rapid rise in the 3P phase shifts at low energies, calculated for e--alkali atom scattering by Karule (1965), is caused bya shape resonance. Further discussion of some of the close coupling results for shape and closed channel resonances will be given in Section IV. Although we have been concerned so far in this section with the close coupling approximation, we have not made any assumptions about P and Q in deriving Eq. (41), except that they are projection operators satisfying Eq. (39). It may often be more convenient to use a different basis than that provided by the target eigenstates to represent the space Q (Gailitis, 1964, 1965). Recently, Burke and Taylor (1966) used a new set of basis functions to span the space Q for the two electron system. A typical term was
where pi, qj, and sj are integers, and GY is an angular function. Obviously, such a basis will represent electron-electron correlation much better than a few closed channel terms from expansion (38), which has been found in certain circumstances to be slowly convergent (Burke, 1963). Recently, the solution of the close coupling equations has been shown to satisfy certain important minimum principles if all open channels are included in the P expansion (Hahn et al., 1962, 1963, 1964a,b; McKinley and Macek, 1964; Sugar and Blankenbecler, 1964; Gailitis, 1964; Hahn and Spruch, 1967). Provided the total energy is below the spectrum of Q H N + I Q ,then the optical potential term in Eq. (41) is negative definite, and expanding the space Q , either by the addition of terms in the expansion (38) or by terms of the type (48), can only make this potential more attractive. Thus, as the space Q is expanded, the corresponding phase shift will increase, and we find, for example for the two electron system, that 6(ls) < 6(ls
+ 2s) < 6(ls + 2s + 2p) c ... < G(exact),
(49)
when the energy is below the n = 2 threshold. The notation in Eq. (49) defines the states included in the close coupling approximation used to calculate the phase shift. Equation (49) will also hold in the closed channel resonance region, provided the phase shift is defined by continuity from below all the resonances. We now see that the minimum principle gives a restriction on the position and width of a resonance, but the value obtained for the position is only an upper bound in the limit of zero resonance width.
RESONANCES IN ELECTRON SCATTERING
20 1
If we omit the open channels from the expansion and solve Eq. (43), we obtain the eigenvalues E,, which are related to the resonance energies by E,
= E,
+ A,.
(50)
The resonance shift A, is obtained by expanding the denominator of Eq. (46) to yield an expression analogous to Eq. (32). We find
which can be either positive or negative. Provided A, is small, which is usually the case when Tsis small (since Eqs. (47) and (51) involve similar integrals), then E, provides a useful approximation to the resonance energy. A RayleighRitz variational method, based on Eq. (43) and yielding upper bounds on the E , has been carried out for the two-electron system by O’Malley and Geltman (1965), Altick and Moore (1965), and Bhatia et al. (1967). If the resonance state trial function is not orthogonal to the open channel space, then the choice of wave function can sometimes be quite an art (Temkin, 1966). However, good results can be obtained, although the result is no longer a bound. Also, since the wave function includes some open channel component, some allowance for the resonance shift in Eq. (50) is made. Calculations of this type have included those by Holoien and Midtdal (1955), Holoien (1958, 1961), Midtdal (1965), Propin (1964), Sewell (1965), Lipsky and Russek (1966), and Weiss (1967). Unlike the close coupling approximation which determines all the resonance parameters, the solution of Eq. (43) or its equivalent only gives an approximate resonance position. To determine the resonance width it is necessary to use the resonance state wave function in an expression analogous to Eq. (47). This is the basis of calculations by Wu (1934), Bransden and Dalgarno (1953a,b; 1955), Propin (1960), and Cooper (1964). Since this procedure is not based on a bound principle, the results are not always too accurate. A recent calculation in which the position and width are combined into a single variational procedure has, however, been discussed by Miller (1 966).
E. MOLECULAR RESONANCE THEORY Reactions such as dissociative attachment, defined by Eq. (6), and its inverse, called “ associative detachment,” are known in many cases to proceed through the formation of intermediate compound negative molecular ion states. In this section we will discuss recent theories developed by Bardsley et al. (1964; 1966a,b), O’Malley (1966, 1967), Chen (1967), and Herzenberg (1967) to treat such processes.
P . G. Burke
202
In the absence of the nuclear motion, an electron scattering on a molecular system would form closed channel and shape resonances just as for electron atom scattering. The nuclear motion, however, will now cause a resonance state to broaden into a continuous band of states if the compound molecular ion potential energy curve is repulsive, or form a set of related discrete states if the potential energy curve is attractive. We will find that when the electronic state is resonant, a complex potential energy curve can be defined, and it is this complex curve that is used in the Born-Oppenheimer approximation to describe the nuclear motion. In the energy region where the velocities of the nuclei are small compared with those of the electrons, we can write the total wave function for the molecular system as
$(X R) x
R)Xny(R),
(52)
where X denotes all the electronic coordinates, R is the internuclear separation vector (we restrict our consideration for simplicity to diatomic molecules), and n and y denote the electronic and nuclear states, respectively. The electronic wave function $,,(X, R), which depends parametrically on R, then satisfies Cffe,(X,
R) -
K(R)l$n
= 0,
(53)
where He, is the electronic part of the complete Hamiltonian. For normal bound states we require a solution of Eq. (53) that dies off exponentially. On the other hand, for resonance states we use the Siegert boundary condition
where r is the radial coordinate of the outgoing electron, 4,, is a wave function containing the coordinates of the residual molecule, and k,,’ is the complex energy of the outgoing electron. This energy is, in appropriate units, equal to W,,(R)- E,,(R), where E,,(R) is the energy of the residual molecule. In order to calculate the state I),, satisfying the boundary condition (54), it is not possible to use the Rayleigh-Ritz variational principle. This would just give the lowest real energy eigenvalue for the symmetry considered (Taylor and Harris, 1963). Herzenberg and Mandl (1963) have developed a variational principle incorporating a complex boundary condition and have used it to predict resonances in e--H scattering. Kwok and Mandl (1965) have also applied it to e--He scattering. They show that the S-wave solution of Eq. (7), defined by the Siegert boundary condition (1 I), satisfies the variational principle S ( D / N ) = 0,
(55)
RESONANCES IN ELECTRON SCATTERING
203
where
N[u] =
c
u 2 ( r ) dr.
The value of DIN at the stationary point is k Z ,which is held fixed at a complex value during the variation. An iterative method for k2 is required to ensure that the value inserted for k Z in Eq. ( 5 5 ) equals the value obtained from D j N after the variation. Using this variational principle, Bardsley et al. (1966a) obtained the potential energy curves given in Fig. 10 for the 'ZU+ and 'Zg+ states of H,-. The 'Xuf state corresponds to shape resonance electron scattering on the ground state of H, . The 'Zg+ state, on the other hand, is weakly coupled to the ground state, and since it lies above the 3Cu+repulsive state of H, , except at large R,and is found to be strongly coupled to this state, it corresponds to shape resonance scattering on the excited state. Further discussion of some
:-L
\
I
R, atomic
units
FIG. 10. Potential energy curves for the 'Zg+and 3C,+ states of HZ, together with the real and imaginary parts of the potential energy curves for the 'Xu+ and 'C,+ states of Hz-calculated by Bardsley et al. (1966a).
P. G . Burke
204
of these points has been given by Sommerville (1966), Taylor (1967), and Taylor et al. (1966). Recent work by Eliezer et al. (1967) has indicated that the ,Xg+ curve of H,- may cross the 3Xu+ curve of H, in the Franck-Condon region and only gives rise to appreciable electron autoionization for smaller values of R. In the Franck-Condon region for excitation from the ground state of H, , Bardsley et al. find that the width of the ,Xg+ state for electron emission is between 1 and 2 eV. However, there is a chance that the electron will remain attached long enough for the H atom and the H - ion to move apart beyond 6 au where the electron decay width becomes zero. This results in dissociative attachment and accounts for the first peak in Rapp et al.'s data shown in Fig. 5. The 'Xu+ state has a broader width for electron emission and thus gives rise predominantly to elastic electron scattering and vibrational excitation. We will now derive an equation for the nuclear motion in associative detachment following a recent discussion given by Herzenberg (1967). If only one resonance state is important, we can assume that the total wave function can be well approximated by Eq. (52). Multiplying the Schrodinger equation
( H - E)$ = 0
(56) by I/,,* and integrating over all electronic coordinates, assuming fixed nuclei, we obtain
h2
-
-(2vR 2M
Xny
*
f I/n*VR $n dX + Xny f $n*vR2$n f I$n12 dX
dX) (57)
If I/"is a bound state, the integrals over the electronic coordinates in Eq. (57) can be extended over all space. For a resonance state, we take the integrals over the inner region excluding the exponentially increasing tail. Provided the width of the resonance is not too large, and thus the state has a welldefined physical meaning, the value of the wave function on the surface of the integration is small. The terms on the right-hand side of Eq. (57) are neglected in the usual Born-Oppenheimer approximation. Herzenberg shows that, for a resonance state, these terms are again small and can be neglected. The equation describing the nuclear motion can then be written
(- & VR2 + Wn(R) - E )xnr(R)
= 0.
The potential Wn(R) is complex and leads to a loss of flux from the beam
RESONANCES IN ELECTRON SCATTERING
205
during the collision. This corresponds to the processes where the electron escapes, leaving the nuclei bound in a molecule (associative detachment). If the compound molecular ion is formed by electron collision, Eq. (58) must be modified by a source term corresponding to the coupling of the electron molecule channel. In order to calculate the probability of associative detachment, we make a partial wave decomposition of Eq. (58). The cross section for associative detachment into all final states is then
where kN is the wave number of the nuclear motion, and 6, is the complex phase shift. In general, the cross section must be weighted by the probability that the incident particles move along the potential curve described by Eq. (58). For H - ions incident on H atoms, the nuclei will approach each other half the time along the ’Xu+ curve, and half the time along the *Zg+ curve. The ’Xu+ curve leads to associative detachment, however the *Zg+ curve leads mainly to nonassociative detachment (Herzenberg, 1967;Dalgarno, and Browne, 1967; Bardsley, 1967). The term e-’Irna1in Eq. (59) has a very important physical interpretation. Using the WKBJ approximation for the phase shift, Bardsley et al. (1964) showed that this term could be written as Rs
exp[ - 2
jR,
T(R’) dR‘ hU(R’)
+
‘1
’
where R, is close to the turning point evaluated for the real part of the potential, R, is the point beyond which the electron is stabilized, v(R) is the nuclear velocity at R, and q is a correction term which allows for detachment close to the turning point. Since rjh is the autoionization probability, and dR/z(R) is the time spent by the nuclei in traversing a distance dR, Eq. (60) is the probability that the electron will not be detached in the inward and outward trajectories, and is called the “ survival probability.” The interaction of two complex potential energy curves has recently been considered by Mandl (1966). He has shown that noncrossing rules apply for complex curves, similar to the Wigner-Von Neumann rule for real curves. However, the real parts of the curves can cross provided their imaginary parts are different at the crossing point. This may be important in deciding which final states will be reached in heavy particle collisions. It appears, in conclusion, that very many associative detachment reactions of negative ions (e.g. Fehsenfeld et al., 1966) may be controlled by these complex potential energy curves.
P. G. Burke
206
F. EFFECTIVE RANGETHEORY We now discuss briefly the analytic behavior of the S-matrix in the neighborhood of thresholds for new reactions and the connection with bound states and resonances. Blatt and Jackson (1949) and Bethe (1949) showed that, for single channel scattering by a short range potential, the low energy phase shift could be parametrized according to k21+ 1
cot Sf(k2)= -u1-'
+ +rolk2+ 0(k4),
(61)
where al is the scattering length and ro, is the effective range. From the definition of the S-matrix it follows that a pole in S occurs when cot 6 = i. It is a simple matter, therefore, to obtain the bound state or resonance parameters knowing a, and rol. O'Malley et a/. (1961) and Levy and Keller (1963) have generalized Eq. (61) for potentials V(r) r-'. For the important case s = 4, O'Malley et al. show that, for 12 1,
-
k2 cot 6,(k2) = 8(1+ +)(1+ +)(1- +)/nu + higher order terms,
(62)
where u is the polarizability of the target. If CI is large, then the rapid linear rise of the phase shift with energy close to threshold, predicted by Eq. (62), may continue through TL radians to give a shape resonance. This, in fact, was found to happen for electron scattering on the 2 3 S state of He, where a is %313uo3 (Burke et ul., 1966a). Ross and Shaw (1961) have generalized the Blatt-Jackson formula to the situation where several channels are coupled by short range potentials. They show that the T-matrix ( = S - 1) can be written
where the diagonal matrices k and / are defined by Eq. (18). The matrix M is real and symmetric on the real energy axis and free of threshold branch cuts, although it may have poles. M, which is the analog of k2'+' cot 6 in Eq. (61), can therefore be expanded according to
M=Mo+Mlk2+M2k4+***, (64) where Mo , MI, . . . are real symmetric energy independent matrices. If T is known, either from experiment or from theoretical calculations, above a threshold, M can be calculated from Eq. (63) and then extrapolated using Eq. (64) to give T below threshold. Damburg and Peterkop (1962) have used Eqs. (63) and (64) to extrapolate the 1s-2s close coupling solution of e--H above the n = 2 threshold to predict a ' S resonance just below the threshold.
RESONANCES IN ELECTRON SCATTERING
207
This resonance had previously been found by direct solution of the 1s-2s equations below the n = 2 threshold (Smith et al., 1962). The presence of long range potentials will modify Eq. (64). The changes required are, however, only known in special cases. Gailitis and Damburg (1963a,b) have considered the important case when degenerate channels are coupled by a dipole potential, as in e--H scattering in excited states. Following Seaton (1961), Gailitis and Damburg diagonalized the matrix of coefficients of the r - 2 potential, including the angular momentum, by an orthogonal transformation AT[l(l
+ 1) + a ] A = h(h + 1).
(65)
If all long range potentials except the degenerate dipole potentials, whose coefficient is denoted in Eq. (65) by the matrix a, are neglected, then A also diagonalizes the asymptotic form of the coupled radial Schrodinger equations. In this new representation Eq. (63) is applicable, but with I replaced by h. It follows that h determines the threshold behavior of the cross section. From an explicit calculation (Burke, 1965), it is possible to show that at all excited thresholds in e--H scattering some of the eigenvalues, &(Ai + 1) in Eq. (65), are less than -*, and thus the corresponding values of l i are complex. From Eq. (63), the cross section to an excited state involves k'+%, and it therefore starts finite in violation of the Wigner (1948) law for short range potentials. Further, Landau and Lifshitz (1958) show that potentials with an attractive tail -ar-2, with CI > +, support an infinite number of bound states. This is the origin of the closed channel resonances in e--H discussed in Sections II,B and IV. Closed channel resonances due to a similar mechanism are predicted in p-H and e+-H scattering (Mittleman, 1966). Also, bound states supported by the dipole field are predicted in the scattering of electrons by polar molecules when the dipole moment is greater than 1.63 x lo-'* esu-cm (Mittleman and Myerscough, 1966; Levy-Leblond, 1967; and Crawford and Dalgarno, 1967). For electron scattering by a positive ion, which for a singly charged ion corresponds to the continuum spectrum of a neutral atom, the long range force is dominated by the Coulomb potential V(r) -2z/r, where z is the ionic charge. We obtain the usual bound state spectrum N
-
(66)
where n has integer values, and d,,, is the quantum defect defined by Ham (1955). The quantum defect is a measure of the departure of the potential from pure Coulomb at small r, and Ham showed that, in general, it is a slowly varying function of the energy. Seaton (1955, 1958) showed that the
208
P. G. Burke
phase shift for single channel scattering at positive energies can be related to the extrapolated quantum defect by
where q = - z / k and d(k2)=& [defined by Eq. (66)] at the negative bound state energies. This equation is the analog of Eq. (61) when a Coulomb interaction is present. In the limit k2 -+ 0, we obtain 6,(0) = nd(0). Shape resonances do not occur readily in electron-ion scattering because the Coulomb tail causes states, which would otherwise be resonant in the absence of the Coulomb interaction, to be bound. However, there is a wealth of closed channel resonances corresponding to Rydberg series of states convergent on each excited state of the ion. This is illustrated by the photon absorption data discussed in Section II,E and illustrated in Fig. 3. Bely et al. (1964) and Seaton (1966a,b) have generalized the extrapolated quantum defect approach to coupled channels, while Gailitis (1963) has extended the Ross-Shaw formula to allow for Coulomb attractive potential tails. Both these approaches are essentially equivalent and allow extrapolations to be carried out through thresholds. An important result proved in this work, and also discussed by Baz (1959) and Newton and Fonda (1960), is that the total cross section, averaged over resonances below threshold, is continuous through a threshold for a new reaction when Coulomb attractive forces are present. Recently, these many channel extrapolation methods have been applied to e--He+ scattering by Bely (1964, 1966) and by Burke et al. (1964a), to e--Ca+ scattering by Moores (1966), and to e--Be+ scattering by Moores (1967). The polarization forces do not change the structure of the resonance series in this case (Gailitis, 1963), but the degenerate dipole terms must be included to get accurate extrapolations in e--He+ (Bely, 1966).
IV. Further Results and Conclusions In this section we give some recent results obtained for resonances in e--H and e--He+ scattering. The reader is referred to reviews by Burke (1965) and by Smith (1966) for more complete tables of earlier work. Finally, we draw some general conclusions from these results. The positions E, and widths r of the lowest lying closed channel resonances below the n = 2 threshold for e--H and e--He+ scattering are given in Table I. As discussed in Section III,F, each resonance is a member of an infinite series converging onto the n = 2 threshold. Theoretically, the most accurate calculations for E, and r have been made by expanding the total
TABLE IRESONANCES IN HOMG. State
E,
HHHHHH-
IS
He He He He He He He
(2s’)’S 57.832 (2p’)’S 62.161 (2s3s)’S 62.959 ( 2 ~ 3 ~ ) ~ 62.616 s (2s2p)’P 60.190 ( 2 ~ 2 p ) ~ P 58.304 (2p’)’D -
’S ’S
‘P 3P ID
9.559 10.178 10.149 10.178 9.727 -
LR E, +A’
BTP
-
9.557 10.177 10.146 9.731
-
-
60.27 58.38 -
Er
-
57.817 62.063 62.953 62.611 60.154e 58.293 -
AND IN
He
BELOW THE
cc E,
cc
9.575 10.178 10.151 10.179 9.768 10.160 57.865 62.808 63.009 62.621 60.269 58.360 60.025
n
=2
THRESHOLD” Experiment
ccc
CCC
E.
r
5.43-’ 2.41-3 1.89V5 2.26-5 7.97-3 7.8-3
9.560 10.178 10.150 10.177 9.740 10.125
4.75-’ 2.19-3 2.06-4 4.50-5 5.94-’ 8.8-3
1.41-’ 1.88-’ 3.25-’ 1.81-4 4.38-’ 1.06-’ 7.32-’
57.842 62.134 62.975 62.615 60.149 58.317 59.91 I
1.24-1 7.3-3 3.63-’ 2.CW4 3.88-* 8.99-3 6.62-’
r
A 3-3
1-3
3-3 -1-3 1.3-’ -
2.5-’ 7.1-’ 2.2-’ 4-3 -5-3 2.4-’ -
E, 9.56(a) -
r 4-’(a) -
a m
r-
B2:
F
-
-
3
-
]-’(a) -
2
9.73(a) -
57.82(b) 62.15(b) 62.95(b) -
60.14(c) 58.34(b) 60.0(d)
-
3.8-’(c) -
All quantities are given in electron volts. The superscripts give the power of 10 by which the number is to be multiplied. In H-, the energy is above the H ground state. In He, the energy is above the He ground state. The He ground state binding energy is taken to be 24.5867 eV, and R , = 13.60535 is used in all conversions. References: O’MG (OMalley and Geltman, 1965), LR (Lipsky and Russek, 1966), BTM (Bhatia et at., 1967), cc (close coupling 3 states), ccc (close coupling plus correlation), a (McGowan, 19661, b (Rudd, 1965), c (Madden and Codling, 1965), d (Simpson et at., 1964), e (Bhatia and Temkin, 1967).
6 m
K0
g v1
0
5
$
2
0
210
P. G . Burke
wave function in terms of the closely coupled Is, 2s, and 2p eigenstates of the target, together with up to twenty correlation terms defined by Eq. (48). These results are labeled ccc (close coupling plus correlation) in the table. In order to obtain E, and r, the relevant equations, which have been derived recently (Burke and Taylor, 1966), are solved at a set of energies through each resonance to yield the phase shift. These phase shifts are then fitted to the resonance form Eq. (15). In columns labeled cc (close coupling), for comparison we give results obtained including the Is, 2s, and 2p states in the close coupling expansion. Apart from the (2p')'S resonance in e--He+ scattering, there is good agreement between the two calculations, indicating that the three-state close-coupling calculation is reliable at these energies. The effect of the minimum principle discussed in Section II1,D is also apparent from these results. Lipsky and Russek's (1966) results were obtained using an expansion in He' eigenstates including some continuum 1s functions, and therefore some part of the resonance shift A [see Eq. (50)].They showed that the (2p')'S resonance is strongly coupled to the (3d')'S state, which is omitted from the 1s-2s-2p close coupling calculation. Bhatia et al. (1967), and previously O'Malley and Geltman (1965), have obtained variational upper bounds for E, defined by Eq. (43), where Q projects onto the excited states of the target. Using their results for E, and the ccc results for E,, we can calculate the resonance shift A. This is also given in Table I. In most cases A is positive, which corresponds to an overall repulsion from the lower lying continuum. However, in the 'P cases A is found to be negative. This confirms two previous calculations in He (Burke and McVicar, 1965; Altick and Moore, 1966). Close coupling calculations have recently been carried out for e--H scattering including six states, Is, 2s, 2p, 3s, 3p, and 3d, by Burke et al. (1966b, 1967a). Six closed channel resonances, caused by the degenerate dipole interaction, have been found below the n = 3 threshold. Using a multichannel resonance fitting program based on Eq. (33), Macek and Burke (1967) have obtained the resonance parameters given in Table 11. Since L and S are good quantum numbers, there are four open channels (three for L = 0) available for the decay of these resonances. The partial width amplitudes, defined by Eq. ( 3 9 , are gIs(decay into the 1s channel), gzs(decay into the 2s channel), g2p+ (decay into the 2p channel with l2 = L l), and gzp- (decay into the 2p channel with Iz = L - l), where 1, is the angular momentum of the incident electron. Also, q and pz = a,/(o, ob),defined by Eq. (3), are given in Table I1 for the Is-2s and 1s-2p transitions. In order to understand why both q and ob in Table I1 are small, Macek and Burke (1967) transformed to the representation defined by Eq. (65) where the degenerate dipole potential is diagonal. It was found that each resonance, in this representation, decays mainly to one state at the n = 2 threshold, and
+
+
TABLE I1 RESONANCES IN H -
BELOW THE
n
=3
THRESHOLD'
State
E,
r
191s1
Iszsl
lgzp+l
192p-I
4lS,ZS
'S 'S 'P 3P
11.733 12.037 11.915 11.764 12.048 11.819
3.88-2 8.53-3 3.83-' 4.83-2 7.13-3 4.93-2
0.355 0.372 0.186 0.129 0.133 0.217
0.641 0.611 0.304 0.737 0.722 0.647
0.680 0.698 0.505 0.412 0.484 0.407
-
0.555 0.695 -0.421 0.421 0.691 -0.229
3P
'D
-
0.786 0.520 0.476 0.607
z
p:s.zs 0.974 0.967 0.996 0.9996 0.999 0.976
41s.ze
pL.2,
0.602 0.866 -0.386 0.283 0.500 -0.169
0.958 0.956 0.987 0.997 0.995 0.927
The resonance positions and widths are in electron volts. The other notation is explained in the text.
a
v,
2
P. G . Burke
212
this state is also the state most strongly excited from the ground state. To a good approximation, therefore, it is found that the S-matrix relevant to the production and decay of the resonancesis two dimensional. In such a situation it is relatively easy to show that q12 is small and -0 where 1 and 2 are the channel labels of the two-dimensional transformed S-matrix (see also Section II1,C). The effect of these resonances on the total 1s-2s cross section is shown in Fig. 1 1 . Also shown are the results of the three-state 1s-2s-2pclose coupling calculation and the three state plus twenty correlation term calculation. The closed channel n = 3 resonances depress the cross section above 11.6 eV, and this depression continues above the n = 3 threshold. The depression is severe enough to cause a peak at 11.6 eV in agreement with experiment (Lichten and Schultz, 1959; Stebbings et af., 1960; Hils et al., 1966), however the last two experiments, which normalize to the Born approximation above 300 eV, find a cross section x0.11nao2 at the peak. This difference of normalization with theory is difficult to understand in view of the good agreement between the three calculations below 11.6 eV. A further important feature shown in Fig. 1 1 is a ' P shape resonance found just above the n = 2 threshold. This has now been shown to be almost a pure
-
n=2
n=3
n=4
*<
6 states
3 stotes
6 states Resonances
IS
I
3P
I
'D
I
'P IS ond 'P I
I
I
-
0.0 0.85 Electron energy rydbergs 10.5 II 11.5 Electron energy eV
-
12
I 0.90
125
FIG.11. The effect of resonances on the 1s-2s cross section in e--H scattering, calculated close to the excitation threshold.
RESONANCES IN ELECTRON SCATTERING
+
213
eigenstate of 1(f + 1) a defined in Eq. (65), with eigenvalue ,I(A + 1) = 2. It is probably this angular momentum barrier, together with the short range nuclear attraction, that provides the mechanism for the resonance. Detailed calculation using the three state plus twenty correlation term approximation gives E, = 10.22204 eV and =0.0151 eV. The width is unusually narrow for a shape resonance, but is understandable in terms of the phase space factor k2'+' with f = 1, where k is measured relative to the n = 2 threshold. This strongly inhibits the decay into the n = 2 channels and it follows directly from the Blatt-Jackson formula, given by Eq. (61), that close to threshold the position k, of a resonance must satisfy Im k,/Re k, = - tan[n/2(21+ l)]. Thus, Im k , , or the width, gets smaller both as I increases and as Re k, decreases. If the potential were slightly more attractive, this resonance would lie below the n = 2 threshold and would become a closed channel resonance. We remark here that the existence of resonances lying close to threshold and dominating the threshold behavior of cross sections will be a rather general feature of transitions involving excited states. We must therefore be very careful when discussing threshold laws that such effects are taken into consideration. We now discuss briefly the connection between the results found for e--H and results found for e--He+. We have seen that the long range dipole interaction determines the linear combination of degenerate levels which is excited and into which the resonance decays. On the other hand, Cooper et al. (1963), in their analysis of the autoionizing states in He (Madden and Codling, 1963), found that the strength of the coupling with the continuum is determined by the inner region of the wave function, and they introduced the and " -" terminology to describe the results [see Section II,D and also Macek (1966)l. A detailed analysis of the resonancesin e--H and e--He+ scattering shows that both the inner and the outer regions of the wave function must be considered simultaneously to get the full picture. The state with the lowest lying eigenvalue A(,I + 1) in Eq. (65) will lie lowest in energy both in the singlet and triplet states (since the long range forces do not depend on exchange), however, in the 3P case for example, this state will correspond to a "+" configuration while in the 'P case it will correspond to "configuration. This explains why the closed channel 'P resonance in e--H scattering below the n = 2 threshold given in Table I is narrow, while the 3P resonance is broad. A detailed analysis of the results obtained by Burke and McVicar (1965) for the 'P and 3P resonances below the n = 2 threshold in e--He+ scattering confirms this picture. It is expected that this balance between the short range electron-electron correlation forces and the long range multipole forces will be of vital importance in many resonance situations in atomic and molecular scattering and its elucidation will provide one of the most stimulating and rewarding fields of endeavour for the future.
"+"
79
214
P. G . Burke
ACKNOWLEDGMENT The author wishes to thank Dr. A. Joanna Taylor for her help in calculating the previously unpublished results given in Table I.
REFERENCES Afrosimov, V. V.,Gordeev, Y u. S., Panov, M. N., and Fedorenko, N. V. (1965). Soviet Phys. -Tech. Phys. (English Trunsl.) 9, 1248. Altick, P. L., and Moore, E. N. (1965). Phys. Rev. Letters 15,100. Altick, P. L., and Moore, E. N. (1966). Phys. Rev. 147,59. Andrick, D., and Ehrhardt, H. (1966). Z . Physik 192,99. Arthurs, A. M., and Dalgarno, A. (1960). Proc. Roy. SOC.(London) A256, 540. Auger, P. (1925). J. Phys. Radium 6,205. Baranger, E., and Gerjuoy, E. (1957). Phys. Rev. 106, 1182. Baranger, E., and Gerjuoy, E. (1958). Proc. Phys. SOC.(London) 72,326. Bardsley, J. N. (1967). Proc. Phys. SOC.(London)91,300. Bardsley, J. N., Herzenberg, A., and Mandl, F. (1964). In “Atomic Collisions Processes” (M. R. C. McDowell, ed.), p. 415. North-Holland Publ., Amsterdam. Bardsley, J. N., Herzenberg, A., and Mandl, F. (1966a). Proc. Phys. SOC.(London) 89,305. Bardsley, J. N., Herzenberg, A., and Mandl, F. (196613). Proc. Phys. SOC.(London) 89, 321. Barker, R. B., and Berry, H. W. (1966). Phys. Rev. 151, 14. Bates, D. R. (1962). In “Atomic and Molecular Processes” (D. R. Bates, ed.), p. 550. Academic Press, New York. Bates, D. R., and Massey, H. S. W. (1954). Phil. Mug. 45, 1 1 1 . Baz, A. I. (1959). Zh. Eksperim. Teor. Fiz. 36, 1762 [Soviet Phys.-JETP (English Transl.) 9, 1256 (1959)l. Bely, 0. (1964). Ann. Astrophys. 27,599. Bely, 0. (19661. Proc. Phys. SOC.88,833. Bely, O.,Moores, D., and Seaton, M. J. (1964). In “Atomic Collision Processes” (M. R. C. McDowell, ed.), p. 304. North-Holland Publ., Amsterdam. Bernstein, R. B., Curtiss, C. F., Imam-Rahajoe, S., and Wood, H. T. (1966). J. Chem. Phys. 44,4072. Berry, H. W. (1961). Phys. Reu. 121,1714. Berry, H.W. (1962). Phys. Rev. 127, 1634. Berry, R. S. (1966). J. Chem. Phys. 45, 1228. Bethe, H. A. (1949). Phys. Rev. 76,38. Beutler, H.(1935). Z. Physik 93, 177. Beutler, H., Deubner, A., and Junger, H. 0. (1936). Z . Physik 98,181. Bhatia, A. K., and Temkin, A. (1967). Abstracts, 5th Intern. Conf.Phys. of Electron. Atomic Collisions Leningrad, p. 270. Nauka, Leningrad. Bhatia, A. K., Temkin, A., and Perkins, J. F. (1967). Phys. Rev. 153,177. Rlatt, J. M., and Jackson, J. D. (1949). Phys. Rev. 76, 18. Boersch, H., Geiger, J., and Stickel, W. (1964). Z. Physik 180,415. Bransden, B. H., and Dalgarno, A. (1953a). Proc. Phys. SOC.(London) A66, 904.
RESONANCES IN ELECTRON SCATTERING
215
Bransden, B. H., and Dalgarno, A. (1953b). Proc. Phys. SOC.(London) A66, 911. Bransden, B. H., and Dalgarno, A. (1955). Proc. Phys. SOC.(London) A69, 65. Breit, G., and Wigner, E. (1936). Phys. Rev. 49,519. Brenig, W., and Haag, R. (1959). Fortschr. Physik 7, 183. (English translation, Atomic Energy Research Establishment Harwell Rept. T1066.) Briglia, D. D., and Rapp, D. (1965). Phys. Rev. Letrers 14,245. Burke, P. G. (1963). Proc. Phys. SOC.(London) 82,443. Burke, P. G. (1965). Advan. Phys. 14,521. Burke, P. G., and McVicar, D. D. (1965). Proc. Phys. SOC.(London) 86, 989. Burke, P. G., and Moores, D. (1968). To be published. Burke, P. G., and Schey, H. M. (1962). Phys. Rev. 126, 147. Burke, P. G., and Smith, K. (1962). Rev. Mod. Phys. 34, 458. Burke, P. G., and Taylor, A. J. (1966). Proc. Phys. SOC.(London) 88, 549. Burke, P. G., McVicar, D. D., and Smith, K. (1963). Phys. Rev. Letters 11, 559. Burke, P. G., McVicar, D. D., and Smith, K. (1964a). Phys. Letters 12,215. Burke, P. G., McVicar, D. D., and Smith, K. (1964b). Proc. Phys. SOC.(London) 84, 749. Burke, P. G., Cooper, J. W., and Ormonde, S. (1966a). Phys. Rev. Letters 17,345. Burke. P. G., Ornmonde. S., and Whitaker, W. (1966b). Phys. Rev. Letfers 17, 800. Burke, P. G., Ormonde, S., and Whitaker, W. (1967a). Proc. Phys. SOC.(London) 92,319. Burke, P.G., Taylor, A. J., and Ormonde, S. (1967b). Proc. Phys. SOC.(London) 92,345. Burke, P. G., Taylor, A. J., Cooper, J. W. and Ormonde, S. (1967~)Abstracts, 5th Intern. Conf. Phys, of Electron. Atomic Collisions, Leningrad, p. 376. Nauka, Leningrad. Burke, P. G., Taylor, A. J., Ormonde, S., and Whitaker, W. (1967d). Abstracts, 5th Intern. Conf. Phys. of Electron. Atomic Collisions, Leningrad, p. 368. Nauka, Leningrad. Chamberlain, G. E. (1965). Phys. Rev. Letters, 14,581. Chamberlain, G. E. (1967). Phys. Rev. 155,46. Chamberlain, G. E., and Heideman, H. G. M. (1965). Phys. Rev. Letters 15, 337. Chen, J. C. Y. (1964a). J. Chem. Phys. 40,3507. Chen, J. C. Y. (1964b). J. Chem. Phys. 40,3513. Chen, J. C. Y. (1966). Phys. Rev. 146,61. Chen, J. C. Y. (1967). Phys. Rev. 156, 12. Codling, K., and Madden, R. P. (1964). Phys. Rev. Letters 12, 106. Codling, K.,Madden, R. P., and Ederer, D. L. (1967). Phys. Rev. 155,26. Conipton, K. T., and Boyce, J. C. (1928). J. Franklin Inst. 205,497. Cooke, G.R., and Metzger, P. H. (1964). J. Opt. SOC.Am. 54,968. Cooper, J. W. (1964). In “Atomic Collision Processes” (M. R. C. McDowell, ed.), p. 595. North-Holland Publ., Amsterdam. Cooper, J. W., Fano, U., and Prats, F. (1963). Phys. Rev. Letters 10, 518. Craggs, J. D., and Massey, H. S. W. (1959). In “Handbuch der Physik” (S. Fliigge, ed.), Vol. 37,p. 314. Springer, Berlin. Crawford, O., and Dalgarno, A. (1967). Chem. Phys. Letrers 1, 23. Dalgarno, A., and Browne, J. C. (1967). Astrophys. J . 149,231. Dalgarno, A., and McDowell, M. R. C. (1956). Proc. Phys. SOC.(London) A69, 615. Damburg, R., and Peterkop, R. (1962). Proc. Phys. SOC.(London) 80, 1073. Demkov, Y. N. (1965). Phyr. Letters 15,235. Demkov, Y. N., and Drukarev, G. F. (1965). Abstracts, 4th Intern. Conf. Phys. of Electron. Atomic Collisions, Quebec, p. 52. Science Bookcrafters, Hastings-on-Hudson, New York. Dibeler, V. H., Reese, R. M., and Krauss, M. (1965). J . Chem. Phys. 42,2045. Doolittle, P. H., and Schoen, R. I. (1965). Phys. Rev. Letters 14,348.
216
P.G . Burke
Eden, R. J., and Landshoff, P. V. (1964). Phys. Rev. 136, B1817. Eden, R. J., and Taylor, J. R. (1964). Phys. Rev. 133, B1575. Edwards, A. K., and Rudd, M. E. (1967). Abstracts, 5th Intern. Conf. Phys. of Electron. Atomic Collisions, Leningrad, p. 134. Nauka, Leningrad. Ehrhardt, H., and Meister, G. (1965). Phys. Letters 14, 200. Ehrhardt, H., and Willmann, K. (1967a). Z. Physik 203, 1. Ehrhardt, H., and Willmann, K. (1967b). Z. Physik 204,462. Eliezer, I., Taylor, H. S., and Williams, J. K. (1967). J. Chem. Phys. 47, 2165. Everhart, E., and Kessel, Q. C. (1965). Phys. Rev. Letters 14, 247. Fano, U. (1935). Nuovo Cimento 12, 154. Fano, U. (1961). Phys. Rev. 124, 1866. Fano, U., and Cooper, J. W. (1965a). Phys. Rev. 137, A1364. Fano, U., and Cooper, J. W. (1965b). Phys. Rev. 138, A400. Fano, U., and Lichten, W. (1965). Phys. Rev. Letters 14, 627. Fano, U., and Prats, F. (1963). Proc. Nail. Acad. Sci. India Sect. A A33, 553. Fehsenfeld, F. C., Ferguson, E. E., and Schmeltekopf, A. L. (1966). J. Chem. Phys. 45,1844. Fender, F. G., and Vinti, J. P. (1934). Phys. Rev. 46,77. Feshbach, H. (1958). Ann. Phys. (N. Y.) 5, 357. Feshbach, H. (1962). Ann. Phys. (N. Y.) 19, 287. Feshbach, H. (1964). Rev. Mod. Phys. 36, 1076. Fisk, J. B. (1936). Phys. Rev. 49, 167. Fleming, R. J., and Higginson, G. S. (1963). Proc. Phys. SOC.81, 974. Fonda, L., and Newton, R. G. (1960). Ann. Phys. (N.Y.) 10,490. Franzen, W . , and Gupta, R. (1965). Phys. Rev. Letters 15, 819. Gailitis, M. K. (1963). Zh. Eksperim. i Teor. Fir. 44, 1974 [Soviet Phys.-JETP (English Transl.) 17, 1328 (1963)l. Gailitis, M. K. (1964). Zh. Eksperim. i Teor. Fir. 47, 160 [Soviet Phys.-JETP (English Transl.) 20, 107 (1965)l. Gailitis, M. K. (1965). In “Atomic Collisions,” Vol. 111 (V. Ia Veldre, ed.) Riga. (Transl. Rept. No. 3, Joint Institute for Laboratory Astrophysics, Boulder, Colorado, p. 129, 1966.) Gailitis, M. K., and Damburg, R. (1963a). Proc. Phys. SOC.82, 192. Gailitis, M. K., and Darnburg, R. (1963b). Zh. Eksperim. i Teor. Fiz. 44, 1644 [Sovier Phys.-JETP (English Transl.) 17, 1107 (1963)l. Carton, W. R. S. (1966). Advan. Atomic Mol. Phys. 2, 93. Gilmore, F. R. (1965). J . Quant. Spectr. & Radiative Transfer 5, 369. Goldberger, M. L., and Watson, K. M. (1964). Phys. Rev. 136, B1472. Golden, D. E., and Bandel, H. W. (1965a). Phys. Rev. 138, A14. Golden, D. E., and Bandel, H. W. (1965b). Phys. Rev. Letters 14, 1010. Golden, D. E., and Nakano, H. (1966). Phys. Rev. 144, 71. Hahn, Y., and Spruch, L. (1967). Phys. Rev. 153, 1159. Hahn, Y., O’Malley, T. F., and Spruch, L. (1962). Phys. Rev. 128,932. Hahn, Y., O’Malley, T. F., and Spruch, L. (1963). Phys. Rev. 130, 381. Hahn, Y., O’Malley, T. F., and Spruch, L. (1964a). Phys. Rev. 134, B397. Hahn, Y., O’Malley, T. F., and Spruch, L. (1964b). Phys. Rev. 134, B911. Ham, F. S. (1955). Solid State Phys. 1, 127. Heideman, H. G. M., Kuyatt, C. E., and Chamberlain, G. E. (1966a). J. Chem. Phys. 44, 355. Heideman, H. G. M., Kuyatt, C. E., and Chamberlain, G. E. (1966b). J. Chem. Phys. 44,440.
RESONANCES IN ELECTRON SCATTERING
217
Heisenberg, W. (1943). Z . Physik 120, 513, 673. Herzberg, G. (1950). ” Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules.” Princeton, New Jersey. Herzenberg, A. (1967). Phys. Rev. 160, 80. Herzenberg, A., and Lau, H. S. M. (1967). Abstracts, 51h Intern. Conf. Phys. of Electron. Atomic Collisions, Leningrad, p. 261: Nauka, Leningrad. Herzenberg, A., and Mandl, F. (1962). Proc. Roy. SOC.(London) A270,48. Herzenberg, A., and Mandl, F. (1963). Proc. Roy. SOC.(London) A274,253. Hils, D., Kleinpoppen, H., and Koschmieder, H. (1966). Proc. Phys. SOC.89, 35. Holoien, E. (1958). Proc. Phys. SOC.71, 357. Holoien, E. (1961). Phys. Norvegica 1, 53. Holoien, E., and Midtdal, J. (1955). Proc. Phys. SOC.(London) A68, 815. Holt, H. K., and Krotkov, R. (1966). Phys. Rev. 144, 82. Hu, N. (1948). Phys. Rev. 74, 131. Humblet, J. (1952). Mem. SOC.Roy. Sci. Liege 12, No. 4. Humblet, J., and Rosenfeld, L. (1961). Nucl. Phys. 26, 529. Karule, E. M. (1965). In“Atomic Collisions,” Vol. 3 (V. Ia. Veldre, ed.) Riga. (Transl. Rept. No. 3, Joint Institute for Laboratory Astrophysics, Boulder, Colorado, p. 29. 1966.) Kessel, Q. C., Russek, A., and Everhart, E. (1965). Phys. Rev. Lerters 14, 484. Khvostenko, V. I., and Dukel’skii, V. M. (1957). Zh. Eksperim. i Teor. Fiz. 33, 851 [Soviet Phys.-JETP (English Transl.) 6, 657 (1958)l. Kleinpoppen, H., and Raible, V. (1965). Phys. Letters 18, 24. Kruger, P. G. (1930). Phys. Rev. 36, 855. Kuyatt, C. E., Mielczarek, S. R., and Simpson, J. A. (1964). Phys. Rev. Letters 12, 293. Kuyatt, C. E., Simpson, J. A., and Mielczarek, S. R. (1965). Phys. Rev. 138, A385. Kuyatt, C. E., Simpson, J. A,, and Mielczarek, S. R. (1966). J. Chem. Phys. 44,437. Kwok, K. L., and Mandl, F. (1965). Proc. Phys. SOC.86, 501. Landau, L. D., and Lifshitz, E. M. (1958). “Quantum Mechanics.” Pergamon Press, Oxford. Lane, A. M., and Thomas, R. G. (1958). Rev. Mod. Phys. 30, 257. Lassettre, E. N. (1965). J. Chem. Phys. 43, 805. Levy, B. R., and Keller, J. B. (1963). J . Math. Phys. 4, 54. Ltvy-Leblond, J. (1967). Phys. Rev. 153, I. Lichten, W., and Schultz, S . (1959). Phys. Rev. 116, 1132. Lipeles, M., Novick, R., and Tolk, N. (1965). Phys. Rev. Letters 15, 815. Lipsky, L., and Russek, A. (1966). Phys. Rev. 142, 59. Ma, S. T., and Wu, T. Y. (1 936). J. Chinese Chem. SOC.4, 345. Macek, J. H. (1966). Phys. Rev. 146, 50. Macek, J. H.,and Burke, P. G. (1967). Proc. Phys. SOC.(London) 92,351. Madden, R. P., and Codling, K. (1963). Phys. Rev. Letters. 10, 516. Madden, R. P., and Codling, K. (1964). J . Opt. SOC.Am. 54, 268. Madden, R. P., and Codling, K. (1965). Astrophys. J . 141, 364. Madden, R. P., and Codling, K. (1966). “Autoionization” (A. Temkin, ed.), p. 129. Mono Book Co., Baltimore, Maryland. Maier-Leibnitz, H. (1935). Z. Physik. 95, 499. Mandl, F. (1966). Proc. Phys. SOC.(London) 87, 871. Martin, A. (1964). Progr. Elem. Particle Cosmic Ray Phys. 8, 1. Massey, H.S. W. (1949). Rept. Progr. Phys. 12, 248. Massey, H. S. W. (1950). “Negative Ions,’’ 2nd ed. Cambridge Univ. Press, London and New York.
218
P. G. Burke
Massey, H. S. W. (1962). Private communication. Massey, H. S. W., and Burhop, E. H. S. (1952). “ Electronic and Ionic Impact Phenomena.” Oxford Univ. Press (Clarendon), London and New York. Massey, H. S. W., and Mohr, C. B. 0. (1932). Proc. Roy. Soc. (London) A136, 289. Massey, H. S. W., and Mohr, C. B. 0. (1935). Proc. Cambridge Phil. Soc. 31, 604. Massey, H. S. W., and Moiseiwitsch, B. L. (1954). Proc. Roy. SOC.(London) A227, 38. McEachran, R. P., and Fraser, P. A. (1963). Proc. Phys. SOC.(London)82, 1038. McGowan, J. W. (1966). Phys. Rev. Letters 17, 1207. McGowan, J. W., and Fineman, M. A. (1965). Phys. Rev. Letters 15, 179. McGowan, J. W., Clarke, E. M., and Curley, E. K. (1965). Phys. Rev. Letters 15, 917. McGowan, J. W., Fineman, M. A., Clarke, E. M., and Hanson, H. P. (1967). To be published. McKinley, W. A., and Macek, J. H. (1964). Phys. Letters 10, 210. Menendez, M. G., and Holt, H. K. (1966). J . Chem. Phys. 45, 2743. Midtdal, J. (1965). Phys. Rev. 138, A1010. Mies, F. H. (1967). Abstracts, 5th Intern. Conf. Phys. of Electron. Atomic Collisions, Leningrad, p. 265. Nauka, Leningrad. Miller, W. H. (1966). Phys. Rev. 152, 70. Mittleman, M. H. (1966). Phys. Rev. 152, 76. Mittleman, M. H., and Myerscough, V. P. (1966). Phys. Letters 23, 545. Mdler, C. (1945). Kgl. Danske Videnskab. Selskab. Mat. Fys. Medd. 23, No. 1. Moores, D. L. (1966). Proc. Phys. SOC.88, 843. Moores, D. L. (1967). Proc. Phys. SOC.91, 830. Mott, N. F., and Massey, H. S. W. (1965). “Theory of Atomic Collisions,’’ 3rd ed. Oxford Univ. Press, London and New York. Newton, R. G . , and Fonda, L. (1960). Ann. Phys. (N.Y.) 9, 416. Normand, C. E. (1930). Phys. Rev. 35, 1217. Nussenzveig, H. M. (1959). Nucl. Phys. 11, 499. Olrnsted, J., 111, Newton, A. S., and Street, K. Jr., (1965). J . Chem. Phys. 42, 2321. O’Malley, T. F. (1966). Phys. Rev. 150, 14. O’Malley, T. F. (1967). Phys. Rev. 155, 59. O’Malley, T. F., and Geltman, S. (1965). Phys. Rev. 137, A1344. O’Malley, T. F., Spruch, L., and Rosenberg, L. (1961). J. Math. Phys. 2, 491. Peierls, R. E. (1959). Proc. Roy. Soc. (London) A253, 16. Peterkop, R., and Veldre, V. (1966). Advan. Atomic Mol. Phys. 2, 264. Pietenpol, J. L. (1961). Phys. Rev. Letters 7, 64. Priestley, H., and Whiddington, R. (1935). Proc. Leeds Phil. Lit. Soc. Sci. Sect. 3, 81. Propin, R. Kh. (1960). Opt. Spectry ( U S S R ) (English Transl.) 8, 158. Propin, R. Kh. (1964). Opt. Spectry ( U S S R ) (Enqlish Transl.) 17, 332. Ramsauer, C., and Kollath, R. (1929). Ann. Phys. (Paris) 3, 536. Rapp, D., Sharp, T. E., and Briglia, D. D. (1965). Phys. Rev. Letters 14, 533. Rice, 0. K. (1933). J. Chem. Phys. 1, 375. Ross, M., and Shaw, G. (1961). Ann. Phys. ( N . Y.) 13, 147. Rudd, M. E. (1964). Phys. Rev. Letters 13, 503. Rudd, M. E. (1965). Phys. Rev. Letters 15, 580. Rudd, M. E., and Lang, D. V. (1965). Abstracts, 4th Intern. Con$ Phys. of Electron. Atomic Collisions, Quebec, p. 1 17. Science Bookcrafters, Hastings-on-Hudson, New York. Rudd, M. E., Jorgensen, T., Jr., and Volz, D. V. (1966). Phys. Rev. Letters 16, 929. Samson, J. A. R. (1963). Phys. Rev. 132 2122. Samson, J. A. R. (1966). Advan. Atomic Mol. Phys. 2, 178.
RESONANCES IN ELECTRON SCATTERING
219
Schulz, G .E. (1965). Abstracts, 4th Intern. Cot$ Phys. of Electron. Atomic Collisions, Quebec, p. I 17. Science Bookcrafters, Hastings-on-Hudson, New York Schulz, G. J. (1959). Phys. Rev. 113, 816. Schulz, G. J. (1962). Phys. Rev. 125, 229. Schulz, G. J . (1963). Phys. Rev. Lctters 10, 104. Schulz, G. J . (1964a). Phys. Rev. Letters 13, 583. Schulz, G. J. (1964b). In “Atomic Collisions Processes” (M. R. C. McDowell, ed.), p. 127. North-Holland Publ., Amsterdam. Schulz, G. J . (1964~).Phys. Rev. 135, A988. Schulz, G. J., and Asundi, R. K . (1965). Phys. Rev. Letters 15, 946. Schulz, G. J., and Fox, R. E. (1957). Phys. Rev. 106, 1179. Schulz, G. J., and Philbrick, J. W. (1964). Phys. Rev. Letters 13, 476. Seaton, M. J. (1953). Phil. Trans. Roy. Soc. London, Ser. A 245, 469. Seaton, M. J . (1955). Compt. Rend. 240, 1317. Seaton, M. J . (1958). Monthly Notices Roy. Astron. SOC.118, 504. Seaton, M. J . (1961). Proc. Phys. SOC.(London) 77, 174. Seaton, M. J. (1966a). Proc. Phys. Soc. (London) 88, 801. Seaton, M. J. (1966b). Proc. Phys. SOC.(London) 88, 815. Sewell, K. G. (1965). J. Opt. SOC.Am. 55, 739. Shenstone, A. G . , and Russell H. N. (1932). Phys. Rev. 39, 415. Siegert, A. J. F. (1939). Phys. Rev. 56, 750. Silverman, S. M., and Lassettre, E. N. (1964). J . Chem. Phys. 40, 1265. Simpson, J . A. (1964a). Rev. Sci. Instr. 35, 1698. Simpson, J . A. (1964b). In “Atomic Collisions Processes ” (M. R. C . McDowell,ed.), p. 128. North-Holland Publ. Amsterdam. Simpson, J. A., and Fano, U. (1963). Phys. Rev. Letters 11, 158. Simpson, J. A., Mielczarek S. R., and Cooper, J. W. (1964). J . Opt. Soc. Am. 54. 269 Simpson, J. A., Chamberlain G. E., and Mielczarek, S. R. (1965). Phys. Rev. 139, A1039. Simpson, J. A., Menendez M. G., and Mielczarek (1966). Phys. Rev. 150, 76. Smith, K. (1966). Rep!. Progr. Phys. 29, pt. 11, 373. Smith, K., McEachran, R. P., and Fraser, P. A. (1962). Phys. Rev. 125 553. Smith, K., Henry, R. J. W., and Burke, P. G. (1967). Phys. Rev. 157, 51. Sommerville, W. B. (1966). Proc. Phys. SOC.89, 185. Stebbings, R. F., Fite, W. L., Hummer, D. G., and Brackman, R. T . (1960). Phys. Rev. 119, 1939. Sugar, R., and Blankenbecler, R. (1964). Phys. Rev. 136, B472. Taylor, H. S . (1967). Proc. Phys. Soc. 90, 877. Taylor, H. S., and Harris, F. E . (1963). J . Chem. Phys. 39, 1012. Taylor, H. S., and Williams, J. K. (1965). J. Chem. Phys. 42, 4063. Taylor, H. S., Nazaroff G. V., and Golebiewski, A. (1966). J. Chem. Phys. 45 2872. Temkin, A. (1966). “Autoionization” (A. Temkin, ed.) p. 55. Mono Book Co., Baltimore, Maryland. Weidenmiiller, H. A. (1964). Ann. Phys. ( N . Y.) 28, 60. Weiss, A. W. (1967). Private communication, t o be published. Westin, S. (1946). Kyl. Norske Videnskab. Selskabs Skrijter No. 2. Wheeler, J . (1937). Phys. Rev. 52, 1107. Whiddington, R., and Priestley, H. (1934). Proc. Roy. SOC.(London) A145,462. Wigner, E . P. (1948). Phys. Rev. 73, 1002. Wilson, W. S. (1935). Phys. Rev. 43, 536. W u , T . Y . (1934). Phys. Rev. 46,239.
This Page Intentionally Left Blank
RELA TIVISTIC INNER SHELL IONIZATION C. B. 0 . M O H R Department of Theoretical Physics, University of Melbourne Victoria, Australia
Introduction .................................................... 221 Relativistic Wave Functions .................... , . . . . . . . . . . . . . . . .,221 Inner Shell Energies .................. ..................224 K Ionization by Electrons ....................................... .226 A. Theoretical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 B. Results and Discussion ............................. 229 V. Ionization by Protons .... VI. Ionization by Photons . . . . . . . . . . . . . . . . 233 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .235
I. 11. 111. IV.
I. Introduction Although the quantum mechanical treatment of excitation and ionization of atoms by particles began nearly 40 years ago, much effort is still being made to improve the accuracy of calculated cross sections, even for the simplest atoms, hydrogen and helium. In treating inner shell ionization of heavy atoms, electron correlation effects are less important, but relativistic effects are considerable even for nonrelativistic energies of impact. Much work has been done in the last few years on relativistic inner shell ionization. Agreement between theory and experiment is fairly good for ionization by photons, less good for protons and electrons. The agreement needs to be improved and the work extended to higher energies. Lack of agreement is due, not to inaccuracy of wave functions, for these are now known to high accuracy, but to the difficulty of taking account of the complicating factors which are present, particularly when electrons are incident.
11. Relativistic Wave Functions The calculation of ionization probabilities requires knowledge of bound state and continuum wave functions of the necessary accuracy. The test of accuracy of bound state wave functions is the closeness of agreement of the 221
222
C. B. 0. Mohr
associated bound state energies with experimental ionization energies. Highly accurate wave functions are also necessary in accounting for relativistic effects in many-electron hyperfine structure (Coulthard, 1967a). For a single electron in a spherically symmetric field the four-component Dirac wave functions corresponding to states with j = I + and j = I - $, respectively, may be written
+
where X[rnis the total angular momentum eigenfunction given by 1/2
x:’,, = 1 C(L 3 , j ; nz - c,c)X,rn-u(Q, 4 ) ~ 1 / 2 , ~ , (2) a=-1/2 the spin functions x being written x,,2,1/2= (A), x1/2,-1/2= (7). Here F is the “ large component which tends to the Schrodinger wave function at low energies, and G the “small component” which tends to zero. The i is introduced before F i n (1) to make both F and G real for bound states and stationary continuum states. Differences in sign occur throughout the literature through arbitrary choice of the signs associated with F and G. Choice of the sign associated with the terms in the Dirac Hamiltonian determines whether components 1 and 2 or components 3 and 4 are the large components. For a complex atom, assuming no correlation between the motion of different electrons, the complete wave function for the system may be written as a product of single-particle Dirac functions, taking an antisymmetric combination of such products to allow for exchange. Then using the variational principle to minimize the total energy gives the Hartree-Fock selfconsistent field equations, which are a set of coupled integro-differential equations involving F, G, and their radial derivatives for each occupied state (Grant, 1961). The formulation has been extended to the case of negative ions with one electron in a continuum state (problem of electron scattering by a complex atom with exchange) (Walker, 1968). The solution of the Hartree-Fock equations requires the use of a fast digital computer, and has been carried out in this Department for the Hg atom (Coulthard, 1967b) and for the scattering of electrons by Hg (Walker, 1968). Less accurate boundstate calculations, sufficient for most purposes, have been made for many atoms by a number of workers (e.g. Herman and Skillman, 1963). Thus, accurate wave functions may be obtained for bound state and ejected electrons in numerical form. Analytic one-electron wave functions may be obtained by neglecting exchange and assuming a pure Coulomb field. We then have a pair of coupled differential equations of the first order in F(r) and G(r), whose solution may be expressed in terms of a linear combination of two confluent ”
RELATIVISTIC INNER SHELL IONIZATION
223
hypergeometric functions (Bethe and Salpeter, 1957a). For bound states the power series terminates. For continuum states the series is infinite, and, also, we have a sum of wave functions (1) over all orders 1 of partial waves from 0 to co. We refer to these later as Coulomb wave functions. The effect of inner screening may be taken into account by using a slightly reduced value of Z (Slater, 1930): Slater's rules give 2 - 0.30 for a K electron, but a relativistic treatment gives Z - 0.37 (Asaad, 1960). The effect of outer screening may be allowed for by adding a constant term to the Coulomb potential (Brenner and Brown, 1953). For a K electron in a Coulomb field, C(r)/F(r)is constant for all Y, and
F(r)/r = Nr" exp( - 2 r / u o ) ,
(3)
where s = (1 - 22/1372)'12- 1 ; for 2 = 80, s = -0.19. Thus, the Dirac wave function rises more rapidly than the Schrodinger wave function (for which s = 0) as Y decreases toward the nuclear radius. For 21137 small, (G/FI N +(Z/137), s N 0, and on inserting the explicit values of Y o , o , Y , ,o, Yl and Yl ,_1 , which occur for the case of j = I with 1 =0, it is easily seen that we may write the Dirac K electron wave functions in the following form, for m = and m = -*, respectively:
++
+
1
0 l h l a 2 m c i aZ
+i
\
1 *K,-1/2
=
. i h i a IN -- - - - i 2 m c i (ax
y:)
*K"
1 h l a
where t,bKs = N exp( - Z r / a o ) is the Schrodinger wave function. The forms (4) have been referred to as Darwin wave functions, for want of a better designation, since they are similar to the well-known plane wave solutions first derived by Darwin (1928), now to be quoted.
C. B. 0.Mohr
224
A simple approximate solution for continuum states of an electron in the field of an atom is the free-particle solution, which for m = 3 and m = -3 takes the respective forms :
+
where D = (mc2 E)/c, E is the kinetic energy plus the rest mass energy mc2, and &s is the Schrodinger plane wave solution = exp(ip
*
r/h).
(6)
Here p z , etc. denote the operators (h/i)(d/dz),etc., and for low velocities, D N 2mc when the spinors in (5) become identical with the spinors in (4). The corresponding solution for the continuum state of an (ejected) atomic electron in the field of the ion is (9,with given most conveniently by the Sommerfeld integral representation of the Coulomb wave function. The effective nuclear charge taken in the integral is that for the shell from which the electron came, although outside the atom the electron moves in a field of unit charge (Arthurs and Moiseiwitsch, 1958). The plane wave and Coulomb wave forms for &', of course, include all partial waves, so that the use of these Darwin wave functions avoids partial wave expansions in the calculation of matrix elements, but the approximation is good only for ( Z /137)2 < 1.
III. Inner Shell Energies The accuracy of calculated energy eigenfunctions and eigenvalues may be tested by comparison of calculated and observed electron binding energies. Observed binding energies have been obtained from the frequencies of emission lines and the frequency of one of the absorption edges, usually the L,,, edge. A comprehensive list of energy levels of nearly all the elements has been obtained in this way by Sandstrom (1957), but no error estimates are given. The absorption edges have a complicated structure, and the method is not as accurate as it was once thought to be. Recently, a more accurate method of electron spectroscopy has been adopted, in which a beta spectrometer is used to measure the energy of the photoelectrons ejected by a known x-ray line emission. The use of this method has shown (Fahlman et al., 1965) that some previous results for light elements obtained by x-ray absorption are seriously in error. The new method has not yet been applied to heavy elements.
225
RELATIVISTIC INNER SHELL IONIZATION
To indicate the accuracy of different methods of calculating inner shell energies of heavy atoms, we shall consider only the K-shell of the element Hg (2 = 80), which has been studied more than others. The following experimental values of the K ionization energy of Hg are available:
6107 k 0.6 Ry Ry 6121.7
(Saxon, 1954), (Sandstrom, 1957).
The following theoretical values are easily obtained : nonrelativistic, Coulomb field (2 = 80) relativistic, Coulomb field (Dirac equation) relativistic, with inner screening (Z,,, = 79.7) relativistic, with inner and outer screening
6400 Ry, 7068 Ry, 7012 Ry, 6098 Ry.
The last value, obtained without consideration of the detailed structure of the outer shell (Bethe and Salpeter, 1957b), is already close to the observed value, but this is fortuitous. A more accurate calculation must use the selfconsistent field method, with or without approximations. Naturally, the use of a perturbation method to obtain a relativistic correction to Hartree-Fock values is less satisfactory than a full relativistic Hartree-Fock treatment, and the Slater method of approximating the effect of exchange in a relativistic Hartree treatmdnt (“ Hartree-Slater”) is less satisfactory than a full relativistic Hartree-Fock treatment. A discussion of such methods, and comparison of results obtained with them, is given by Liberman et al. (1965). The most accurate results for the K-shell energy eigenualue for Hg are: relativistic Hartree-Slater (Liberman et al., 1965) relativistic Hartree-Fock (Mayers et al., 1959)
61 30 Ry, 6152 Ry.
However, the equality of the energy eigenvalue and the binding energy (Koopman’s theorem) is only approximate, for when a bound electron is removed the other electrons slightly change their energies. An approximate calculation (Mayers et al., 1959) of the rearrangement energy (which is of order mc2/137’ = 2 Ry) gives + 4 Ry for the case of the K ionization of Hg. A more satisfactory way of obtaining the binding energy is to calculate the total energy of all the electrons in Hg and the total energy of the ion Hg’, and subtract one from the other. Since two nearly equal quantities are being subtracted, the total energies must be calculated as accurately as possible. In this way the following value has been obtained for the binding energy: relativistic Hartree-Fock (Coulthard, 1967b) where the error estimate is stated to be
“
6145
8 Ry,
certainly excessive.”
C. B. 0. Mohr
226
Corrections must now be applied whose approximate values are estimated (Mayers et al., 1959) to be as follows: magnetic interaction of the K electrons Lamb shift finite size of nucleus correlation in angle of the K electrons
- 15
Ry, -38 Ry, -4 RY, ?
Applying these corrections to the best theoretical binding energies brings them below the experimental values. Still more accurate work is required to see if the discrepancy lies mainly in the experimental or in the theoretical results (particularly for the Lamb shift).
N. K Ionization by Electrons A. THEORETICAL
The probability per unit time for transition of the system, atom plus incident electron, from initial state i to final statefis given by
d P = (27-m l
(7)
where p is the number of final states per unit energy interval in the continuum for the ionized electron. The cross section for scattering of the incident electron naturally involves the same squared matrix element. V is the potential energy of interaction of the incident and target electron. For relativistic electrons in the field of a heavy nucleus, V is given, to the order of e2/4ntic, by (Brown, 1952): V(r12) = (e2/r12)(l
- a1 * a,)ex~(iIEi - E ~r12)r I
(8)
where we take ti = c = 1 here and later to simplify the formulas and a has as Cartesian components the Dirac matrices, a,, a,,, and a,. The term a1 a2 arises from the magnetic interaction of the two electron spins. Up to the present, no calculations of relativistic K ionization have attempted to include the complication of electron exchange. Interference between scattered and ejected eletrons decreases as the energy of the incident electron decreases (Mott and Massey, 1965a), and Perlman (1960a) has roughly estimated that for an incident energy ten times the K ionization energy in Hg, the contribution of exchange to the total cross section is less than 5%. Empirical evidence for little exchange is provided by the equality, within the experimental errors of 10 to 20%, of K ionization cross sections for electrons and positrons with energies between 0.24 and 1.13 MeV on Zr, Sn, W, and Pb (Hansen et al., 1964).
-
227
RELATIVISTIC INNER SHELL IONIZATION
The matrix element for the transition is therefore
[dft(1)$ft(2~3){V(r12) -I- V(r13)}di(1)$i(2>3) drI dr2 dr3
9
(9)
where 1 denotes the incident electron, and 2 and 3 denote the K-shell electrons. Existing calculations have used the plane wave Born approximation to avoid extreme complexity, taking Dirac plane waves (5) with (6), with pi and pf in + i ( l ) and 4f(l), respectively. If the column matrix in (5) is denoted by a, so that at is a row matrix, substituting for 4 i , c j f , and V in (9) and integrating over rl gives, on putting q = pi - ps:
1 exp(iq - r,) 3
47re2NsNi{q2- (Ei - E f ) 2 } - ’
v=2
The antisymmetrized t+hi and $f have the form
Substituting these in (10) gives a number of terms involving the following single-particle matrix elements with Dirac wave functions :
[
4 C t (r)$K(r)exp(iq
Sgct(r)a$,(r)exP(iq
*
r, *
r) dr.
( W
(12b)
Perlman showed it was possible to evaluate (12) bytakingfor$, theCoulomb wave function (3), for $c the partial wave expansion for Dirac Coulomb waves, and for exp(iq * r) the partial wave expansion. The angular integrations are readily performed, and the radial integrals may be expressed as a sum of several slowly convergent complex infinite series. However, he found that carrying out the subsequent calculations was a formidable task. He therefore performed the radial integrations numerically after computing Dirac Coulomb wave functions for the ejected electron and numerical values of the Dirac K electron wave function (3), and using tables of spherical Bessel functions (which are involved in the partial wave expansion of exp(iq r)). A considerable number of integrations was required, and K ionization cross sections for Hg were obtained within an error of 5%. At that time, relativistic self-consistent field wave functions were not available. For light atoms and low energies one may use the simpler Darwin wave functions (4) for $ K , and (5) for &, taking for &’ the Sommerfeld integral representation of a Coulomb wave. Then the Dirac matrix elements (12) are
-
C . B. 0. Mohr
228
expressible in terms of a number of products of Schrodinger-type matrix elements E , ex’, ey‘, and eZ’, given by e(q) =
e,’(q) =
s
&’*
J
$Ks
-
exp(iq r) dr,
0
- $Ks exp(iq r) dr.
These integrals were evaluated analytically by Perlman to obtain “semirelativistic” K ionization cross sections for Ni and Hg for comparison purposes. Imposing the more restrictive conditions (Z/137)2 4 1 and qa,/137 6 1 for those values of q contributing significantly to the ionization cross section, leads to a still simpler expression for the Dirac matrix element (9) which involves the following term (Mnrller, 1932): { q 2 - (Ei
- E,)’}-’
{E
+ iPez’},
(14)
where fi = vjc. This expression has been evaluated numerically by Arthurs and Moiseiwitsch (1958). If one takes V = e2/r12in place of (8), the term (Ei - E,)2 does not appear in (14), and at low energies the magnetic interaction term ljeZ‘ is small compared with the Coulomb interaction term e, which is the usual nonrelativistic matrix element (Mott and Massey, 1965b). The numerical evaluation of E was first carried out by Burhop (1940) for K- and L-she!l ionization of Ni, Ag, and Au. The calculations of Arthurs and Moisewitsch extend only up to 20 times the ionization energy. Those of Perlman for Hg extend up to 1.5 MeV, beyond which convergence difficulties greatly increase the labor of calculation. It would be of great interest to have accurate results at higher energies, but here one has only the time-honored work of Mnrller and Bethe, which leads to the following form for the cross section cr for ionization of a particular shell (Mott and Massey, 1965c),
where the constant Z is the so-called mean ionization energy of the shell, and the constant A involves the square of the matrix element for electric dipole transitions, and arises from the term iq r in the series expansion of exp(iq r) in (13a). The approximations made in deriving this result include the assumption that Z/137 is small, but even for heavy elements, at high energies one expects an increase of cr with energy, as indicated by the presence of the logarithmic term in (15); for it is due partly to the Lorentz extension of the
-
RELATIVISTIC INNER SHELL IONIZATION
229
transverse field (which increases distant collisions), and partly to increase in maximum energy transfer (which increases close collisions). The relativistic rise is reduced through the polarization of the medium by the moving charge (density effect), the reduction setting is just a little beyond the minimum in the cross section (Sternheimer, 1952). The magnitude of the rise obtained in any experiment depends on the upper limit of momentum transfers observed in the experiment. The methods adopted in deriving (15) have been used to derive well-known expressions for specific primary ionization and rate of energy loss, but the expressions will not be reliable in estimating the contribution from the inner shells of heavy atoms, though this contribution will be largely swamped by the contribution from the outer shells. The relativistic theory of specific primary ionization and stopping power has been formulated and applied to helium (Perlman, 1956). Also, results for relativistic K ionization have been used to obtain values of the relativistic K-shell stopping power of Hg (Perlman 1960b), and these turn out to be markedly different from the nonrelativistic values.
B. RESULTS AND DISCUSSION Measurements of K ionization cross sections fall into two main groups: those made before 1947, using an ionization chamber to measure the intensity of the x rays produced; and those made since 1964, using a NaI scintillation spectrometer. The earlier measurements were made on 28Ni and 47Ag at energies up to 0.18 MeV. They have been discussed (Motz and Placious, 1964) and compared with the nonrelativistic calculations of Burhop and the semirelativistic calculations of Arthurs and Moiseiwitsch. For 28Ni, the experimental values fit the curve of cross section versus energy obtained by Arthurs and Moiseiwitsch, but not that obtained by Burhop. For 47Ag, the experimental values do not fit the theoretical curves of either (though the differences are not large), so that the effect of relativity is clearly observed for Z = 47, even for electron energies much smaller than me2. Recent measurements have been made by Motz and Placious (1964) for 5oSn and 7 9 A ~by , Hansen et al. (1964) for 4oZr9 5oSn, 74W, and 82Pb, by Hansen and Flammersfeld (1966) for 47Ag, 5oSn, 74W, 7 9 A ~and , szPb, and by Rester and Dance (1966) for 47Ag, 5oSn, and 7 9 A ~for , energies in the range 0.1-2 MeV. The values obtained by these observers for 5oSn and 7 9 Aare ~ shown in Fig. 1, together with the theoretical values of Arthurs and Moiseiwitsch for these two elements, and of Perlman for ,,Sn and 80Hg. The values shown for Hansen et al. for 7 9 A at ~ the higher energies were obtained by interpolation using their values for Z = 40, 50, 74, and 82.
C . B. 0.Mohr
230
--
I -
- 60 T E -n w
1
I
T
I
I
I
T
i
C
0 * 0
w
40~ 0
c
P
-
0
N
0 C -
Y
T
I
20.
05
10
I!
Electron energy (MeV)
FIG.1. Illustrating the variation of K ionization cross section with energy of the incident electron for and 7 9 A ~The . experimental values are shown by points (Hansen and Flammersfeld, 1966), circles (Motz and Placious, 1964), and triangles (Rester and Dance, 1966). The theoretical values are shown by full lines [semirelativistic (Arthurs and Moiseiwitsch, 1958)], crosses [semirelativistic (Perlman, 1960a) for soHg], and a broken line [fully relativistic (Perlman, 1960a) for 80Hg].
The experimental values of Rester and Dance for Sn agree with those of Motz and Placious where they overlap, and both agree with the calculated semirelativistic values of Arthurs and Moiseiwitsch; and this is a reasonable result, since (Z/137)2for Sn is 0.13. But the results of Hansen et al. are considerably different, and this makes them suspect. Moreover, if such large errors can occur in one set of the recent experiments, can they not occur in the others? For Au the results of Motz and Placious agree best with the calculations of Arthurs and Moiseiwitsch, and the results of Rester and Dance best with the fully relativistic values of Perlman; and the results of Hansen et al. are again considerably different. There is, however, no good reason why the experimental results should agree with the semirelativistic calculations of Arthurs and Moiseiwitsch, since (Z/137)2 for Au is 0.33, which is no longer small compared with 1, so that any such agreement is fortuitous. Thus, the semirelativistic calculations of Perlman using a more accurate formula give
RELATIVISTIC INNER SHELL IONIZATION
23 1
values much greater than those of Arthurs and Moiseiwitsch. The fully relativistic values of Perlman required computations of considerable complexity, some of which were performed with the aid of a computer which would now be regarded as slow and of low capacity. Further calculations for the other values of Z, and also for higher energies, are highly desirable. Some calculations are now in progress in this Department. The form of the relativistic increase in ionization with energy, as given by (15) for light elements with due modification for the density effect, is supported by measurements of grain density of tracks in nuclear emulsions (e.g. Stiller and Shapiro, 1953). All of the calculations are based on the plane wave Born approximation, whose validity in this problem is perhaps open to doubt. Only a few of the partial waves comprising the incident and scattered waves overlap appreciably with the K-shell wave function, and the phase shifts of these waves are never small compared with 1. Using an approximate method (Mohr and Tassie, 1954) of calculating the phase shifts q1 for a screened Coulomb field V = -Ze2exp( -h ) / r , we find that
= (W37P) ln(137PylXl + 3)),
(16) where y = (1 - P2)-1/2. It follows that, as the energy increases falls to a minimum and then rises slowly, the rise being due to relativity. For Z = 80, A N 310, and qo , q l , and q z have minimum values of about 3.5, 2.6, and 1.7, respectively. We have, however, overstated the case for using the distorted wave Born approximation. The criterion is not the magnitude of the phase shifts of the waves at large distances, but at distances in the region of the K-shell where they are appreciably less; but even here the phase shifts are of the order 1 for the first couple of partial waves. It may well be that the differential cross section for K ionization is more sensitive to wave distortion than the total cross section, and only the latter is measured. Hansen et al. find that the total cross section for heavy elements is the same for electrons and positrons within the experimental errors of between 10 and 20%, suggesting that the effect of distortion on the total crosssection does not exceed this amount. The calculation of accurate K ionization cross sections for relativistic electrons on heavy elements is clearly full of complications and difficulties, but the results provide a useful test of the interaction between relativistic electrons as given by (8). 'I1
V. Ionization by Protons Experiments on inner shell ionization by protons have been confined to energies below 3 MeV, for which v/c < 1. It might be expected, therefore, that the nonrelativistic theory would apply. In place of (8) one has the simpler
232
C. B. 0. Mohr
expression V(r,,) = e2/rI2,and antisymmetrization is no longer necessary as it was for incident electrons. Although the proton waves are strongly distorted, a large number of partial waves is involved, and it turns out that the use of the plane wave Born approximation seems reasonably justified for large Z (Henneberg, 1933). Then the transition matrix element in (7) reduces to E ( q ) as given by (13a) with Schrodinger Coulomb wave functions, and is evaluated by expanding in partial waves. However, the corresponding relativistic matrix element (12a) with Dirac Coulomb wave functions has been found (Jannik and Zupancic, 1957) to give K ionization cross sections larger than the nonrelativistic ones by a considerable factor in the case of heavy elements. For 2 to 3-MeV protons on *,Pb, this factor has values 4.3-3.5. Similar differences may be expected to occur between the relativistic and nonrelativistic K stopping power of heavy elements for protons. Experiments on K ionization by protons in the above energy range, using a NaI scintillation counter for detection of the K x ray photons (Merzbacher and Lewis, 1958), gave cross sections for Ta and U which agreed fairly well with the nonrelativistic (Born approximation) values. Earlier experiments (Lewis et al., 1953) gave cross sections for Au and Pb which were larger than the nonrelativistic by factors which increased from about 2 to 4 between 2 and 3 MeV, and which were thus in rough agreement with the relativistic (Born approximation) values ; but these experiments are stated (Merzbacher and Lewis, 1958) to be “less reliable.” More recent experiments (Khan el al., 1965) on K ionization using proportional counters have been confined to light elements, for which the observed cross sections differ from the nonrelativistic values by a factor of nearly 2 at 1.9 MeV, and are less by factors which increase to about 20 as the energy decreases to 30 keV. (It is to be noted that the experimental cross sections decrease with decreasing energy by several orders of magnitude in this energy range.) However, the absolute magnitude of the experimental cross sections depends on the value adopted for the fluorescent yield, and this quantity is not known with any accuracy for some light elements. Thus, for Al, Khan et al. (1965) had to choose a mean fluorescent yield from experimental values of 0.008, 0.038, and 0.045. Khan et al. (1965) have obtained cross sections for L ionization of the elements 2 = 60-67 by 0.5-1.7 MeV protons which agree reasonably well with those given by the nonrelativistic theory. But Khan et al. (1966) have obtained L ionization cross sections for Cu which are only about one-tenth of the theoretical ones throughout the range 0.025 -1.7 MeV, though part of the discrepancy may be due to inaccuracy in the value adopted for the fluorescent yield. What differences in the theoretical values may be produced by the use of relativistic L-shell wave functions is not known.
RELATIVISTIC INNER SHELL IONIZATION
233
Existing calculations have not given a satisfactory account of K- and L-shell ionization of light elements by protons, the use of the Born approximation being least justified for light elements and low energies. A more accurate method will decrease the calculated cross sections, and so reduce the discrepancy with experiment. The use of relativistic K-shell wave functions considerably increases the K ionization cross section for heavy elements, but this effect has yet to be confirmed by experiment.
VI. Ionization by Photons The differential cross section (relativistic) for photoionization involves the square of the matric element
s
dcta * e exp(ik * r)$o dr,
(17)
where t,bo is the Dirac wave function for the initial or bound state of the electron, k denotes the momentum of the incident photon, and e is a unit vector specifying the direction of polarization of the photon. The form (17) is exact, except for neglect of radiative corrections. The retardation term exp(ik r) is associated with the electromagnetic wave, and the complications of wave distortion and of integrating over the separate coordinates of incoming and scattered particle do not arise as they do for incident electrons and protons. The incident photon, in effect, possesses a wave function exp(ik * r), and has a zero-range interaction with the electron. The problem of photoionization is, therefore, formally a simpler one than that of ionization by a material particle, and accordingly one expects better agreement between theory and experiment. The nonrelativistic ionization cross section involves (17) with Schrodinger Coulomb wave functions and with p in place of a essentially because p and a occur, respectively, in the nonrelativistic and relativistic expressions for the current density. The cross section was obtained in closed form for the Kand L-shells by Stobbe (1930) by neglecting retardation, i.e. by putting exp(ik r) = 1, and this takes into account only electric dipole transitions. Retardation has been taken into account in subsequent calculations, usually relativistic, by expanding exp(ik * r) and in partial waves. Not long after Stobbe's work came a group of relativistic calculations for the K-shell. The first was due to Sauter (1931), who made the approximation of taking (12 - (Z/137)2)'/2= 1 in the Coulomb radial wave functions for t,bK and dct, and obtained closed formulas. Hulme et al. (1935) made numerical calculations for a few particular energies and elements. Hall (1936)
-
-
&'
234
C . B. 0. Mohr
obtained the high energy limit for arbitrary Z . On the basis of these limiting and special cases, Grodstein (1957) used extrapolation procedures to obtain values of the cross section for arbitrary energy and Z . These results are generally in good agreement with experiment. Since then, many other calculations have been carried out, nearly all of them for a Coulomb field, and these are discussed briefly in the more recent and accurate work to which we now refer. The most complete work is that of Pratt et al. (1964), who have made numerical calculations of $ K and & to obtain K-shell cross sections and photoelectron angular distributions for several photon energies between 0.2 and 2 MeV, and Z values between 13 and 92, permitting interpolation throughout these ranges. Similar calculations for the K-shell have been made by Hultberg et al. (1962) for energies between 0.12 and 0.5 MeV for Z = 82 and 92, and by Alling and Johnson (1965) for the K-/L-shells for energies between 0.08 and 1.33 MeV for Z = 82 and 92. The total cross sections obtained in these three recent calculations agree closely amongst themselves where comparison is possible. However, the results fall significantly below the extrapolated values in the NBS tables of Grodstein and the experimental values, the discrepancy increasing from a few percent at 1 MeV to as much as 25% at 2 MeV. There seems to be no theoretical explanation of this remarkable discrepancy. It cannot arise from screening, for this reduces the calculated cross section (Hall and Sullivan, 1966) and so increases the discrepancy ; also, this reduction decreases with increasing Z , and is only about 2% for heavy elements. Much larger discrepancies occur between K-shell theoretical and experimental differential cross sections, but here the experimental data are meager, many more being needed to explore the situation. For heavy elements, the nonrelativistic theory is, of course, seriously in error, the Sauter differential cross sections differing from those of Pratt et al. at some angles by a factor of 2. The ratios of the total cross sections for 9 2 U for the L , , L,, , and L,,, subshells and the K-shell, calculated by Alling and Johnson for energies up to 1.33 MeV, differ from experimental values in this energy range by 10 to 20%, but experimental errors are often nearly as large. Their calculated values for the ratio of the sum of the L,, and L,,, cross sections to the L, cross section differ from Stobbe’s nonrelativistic values by a factor which increases from 1.6 at 0.08 MeV, to 8.2 at 1.33 MeV. It is clear that the nonrelativistic theory of inner shell photoionization is completely inadequate for heavy elements. The relativistic theory gives much more satisfactory results, but there remains unexplained the discrepancy between the recent calculated results on the one hand, and the experimental results and the earlier calculated results on the other.
RELATIVISTIC INNER SHELL IONIZATION
235
REFERENCES Ailing, W. R., and Johnson, W. R. (1965). Phys. Rev. 139, A1050. Arthurs, A. M., and Moiseiwitsch, B. L. (1958). Proc. Roy. Soc. (London) ,4247, 550. Asaad, W. N. (1960). Proc. Phys. SOC.76, 641. Bethe, H. A., and Salpeter, E. E. (1957a). In “Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 35, Sec. 14 and 15. Springer, Berlin. Bethe, H. A., and Salpeter, E. E. (1957b). In “Encyclopedia of Physics” (S. Flugge, ed.), Vol. 35, p. 173. Springer, Berlin. Brenner, S., and Brown, G. E. (1953). Proc. Roy. SOC.(London) A218,422. Brown, G . E. (1952). Phil. Mag. 43,467. Burhop, E. H. S. (1940). Proc. Cambridge Phil. SOC.36,43. 90, 615. Coulthard, M. A. (1967a). Proc. Phys. SOC. Coulthard, M. A. (1967b). Proc. Phys. SOC.91, 44. Darwin, C. G. (1928). Proc. Roy. SOC. (London) A118, 654. Fahlman, A., Hamrin, K., Nordberg, R., Nordling, C., and Siegbahn, K. (1965). Phys. Rev. Letters 14, 127. Grant, I. P. (1961). Proc. Roy. SOC.(London) A262, 555. Grodstein, G. W. (1957). Natl. Bur. Std. (US.), Circ. 583. Hall, H. (1936). Rev. Mod. Phys. 8, 358. Hall, H., and Sullivan, E. C. (1966). Phys. Rev. 152, 4. Hansen, H., and Flammersfeld, A. (1966). Nucl. Phys. 79, 135. Hansen, H., Weigmann, H., and Flammersfeld, A. (1964). Nucl. Phys. 58, 241. Henneberg, W. (1933). Z. Physik 86, 592. Herman, F., and Skillman, S. (1963). “ Atomic Structure Calculations.” Prentice-Hall, Englewood Cliffs, New Jersey. Hulme, H. R., McDougall, J., Buckingham, R. A. and Fowler, R. H. (1935). Proc. Roy. SOC. (London) A149, 131. Hultberg, S., Nagel, B., and Olsson, P. (1962) Arkiv Fysik 20, 555. Jannik, D., and Zupancic, C. (1957). Kgl. Danske Videnskab. Selskab, Mat. Fys. Medd. 31, No. 2. Khan, J. M., Potter, D. L., and Worley, R. D. (1965). Phys. Rev. 139, A1735. Khan, J. M., Potter, D. L., and Worley, R. D. (1966). Phys. Rev. 145, 23. Lewis, H. W., Simmons, B. E., and Merzbacher, E. (1953). Phys. Rev. 91, 943. Liberman, D., Waber, J. T., and Cromer, D. T. (1965). Phys. Rev. 137, A27. Mayers, D. F., Brown, G. E., and Sanderson, E. A. (1959). Phys. Rev. Letters 3, 90. Merzbacher, E., and Lewis, H. W. (1958). In ‘‘ Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 34, p. 166. Springer, Berlin. Mohr, C. B. O., and Tassie, L. J. (1954). Proc. Phys. SOC.A67, 711. Mgller, C. (1932). Ann. Physik 14, 531. Mott, N. F., and Massey, H. S. W. (1965a). “The Theory of Atomic Collisions” 3rd ed., p. 492. Oxford Univ. Press, London and New York. Mott, N. F., and Massey, H. S. W. (1965b). “The Theory of Atomic Collisions” 3rd ed., p. 477. Oxford Univ. Press, London and New York. Mott, N. F., and Massey, H. S. W. (1965~).“The Theory of Atomic Collisions” 3rd ed., p. 815. Oxford Univ. Press, London and New York. Motz, J. W., and Placious, R. C. (1964). Phys. Rev. 136, A663. A69, 318. Perlman, H. S. (1956). Proc. Phys. SOC.
C.B. 0.Mohr Perlman, H. S. (1960a). Proc. Phys. Soc. 76, 623. Perlman, H. S. (1960b). Proc. Phys. Soc. 76,433. Pratt, R. H., Levee, R. D., Pexton, R. L., and Aron, W. (1964). Phys. Rev. 134, A898. Rester, R. H., and Dance, W. E. (1966). Phys. Rev. 152, 1. Sandstrom, A. E. (1957). In “Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 30, p. 78 Springer, Berlin. Sauter, F. (1931). Ann. Physik 11,454. Saxon, D. (1954). Thesis, Univ. of Wisconsin, Madison, Wisconsin (unpublished). Slater, J. C. (1930). Phys. Rev. 36, 57. Sternheimer, R. M. (1952). Phys. Reu. 88, 851. Stiller, B., and Shapiro, M. M. (1953). Phys. Rev. 92, 735. Stobbe, M. (1930). Ann. Physik, 7, 661. Walker, D. W. (1968). To be published.
RECENT MEASUREMENTS ON CHARGE TRANSFER J. B. HASTED Department of Physics, University College
London, England Introduction .................................................. 237 Total Cross Sections for the Symmetrical Resonance Process . . . . . . . ,237 Total Charge Transfer Cross Sections for Unlike Ions and Atoms ... ,242 Differential Scattering with Capture ............................. .243 Pseudocrossing of Potential ................246 Molecular Charge Transfer . . . . . . . . . . . . . .248 Experimental Techniques . ...................249 Role of Excited Species . . . . . . . . . . . . . . . . . ..254 A. Experiments with Excited and Ground State Ions ............... ,254 B. Processes Terminating in Excited Species ........... IX. Miscellaneous Topics ........................................... .259 A. Ionization with Capture ..................................... .259 B. Radiative Charge Transfer ................................... ,261 C. Two Electron Capture Processes ............................. .261 References ..................................................... .263
I. 11. 111. IV. V. VI. VII. VIII.
I. Introduction This article discusses recent experimental research carried out on charge exchange collisions at low and moderate impact energies. The intention is to concentrate on the features which are closest to the center of understanding in terms of quantum theory.
II. Total Cross Sections for the Symmetrical Resonance Process
At intermediate impact velocities, there exist valuable generalized treatments using semiempirical orbitals (Firsov, 1951; Rapp and Francis, 1962). The derived cross section 0 at impact velocity u has the general form
where a and b are constants. Little relevant laboratory work has been done since the comparison with experimental data made by Rapp and Francis (1962). However, there are still discrepancies between different measurements, 237
238
J. B. Hasted
sufficientto make the comparison a complex matter. One important source of error arises in the McLeod pressure gauge which is almost always used as an absolute standard. Ishii and Nakayama (1961) have pointed out that the mercury vapor stream in the gauge acts as a diffusion pump, so that the measured pressure p m is low by an amount Ap given by dp/pm= 0.905rpH,(T/D)'/2
(2) where pHgis the vapor pressure (Torr) of mercury, r the inside radius of the tube connecting to the gauge, D the gas diffusion coefficient, and T the gas temperature. This formula has been verified experimentally within limits. Measurements can now be made with commercial refrigerated gauges with pHg so low that Ap is negligible. The early symmetrical resonance charge transfer measurements should be repeated. It may be noted that the errors are, in general, higher for the heavier gases.' As McLeod gauge tube diameters have tended to be larger in recent years, some old measurements may be the best available. It is possible to rearrange the Firsov and RappFrancis formulations in such a way that a direct comparison may be made between them (Lee, 1967). This arrangement leads to the graphical representation of cross-section functions shown in Fig. 1, in the form yp,(log,, v), where the cross section is a = +npI2
(3)
y = (Ei/13.6)'/2
(4)
with2
Ei being the ionization potential in electron volts. The symmetrical resonance cross section at a given impact velocity is inversely proportional to the ionization potential in the Rapp-Francis, but not in the Firsov formulation. This dependence appears in the logarithmic plot shown in a previous article (Hasted, 1962). In the Firsov formulation the cross sections of low ionization potential atoms are higher than in the RappFrancis formulation. A search for systematic deviation shows that cross-section functions for high ionization potential E j agree with theory within the experimental error; but as the ionization potential decreases the measured cross sections become progressively larger than predicted. The form of Eq. (1) is in agreement with experiment. The position with regard to the parameter b is uncertain. However, the experimental parameter a, for low ionization potential atoms, is undoubtedly larger than given by the Rapp-Francis or even the Firsov formulation. This judgment is based on a number of experiments (Kushnir et a[., An error of 50% is possible. Note that y (Rapp and Francis, 19621, which is used here, is equal to y-' (Firsov, 1951).
RECENT MEASUREMENTS ON CHARGE TRANSFER
239
16 -
14 -
12 -
I
10-
0
n
8-
6-
4-
FIG. 1 . The function ypl(log,, v ) , which leads directly [Eqs. (31, (4)] to symmetrical resonance charge transfer cross sections in both Firsov (thin line) and Rapp-Francis (thick line) formulations. In the Firsov formulation, the function is dependent on y , while in the RappFrancis formulation, it is not.
1959; Marino et al., 1962; Chkuaseli et al., 1963; Edmonds and Hasted, 1964; Palyukh, 1967). The conflict with theory is not obscured by the considerable discrepancies existing between the measurements. In electron capture by an atom from its negative ion, the electron moves in the relatively weak fields of two atoms. The effect of this is seen, in Fig. 3, to result in enhanced cross sections even for nonresonance collisions, since the small value of y not only raises the cross section by virtue of the relative invariance of y p I with y (bearing in mind that the cross section is trip,'), but also by virtue of the fact that in the Firsov formulation ypI is actually large for small values of y. Use of this formulation has enabled the electron affinities E, (eV) of alkali metals to be inferred from the charge transfer cross sections of their negative ions in their own vapor (Bydin, 1964). The inferred values are: Na, 0.41 eV; K, 0.22 eV; Rb, 0.16 eV; Cs, 0.13 eV. However, the possible errors are rather large. The affinities derived from collisional detachment measurements (Bydin, 1966) using the theory of Smirnov and Firsov (1964) should be more accurate.
J. B. Hasted
240
0
lo4 2
5
to5
2
5
lo6 2
5
10'
2
5
lo8
V Icrn/sec)
FIG.2. Symmetricalresonance charge transfer and mobility data displayed as o'/*(logl0 v). Broken lines represent calculations from Rapp-Francis (1962) formulation and from Eq. (5). Experimental data are represented as full lines and points. (1 967) A M Mahadevan G H Gilbody and Hasted (1956) (1957) (1954, 1956) F Fedorenko et al. BC Biondi and Chanin (1966) FS Flaks and Solov'ev (1958) P Patterson (1955) (1953) D Dillon et al. Ziegler 0 Z G Ghosh and Sheridan (1 957) Nichols and Witteborn (1966) 0N (1959) (1964) KLS,S Kushnir et al. E Edmonds and Hasted
An interesting problem arises in symmetrical resonance charge transfer at very low impact energies where the approximation of rectilinear motion is unsatisfactory (Edmonds and Hasted, 1964). As the impact parameter decreases, at a fixed impact velocity u, the polar scattering angle increases until a critical impact parameter p c is reached, when the system passes into stable orbiting. Within this impact parameter, inward spiraling orbits occur. Rapp and Francis (1962) have proposed that for sufficiently small impact velocities p c > p1 [of Eq. (3)], so that for p < p c the system will spiral inwards
24 1
RECENT MEASUREMENTS ON CHARGE TRANSFER
until p < p l , when the probability of charge transfer becomes one-half. The cross section is thus
a = +zp, =
-(-)
ne a U
'I2
(5)
P
This may be written a=
4.6 x lo-''
(;)'I2
cm2
U
where the velocity vis in centimeters per second, the polarizability a is in atomic units, and the reduced mass p is on the chemical scale. At impact velocities such that p1 > p c , the cross section remains approximately +npI2.The extent to which the experimental data justify these proposals is shown in Fig. 2. The charge transfer cross section is displayed in the form a'12(v), so that a linear fall is expected on the basis of Eq. (1). Reduced mobility measurements (Biondi and Chanin, 1954; Patterson, 1966) at low field strength to pressure ratio, but variable gas temperature, are converted to average diffusion cross section and thence to charge transfer cross section (Dalgarno et al., 1958). Beam measurements have, in general, been selected only from among the most recent. However, those due to Gilbody and Hasted (1956) have been included because the McLeod gauge employed is known and the laboratory temperature can be estimated so that fractional pressure corrections may be made.
0
106
2
4
7
10'
Velocity
2
4
7
108
(crnlsec)
FIG. 3. Negative ion nonresonant total charge transfer data (Snow,1966). 0 , O - H; 0 ,H - 0 ; 0, C - H ; 8 , C - 0 .
J. B. Hasted
242
The interest of the displayed a”2(u) functions lies not so much in the agreement with theory, but in the demonstration that orbiting can usually only contribute significantly at low temperatures. Charge transfer experiments using molecular ions and their parent neutral molecules are not measurements of the pure symmetrical resonance process, because neither the vibrational state distributions of the ion nor of the molecule formed from it are specified. Detailed analysis will not be possible until vibrational state population diagnosis is achieved. In the meantime, it would be unwise to expect the general theoretical treatment to be precisely applicable to the experiments. Recent measurements include those for N2’N2 by Nichols and Witteborn (1966). Molecular charge transfer processes at very low energies will be considered in Section VI.
III. Total Charge Transfer Cross Sections for Unlike Ions and Atoms Considering only the initial and final states of a colliding ion and unlike atom, approximate but realistic calculations of the total charge transfer cross section can be made. In general, the adiabatic criterion (Massey, 1949) applies fairly well. According to this, the probability of charge transfer is small if the time of transition h/lAE I is much shorter than the time of collision a/v (where A E is the energy separation of the states A’B and B’A at infinite nuclear separation, v is the impact velocity, and a is the “adiabatic parameter”). When the two times are comparable, the probability can be large. A successful experimental test was made (Hasted, 1951) with the arbitrary assumption that the cross section function rises with increasing v to a maximum value when h/lAEl is equal to a/u. For a large number of typical single charge transfer processes, it was found that this assumption was consistent with an invariant adiabatic parameter equal to 7 A. The application of this “ adiabatic maximum rule” to other types of inelastic heavy particle collision, such as ionization, is reasonably successful, although for different reasons (Hasted and Lee, 1962; Hasted, 1964). Since the energy separation of the initial and final potential curves does not remain constant during the collision, the replacement of AEm calculated at finite internuclear distance by a value suitably averaged over the collision region, results in improved correspondence with the data (Lee and Hasted, 1965). The effect may be very striking. Thus the cross section for
=
He2+
+ H+He+(2s, 2p) + H + ,
(7) which is in exact energy resonance, passes through a maximum at about 30 keV (Fite et al., 1962).
RECENT MEASUREMENTS ON CHARGE TRANSFER
243
The general intermediate impact velocity features of the unlike ion-atom charge transfer function are reasonably consistent with the results of a set of approximate, but generally applicable, calculations due to Rapp and Francis (1962). Tables of functions from which any cross section can readily be derived are available (Lee and Hasted, 1965). The predicted cross section functions are v4 dependent at very low energies, and rise to maxima when the collisions cease to be adiabatic. A relation yielding v,,, , the impact velocity for maximum cross section, similar to the adiabatic maximum rule, but containing no empirical adiabatic parameter, can be deduced (Lee and Hasted, 1965) from the RappFrancis equations, as follows : -
uiaX==2.25 x
lOI4
Y
2.5
+ 2 log,,
)
Ei (cm/sec>Z IAEI
(8)
The dependence of u,,, on y might possibly be inferred from recent negative ion data (Snow, 1966), some of which are displayed in Fig. 3. Here the apparent adiabatic parameter a is about I8 A. Further deductions concerning the dependence of v,,, on AE have recently been presented by Drukarev (1967). In interpreting intermediate velocity charge transfer collisions, the two atomic systems should be considered in terms of the states of the quasimolecule they compose. The comparison of O'H, H'H data (Stebbings et al., 1964) is consistent with this approach, and so are rare gas data taken at high energies (Lee and Gilbody, 1964; Gilbody et al., 1963). It is even possible to make inferences concerning the type of angular momentum coupling applicable to the quasimolecule (Edmonds and Hasted, 1964).
IV. Differential Scattering with Capture On the two-state impact parameter approximation, the probability Po of symmetrical resonance charge transfer considered as a function either of p or of v, oscillates between zero and unity (cf. Bates and McCarroll, 1962). The oscillation is thus observable both in collisions at fixed polar scattering angle and varying impact velocity, and in collisions at fixed impact velocity and varying polar scattering angle. Such experiments were first carried out by Everhart and his colleagues (Ziemba et al., 1960; Everhart et al., 1964). Some important data are illustrated in Figs. 4a and 4b. For collisions between unlike ions and atoms, the maxima of the oscillations are not constant at unity, but decrease with increasing p (Fig. 4b). Even for symmetrical resonance collisions, the probabilities do not reach zero or unity. This is because of wave effects (Massey and Smith, 1933; Smith, 1964) and because of coupling to other states (Bates and Williams, 1964).
N P P
FIG.4. Data for differential scattering with capture. (a) Symmetrical resonance capture probability for H+ on H at 0=3", as a function of 0-l. (b) Some resonance and nonresonance capture probabilities. (c) Total charge transfer cross section functions for cesium and rubidium (Perel et ul., 1965). 0 , Rb+ C S ; 0, Cs+ Rb; x, Rb+ Rb; 0, Cs+ Cs. 1
0
1
1
1
1
1
100 Reclprocol veloclly.
FIG.4a.
1
1
I
200
IdB sec/rneler
I
I
I
300
RECENT MEASUREMENTS ON CHARGE TRANSFER
C
-
245
246
J. B. Hasted
Under certain circumstances, oscillating behavior is found in total charge transfer collision functions, both symmetric and asymmetric (Perel et al., 1965; Smith, 1966). Typical data are shown in Fig. 4c. Everhart's experiments on differential scattering with capture involve a measurement, at the same polar scattering angle, of both charged and neutral components. However, it is possible to obtain information about charge transfer (except in the adiabatic region) from measurements only of the differential scattering of the charged particles (Lorents and Aberth, 1965; Marchi and Smith, 1965). The structure in the scattering functions arises from rainbow effects, from nuclear interchange, and from interference between waves scattered from the symmetric and antisymmetric potentials of the quasimolecular ion.
V. Pseudocrossing of Potential Energy Curves When the initial and final interaction energy functions approach each other and pseudocross, at nuclear separation R, , then anomalous large total cross sections at low impact velocities are to be expected. Such a situation is found in exothermic single electron capture by a multiply charged ion: An+ + B+A(n-I)+ + B +
(9) for which the Coulomb repulsion between the products greatly exceeds the polarization attraction between the collidants. Even in such processes as H+
+ Xe +H(ls) + Xe+
(10) pseudocrossing is found. It is possible that antibonding interaction curves might take part in pseudocrossing and produce large cross sections at low impact energies. The earliest approximation by which pseudocrossing was treated is that due to Landau (1932) and Zener (1932). Using this approximation and simple wave functions, calculations have been carried out on some processes which have been studied in the laboratory (Hasted et al., 1964; Flaks and Ogurtsov, 1963). The comparison of pseudocrossing energy separations A U deduced from experiment, with those calculated from the wavefunctions, is not entirely satisfactory, as can be seen from Table I. This may be associated with the fact that in very few of these processes does the active electron start and finish in an s orbital (cf. Bates, 1960). For some years it has been recognized that the Landau-Zener approximation has certain defects (Bates, 1960; Mordvinov and Firsov, 1960). Another approach (Bates et al., 1964) is to solve numerically the equations of the twostate approximation. It is found that the probability of transition P oscillates with varying p and hence varying scattering angle 6 (Fig. 5). Calculations of
247
RECENT MEASUREMENTS ON CHARGE TRANSFER
TABLE I COMPARISON OF EXPERIMENTAL AND CALCULATED ENERGY DIFFERENCES AT PSEUDOCROSSING Collision
R, (a.u.)
AUcalc (eV)
AU exptl (eV)
Ar2+Ar Kr2+Kr Xe2+Xe Kr2+Ne ArZ He N2+He Kr3+He Kr3 Ne Xe3+Ne Xe4 Ne
3.35 3.80 4.37 9.1 8.90 5.40 4.70 3.80 5.35 3.80
1,75 1.60 0.93 0.03 0.0032 0.27 1.7 0.76 0.86 1.17
1.4 1.1 0.94 0.35 0.41 0.75 1.17 1.6 1.6 5.4
+
+
+
080
200175 150 ( 2 5 I
r
I
I
-
0 50
075
100 1
0 25 I
Angle in degrees
O*'+Ne-tO'+Ne*
-
E = 2600 eV
050-
-
030-
-
o 15
16
10 20 30 40 50 60 700, P17
18
19
20
21
22a,
I
I
23
24
25
FIG. 5. Probability of capture as a function of impact parameter, P(p) deduced from differential scattering measurements OZ+Ne,at impact energy 2600 eV (Alam ef al., 1967). Vertical arrow indicates crossover at 2 . 0 2 ~inset, ~ ; P(p) calculated by Bates et al. (1964) for BeZ+Hat 102.75eV energy.
J. B. Hasted this type lead to cross sections of the same form as those derived from Landau-Zener approximation, but larger. However, they may yield a second maximum in the cross section at higher energies than the Landau-Zener maximum. In general terms, this is in harmony with multiply charged ion experiments [such as Kr3+Ne(Hasted,1962)] in which a high energy maximum is observed, similar in form to those in noncrossing unlike ion-atom collisions. Figure 5 shows P ( p ) functions for 0 ” N e single electron capture at two different impact energies. With the resolution of this experiment, it is not certain that the minima are not close to zero, as would be implied in calculations of Bates et al. (1964). The 0’’ ion beam contains a proportion of ‘D excited states, which will contribute with pseudocrossing at R, = 1.7a0, thus causing a second feature in the P ( p ) functions superposed on the principal feature at R, = 2.02a0.
VI. Molecular Charge Transfer Processes at Low Energies The interpretation of molecular charge transfer processes by theory is not possible unless the vibrational as well as the electronic states of the reactants and products are known. This information is rarely available. Exothermic molecular charge transfer processes have large cross sections over a wide impact energy range, because it is usually possible for the excess internal energy to be converted into vibrational and rotational energy. Thus near-resonance behavior can often be assumed. Similar reasoning can be applied to the dissociative charge transfer process A
+ BC -+ A + B+ + c.
(1 1) Instructive examples of such processes are the collisions (Stebbings et a]., 1963) of helium ions with oxygen and nitrogen, producing, even at impact energies as high as a few kilovolts, a great preponderence of atomic over molecular ions. The apparent accidental resonances are attributed to the processes He+ + N,+He + N,+ (C2C,+, u = 3) + 0.28 eV (12) He+ + 0,+ He + 0, (c4Zu-) -t0.02 eV. (13) These are followed by predissociation. Slow collisions between molecular ions and atoms and molecules can result in chemical interchange processes of the type +
+
A++BC-,AB++C. (14) These processes are also difficult to treat theoretically. Ion-atom interchange processes are not the subject of this article, but not only are the experimental methods for their study very similar to those used for charge transfer, but also the distinction between the two types of process is not always well defined.
RECENT MEASUREMENTS O N CHARGE TRANSFER
249
VII. Experimental Techniques The reaction products in ion-molecule collisions may have appreciable kinetic energy. Where product mass analysis is not required, the collection of the slowest charged products is relatively simple, but to focus these upon a small mass-spectrometer entrance slit is more difficult. I n older experiments of this type, an ion beam was directed through a gas in the presence of a uniform electric field transverse to the beam. The mass-spectrometer slit was located downfield from the beam, and two conditions were deemed necessary for a completely efficient collection at the slit to have been achieved: (1) independence of mass-analyzed ion current from variation of electric field (" saturation conditions ") ; (2) independence of mass-analyzed ion current from variation of entrance and exit slit widths (" flat-topped peaks"). At high impact energies, these conditions may quite readily be satisfied. But the problem becomes more difficult as the difference between energies of fast primary beam and slow charged product decreases. There is a case for setting up a ray-tracing computer program, and also for combining total crosssection measurements with measurements of product angular distributions In ion-molecule reactions the distinction between fast and slow collision products may be obscured. Successful use has been made of the strong focusing electrostatic quadrupole lens (Giese and Maier, 1963). The " flat-topped peak" condition must be satisfied both for the entrance and the exit slits of a sector magnetic mass spectrometer. The former tests the collision chamber optics and the latter the mass-spectrometer optics. A relevant difficulty is that the sensitivity of certain particle multipliers varies form point to point of particle impact on the first dynode. When a mass-spectrometer slit is widened for these tests, the resolution may inadvertently be lowered to the point where ionic products of different mass number are colcleted, thus compromising the flat-topped peak condition. It is necessary to search at high resolution, and then to conduct the experiment under flattopped peak conditions. A mass spectrometer for which this can be achieved without mechanical adjustment is the quadrupole mass filter. The most promising technique applicable to high and medium impact energies is the spectrometer utilizing crossed electric and magnetic field. This has proved successful for the analogous electron impact problem (Schram et al., 1966). The magnetic field is parallel to the impacting beam, and slow collision products emerging at any azimuthal angle will be focused onto a collector placed parallel to the beam, provided that the mass number bears the correct relation to the field intensities.
250
J. B. Hasted
At impact energies of a few electron volts these techniques cannot be operated satisfactorily, and recourse is often made to electronic pulse techniques for total cross-section measurements (retaining steady beams only for differential measurements). A pulsed extraction field will not interfere with the path of a pulsed ion beam provided that the timing and length of the pulses are suitable. For complex molecules at low energies such techniques would be essential, and reference may be made in particular to nanosecond techniques (Matus et al., 1967) for ion-molecule reactions. The original studies of ion-molecule reactions (Stevenson and Schissler, 1955; Talrose and Lyubimova, 1952) were made by passing a magnetically confined electron beam through a gas in the presence of a transverse electric field produced by the " repeller " in a conventional electron-impact massspectrometer source. The ions, mostly produced with only thermal energy, are continuously accelerated throughout the collision region. In general, the true cross-section function is not obtained, even though the repeller potential is varied. A significant advance was made (Talrose and Frankevich, 1960) by pulsing the electron beam, removing the steady transverse electric field, and applying a delayed pulse to the repeller; thus the ions diffuse thermally until the reaction products are extracted. Over a limited range the reaction product flux is directly proportional to the delay time, and a true thermal energy reaction rate can be deduced. Attempts have been made to combine the advantages of the pulse technique with the facility of varying the ion velocity, using a steady repeller potential, but it is clear that this compromise retains the disadvantages of the original steady potential technique. However, with sufficient detection sensitivity and time resolution, this difficulty can be surmounted (Matus et al., 1967) by applying a pulsed repeller potential for accelerating the ions. This pulse is applied directly after the electron beam pulse, and after its completion the ions move in Newtonian fashion in a field-free region. Their energy distribution is therefore relatively narrow, and many of the inherent defects of the previous " mass-spectrometer source " techniques are removed. Analysis of the dependence of the product ion current on the time delay of the third pulse (applied to an electrode outside the chamber) allows information to be obtained about the momentum transfer in the collision. But at present the width of the electron beam, and the relatively undeveloped state of the apparatus, limit the energy range of the technique. However, pulsed techniques have not been widely used for charge transfer studies, for which the tradition stems from the passage of a mass-separated ion beam through a collision chamber. Confinement of the collision region to a short parallelepiped rather than a long track is often an aid to the ion optics of the system, and may be achieved with the aid of crossed-beam techniques (Stebbings, 1968). Collisions of the ion beam with background gas can be minimized if mechanical interruption of the molecular beam is associated with
RECENT MEASUERMENTS ON CHARGE TRANSFER
25 1
phase sensitive detection of the collision products. The complexities of calculating the impact velocity distribution from the velocity distributions of the two beams are only avoided when the ion beam velocity greatly exceeds that of the molecular beam. There is an effective lower limit, set by space charge, surface contact potential variation, etc., to the energy of an ion beam with which charge transfer experiments can be carried out. Although this limit is at present in the region 2-3 eV, it is possible to perform simple symmetrical resonance total cross section measurements in a magnetic field using atomic ions and targets at energies as small as 0.1 eV (Bullis, 1965); in this case the surfaces of the entire electrode system are coated (by deposition or electrolysis) after assembly, in order to avoid variations of contact potential. But it will not be easy to extend these techniques to experiments involving mass analysis of collision products. Even at 2-3 eV the available primary ion beam intensities (lo-’A) limit the experimental capability. There are still two rival approaches to the problem of production of low energy beams, retardation, and momentum analysis of unaccelerated ions emerging from a source. There has been no real breakthrough in either approach, so that routes must be found round this energy barrier. One such route lies in the nanosecond pulse techniques described above. Four other routes exist: (1) ion cyclotron resonance spectroscopy (Wobschall, 1965; Anders et al., 1966); (2) retardation of ion beam by means of an electrostatic mirror (Schlier, 1967); (3) the merged beam method (Trujillo et al., 1966); (4) the drift tube method (Kaneko et al., 1966). We shall only discuss the last. It has developed from the theoretical and experimental work done on the analysis of ions drifting in gases under the action of uniform electric fields (Wannier, 1951 ; Allis, 1956; Dalgarno el al., 1958). Under “constant mean free path” conditions the mean energy of these ions is simply related to their drift velocity ud:
E
= - f m+ ud2
+ 3m,vdz + 3kT,
(15)
where the ion and gas molecule random velocities are respectively u+ and ug . The drift velocity can be measured by well-established electric shutter techniques, and is a function only of X / p , the ratio of field strength to p r e ~ s u r e . ~ In order to set up the collision events, this parameter is maintained constant, although buffer gas pressure may be varied. Buffer gas pressures of the order More strictly, to the number density n. The quantity X/n is sometimes described in terms of units known as Townsends, equal to lo-’’ volt cm2 molecule-’.
252
J. B. Hasted
of 1 Torr are maintained in the collision chamber, in order to minimize the radial and axial diffusion of the ions. The latter produces negligible effects, and the former introduces a correction factor which is treated in the original paper (Kaneko et al., 1966). An ion beam is injected into the collision chamber, which contains a buffer gas chosen for its inactivity to low energy inelastic processes of ground state ions. Helium has several advantages. Excited ions can be “filtered ” in the buffer gas in the manner described in Section VII1,A. Along the axis of the chamber a uniform electric field is maintained, and at the exit there is a sampling orifice, followed by mass spectrometer and detector. Measured traces of reactant gas are introduced and normally these will not affect the drift process appreciably. The cross section for conversion of ion A’ to B + is related to the sample currents ZA and I,:
where ud and v, are, respectively, drift and random velocities of A’, whilst 1is the length of the drift space, and no the density of reactant gas. Tests are normally made to ensure that the beam injected thermalizes to its normal drift conditions before the first drift velocity measurement grid is reached. For very small cross sections, it is possible to use the reactant gas as its own buffer. Cross sections as small as cm2 can be studied. At thermal energies it is possible to measure rate coefficients for charge transfer in the afterglows of electric discharges. The first experiment of this type was carried out by Dickinson and Sayers (1960), who sampled the time dependence of atomic ion densities in the afterglow of a pulsed radio frequency discharge. Since atomic ions cannot recombine with electrons in weak discharges except radiatively, the only serious competitive decay process for these ions is diffusion (ambipolar constant D,, diffusion length A). The decay is governed by the equations I, = I, exp(-t/z) 7-l
= (D,/A2)
+ n, noaij
where the cross section CJ is appropriate to a mean velocity of impact 6, and the symbols n represent number densities, The diffusion is minimized by conducting the experiment in a relatively high pressure of gas C which is chosen to be unreactive with the atomic ion A’. The diffusion coefficient can usually be related to the known ionic mobility of A + in C, and the rate constant separated by variation of the partial pressure of B. The exponential variation of A + density with time is observed over perhaps two orders of magnitude. The ion density is taken to be proportional to the flux of ions emerging from a metal
RECENT MEASUREMENTS ON CHARGE TRANSFER
253
orifice at wall potential exposed to the afterglow plasma, and backed by a mass spectrometer and detector. More complicated kinetic situations are treated (Fite el al., 1962) by monitoring not only of the time decay of A', but the time growth of the ionic species formed in the reaction. The principal disadvantage of the time-dependent afterglow is that during the excitation period the gas molecule can receive internal energy in a form which lasts as long as the positive ions. Thus the specification of reactants is not so exact as is desirable. A degree of refinement is added by converting the afterglow into a flowing system, after the manner used in chemical kinetics (Fehsenfeld et al., 1966). A mechanical pump is used to pump large volumes of gas along a wide tube; the speed may be as high as 500-1000 sec-' at 1 Torr, and care is taken to preserve laminar flow. A buffer gas maintains the flow, so that the different reactants can be introduced at different points in the flow, without interference. Ions can either be produced by means of an electric discharge, or by another arrangement discussed below; or they can be produced in the afterglow tube itself by Penning ionization from metastable atoms, produced at an earlier point in the afterglow tube:
+
He" 2 3S Ar+He Is
+ Ar+ + e.
(19)
In the flowing system the excitation energy applied to gas flowing through one introduction tube can be made to have little effect in exciting a reactant gas introduced to the main flow through a different tube. In the flowing afterglow system the sampling device, such as an orifice with pumped mass-spectrometer and particle detector, may be capable of being traversed along the tube axis, so that if the flow velocity is known, the time dependence of a species density may be calculated from its variation along the tube axis. The mass flow velocity is measured by measuring change of pressure in a gas reservoir, and is monitored by a pressure transducer. In principle, it is not necessary to traverse the detector, provided that the rate of introduction of the reactant gas can be varied and measured. Since the problems of raising gas temperature in a flowing afterglow system are formidable, this type of measurement is effective essentially only at room temperature. The great advantage of the flowing technique lies in the ability to apply electric discharge (or other) ionizing power to the important component A without exciting or ionizing the species B and C . But in view of the existence of laminar flow and the slow thermalization of electrons, it is advisable to pulse the electrodischarge even in a flowing system. Detailed description of the flowing afterglow technique has been given by Stebbings (1968). Some discussion of the relative advantages of different techniques of producing ionization in a gas is necessary. An ideal source of ionization would produce entirely " parent" ions from a molecular gas, whilst leaving the
254
J . B. Hasted
neutral molecules completely unexcited. The undesirable vibrational excitation of molecules is supposed (Schulz, 1959, 1962; Haas, 1957) to take placevia virtual negative ion states, and since these are present in a large number of molecules at energies of a few electron volts (Boness and Hasted, 1966), it follows that continued heating of the electrons in a discharge is to be avoided. Thus radio-frequency and microwave discharges are undesirable, even (although less so) in the pulsed form. Less undesirable are breakdown pulses applied to metal electrodes, and in particular the brush cathode investigated by Persson (1965) has been found to eliminate all components from the electron energy distribution except the thermal component and a group of electrons possessing energies given approximately by the cathode-anode voltage. Superior even to the pulsed brush cathode discharge are fluxes of very high energy electrons and of ultraviolet photons. Electrons of perhaps 50 keV energy can readily be directed through very thin metal windows dividing the gas inlet from a high vacuum. They discriminate in favor of those excited states whose oscillator strengths for transition from the ground state are largest (resonance states). Thus even if there are unwanted species present, their proportions can be estimated. Ultraviolet photons are still more refined, but difficult to make available. There are a number of sources which provide ultraviolet continua capable of ionizing many molecules, but too energetic to allow many long-lived excited states of the neutral molecule to be produced. However, such sources cannot be separated from the gas inlet by any sort of solid window, so that very fast pumping, and possibly a flowing gas window, must be used. Usually only pulsed operation is possible, otherwise the intensity will be insufficient. Screening of detectors from the active ultraviolet photons is important.
VIII. Role of Excited Species A. EXPERIMENTS WITH EXCITED AND GROUND STATE IONS
Several experimental investigations have been carried out using atomic ions produced in ion sources capable of eliminating most of the long-lived excited ions in the beam (Hasted, 1954; Amme and Utterback, 1964). Most ion sources (conventional electron impact, oscillating electron, radio frequency discharge, etc.) are inferred to produce a proportion of electronically excited ions whose lifetimes are sufficiently long for the excitation to persist right up to the moment of collision. In general the larger the internal energy of the excited ion, the lower the proportion that is likely to be produced. However, it is not possible at this stage to make reliable deductions of this proportion from theoretical analysis of the source; certain indirect inferences may be
RECENT MEASUREMENTS ON CHARGE TRANSFER
255
made from the experimentally determined charge transfer cross-section functions, provided that comparison can be made between conventional ion source data and “ground state ion source” data. Ion beams can be produced with ions all in the ground state by means of: (1) surface ionization; (2) monochromated ultraviolet radiation ; (3) controlled energy electron fluxes. Surface ionization is, of course, limited to atoms of very low ionization potential, that is, to the alkali atoms. The first experimental comparison of inelastic cross sections obtained with ground state positive ions and with conventional ion sources was made (Fogel et al., 1959) using Li’ produced by surface ionization. The production of monochromated photon fluxes of sufficiently high energy to ionize gases requires the use of vacuum ultraviolet techniques and grazing incidence monochromation. For this reason these ion sources are unlikely to be as widely accessible as those using electrons of controlled energy. Ionization functions by electrons are zero at threshold, whereas those for ionization by photons are finite at threshold. When ionization yielding a number of excited levels of the ion is possible, one might expect a piecewise linear function for electron impact, and a step function for photons. Thus electrons are inferior but convenient tools for the production of ground state ion fluxes. A compromise must be struck between the maximization of ion flux for the subsequent collision experiment and the optimization of the resolution of the electron energy selection. Where the lowest long lifetime electronic excited state of the ion lies several electron volts above the ground state, thermionic electron energy distributions are sufficiently narrow to allow of effective elimination of the excited states; a conventional or specially adapted electron impact ion source is adequate (Stebbings et al., 1966; Bohme et al., 1967), and will naturally produce a greater ion flux than a source using momentum-analyzed electrons. The latter (Scott and Hasted, 1964; Hussain and Kerwin, 1965) are necessary in experiments where the lowest long-lifetime state of the ion lies within about 1 eV of the ground state. The dependence of the charge transfer cross section O + N z upon the ion source electron energy is shown in another article in this volume (Stebbings, 1968). Momentum analyzed electrons are not used. The O’N, experiment yields a large cross section for the production of Nzf by excited 0’ ’0, while the cross section for the ground state ion is small. Elimination of excited ions from the beam can be achieved by passing it through nitrogen gas. The higher the ion beam energy, the less serious the effect of elastic scattering. The filtering technique is indispensable when the threshold ionization function is such that ground state ions cannot easily be produced without momentum-analyzed electrons. However, a Nier-Bleakney electron impact source
256
J . B. Hasted
with capillary gas feed (Edmonds and Hasted, 1964) is adequate (Bohme et al., 1967) for the production of ground state 0 ' beams. Figure 6 shows the dependence of 0 ' intensity upon electron energy for ion source gas CO; the curve follows closely recent mass-spectrometric investigations (Cuthbert et al., 1966), and allows the electron energy scale to be corrected to the sharp ' 2D appearance. The 0 ' 'D Ar charge transfer is 1.17 eV break at 0 exothermic, while the ground state process Of 4SAr is 2.15 eV endothermic; at 0.8 eV impact energy, the latter cannot take place. Figure 6 also shows the electron energy dependence of the Ar + intensity produced in 0.8-eV collisions. Under these circumstances it is possible to extract reliable ground state O f collision cross sections from data obtained several electron volts below 0 ' D appearance. The figure gives a good indication of the capabilities and limitations of this approach.
'
Laboratory electron energy lev)
I
I
20 I
30
25 I
35
I
1 0
9-
-VI
'E
a
7-
E
-a
3-
20
25
Electron energy ( e V ) normalized to ' 0
30
35
onset of 0'
FIG. 6. Ground state ion source data: appearance potential functions of O f ions from CO by electron impact (Bohme et al., 1967); 360 pA electron current: pressure < 1 x 10Tom. Open circles represent production of Art ions by inelastic O + *DAr collisions at 0.8 eV impact energy; the yields of ions below O + '0 threshold can be regarded as background.
RECENT MEASUREMENTS ON CHARGE TRANSFER
257
Investigations using ions produced by collisions with electron beams, which have been energy selected in analyzers, were applied (Scott and Hasted, 1964; Hussain and Kerwin, 1965) to the symmetrical resonance charge transfer collisions of singly-charged heavy rare gas ions. The lowest state of these ions can exist with inner quantum number J = +, 3, the J = 3 being, for example, 0.67 eV the higher in Kr. Thus the possibility arises of the ions at the commencement and termination fo the collision being:
At sufficiently low impact energies, the processes involving energy defects should be unlikely, whilst the symmetrical processes would be expected to have large cross sections (Section 11). But in the initial experiments at 250 eV impact energy (Scott and Hasted, 1964) the ratio of J = 3 to J = 4 cross sections were found to be 10.1. In subsequent experiments (Hussain and Kerwin, 1965) using a finer resolution of electron energy, the ratio was found to increase with increasing impact energy and pass through a maximum value 0.6 at 700 eV. No quantitative explanation of this interesting situation has been evolved.
B. PROCESSES TERMINATING IN EXCITED SPECIES Charge transfer processes terminating in excited states, whether of the ionized or neutral species or both, may be investigated experimentally by monitoring the radiation emitted from the collision region, provided that the states in question are not metastable. When they are, the experimental techniques are more difficult, but in cases such as H 2s it is possible to quench the state by an electric field, so that the Lyman-cr radiation from H 2p can be monitored. The experimental data for the processes H++H+H2s+H+
(21)
H C+ N e + H 2 s + N e f (22) are displayed in Figs. 7 and 8. The neon cross-section function exhibits two maxima, of which one falls at the same energy as that of the H'Ne total charge transfer cross section (Stedeford and Hasted, 1955) which is dominated by the H 1s product. Such processes are suitable for the investigation of coupling effects. The contribution of the path H'Ne+ H 1s Ne' + H 2s Ne' is significant. Rare gas studies made by Geballe and his collaborators (Jaecks et al., 1965) and by Ankudinov et al. (1965) serve to emphasize its significance.
258
J. B. Hasted
Following the work of Love11 and McElroy (1965), various workers have attempted atomic multi-state approximation calculations. Poluektev and Presnyakov (1967) take the overall probability P of transition from initial state (0) to final state (2), including the contribution of intermediate state (l), as (23) P = P,, $POlP,, .
+
The results of their calculations are displayed in Fig. 7. In passing, we note that the most recent experiments of Ankudinov et al. (1967) are conducted with the collision products exposed to electric fields sufficient to cause Stark separation of the substates of the H 3s level, which decay producing Ha and Lyman+ radiation, the observed relative proportions of which are dependent on the electric field.
FIG.7. Total cross sections for production of H 2s in proton collisions with Ne: full line, experimental data of curve 1, Jaecks et al. (1965) and curve 2, Ankudinov et at. (1965); broken line, calculations of Poluektev and Presnyakov (1967).
Calculations of cross sections for process (22) have been made by Wilets and Gallaher (1966, 1967) using Sturmian eigenfunctions. Comparison with experiment is made in Fig. 8. Recent experiments using optical techniques for the detection of excited species (Lipeles et al., 1965; Lorents et al., 1966) have shown that certain rare gas charge transfer processes terminating in excited states are not improbable even at impact energies as low as several electron volts. Pseudocrossing of potential energy curves may contribute to such collisions. It is reported that the total emission function is sometimes oscillatory with impact energy.
RECENT MEASUREMENTS ON CHARGE TRANSFER
259
E
0
20
I
I
40
I I
I
60
t--f---J100
00
Elab( k e V )
FIG.8. Total cross sections for production of H 2s in proton hydrogen atom collisions; points with error bars, experiments of Stebbings et al. (1965); broken line, calculations of Bates and Williams (1964); dotted line, calculations of Bell and Skinner (1962); full line with points, calculations of Wilets and Gallaher (1967).
The possibility must be considered that coupling to intermediate atomic states contributes frequently to total capture cross sections in the far adiabatic region. It often happens that such cross sections are larger than is expected on an atomic two-state approximation. The u4 dependence predicted by Rapp and Francis (1962) is not apparent from the data, although this may, in part, be due to the experimental difficulties inherent in measuring very small cross sections. An outstanding example is the exothermic collision (24) for which the cross section (Fig. 9) is as large as lo-’’ cm2 at 4-eV impact energy (Bohme et al., 1967). The neon ions used in these experiments were produced by controlled energy electron impact, so that they could include only the ground states (J = 4, 3). Differential scattering experiments show little indication of curve crossing. There remains the possibility of coupling to excited states, for example to Ne’(3s “P),which can undergo the accidentally resonant process Ne++Ar-+Ne+Ar+
+
+
Ne+(3s“P) Ar(3p6 IS)+ Ne(3s[l&]) Ar+(3d4D).
(25)
IX. Miscellaneous Topics A.
IONIZATION WITH CAPTURE
Ionization with capture is a collision process which may be represented typically as A + + B + A + B2++ e 10/02. (26)
J. B. Hasted
260
“I 2
L
10-l~
0.01
FIG.9. Total cross sections (Bohme er al., 1967) for Ne+Ar charge transfer. Full circles indicate experiments carried out with ion beams containing excited states; open circles indicate experiments carried out with ground state ions. Full line represents the difference, presumably dominated by process 25. Circles with centers represent earlier data of Stedeford and Hasted (1955).
Let us consider how far such collision cross sections differ from, and perhaps exceed, those for multiple ionization in which no capture takes place A+
+ B + A + + B3++ 2e
10/12.
(27)
It is of particular importance because the statistical theory of Russek and Thomas (1958) is reasonably satisfactory in interpreting multiple ionization collisions, but takes no account of any inclusion of capture. Processes (26) and (27) can only be distinguished if the charge state of the primary beam is determined simultaneously with that of the slow ionized product, possibly by detection with coincidence. This was pointed out nearly ten years ago (Hasted, 1960), but the first measurements directed at measuring the cross section have only been reported recently (Afrosimov, 1967). The coincidence technique has been developed at Leningrad for the study of the conversion of energy from kinetic to internal in the multiple ionization process, but here it is combined with extraction from the collision chamber using transverse electric field. Transferred momentum, which might seriously complicate this technique, is minimized in the first experiments by the use of the proton as projectile and rare gases as targets.
RECENT MEASUREMENTS ON CHARGE TRANSFER
26 1
It has been found that the cross sections for ionization with capture 10/02 and 10/03 are indeed much larger than the corresponding ionization cross sections 10/n2 and 10/n3, n = 1, 2, . . . . The cross-section functions also maximize at much lower energies, as would be expected on the basis of the adiabatic maximum rule. No detailed comparison with a wide variety of processes is possible at this stage, but it is of interest that the adiabatic parameter for 10/03 is smaller than that for 10/02. It has been proposed (Parilis, 1967) that the ionization with capture process is dominated by capture into autoionizing states and so may be treated by theory similar to that for pure capture processes.
B. RADIATIVE CHARGE TRANSFER The process
in which electromagnetic radiation carries away the excess energy, can, under some circumstances, be more likely than the similar process in which no radiation is produced. Radiative charge transfer can only arise if the quasimolecule of the collision is formed in a state which can radiate to another state, which in turn can dissociate in the required manner. Daly and Powell (1966) have recently made some studies in helium using the trapped-ion technique (Baker and Hasted, 1966). They observed the process
+
He2+ He + He+
+ He+
(+ hv)
(29)
at thermal energies. The nonradiative process would be exothermic by 5.2 eV. Even allowing for pseudocrossing and for the possibility of the collision products being in excited states, it is likely to have a much smaller cross section than that for the radiative process which has been calculated by Allison and Dalgarno (1965). Daly did not attempt to detect radiation.
C. Two ELECTRON CAPTURE PROCESSES Two electron capture 10/12 by singly charged ions has been observed for some years in beam experiments, principally those conducted by Fogel and his colleagues [references in Kozlov and Bondar (1966)l. It is not necessary that the negative ions be formed in their ground states in these processes. A proportion of fast positive ions passing through a gas will be converted into
J. B. Hasted
negative ions, and will be observed as such provided that their lifetimes are sufficiently long for them to live until collected, or at least momentumanalyzed. The lifetimes of the negative ion states involved in the elastic scattering of electrons by atoms (Burke and Schey, 1962; Schulz, 1964) are, in general, far too short for this condition to be satisfied. Some of the importance of two electron capture processes lies in their use in double electrostatic generators (Jorgensen, 1965);positive ions are accelerated, converted into negative ions, and further accelerated in the same electric field so that the energy achieved is twice that available in the conventional electrostatic generator. Typical cross-section functions (Kozlov and Bondar, 1966) for two electron capture processes are shown in Fig. 10. Over a range of one and a half orders of magnitude, the cross section is proportional to the exponential of -l/v (Hasted, 1959). However, the low energy rate of rise is less rapid. The capture of two electrons by a doubly charged ion has been observed by Islam et al. (1962).
FIG.10. Total cross section functions for two electron capture by protons. Open circles, Kozlov and Bondar (1966); closed circles and open squares, Fogel er af. (1959), see Kozlov and Bondar (1966). Vertical lines indicate thresholds. Energy in keV.
RECENT MEASUREMENTS ON CHARGE TARNSFER
263
REFERENCES Afrosimov, V. V. (1967). Abstracts. Proc. Intern. Conf. Phys. Electron. At. Collisions, 5th, 1967. Akad. Nauk, Leningrad. Alam, G. D., Bohme, D. K., Hasted, J. B., and Ong, P. P. (1967). Abstracts. Proc. Intern. Conf. Phys. Electron. At. Collisions, 5th, 1967. Akad. Nauk, Leningrad. Allis, W. P. (1956). In “Handbuch der Physik” (S. Fliigge, ed.). Springer, Berlin. Allison, D. C. S., and Dalgarno, A. (1965). Proc. Phys. SOC. (London) 85, 845. Amrne, R. C., and Utterback, N. G. (1964). Atomic collision processes. Proc. Intern. Conf. on Phys. Electron. At. Collisions, 3rd, 1964, p. 847. North-Holland Pub]., Amsterdam. Anders, L. R., Beauchamp, J. L., Dunbar, R. C., and Baldeschwieler, J. D. (1966). J. Chem. Phys. 45, 1062. Ankudinov, V. A., Bobashev, S. V., and Andreev, E. P. (1965). Zh. Eksperim. i Teor. Fiz. 48, I , 40. Ankudinov, V. A., Bobashev, S. V., and Andreev, E. P. (1967). Abstracts. Proc. Intern. Conf. Phys. Electron. At. Collisions, 5th, 1967. Akad. Nauk, Leningrad. Baker, F. A., and Hasted, J. B. (1966). Phil. Trans. Roy. SOC.(London) A261, 33. Bates, D. R. (1960). Proc. Roy. SOC.A257, 22. Bates, D. R., and McCarroll, R. (1962). Advan. Phys. 11, 39. Bates, D. R., and Williams, D. A. (1964). Proc. Phys. SOC. (London) 83, 425. Bates, D. R., Johnson, H. D., and Stewart, A. L. (1964). Proc. Phys. SOC.(London) 84, 517. Bell, R. J., and Skinner, B. G. (1962). Proc. Phys. SOC.(London) 80, 404. Biondi, M. A., and Chanin, L. M. (1954). Phys. Rev. 94,910. Biondi, M. A., and Chanin, L. M. (1956). Phys. Rev. 106,473. Bohme, D. K., Nakshbandi, M. M., Ong, P. P., and Hasted, J. B. (1967). Proc. Intern. Conf. Ionization Phenomena Gases, 7rh, Vienna,1967. North-Holland Publ., Amsterdam, Boness, M. J. W., and Hasted, J. B. (1966). Phys. Letters 21, 526. Bullis, R. H. (1965). Abstracts. Proc. Intern. Conf. Phys. Electron. At. Collisions, 4rh, Quebec, 1965, p. 263. Science Bookcrafters, New York. Burke, P. G., and Schey, H. M. (1962). Phys. Rev. 126,147. Bydin, Yu. F. (1964). Zh. Eksperim. i Teor. Fiz. 46,1612. Bydin, Yu. F. (1966). Zh. Eksperim. i Teor. Fiz. 50,35. Chkuaseli, D., Gouldamachvili, A. I., and Nikoleychvili, U. D. (1963). Proc. Intern. Conf. Ionization Phenomena Gases, 6th, 1963, p. 475. SERMA, Paris. Cuthbert, J., Farren, J., Prahallada Rao, B. S., and Preece, E. R. (1966). Proc. Phys. SOC. (London) 88,91. Dalgarno, A., McDowell, M. R. C., and Williams, A. (1958). Phil. Trans. Roy. SOC.(London) A250,411. Daly, N. R., and Powell, R. E. (1966). Proc. Phys. SOC.(London) 89, 281. Dickinson, P. H. G., and Sayers, J. (1960). Proc. Phys. SOC.(London) 76, 137. Dillon, J. A., Jr., Sheridan, W. F., Edwards, H. D., and Ghosh, S. N. (1955). J. Chem. Phys. 23, 776. Drukarev, G. F. (1967). Soviet Phys. JETP (English Transl.) 52,498. Edmonds, P. H., and Hasted, J. B. (1964). Proc. Phys. SOC.(London) 84, 99. Everhart, E., Helbig, H. F., and Lockwood, G. J. (1964).Proc. Intern. Conf.Phys. Electron. At. Collisions, 3rd, Univ. CON.,London, 1963. North-Holland Publ., Amsterdam.
264
J. B. Hasted
Fedorenko, N. V., Afrosimov, V. V., and Kaminker, D. M. (1957). Soviet Phys.-Tech. Phys. (English Transl.) 26, 1861. Fehsenfeld, F. C., Schmeltekopf, A. L., Goldan, P. D., Schiff, H. I., and Ferguson, E. E. (1966). J. Chem. Phys. 44,4087, 3022,4095; 45, 25, 404. Firsov, 0. B. (1951). Zh. Eksperim. i Teor. Fiz. 21, 1001. Fite, W. L., Rutherford, J. A., Snow, W. R., and van Lint, V. (1962). Discussions Furuday SOC. 33,264. Flaks, I . P., and Ogurtsov, G. N. (1963). Soviet Phys. JETP (EngIish Transl.) 33, 748. Flaks, I. P., and Solov’ev, E. S. (1958). Soviet Phys.-Tech. Phys. (English Transl.) 3, 564. Fogel, Ya. M., Kozlov, V. F., Kalymkov, A. A., and Muratov, V. I. (1959). Zh. Eksperim. i Teor. Fiz. 36, 929. Ghosh, S. N., and Sheridan, W. F. (1957). J. Chem. Phys. 26,480. Giese, C. F., and Maier, W. B. (1963). J. Chem. Phys. 39, 197. Gilbody, H. B., and Hasted, J. B. (1956). Proc. Roy. SOC.A238, 334. Gilbody, H. B., Hasted, J. B., Ireland, J. V., Lee, A. R., Thomas, E. W., and Whiteman, A. S. (1963). Proc. Roy. SOC.274,40. Haas, R. (1957). Z. Physik. 148, 177. Hasted, J. B. (1951). Proc. Roy. Sac. A205,421. Hasted, J. B. (1954). Proc. Roy. SOC.A222, 74. Hasted, J. B. (1959). J. Appl. Phys. 30, 25. Hasted, J. B. (1960). Nutl. Acud. Sci. Nutl. Res. Council. Rept. No. 29. Hasted, J. B. (1962). In “Atomic and Molecular Processes” (D. R. Bates, ed.), p. 696. Academic Press, New York. Hasted, J. B. (1964). “ Physics of Atomic Collisions.” Butterworths, London. Hasted, J. B., and Lee, A. R. (1962). Proc. Phys. SOC.(London) 79, 1049. Hasted, J. B., Lee, A. R., and Hussain, M. (1964). Proc. Intern. Conf. Phys. Electron. At. Collisions, 3rd, Univ. CON.,London, 1963. North-Holland Publ., Amsterdam. Hussain, M., and Kerwin, L. (1965). Abstracts. Proc. Intern. Conf Phys. Electron. At. Collisions, 4th, Quebec, 1965, Science Bookcrafters, New York. Ishii. H., and Nakayama, K. (1961). Trans. Natl. Vacuum Symp. 8, 518. Islam, M., Hasted, J. B., Gilbody, H. B., and Ireland, J. V. (1962).Proc. Phys. SOC.(London) 79, 1118. Jaecks, D., van Zyl, B., and Geballe, R. (1965). Phys. Rev. A137, 340. Jorgensen, T. (1965). Phys. Rev. 140, A1481. Kaneko, Y.,Megill, L. R., and Hasted, J. B. (1966).J. Chem. Phys. 45, 3741. Kozlov, V. F., and Bondar, S. A. (1966). Zh. Eksperim. i Teor. Fiz. 50, 297. Kushnir, R. M., Palyukh, B. M., and Sena, L. A. (1959). Bull. Acad. Sci. USSR Phys. Ser. (English Transl.) 23,995. Landau, L. (1932). Phys. Z. Sowjet union 2,46. Lee, A. R. (1962). Thesis, Univ. of London, London. Lee, A. R. (1967). Private communication. Lee, A. R., and Gilbody, H. B. (1964). Proc. Intern. Conf. Phys. Electron. At. Collisions, 3rd, Univ. Coil., London, 1963. North-Holland Publ., Amsterdam. Lee, A. R., and Hasted, J. B. (1965). Proc. Phys. SOC.(London) 85, 673. Lipeles, M., Novick, R., and Tolk, N. (1965). Phys. Rev. Letters 15, 815. Lorents, D. C., and Aberth, W. (1965). Abstracts. Proc. Intern. Conf. Phys. Electron. At. Collisions, 4th, Quebec, 1965, p . 296. Science Bookcrafters, New York. Lorents, D. C., Aberth, W., and Hesterman, V. W. (1966). Phys. Rev. Letters 17, 849. Lovell, S. E., and McElroy, M. B. (1965). Proc. Roy. SOC.A283, 100.
RECENT MEASUREMENTS ON CHARGE TRANSFER
265
Mahadevan, P. (1967). Abstracts. Proc. Intern. Conf.Phys. Electron. At. Collisions, 5th. 1967. Akad. Nauk, Leningrad. Marchi, R. P., and Smith, F. T. (1965). Abstracts. Proc. Intern. Conf. Phys. Electron. At, Collisions, 4th, Quebec, 1965, p. 296. Science Bookcrafters, New York. Marino, L. L., Smith, A. C. H., and Caplinger, E. (1962). Phys. Rev. 128,2243. Massey, H.S. W. (1949). Rept. Progr. Phys. 12,248. Massey, H. S. W., and Smith, R. A. (1933). Proc. Roy. SOC.A142, 142. Matus, L., Hyatt, D. J., and Henchman, M. J. (1967). J . Chem. Phys. 46,March 15. McCarroll, R. (1961). Proc. Roy. SOC.A264, 547. McClure, G. W. (1966). Phys. Rev. 148,47. McElroy, M. B. (1963). Proc. Roy. SOC.A272,542. Mordvinov, Yu. P., and Firsov, 0. B. (1960). Soviet Phys. JETP (English Transl.) 12,301. Nichols, B. J., and Witteborn, F. C. (1966). NASA Tech. Note T N D-3265. Palyukh, B. M. (1967). Abstracts. Proc. Intern. Conf. Phys. Electron. At. Collisions, Sth, 1967. Akad Nauk, Leningrad. Parilis, I. (1967). Abstracts. Proc. Intern. Conf.Phys. Electron. At. Collisions, 5th, 1967. Akad Nauk, Leningrad. Patterson, P. L. (1966). JILA Rept. No. 87. Univ. of Colorado, Boulder, Colorado. Perel, J., Vernon, R. H., and Daley, H. L. (1965). Phys. Rev. 138,A837, 336. Persson, K. B. (1965). J . Appl. Phys. 36,3086. Poluektev, I. A., and Presnyakov, L. P. (1967). Abstracts. Proc. Intern. Con$ Phys. Electron. At. Collisions, 5th, 1967. Akad Nauk, Leningrad. Rapp, D., and Francis, W. E. (1962). J. Chem. Phys. 37,2631. Russek, A., and Thomas, M. T. (1958). Phys. Rev. 109,2015. Schlier, C. (1967). Reported at Spring meeting of German Phys. SOC. Schram, B. L., Adamczyk, B., and Boerboom, A. J. (1966). J. Sci. Instr. 43, 638. Schulz, G. J. (1959). Phys. Rev. 116, 1141. Schulz, G. J. (1962). Phys. Rev. 125,229. Schulz, G.J. (1964). Proc. Intern. Conf. Phys. Electron. At. Collisions, 3rd, 1964. NorthHolland Publ., Amsterdam. Scott, J. T., and Hasted, J. B. (1964). Mass-Spectry. Symp., Paris, 1964. Inst. of Petroleum, ASTM, Philadelphia, Pennsylvania. Smirnov, B. M., and Firsov, 0. B. (1964). Zh. Eksperim. i Teor. Fiz. 47,232. Smith, F.J. (1964). Proc. Phys. SOC.(London) 84, 889. Smith, F. J. (1966). Phys. Letters 20,271. Snow, W. R. (1966). Thesis, Univ. of Washington, Seattle, Washington. Stebbings, R. F. (1968). This volume. Stebbings, R. F., Smith, A. C. H., and Ehrhardt, H. (1963). J. Chem. Phys. 39,968. Stebbings, R. F., Smith, A. C. H., and Ehrhardt, H. (1964). J. Geophys. Res. 69,2349. Stebbings, R. F., Young, R. A., Oxley, C. L., and Ehrhardt, H. (1965). Phys. Rev. 138, 1312. Stebbings, R. F., Turner, B. R., and Rutherford, J. A. (1966). J. Ceophys. Res. 71,771. Stedeford, J. B. H., and Hasted, J. B. (1955). Proc. Roy. SOC.A227,466. Stevenson, D.P., and Schissler, D. P. (1955). J. Chem. Phys. 23, 1353. Talrose, V. L., and Frankevich, E. L. (1960). Russ. J. Phys. Chem. (English Transl.) 34, 1275. Talrose, V. L., and Lyubimova, A. K. (1952). Dokl. Akad. Nauk SSSR 86,909. Trujillo, S. M., Neynaber, R. H., and Rothe, E. W. (1966). Rev. Sci. Instr. 37, 1655. Wannier, G.H. (1951). Phys. Rev. 83,281. Wilets, L.,and Gallaher, D. F. (1966). Phys. Rev. 147, 13.
266
J. B. Hasted
Wilets, L., and Gallaher, D. F. (1967). Abstracts. Proc. Intern. ConJ Phys. Electron. At. Collisions, Sth, 1967. Akad Nauk, Leningrad. Wobschall, D. (1965). Rev. Sci. Instr. 36,466. Zener, C.(1932). Proc. Roy. SOC.A137, 696. Ziegler, B. (1953). Z . Physik 136, 108. Ziemba, F. P., Lockwood, G. J., Morgan, G. H., and Everhart, E. (1960). Phys. Reo. 118, 1552.
MEASUREMENTS OF ELECTRON EXCITA TION FUNCTIONS D. W . 0 . HEDDLE and R . G . W. KEESING Physics Department, University of York Heslington, York, England
I. Introduction ................................................... 11. The Excitation Equilibrium ..................................... A. A Simplified Picture ........................................ B. Cascade Population .......................................... 111. IV. V. VI. VII. VIII.
C. Pressure Dependent Secondary Processes ...................... The Angular Distribution of the Light ........................... Simultaneous Ionization and Excitation .......................... High Resolution Measurements ................................. Time-Resolved Measurements ................................... Related Measurements ......................................... Comparison of Observations ................................... References ....................................................
267 .269 .269 270 .273 .278 .281 .284 .289 .292 .294 296
I. Introduction The first demonstration that in collision with an atom an electron could transfer all its kinetic energy and raise the atom to an excited state was made by Franck and Hertz (1914) very soon after their renowned energy-loss experiment. Their experimental system was very simple and is shown in Fig. 1. Electrons from a hot filament F were accelerated to an anode A through mercury vapor at a pressure of about 1 Torr. The spectrum of the light emitted from the system was photographed and it was found that no light (except thermal radiation from the filament) was emitted if the anode potential was below 4.9 V, but at slightly higher potentials the ultraviolet resonance line corresponding to the 6 3P,-6 ' S o transition was produced and even though this could be overexposed no other lines were detectable. A considerable amount of experimental work was done in the period 1925-1935 on many atomic and molecular systems. The experimental arrangements were simple: an electron gun was used to produce a beam of electrons in a collision chamber containing the gas under study. Light from the collision chamber was collected (almost invariably at 90" to the electron 267
268
D . W. 0 .Heddle and R. G . W . Keesing
beam) and the wavelengths of interest selected by a spectrometer or filters. An electron gun typical of those used in Jena where much important work was done in shown in Fig. 2a. At this time only photographic detection of the light was practicable and exposure times were long, often of several hours. Many experiments were consequently made in which the gas pressure or the
FIG.1. The excitation tube of Franck and Hertz (1914).
(0)
FIG. 2. Electron collision chambers. (a) Hanle (1929), (b) Jongerius (1961), and (c) Simpson and Kuyatt (1963).
MEASUREMENTS OF ELECTRON EXCITATION FUNCTIONS
269
electron beam current were really much too large for a simple interpretation of the observations to be made. However it became clear that the electron excitation functions of states of different character could be very dissimilar. Massey and Mohr (1931) quickly realized the importance of this and showed that one would expect, on theoretical grounds, excitation functions for the 'P and 3P states of helium which were very different from each other, but similar to those found by experiment both in helium (Lees, 1932) and in cadmium (LarchC, 1931). In the past ten years an increasing amount of experimental work has been carried out with, generally speaking, a better understanding of the problems of interpretation. The development of photomultiplier tubes and the application of electron optical considerations to the experimental systems have meant that experimental conditions could be such that a more direct relationship exists between the observed light output and the electron excitation cross section itself. While measurements have been made on all the rare gases, the alkalis and several other gases, detailed analyses of the observations have only been made in the cases of helium and mercury and we will consequently refer specifically to these gases more frequently than to others in this paper.
II. The Excitation Equilibrium A. A SIMPLIFIED PICTURE The equilibrium concentration of atoms in a given state does not in general depend only on the rate of electron excitation to that state and of radiative decay from it though this can be a very useful model to use in the expression of experimental results. Consider the schematic energy level diagram of Fig. 3. The volume rate of production of atoms in the statej by electron impact is given by Ng N , vQj where Ng and N , are the concentrations of atoms in the ground state and of electrons, respectively, u is the relative velocity (which
I
I
FIG.3. Schematic energy level diagram.
270
D. W. 0. Heddle and R. G. W. Keesing
will be predominantly the electron velocity, though the atomic velocity may have a significant effect in limiting the energy resolution with which measurements may be made (see Section V ) )and Q, is the cross section for excitation of the state j from the ground state. It is more convenient to work in terms of the electron beam current Z and the production rate is then NZQ,/eS where e is the electronic charge and S the cross-sectional area of the beam. The volume rate of radiative depopulation is N, A, where A, = x k A , and A j , is the Einstein transition probability for a spontaneous transition from statej to a lower state k. Only the radiation corresponding to a given transition is commonly observed. Let $jk be the rate of emission of photons corresponding to the transition from statej to state k per unit length of electron beam, then $jk
= NjAjkS.
If no other processes contribute to the population or depopulation rate then at equilibrium we have
and hence Q.=-Aj
4jk
A j , NJle'
The assumption that there are no other populating processes is usually false, but the form of Eq. (2) is still useful in defining an apparent cross section which we will call Q,'.
B. CASCADE POPULATION Some atoms will be excited to states i of higher energy than s t a t e j and radiative transitions i - j may occur. We call these cascade transitions and they populate the state j at a rate = x i N i A i j S per unit length of electron beam. If we can measure these $ i j then we know the cascade contribution to the statej. This is not often possible. For some transitions in the heavy rare gases, in mercury, and for the first P levels of the alkalis some measurements can be made. The contribution of transitions from the ' S and 'D levels of rubidium to the 5 'P levels are shown in Fig. 4 (Zapesochny and Shimon, 1966b). In many cases the cascading transitions are at too long a wavelength for detection with high sensitivity, but then there will most probably be transitions such as i I which lie at shorter wavelengths, in the same spectrum region as transitions j + k and 4ii will be measurable.
xi$ij
MEASUREMENTS OF ELECTRON EXCITATION FUNCTIONS
27 1
--------a \ ------__
.------t
C
20
10
30
Electron energy (volts)
FIG.4. (a) The observed excitation function of the rubidium resonance lines (Zapesochny and Shimon, 1966b), (b) The cascade population component,and (c)The electron excitation component.
The
+ij
can then be calculated from Ai j
4ij
7
Ail
+it
We can include the effect of cascade population in Eq. (1) by writing for the population rate I dil Aij N Q .- per unit volume ' eS i S Ail when we have
+
which we can write as with J j =
xiJ i and
Q . = Q.' I
Jij
J
- J J.
Aij dil =Ail N , l / e '
By analogy with Eq. (2) Q : = -A- ,
dil
A , N , I/e*
From Eqs. (4) and (5)
-
J . . = Qi' Ail 1J
Ai
212
D . W. 0 . Heddle and R. G . W . Keesing
We have only to measure the apparent cross section for excitation of the state i and (given the branching ratio A i j / Ai) we have the cascade correction to that of state j . The processes populating state i are quite unimportant and there is no need to determine the true cross section Q i . In the case of atoms which show series (as opposed to multiplet) spectra the total cascade contribution may be calculable from observations of the first few members of the cascading series. It appears that the excitation functions of different members of the same series are very similar in form: Fig. 5 shows the excitation functions of members of the diffuse series of cesium (Zapesochny and Shimon, 1964). Frost and Phelps (1957) suggested that the maximum cross sections for states of a given series might be related to the principal quantum number n by QmaXccn-'where c1 is a constant for a
J 5
1
10
I
I
I
25 Elrctrm rmqy in vonr 15
20
I
33
FIG. 5. Excitation functions of the diffuse series of cesium (Zapesochny and Shimon, 1964). The principal quantum numbers are shown.
MEASUREMENTS OF ELECTRON EXCITATION FUNCTIONS
273
given series. Gabriel and Heddle (1960) suggested that the effective principal quantum number n* might be more appropriate (n* is defined as 1/E'/' where E is the term value in rydbergs and n - n * is the quantum defect). If such a relationship can be shown to apply it gives a means of extrapolation to find the total cascade contribution. Zapesochny has found that the n-" relationship holds for the alkalis, but that GI takes a wide range of values for different atoms and series. If the (n*)-"' relationship is used the values of a* for different series in any one atom are much more similar though differences remain between atoms. The position is summarized in Table I. In the case of TABLE I VALUESOF THE EXPONENTS u AND u* IN THE
RELATIONSHIPS Qmax
em.,cc n-'
AND
cc @*I-"'
Atom
Series
u
U*
cs
S
8 6 5.2 4.1
3.1 5.5 3.1 3.9
P D F K
Na
S P D
6 10
4
4.1 9 3.9
S P D
6.9 16 5.2
5.2 16 5.2
helium and the alkalis cascade population is rarely more than 10% of the direct electron excitation unless other pressure dependant secondary processes are also important. The situation is very different in atoms like mercury where a large part of the excitation may be via cascade. Jongerius (1961) has made a very full study of mercury and his results for the 6 3P1-6 ' S o line are shown in Fig. 6 . The onset of cascade population above the threshold (7.73 eV) for excitation of the 7 3 S , state is clearly shown. In this case the intensities of nine cascading transitions (some unresolved) were measured. A very important advance in the measurement of cascade population has recently been made using time resolution techniques; these will be discussed in Section VI.
C. PRESSURE DEPENDENT SECONDARY PROCESSES There are two important processes which affect the population of excited states in a pressure dependant fashion.
274
D. W. 0. Heddle and R. G. W. Keesing
J
8
12
16
20
Electron energy (eV)
FIG.6. The excitation function OF the 2536 A line of mercury showing the part played by cascade population (Jongerius, 1961).
(1) The state of excitation of an atom may be changed in a collision with another particle. This other particle will be an atom in the ground state in any experiment in which there is any hope of extracting electron excitation data. Helium is the only system for which a serious study has been made and even here the analysis can be very complicated. The principal observation is that the D-states, both singlet and triplet, are excessively populated at high pressure. The degree of excess population being greatest for electron energies around 100 eV. This suggests that the increase is due to a transfer of excitation from atoms in ' P states because the excitation function of these states has a broad maximum at 100 eV. The excitation cross section is also large and the density of atoms in ' P states is much greater than in other states at this energy. This explanation was first offered by Lees and Skinner (1932). Maurer and Wolf (1934) and Wolf and Maurer (1940) produced excitation to 'P states only, by absorption of resonance radiation, and found that lines from all levels were emitted. They assumed that excitation was transferred in collisions such as
+
He(n 'P) He( 1 S)--+ He( 1 * S)
+ He(nX)
(7) where He(nX) is any other state of principal quantum number n. Heron el al. (1956) showed that the 3'P state was not significantly depopulated by collisions at pressures up to 50 mTorr. Gabriel and Heddle (1960) showed that
MEASUREMENTS OF ELECTRON EXCITATION FUNCTIONS
275
an analysis based on direct transfer into the observed states gave results which depended on the experimental conditions and concluded that direct transfer was not the process populating the 3 0 states. They showed that the concentration of helium atoms in the various 'P states depended very little on the principal quantum number and consequently collisional transfer from high 'P states followed by radiative transfer (cascade) to the lower D states was the probable population process. St. John and Fowler (1961) suggested that transfer into F states followed by cascade would be important and Lin and Fowler (1961) showed that in collisions of the type represented by Eq. (7) a change in azimuthal quantum number L of k2 would be favored. In order to make analysis of the observed data practicable, St. John and Nee (1965) made the additional assumptions that the cross sections for P + F state transfer increase as n or nz and that the 'F2 and 'F4 states are not involved. They were then able to solve the transfer problem. While these assumptions may be questioned it is certain that no analysis of the problem of excitation transfer from a large number of 'P states is practicable without some simplifying assumption or some additional measurements. Direct observation of the cascading transitions is difficult because they lie in the infrared. The 6F-3D transitions are perhaps the most accessible being at 10915 k 28, which is detectable using a S 1 photocathode, but resolution of the singlet and triplet lines might be difficult. Kay and Hughes (1967) have made a time-resolved study in which they identify the cascade population of the D states by its decay rate and show that the F states are indeed of great importance. We will discuss their work further in Section VI. Heddle (1962) has suggested that production of atoms in a single 'P state by absorption of monochromatic resonance radiation would give data which might be analyzed more easily. Preliminary observations (Samuel, 1965) have indicated that this method is practicable but further work is necessary. The fact that direct collisional transfer from the 3 'P state to the 3 'P state does not occur has been demonstrated by Teter and Robertson (1966). They irradiated a discharge in helium with 5016 8, light from another discharge. Absorption by atoms in the 2 ' S metastable state increased the population of the 3 'P state by 14 %. The populations of the 3 ' S and 3 'P states were changed by less than 0.1 %. The population of the 3 '0 state increased by 1.2% which they attribute to pumping by 5876 A light to which their filter was slightly transparent. The population of the 3 D state increased by 5.9 %. It is not possible to say whether excitation transfer is occuring in this case as a result of collisions with atoms or with electrons: the conditions in a discharge are very different from those in an electron excitation collision chamber. (2) If the statej is optically connected with the ground state then as well as transitions j + k there will be transitions j+g. In many cases these latter
276
D. W . 0 . Heddle and R. G . W.Keesing
transitions give lines which lie in the far ultraviolet and are difficult to observe with high sensitivity, but in other cases these resonance Zines lie in the observable spectrum region. This is the case for the alkalis and certain other elements. Resonance radiation can be absorbed by atoms in the ground state and if this occurs within the collision chamber it is an additional source of excitation and one which may be important beyond the limits of the electron beam. Light from outside the electron beam was first noted in an excitation experiment by Lees and Skinner (1932), who correctly accounted for it. The transfer of resonance radiation has been frequently discussed and the work of Holstein (1947, 1951) is important. The effect of repeated absorption and reemission of resonance radiation on electron excitation experiments has been analyzed by Phelps (1958). The fraction of resonance photons which escape from the collision chamber g has been calculated by Phelps as a function of the optical depth at the line center. This is shown in Fig. 7 for a cylindrical collision chamber and a Doppler broadened line. For a line broadened by Doppler effect the absorption coefficient of the line center is given by A3N0 ojA , k o = T - 871 wg v where ogand w j are the statistical weights of the g and j states respectively and u is the thermal velocity. The effect of absorption will clearly depend on whether the resonance line or that corresponding to thej-t k transition is observed. The resonance lines of the alkalis have been observed by Zapesochny and Shimon (1966b) in an excitation system in which the distance from the electron beam to the window could be varied. Their observations of the 8521 8, line of cesium at two
k0p
FIG.7. The fractiong of resonance photons which are not absorbedin a collision chamber of effective radius p. The absorption coefficient ko is defined by Eq. (8).
MEASUREMENTS OF ELECTRON EXCITATION FUNCTIONS
277
pressures are shown in Fig. 8. By extrapolation to zero absorption they were able to determine the excitation cross section. Helium is the only other gas for which systematic observations of the effect of resonance line absorption have been made. In this case it is the nonresonance n 'P-2 ' S transitions which are observed. It is a simple matter to modify Eq. (1) to allow for additional population at a rate NjAj,(l - g) and obtain the following relationship between the apparent and true excitation cross sections :
The variation of Q f 3 i p with pressure found by Heddle and Lucas (1963) is shown in Fig. 9. The full curve is that given by Eq. (9) using the g values of Fig. 7 with an effective collision chamber radius of 0.6 cm. The collision chamber was the same as that used by Gabriel and Heddle (1960) for which an effective radius of 0.5 cm was found from measurements at high pressures, and was 2.4 cm i d . and 2.0 cm long. These results demonstrate the need for measurements on resonance lines to be carried out at a very low pressure; even at Torr the cross section will be 10% high if a correction is not made. If a larger collision chamber is used the working pressure must be correspondingly reduced. Zapesochny and Feltsan (1965) show an intensitypressure curve from which it appears that at lo-* Torr their apparent cross section is about 40 % greater than the true value, from which we deduce that their collision chamber has a effective radius of 1.6 cm. Their observations were made at a pressure of 4 to 5 x Torr.
FIG.8. The intensity of the 8521 A line of cesium as a function of the pressure and absorbing path (Zapesochnv and Shimon, 1966b).
D. W. 0. Heddle and R. G. W. Keesing I
I I I lo I I
I I I I
I
105
3
la4 Pressure in Torr
i2
FIG.9. The effect of absorption of resonance radiation on the apparent cross section for excitation of the 3 'P state of helium (Heddle and Lucas, 1963).
III. The Angular Distribution of the Light In Eq. ( 2 ) $ j k is the total rate of photon emission per unit length of electron beam. It is normally possible to measure the flux emitted into some solid angle d o in a direction 8 with respect to the electron beam and we have to determine
s
4j k = 4 j k ( @
(10)
dm
from our observations. It is convenient to work in terms of (hjk(ll) and $ j k ( l ) which we define as the photon fluxes emitted into unit solid angle perpendicular to the electron beam with electric vectors parallel and perpendicular to the electron beam, respectively. The photon flux 4 j k ( e ) is then given by 4 j k ( @ = 4 j k ( l l ) sin2 e On substitution into Eq. (10) we have
+
ojk(l)(1
+ cos2 e).
The light emitted in any direction is polarized. We define a polarization factor Pjt by
279
MEASUREMENTS OF ELECTRON EXCITATION FUNCTIONS
Equations (10)-(12) give
1 from which it appears that observations made at an angle 8, such that cos' OM = 3 can give information on the total photon flux regardless of the state of polarization of the light. Fite and Brackman (1958) observed excitation of Lyman a in electron-hydrogen atom collisions using a photon counter at this angle (which is 54.7'). They also made some measurements of the angular distribution though they present these as a polarization function and not as an integrated cross section. It is, in general, easier to measure the degree of polarization of the light rather than its angular distribution. The equivalence has been demonstrated very nicely by McFarland and Soltysik (1963). They observed excitation to the 4 l O state of helium using an interference filter to isolate the light. The observations required correction for the fact that the volume of excited gas observed depends on the angle (upproximutely as csc 0) and this they achieved by measuring also the angular distribution of the light from the 4 3S-2 3P transition which is emitted isotropically. They found that the angular distribution of light from the 4 0 - 2 ' P transition agreed with that expected from their own measurements of the polarization factor. Any measurements of polarization have to be corrected for polarization sensitivity in the measuring system. This can be measured using a source of unpolarized light (and this must be checked) such as a glow discharge. One must expect a high degree of polarization sensitivity in a multiprism spectrometer because of the many oblique reflections. It is convenient to introduce cross sections Qj(ll) and Q j ( l ) defined by Eq. (2), but with the total photon flux replaced by the fluxes of polarized photons 4n4jk(\\) and 4n4jk(l). From the total photon flux emitted perpendicular to the electron beam we calculate a cross section Qj(90) given by
'
Qj(90) = Qj(II) + Q j ( l ) * The excitation cross section which, by Eq. (I 1) takes account of the angular distribution of the emission, is given by Qj
=
SCQj(II) + 2Qj(J-)l.
The following relationships are occasionally useful in the analysis of observations or the comparison of different authors' work: 2Qj = Qj(901C3 - p j l ,
(14a)
280
D . W . 0 . Heddle and R . G . W. Keesing
[:1 3 7
2Qj(l) = Qj(90)[1 - P j ] = 2Qj -
P j is a convenience rather than a quantity of physical significance. The cross sections Qj(ll) and Qj(L) are, however, significant because they are related (Percival and Seaton, 1958) to the cross sections for excitation of the various Zeeman levels of the statej. Most experimental cross sections are presented as Qj(90), though unless the polarization sensitivity of the measuring system has been allowed for they are not Qj(90), but some weighted sum of Qj(ll) and Qj(L). Many authors " correct "their observations usingpublished polarization factors. Even if these factors applied to their experimental conditions, such correction is not meaningful unless the polarization sensitivity is measured and allowed for. Comparison of experimental and theoretical excitation functions is complicated by the fact that Qj is normally the quantity calculated. There are a few instances where the cross sections for excitation of Zeeman levels have been calculated (Massey and Moiseiwitsch, 1960) and here a comparison with polarization measurements can also be made. If the population of the state j is significantly by means other than direct electron excitation the polarization factor will be reduced. In the particular case in which the absorption of resonance radiation leads to an increase in Qj(90) Eq. (14d) shows that the polarization factor decreases in such a way as to keep the product Pi Qj(90) constant and equal to the difference in the excitation cross sections Qj(n) and Qj(L). The increase in Qj(90) is not without limit but only by a factor (from Eq. (9)) of A j / ( A j- Aj,) and so the polarization does not fall to zero. This provides us with a means of measuring the polarization of resonance lines while not requiring the polarization measurement, which involves a difference of photon fluxes, to be made at the lowest pressures. Until recently the measured polarization factors of almost every spectrum line studied tended to zero as the electron energy was reduced to the threshold value though it might be significantly different from zero at higher energies. In certain cases this is to be expected : in Fig. 10 we show the polarization factors found by Federov and Mezentsev (1965) for the 7 3 S , 6 3 P , , , triplet in mercury. Electron excitation is expected to lead to the emission of unpolarized light, but cascade population from higher' 3P levels leads to polarization. In the majority of cases, however, a finite polarization factor is expected at threshold and the experimental evidence is suspect. Improvement in electron energy resolution has led to the demonstration that many polarization functions have a minimum close to the threshold, but
MEASUREMENTS OF ELECTRON EXCITATION FUNCTIONS
28 1
r I0
t
t 6
9 I
1
\
10
II
12
I
I
I
13 I
Electron energy (volts)
FIG.10. The polarization of light from the 7 ' S state of mercury (Federov and Mezentsev, 1965). (a) 4047 A, (b) 5461 A, and (c) 4358 A.
rise again at the lowest energies. The polarization of light from the 4 ' 0 - 2 ' P transition of helium at 4922 A is shown in Fig. 11. Hafner and Kleinpoppen (1965, 1967) have measured the polarization factors near threshold for the resonance lines of the lithium isotopes and of sodium. Their results are shown in Fig. 12. Kleinpoppen and Neugart (1967) have shown that the threshold polarization factors are in excellent agreement with the polarization factors of resonance scattered plane polarized light and with values calculated by Flower and Seaton (1967).
IV. Simultaneous Ionization and Excitation Provided the electron energy is high enough an atom may be raised from its ground state to an excited state of its positive ion. Observations of this type of process have been made concurrently with simple excitation measurements since 1930 and data for the alkalis and the rare gases are available. In general the excitation functions show broad maxima at energies of 2 to 4
++
0
o
E
+
401 'Ot
I
I
24
1
I
25
1 -
26
I
1
27
Electron energy (volts)
FIG.11. The polarization of the 4 '0-2 'P line of helium. (+) McFarland (1964), ( 0 ) Heddle and Keesing (1967a), and(-)Federov and Golovanevskaya (1966).
L I
FIG.12. The excitation and polarization functions of the resonance lines of (a) lithium-6, (b) lithium-7, and (c) sodium (Hafner and Kleinpoppen, 1965, 1967).
MEASUREMENTS OF ELECTRON EXCITATION FUNCTIONS
283
times the threshold energy and the cross sections are small compared with those for the simple excitation of the neutral atom. More observations have been made on the 4686 8, line of helium than any other and these are shown in Fig. 13. The three most recent measurements are in excellent agreement. The values at the maximum are 10.6 x cm2 (Zapesochny and Feltsan, 1965) and 9.8 x cm2 (St. John and Lin, 1964).' The polarization function of this line has been measured by Haidt and Kleinpoppen (1966) and by Elenbaas (1930) though the latter measurement was made at such a high pressure that secondary processes must reduce its significance.
Ir
1
I00
I
I
150
I
I
200
I
I
250
Electron energy (volts)
em,,
FIG. 13. The excitation functions (from He(1 ' S ) ) of the 4686 A line of He+ (a) = 8.1 x cmz Thienie (1932), (b) Elenbaas (1930), (c) Zapesochny and Feltsan (1963), (d) em,,= 8.4 x lo-'' cmz Hughes and Weaver (1963), (e) = 9.8 x cm2 St. John and Lin (1964), (f) Haidt and Kleinpoppen (1966), and (g) em.,= 10.6 x lo-*' cm2 Zapesochny and Feltsan (1965).
em.,
* We are indebted to Dr. Lin for informing us of this revised value.
284
D. W. 0. Heddle and R. G. W. Keesing
There has recently been a revival of interest in simultaneous ionization and excitation because these processes are believed to be important in producing population inversion in argon and krypton lasers. The cross sections for excitation of several levels of Ar' relevant to the argon laser have recently been measured by Bennett et al. (1966) and Hammer and Wen (1967) and their values for the 4p2P,,2 state are shown in Fig. 14 together with those of Fischer (1933). Fischer's absolute calibration would appear to be seriously in error, and the agreement between the two recent measurements is very poor. Both were made under conditions of rather high pressure and very high current: [several hundred milliamperes in the case of Hammer and Wen (1967)l and we do not feel that they represent the behavior of only the simple process.
Electron energy (volts)
FIG.14. The excitation functions of the A + 4p 2P,,z state. (a) Fischer (1933), (b) Bennett el
al. (1966), and (c) Hammer and Wen (1967).
V. High Resolution Measurements Electron guns of the type shown in Fig. 2a can give electron currents of order 1 mA, but the energy spread in the exciting electrons is not small (several electron volts) because of field penetration. This type of electron gun was used almost exclusively before 1950 and consequently only the most prominent structural features of excitation functions were known until then. Several authors then developed electron guns in which field penetration was
MEASUREMENTS OF ELECTRON EXCITATION FUNCTIONS
285
much reduced: that of Jongerius (1961) and Smit (1961) is shown in Fig. 2b. These authors made observations with electron energy spreads of little more than 3- eV. Jongerius (1961) made an extensive study of the excitation functions of mercury which show a considerable amount of structure (see Figs. 6, 10, and 17). He measured the collision chamber potentials at which maxima occurred in the 5461 and 3650 A lines as functions of electron beam current and mercury vapor pressure and was able to show that the electron energy differed significantly from that given by the collision chamber potential except for currents of less than a few microamperes and that the difference changed abruptly (by about If V at 26pA) at the ionization potential. This is due to a depression of potential in the electron beam by the electron space charge which is neutralized by positive ions at higher energies to an extent dependant on the pressure, but which is fairly complete at pressures of 1 to 10 mTorr. The central potential depression is easily calculable for a cylindrical beam of radius r passing axially down a cylindrical electrode of radius R and is 6 V = 3 x 10-2iE-"2 ln(R/r) volts where i is the current in microamperes and E is the electron energy in electron volts. There will be a further potential spread across the electron beam which for a uniform beam is given by A V = 1.5 x 10-2iE-'/2 volts. The potential difference between the electron beam and the collision chamber is not in itself the cause of serious difficulty. There will in any case be a correction to the electron energy because of the contact potential difference between the cathode and the inside of the collision chamber. It is not unusual or alarming for thecollision chamber potential at which the onset of excitation occurs to be several volts different from the excitation potential of the state under study. The danger lies in the change of potential difference at the ionization potential for at this point a small change in collision chamber potential can cause a much larger change in electron energy and an excitation function, which is (for example) a linearly rising function of electron energy will appear to have a " shoulder " if presented against collision chamber potential. The situation might be eased by adding some gas of lower ionization potential to that being studied so that there is no serious discontinuity of energy, but we are unaware of any deliberate attempt to do this. It is probably much safer to work at a current low enough (a few microamperes) to reduce the effect to insignificance. In recent years more use has been made of electron optical theory to design electron guns. An example is shown in Fig. 2c (Simpson and Kuyatt, 1963). The collision chamber and electron beam collector are not depicted : the collection of electrons is discussed in detail by Kuyatt (1 968). Even in cases where there is no marked structure in the excitation function the use of too high a current or pressure can give misleading results. Figure 15 shows the effect on the excitation function of the D-lines of sodium (Zapesochny and Shimon, 1962).
D. W. 0 .Heddle and R . G. W. Keesing
286
b
.C
C 0
5
10
15
20
25
Electron energy (volts)
FIG. 15. The effect of excessive current and pressure on the excitation functions of the sodium D-lines (Zapesochny and Shimon, 1962). (a) p = 5 x lov4Torr, current density = Torr, current density = 1.5 x A/crnZ, and (c) A/cmZ, (b) p = 1 x 1.5 x Torr, current density = 10 x A/cm2. p =5 x
Heddle (1967) has discussed the effect of electron space charge and its neutralization by positive ions with particular reference to structure near the excitation threshold. The presence of detailed structure in electron excitation functions is of considerable interest and importance. Bates et al. (1950) emphasize the differing sharpness of excitation functions at low energies and Massey and Moiseiwitsch (1954) find an extremely sharp peak at the threshold for excitation of the helium 2 3 S state. In some cases threshold peaks have been observed with electron energy inhomogeneity of up to I eV, but there are probably many instances where a narrow peak has been overlooked because of inadequate energy resolution. The excitation functions of states of atomic hydrogen are of particular interest because of the comparative simplicity of the atom. Measurements are complicated by the fact that hydrogen is normally molecular and one has consequently to work with beams of hydrogen atoms produced by dissociation in a furnace or discharge. Fite and Brackmann (1958) were the first to measure the excitation function of the 2P state. The region near the threshold was studied by Chamberlain et al. (1964) who found evidence of structure and unfolded their measured electron energy distribution from their observed data and showed that there appeared to be a peak immediately above the threshold (Fig. 16). They used an electron gun similar to that of Fig. 2c with an axial hemispherical electron velocity analyzer (Simpson, 1961). A threshold peak of this general form has been predicted by Damburg and Gailitis (1963).
MEASUREMENTS OF ELECTRON EXCITATION FUNCTIONS
d
287
/
I 12 0
Collision chamber potential (volts 1
FIG.16. The excitation function of the Lyman c( line of hydrogen near threshold (Charnberlain et al., 1964); (0)observed data points, (-) unfolded excitation function.
Several attempts have been made to use the retarding potential difference technique of Fox et al. (1955) and in certain cases a detailed structure of sharp peaks has been seen. We believe, however, that these d o not represent the true excitation functions as such, because the technique gives an output which depends not only on the cross section, but on its derivative with respect to electron energy. (Heddle, 1964; Marmet, 1964.) Electron monochromators have been used for some years to study ionization, elastic scattering and in energy loss spectroscopy. Their design has been discussed in detail by Kuyatt and Simpson (1967). Zapesochny and Shpenik (1966) have used a 127" cylindrical monochromator to study the low energy portions of the excitation functions of certain states of helium, zinc, cadmium, mercury, sodium, and potassium with electron beams homogeneous in energy (to 90%) within 0.1 t o 0.5 eV. Their results for the 5461 A line of mercury (made with 0.15 eV energy spread) are shown in Fig. 17. Much of the structure seen in this excitation function is due to cascade population from higher states. This will be discussed further in Section VI. We have used a hemispherical electron monochromator to study excitation near threshold of a number of states of helium. Our results for the 4 3S and 4 ' S states made with a n energy spread of 0.1 eV are shown in Fig. 18 together
288
D. W. 0. Heddle and R. G. W. Keesing
Electron energy (volts)
FIG. 17. The excitation of the 5461 A line of mercury observed by Zapesochny and Shpenik (1966) with energy resolution of 0.15 eV.
Electron energy (volts)
FIG. 18. Excitation functions in helium, (a)-(e) 4 3Sand (f)4 ' s . (a) Heddle and Lucas (1963), resolution > I eV, (b) Yakhontova (1959), resolution 0.8 eV, (c) Smit ei al. (1963), resolution 0.4 eV, (d) Zapesochny and Shpenik (1966), resolution 0.3 eV, (e) and (f) Heddle and Keesing (1967b), resolution 0.13 eV.
MEASUREMENTS OF ELECTRON EXCITATION FUNCTIONS
289
with several other results for the 4 3 S state. An interesting feature is that the second maxima in the 4 3 S and 4 ‘ S excitation functions coincide in energy at approximately 23.90 eV and cannot be due to cascade population. These maxima appear to indicate the presence of a resonance in the He-e system. The energy resolution attainable in an excitation experiment depends not only on the spread in electron velocity, but on the motions of the gas atoms. A beam of perfectly monoenergetic electrons will appear to have a Gaussian energy distribution, of half width A E given by (Bethe, 1937)
AE = 7.23 x 1 0 - 4 ( ~ ~ / ~ ) 1 / 2 .
VI. Time-Resolved Measurements The radiative decay of atoms in excited state affords a means of measuring radiative transition probabilities. There have been a number of experiments in which electron impact has been used to produce the excited atoms. These have been reviewed by Bennett et al. (1965). In a number of cases the spectrum line intensities have been found to decay in a fashion more complex than exponentially with a single time constant. An example of such a decay is given in Fig. 19 (Osherovich and Verolainen, 1966) which shows observations made on the 3610 A line of cadmium (5 D,-5 3P2)at electron impact energies of 7.9 and 8.7 eV. In the former case a simple exponential decay is found with a time constant of 1.6 x sec, but in the latter case the decay is more complex and is resolved into the sum of two exponentials the second one
I? 1 1 1 1 1 0 24
72
I20
168
216
264
Erne n sac
FIG.19. The decay of excitation in the 3610 A line of cadmium (Osherovich and Verolainen, 1966). (a) excitation energy 8.7 eV, and (b) excitation energy 7.9 eV.
290
D.W. 0 .Heddle and R . G . W. Keesing
having a time constant of 11.5 x lo-’ sec. The excitation function of this line is shown in Fig, 20 (Zapesochny and Shevera, 1963). It appears from this that between 8 and 9 eV population by cascade transitions begins : the energy of the 4 3F4 state is 8.15 eV and atoms in this state can only radiate to the 5 3 D , state. The wavelength of the transition is 16,482 A.
II
I
10
I
I
15 20 Electron energy (volts)
I
FIG. 20. The excitation function of the 3610 8, line of cadmium (Zapesochny and Shevera, 1963).
Consider the population of a state j by electron impact and by cascade from a single higher state i d N j / d t = N,(I/e)Qj
+ NiAij - N j A j .
Suppose that equilibrium is established at t = 0 and that the electron current is then stopped. The equilibrium concentration of atoms in state j is given by N:
=
[N,(I/e)Qj
+ NioAij]/Aj.
For t > 0 and N i = Nioe-Air N j = Ce-AJ‘+ De-Al‘ where C = N; - D and D = NioAij/(Aj- A i ) . The two exponential decay constants of the upper curve of Fig. 19 are thus identified with the transition probabilities for radiative decay of the 5 D, and 4 3F4 states of cadmium. Consider now the growth of population in state j when the electron beam is first switched on: N J. = N J.‘[I - e-AJ’- Fe-”l’]
MEASUREMENTS OF ELECTRON EXCITATION FUNCTIONS
29 1
where
F=
Qi A j Aij
AdAj - A i ) ( Q j + ( Q i A i j / A i ) )
if state i is only populated by electron excitation. If A i A i j and A j >> A i N
Qi
F% (Qj
+ Qi)
'
Observations made of the photon flux emitted in a short (< l/Aj) time after the electron beam is switched on will give a result which depends largely on the cross section Qjand observations made after a time delay >l/Aj after the electron beam is switched off will give information on the population via cascade transitions. Bogdanova and Marusin (1966) have studied three mercury lines using this approach. Some of their observations on the 5461 A line are shown in Fig. 21. Curve (a) shows the excitation function found under conditions of steady excitation: it is clearly the same as that of Fig. 17, but with lower energy resolution. Curve (b) was taken 0.15 psec after the end of
Electron energy (volts)
FIG.21. The excitation function of the 5461 8, line of mercury (Bogdanova and Marusin, 1966). (a) Steady excitation; (b) 0.15 psec after the end of the exciting pulse; and (c) the first 0.1 psec of the exciting pulse.
292
D . W. 0 .Heddle and R . G . W. Keesing
the exciting pulse and illustrates the cascade contribution ; curve (c) was taken during the first 0.1 psec of the exciting pulse and shows the direct electron excitation. The subsidiary maximum near 11 eV also occurs in the excitation function of the 2536 8, line and in the ionization function (Hickam, 1954). There is an autoionizing level of configuration 5d9 6s2 6p 3P1in mercury at an energy of 10.95 eV. It is possible that it may be responsible for the observed feature. Lee (1939) has measured an excitation function for this state by observing electrons scattered in the forward direction with this energy loss. His results are shown in Fig. 22.
Electron energy (volts)
FIG.22. The excitation of the 5d 6sz 6p 3P1 state of mercury (Lee, 1939).
Kay and Hughes (1967) have made an extensive study of decay rates in helium. They have found that the decay of excitation in the 3 0 states at a pressure of 34 mTorr showed several exponential components which they identify as due to cascade population from F states. They found in addition that the higher (n 2 4) ' P states were collisionally coupled to the F states and were able to show that excitation is transferred from ' P states to F states in collisions of the type represented by Eq. (7). For n = 4 they found that transfer was mainly into the ' F state, but for n = 6 the 3Fand 'F states were populated in the ratio of (3.0 k 1.6): 1. The strongest component in the cascade appears to be the 6F-30 transition at 10,915 A which, as mentioned in Section II,C might be detectable.
VII. Related Measurements There are a number of experimental methods of studying electron excitation which do not involve the measurement of light. The threshold region of an inelastic process can be studied using the trapped electron method (Schulz, 1959) in which the electrons which have lost energy in a collision are collected. This technique has been applied by Korchevoi and Przhonskii (1966) to
MEASUREMENTS OF ELECTRON EXCITATION FUNCTIONS
293
measure excitation to the 6'P states of cesium. Their result is shown in Fig. 23 together with the absolute optical measurements of Zapesochny and Shimon (1966a) which required a careful extrapolation to remove the effects of self-absorption (see Fig. 8). It is difficult to use the trapped electron method at energies where more than a single inelastic process occurs, but it does give the total (i.e., over all angles) cross section. Energy analysis of scattered electrons enables one to measure the excitation function of any state which is sufficiently separated in energy from its neighbors. An angular distribution measurement must be made if the total cross section is required. The differential (forward scattering) cross section for
1
2
3
4
5
Electron energy (volts)
FIG.23. Excitation of the 6 2Pstate of cesium. (a) Trapped electron method (Korchevoi and Przhonskii, 1966) and (b) optical method (Zapesochny and Shimon, 1966a).
294
D. W. 0. Heddle and R. G. W. Keesing
excitation of the E C ' ,' state of nitrogen as found by energy loss measurements using a high resolution hemispherical electron monochromator (Heideman et al., 1966) is shown in Fig. 24. The excitation function for metastable state production in nitrogen (Olmsted et a/., 1965) is also shown in this figure. Excitation of the E-state leads to metastable production because the state decays radiatively to the A 3C,+ long-lived state with the emission of a band spectrum at wavelengths near 2500 A. The excitation function of the E-state has not yet been observed optically.
Electron energy (volts)
FIG. 24. Excitation in nitrogen. (a) Metastable production (Olmstead et al., 1965) and (b) 11.87 eV energy loss (Heideman et al., 1966).
VIII. Comparison of Observations Despite the large amount of work reported, the agreement between different observers is often poorer than one might expect. In some cases this is because the excitation functions show a structure on a scale comparable with the electron energy resolution : in others the excitation is strongly affected by pressure dependent processes, but there are many cases where such considerations do not apply. An example is the 4 ' S state of helium. This is studied by observing the 4lS-2'P transition at 5047 A a wavelength at which low noise photomultiplier tubes of reasonable quantum yield are readily available. The
295
MEASUREMENTS OF ELECTRON EXCITATION FUNCTIONS
light is emitted isotropically and is consequently unpolarized, so any polarization sensitivity in the instrument is quite unimportant. Population by processes other than direct electron excitation are of little importance; the greatest contribution will be by cascade from 'P states and this will be pressure dependent because of the enhanced 'P population a t high pressure as a result of resonance line absorption. However it can only amount to a few percent. The nearest line in the helium spectrum is at 5016 A and while this may well be much more intense there should be no problem in eliminating it. The results of four recent measurements of the excitation function of the 4 ' S state of helium are shown in Fig. 25. They are not to scale though the absolute values at the main maximum differ by very little. The agreement in the form of the excitation function is not really as good as one might expect and it is perhaps not surprising that for transitions which emit polarized light the agreement is worse. To a certain extent the disagreement is due to the use of unsuitable experimental conditions. We believe that this is the direct result of the fact that much of the definitive work is not well known. Consequently a great deal of experimental effort has been expended in repeating the errors of thirty years
g C
aE
-%
0
a"
I
40
I
I
60
80
I
I
100 20 40 Electron energy (volts)
I
60
I
80
I
200
I
I
FIG.25. The excitation function of the 4IS state of helium. (a) em,,= 2.5 x cm2 (Zapesochny and Felstan, 1965), (b) em,,= 2.4 x cmz (St. John et al., 1964), cm2 (Heddle and Lucas, 1963), and (d) = 2.0 x cm2 (c) = 2.6 x (Yakhontova, 1959).
em,,
em,,
296
D. W. 0. Heddle and R. G. W. Keesing
ago with modern electronic techniques. The work of the Joint Institute for Laboratory Astrophysics in maintaining an information center to collect reports and disseminate information to collision physicists deserves the widest publicity and the highest praise. However, we are encouraged by the quality of some of the most recent papers to hope that a closer agreement between the results of different workers will be attained.
REFERENCES Bates, D. R., Fundaminsky, A., Leech, J. W., and Massey, H. S. W. (1950). Phil. Trans. Roy SOC.(London) Af43,93. Bennett, W. R. Jr., Kindlmann, P. J., and Mercer, G. N. (1965). Appl. Opt. Suppl. Chem, Lasers p. 34. Bennett, W . R. Jr., Mercer, G. N., Kindlmann, P. J., Wexler, B., and Hyman, H. (1966). Phys. Rev. Letters 17,987. Bethe, H. A. (1937). Rev. Mod. Phys. 9,69. Bogdanova, I. P., and Marusin, V. D. (1966). Opt. Spectry (USSR) (English Transl.) 21, 148. Chamberlain, G. E., Smith, S. J., and Heddle, D. W. 0. (1964). Phys. Rev. Letters. 12, 647. Damburg, R., and Gailitis, M. (1963). Proc. Phys. SOC.82, 1068. Elenbaas, W. (1930). Z. Physik 59,289. Federov, V. L., and Golovanevskaya, L. E. (1966). Opt. Spectty (USSR) (English Transl.) 20, 419. Federov, V. L., and Mezentsev, A. P. (1965). Opt. Spectry (USSR) (English Transl.) 19,5. Fischer, 0. (1933). Z. Physik 86,646. Fite, W. L., and Brackmann, R. T. (1958). Phys. Rev. 112,1151. Flower, D. R. and Seaton, M. J. (1967). Proc. Phys. SOC.(London) 91, 59. Fox, R. E., Hickam, W. M., Grove, D. J., and Kjeldaas, T. (1955). Rev. Sci. Instr. 26, 1101. Franck, J., and Hertz, G. (1914). Ber. Deut. Phys. Ges. 16,512. Frost, L. S., and Phelps, A. V. (1957). Westinghouse Res. Lab., Res. Rept. 6-94439-6-R3. Gabriel, A. H., and Heddle, D. W. 0. (1960). Proc. Roy. Soc. (London) A258, 124. Hafner, H.,and Kleinpoppen, H. (1965). Phys. Letters 18,270. Hafner, H., and Kleinpoppen, H. (1967). 2. Physik, 198,315. Haidt, D.,and Kleinpoppen, H. (1966). 2. Physik 196,72. Hammer, J. M., and Wen, C. P. (1967). J. Chem. Phys. 46, 1225. Hanle, W. (1929). Z. Physik 56,94. Heddle, D.W. 0. (1962). J. Quant. Spectry & Radiative Transfer 2,349. Heddle, D. W. 0. (1964). Proc. Symp. Aromic Collision Processes in Plasmas, Culham, AERE-R4818, p. 88. Heddle, D. W. 0. (1967). Proc. Phys. SOC.90,81. Heddle, D. W. O., and Keesing, R. G. W. (1967a). Proc. Roy. SOC.(London) A299,212. Heddle, D. W. O., and Keesing, R. G. W. (1967b). Proc. Phys. SOC.,(London) 91,510.
MEASUREMENTS OF ELECTRON EXCITATION FUNCTIONS
297
Heddle, D. W. O., and Lucas, C. B. (1963). Proc. Roy. SOC.(London) A271, 129. Heideman, H. G. M., Kuyatt, C. E., and Chamberlain, G. E. (1966). J. Chem. Phys. 44, 355.
Heron, S . , McWhirter, R. W. P., and Rhoderick, E. H. (1956). Proc. Roy. Soc. (London) A234, 565. Hickam, W. M. (1954). Phys. Rev. 95, 703. Holstein, T. (1947). Phys. Rev. 72, 1212. Holstein, T. (1951). Phys. Rev. 83, 1159. Hughes, R. H., and Weaver, L. D. (1963). Phys. Rev. 132, 710. Jongerius, H. M. (1961). Philips Res. Rept. Suppl. No. 2. Kay, R. B., and Hughes, R. H. (1967). Phys. Rev. 154, 61. Kleinpoppen, H., and Neugart, R. (1967). Z . Physik 198, 321. Korchevoi, Yu. P., and Przhonskii, A. M. (1966). Soviet Phys.-JETP 23,208. Kuyatt, C. E. (1968). In “Methods of Experimental Physics” (L. Marton, ed.), Vol. 7A, Academic Press, New York, to be published. Kuyatt, C. E., and Simpson, J. A. (1967). Rev. Sci. Instr. 38, 103. Larche, K. (1931). Z . Physik 67, 440. Lee, A. H. (1939). Proc. Roy. Soc. (London) A173, 569. Lees, J. H. (1932). Proc. Roy. SOC.(London) A137, 173. Lees, J. H., and Skinner, H. W. B. (1932). Proc. Roy. SOC.(London) A137, 186. Lin, C. C., and Fowler, R. G. (1961). Ann. Phys. (N.Y.) 15,461. McFarland, R. H. (1964). Phys. Rev. 133, A986. McFarland, R. H., and Soltysik, E. A. (1963). Phys. Rev. 129, 2581. Marmet, P. (1964). Can. J . Phys. 42, 2120. Massey, H. S. W., and Mohr, C. B. (1931). Nature 127, 234. Massey, H. S. W., and Moiseiwitsch, B. L. (1954). Proc. Roy. SOC.(London) A227, 38. Massey, H. S. W . , and Moiseiwitsch, B. L. (1960). Proc. Roy. SOC.(London) A258, 147. Maurer, W., and Wolf, R. (1934). Z . Physik 92, 100. Olmstead, J., 111, Newton, A. S., and Street, K., Jr. (1965). J. Chem. Phys. 42, 2321. Osherovich, A. L., and Verolainen, Ya.F. (1966). Soviet Phys.-Doklady (English Transl.) 10, 951. Percival, I. C., and Seaton, M. J. (1958). Phil. Trans. Roy. SOC.(London) A251, 113. Phelps, A. V. (1958). Phys. Rev. 110, 1362. St. John, R. M., and Fowler, R. G. (1961). PhyT. Rev. 122, 1813. St. John, R. M., and Lin, C. C. (1964). J. Chem. Phys. 41, 195. St. John, R. M., and Nee, T.-W. (1965). J. Opt. SOC.Am. 55, 426. St. John, R. M., Miller, F. L., and Lin, C. C. (1964). Phys. Rev. 134, A888. Samuel, M. J. (1965). Private communication. Schulz, G. J. (1959). Phys. Rev. 116, 1141. Simpson, J. A. (1961). Rev. Sci. Instr. 32, 1283. Simpson, J. A., and Kuyatt, C. E. (1963). Rev. Sci. Instr. 34, 265. Smit, C. (1961). Dissertation, Univ. of Utrecht, Utrecht, Holland. Smit, C., Heideman, H. G. M., and Smit, J, A. (1963). Physica 29 245. Teter, M. P., and Robertson, W. W. (1966). J. Chem. Phys. 45, 2167. Thieme, 0. (1932). Z . Physik 78, 412. Wolf, R., and Maurer, W. (1940). Z . Physik 115, 410. Yakhontova, V. E. (1959). Vestn. Leningr. Univ. Ser. Fiz. i Khim. 14, 27. Zapesochny, I. P., and Feltsan, P. V. (1963). Bull. Acad. Sci. USSR, Phys. Ser. (English Transl.) 27, 1015. Zapesochny, 1. P., and Feltsan, P. V. (1965). Ukr. Fiz. Zh. 10, 1187.
298
D. W. 0 . Heddle and R . G . W. Keesing
Zapesochny, I. P., and Shevera, V. S. (1963). Bull. Acad. Sci. USSR,Phys. Ser. (English Transl.) 27, 1018. Zapesochny, 1. P., and Shimon, L. L. (1962). Opt. Spectry (USSR)(English Transl.) 13, 355. Zapesochny, I. P., and Shimon, L. L. (1964). Opt. Spectry (USSR) (English Transl.) 16, 504. Zapesochny, I. P.,and Shimon, L. L. (1966a). Soviet Phys.-Doklady (English Transl.) 11,44. Zapesochny, I. P., and Shimon, L. L. (1966b). Opt. Specrry (USSR) (English Transl.) 21, 155. Zapesochny, I. P.,and Shpenik, 0. B. (1966). Soviet Phys.-JETP (English Transl.) 23, 592.
SOME NEW EXPERIMENTAL METHODS IN COLLISION PHYSICS R . F. STEBBINGS Department of Physics, University College London, England
I . Introduction
..................299
..........
11. Flowing Afterglows . . . . ........................ 111. Merged Beams .................................................
300 .304
Ion Beam Measurements A. Angular Distributions i B. Angular and Energy Distributions of the Ejected Electrons ....... .313 C. Kinetics of Rearrangement Collisions ......................... .314 D. Excited States in Ion Beams ................................. .316 V. Electron Beam Measurements ...................... A. Studies with Monochromators . . . . . . . . . . . . . . . . . . B. Dissociative Ionization. The Angular and Energy Distributions of the Product Ions ........................................ .320 C. Collisions with Positive Ions ................................. ,321 VI. Photoelectron Spectroscopy . . . . . . . . . . . . . . . . . . ..324 VII. Metastable Atom Measurements ................................. .327 References ................................................... .329
IV.
I. Introduction In recent years an enormous growth in activity in the field of experimental atomic physics has been experienced. Great diversity in measurement has been achieved, and many new and unexpected phenomena have been observed. The advances have, in many instances, been linked to technological improvements ; in, for example, high vacuum technique, particle detectors, and data handling and processing, while the development of new and imaginative experimental techniques has allowed experimentation in many areas previously inaccessible to measurement. In this chapter some of the developments that have taken place within the last decade are discussed ; the emphasis rests primarily on the experimental method rather than upon the purpose or significance of the measurements which, in many instances, provide the motivation for other chapters in this book. It is not the intent to catalog all new experimental developments, but rather to consider a few that have had, or show promise of, particular significance in the development of this field. 299
300
R.F. Stebbings
11. Flowing Afterglows Considerable impetus to the study of thermal energy ion-neutral reactions was provided by the development of the flowing afterglow technique. Prior to this, much of the pertinent information had been derived from studies of pulsed stationary afterglows in which a discharge is struck in a cell filled with an appropriate gas or mixture of gases, and the behavior of the ionized gas is observed as a function of time following the termination of the discharge. This latter technique has been exploited by a number of workers (Fite et al., 1962; Langstroth and Hasted, 1962; Sayers and Smith, 1964) and useful data on a number of reactions have been obtained. However, this procedure does not appear to be suitable for the general investigation of thermal energy reactions because of a number of inherent limitations. Foremost among these is the lack of control or knowledge of the states of either the ions or the neutral particles populating the afterglow, while in addition, the method has not proved suitable for the study of chemically unstable neutral particles or negative ions. These limitations are in large measure overcome by the flowing afterglow technique, the essential features of which are illustrated in Fig. 1, which shows schematically the apparatus used by Ferguson and his colleagues at ESSA (Ferguson et al., 1965; Goldan et al., 1966). Helium gas is admitted at one end of a Pyrex tube 1 meter in length and 8 cm in diameter at a flow rate of about 200 atmospheric cm3 sec-' and flow velocity of 104cmsec-'. The tube is exhausted by a large Roots pump backed by a mechanical fore pump providing a pumping speed of 500 liters sec-', at pressures between about 10 and Torr. A discharge is established between a cylindrical cathode and a small wire anode which are placed close to the point of entry of helium into the tube. The discharge may
FIG.1. The ESSA flowing afterglow reaction system.
SOME NEW EXPERIMENTAL METHODS
30 1
be operated in either a dc or an ac mode, and produces ion densities of order 10" cm-3 together with a comparable density of He(2 3 S ) metastable atoms. Neutral reactants are then introduced, via small glass jets, into the helium afterglow, which extends downstream from the discharge region. The ion composition in the afterglow is ascertained at the end of the reaction zone by means of a frequency scanned quadrupole mass spectrometer, followed by a windowless particle multiplier or Faraday cup and electrometer. The decrease in primary ion current and the increase in the currents of reaction product ions are observed as the neutral reactant is admitted to the tube. The reaction rate constants are then calculated as follows. For the reaction A + + B + products, the rate constant K is defined by
dCA'1 -= -K[A+][B]. at Over the range of observable A + decrease, [ B ]% [A'], and [ B ] may be considered constant throughout the reaction zone. Then In([A']/[A+],)
= -Ks[B]
when [ A ' ] , is proportional to the A + ion signal before addition of B, and T, the reaction time, is simply the length of the reaction zone divided by the measured gas flow velocity. Thus the slope of a logarithmic plot of primary ion current against the concentration of the added neutral reactant, when divided by T, gives K. Corrections are made to this analysis to allow for the diffusion and incomplete mixing of the neutral gas after its introduction through a small nozzle. The great elegance and versatility of this technique lie primarily in the separate control that can be exercised over the ions and neutrals prior to reaction, and also in the fact that the atomic processes occurring in the afterglow are susceptible to many diverse forms of investigation. Reactions of helium ions are investigated by introducing the neutral gas of interest into the helium afterglow and observing the loss of helium ions as a function of the concentration of the added constituent. When molecular nitrogen is introduced into a helium afterglow, N + ions are formed through the reaction He+(lS) + N 2 ( X ' x , + ) + H e ( l S ) +N+(3P)+N(4S)
which has a rate constant of 1.7 x cm3 sec-'. Experimental data for this reaction are shown in Fig. 2. Nitrogen molecular ions are also observed and apparently result primarily from the slower reaction
+
He(2 3S) Nz+ H e
+ N,
+
rather than from He+ + N, collisions. This was demonstrated by applying 2450-Mc sec-' microwave power to the afterglow in order to increase the
302
R. F. Stebbings
electron temperature, and therefore the ambipolar diffusion of the helium ions and electrons to the tube wall. The N f signal was observed to decrease correspondingly while the N, signal remained unchanged, and therefore originated from an uncharged species, unaffected by diffusion, rather than from He+. +
3x
10-8
24
18 15
He++ Np+
He t N + N +
12 9
6
t
=
?
0 3 C
-
3x
24 18
15 I2
9
6
3 x 10 -I0
0
0.4
08
N,
12
16
20
24
26
flow (relative scale)
FIG.2. Ion currents as a function of N2 added t o a helium afterglow.
In general, the presence of both He' and He(2 3 S ) in a helium afterglow will lead to confusion, but in the flowing system one or the other may be removed by the addition of an appropriate contaminant. Thus argon may be introduced to destroy He(2 3 S ) selectively through Penning ionization, while O,, N,, and CO react an order of magnitude more rapidly with He+ than with He(23S), and may therefore be used to eliminate He+. In a stationary time-dependent afterglow this procedure is impracticable. Quite apart from providing an environment for the investigation of He+ reactions, the helium afterglow is used to provide a highly versatile source of secondary ions. Thus He+ reacts rapidly to produce 0' from 0, , N from +
SOME NEW EXPERIMENTAL METHODS
303
N,, C + from CO, while He(23S) rapidly ionizes most gases in Penning collisions. These secondary ions may then be observed in reaction with neutral gases introduced still further downstream. It has been increasingly appreciated over the past few years that the states of particles undergoing collision are frequently crucial in determining the magnitude of the cross sections or reaction rates. In the flowing afterglow, as distinct from the static arrangement, the states of the reactants are often known. The ions, for example, even if they are formed in excited states, will almost certainly be in their ground electronic state when they arrive at the position where the neutral gas is introduced, because of superelastic collisions with electrons. In some cases, of course, energy considerations dictate that the ions are only formed in the ground state, as for example in He+ + N, -+ N'(3P). The fact that excited ions experience superelastic collisions with electrons represents both an advantage and a limitation in that while the ion states are known, they are not susceptible to variation, and it would appear that alternative methods must be employed for the investigation of thermal energy collisions of electronically excited ions. For molecular ions the state of vibrational and rotational excitation is also of concern. In the case of N,' ions, optical measurements have demonstrated that the He(2 3S)-N2 Penning reaction leads primarily to N,+(B 'Xu+), which radiates to N2'(X ,C,+). Quantitative measurements of this radiation show that most of the N,+ ions end up in the ground vibrational state (Schmeltekopf et al., 1967). In the general case, however, it is to be expected that the molecular ions will be vibrationally excited. The excitation of the neutral reacting species is also influential in determining the course of reaction. In that the neutral gas is introduced into the tube downstream from the discharge region it is always possible, in the case of stable gases, to study the reactions of ground state atoms and molecules. In certain instances it is also possible to excite the neutral gas in a controlled way. Thus nitrogen may be vibrationally excited by running it through a weak discharge prior to its introduction into the afterglow tube. The rate constant for the reaction 0"+ N , + N O + + N
has been investigated and found to be extremely dependent upon the N, vibrational state (Schmeltekopf et al., 1967). Discrepancies between earlier time dependent afterglow measurements of the 0' N, + NO' rate constant may well have arisen from varying degrees of N, vibrational excitation resulting from differing discharge conditions. Atomic oxygen and atomic nitrogen may also be introduced into the flowing afterglow and their reaction with various ions investigated. N atoms are produced by discharging N, prior to its introduction into the flow tube. Only
+
304
R.F. Stebbings
weak dissociation-a few percent-is achieved, and investigations are limited to those reactions where the presence of the undissociated N, will not mask the effects resulting from the N atoms, The N-atom concentration is determined by the addition of NO, giving N
+ NO+Nz + 0.
This titration has a visible end point, and measurement of the corresponding NO flow rate gives the N-atom flow rate prior to NO addition. This procedure also provides a known flow of 0 atoms which may be varied in a controlled way from zero up to the flow rate of the N atoms. Reactions investigated in this way include (Ferguson el al., 1965) Oz+ + N + N O + $0
The interest in thermal energy reactions is certainly not confined to those involving positive ions. Reactions of negative ions are also of aeronomic interest and may be readily studied in the flowing afterglow configuration. This is in contrast to the limited success in negative ion studies with time dependent afterglows which were subject to an unresolved sampling problem associated, apparently, with plasma polarization. The ESSA group have investigated a number of negative ion reactions involving charge transfer, charged rearrangement, and associative detachment (Fehsenfeld et al., 1966). These latter collisions, of which 0-+ N O -+ NOz
+e
is an interesting example, constitute a different class of reaction in that the charged product is an electron rather than an ion, and either the electron or the molecule is detected in order to observe the occurrence of reaction. In concluding this section it should be pointed out that the use of flowing afterglow techniques is by no means limited to the ESSA group. Robertson, Kaufman, Schiff, and Young, and their respective colleagues, to name but a few, have all made valuable contributions in this area of research (Young and Sharpless, 1963; Collins and Robertson, 1964; Morse, and Kaufman 1965; Phillips and Schiff, 1965).
111. Merged Beams Afterglow studies are likely to be restricted in the immediate future to temperatures below about 600°K (+ eV) and at higher energies alternative techniques are required. Considerable success has been achieved using mass spectrometer ion sources (Giese, 1966) and mobility tubes (McDaniel et al.,
SOME NEW EXPERIMENTAL METHODS
305
1962) in the study of ion-molecule reactions within the energy region from thermal to a few electron volts, while at higher energies ion beam techniques are dominant. Quite recently an entirely new technique for the investigation of low energy ion-neutral collisions has been developed by Trujillo and his colleagues (1966). In this system two beams of'particles having laboratory energies in the kiloelectron volt range are caused to travel in the same direction along a common axis. When their laboratory energies are made nearly equal, the resulting energy of interaction between the particles in the two beams becomes extremely small. The interaction energy W (i.e. the relative energy in the center-of-mass coordinates) is given by
when p is the reduced mass and V 2 and V , are the laboratory velocities of the particles in the two beams. If E 2 , E , and m,, m , are the corresponding kinetic energies and masses of the particles
= j [ E i / 2- E i i 2 I 2
if m , = m 2 = m .
When the energy difference between the two beams, AE, is small compared to the beam energies, and E 2 z E , M E,
W z AE2/8E. Trujillo et al. define an energy deamplification factor
D = A E / W M 8ElAE 4 1. Thus the interaction energy W is small compared to the difference in the laboratory energies of the particles for large laboratory energies. For example, if El = 5000 eV, E, = 5100 eV, then A E = 100 eV and the energy of interaction W % 0.25 eV. It may further be shown that a variation of SE in A E gives approximately a variation 6 W in W where
SW 2 S E -2SE -=-W
WD
AE'
For an energy spread (full width at half maximum) of 1.5 eV in each beam (6E = & 1.5 eV), the corresponding spread 6 Win W is f0.0075 eV. Thus the technique permits the investigation of reactions at very low interaction energy, whilst retaining the advantages of a high energy experiment in terms of beam manipulation. The technique is, in principle, suitable for the general study of low energy two body reactions in that either of the two beams may be neutral or charged.
306
R.F. Stebbings
The apparatus developed by Trujillo and his colleagues for the study of ionneutral reactions is shown in Fig. 3. The Ar' ions are extracted from an electron bombardment source S , , mass analyzed, and focused at an energy E2 into a gas cell where charge transfer converts some of them into argon atoms having the same laboratory energy. The mixed (charged and neutral) beam which emerges from the chamber is mechanically chopped at 100 cps before the ions are removed by a transverse electric field. The neutral beam continues and, during passage through the second magnet, is merged with an ion beam extracted from a second identical ion source S, and having an energy El < E, . Charge changing collisions then occur
+
Ar+(EI) Ar(&) -+A@,)
+ Ar+(E2)
between the particles in the two merged beams until they enter the field of the demerging magnet which deflects the charged particles into a lens system which, by retardation, separates the slower primary ions from the more energetic secondaries. Prior to phase sensitive detection, the secondary ions are passed through a low resolution hemispherical electrostatic energy analyzer to prevent fast neutrals, resulting from change transfer with the background gas, from reaching the detector. The experiment consists of determining S, the number of Ar+ ions generated per second in the interaction region. The corresponding cross section Q is related to S by
Q = (S/I)(E,E,/mW)'/2 where I =
b
JIJz d x dy dz
and the integral is performed over the volume of the beams in the interaction region where J , and J , are the beam fluxes of the particles of energy E , and E, . In order to evaluate I-called by Trujillo et al. the overlap integral-the profiles of the two primary beams are determined along the length of the interaction region, and to eliminate end effects, S is determined as the difference between the signals obtained for two values of the interaction length. To date two experiments of this type have been reported, both for symmetric resonant change transfer. Trujillo et al. have published data for Ar+-Ar, while Belyaev et al. (1966) have reported measurements of H+-H at 15.9 eV. In the latter experiment only one ion source was used. The mixed proton and H-atom beam emerging from a charge transfer cell entered a region of higher potential when the protons were decelerated, thereby producing the required energy difference between the two beams.
SOME NEW EXPERIMENTAL METHODS
FIG.3. Schematic diagram of merging beams apparatus.
307
308
R. F. Stebbings
A particularly valuable feature of this approach to the study of ionic reactions is that by the use of an appropriate ion source, the states of the ions are subject to some control. It is normally straightforward, for example, to obtain beams of ground state ions, although beams of ions in a single excited state are not yet readily obtainable. However, when one of the two beams in a merged beam experiment is comprised of neutral particles formed through change transfer, they will, in general, be present in both the ground state and a variety of excited states. In the case of H-atom beams all excited atoms will either decay rapidly or may be readily quenched giving a pure ground state H-atom beam. For rare gases the relative abundancies of excited atoms may well be quite small, but in general it is to be expected that the excited state population may be high and, in the case of molecular beams, vibrational and rotational excitation must also be anticipated. In that the low energy cross sections under investigation are likely to be critically dependent upon the state of excitation of the neutral, as well as the ionic, species, considerable care will need to be exercised in such studies. At present the techniques necessary to analyze an energetic neutral beam in terms of the excited species are not developed. It would seem certain, however, that the merged beam technique will find increasing application in the future, and indeed experiments are currently under way to investigate ion-ion mutual neutralization in this way. Electron-ion recombination may also be studied, although when ions and electrons move at comparable velocities their laboratory energies are quite disparate, and it will be necessary to operate at rather high ion energies in order that the energy of the electrons shall not be unduly small.
JY.Ion Beam Measurements Until the last few years the experimental study of ion-atom collisions, using beam techniques, was primarily concerned with measurements of the gross production of slow ions and electrons. Recent refinements of technique have, however, opened up new avenues of investigation. Measurements of the angular, charge, and energy distribution of the heavy products of collision have provided information on the kinetics of the collision, while optical and allied measurements have allowed identification of the states of the product particles. The energy and angular distributions of the secondary electrons have been determined in a limited number of cases, and progress has been made in determining the influence of the state of the primary ion upon the course of the reactions. Coincidence measurements have been used (Afrosimov et al., 1964, Everhart and Kessel, 1965) to permit simultaneous determination of the energy, charge state, and scattering angle of both the projectile and the recoil particle following a violent collision.
SOME NEW EXPERIMENTAL METHODS
309
A. ANGULAR DISTRIBUTIONS IN ELASTICAND INELASTIC SCATTERING Much valuable information on the forces between interacting particles has been extracted from measurements of the total elastic scattering of beams of high energy projectiles by gases. Notable among this work was that of Amdur and Simons and their co-workers which has been discussed in detail by Mason and Vanderslice (1962). Interpretation of these data is complicated, however, by the fact that several different forms of the interaction potential V(r) can reproduce a given set of measured cross sections, and it is necessary to have some prior knowledge of V(r) if a unique solution is to be obtained. This difficulty is removed if the angular distribution of the scattered particles is measured, since V ( r ) may then be calculated without assumption as to its form other than that it be a monotonically decreasing function. Everhart and co-workers (Ziemba and Everhart, 1959; Ziemba et al., 1960) in the USA, and Fedorenko and co-workers (Fedorenko, 1954; Fedorenko. et al., 1960) in the USSR, pioneered angular distribution studies of this type. In similar studies involving scattering of He' at lower energies, Lorents and Aberth (1965) used the apparatus shown schematically in Fig. 4.The He' ions are produced by low energy electron impact and are extracted to form a beam, with an energy spread of about 2eV, which emerges into the main experimental chamber through a rectangular slit. An identical collimating slit is located downstream at the entrance to the scattering chamber which is constructed of three concentric cylindrical elements. The exit slit is located in the outer element which may be rotated together with the detector assembly about the axis of the gas cell. Ions which are scattered through this slit are energy analyzed with a 127"cylindrical electrostatic analyzer, of mean radius 5.5 cm, prior to detection with a Bendix multiplier. The differential scattering cross section a(0) is then given by
when Z(0) is the ion current detected at a scattering angle 0, I, is the incident ion current, n the atom density in the collision chamber, and /Ax(e) o dx is the geometrical factor obtained by integrating the scattering solid angle w as seen by an element dx of the primary beam path along Ax, the total length along the beam path that contributes the scattered signal. The elastic differential cross sections obtained between 20 and 600 eV are shown in Fig. 5. The observed structure exhibits three distinct features which are attributed to various interference phenomena. The prominent sinooth oscillations result from interference between the waves scattered from the lowest gerade and ungerade potentials describing
3 10
R.I;: Stebbings
FIG.4. Schematic diagram of apparatus for determining the angular and energy distributionsof scattered ions.
0
3
6
9
I2 15 I8 21 24 27 30 33 36 39 8(degrees) lab coordinates
FIG. 5. Elastic differential scattering cross sections for He+ on He at incident energies from 20 to 600eV. The proper cross section scale is identifiedat 10-’4cmz by theintersection of a horizontal line with each curve.
R. F. Stebbings
312
18
20
22 24 26 Energy loss (eV)
28
FIG.6. Energy loss profile of 600-eV He+ ions scattered inelastically from He atoms at a number of angles. The energy spread of the incident beam is indicated by the profile of the elastically scattered ions at 1.6", which is displaced by 20.0 eV.
the He2+ ion. These are predicted by the impact parameter theory and were first observed by Ziemba and Everhart (1959). The increase of the cross sections above the upper envelope of the oscillations, which is observed at low energies for small angles, is attributed to rainbow scattering: The small oscillations, which are superimposed upon the major oscillations at large angles at high energies, are attributed to an interference between direct scattering at 8 in the center-of-mass system, and scattering with charge exchange at ll - 8. This has been experimentally verified by scattering
SOME NEW EXPERIMENTAL METHODS
313
4He+ from 3He where, because of the absence of nuclear exchange symmetry, these secondary oscillations do not occur. This work, which was concerned primarily with elastic scattering, has been extended by Lorents et al. (1966) to include investigation of the differential cross sections for excitation of the 2 3 S and higher states of He by 600-eV He' ions. The apparatus is similar to that used for the elastic scattering measurements except that the primary beam collimating apertures were made smaller to improve the angular resolution at small scattering angles. The scattered ions were also retarded to 90 eV or less prior to analysis to improve the energy resolution. Energy loss profiles were obtained at angles between 0.5" and 4", and a representative sample is shown in Fig. 6. The resolution is sufficient to distinguish the scattered ions that have excited the target atoms to 2 3 S from those that have excited the other N = 2 states. Perhaps the most remarkable feature about these curves is the abrupt change in the relative magnitudes of the different excitation processes that occurs with small changes in angle, suggesting that the excitation of a given state is an oscillatory function of angle. The angular dependence of the 2 3 S cross section was determined by monitoring, as a function of angle, those ions that had suffered an energy loss of 19.96 eV, and integration of these angular data yielded a value for the total cross section for 2 3Sexcitation by 600-eV He' ions of 6.9 f 3.5 x lo-'* cm'.
B. ANGULAR AND ENERGY DISTRIBUTIONS OF THE EJECTED ELECTRONS Considerable attention has been focused upon the photons and electrons resulting from the impact between heavy particles. Optical methods have been widely used to determine the states of the charged and neutral collision products, the photon wavelengths being most commonly determined through the use of optical spectrometers, although suitably filtered multipliers and photon counters have also been employed. When long lived excited products are formed, as in H+
+ Cs
+
H(2s)
+ Cs'
alternative detection schemes are required. Some experimental results are also available on the energy and angular distributions of the free electrons produced in heavy particle collisions. Differential cross sections for ejection of secondary electrons at various angles and energies were measured by Kuyatt and Jorgensen (1963) for hydrogen bombarded by protons. This work was extended by Rudd and Jorgensen (1963) and by Rudd and Lang (1965), who used a parallel plate electrostatic energy analyzer with a resolution of 0.25 eV to observe the energy distribution of the electrons ejected at 160" from various gases when
R . F. Stebbings
314
bombarded by positive ions from a Cockcroft-Walton accelerator. In addition to the continuous spectrum of electrons, they observed considerable structure in certain energy regions, which is due to autoionization of highly excited states of the target gases. Many of the levels associated with the structure are observed also in ultraviolet absorption (Madden and Codling, 1963, 1964) and in electron scattering experiments (Silverman and Lassetre, 1964 and Simpson et ~ l . 1964). , Part of the spectrum observed by Rudd and Lang in the bombardment of helium by protons and H2+ ions is shown in Fig. 7. The scale labeled excitation energy differs from that labeled electron ejection energy by the ionization potential of helium plus a small correction term which allows for contact potentials and space charge. The peaks result from autoionization of doubly excited helium states having the indicated electron configurations. The 2s2p3P peak in the H2+data arises from a collision involving electron exchange, and hence is absent from the proton data. 50
-
40
Y2
! 30
r
f 20
U
s
r
Excitation energy ( e V )
1
32
57
58
33
59
34 Electron election energy ( e V )
60
35
36
FIG.7. Energy spectrum of the electrons ejected from helium by Hz+and H+.
c. KINETICSOF REARRANGEMENT COLLISIONS At low impact energies collisions between heavy particles may lead to rearrangement, as in N + +Oz-+NO+ f O .
Processes of this type have been studied in afterglows, drift tubes, ion sources, and in beam experiments, the principal information acquired being the total collision cross section and its variation with energy and temperature. The
SOME NEW EXPERIMENTAL METHODS
315
details of the collision kinematics and dynamics are not revealed since they may only be exposed through measurements of the angular and energy distribution of the reaction products. The first measurements bearing on this problem were reported by Turner et al. (1965) who investigated, in a crossed beam apparatus, the angular distribution of the N2D+ ions resulting from the reaction Nz+ + D2 4NZD+ + D
for N 2 + ions in the energy range 7.5 to 57.5 eV. Conservation of energy and momentum dictates that the N2D+ ions are confined to a narrow cone whose axis is only slightly displaced from that of the primary ions. There are, in consequence, two possible center-of-mass scattering angles and, therefore, two values for the laboratory energy of the scattered ion corresponding to each laboratory scattering angle. The scattered ions were sampled with a quadrupole mass filter which, providing the ion transit time is sufficiently long, discriminates singly charged ions solely on the basis of their masses, irrespective of their energy or momentum. Thus both energy groups of the secondary ions were simultaneously measured. Differential cross sections were obtained as a function of laboratory scattering angle at various ion energies. From a comparison of the observed angular distributions with those computed on the basis of different scattering models, it was concluded that the results were consistent with the formation of an activated complex whose lifetime against dissociation was at least comparable to its rotational period. The angular resolution of the equipment was not, however, sufficiently high that stripping or rebound mechanisms (Herschbach, 1966) could be excluded. This reaction has since been investigated further by Doverspike et al. (1966) who determined both the angular and energy distributions of the product ions. They directed a mass analyzed and velocity selected N,' beam into a collision chamber containing the target gas. The secondary ion detection system, which could be rotated about the center of the scattering region, comprised a 127" electrostatic velocity selector followed by a quadrupole mass spectrometer. When the primary ion energy was below 70 eV, Doverspike et al. observed two peaks in the product ion energy distribution. Accurate measurement of the flux of the lower energy group was difficult, but its contribution was about 20% of that of the higher energy group at the lowest collision energy of 4.1 eV, and less than 2 % above 50 eV. These results imply that at the lowest energies the reaction proceeds via an activated complex, although at the higher energies the absence of a low energy group indicates that a stripping process may be operative. In neither experiment is the angular resolution sufficient to expose any fine structure that may be present, and further detailed work in this area is clearly needed.
316
R.F. Stebbings
D. EXCITED STATES IN ION BEAMS It is well known that the manner in which ions react in collision is dependent upon their state of excitation. This is most easily seen from the variation in the magnitude of a particular collision cross section which is observed, at fixed ion energy, as the energy of the ionizing electrons in the ion source, and therefore the state composition of the resulting ion beam is changed. Typical of such measurements are the data of Stebbings et al. (1966) shown in Fig. 8 for the reaction O+ + N z + O + N z + .
When the electron energy is below the threshold for O'('D), the primary ion beam is pure ground state O'(4S), and the N,' production is zero. As the electron energy is raised above the O'('D) threshold some N,' production is observed, and with further increase in electron energy, a rapid increase in the N,' production cross section results from the increasing fractional abundance of O'('0) ions in the primary beam. Data of this type, though useful, are of limited value unless the state composition of the primary ion beam is also known as a function of the electron energy, since only then may the individual cross sections appropriate to discrete ion states be evaluated. A technique has been developed for this purpose which may be understood as follows. A beam of ions o f type 1 passing through a scattering gas is attenuated according to I, = Zloexp(-nQlx)
when Zlo is the initial current, Z, is the current unscattered after passage through a distance x of the scattering gas whose number density is n, and Q is the total scattering cross section. A plot of In I, against n gives, in Fig. 9, a straight line from the slope of which Q, may be obtained. For a beam of ions of type 2 where Q, > Q,, a similar line, 2, of greater (negative) slope, is obtained while the attenuation of a mixed beam of type 1 and type 2 ions has the form of line 3. At sufficiently large pressures ions of type 2 are almost totally eliminated from an initially mixed beam, and the subsequent attenuation will be indistinguishable from that for the beam of type 1 ions. A linear extrapolation of this portion of the attenuation plot to zero pressure then gives Zl0, the incident flux of ions of type 1, as intercept. ZZo, the incident flux of type 2 ions, is then simply the difference between Zlo and the total incident flux. This procedure was developed by Stebbings et al. (1966) to investigate the ' beam, and extended by Turner et al. (1966) in a study composition of an 0 of 0,'. In each case the scattering gases were selected so that the ground state
SOME NEW EXPERIMENTAL METHODS
317
14 1
a 12
0
a
10
8 N
E
0
P
6
-
0.
0
20
0
100 eV ions
eV ions
b
4
2
n 0
20
40
I
1
60
80
I
I
120
140
I
100
Electron energy ( e V
1
I
160
I80
200
1
FIG. 8. Dependence of the cross section for N2+production in O+-N2collisions upon the energy of the electrons used in the ion source to produce O + from molecular oxygen. Data are given for two ion energies, 20 eV and 100 eV. The threshold energies for O+(4s) and O + ( 2 D are ) indicated.
I
1
I
2
I
I
3
4
.
5
I
I
I
6
7
1
I
8
9
I
1
0
Pressure
FIG.9. Attenuation of differently constituted ion beams.
(type 1) ions experienced only elastic scattering, while the excited (type 2) ions also suffered appreciable inelastic scattering. At present this technique has been applied to the analysis of ion beams in which only two states appear to be present in significant and comparable amounts. It is not yet possible to analyze, in this way, .beams that contain several different species or to identify components present in only small fractional abundancies. Nonetheless, the procedure appears to be capable of considerable refinement and may well
R. F. Stebbings
318
prove suitable for the general study of ion beam composition. In Table I the ' ions are given provisional results published by Stebbings et al. (1966) for 0 together with the results of Turner et al. (1966) for 02+ions. ' data may then be coupled with the data shown in Fig. 8 to obtain These 0 the cross sections for the excited ions. Stebbings et al. conjecture that the principal process contributing to the N,' production is
+
+
O + ( 2 D ) N 2 ( X ' & ) ~ =+ o O(3P) N z + ( A'nu)
cm2.
for which the cross section at 100 eV is about 28 x TABLE I Energy of electrons bombarding O2in the ion source (eV) ~~
~
16 21 25 30 50 100
O + ground state fraction ~~
O + excited state fraction
0 2 +ground state fraction
Oz+ excited
0
1 0.88
0.25
-
0 0.12
0.34 0.44
0.69 0.74
0.31 0.26
state fraction
~
1
-
0.75 0.66 0.56
-
-
V. Electron Beam Measurements A. STUDIES WITH MONOCHROMATORS The course of electron impact studies within the past few years has been influenced considerably by the introduction of improved forms of electron monochromators. There are currently four main types in use, of which perhaps the most widely used is that based on a device described by Clarke (1954) and improved by Marmet and Kerwin (1960). In essence the system is a 127" sector cylindrical electrostatic deflector which produces a focused ribbon beam of electrons. Another form of electrostatic monochromator, used by Kuyatt and Simpson (1967) and by Meyer et al. (1965), utilizes 180" deflection between concentric hemispheres. This configuration provides focusing in two directions and may be used advantageously in conjunction with an axially symmetric lens system to form a narrow pencil beam of electrons. In the device used by Boersch et al. (1962) the electrons pass normal to crossed electric and magnetic fields whose strengths are adjusted to give no deflection to electrons of the required energy, while a modified Ramsauer technique employing deflection in a magnetic field has been used by Golden and Bandel (1965).
319
SOME NEW EXPERIMENTAL METHODS
These devices have been used for a variety of experiments involving elastic and inelastic scattering. One particularly valuable contribution of high resolution electron spectrometry has been the experimental verification of the theoretically predicted resonances in the scattering of electrons by atomic and molecular systems. This work is treated in detail by Burke elsewhere in this book. Another problem of considerable theoretical significance that has been investigated with these devices is the precise form of the ionization efficiency curve for atomic hydrogen near threshold. Many theoretical approximations predict that the cross section just above threshold is proportional to (E, - IP)" where E, is the electron energy, IP is the ionization potential, and n ranges from 1 to 1.5 in the various approximations. Pertinent experimental data shown in Fig. 10 have recently been acquired by McGowan and coworkers (1967) using a 127" analyzer with a resolution of 0.05 eV (full width at half maximum). They find that for the first 0.03 eV above threshold, the (E, 0 020
- IP) for n = I00 ( e v )
0
01
I
I
03
02
H ( l s ) + e-, H + + 2 e Ionization threshold laws
A,
0 05eV
- 0 015
N
-g -0 C
u
Y,
0010
Y)
P
Threshold n = I 127 and I 50
-g c
5!
0 -
0 005
0
0
01 (E,-IP)
0 2
0 3
for n = I127 and l 5 0 ( e V )
FIG. 10. The measured ionization cross section for H(1s) near threshold shown with calculated cross sections. A good fit between the experimental data, shown as open circles, and the n = 1.127 calculated curve is obtained if the energy scale is displacedfrom that associated with the linear curve by -0.03 eV.
R.F. Stebbings
320
form of the ionization efficiency curve is not linear but may be reproduced most satisfactorily when their measured energy distribution is folded into a 1.127 power law as predicted by Wannier (1953). For the next 3 eV the ionization efficiency curve is essentially linear. It had been customary in earlier experiments to establish the electron energy scale by setting the linear extrapolation of this part of the curve at 13.6 eV, but because of the nonlinearity observed at threshold, McGowan et al. conclude that this procedure is in error by 0.03 eV. Support for this conclusion is provided by the better agreement between experiment and theory in the positions of the 'S and 3P elastic resonances in H, and the autoionizing structure near the Hz ionization threshold that results from a displacement of the energy scale by 0.03 eV.
B. DISSOCIATIVE IONIZATION. THEANGULAR AND ENERGY DISTRIBUTIONS OF THE PRODUCT IONS The energy and angular distributions of the fast protons resulting from the dissociative ionization of the Hzby electrons with energies up to 1500 eV have been determined by Dunn and Kieffer (1963) using the apparatus shown schematically in Fig. 1 1. The energy distributions they obtained are, unlike the earlier data which are summarized by Stevenson (1960), consistent with the predictions of the Franck-Condon rule. Their measurements of the angular distribution of 8.6 eV protons for various electron energies are shown in Fig. 12. The solid lines are obtained by empirical choice of P(E, E ) in the expression
z@(E,8) = zg0.=(~, &)[I + P(E, &)
COSZ
el
C
FIG. 11. Schematic drawing of the apparatus used by Dunn and Kieffer. The electron gun G is attached directly to the cylindrical scattering chamber S which can be rotated about its axis. Ions formed at the center of the scattering chamber drift out through a slot in S through the lens system L1, L2,L 3 ,L4 into the spectrometerA, which focuses ions with the appropriate momentum into the exit slit El.
SOME NEW EXPERIMENTAL METHODS
32 1
which has the form encountered in electric dipole radiation processes. E and E refer to the ion and electron energies. The measurements clearly show the change from near isotropy at high electron energies to marked anisotropy near threshold. The lack of forward-backward symmetry is thought to be associated with asymmetries in the collision chamber. They account for these observations by noting that, following a transition to an antibonding state of the molecular ion, dissociation typically occurs in a time short compared to the period of molecular rotation. The dissociation products, therefore, separate along the line containing the axis of the molecule at the instant of collision. The angular distribution of the products is then a direct indication of the varying probability of dissociative ionization with orientation of the molecule relative to the direction of the incident electron. It is evident that the observed anisotropies necessitate extreme caution in measurements of the cross sections and energy distributions of dissociation products.
'0
350
20
330
40
310
60
290
80
270
100 120 140 160 I80 250 230 210 190
Angle (degrees )
FIG.12. Angular distribution of 8.6 eV protons for various electron energies. The data are corrected for the variation in the scattering volume with angle by multiplying by lsin 81. Data taken at angles symmetric to the electron beam axis (left-right) are averaged and the curves normalized to unity at 90" (270").
C. COLLISIONS WITH POSITIVE IONS Inelastic collisions between electrons and positive ions are of considerable astrophysical interest, while reactions involving the simpler ions, such as He+, are also theoretically important and have been treated in a number of approximations.
322
R.F. Stebbings
Experimental study of this class of reaction was not successfully undertaken until Dolder et al. (1961) carried out measurements of the electron impact ionization of He' in a crossed beam experiment. Their apparatus is shown schematically in Fig. 13. Ions from the source S were accelerated to an energy of 5 keV before mass analysis in an electromagnet M I , which steered the He' beam through the interaction space B, where it was crossed at right angles by electrons passing from the gun G to the Faraday cup C , . The He2' ions produced by electron impact travelled together with the primary He' ions to the analyzer magnet M 2 , which deflected the He' ions into the collector C , and the He2+ ions into C2 which was connected to a vibrating reed electrometer. He2+ ions were also formed in stripping collisions between the He' ions and background gas, which was subject to pressure variations due to the evolution of gas by the impact of the electron beam on the collector surfaces. To prevent the introduction of error due to this evolved gas, both charged beams were pulsed with a time period short compared with the pumping time constant of the interaction region. The HeZ' current resulting from e-He+ collisions was then determined as the difference between the currents observed when the two beams were in coincidence and in anticoincidence. The two modes were alternated every 30 sec.
60 cm
t
FIG.13. Schematic views of the apparatus used by Dolder et al. (1961).The upper diagram shows the side elevation and the lower the plan view: S ion source; L lens; dl and d2 deflector plates used to align the He+ beam; M I and M z electromagnets; B interaction space; G electron gun; C3 electron collector; CI, C 2 , C, ion collectors; A l dc amplifier; A Z vibrating reed electrometer; R1 and R2 pen recorders.
SOME NEW EXPERIMENTAL METHODS
323
This period was sufficiently long so that short term fluctuations of the He2+ current were averaged by the electrometer, but short enough so that errors due to slow changes of experimental conditions were avoided. The results are shown in Fig. 14. The cross sections for the single ionization of a number of other ions have since been determined using the crossed beam technique and are discussed b y Kieffer and Dunn (1966). The excitation reaction e
+ He+(ls)+e + He+(2s)
has also been investigated (Dance et al., 1966) while, more recently, results for the dissociation reaction e+H,++e+H+ + H
have been reported (Dunn and Van Zyl, 1967; Dance e l al., 1967). The major difficulty encountered in these crossed charged beam experiments results from space charge interaction between the two beams. The energetic target ion beam invariably produces a background current at the detector due to its interaction with the residual gas. Deflection of this beam by the space charge field of the electrons may cause a change in the background current that is indistinguishable from true signal current. This effect is generally
I I I J I I I / I I I I I I I I 3.0
9.6
2.0
,
2.5
Loq,, E ( e V )
FIG. 14. Cross section for ionization of He+ by electron impact.
324
R . F. Stebbings
investigated by looking for variation of the measured cross section with ion beam energy or by looking for signal current below the threshold for the electron-ion reaction. These crossed beam measurements are now being complemented by techniques (Baker and Hasted, 1966) whereby ions are spatially confined by a multipole trap or by electron space charge, and are then subjected to electron bombardment.
VI. Photoelectron Spectroscopy When an atom or molecule is photoionized, the excess of the photon energy over the energy of the ionized state appears as kinetic energy of the charged products. Conservation of momentum requires that, to about 1 part in lo5, this energy is all acquired by the photoelectron. Measurement of the energy distribution of the electrons resulting from ionization by photons of well defined energy, thus provides detailed information on the ionization potentials of the target system and the related ionization cross sections. Prior to the use of this technique, information on higher ionization potentials had been largely derived from the use of threshold photon and electronimpact techniques together with observations of fluorescence resulting from decay of excited ions to lower states. These threshold techniques suffered from the lack of suitably intense sources of monoenergetic photons or electrons, and also from ambiguities due to alternative modes of ion production such as autoionization and ion pair production. Photoelectron spectroscopy appears to have been first applied by Kurbatov et al. (1961), although a number of groups are now actively engaged in studies of this nature. Turner and his colleagues have obtained the photoelectron spectra of a number of gases. In an early form of their apparatus (Al-Joboury and Turner, 1963), the neutral gas was ionized by a beam of helium resonance radiation (584 A, 21.21 eV) from a discharge source. The energies of the resulting electrons were determined by retardation in an electrostatic field between two cylindrical grids which were coaxial with the light beam. The electrons which penetrated this retarding field were collected on an outer cylinder, biased to prevent the collection of positive ions. The photoelectron spectrum was then obtained as the derivative of the collector current with respect to the retarding potential. Similar measurements have been reported by Schoen (1964), who found that for radiation in the wavelength range 500-700 A more than half the ions formed in N 2 , 02,and CO were in excited states. A major limitation of this cylindrical retarding potential system is that it determines only the component of electron energy normal to the photon
SOME NEW EXPERIMENTAL METHODS
325
beam, and the system is therefore highly sensitive to the angular distribution of the electrons as well as to their kinetic energies. In consequence, the peaks in the resulting photoelectron energy spectra exhibit certain characteristic asymmetries, and although the vibrational components are seen, their separation is inadequate to allow accurate comparison of their relative intensities. In an improved arrangement used by Turner and May (1966) the photoelectrons emitted in a small arc (&7"), at right angles to a monochromatic beam of 21.21 eV photons, are energy analyzed in a 180" magnetic analyzer which has a resolving power EjAE of about 40. At the lower energies this represents a considerable improvement over the earlier retarding potential arrangement, and as an additional benefit the peaks are nearly symmetric in form. In that the probabilities for vibrational and electronic excitation are separable, the intensities in the vibrational fine structures can be directly related to the corresponding Franck-Condon factors. Thus the electron flux associated with a given vibrational component of the electron spectrum will be proportional to the product of c i ,the cross section for ionization to the electronic state in question, and a Franck-Condon factor. The FranckCondon factors may, in consequence, be estimated from the relative heights of the vibrational components in an electron energy spectrum. Correction must first be made, however, for the variation of c i with electron energy. In addition, because the spectra are obtained by scanning the magnetic field, the electron energy bandwidth increases with electron energy, and the heights of the vibrational peaks must be corrected by dividing them by the square root of the corresponding electron energy. The results of Turner and May for O2 are shown in Fig. 15. The first band 02'(X2n,)is not well resolved because of the relatively large kinetic energy of the electrons. The 0 c 0 component occurs at an ionization potential IP = 12.08 eV, and four additional components with a mean spacing of 0.22 eV are seen. Lines of length proportional to the calculated Franck-Condon factors (Hallmann and Laulicht, 1965) are shown at each of the observed peaks. The second band is ascribed to a transition to the 411u state of the ion, and the heights of 15 vibrational peaks observed in this band are in excellent agreement with the calculated FranckCondon factors. The 'IT, state IP = 17.18 eV is not observed, but the vibrational structure of the higher 4Xe- state is moderately well resolved. The identity of the fourth band whose 0 c 0 component appears at 20.29 eV is uncertain, although Turner and May conjecture that it may be a 4Custate. The vibrational peaks are almost totally resolved, and their positions and experimentally derived Franck-Condon factors are given in the table. Franck-Condon factors have also been obtained experimentally for a number of gases, including 0 2 by , Berkowitz et al. (1967) and by Puttkammer and Spohr (1967). In the experiment of Berkowitz et al. shown in Fig. 16 a beam of 21.21 eV photons crosses a molecular beam, and the angular and
R. F. Stebbings
326 02+, 4 Z
o,+,~z;
~;,4n,
v' 0 I 2 3 4 5 6
u' 0
IP 1817 I 1833 2 1845 3 1858 4 18 71
u'
IP FCF 2029 18 2042 24 2055 22 2067 16 2078 12 2088 04 2096 03
0 I
2 3 4 5 6 7
0,C.2n9-0, x329-v"='
IP 16 12 1626 1637 1649 1660 1672 1683 1693
IP 1208 12 32 12 54 12 73
ur
0 I
2 3
u
v1
\
v)
c
0 0
1000
1
I
1
1
I
1
1
1
I
2
3
4
5
6
7
8
3
1 9 10
Electron energy ( e V )
FIG.15. Photoelectron spectrum of oxygen excited by helium resonance radiation obtained by Turner and May (1966). The ionization potentials for the vibrational levels associated with the four electronic states are indicated.
FIG. 16. Apparatus used by Berkowitz et al. (1967) to measure photoelectron angular and energy distributions.
SOME NEW EXPERIMENTAL METHODS
327
energy distributions of the resulting electrons are determined within the angular range 30-130" with an energy resolution of about 40 mV. A slight preference for electron ejection along the direction of the light propagation is found in the formation of the electronic ground state 02'(X217,). A similar behavior is observed in the formation of NO+(X'C+) which also involves the ejection of an electron from a l7, orbital. The great wealth and directness of the information which may be acquired in this way make photoelectron spectroscopy a powerful new technique. In addition, it may be conjectured that ion sources utilizing photoionization will find increasing application in ion beam experiments in view of the detailed information that is gained about the state composition of the resulting ion beam.
VII. Metastable Atom Measurements Many recent investigations of excited atoms and molecules have centered upon the Penning ionization process A*+B+A+B++e
whereby an electronically excited atom or molecule A* may ionize an atom or molecule B when the excitation energy of A* exceeds the ionization potential of B. Cermak (1966a) has made use of this mechanism to determine the excitation energies of many long lived excited molecular states. In his experiment the neutral gas, whose excited states are of interest, is admitted from a multi-channel source into the excitation region where it is crossed by an electron beam. Electric fields confine the charged particles to this excitation region, but some of the resulting long lived excited neutrals pass into an ionization region into which various gases-alled detector molecules-are admitted through another multi-channel tube. By using detector molecules B of successively decreasing ionization potential, the various long lived excited states of A may be distinguished according as their excitation energy exceeds or falls below the ionization potential of a given target molecule. Complications due to associative ionization A*
+ BC+
ABC+
+e
may be avoided through mass analysis of the product ions. In a more recent version of this apparatus, the kinetic energy of the electrons resulting from Penning ionization has been determined by a Lozier stopping-potential method. Cermak (1966b) investigated the reactions He(2 IS,2 3S)+ A
+ He(1
IS)
+A+ + e
in this way and identified two groups of electrons differing in energy by 0.8 eV, which is the energy separation of the 2 3 S and 2 ' S states (19.81 and 20.61 eV).'
R. F. Stebbings
328
The individual electron excitation functions for the two metastable levels were determined from the fluxes of the electrons of energy 4.06 eV( = 19.81 15.75 eV) and 4.86 eV(=20.61 - 15.75 eV), obtained as the energy of the exciting electrons was varied. Collisions between the metastable helium atoms and a variety of molecular gases have also been investigated in a similar manner (Cermak 1966~).Typical of this work are the data for nitrogen shown in Fig. 17 which give the variation, with stopping potential, of the electron current to the detector together with the differential of this curve. Peaks la, 2a, 3a are associated with ionization of N2 by 2 ' s atoms and peaks lb, 2b, 3b with 2 3 S atoms. They correspond to the processes N2+(X2C,+ He(2 'S, 2 3S) N2+ N2+(A211.)
+
"2+(B
+ He(1 ' S ) + e.
2C,+
The onsets of peaks 2b and 3b are separated from the onset of peak l b by 1.1 and 3.3 eV respectively, which are in close agreement with the spectroX 2 C , + and B2C,+- X2X,+ scopically determined values for the A 211useparations. The peaks in the ground state and the B2X,+ state are narrower than those for A211,, suggesting that more vibrational levels are excited in the latter processes. The energy resolution is inadequate to allow detailed comparison of the excited state population resulting from metastable atom impact with that for
Stopping potential V (volts)
FIG. 17. Spectrum of electrons ejected from N2 by helium metastable atoms. Curve 1 is the collector current, curve 2 is the differential of curve 1 with respect to the stopping potential.
SOME NEW EXPERIMENTAL METHODS
329
photoionization, although Cermak concludes that ionization of N, , CO, COS, and CO, occurs probably in a Franck-Condon transition, and that all energetically accessible ionic states are populated. Ionization of NO, however, appears to proceed through preionization of an HeNO complex, in that electronically excited states of NO? are not populated.
REFERENCES Afrosimov, V. V., Gordeev, Yu. S., Panov, M. N., and Fedorenko, N. V. (1964). Zh. Tekhn. Fir. 34, 1613. Al-Joboury, M. I., and Turner, D. W. (1963). J. Chem. SOC.,p. 5141. Baker, F. A., and Hasted, J. B. (1966). Phil. Trans. Roy. SOC.(London) 261, 33. Belyaev, V. A., Brezhnev, B. G., and Erastov, E. M. (1966). JETP Letters (English Transl.) 3,207. Berkowitz, J., Ehrhardt, H., and Tekaat, T. (1967). 2.Physik. 200,69. Boersch, H., Geiger, J., and Hellwig, H. (1962). Phys. Letters 3,64. Cermak, V. (1966a). J. Chem. Phys. 44,1318. Cermak, V. (1966b). J . Chem. Phys. 44,3774. Cermak, V. (1966~).J. Chem. Phys. 44,3781. Clarke, E. M. (1954). Can. J. Phys. 32,764. Collins, C. B., and Robertson, W. W. (1964). J. Chem. Phys. 40, 701. Dance, D. F., Harrison, M. F. A., and Smith, A. C. H. (1966). Proc. Roy. SOC.A290, 74. Dance, D. F., Harrison, M. F. A., Rundel, R. D., and Smith, A. C. H. (1967). Proc. Phys. SOC.92,577. Dolder, K. T., Harrison, M. F. A., and Thonemann, P. C. (1961). Proc. Roy. SOC.A264 367. Doverspike, L. D., Champion, R. L., and Bailey, T. L. (1966). J. Chem. Phys. 45,4385. Dunn, G. H., and Kieffer, L. J. (1963). Phys. Rev. 132,2109. Dunn, G. H., and Van Zyl, B. (1967). Phys. Rev. 154,40. Everhart, E., and Kessel, Q. C. (1965). Phys. Rev. Letters 14,247. Fedorenko, N. V. (1954). Zh. Tekhn. Phys. 24, 784. Fedorenko, N. V., Filippenko, L. G., and Flaks, I. P. (1960). Soviet Phys. Tech. Ph. (English Transl.) 5 , 45. Fehsenfeld, F. C., Ferguson, E. E., and Schmeltekopf, A. L. (1966). J. C. R . 45, 1844. Ferguson, E. E., Fehsenfeld, F. C., Goldan, P. D., and Schmeltekopf, A. L. (1965). J . G . R . 70, 4323. Fite, W. L., Rutherford, J. A., Snow, W. R., and van Lint, V. A. J. (1962). Discussions Faraday SOC.33,264. Giese, C. F. (1966). Ado. Chem. Phys. 10, 247. Goldan, P. D., Schmeltekopf, A. L., Fehsenfeld, F. C., Schiff, H. I., and Ferguson, E.E. (1966). J. Chem. Phys. 44,4095. Golden, D. E., and Bandel, H. W. (1965). Phys. Rev. 138,14. Hallmann, M.,and Laulicht, I. (1965). J. Chem. Phys. 43,1503. Herschbach, D.R. (1966). Advan. Chem. Phys. 10, 319. Kieffer, L. J., and Dunn, G . H. (1966). Rev. Mod. Phys. 38, 1 . Kurbatov, B. L., Vilesov, F. I., and Terenin, A. N. (1961). Soviet Phys. " Doklady" 6, 490,883.
R.F. Stebbings
330
Kuyatt, C. E., and Jorgensen, T. (1963). Phys. Rev. 130, 1444. Kuyatt, C. E., and Simpson, J. A. (1967). Rev. Sci. Instr. 38, 103. Langstroth, G. F. O., and Hasted, J. B. (1962). Discussions Furuduy SOC.33, 298. Lorents, D. C., and Aberth, W. (1965). Phys. Rev. 139, 1017. Lorents, D. C., Aberth, W. and Hesterman, V. W. (1966). Phys. Rev. Letters 17, 849. Madden, R. P., and Codling, K. (1963). Phys. Rev. Letters 10, 516. Madden, R. P., and Codling, K. (1964). Phys. Rev. Letrers 12, 106. Marmet, P., and Kerwin, L. (1960). Can. J. Phys. 38, 787. Mason, E. A., and Vanderslice, J. T. (1962). In “Atomic and Molecular Processes” (D. R. Bates, ed.). Academic Press, New York. McDaniel, E. W., Martin, D. W., and Barnes, W. S. (1962). Rev. Sci. Instr. 33, 2. McGowan, J. W., Finernan, M. A., Clarke, E. M., and Hanson, H. P. (1967). Gen. At. Rept. GA-7387 Pt. 1 ; Phys. Rev.. To be published. Meyer, V. D., Skerbele, A., and Lassettre, E. W. (1965). J. Chem. Phys. 43, 805. Morse, F. A., and Kaufman, F. (1965). J . Chem. Phys. 42, 1785. Phillips, L. F., and Schiff, H. I. (1965). J . Chem. Phys. 42, 3171. Puttkammer, E., and Spohr, R. (1967). Z. Nuturforch. To be published. Rudd, M. E., and Jorgensen, T. (1963). Phys. Rev. 131, 666. Rudd, M. E., and Lang, D. V. (1965). Proc. Intern. Con5 Phys. Electron. At. Collisions, 4rh, 1965. Science Bookcrafters, New York. Sayers, J., and Smith, D. (1964). Discussions Furuduy SOC.37, 167. Schmeltekopf,A. L., Fehsenfeld, F. C., Gilman, G. I., and Ferguson, E. E. (1967). Planetary Space Sci. 15, 401. Schoen, R. I. (1964). J. Chem. Phys. 40, 1830. Silverman, S. M., and Lassettre, E. W. (1964). J. Chem. Phys. 40, 1265. Sirnpson, J. A., Mielczarek, S. R.; and Cooper, J. (1964). J. Opt. SOC.Am. 54, 269. Stebbings, R. F., Turner, B. R., and Rutherford, J. A. (1966). J. Geophys. Rec. 71, 771. Stevenson, D. P. (1960). J. Am. Chem. SOC.82, 5961. Trujillo, S. M., Neynaber, R. H., and Rothe, E. W. (1966). Rev. Sci. Instr. 37, 1655. Turner, D. W., and May, D. P. (1966). J. Chem. Phys. 45,471. Turner, B. R., Fineman, M. A., and Stebbings, R. F. (1965). J. Chem. Phys. 42,4088. Turner, B. R., Compton, D. M. J., and McGowan, J. W. (1966). Gen. At. Rept. GA-7419. Wannier, G. H. (1953). Phys. Rev. 90, 817. Young, R. A., and Sharpless, R. L. (1963). J. Chem. Phys. 39, 1071. Ziemba, F. P., and Everhart, E. (1959). Phys. Rev. Letters 2, 299. Ziemba, F. P., Lockwood, G. J., Morgan, G. H., and Everhart, E. (1960). Phys. Rev. 118, 1552.
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE M . J . SEATON Department of Physics, University College London, England
................................................... ......................................... General Theory ............................................. Hydrogenic Systems .........................................
I. Introduction
.331 ,332 A. .332 B. .334 C. Nonhydrogenic Systems ....................................... .346 111. The Forbidden Lines ........................................... .356 A. Historical Introduction ........................................ 356 B. Expressions for Level Populations and Line Intensities ............358 .361 C. Calculations of Collision Strengths ............................. References ..................................................... .378 11. Recombination Spectra
I. Introduction The study of physical processes in gaseous nebulae has led to many pioneering investigations in atomic physics, plasma physics, and radiative transfer theory. The present article is concerned with atomic collision processes in ionized nebulae, characterized by bright emission lines in the visible spectrum. There are two main types, the diffuse nebulae, such as Orion, which are of irregular shape and contain a number of hot stars, and the planetary nebulae, which are more regular and contain a single hot star. So far as atomic processes are concerned these two types are very similar. Typical values for the electron density, N, , are generally in the range lo3 to lo4 cm-’, but values as high as lo6 to lo7 cm-3 occur in certain dense planetaries and values as low as 1 electron cm-3 occur in diffuse nebulae. The chemical compositions are such that for every 10,000 atoms of hydrogen there are about 1500 atoms of helium and 10 atoms of all other elements. The electron temperatures T, are of order lo4 O K . The primary physical process is photoionization of ground state atoms and ions by ultraviolet stellar radiation. Most of the free electrons are produced by ionization of hydrogen, H ( l s ) + h v - + H ++e.
33 1
(1)
M. J. Searon
332
Electron-electron collisions are very effective in setting up a Maxwellian distribution for the free electrons. Radiative recombination can take place to any excited state, H++ e + H(nl) + hv' (2) and processes of recombination and cascade produce the observed spectrum lines of H I, He I, and He 11. The free electrons can also produce collisional excitation of low-lying metastable states. The 02+ion has a ground state ls22s22p23P and metastable states ls22s22p2 D and 'S. The inelastic collision process 03+(3k')+e+02+(1D)+e (3) is followed by the radiative transition 2p2 D 3P, which produces the green " nebulium " lines at 114959, 5007 A; these are the strongest observed lines in many nebulae. Similar forbidden transitions occur in other ions, such as O', N', Ne2+,S', S2', and Ar2'. Many of the basic ideas required for an understanding of the physics of gaseous nebulae were first developed about 40 years ago by I. S. Bowen, D. H. Menzel, and H. Zanstra. A great deal of subsequent effort has been devoted to obtaining accurate observations and to calculating accurate reaction rates for atomic processes. It is now possible to make detailed quantitative interpretations of nebular spectra and to obtain results which are of very general astrophysical importance. Thus, for example, the abundances of a number of chemical elements are known more accurately for gaseous nebulae than for any other astronomical object. For general accounts of problems of gaseous nebulae we refer to the books by Vorontsov-Velyaminov (1948), Wurm (1954), Dufay (1954), Aller (1956), Pikelner (1961), and Gurzadian (1962) and to review articles by Seaton (1960a), Osterbrock (1964), Pottasch (1965), and Dieter and Goss (1966).
II. Recombination Spectra A. GENERAL THEORY Consider a radiation field with intensity Yv;Yvdv dw is the radiant energy crossing unit normal area per unit time in the frequency range dv and in the solid angle dw. With N ( X ( i ) ) atoms per unit volume in level i, the number of photoionizations, X ( i ) + h v + X + +e, (4) per unit volume per unit time is
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
333
where vi is the threshold frequency and av,ithe photoionization cross section. In gaseous nebulae the radiation intensity is less than the intensity of a blackbody at the electron temperature, B,(T,), by a factor of order Stimulated recombination may therefore be neglected. The number of spontaneous recombinations
(6)
X+ +e+X(i)+hv
per unit volume per unit time is N , N(X(i))ai(Te)where
and where
=
(f)
'I2
e-h(v
Wih3V2aV,i
-vi)/kTe
w,~~(rnkT,)~/*
In Eq. (8) mi is the statistical weight of level i (for nl states of hydrogen wnl= 2(21+ 1)) and w + is the statistical weight of the recombining ion. Let
,4i,i,be the probability per unit time for the radiative transition X ( i ) + X(i')
+ hv.
(9)
If it is assumed that the populations of the excited states are determined by the capture and cascade processes (6) and (9), the equilibrium equations are N , N + ai
+ c> Nit. i" i
=Ni
c
,
i'
where N + = N ( X + ) , N i = N ( X ( i ) ) , and where we take i" > i to mean that the excitation energy of level i" is greater than that of level i. The level populations obtained on solving Eqs. (10) are proportional to N , N + . They are sometimes expressed in terms of factors bi which provide a measure of the departures from the thermodynamic equilibrium values, as given by the Saha equation:
In considering the recombination spectrum it is convenient to introduce efectioe recombination coefficients ciiVit for the spectrum lines, such that cii,i, N + N , is the total number of quanta per unit volume per unit time emitted in the i + i' line. Another useful quantity is mic, which is such that aicN, N + is the total number of atoms entering level i due to direct capture on i and to capture on higher levels followed by cascade to i.
M . J. Seaton
334 B. HYDROGENIC SYSTEMS
For hydrogenic systems the transition probabilities (Green, et al., 1957) and recombination coefficients ai (Burgess, 1964) are known exactly. A great deal of effort has been devoted to the problem of solving the capture-cascade equations and to interpreting the H I and He I1 spectra observed in nebulae.
1. Calculations Neglecting Orbital Angular Momentum Quantum Numbers
In all of the earlier work an implicit assumption was made concerning the populations of degenerate states. For a given value of the principal quantum number n it was assumed that the populations of the quantum states nl were proportional to the statistical weights, wnI= 2(21+ 1); if this is true we have N,, =
+
(21 1) n2 Nn ~
Calculations assuming Eq. (1 2) were made by Plaskett (1928), Carroll (1930), and Cillit (1932) for finite numbers of levels, and by Baker and Menzel (1938) for an infinite number. Techniques which may be used to solve Eqs. (10) for an infinite number of levels are discussed by Seaton (1959), who shows that the method first employed by Plaskett is very convenient. This paper also corrects some numerical errors in the work of Baker and Menzel. Baker and Menzel introduced the important distinction between nebulae which are optically thin (case A) and optically thick (case B) in the Lyman lines. For case B it is assumed that radiative transitions to the ground state are exactly balanced by reabsorptions from the ground state. The only modification required in Eqs. (10) is to omit, in the sum on the right-hand side, the transition probabilities to the ground state. It is to be expected that case B applies for most nebulae. In the earliest work on recombination spectra it was hoped that electron temperatures could be deduced from comparisons of observed and calculated relative intensities. It turns out that this cannot be done, since the relative intensities are insensitive to T, . Fortunately, T, can be determined rather well by other methods (see Section 111). Comparisons of observed and calculated relative intensities enable us to obtain a value for the amount of interstellar reddening. If Fo(A) is the observed flux due to a spectrum line at wavelength A and Fc(A) the flux which would be observed in the absence of reddening, then
log Fc(A) = log F&)
+ c[1 + f(4]
(13)
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
335
where f ( A ) is a known function. This function, as tabulated by Seaton (1960a), is such thatf(1) = 0 at the wavelength 4861 8, of the HP line and f(1)= - 1 in the limit of 1 -,co.The constant c in Eq. (13)is then the logarithmic extinction at HP. For a well-observed bright planetary nebula we may expect to have 40 or more observed lines in the recombination spectra of H I, He I and He I1 and we have to consider whether it is possible to find a value of c which brings all of the observed relative intensities into agreement with the calculated values. A further check may be obtained by considering the emission at radio wavelengths (see Section II,B,7). In Table I we give the relative intensities Z, of H I and He I1 lines observed by Minkowski and Aller (1956) for the bright planetary NGC 7662, the TABLE I RELATIVE INTENSITIES OF H I AND HE 11 LINESIN PLANETARY NEBULA NGC 7662
THE
HI, n + 3 11 12 13
6.2 2.8 2.0
3.2 1.5 1.1
1.5 1.2 1.o
269 100 51 28.7 10.3 7.8 6.3 5.2 4.7 3.1 2.1 1.8 1.3 1.o
257 100 49.8 29.0 9.0 6.7 5.0 4.0 3.2 2.1 1.4 1.2 1.o 0.9
53
53
H I , n-2 3 4 5
6 9 10 11 12 13 15 17 18 19 20
390 100 4
4
23.2 7.7 5.7 4.6 3.8 3.4 2.2 1.5 1.3 0.9 0.7
He 11, n -+ 3 4
51
He 11, n + 4 7 9 11
17
6.0 2.4 1.4 0.6
5.2 2.6 1.7 0.8
5.2 2.8 1.6 0.5
336
M . J. Seaton
intensities Z, corrected for reddening, and the intensities IR calculated by Seaton (1959) for Case B, T, = 1.5 x lo4 "K. The H I intensities are relative to HB, 4 -,2, and the He I1 intensities are relative to the intensity of the 4 -,3 line. 2. Calculations Including Orbital Angular Momentum Quantum Numbers
Although the observational results of Table I show reasonable agreement with calculations made under the assumption that the populations of the nl states are given by Eq. (12), it is clearly desirable that the validity of this assumption should be further examined. The first calculations taking account of the individual nl states were made by Searle (1958), who solved for all nl states with n < 10, and by Burgess (1958) who solved for all nl states with n < 12 and added corrections, as calculated by Baker and Menzel, for n > 12. Finally, the full equilibrium equations (10) were solved exactly by Pengelly (1964), taking all values of n and 1 into account. In Table I1 we give (for T, = 1 x lo4 OK and cases A and B) the values of b, obtained by Seaton (1959) in calculations assuming (12) and values of bnl obtained by Pengelly not assuming (12). If (12) is valid, for each value of n, b,, is independent of 1. It is seen that this is far from being the case for Pengelly's results. We note that, for case A, b,, is small for 1 = 1, due to the large transition probabilities for transitions to the ground state. An interesting feature of Pengelly's results is that the b,ls are large for the states of largest
COEFFICIENTS6, AND
TABLE I1 b., FOR HYDROGEN, T, = 1 x lo4 "K
Case A
0.0369 0.0910 0.145 0.193
0.607 0.809 0.947 1.04
0.0265 0.0518 0.0706 0.0842
0.119 0.210 0.276 0.335
1.01 1.12 1.19 1.24
0.265 0.320 0.384 0.429
0.0839 0.143 0.184 0.214
0.269 0.332 0.372
0.460 0.524
0.599
0.269 0.332 0.372
0.460 0.524
0.599
Case B
0.0855 0.146 0.189 0.221
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
337
angular momentum, 1 = n - 1. Captures on states of large angular momentum tend to cascade down to n, 1 = n - 1. From these states the only possible radiative transitions are to n' = n - 1, I' = n' - 1. In consequence the lines n + n' = n - 1 are stronger when (12) is not assumed. In Table I11 we give relative intensities for some H I lines (relative to 100 for HB) and some He I1 lines (relative to 100 for He I1 4 + 3), assuming (12) TABLE I11 CALCULATEDRELATIVEINTENSITIES OF H I
AND
HE 11 LINES'
Intensity
n
Assuming (12)
Not assuming (12)
3 4 5 6 7 8 9 10 11 12 13 14 15
27 1 100 50.6 29.8 19.2 13.2 9.5 7.1 5.4 4.2 3.4 2.8 2.3
287 100 46.6 25.6 15.8 10.5 7.3 5.3 4.0 3.0 2.4 1.9 1.5
100
100
HI, n+2
He 11, n +3 4
HeII,n+4 5 6 7 8 9 10
21.9 14.6 9.9 7.3 5.4 4.2
23.7 13.4 8.0 5.3 3.6 2.6
6 7 8 9 10
6.8 5.0 3.7 2.8 2.1
8.3 5.2 3.4 2.4 1.7
He 11, n --f 5
' Case B, T, = 1 x lo4 OK.
M. J . Seaton
338
and not assuming (12). Comparisons made by Pengelly (1964) and by Kaler (1964) show that the relative intensities calculated assuming (12) agree better with observations than the calculations not assuming (12). In Table IV we give effective recombination coefficients for HQ and for He I1 4 -+ 3. The two calculations are seen to give similar results for HQ but markedly different results for the He I1 line. TABLE IV
EFFECTIVE RECOMBINATION COEFFICIENTS FOR HYDROGENIC IONS' 10'44~p) 10-4 x T, Assuming (12) Not assuming (12) N. N lo4 ~ 1 1 1 a
1
2
1
2
3.00 3.07
1.60
3.02
1.60
20.8 37.8 29.4
11.8 18.0 14.8
=
~ ~
1 0 1 4 4 11, ~ ~4 -+ 3)
1.64
an,",in cm3 sec-'.
3. Collisional Redistribution of Angular Momentum
Equation (12) will be valid if collisional processes leading to a redistribution of angular momentum are faster than radiative processes. Pengelly and Seaton (1964) considered the redistribution of angular momentum due to electron, proton, and a-particle impacts, H(nl)
+e
-+
H(nl f 1)
+e
(14)
They found that these processes have large cross sections when n is large. In order to obtain finite cross sections it is necessary to introduce a cutoff in the impact parameter; this may be determined by Debye shielding or by the fact that the atom may radiate during the collision. It is found that, for electron densities of lo4 cm- typical of planetary nebulae, the probability of collisional redistribution of angular momentum is greater than radiative transition probabilities for n 2 15 in H and n 2 22 in He+. This redistribution among the higher states can have a significant effect on the intensities of the transitions between the lower states. This effect is illustrated in Table IV, where we include effective recombination coefficients calculated by Pengelly (unpublished) taking explicit account of the redistribution of angular momentum at densities of lo4 cm-3.
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
339
4 . Collisional Redistribution of Energy For large electron densities, or for large values of the principal quantum number n, one has to consider collisional processes which redistribute energy in addition to the processes (15) which redistribute angular momentum. The capture-cascade equations (10) cannot be exact for highly excited states, since at the series limit there is a clear discontinuity in the physical assumptions which are made. For the free electrons it is assumed that redistribution of energy in electron-electron collisions is much faster than radiative and inelastic collision processes, and hence that a Maxwell distribution is set up. This distribution is used in calculating the recombination coefficient cli in Eqs. (10). On the other hand it is assumed in (10) that, for all of the bound states, radiative processes are much faster than collisional processes which redistribute energy. In Fig. 1 we plot values of b, obtained (Seaton, 1964) from a solution of (10) as a function of An, the energy per reciprocal centimeter measured from the series limit; do = - 109737/n2. For the continuum, A 6 > 0, the assumption of a Maxwell distribution implies that b is equal to unity. It is seen that the b, factors calculated from (10) have a discontinuous behavior at the series limit.
L
n 60
50 I
80
loo
150
4)
I
I
I
I
I
10
0.9 b”
071
I
-60
I
-50
I
-40
I
- 30
I
-20
I
- 10
1
0
I
Au. cm-‘
FIG. 1 . Factors b. for hydrogen calculated neglecting collisional redistribution of energy (curve I) and allowing for collisional n + n f 1 transitions and for collisional ionization and its inverse (curve 11). Calculations for T. = 1 x lo4 O K , N. = 1 x lo4 ~ r n - ~ .
M. J. Seaton
340
Some approximate calculations allowing for collisional redistribution of energy have been made by Seaton (1964). Account is taken of transitions between neighboring levels, H(n)
+ e zz H(n + 1) + e ,
(16) for which fairly accurate cross sections are available (Saraph, 1964), and for collisional ionization and its inverse, (17) for which crude classical estimates were employed. No account was taken of collisional n --in’ transitions, with n’ # n i-1. The results obtained in these calculations are also shown in Fig. 1. According to these results, collisional redistribution of energy is important only for large values of n, n 2 40 at N, = lo4 cm-’. In this region the Balmer lines may not be resolved but will appear as a continuum, the observed continuum intensity being proportional to b, . The shape of the continuum in the region of the Balmer limit is modified by collisional processes : without collisional redistribution there is a sharp intensity drop at the limit but with redistribution there is a more gradual transition extending over about 5 A. The effect should therefore be clearly observable on high dispersion spectra. The calculation of populations of highly excited states is also of interest for the interpretation of the radio spectrum (see Section II,A,8). H(n) + e
H+ + e + e ,
5. Further Comparisons with Observed H I and He II Line Intensities The comparison between observed and calculated H I and He I1 intensities given in Table I is typical of the results obtained for a number of nebulae. The agreement is as good as could be expected when account is taken of the probable observational errors and of uncertainties in the theory. Anomalous results have been reported for the bright planetary NGC 7027 (Aller et al. 1955; Aller and Minkowski, 1956). It was suggested by Seaton (1960b) that these might be due to calibration errors, such that the intensities of all infrared lines were underestimated and the intensities of all weak lines were overestimated. Subsequent observational work has confirmed the correctness of the first of these suggestions, but not the second: it appears that the weak recombination lines from highly excited states (n 2 IS) have observed intensities which are larger than the calculated intensities (Kaler, 1964, 1966). No satisfactory explanation of this result has yet been found. Attempts have been made to test the theory by considering measurements of high accuracy for the first three Balmer lines, HE, HB, and Hy (Osterbrock et al., 1963; O’Dell, 1963; Osterbrock, 1964). The procedure used by these authors is to make a “ colour-colour ” plot of log(HB/Hy) against
34 1
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
log(Ha/HP). An example of such a plot is shown in Fig. 2. The open circle represents the point obtained from the calculations of Burgess (1958) for T, = 1.5 x lo4 OK, assuming collisional redistribution of angular momentum for n > 12 but not for n < 12.If the emission ratios are given correctly by this theory, the effect of interstellar reddening will be to move the observed ratios along the reddening line in Fig. 2. A further possible effect in denser nebulae is self-absorption in the Balmer lines (Pottasch, 1960a,b; Capriotti, 1964). When this effect occurs, and reddening is neglected, the observed points should lie on the self-absorption line shown in Fig. 2.
0.61
I
I
04
0.8
0.6
3
log Ha/W
FIG.2. A
"
colour-colour " plot for Hcr, H j , and Hy in planetary nebulae.
Results are plotted for five planetaries. For NGC 6826, 7662, IC 418, and NGC 7027 there is reasonable agreement with theory, assuming self-absorption to be negligible, although Osterbrock considers that the deviations for NGC 6826 and 7662 are rather larger than would be expected from observational errors. For the planetary nebula VV 8, which has an exceptionally high density ( N , 2 lo6 cm-3), there is evidence for self-absorption effects. 6. Continuum Emission at Optical Wavelengths
Continuous emission in planetaries arises from recombination, free-free transitions X+ + e + X + +e+hv, (18)
342
M . J. Seaton
and two-quantum emission from H(2s), H(2s) + H(ls)
+ hv + hv'.
(19) In diffuse nebulae scattering by grains may give a further contribution to the observed continuum. In comparing observed and calculated continuum intensities one has to consider the energy distribution in the continuum, the strength of the continuum relative to the strength of the lines, and the magnitude of the discontinuity at the Balmer limit. At densities in planetaries the probability for collisional transitions of the type H(2s)
+ H + +H(2p) + H+
(20) is comparable with the probability for two-quantum emission, and in consequence the Balmer discontinuity depends on electron temperature and on electron density. It appears that calculated results for the continuum are in reasonable agreement with the available observational results. This subject is discussed in more detail in a previous review article (Seaton, 1960a). 7 . Continuum Emission at Radio Wavelengths
The continuum emission at radio wavelengths is due to free-free transitions. The emissivityj, (such thatj, dv is the emission per unit volume per unit solid angle) is given by an expression of the form j v = Gv(T'P" N +
(21)
(Oster, 1961; Brussard and van de Hulst, 1962). If df is an element of length the transfer equation is
d$V
dl = j , - I C ~ Y , , , where K , is the absorption coefficient. So long as the electrons have a Maxwellian velocity distribution, j , and K , satisfy the Kirchhoff relation j v = ~v &(Te),
(23)
where
is the intensity of blackbody radiation. The transfer equation may therefore be written
dJ,ldr, = B, - 9,,
(25)
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
343
where dr, = ti, dlis the element of optical depth. Integrating (25), the observed intensity is given by
9,= B,(1 - e-'.),
(26)
where z, is the optical depth in the line of sight. The function G, is such that ti, varies as the square of the wavelength. Two cases arise. For shorter wavelengths it is found that, for most observed nebulae, T, G 1 and
9,N B,z,
=
s
j , dl.
(27)
The ratio of radio emission to hydrogen line emission in the visible can then be calculated from recombination theory. Since there is no interstellar absorption at radio wavelengths, the observed ratios can be used to determine the reddening c. The values obtained (Terzian, 1966) are generally in good agreement with values deduced from the intensity ratios of the lines in the visible. At longer wavelengths one has z, B 1 and 4, N B, . From the observed surface brightness at these wavelengths the electron temperature can be deduced. Values obtained (Terzian, 1965) are in agreement with values deduced from relative intensities of forbidden lines. The thermal processes which we have considered do not produce the observed radio emission in all nebulae. In the Crab nebula, for example, the emission is mainly due to synchrotron radiation. However, thermal emission appears to dominate in planetary nebulae and diffuse emission nebulae.
8. Line Emission at Radio Wavelengths The 21-cm line, which arises from transitions between the hyperfine structure levels of the hydrogen ground state, is emitted in nebulae containing neutral hydrogen. In the present section we consider lines of a different type, due to transitions between highly excited states populated by recombination in ionized nebulae. The possibility of detecting lines due to transitions n + 1 -+n in hydrogen where n 100, was first suggested by Kardashev (1959). Transitionsn + 1 -+ n are referred to as na lines and n 2 -+ n as nfi lines. The following hydrogen lines have been observed: 104a (Dravskikh et al., 1965); 109a (Hoglund and Mezger, 1965); 90a (Sorochenko and Borodzich, 1965); 156a and 158a (Lilley et al., 1966a,b); 166a (Palmer and Zuckerman, 1966); and 125a and 166a (McGee and Gardner, 1967). Hydrogen nfi transitions (Gardner and McGee, 1967) and helium na lines (Lilley et al., 1966b) have also been observed.
-
+
344
M. J. Seaton
The emission is calculated on solving a transfer equation of the type (22) where we now takej, and IC, to be the total emissivities and absorption coefficients for the continuum and for the line: j , = j,'+jy',
IC,= K,C
+~y',
(28)
where superscripts c and I denote continuum and line. For the transition n' + n the line emissivity is jy' = N,, &n - hvcp, ,
471
where Ant,, is the spontaneous transition probability and cp, a normalized profile factor, Scp, dv = 1. The broadening of the lines is considered by Griem (1967), who shows that ion broadening is negligible and that broadening by electrons can be calculated in the impact approximation. Taking N , = lo3cm-3 for diffuse nebulae, T, = 1 x lo4, and a " Doppler" temperature TD = 3 x lo4 (larger than the electron temperature in order to allow for nonthermal turbulent motions), Griem obtains for the ratio of collision half-width to Doppler (lie) width: 0.4 for n = 170, 0.1 for n = 140, 0.02 for n = 110 and 0.005 for n = 90. It is therefore permissible, for many of the observed lines, to take cp, to be given by the Doppler formula, c c p v = - VO
p
();
-pcqv - vo)2
'I2
exp[
V02
1 9
where B = (M/2kT,) and vo = v,.,.This is in good agreement with observations. Let us put IC,'= N , X , - N,,, Y, , (3 1) where the first term corresponds to absorption from level n and the second to stimulated emission from level n'. Expressions for the atomic coefficients, X, and Y , , may be obtained by considering conditions of thermodynamic equilibrium. Using the Kirchoff relation j,' = IcJB,,the Planck relation (24), the expression (29) for jy', and the Boltzmann relation N,, = (w,,/w,,)N,,exp( - hv/kT),
we obtain K,1 = c2
2hv3
[5 N , - N,.] !2471 hvcp, .
Provided that one uses correct values for N , and N , , , this relation remains valid for conditions which do not correspond to thermodynamic equilibrium. It is usual to express the transition probability in terms of the oscillator
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
strength,f,,,,. Using Eq. (1 I), the expression for
K:
345
may be written
Expressions for f,.,, are given by Kardashev. For the radio-frequency lines, for 109a at lo4 O K ) and the quantity hv/kTe is very small (hv/kTe= 2.4 x one may therefore put
Since b,
N
6,.
N
1, the expression for the absorption coefficient reduces to
The expression for
K,'
used by Kardashev,
was obtained assuming thermal equilibrium, b, = b,, = 1. The importance of departures from thermal equilibrium have been emphasized by Goldberg (1966). From the results of Seaton (1964), as given in curve 11 of Fig. 1, Goldberg obtains bllo - bIo9= 7 x which is larger than (hv/kTe)= It follows that the stimulated emission term in the expression for 2.4 x K,' is larger than the absorption term, and hence that K,' is negative. This leads to an amplification in the line intensity (maser action). The observed intensities of nu lines tend to be larger than the intensities predicted using the thermal equilibrium theory of Kardashev, and in better agreement with the nonequilibrium theory of Goldberg. It has been seen that, in calculating b, , it is necessary to allow for radiative processes and for collisional redistribution of energy. The factors are sensitive to electron density, approaching unity in the limit of high density and approaching the purely radiative case (curve I of Fig. 1) in the limit of low density. In order to make further comparisons with the radio observations it is desirable that improved calculations should be made. The profile factors cpv should be calculated using Griem's theory and the level populations should be calculated, as functions of electron density, using improved cross sections and taking all relevant collision processes into account. Some recent new calculations have been made by Hayler (1967) and by McCarroll (1968).
M . J . Seaton
346
C . NONHYDROGENIC SYSTEMS
1. Atomic Data In calculating recombination spectra we are concerned with atomic systems having a single valence electron in excited states. Transition probabilities can be calculated using the method of Bates and Damgaard (1949) and photoionization cross sections using the method of Burgess and Seaton (1960a). Improved tables required for the photoionization calculations have been published by Peach (1967b). A good check on the accuracy of the photoionization data can be obtained from a comparison of observed and calculated total absorption coefficients. In conditions of thermodynamic equilibrium the bound-free absorption coefficient is
where AEi is the excitation energy of level i and where the summation is over all states with v i < v. The total absorption coefficient is obtained on adding to Eq. (37) a contribution from free-free transitions (Peach, 1965, 1967a). Comparisons of calculated (Peach, 1967c) and observed (Boldt, 1959a,b) total absorption coefficients for No and 0 ' are given in Figs. 3 and 4.
12,500'K
12,000'K
6
I I,5OO0K 0°K I1,00O0K 10,500'K Wavelength in
FIG. 3. Continuous absorption coefficients of No in conditions of thermodynamic equilibrium. Dashed curves give calculated results (Peach, 1967c) and full line curves give experimental results (Boldt, 1959a).
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
347
15 -
Wavelength in 1
FIG.4. Continuous absorption coefficients of 0 ' in conditions of thermodynamic equilibrium. Dashed curves give calculated results (Peach, 1967c) and full line curves give experimental results (Boldt, 19596).
2. The He I Recombination Spectrum
Helium recombination coefficients have been tabulated by Burgess and Seaton (1960b) and calculations of the He I recombination spectrum have been made by Mathis (1957), Seaton (1960a), Pottasch (1961), and Pengelly (1963). Complications arise from the fact that the populations of the metastable levels, 2 ' S and 2 3S, may be large. We first consider transitions of the type nd --t 2p for which these complications do not have to be considered and for which the radial wavefunctions for the upper levels are closely hydrogenic. Cascade transitions to nd states come mainly from hydrogenic nl states with 1 > 2. Allowing for statistical weight factors we have a(")(HeI, n 3D) 'v Za(")(H I, nd)
and a")(He I, n ' D ) 'V &a"'(H I, nd)
where
dC)is the recombination coefficient allowing for cascade (see Section
11,A). Table V gives a comparison of observed relative intensities for NGC
7662 with intensities calculated by Seaton (1960a) using these relations and the values of d''(H1, nd) obtained from the work of Burgess (1958). The agreement is seen to be satisfactory. Table VI gives effective recombination coefficients for the lines 15876, 3 D + 2 3P and 14471, 4 D 2 3P, in three different approximations: (a) --f
M . J. Seaton
348
TABLE V RELATIVE INTENSITIES OF HE I nd+ 2p LINES IN NGC 7662"
He I n 3D -+ 2 3P
He 1 n l D
-+
3 4 5 6
12.6 5.5 3.6 2.2
12.6 5.1 2.6 1.5
3 4 5 6
4.0 1.3 0.7 0.4
3.7 1.5 0.8 0.4
2'P
Observed intensities,I , , corrected for reddening as in Table I and relative to I,(HP) = 100. Calculated intensities In normalized to observed intensities for 3 3 D z3P. -+
TABLE VI EFFECTIVE hCOMBINATION COEFFICIENTS FOR HE I LINES 10-4~,
1
2
10I4a(3 3 D + 2 3 P )
3.95" 4.87b 5.21"
1.79" 2.17b 2.30'
10I4a(4 D + 2 3 P )
1.35'
0.644'
Using hydrogenic data of Burgess (1958), assuming (12) for n > 12. * Using hydrogenic data of Pengelly (1964), not assuming (12). Nonhydrogenic calculations of Pengelly (1963), not assuming (12).
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
349
using the values of Burgess (1958) for cr'"'(H I, nd); (b) using the values of Pengelly (1964) for d"(H I, nd); (c) results obtained by Pengelly (1963) using accurate helium recombination coefficients and transition probabilities. For (b) and (c) no allowance is made for collisional redistribution of angular momentum. Approximation (c) gives the intensity ratio 1(15876)/1(14471) to be 2.9 at 1 x lo4 "K and 2.7 at 2 x lo4 "K.The mean value for seven nebulae observed by O'Dell (1963), corrected for reddening, is 2.5. Neglect of collisional redistribution will lead to 4 3 D + 2 3 P ) being slightly overestimated but will have less effect on 4 4 D + 2 ' P ) . Effects arising from the metastability of 2 ' S have been discussed by Osterbrock (1964) and by Capriotti (1967). In Fig. 5 we give an energy level diagram for the lower helium triplet states, and in Table VII we give relative intensities of triplet lines calculated by Pottasch (196 l), allowing for selfabsorption, together with observed relative intensities for three planetaries. In the limit of zero optical depth, to = 0, the calculations of Pottasch are in good general agreement with those of Pengelly. Absorption of 13889 can be followed by reemission of 13889 or by the cascade process 3 3P + 3 3 S + 2 3P -P 2 ' S . The effect is to reduce the intensity of 13889 and to increase the intensity of 17065 and 110,830. It is seen from Table VII that this process explains satisfactorily the observed intensities of 13889 and 27065, but does not explain the very large observed intensities of 110,830. It should be noted that absorption of 110,830 can only be followed by reemission of 210,830.
'
5s'
'
5 3P
--
4P '
1-
I
FIG.5. Triplet levels of helium.
M . J . Seaton
350
TABLE VII
RELATIVE INTENSITIES FOR HE I TRIPLET LINES Relative intensity Calculated allowing for self-absorption"
Observed
Line
~0=0.0
1.5
5.0
15.0
IC2149
IC418
IC4997
z3P-+2 3S,h10830 3 3P 2 3S, h3889 3 3S .+ 2 =P,A7065 3 3 D + 2 3P,h5876 43D-+23P,h4471
1.58 1.08 0.18 1.00 0.39
1.69 0.75 0.38 1.00 0.39
1.84 0.32 0.67 1.00 0.38
1.91 0.10 0.82 1.00 0.38
3.32 0.74 0.34 1.00 0.39
14.20 0.48 0.47 1.00 0.34
7.10 0.38 0.59 1.00 0.44
-+
~~~
' - r 0 is
~
the central optical depth for the A3889 line.
In order to explain the observed 210,830 intensities, effects of collisional excitation from 2 3 S must also be taken into account (Mathis, 1957; Pottasch 1961; Osterbrock, 1964). The population of 2 3 S is determined by: (a) capture and cascade, (b) 2 3S -,1 ' S transitions with two-photon emission, (c) collisional transitions to singlet states, and (d) photoionization by stellar quanta and by La quanta produced in the nebula. In calculating the 2 3Spopulation we do not include collisional transitions to other triplet states since these are followed by cascade back to 2 3S. Taking account of the above processes, the level population is given by N , N + a"'(2
3s)= N ( 2 3 S ) { A+ q d N , + c},
(38)
sec-' according to Mathis, where A is the two-photon probability, 2.2 x qdN, is the collisional deactivation probability, and C=
lj" hv
a,(2 3S) dv dw
(39)
is the photoionization probability. In the absence of collisional excitation, the number of atoms entering 2 3P is N , N+d"(2 3 P ) . The number of collisional excitations to 2 3P is N ( 2 3S)q(23 S 42 ' P ) N , . The effect of collisions is there fore to increase the intensity of 110,830 by a factor
The observed factor varies between 1 and 10 (see Table VII). Using estimated cross sections, and neglecting photoionization, Osterbrock obtained P 100
-
ATOMIC COLLlSlON PROCESSES 1N GASEOUS NEBULAE
351
and concluded that photoionization had to be included in order to obtain a more reasonable value. Greatly improved cross sections have recently been obtained by Burke et al. (1967j who have solved the coupled integro-differential equations for the collision problem including all n = 2 states.' Table VIII gives reaction rates calculated from these cross sections. The Table includes values of &)(2 3S) ~ ( " ( 23P), and ~ ( 3qq(2 2 3 s +2 3 ~ ) P,=1+ (41) c('"(2 3P)q, 9
where qd = q(2 3 S -+ 2 'Sj + q(2 3 S + 2 ' P ) : in the limit of high density, N e $ ( A + C)/qd, two photon emission and photoionization may be neglected and P = P,. It is seen that the value of P , obtained using the cross sections of Burke et al. is much smaller than the value estimated by Osterbrock, and is comparable in magnitude with the factor deduced from observations. The question of whether photoionization plays an important role in depopulating 2 3 S can be decided only on making a more detailed study of physical conditions in individual nebulae. TABLE VIII REACTIONRATESREQUIRED FOR THE CALCULATION OF HELIUM LINEINTENSITIES
10-4Te 1.0
1.5
2.0
3.1 1.4 20 79 9.9 21.0 14.4 7.4
3.5 2.4 38 122 10.1 15.0 10.0 10.6
3.4 3.1 53 153 10.1 12.0 7.8 13.4
3. Determination of Helium Abundances
The total number of quanta emitted by a nebula in the recombination line i -+ i' of an atom X is
I am indebted to Dr. Burke and his collaborators for providing these results in advance of publication.
M . J. Seaton
352
where the integral is over the volume of the nebula. Assuming the electron temperature to be constant, ai,i,can be taken outside of the integral and the observed intensity is then proportional to the quantity {A?}, defined as
{X'}
=
s
N ( X + ) N , dK
(43)
Using effective recombination coefficients from Tables IV and VI we obtain
and Z,(He ZZ, 14686) { He2'} -- 0.102 {H+l
I,(HP)
(45)
The He/H abundance ratio, by numbers of atoms, is usually calculated as N(He) -N(H) -
{He'}
+ {He"} {H+}
It is here assumed that there is no neutral helium in the region in which hydrogen is ionized. This assumption is justified for many nebulae but not for certain low excitation planetaries (Harman and Seaton, 1966). Typical abundance results are that N(He)/N(H) equals 0.16 for planetaries (Harman and Seaton, 1966), 0.12 for diffuse nebulae in the galaxy (Aller and Liller, 1959; Mathis, 1962; Faulkner and Aller, 1965), and 0.08 for diffuse nebulae in the Magellanic clouds (Faulkner and Aller, 1965).
4. Recombination Spectra of 0 III, 0 IV, and 0 V Lines due to optically allowed transitions in 0 111, 0 IV, and 0 V are observed in NGC 7027 and various other nebulae. The strongest 0 I11 lines are due to the Bowen fluorescent mechanism of absorption of He I1 La, 1303.780, in the transition 02+ 2p2 3P2 + 2p 3d3P2, 1303.799, but this mechanism does not account for all of the observed 0 I11 lines. Calculations of the recombination spectra of 0 111, 0 IV, and 0 V have been made by Burgess and Seaton (1960~). Table IX gives results for abundances of hydrogen, helium ions, and oxygen ions in NGC 7027. The helium abundances are obtained using Eqs. (38) and (39), the abundances of Oo, O', and 02+ are obtained from forbidden lines (see Section 111), and the abundances of 03', 04+ , a nd 0'' from the calculations of recombination spectra. Consideration of the ionization equilibrium casts doubt on the correctness of these results. The ionization potential
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
353
TABLE IX ION ABUNDANCES I N NGC 7027" Ion
H+
Relative abundance 100,000
He HeZ +
+
10,900 5,100
O0
0.4 1.O 15 64b 28b 8b
Of OZ
+
0 3
+
04+ 0 5
+
Helium ion abundances from Seaton (1960a) (corrected using improved recombination coefficients of Tables IV and VI), oxygen ion abundances from Burgess and Seaton (1960~). , and 0 5 +are b T h e abundances of 0 3 +04+, obtained on interpreting weak permitted lines of 0 111, 0 IV, and 0 V using recombination theory. This is discussed further in the text.
of 0 2 +(54.89 eV) is close to the ionization potential of He' (54.40 eV) and we therefore expect that, in the region in which helium is doubly ionized,2 the oxygen will be ionized beyond 0 2 +To . a good approximation we should have
+
{O+> { 0 2 + > {He"} - {03'> + {04+} { 0 5 +* }
We+} --
+
(47)
From the results in Table VII we obtain the value 2.1 for the ratio on the left of (47) and 0.16 for the ratio on the right. The helium ion abundances, and the oxygen ion abundances from forbidden lines, should be reliable. We are therefore led to question the correctness of the interpretation of the weak permitted lines of 0 I l l , 0 IV, and 0 V using recombination theory.
5 . Recombination Spectra of C 11, C 111, and C IV l t has generally been assumed that weak lines of C 11, C 111, and C IV observed in nebulae are excited by radiative recombination. Calculations of the recombination spectra have been made by Pengelly (1963). From the The boundary between singly and doubly ionized helium should be sharp.
354
M . J . Seaton
observed C 11, C 111, and C IV intensities in four planetaries, N G C 2392, 7662, 7027, and 7009, he obtains carbon to hydrogen abundance ratios, by numbers of atoms, of about 5 x These are much larger than the C/H. ratios, of about 2 x obtained for stellar atmospheres (Aller, 1961), While one cannot exclude the possibility that these nebulae are carbon rich, it should be noted that the interpretation of forbidden line spectra does not suggest that they have unusually large abundances of oxygen or nitrogen. We have seen that the interpretation of weak permitted lines of 0 111, 0 IV, and 0 V using recombination theory gives ion abundances which appear to be much too large, and it appears that a similar situation may arise for C 11, C 111, and C IV.
6 . Excitation by Absorption of Stellar Radiation in Spectrum Lines The weak permitted lines of oxygen and carbon ions have excitation energies which are much too large for collisional excitation to be of importance. Seaton (1968) has considered the possibility that these lines are excited by absorption of stellar radiation in resonance lines. Let J V , , , ~ ( X + ~=) (total number of ions X m +entering excited states per unit time due to line absorption) and JVca,,(X'm) = (total number of ions
A?"+ entering excited states per unit
time due to capture). in N G C 7027. We estimate the ratio of these two quantities for 02+ It is convenient first to obtain some formulas for hydrogen. With { H + } defined by Eq. (43) we have
JVcapt(HO) = .B(H0){H'>,
(48)
where cr,(Ho) is the hydrogen recombination coefficient summed over all excited states. For a nebula which is optically thick at the Lyman limit, captures on the hydrogen ground state produce ionizing quanta which are reabsorbed. The number of captures on hydrogen excited states is then equal to the number of ionizing stellar quanta absorbed,
where vH is the frequency of the Lyman limit, T, the Lyman continuum optical depth, and L , the stellar luminosity ( L , dv ergs sec-'). The presence of [O.I] lines shows that NGC 7027 is optically thick at the Lyman limit and hence that Eq. (49) is valid (Seaton, 1960a).
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
355
Let us consider
we obtain Using a hydrogenic approximation for the excited states of 02+ CcB(02+)/Ccg(Ho)= 13. We assume {04+} and { 0 5 +to) be small compared with { 0 3 + }and, using Eq. (47) and data from Table IX, obtain3
The number of stellar quanta absorbed in a spectrum line i is
where z,(i) is the optical depth in the line. This equation defines the equivalent widrh of the line, ( A v ) ~Using . (49) we obtain Ci (Lv/hv)i(Av)i Nabs(02 ' Ncapt(Ho)= Jy", (L,/hv)(l - e-'") dv
(54)
and, making the assumption that the star radiates as a blackbody, so that L, is proportional to B,(T,), this reduces to N a b s ( O 2+)
Ncap1(Ho)= J&kTS
xi [v3(eu- l)-l]i(Av/v)i v2(eu- l ) - ' ( l - e-'") dv'
(55)
where v = hvjkT,, and where the integral in the denominator is tabulated by Hummer and Seaton (1963) as a function of T, and T(H), the optical depth at the Lyman limit. Seaton (1968) expresses the optical depths of the 0 I11 resonance lines in terms of the optical depth, r(02+), at the threshold of the 02+ ionization continuum. Burgess and Seaton (1960~)obtain ~ ( 0 ~ = ' ) 1.8. So long as the optical depth at the line center is much larger than unity, the equivalent width is largely determined by the Doppler width. Consistent with measurements of Osterbrock et a/. (1966) the Doppler temperature for 0'' is taken to be 5 x lo4 OK. Final results for the ratio ~ a b s ( O Z + ) / ~ c a p t ( 0 2 + ) , calculated from Eqs. (52) and ( 5 9 , are given in Table X for various values of T, and z(H). The value of T, for NGC 7027 is not known exactly, but is known to be within the range 1 x lo5 to 2 x lo5 OK (Seaton, 1968). abundance in Table IX was obtained on assuming that the 0 I11 lines are The 0 3 + produced by recombination. It is now assumed that this is not correct.
M . J . Seaton
356
TABLE X RATIOOF NUMBER OF EXCITATIONS OF 0’’ LINEABSORPTION TO NUMBER BY RADIATIVE CAPTURE. DATAFOR NGC 7027
BY
dH)
T, = 1 x lo5
1 5 10
12 4 3 2
a3
T, = 2 x 1 0 5 48
16 11 5
It was seen in Section II,C,4 that the interpretation of permitted 0 I11 lines as recombination lines gave 03’ abundances an order of magnitude too large. It is now seen that the number of 0” excitations by line absorption may be an order of magnitude greater than the number of excitations by recombination. The absorption mechanism therefore appears to be capable of explaining the observed 0 111 intensities, and it may be expected that it can also account for the observed intensities of other weak permitted lines, such as those of 0 IV, 0 V, C 11, C 111, and C IV. The number of stellar quanta absorbed by any ion in resonance lines is very insensitive to the ion abundance, since the equivalent widths are determined mainly by the Doppler effect. For hydrogen the number of excitations by recombination will exceed the number by line absorption by about two orders of magnitude.
111. The Forbidden Lines A. HISTORICAL INTRODUCTION
The identification of the “ nebulium lines, first observed by Huggins (1864), remained one of the major unsolved problems of astrophysics until Bowen (1928) showed that they were due to transitions within the ground configurations of ions such as O’, 02+, and 3 ’ . They are strictly forbidden for electric dipole radiation but not as electric quadrupole and magnetic dipole transitions. Bowen realized that these lines could be excited by electron impact and could be very intense if the density was sufficiently low for collisional deexcitation to be of minor importance. In the revised edition of their textbook, Russell et al. (1945) remarked that nebulium had “vanished into thin air.” All of the forbidden lines in the visible spectrum are due to transitions within configurations 2pq and 3pq, with q = 2, 3, or 4. The terms in these configurations, in order of increasing excitation energy, are ’P, D , and ‘sfor ”
‘
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
357
q = 2 and 4, and 4S, D, and ’P for q = 3. As a consequence of the transition probabilities being small, self-absorption does not occur for the forbidden lines. The processes responsible for producing these lines are therefore much simpler than the processes of line formation in stellar atmospheres. Given good observations and good atomic data, a great deal of valuable astrophysical information can be obtained from the interpretation of the forbidden line intensities. Cross sections for excitation of the forbidden lines by electron impact would be very difficult to measure experimentally, and a great deal of effort has therefore been devoted to making accurate quantum mechanical calculations. In all of the earlier work, calculations were made using perturbation theory, that is, assuming weak coupling between the initial and final states. Since a change of spin is involved for many of the important transitions, full allowance must be made for electron exchange. This means that the wave functions must be fully antisymmetric in the coordinates of the atomic electrons and the colliding electron. Calculations for 0’’ were made by Hebb and Menzel (1940) using Coulomb waves, and calculations for 0’ were made by Yamanouchi et al. (1940) using distorted waves. Although subsequent work has shown that these calculations did not give accurate numerical results, it must be emphasized that they represented important pioneering investigations in atomic collision theory. Hebb and Menzel (1940) introduced the parameter R ( i , j ) , which Seaton (1955a,b) subsequently suggested should be termed the collision strength. It is defined by
where Q ( i - j ) is the cross section, w i the initial statistical weight, and k , = mvi/h, where ui is the velocity of the incident electron. Since k , has the dimensions of a reciprocal length, R(i, j ) is dimensionless. From reciprocity theorems it follows that R is symmetrical, R(i,j) = R(j, i). When atomic units are used, ki2 is numerically equal to the energy of the incident electron in Rydbergs (13.60 eV) and Q(i - j )
=R ( i ’ j j nao2
wi ki
( k i 2in Rydbergs),
(57)
where a, is the Bohr radius. The use of Coulomb waves in the calculations of Hebb and Menzel gave the correct threshold law for electron excitation of positive ions. This is such that R remains finite at threshold. In many cases the R’s vary rather slowly as functions of energy and may be treated as constants.
358
M . J, Seaton
In all of the earlier work on 0' and 0' it was found that the dominant contributions to the cross sections come from states of the colliding electron having angular momentum 1 = 1 (p-waves), but it was not realized at the time that conservation conditions set an upper limit on the contribution from each angular momentum state. These conditions are not satisfied automatically in the forms of weak coupling approximations which were employed, and it was pointed out by Bates, Fundaminsky, Leech, and Massey (1950) that the results obtained violated conservation conditions by quite large factors. At about the time that this paper was published, H. S . W. Massey interested me in the problem of developing improved methods for the calculation of these cross sections. It was clearly necessary to have a method which allowed for the possibility of strong coupling in exchange collisions. The Hartree-Fock method had been used extensively in calculating wave functions for bound states of atomic systems and also for elastic scattering problems. Morse and Allis (1933) had treated elastic scattering by H and He, and calculations for elastic scattering by positive ions, of interest for the determination of photoionization cross sections, had been made by Bates and Massey (1941) and by Bates and Seaton (1949). A more general formulation of Hartree-Fock theory was developed and was first applied to excitation of 0' (Seaton, 1953a). Results obtained in subsequent calculations for a number of positive ions (Seaton, 1953b, 1955a,b, 1958) have been used extensively for the determination of temperatures, densities, and chemical composition of gaseous nebulae. The generalized Hartree-Fock theory (often referred to as the closecoupling approximation) involves the solution of systems of coupled integrodifferential equations. The first calculations on the forbidden lines were done entirely on desk machines. With the development of automatic computing techniques the Hartree-Fock, or close-coupling method, has been applied to a great many different collision problems (see review articles by Burke and Smith, 1962; Heddle and Seaton, 1964; Moiseiwitsch, 1968). Improved calculations for the forbidden lines are being made by a group at University College London, in collaboration with Dr. S . J. Czyzak and Mr. T. K. Krueger (Shemming, 1965; Saraph et al. 1966; Czyzak and Krueger, 1967; Czyzak et al., 1967; Czyzak el al., 1968) and some independent calculations are being made by Smith et al. (1966). Further details on the formulation and on results obtained to date will be given in Section 111, C. +
B. EXPRESSIONS FOR LEVELPOPULATIONS AND LINEINTENSITIES We consider an ion with energy levels Ei, i = 1,2, 3, . . . ,where i = 1 is the ground state and where E j > Eifor j > i. The radiative transition probabilities are denoted by A(i +j ) and the collisional transition probabilities by
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
359
q ( i + j ) N , where f.0
q(i -,j ) =
Jo
Q(i
-,j ) u , f ( u i , T,) dui ,
wheref(u, T ) is the Maxwell distribution function normalized to
Jfhw u = 1, and u iis the velocity of the incident electron. Expressing Q(i - j ) in terms of R(i,j) the probability of deexcitation, in cm3 sec-', is q ( j + i)
Y(j, i )
8.63 x =
w j Tki2
( j > 9,
(59)
where T, is in degrees Kelvin and Y ( j , i) = j:Cl(j, i) exp( - 2 mv .2 ) d(13f;ll).
2kT,
2kT,
If the energy variation of R can be neglected, Y ( j ,i ) = Q ( j , i).The probability of excitation is q(i
-
j)=w 2. q ( j
mi
-
(-
i) exp
'j,ieEi) ~
( j > i).
The transition probabilities for the forbidden lines may be as large as 10 sec-' or as small as sec-' [compilations of these transition probabilities are given by Aller (1956) and by Garstang (1968)l. The collision strengths are of order unity and, for temperatures of order lo4 OK, the deexcitation probabilities are of order 10-4sec-' for N , = lO4cmP3.It is seen that, in general, both collisional and radiative deexcitations have to be taken into account. The total probability for the i -J transition is d j i = q(i + j ) N e
+ A(i
-+
j)
( j # i),
(62)
where it is to be understood that A(i - j ) = 0 for i < J . With N i ions per unit volume in level i, the equations defining a steady state are d i jN j =
1d j i N i .
j#i
j#i
It is convenient to have simple explicit expressions for the solutions of these equations. Putting dji we have
=
- C dj i
9
j#i
1dij N j = 0. j
(64)
M. J. Seaton
360
,
Let P, be the cofactor of dl , in the determinant
D=
dll
d12
d13
d21
d22
d23
d31
d32
d33
* * *
* * * *
Using a well-known theorem for determinants, we have
C d,,jPj = D
1di,jPj= 0
and
for i # 1.
(67)
i
j
Using (64) it is readily shown that D
=0
C d,, P, = 0
and hence that (all i).
j
The solutions of Eq. (65) may be written
where
S=CPj
and
j
N=CNj, i
For an ion with three levels we have
It is of interest to consider two limiting cases. If N , is so small that collisional deexcitation can be neglected we have N2A(2+ 1) = N ; ( q ( l - 2 )
2) + A(3q(l+ 2)3)A(3 + A(3 + 1))Ne 4
-+
(72)
and N3(A(3 -+ 2) f A(3 -+ 1)) = Nlq(l
-+
3)Ne;
(73)
level 3 is populated only by excitation from level 1, and level 2 is populated by excitation from 1 and cascade from 3. If, on the other hand, N , is so large that radiative deexcitation can be neglected we obtain a Boltzmann distribution,
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
361
The energy emitted in a line i - j is (Ei - E j ) N jA(i + j ) , and the ratio of the intensities of two lines due to the same ion is
I(i - j ) -I(i’ + j ‘ )
- (Ei - Ej)P, A(i +j )
(Ei. - Ej,)Pi.A(i’ - j ’ )
(75)
’
In general, such ratios depend on T, and on N , and from observations of two or more ratios both T, and N , may be determined. The ratio of the intensity of a forbidden line of an ion X m + to the intensity of HP is
from which the ion abundance may be determined relative to the H + abundance.
c. CALCULATIONS OF COLLISION STRENGTHS 1. Formulation of the Collision Problem We consider electron collisions with an ion containing N electrons and having a nuclear charge 2.The charge on the ion is z = Z - N . We denote the antisymmetric ion wave functions by $ ( y ~ , L M s , ~ I, 192, , .* . N )
(77)
9
where 1,2, . . . , N indicates the space and spin coordinates of the atomic electrons, and we introduce one-electron functions
where xm, is a spin function, Y f ma, spherical harmonic, and F a radial function. From (77) and (78) we may form the vector-coupled functions $(ySlL1ZSLMsML I 1,2, . . ., N ; N .t1)
x $(YS1L1Ms,M,, I 192,
.. . , N ) q ( l m ,
IN
+ I),
(79)
and from (79) we may form the fully antisymmetric functions $(l, 2,
. ..,N
(-1)N+1
+ 1) = ( N + 1)l”
N+l
1 (-l)i$(l,
i=l
2, ..., i - 1,
i + 1, . . . , N
+ 1 ; i). (80)
M . J . Seaton
362
Using c1 for the set of quantum numbers S I L , l S L M s M L we , put Y, =
C I)(. I 1, 2, ..., N + 1)
(81)
a
and we denote the radial functions by Fa,a,. It should be noted that o! specifies the function I)(.) in which the radial function Fa,a, occurs, and that a’ specifies a boundary condition which is to be imposed. We now take the radial functions to be such that ‘Fa,is a solution of the Schroedinger equation
( H - E)Ya*= 0,
(82)
where H is the Hamiltonian for the complete ( N + 1)-electron problem. The radial functions may be taken to have asymptotic form
-
+ (cos x,)RaP,),
ka-”2{(sin x,)
(83)
where
E
= E,
+ ka2
(Rydberg units),
(84)
+ (z/k,) ln(2kar ) + arg r(Za + 1 - iz/ka).
(85)
and where E, is the ion energy and x, = k, r - $Za
71
This defines the reactance matrix R, which is real and symmetric and diagonal in S, L , M,, and M L . Further, the matrix elements R(yS,L1lSLM,ML, y’Sl‘Ll’I‘SLM, M L ) are independent of M , , M L . The scattering matrix is S = ( 1 iR)(1 - iR)-’ and the transmission matrix is T = 1 - S. The collision strength is
+
%JSlL l ?Y’SI’L, ’)
= 4 C(2S II’SL
+ 1)(2L + 1) IT(yS,LIISL,y’Sl’Ll’l’SL)12. (86)
2. Variational Principles
In practice, exact solutions of the Schroedinger equation (82) cannot be obtained. Approximate calculations are best made using a variational principle. Of the various forms which have been employed we consider only the one which has been found to be most convenient in recent work (Saraph et al. 1966). In matrix notation, (83) may be written
F
-
k-1/2{sin x
+ (cos x)R},
(87)
where it is to be understood that quantities in italic type, without subscripts, represent diagonal matrices. It sometimes happens that the elements of R are
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
363
large, in which case (87) is not convenient. We therefore introduce functions
G = F(cos z - (sin z ) p ) with asymptotic form
G
-
=
[sin
where
R
k-'/'(sin(x 5
+ z) + (cos(x + r ) ) p } ,
+ (cos z)p][cos
T
(89)
- (sin 7 ) p l - l .
(90)
The phases za are chosen to be such that p is not large. Consider wave functions Yatcontaining radial functions Ga,arwith asymptotic form (89), and functions Y a ,+ 6Ya, containing radial functions Ga,a,+ 6Ga,,, with asymptotic form
G
+ 6G
-
k-'/'(sin(x
+ 2) + (cos(x + z))(p + Sp)},
(91)
where it should be noted that the phases z are not varied. Defining matrices L and L + 6L with elements
La??,a, = (Ya,,1H- EJY',.)
+
+
(92)
+
La,,,ar 6La?r,a< = (Yau 6YawlH- qYa* 8Ya' )
9
(93)
we obtain the variational principle that, for small variations about the exact functions, S{p - L} = 0 (94) where H is in Rydberg Units. If the variational condition (94) is imposed for all possible variations of the functions, then these functions must be exact solutions of the Schroedinger equation. In practice we impose certain restrictions on the functions such as, for example, retaining only a finite number of terms in the expansion (81). In this case the " best" functions may be determined by requiring that (94) should be satisfied for all variations consistent with these restrictions. Another use of the variational principle is in obtaining corrections to the p matrix. Suppose that we have approximate functions 'I" containing matrices pf. An improved estimate for p is p K = pf
- L'.
(95)
The error in pK is of quadratic order in the errors in the wave functions. 3. Reduction of Integrals and Derivation of Radial Equations
In all of the earlier work the expressions for the elements of the matrix L were simplified on making use of the ion Schroedinger equation, and approximate ion functions were then substituted into these simplified expressions. It
M . J. Seaton
364
was shown by Bates et al. (1950) that the inconsistency in this procedure leads to reciprocity failures (the " post-prior " discrepancy). In more recent work the difficulty has been avoided by evaluating L exactly, using only the equations actually satisfied by the approximate ion functions employed. In practice Hartree-Fock ion functions are used. The algebraic reductions are greatly simplified on putting GAr) = GLo'(r)
+ Cn pn,l=(r)Cn,a >
(96)
where the functions Pn,l are radial functions for the ion and where the coefficientsCn,,are such that the functions Gio)are orthogonal to all functions P n , , with 1 = I,. On integrating by parts it may be shown that imposition of the variational condition (94) is equivalent to imposition of the condition (6Ya,,lH- EJY',,) = 0.
(97)
If one retains only a finite number of terms in the expansion (81), and requires that the variational condition should be satisfied for all possible variations in the radial functions, one obtains a set of radial equations N;
+
')](If - E ) Y ( l , 2, . . . , N
+ 1) d V i + l = 0,
(98)
where the integration is over all coordinates except r N + l .These equations have to be satisfied for all values of c1 retained in the expansion (81). It should be noted that the form of the equations does not depend on the boundary conditions to be imposed and that specification of these conditions is therefore omitted in Eqs. (96) and (98). Equations (98) may be written
x
[$(a'
I 1,2, . .., N ; N
U'
-N$(a'I1,2
+ 1)
,..., N - l , N + l ; N ) ] d V , + l = O .
(99)
After all reductions have been made, the final form of the radial equations is
la(la
+ 1)
[$-- r2
2.2 r
(Uu,a,- W,,,,)Fa,(r) = 0, (100)
where the Ua,,, are potential operators and the Wa,a# are exchange operators. Full details of the reductions and of the form of the radial equations are given by Shemming (1965).
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
365
It is of interest to consider the behavior of the solutions along isoelectronic sequences. In the limit of Z + 03, the ionic radii behave as l/Z and the ion energies behave as 2'. Equations (100) are conveniently written d2
la(/,
+ 1)
1
2
P2
P
(u,,,,- w,,,,)F~,= 0,
(101)
where
($) , 2
p=zr,
Ea=
and (Ua,,,
- ~ u , u , )= ( u a , u ,
-- Wa,a*)lz.
(103)
It should be noted that Ua,,,and W,,a,behave like (1/r12)= ( z / p 1 2 )and , hence thgt u,,~,and w,,,. remain finite in the limit of z -, co. A complex is defined as a set of principal quantum numbers (Layzer, 1954). Thus, for example, the configurations 2s22p2,2s2p3, and 2p4 all belong to the the complex 24. As z + co,the differences in the parameters E, , E,
- E,.
1
= - (Eu,- E a h Z2
(104)
remain finite if a' and CI belong to different complexes but go to zero as l / z if they belong to the same complex. 4. Approximations in the Collision Problem
In all of the work which has been done so far, the approximation has been made of retaining in the expansion (81) only those ion states which belong to the ground configuration ns2npq.In the best calculations which have been made, neglect of coupling to other states in the ground complex is likely to be the most serious source of error. In Fig. 6 we show the positions of the first nine terms in the energy level diagrams of ions of the C sequence (N', 02+, and F3+). After the three terms of the 2s22p2 ground configuration we have six terms of the 2s2p3 configuration. If coupling between 2s22p2 and 2s2p3 were allowed for in the collision calculations, polarization of the ion during the collision would be largely taken into account So far, this effect has been neglected. For highly ionized systems it is also desirable to allow for configuration interaction between 2s22p2 and 2p4 in the ion wavefunctions. The three terms in the ground configurations, in order of increasing excitation energy, will be denoted by 1, 2, and 3 (3P, 'D,and ' S for q = 2 and 4, 4S, 2D,and 'P for q = 3). For the transitions 1 - 2 and 1 - 3, which involve
M . J. Seaton
366
2s2p3
2: ;2
FIG.6 . Energy levels for ions in the C isoelectronic sequence.
a change of ion spin, the dominant contributions to the collision strengths come from the p-waves for 2pq and from the d-waves for 3pq. For the 2-3 transitions many more angular momentum states make significant contributions. Calculations of collision strengths have been made in the following approximations. (a) Exact solutions of the coupled equation. It should be noted that, for exact solutions of Eqs. (loo), L' = 0 and pK= p' in (95). (b) The exact resonance (ER) approximation. This is used only for p-waves and is of interest mainly for 2pq ions. Expanding the electron-electron interaction,
where r c is the smaller of rlr r2 and r, the greater. For p-waves we have contributions only from I = 0, 1, and 2. If the A = 2 terms are neglected in calculating ion wave functions and ion energies, all terms in the 2pq configurations have equal energies. If A = 2 terms are also neglected in the collision problem, the equations (100) can be uncoupled. The procedure used in the most recent work (Saraph et al. 1966) is to solve these uncoupled equations and from the solutions to construct the orthogonal functions Gio' in (96), with full allowance for energy differences. The functions (96) are then used to calculate L, with inclusion of I = 2 terms, and the parameters Cn,ain (96) are determined using the variational principle. The final results for p are obtained from (95).
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
367
(c) The distorted wave (D W ) approximation. The distorted wave radial functions are solutions of
where U,',"' is the spherically symmetric (A = 0) part of U,, . They are taken to have asymptotic form
f
N
k-'I2 sin(x
+ T).
(107)
This defines the choice adopted for the phases T in both the ER and the DW approximations. The DW functionsf, are used to construct orthogonal functions GLo' and the p matrix is calculated as in the ER approximation, the coefficients C,,abeing treated as variational parameters. 5. Results of Calculations
The full coupled equations (100) have been solved by Smith et al. (1 967) for the neutral atoms, C, N, and 0. Collision strengths obtained from their calculations are given in Table XI, and comparisons with the earlier work of Seaton (1953a, 1955a,b) for 0 ' are given in Fig. 7. In this earlier work the p-wave contributions were calculated using a rather crude form of the ER approximation, and d-wave contributions to R(2, 3) were calculated using Coulomb functions. The agreement with the results from full solutions of the coupled equations is generally very satisfactory. The largest discrepancy is in the low energy results for R(2, 3) and is due to neglect of contributions from 2p2('D)k2d - 2p2('S)k,s in the work of Seaton.
30
-N .
C
01
kg,
kg,
k:
FIG.7. Collision strengths for neutral oxygen. Full line curves from exact solutions of the coupled equations (Smith, et al., 1967), dashed curves from the earlier work of Seaton (1953a, 1955a,b).
TABLE XI COLLISION STRENGTHS FOR NEUTRAL C, N,
AND
0"
is 0.2 0.3 0.4 0.5 0.6 0.8 1.o
2.94 4.69 5.72 6.42 6.89 7.56 8.03
Smith et al. (1967).
0.01 0.46 0.73 0.90 1.oo 1.14 1.23
0.05 0.40
0.64 0.84 0.99 1.19 1.29
1.01 1.78 2.18 2.46 2.83 3.08
0.21 0.56 0.79 1.07 1.24
0.79 1.69 2.40 3.45 4.13
0.29 0.92 1.40 1.73 1.97 2.28 2.52
0.08 0.15 0.21 0.27 0.32
-
0.14 0.21 0.27 0.38 0.45
4
369
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
As the ionic charge increases the strength of the coupling decreases. The ER approximation should therefore be better for positive ions than for neutral atoms, and for more highly ionized systems the ER and DW approximations should give similar results. Table XI1 gives a comparison of ER and DW results for ions in the 2pz sequence. It is seen that the DW approximation gives poor results for z = 1 but more reasonable results for z 2 2. Tables XIII, XlV, and XV give results for 2p4 ions, q = 2, 3, and 4,calculated using the ER method for p-wave contributions with z < 4, and the DW TABLE XI1 COMPARISON OF THE EXACTRESONANCE AND DISTORTED WAVEAPPROXIMATIONS FOR p-WAVECONTRIBUTIONS TO THE COLLISION STRENGTHS FOR 2pq IONS, k3' = 0 *
N+ O2+
Ne4
+
2.685 1.664 0.690
0.304 0.209 0.088
0.030 0.012 0.003
1.237 1.395 0.649
0.092 0.180 0.084
Saraph et al. (1966). TABLE XI11 COLLISION STRENGTHS FOR 2p2 IONS"
N+
0.0 0.2 0.4
3.050 3.203 3.289
0.342 0.391 0.428
0.376 0.424 0.457
O2
0.0 0.2 0.4
2.391 2.398 2.388
0.335 0.345 0.351
0.310 0.319 0.326
0.0 0.2 0.4
1.376 1.552 1.727
0.218 0.216 0.212
0.185 0.186 0.187
0.0
0.800
0.128
0.117
SlO+
0.0
0.353
0.055
0.065
Zn24+
0.0
0.070
0.010
0.017
+
Ne4
+
Mg6
+
42.8
Saraph et al. (1966).
5.83
12.5
0.004 0.008 0.003
M. J. Seaton
370
method for all other contributions. Tables XVI, XVII, and XVIII give results for 3p4 ions, q = 2, 3, and 4, calculated using the DW method. In all of these calculations, theory energy differences are used. These are calculated from
k I 2 - kZ2= 0.48F2
and
k I 2 - k,' = 1.20F2
for y
=2
k I 2- k Z 2= 0.72 F2
and
k12- k3' = 1.20F2
for y
= 3,
and 4
(108)
where values of F2 are given in Table XIX. TABLE XIV COLLISION STRENGTHS FOR
O+
2p3 IONS'
0.0 0.05 0.10
1.43 1.46 1.48
0.428 0.445 0.462
1.70 1.90 1.99
0.0 0.05 0.10
1.25 1.28 1.29
0.461 0.483 0.500
1.67 1.84 1.91
0.0 0.05 0.10
1.04 1.03 1.01
0.427 0.431 0.428
1.42 1.51 1.52
Na4
0.0 0.05 0.10
0.836 0.799 0.755
0.359 0.346 0.327
1.22 1.14 1.11
Mg5+
0.0
0.652 0.604 0.555
0.289 0.266 0.241
0.942 0.916 0.882
0.0
0.208
0.089
0.340
KrZ9
0.0
0.0353
0.014
0.0691
lim,,,(z2Q)
0.0
F2
+
Ne3
+
+
0.05 0.10
Ar"
+
+
31.6
11.7
66.9
' Czyzak et al. (1967). Note that e3 = (k&)'.
6. Use of Quantum Defect Theory The equations (101) for an electron interacting with an ion X"' in the configuration np4 could be solved for negative values of the parameters E, and hence used to calculate the positions of the energy levels of states
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
TABLE XV COLLISION STRENGTHS FOR 2p4 IONSa
0.00 0.05 0.10
1.34 1.36 1.39
0.147 0.152 0.157
0.193 0* 200 0.206
NeZ
0.00 0.05 0.10
1.27 1.31 1.34
0.164 0.173 0.180
0.188 0.194 0.201
Na3
0.00 0.05 0.10
1.14 1.15 1.15
0.163 0.168 0.170
0.157 0.161 0.163
0.00 0.05 0.10
0.973 0.952 0.919
0.146 0.144 0.141
0.129 0.131 0.133
AIS
0.00 0.05 0.10
0.792 0.750 0,701
0.123 0.116 0.109
0.107 0.108 0.109
AP+
0.00
0.314
0.050
0.051
BrZ7+
0.00
0.054
0.008
0.0122
lirn,,,(zZR)
0.00
42.8
5.83
F+
+
+
Mg4
+
+
Czyzak et a1 (1967). Note that
12.5
= (k3/z)’.
TABLE XVI COLLISION
Ion
3pz
IONS,
n(l,2)
n(1,3 )
n(2,3)
6.312 4.966 I .993 1.192 0.742 0.564 0.286 0.233
1.124 1.068 0.328 0.141 0.090 0.068 0.035 0.029
1.110 0.961 1.030 0.945 0.787 0.667 0.406 0.345
P+ S’
+
c 1 3
STRENGTHS FOR k3’ = 0.0005 a
+
Ar4
+
K5
+
Ca6
+
v9
+
Crlot ~
Czyzak et al. (1967).
37 1
M . J . Seaton
372
TABLE XVII
COLLISION STRENGTHS FOR 3p3 IONS, kS2 = 0.0005
S+
CI2 Ar3+ K4 Ca5 +
+
+
V8+
Fell+
3.065 3.189 1.432 0.751 0.522 0.269 0.155
1.278 1.967 0.645 0.256 0.162 0.083 0.048
6.218 6.639 4.920 4.242 3.654 2.372 1.470
Czyzak and Krueger (1967).
TABLE XVIII
COLLISION STRENGTHS FOR 3p4 IONS, k3'
Ion
c1+ Ar2 K3 Ca4
+
+
+
V7
+
Cr8 Mn9+ Fe'O+ +
Ni12+
= 0.0005
w,2) 3.938 4.745 1.915 0.908 0.403 0.336 0.280 0.235 0.169
Q(2,3) 0.412 0.724 0.296 0.115 0.049 0.040 0.033 0.028 0.020
0.749 0.665 0.681 0.777 0.475 0.394 0.344 0.306 0.231
Czyzak et al. (1967).
TABLE XIX
THE ELECTROSTATIC INTEGRALS Fz Ion N+ O2+
Ne4+ Mg6+ SlO+ ZnZ4+
F2
0.3331 0.4263 0.6080 0.7871 1.1423 2.3768
Ion O+ F2 Ne3+ Na4+ Mgs+ Ar"+ KrZ9+ +
F2
Ion
F2
0.3775 0.4706 0.5621 0.6526 0.7424 1.2761 2.8633
F+ Ne2+ Na3+ Mg4+
0.4217 0.5149 0.6064 0.6971 0.7871 1.2326 2.7323
~
1
+
5
Arlo+ BrZ7+
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
373
np4n,l, in the ion X C r n - ' ) +. The same result can be obtained more easily using extrapolation techniques of generalized quantum defect theory (Seaton, 1966). The R matrices calculated for a number of positive energies can be fitted to analytic expressions in the energy, extrapolated to negative energies, and used to calculate the positions of the bound states. The results obtained using these extrapolation techniques should not differ significantly from those which would be obtained from exact solutions of the system (101) for negative energies. The effective quantum number v, in channel a is defined by E, =
- l/v,2.
(109)
In order to compare with observed levels it is necessary to use the exact experimental values for the energies in the npq configurations, including fine structure. In Table XX we give effective quantum numbers for 0' 2p2np calculated in the ER approximation and differences between calculated and measured effective quantum numbers. We use the usual spectroscopic notation 2p2np, 2p2n>, and 2p2n"p for states in which the dominant contribution to the wave functions come from parent terms 2p2 3P, 'D,and 'S,respectively. For the 2P states there is coupling between all three parent terms and for 2D states there is coupling between the 3P and 'D parent terms. It is seen that the calculated effective quantum numbers are systematically larger than the calculated observed values. This is due to neglect of polarization. The differences are seen to be nearly constant for the different fine structure states of each term. In Table XXI we give differences for the terms, averaged over fine structure, in various approximations. It is seen that the ER approximation is definitely superior to the DW approximation. The amount of available experimental energy level data is not sufficient to enable us to make completely empirical determinations of the R matrices, but it is possible to adjust the calculated matrices so as to obtain an improved agreement with observed levels. In Table XXI we give some results for 0' energy levels obtained on introducing four parameters into the expressions for R, which are adjusted so as to minimize the sum of the squares of the differences between observed and calculated energies. Table XXII gives the p-wave contributions to the collision strengths for six different ions, calculated with adjusted matrices. The adjusted results should be more accurate than the ER results. 7. Resonances in Collision Strengths
All calculations have been made for energies such that all channels are open, k,' 3 0. Values of !2(1,2) are required in the region for which k22 > 0 but k,' < 0. These may be obtained using extrapolation techniques. If
w
4 P
TABLE XX EFFECTIVE QUANTUM NUMBERS FOR O + 2p’np REFERRED TO THE 02+ 2p’ CALCULATIONS IN THE ER AF-PROXIMATION
LIMIT.
3 ’S
2.3959
0.0432
3 4s
2.5502
0.0652
3 zP
2.5524 2.5545
0.0316 0.0326
3 4P
2.4488 2.4495 2.4511
0.0280 0.0279 0.0280
3”P
3.0044 3.0047
0.0639 0.0628
4.4896 4.4933 4.5036
0.0248 0.0257
2.4250 2.4259 2.4273 2.4292
0.0299 0.0299 0.0299 0.0298
3.4501 3.4523 3.4563 3.4622
0.0278 0.0278 0.0278 0.0279
4.461 1 4.4647 4.4732 4.4873
0.0266 0.0269 0.0271
5 4P 4 ’P
3.5862 3.5916
0.0352 0.0365
2.5068 2.5097
0.0330 0.0326
2.9425 2.9431
0.0722 0.0733
4 ‘D
3.5659 3.5749
0.0364 0.0365
5 ’0
4.5462 4.5643
0.0327 0.0327
3°F
2.8787 2.8781
0.0413 0.0408
3”D
3”D
3 4D
4 4D
5 4D
is
r,
!? 9 0 3r
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
TABLE XXI EFFECTIVE QUANTUM NUMBERS FOR O+ 2p2np, NEGLECTING FINE STRUCTURE, REFERRED TO THE 0" 2p2 3P LIMIT
DW
ER
Adjusted
2s
0.0960
0.0432
+0.0078
3 2P 3' 2P 4 2P
0.0644 0.0910 0.0655
0.0323 0.0632 0.0361
-0.0077 -0.0090 -0.0027
'D
0.0671 0.1085 0.0509 0.0581
0.0328 0.0728 0.0365 0.0327
+0.0017 +0.0043 +0.0035 +0.0005
3' ' F
0.1028
0.0410
-0.0170
4s
0.0721
0.0652
+0.0270
3 4P 5 4P
0.0678 0.0520
0.0280 0.0253
-0.0070 -0.0047
3 4D 4 4D 5 4D
0.0752 0.0616 0.0569
0.0299 0.0278 0.0269
-0.0054 -0.0037 -0.0033
Level 3
3 3' 4 5
3
a
'D '0 '0
Czyzak et al. (1968). TABLE XXII
P-WAVE CONTRIBUTIONS TO COLLISION STRENGTHS IN THE EXACTRESONANCE APPROXIMATION (ER) AND USING ADJUSTED MATRICES (A)
nyI ,2)
n q , 3)
ER
A
ER
A
N+ 02+ F3
2.657 1.658 1.045
2.678 1.582 0.733
0.288 0.204 0.131
0.480 0.195 0.083
O+
1.302
1.449
0.381
0.423
Ne2+ Na3+
0.979 0.730
1.001 0.543
0.116 0.088
0.157 0.069
Ion
+
375
M. J. Seaton
3 76
channel 3 was not coupled to channels 1 and 2 we would have bound states in channel 3 in the region of k32 < 0. With coupling between channel 3 and channels 1 and 2 we obtain, in this region, a series of resonances in R(1,2). For most applications we require collision strengths n(1, 2) averaged over these resonances. The formulas required for the calculation of n(1, 2) are given by Gailitis (1963), and some results are given in Table XXIII. These may be compared with the results of Table XI11 for k32> 0. It should be realized that additional resonances in the collision strengths would be obtained if allowance were made for coupling with more highly excited ion states, such as those in configurations nsnp4+'. TABLE XXIII VALUESOF @I, 2) FOR k3' < 0 a
N+
-0.2399 -0.1199 -0.0
0.0 0.1199 0.2399
2.807 2.990 3.147
02+
-0.3069 -0.1535 -0.0
0.0 0.1535 0.3069
2.450 2.482 2.502
Ne4
-0.4377 -0.2189 -0.0
0.0 0.2189 0.4377
1.434 1.461 1.481
O+
-0.1812 -0.0906 -0.0
0.0 0.0906 0.1812
1.364 1.466 1.548
-0.3707 -0.1854 -0.0
0.0 0.1854 0.3707
I .235 1.286 1.330
+
Ne2
+
a
Saraph et al. (1966), Czyzak et al. (1968).
8. Transitions between Fine Structure Levels The intensity ratio of the components, at 13726, 13729, of the 2D - 4S doublet in [0111 is sensitive to N , and provides a valuable means of determining densities in nebulae (Seaton and Osterbrock, 1957). Table XXIV gives collision strengths for transitions between fine structure levels in O', required for the interpretation of this intensity ratio.
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
377
Transitions between fine structure components of the ground terms in npq ions can be of importance in connection with studies of the thermal balance in nebulae. Table XXV gives some results for transitions of the type 3PJ + 3P,, in 2pz and 2p4 ions. TABLE XXIV COLLISION STRENGTHS FOR TRANSITIONS BETWEEN FINE STRUCTURE LEVELS I N 0 , CALCULATIONS FOR k31 = 0" +
Czyzak et al. (1968). TABLE XXV COLLISION STRENGTHS FOR TRANSITIONS WITHIN 3P TERMS. FOR kS2= 0 CALCULATIONS
N+ O2
+
Ne2
+
0.401 0.378 0.185
0.279 0.213 0.131
0.128 0.948 0.527
Czyzak et al. (1968).
9. Summary on Collision Strength Calculations In the present review we have reported results obtained in an extensive new program of calculations. The work is still in progress. It is intended to apply methods of quantum defect theory to many more cases, to obtain for all cases values of n(1, 2) for energies such that k3' < 0, and to calculate further collision strengths for transitions between fine structure states of the ions. Attempts should also eventually be made to obtain improved approximations in which interactions with all states in the ground complex are taken into account. Electron temperatures and densities, and chemical compositions of nebulae, have been determined (Seaton, 1954; Aller, 1956; Seaton and Osterbrock, 1957; Seaton, 1960a; Aller, 1961) using collision strengths obtained in the
378
M . J. Seaton
earlier calculations. The new results for the important 2p4 ions, 02+, O’, and N’, do not differ from the earlier results by large factors. Comparison of calculated and adjusted results (Table XXII) gives a good indication of the accuracy of the calculations for 2p4 ions. The new results for 3p4 ions are certainly much more accurate than any previously available, but it is not possible to estimate their probable accuracy until further work has been done using quantum defect theory.
REFERENCES Aller, L. H. (1956). “Gaseous Nebulae.” Chapman & Hall, London. Aller, L. H. (1961). “The Abundance of the Elements.” Wiley (Interscience), New York. Aller, L. H., and Liller, W. (1959). Astrophys. J. 130, 45. Aller, L. H., and Minkowski, R. (1956). Astrophys. J. 124, 110. Aller, L. H., Bowen, I. S., and Minkowski, R. (1955). Astrophys. J . 122, 62. Baker, J . G., and Menzel, D. H. (1938). Astrophys. J . 88, 52. Bates, D. R., and Damgaard, A. (1949). Phil. Trans. Roy. SOC.London Ser. A 242, 101. Bates, D. R., and Massey, H. S. W. (1941). Proc. Roy. SOC.A177, 329. Bates, D. R., and Seaton, M. J. (1949). Monthly Notices Roy. Astron. SOC.109, 698. Bates, D. R., Fundaminsky, A., Leech, J. W., and Massey, H. S . W. (1950). Phil. Trans. Roy. SOC.London Ser. A 243,93. Boldt, G. (1959a). Z. Physik 154, 330. Boldt, G. (1959b). Z. Physik 154, 319. Bowen, I. S. (1928). Astrophys. J . 67, 1.Brussard, P. J., and van de Hulst, H. C. (1962). Rev. Mod. Phys. 34, 507. Burgess, A. (1958). Monthly Notices Roy. Astron. SOC.118, 477. Burgess, A. (1964). Mem. Roy. Astron. SOC.69, 1. Burgess, A., and Seaton, M. J. (1960a). Monthly Notices Roy. Astron. SOC.120, 121. Burgess, A., and Seaton, M. J. (1960b). Monthly Notices Roy. Astron. SOC.121,471. Burgess, A., and Seaton, M. J. (1960~).Monthly Notices Roy. Astron. SOC.121, 76. Burke, P. G., and Smith, K. (1962). Rev. Mod. Phys. 34,458. Burke, P. G., Cooper, J. W., Ormonde, S., and Taylor, A. J. (1967). Abstracts, 5th. Intern. Conf Phys. Electron. At. Collisions, p. 376. Nauka, Leningrad. Capriotti, E. R. (1964). Astrophys. J. 139, 225. Capriotti, E. R. (1967). Astrophys. J. 150, 95. Carroll, J. A. (1930). Monthly Notices Roy. Astron. SOC.90, 588. Cillie, C. G. (1932). Monthly Notices Roy. Astron. SOC.92, 820. Czyzak, S. J., and Krueger, T. K., (1967). Proc. Phys. SOC.(London) 90, 623. Czyzak, S. J., Krueger, T. K., Saraph, H. E., and Shemrning, J. (1967). Proc. Phys. SOC. (London). 92, 1146. Czyzak, S. J., Krueger, T. K., Martins, P. de A. P., Saraph, H. E., Seaton, M. J., and Shernming, J. (1968). Planetary Nebulae. Proc. 34th. Int. Astr. Union Symp. Reinhold, New York. Dieter, N. H., and Goss, W. M. (1966). Rev. Mod. Phys. 38, 256. Dravskikh, A. F., Dravskikh, Z. V., Kolbasov, V. A., Misezhnikov, G. S., Nikulin, D. E., and Shteinshleiger, V. B. (1965). Dokl. Akad. Nauk. SSSR 163, 332. Dufay, J. (1954). “ Nebuleuses Galactiques et Matiere Interstellaire.” Albin Michel, Paris.
ATOMIC COLLISION PROCESSES IN GASEOUS NEBULAE
379
Faulkner, D. J., and Aller, L. H. (1965). Monthly Notices Roy. Astron. SOC.130, 393. Gaititis, M. (1963). Soviet Phys. JETP (English Transl.) 17, 1328. Gardner, F. F., and McGee, R. X.(1967). Nature 213, 480. Garstang, R. H. (1968). Planetary Nebulae. Proc. 34th. Intern. Astron. Union Symp. Reinhold, New York. Goldberg, L. (1966). Astrophys. J. 144, 1225. Green, L. C., Rush, P. P., and Chandler, C. D. (1957). Astrophys. J. Suppl. Ser. 3, 37. Griem, H. R. (1967). Astrophys. J. 148, 547. Gurzadian, G. A. (1962). “Planetary Nebulae.” Moscow. Harman, R. J., and Seaton, M. J. (1966). Monthly Notices Roy. Astron. SOC.132, 15. Hayler, D. (1967). Astrophys. J . 143, 547. Hebb, M. H., and Menzel, D. H. (1940). Astrophys. J. 92,408. Heddle, D. W. O., and Seaton, M. J (1964). In “Atomic Collision Processes” (M. R. C. McDowell, ed.), p. 137. North-Holland Publ., Amsterdam. Hoglund, B., and Mezger, P. G. (1965). Science 150, 339. Huggins, W. (1864). Phil. Trans. Roy. SOC.London 154, 437. Hummer, D. G., and Seaton, M. J. (1963). Monthly Notices Roy. Astron. SOC.125, 437. Kaler, J. B. (1964). Publ. Astron. SOC.Pacific 76, 23 1. Kaler, J. B. (1966). Astrophys. J . 143, 722. Kardashev, N. S. (1959). Soviet Astron. AJ (English Transl.) 3, 813. Layzer, D. (1954). Monthly Notices Roy. Astron. SOC.114, 692. Lilley, A. E., Menzel, D. H., Penfield, H. ,and Zuckerman, B. (1966a). Nature 209, 468. Lilley, A. E., Menzel, D. H., Penfield, H., and Zuckerman, B. (1966b). Nature 211, 174. Mathis, J. S. (1957). Astrophys. J. 125, 318; 126, 493. Mathis, J. S. (1962). Astrophys. J . 136, 374. McCarroll, R. (1968). Planetary Nebulae. Proc. 34th. Intern. Astron. Union. Symp. Reinhold, New York. McGee, R. X., and Gardner, F. F. (1967). Nature 213, 579. Minkowski, R., and Aller, L. H. (1956). Astrophys. J. 124, 93. Moiseiwitsch, B. L. (1968). Rev. Mod. Phys. To be published. Morse, P. M., and Allis, W. P. (1933). Phys. Rev. 44, 269. O’Dell, C. R. (1963). Astrophys. J . 138, 1018. Oster, L. (1961). Astrophys. J. 134, 1010. Osterbrock, D. E. (1964). Ann. Rev. Astron. Astrophys. 2, 95. Osterbrock, D. E., Capriotti, E. R., and Bautz, L. P. (1963). Astrophys. J. 138, 62. Osterbrock, D. E., Miller, J. S., and Weedman, D. W. (1966). Astrophys. J. 145,697. Palmer, P., and Zuckerman, B. (1966). Nature, 209, 11 18. Peach, G. (1965). Monthly Notices Roy. Astron. SOC.130, 361. Peach, G. (1967a). Mem. Roy. Astron. SOC.71, 1. Peach, G. (1967b). Mem. Roy. Astron. SOC.71, 13. Peach, G. (1967~).Mem. Roy. Astron. SOC.71,29. Pengelly, R. M. (1963). Thesis, Univ. of London, London. Pengelly, R. M. (1964). Monthly Notices Roy. Astron. SOC.127, 145. Pengelly, R. M., and Seaton, M. J. (1964). Monthly Notices Roy. Asrron. SOC.127, 165. Pikelner, S. (1961). “Physics of Interstellar Space” (English Trans.). Foreign Languages Publishing House, Moscow. Plaskett, H. H. (1928). Publ. Dominion Astrophys. Obs. Victoria, B.C., 4, 187. Pottasch, S. R. (1960a). Ann. Astrophys. 23, 749. Pottasch, S. R. (1960b). Astrophys. J. 131, 202. Pottasch, S. R. (1961). Astrophys. J. 135, 385.
380
M . J. Seaton
Pottasch, S. R. (1965). In “ Vistas in Astronomy ” (A. Beer, ed.), Vol. 6, p. 149. Pergamon Press, Oxford. Russell, H. N., Dugan, R. S., and Stewart, J. Q.(1945). “Astronomy,” Vol. 11, p. 838 (revised ed.). Ginn, Boston. Saraph, H. E. (1964). Proc. Phys. SOC.(London) 83, 763. Saraph, H. E., Seaton, M. J., and Shemming, J. (1966). Proc. Phys. SOC.(London) 89,27. Searle, L. (1958). Astrophys. J . 128,489. Seaton, M. J. (1953a). Phil. Trans. Roy. SOC.London Ser A 245,469. Seaton, M. J. (1953b). Proc. Roy. SOC.A2 1 8 , W. Seaton, M. J. (1954). Monthly Notices Roy. Astron. SOC.114, 154. Seaton, M. J. (1955a). Proc. Roy. SOC.A231, 37. Seaton, M. J. (1955b). In “The Airglow and the Aurora” (E. B. Armstrong and A. Dalgarno, eds.), p. 289. Pergamon Press, Oxford. Seaton, M. J. (1958). Rev. Mod. Phys. 30, 979. Seaton, M. J. (1959). Monthly Notices Roy. Astron. SOC.119,90. Seaton, M. J. (1960a). Rept. Progr. Phys. 23, 313. Seaton, M. J. (1960b). Monthly Notices Roy. Astron. SOC.120, 326. Seaton, M. J. (1964). Monthly Notices Roy. Astron. SOC.127, 177. Seaton, M. J. (1966). Proc. Phys. SOC.(London) 88, 801. Seaton, M. J. (1968). Monthly Notices Roy. Astron. SOC.To be published. Seaton, M. J., and Osterbrock, D. E. (1957). Astrophys. J. 125,66. Shemming, J. (1965). Thesis, Univ. of London, London. Smith, K., Henry, R. J. W., and Burke, P. G. (1966). Phys. Rev. 147, 21. Smith, K., Henry, R. J. W., and Burke, P. G. (1967). Phys. Rev. 157,51. Sorochenko, R. L., and Borodzich, E. V. (1965). Dokl. Akad. Nauk. SSSR. 163, 603. Terzian, Y. (1965). Astrophys. J. 141, 745. Terzian, Y. (1966). Astrophys. J . 144, 657. Vorontsov-Velyaminov,B. A. (1948). “Gaseous Nebulae and New Stars.” Moscow. Wurm, K. (1954). “Die Planetarischen Nebel.” Akademie Verlag, Berlin. Yamanouchi, T., Inui, Y.,and Amemiya, A. (1940). Proc. Phys. Math. SOC.Japan, 22, 847.
COLLISIONS IN THE IONOSPHERE A . DALGARNO* School of Physics and Applied Mathematics The Queen’s University of Belfast Belfast, Northern Ireland
I. Introduction .................................................... 381 11. The Slowing Down of Fast Electrons ............................. .382 The Dayglow . . . . . . . . . . . . .................388 111. Electron Cooling Processes 1V. Ion Cooling Processes . . . . . . . . . . . . . . . . . ............ Ion and Electron Ternperatu V. Ion-Molecule Reactions ....................... . . . . . . . . . . . . . . .399 VI. The Slowing Down of Fast Protons .405 References .....................................................
I. Introduction The ionosphere of the Earth is produced mainly by the ionization of the neutral particle constituents of the atmosphere by solar ultraviolet radiation, leading to the production of free electrons and positive ions. The electrons possess initially a broad range of kinetic energies. As they slow down by collisions, the electrons cause excitation of the neutral particles and the resulting luminosity is an important component of the dayglow. The photoelectrons also preferentially heat the electron gas, causing its temperature to rise above that of the neutral particles. At high altitudes, the electron gas cools more efficiently in collisions with the positive ions than with the neutral particles, and the ion gas temperature also rises above the neutral particle temperature. The thermal electron gas is removed by recombination processes, and the recombination of electrons and positive ions is an important source of heating of the neutral particle atmosphere. Electrons can also be removed by attachment processes leading to the formation of negative ions at low altitudes. The positive and negative ions undergo a complex sequence of chemical reactions before the charged species are finally destroyed by recombination, detachment, and neutralization processes. Corpuscular bombardment is also an important source of ionization in the atmosphere, especially at high latitudes where auroral absorption and polar cap absorption events occur. The ionization is necessarily accompanied by the
* Present address: Harvard College Observatory and Smithsonian Astrophysical Observatory, Cambridge, Massachusetts. 381
382
A . Dalgarno
production of luminosity. Most visual auroras are due to electron bombardment, but for some the incident stream contains a large proton component. The auroras associated with polar cap absorption events are due almost entirely to solar cosmic ray protons. Galactic cosmic rays are the predominant source of ionization at low altitudes, but electron bombardment may also be important in the lower atmosphere at middle and low latitudes; the winter anomaly in the D region may be a manifestation of the precipitation of fast electrons into the atmosphere.
II. The Slowing Down of Fast Electrons Fast electrons lose energy by exciting and ionizing the neutral particle constituents in optically allowed transitions. The rate of energy loss per centimeter of path for a high energy electron moving through a gas of number density n cm-3 is given by the Bethe formula (ignoring relativistic effects) 1.87 x lo-'' E d-E-In- e v c m - ' -n dx E I where E is the energy in electron volts and Z is a mean excitation energy characteristic of the gas. For a gas mixture consisting of a number of constituents i with mean excitation energies Z i , (1) must be extended to
dE _ -- - 1.87 x dx E
C ni In -E i
eV cm-'.
Ii
The high atmosphere consists of atomic hydrogen, helium, atomic oxygen, molecular nitrogen, and molecular oxygen. For atomic hydrogen Z = 15.0 eV (Bethe, 1930), for helium I = 42.0 eV (Chan and Dalgarno, 1965), and for molecular nitrogen Z = 82 eV (Dalgarno et af., 1967). Precise values are not available for atomic and molecular oxygen but theory suggests a common value of 94 eV. Measurements in air at ground level yield a mean value for Z of 90 eV. Because of scattering, electrons do not travel along a straight path in the atmosphere, and the calculation of the spatial distribution of the dissipation in energy of an incident electron stream is complicated. The problem has been solved theoretically by Spencer (1959) and by Maeda (1965). For electrons in air, the predictions are in harmony with the measurements of Grun (1957). The results of Grun (1957) between 4 keV and 54 keV can be described by a simple formula for the range r ( E )
r(E) =
- jOE(dE,dx)-' dE.
COLLISIONS IN THE IONOSPHERE
383
Expressing range in atmosphere-centimeters, Grun's result is
ro = 4.57 x 10-6(E/103)7'4p
(3)
where p is the mass density of air in grams per cubic centimeter. Calculations of the altitude distributions of the energy dissipation for a number of incident energies and with various assumptions about the distribution of pitch angles in the upper atmosphere have been carried out by Rees (1963) and by Maeda (1965). The effects of the Earth's magnetic field have not been included so that the calculations apply only to high latitudes where the magnetic lines of force are nearly vertical (Maeda and Singer, 1961 ; Rees, 1963). The primary electrons produce secondary electrons of lower energies. The total ionization can be obtained directly from the calculations of energy deposition because it happens that the mean energy W, which must be expended to produce an electron-ion pair, is nearly independent of the energy of the primary particle for energies greater than a few hundred electron volts (cf. Dalgarno, 1962). For hydrogen W = 36 eV, for helium 46 eV, for molecular oxygen 33 eV, and for molecular nitrogen 36 eV (Valentine and Curran, 1958). The value of W for atomic oxygen is unknown but it is probably about 35 eV also. The production of N,+ molecules in the excited B2XC,+state can also be obtained directly from the primary energy deposition function because the cross section for the process of simultaneous excitation and ionization e+N2(X'C,+)~e+N2+(B2C:.+)+e
(4)
has a variation with electron impact energy which is very similar to that for total ionization e + N,(X'C,+)+e + CN2++ e, (5) where the summation sign indicates all energetically accessible states of N, +,including dissociating states producing N + and ionized states producing Nz +. The efficiency with which N, + is produced in the B ' X u + state per ionizing collision is an important parameter since it provides a quantitative relation between the intensity of the first negative system A ,Xg+ - B ,C,+ of N z +and the energy flux responsible for its production (Omholt, 1959). Thus deactivation of N 2 + ( B 2 C , + ) is insignificant at altitudes above about 50 km and there is no other source of excitation. Although
(6) has a large cross section, it is negligible because of the rapidity with which N, + ions are removed by dissociative recombination and ion-molecule reactions. e + N 2 + ( A 2C:,+)+e+N2+(B2C,+)
384
A . Dalgarno
The total ionization cross section for N, has been measured by Tate and Smith (1932), Fox and Hickam (1954), Lampe (1957), Fite and Brackman (1959), Peterson (1964), Rapp and Englander-Golden (1965), Schram er al. (1965), and Schram et al. (1966). The excitation function for the production of the first negative system (including cascading contributions) has been measured by Stewart (1956), Sheridan et af.(1961), Hayakawa and Nishimura (1964), Davidson and Neil (1965), and Latimer and McConkey (1965). There are substantial discrepancies between the different measurements. Dalgarno et af.(1965) concluded that an efficiency of 0.06 123914 A photons per ionizing event best represented the available data, but their conclusion has been contested byDavidson(1966). Recent measurements by McConkey et al. (1967) and Holland (1967) appear to confirm the value of 0.06, but the possibility that it may be as small as 0.03 should not be dismissed. The ionization cross sections of 0, have been measured by Tate and Smith (1932), Craggs et a]. (1957), Lampe (1957), Fite and Brackman (1959), Rapp and Englander-Golden (1965), and Schram et al. (1965, 1966). They are close in magnitude to those for 0,. The ionization cross sections of 0 have been measured by Fite and Brackman (1959) and Rothe et af (1962). They are about half those of N, and 0,. Given a model atmosphere, the efficiency of production of 3914 8, photons per ionizing collision can be calculated as a function of altitude. AdoptingJhe value of 0.06 for collisions in nitrogen and taking appropriate account of the oxygen content of the atmosphere, we conclude from the observation of 5 rayleighs of 3914 8, radiation above 85 km at midlatitudes in July 1964 by OBrien et af. (1965) that the flux of energetic electrons was about lo-, ergs cm-, sec-' (excluding electrons with energies greater than 40 keV which penetrate more deeply than 85 km). If the altitude profile of the 3914 A luminosity were known, the calculations of Rees (1963) or of Maeda (1965) would allow the determination of the absolute energy spectrum. The analysis of sixteen auroral arcs has been carried out by Belon et al. (1966) using the observed luminosity profiles to yield energy fluxes ranging from 30 to 600 ergs cm-' sec-'. The derived energy spectra may be used to calculate the intensities of other auroral luminosity features, provided the behavior of the low energy secondary electrons can be accurately described. There are no measurements of the energy distribution of the secondary electrons ejected in an ionization collision, though calculations have been performed within the first Born approximation for hydrogen and helium (Burgess, 1960; Sloan, 1965). Calculations for neon (Bates et af., 1947) may be more typical of the heavier atmospheric constituents. Seaton (1959) used the dipole approximation to the first Born approximation (Bethe, 1930) to compute the total ionization cross sections for neon and atomic oxygen, and
COLLISIONS IN THE IONOSPHERE
385
obtained satisfactory agreement with experiment. The dipole approximation may also give an acceptable description of the energy distribution of the ejected electrons. It relates the electron impact cross section for the ejection of an electron of specified energy to the photoionization cross section. Cross sections for N, have been obtained with this procedure by Takayanagi and Takahashi (1966) and it could be applied equally well to atomic and molecular oxygen. The Bethe formula (I) becomes inaccurate at low energies and it cannot be used to describe the energy losses of the photoelectrons produced by solar ultraviolet radiation or of the secondary electrons produced by corpuscular radiation. The various energy loss processes must be examined in greater detail. The inelastic scattering of electrons with energies in the region of 500 eV by N, and 0, has been measured as a function of scattering angle and of energy loss in a series of experiments by Lassettre and his colleagues. They have shown that most of the data can be interpreted in the framework of the first Born approximation which can then be applied to calculate the total excitation cross sections for individual transitions as a function of impact energy. The cross sections so derived should be satisfactory except in the threshold region. Measurements for N, have been reported by Lassettre and Krasnow (1964), Silverman and Lassettre (1965), Geiger and Stickel (1965), Lassettre et al. (1965), and Meyer and Lassettre (1966), and for 0, by Lassettre el a). (1964) and Silverman and Lassettre (1964). The Born approximation cross sections for the excitation of the alIlgstate, the b ' l l , state, and a group of states of N, at 14 eV have been computed by Takayanagi and Takahashi (1966) and the Born approximation for the dissociation of 0, has been computed by Silverman and Lassettre (1964). There are available also direct measurements at various impact energies of the excitation function of the C 'll, state of N, ,the upper level of the second positive system (Thieme, 1932;Langstroth, 1934; Hermann, 1936; Stewart and Gabathuler, 1958 ; Kishko and Kuchinka, 1959; Zapesochnyi and Kishko, 1960; Zapesochnyi and Skubenich, 1966; Jobe et al., 1967), of the cross section for dissociation of N, (Winters, 1966), and of the cross section for simultaneous excitation and ionization to the A ,Ilg state of N 2 + ,the upper state of the infrared Meinel band system (Zapesochnyi and Skubenich, 1966). Avalue of 4 x lo-" cm2 has been obtained by Williams (1935) for the excitation of the It'll, state, the upper state of the first positive system, and Engelhardt et a/. (1964) have obtained an estimate of the cross section for a transition occurring near 6.7 eV, which probably corresponds to excitations of the B311gand A 'Xuf states (Takayanagi and Takahashi, 1966). Larger cross sections for the excitation of the B 'l-4 state have been obtained by Zapesochnyi and Skubenich (1966). For 0, , measurements of the cross section for
386
A . Dalgarno
dissociative attachment have been reported (Rapp et al., 1965b) for a wide range of energies, and the inelastic scattering of electrons with energies between 4.5 and 12.5 eV has been measured by Schulz and Dowel1 (1962) [see also McGowan et al., 19641. No measurements of energy loss processes in atomic oxygen have been performed, but theoretical computations have provided accurate values for the excitation processes
(Smith et al., 1967b) which lead to emission of the oxygen red and green lines. The experimental data on the excitation of electronic levels of N, , 0, , and 0 have been supplemented by a variety of semiempirical procedures including the dipole approximation (cf. Seaton, 1962) and classical approximations (Gryzinski, 1959; Bauer and Bartley, 1965), and detailed tabulations of cross sections have been presented by Green and Barth (1965), Takayanagi and Takahashi (1966), Stolarski and Green (1967), Stolarski et al. (1967), and Watson et al. (1967). Inspection of the cross section data shows that the fast electrons slow down by producing ionization and excitation in optically allowed transitions which produce luminosity mainly in the ultraviolet region of the spectrum. With decreasing energy, optically- forbidden transitions become relatively more probable, and the slowed primaries and the secondary electrons populate metastable states leading, in particular, to the emission of infrared lines of atomic oxygen and of the first positive, second positive, Lyman-BirgeHopfield, and Vegard-Kaplan bands of molecular nitrogen. The band systems of molecular oxygen will not appear with great intensity since a substantial fraction of the collisions leads to dissociation. The lowest excited electronic state of N, lies 6.17 eV above the ground state (Miller, 1966). Below 6.2 eV, vibrational excitation is the principal source of energy degradation in N, . The process e
+ NAO) e + N 2 ( 4 +
(8)
has been studied experimentally by Haas (1957) and by Schulz (1959, 1962, 1964), and related measurements have been made bySchulz and Koons (1966), Boness and Hasted (1966), and Andrick and Ehrhardt (1966). The associated energy loss rates have been computed by Dalgarno et al. (1963), who drew attention to the importance of the process in the upper atmosphere, and by Takayanagi and Takahashi (1966). It is important not only as an energy loss process, but also because the presence of vibrationally excited nitrogen can significantly increase the rate of disappearance of electrons in the F region. Thus Schmeltekopf et al. (1967)
COLLISIONS IN THE IONOSPHERE
387
have found from laboratory measurements that the rate of the ion-molecule reaction O+ +Nz+NO+ + O (9) increased markedly with increasing vibrational temperature. Vibrationally excited nitrogen may be responsible for the decrease in electron density found in red arcs and following high altitude nuclear detonations (Whitten and Dalgarno, 1967). The processes by which vibrationally excited N, molecules are destroyed are not definitely identified. Superelastic collisions with the ambient thermal gas e
+ N2(u).+ e + N&’)
(10)
may be the controlling processes, but atom-atom interchange N
+ Nz(u)
+
N~(u’) N
(1 1) is also a possibility (Dalgarno et af., 1963). The former process is a means of heating the electron gas and the latter a means of heating the neutral particle gas. The presence of vibrationally excited N, should manifest itself in an anomalous vibrational development of the first negative system. There is some evidence that such an anomaly occurs in low latitude, high altitude, red aurora (cf. Dalgarno, 1964a). In the upper atmosphere, the low energy electrons also lose energy in exciting the metastable ‘Sand D states of atomic oxygen, the lower of which has an excitation energy of 2 eV. There are several metastable electron states of 0, , the lowest of which, the a Ag state, lies 1 eV above the ground state. The cross sections are not known with certainty, and neither is that for vibrational excitation of 0 2 .It is to be expected, however, that the cross sections eventually decrease rapidly with decreasing energy, as does the vibrational cross section of N, . Ultimately then, rotational excitation of N, and 0, will be the most efficient energy loss mechanisms in the neutral atmosphere together with the fine-structure transition +
’
e
+ 0 3PJ’e + 0 ’ P J , .
(12)
These loss mechanisms, however, are not significant in slowing down the energetic electrons because throughout the ionosphere above the D region, the energy loss in elastic scattering by the ambient thermal electrons is more efficient. This elastic scattering is an important process for selectively heating the electron gas, and it is primarily responsible for the lack of kinetic thermal equilibrium in the ionosphere (Hanson and Johnson, 1961). Detailed comparison of the efficiencies of the various energy loss processes as a function of altitude and as a function of electron kinetic energy have been presented by
388
A . Dalgarno
Dalgarno et al. (1963). It appears that if the vibrational energy of N, is converted into thermal energy of the electron gas by superelastic collisions, the last 5 eV of the kinetic energy of an electron absorbed in the daytime atmosphere above 120 km is transferred to the electron gas (the mean energy of which is of the order of 0.2 eV). The elastic scattering of the low energy secondary electrons by the ambient thermal electrons absorbs energy which would otherwise appear as metastable energy of atomic oxygen, and it provides an explanation of the anomalously low value of the ratio of the intensities of the oxygen green line at 15577 8, and the nitrogen first negative band at 13914 A observed in most auroras (Dalgarno, 1964a). It is basic also to the explanation of the approximate constancy with altitude of the 5577/3914 intensity ratio (Dalgarno and Khare, 1967). THEDAYGLOW There have been several calculations of the intensities of the emission features of the dayglow in the upper atmosphere, arising from the impact of photoelectrons, of which the most comprehensive are those of Wallace and McElroy (1966), Green and Barth (1967), and Dalgarno et al. (1968). They all assume that energy loss is a continuous process, an assumption which becomes increasingly implausible as the energy decreases. Recently, Stewart (1967) has carried out calculations which recognize the discrete nature of the energy losses. Many observations of the dayglow have been carried out using rocketborne instrumentation. Their interpretation is, in most cases, complicated by the existence of additional excitation mechanisms, such as fluorescence of solar radiation, and by deactivation processes. The second positive band system of nitrogen is of special interest since it is apparently free from such complications (Barth and Pearce, 1966). The C 'nustate is populated by fast electron impact, and it radiates through an allowed transition. The dayglow emission of the 0-0band at 3371 8, has been observed by Barth and Pearce (1966), who obtained a zenith intensity of about 400 rayleighs above 165 km, in harmony with the calculations of Nagy and Fournier (1965). The atomic oxygen resonance triplet at 1302, 1304, and 1306 8, has been observed in the dayglow (Chubb et al., 1958; Donahue and Fastie, 1964; Fastie et al., 1964; Fastie and Crosswhite, 1964; Kaplan et al., 1965; Katyushina, 1965). Theoretical interpretations of the altitude distribution, based upon resonant scattering, appeared to demand an additional source of radiation near 200 km (Donahue and Fastie, 1964) and photoelectron impact was suggested by Dalgarno (1964a, b) and Tohmatsu (1964). There are several uncertainties in the original interpretation and the position remains
COLLISIONS IN THE IONOSPHERE
389
obscure (Kaplan and Kurt, 1965; Donahue, 1965; Tohmatsu, 1965), but detailed calculations show that the photoelectron impact source has a peak value of about 100 cm-3 sec-' near 150 km (Tohmatsu, 1964; Green and Barth, 1967; Dalgarno e t a / . , 1968) which must be taken into consideration. Tohmatsu (1965) claims that the observed intensity ratio of the triplet lines implies an effective temperature of 2000"K, and he suggests that the atomic oxygen levels are in thermal equilibrium with the electron gas. Collisions with neutral particles are far more effective than collisions with electrons in causing fine-structure transitions in the atmosphere, and the suggestion is untenable. Fastie et al. (1964) and Donahue and Fastie (1964) have observed a dayglow feature at 1356 A which they identify as a forbidden atomic oxygen line. Donahue (1965) has argued that a local excitation source must be invoked. Photoelectron excitation is the most plausible (Dalgarno, 1964a; Tohmatsu, 1964) and detailed calculations generally support the suggestion (Green and Barth, 1967; Dalgarno el a/., 1968). Although not detected in the dayglow, an emission feature near 10,830 8, has been observed during twilight (Shefov, 1961, 1963; Scheglov, 1962) and during a solar eclipse (Shuiskaya, 1963). It has been identified by Shefov (1961). as a resonance line of triplet metastable helium, produced by resonant scattering of sunlight. Shefov ( 1962) has suggested that photoelectron impact excitation is the main source of He(2 3 S ) atoms, and Ferguson and Schluter (1962) have pointed out that Penning ionization
+
+
He(2 3S) 0 + He(1 IS) O+
+e
(13)
is the main destruction mechanism. A comprehensive discussion of the problem has been given by McElroy (1965), who confirms the importance of the photoelectron source. The Penning ionization process (13) has been advanced as a mechanism leading to escape of helium from the atmosphere (Ferguson et a/., 1965), but it rests on the unverified assumption that the ground state helium atoms are produced with substantial kinetic energy.' Emission of the green line in the dayglow is due to a number of processes (Walker, 1965), including dissociative recombination (Nicolet, 1964) and photodissociation (Bates and Dalgarno, 1954). The green line has been observed in the dayglow by Wallace and Nidey (1964) and by Wallace and McElroy (1966), who have also calculated the contribution to the intensity from photoelectron impact. Within the various uncertainties, theory and observation are in harmony and establish photoelectron impact as a major source (Wallace and McElroy, 1966). Perhaps the most interesting feature of the dayglow emission spectrum that has been observed is the red line of atomic oxygen at 6300 A, which originates
* Patterson (1967) has argued recently that the mechanism is in any case insufficient.
390
A . Dalgarno
in the metastable D state of atomic oxygen. It has been observed at ground level by Noxon and Goody (1962), Noxon (1964), and by rocket-borne instrumentation by Zipf and Fastie (1963, 1964), Wallace and Nidey (1964), Nagata et al. (1965), and Wallace and McElroy (1966). A theoretical analysis by Dalgarno and Walker (1964) showed that the data implied a source of O(' 0 )atoms in addition to dissociative recombination, photodissociation, and chemical reactions. Detailed calculations by Fournier and Nagy( 1965) and Wallace and McElroy (1966) confirm that photoelectron impact is a major source. Dalgarno and Walker (1964) and Noxon (1964) also concluded that O ' D must be deactivated with high efficiency, and a rate coefficient of 1 x lo-'' cm3 sec-' was proposed on the assumption that 0, was the deactivating species. The high deactivation is necessary to explain not only the altitude profile, but also the total intensity. With the inclusion of the photoelectron source, it becomes necessary to postulate a still more efficient deactivation process, and Wallace and McElroy (1966) argue that N, is the deactivating species and that the rate coefficient is 7 x lo-" cm3 sec-l. Their discussion uses arguments of Hunten and McElroy (1966) to dismiss O2 as an efficient deactivating species. Though there seems little doubt that N, must be efficient, the total neglect of 0, may be premature. Noxon (1964) has obsreved that the intensity of the red line in the dayglow undergoes rapid variations which do not seem to be correlated with variations in other atmospheric parameters that are measured at the ground, and Noxon (1964) and Dalgarno and Walker (1964) have suggested that the increases in intensity may be due to thermal excitation by the ambient electron gas. The source of the enhanced electron temperature has not been identified, but it requires very little energy. Soft corpuscular radiation and electric fields are possible sources. No other dayglow emission has been observed for which photoelectron impact is a major source, but several interesting possibilities are suggested by an examination of auroral spectra, such as the infrared Meinel bands of N 2 + and the infrared atomic oxygen lines at 7774 and 8446 8, (cf. Dalgarno, 1964a; Green and Barth, 1967; Dalgarno et al., 1968).
XII. Electron Cooling Processes The electron gas in the ionosphere is heated by elastic collisions with the photoelectrons or secondary electrons and by superelastic collisions with metastable species and vibrationally excited molecules. Because of the high efficiency of energy transfer in collisions of one electron with another, a Maxwellian velocity distribution is rapidly established, characterized by a
39 1
COLLISIONS IN THE IONOSPHERE
temperature T, which will tend to be greater than the neutral particle temperature T, . The heated electron gas cools by a variety of collision processes. Energy transfer of electrons in elastic collisions with heavy particles in the upper atmosphere has been discussed by Hanson and Johnson (1961), Dalgarno et al. (1963), and most recently by Banks (1966a), who presents formulas for the individual gas components derived from the available elastic cross section data. His results for the energy transfer rates in electron volts per cubic centimeter per second are
dU dt
=
-n,(T, - T,){1.77 x
n(N2)[1 - 1.21 x l o y 4 T,]T,
+ 1.21 x
lo-’’ n(O,)[l
+ 3.6 x
+ 3.74 x + 9.63 x + 2.46 x
lo-’’ n(o)Tb/’
n(H)[1 - 1.35 x
Ta/’]T;/’
T,]Tf/’
n(He)T;/’) (14) where the n’s are number densities per cubic centimeter. The cooling rates in 0 and O2 are somewhat uncertain. The measurements of Sunshine et al. (1967) for atomic oxygen (not all of which was in the ground state) at energies above 0.5 eV are in harmony with the calculations on ground state atomic oxygen of Robinson and Geltman (1967), Myerscough (1967), Mjolness and Ruppel (1967), Garrett and Jackson (1967), and Henry (1967), but large discrepancies exist in the theoretical results at thermal velocities. The shocktube low energy measurements of Lin and Kivel (1959) and Daiber and Waldron (1966), and some of the theoretical work suggest that the cross section may be rising rapidly with decreasing velocity at low velocities, whilst other theoretical studies suggest it is decreasing slowly. It seems best at present to accept (14), recognizing that it may be too large or too small by a factor of two. For O , , the possible error is probably comparable.’ The cooling rate for N, in (14) should be valid for temperatures exceeding 200”K, but it has the incorrect form in the limit of vanishing temperature. A representation of the cross section that is more accurate in the low velocity limit has been given by Dalgarno and Lane (1966). Energy transfer in elastic collisions with N, is in fact never important in the upper atmosphere because rotational excitation of N, is a much more efficient cooling process. Rotational excitation of N, has been studied theoretically by Gerjuoy and Stein (1955), Dalgarno and Moffet (1963), Takayanagi and Geltman (1964, 1965), Geltman and Takayanagi (1966), and Mjolness and Sampson (1964). The computed cross sections are in satisfactory
* The recent work of Hake and Phelps (1967) greatly improves the position.
392
A . Dalgarno
agreement with the analysis of swarm experiments (Engelhardt et al., 1964) in the thermal energy region. Takayanagi (1965a) has calculated the corresponding energy transfer cross sections by averaging the cross sections of Takayanagi and Geltman (1965) over the rotational distribution of the molecule for neutral particle temperatures up to 1000°K. The theoretical model he employs is not valid above about 1 eV where close coupling calculations are probably necessary. Takayanagi did not calculate the average over the electron velocity distribution. The earlier calculations of Dalgarno and Moffett (1962) should be satisfactory for the analysis of upper atmosphere phenomena. As extended by Dalgarno and Henry (1965), they can be represented by the cooling rate
with a possible uncertainty of a factor of 2. Rotational excitation of 0, has been studied by Dalgarno and Henry (1965), Takayanagi (1965b), Geltman and Takayanagi (1966), and Sampson and Mjolness (1966). The corresponding energy transfer cross sections have been computed by Takayanagi (1965b), and the cross sections and their average over the electron velocity distribution by Dalgarno and Henry (1965). Both calculations were based upon the long range model of Dalgarno and Moffett (1963) which, because of the small quadrupole moment of 0, (Bridge and Buckingham, 1964), yields energy transfer rates much less than those for N, at thermal velocities. Experimental investigations of Mentzoni and Rao (1965) show that cooling in O2 proceeds more rapidly than in N 2 . Geltman and Takayanagi (1966) have carried out exploratory calculations which demonstrate that short range interaction effects may be responsible but further study is needed. We recommend tentatively the cooling rate
but the possible error is as large as an order of magnitude. Its influence on electron temperature calculations is not important above 120 km. Above 1500"K, vibrational excitation of N , is a more efficient cooling process than rotational excitation of N , (Dalgarno and Henry, 1965). Takayanagi (1965a) has tabulated the experimental data on energy transfer cross sections but not the average over the electron velocity distribution. We present in Table I the cooling rates due to vibrational excitation of N , used by Dalgarno et al. (1967a). There is considerable uncertainty in the cross sections for vibrational and electronic excitation of 0, by electron impact, and we assume that Eq. (16)
COLLISIONS IN THE IONOSPHERE
393
TABLE I COOLING RATESTHROUGH VIBRATIONAL EXCITATION OF N,"sb ~
~
Te ( O K )
R
200 300 500 I000
I500
3 x 10-19 7 x 10-17 7 x 10-15 2 x 10-13 8 x 10-13
2000
2x
a
~
R 1x 5x 1x 3x 2x 7x
2500
3000 3500
4000 4500 5000
10-12
Rees et al. (1967). dUJdt = --n,n(N,)R
~~
Te (OK)
10-11 lo-" 10-10 10-l0 10-9 10-9
eV c m v 3sec-'.
[with its attendant uncertainty] incorporates vibrational and electronic cooling also. With increasing altitude, cooling of the heated electron gas by neutral particle collisions is dominated by atomic oxygen. Energy transfer by elastic collisions, as expressed in Eq. (14), is probably not significant. Thermal electrons in atomic oxygen are expected to lose energy more efficiently in exciting fine-structure transitions e
+ O(3PJ) e + O(3PJ.).
(17)
+
The 3P0and 3P1levels lie 0.028 and 0.020 eV above the 3P2 level. The importance of the heavy particle analog of (17) in the thermal balance of the neutral atmosphere has been discussed by Bates (1951). No calculations of the cooling rates associated with (17) have been reported, and the process has been ignored in studies of the charged particle temperature equilibrium in the i~nosphere.~ With increasing temperature, cooling by excitation of the metastable D level of atomic oxygen becomes significant. Rees ef a/. (1967) give for the corresponding cooling rate
x (0.406
+ 0.357 x
T, - (0.333
x exp(- 1.37 x 104/Te)- (0.456
+ 0.183 x
+ 0.174 x
x exp( -2.97 x 104/Te)} eV cm-3 sec-'.
T,) T,) (18)
Cooling rates for (17) have now been computed (Dalgarno and Degges, 1968). They confirm that (17) is a major cooling mechanism.
394
A . Dalgarno
Although elastic collisions with neutral particles make only a minor contribution to the cooling of the electron gas, elastic collisions with positive ions play a major role (Hanson and Johnson, 1961; Dalgarno et al., 1963). If M iis the mass of the ion in atomic mass units, the cooling rate is
due - - - -7.7 x 10-6ne ni(Te - Ti) eV cm-3 sec-' dt Mi T:"
(19)
where Ti is the ion gas temperature. Depending upon ionospheric conditions, the coefficient in (19) may vary by 10% (cf. Banks, 1966a). Since the fractional ionization content of the atmosphere increases upwards, (19) shows that cooling to ions for Te less than about 3Ti becomes the most efficient energy loss process at high altitudes, but at higher temperatures cooling is controlled by collisions with neutral particles. It is interesting to note that in the absence of neutral particles, a runaway condition could be established (Hanson and Johnson, 1961; Dalgarno et al., 1963). The dominance of electron-ion cooling at high altitudes means that the ion gas will be heated by the electron gas and the ion temperature Ti may be greater than the neutral particle temperature also (Hanson, 1963; Dalgarno, 1963). Since the lighter ions are more efficient cooling agents, the different ionic components may have different temperatures (Banks, 1967 a, b ; Dalgarno and Walker, 1967).~
IV. Ion Cooling Processes The heated ion gases transfer energy in collisions with the neutral particles. In the F region of the ionosphere, ion cooling is mainly by means of the resonance charge transfer process (204 (Hanson, 1963). At higher altitudes, elastic collisions of 0' with He contribute, and the He' ions are cooled by resonance charge transfer O++0+0+0+
He+ + H e + H e + H e + (20b) and by elastic collisions of He' with 0 (Willmore, 1964). Various formulas for the ion cooling rates have been published (Hanson, 1963; Dalgarno, 1963; Willmore, 1964; Brace et al., 1965). A comprehensive collection, based upon the most accurate collision cross section data, has been presented recently by Banks (1966b), who gives formulas for the energy transfer in elastic collisions of H + , He+, and 0' with H, He, N,, and O , , and for energy transfer in resonant charge transfer collisions of H + , He', O', N + , N2+and 0,' in their parent gases.
COLLISIONS IN T H E IONOSPHERE
395
The results for elastic collisions are based upon the assumption that the scattering is controlled by the long range polarization forces. This assumption yields satisfactory cross sections at temperatures below 300°K (cf. Dalgarno et al., 1958), but it tends to underestimate the energy transfer rate at higher temperatures. The variation with temperature has been discussed qualitatively by Dalgarno (1961), but detailed predictions for the atmospheric ions in atmospheric gases are not possible [with the unimportant exception of H + in He, the cooling rate for which can be derived from recent calculations of the mobility of H + ions in He by Dickinson (1968)l. Within the range of atmospheric ion temperatures, the energy transfer rates given in Table I1 (selected from Banks, 1966b) are probably accurate to within a factor of three. TABLE I1
ELASTICION i
M
R x 1014
O+
He H 0 He 0 H
2.8 3.3 3.5 5.5 5.8 10
H+ He+
a
COOLING R A T E S " b
From Banks (196613). dUi/dt = --ni n(M)(Ti - Tn)R
eV ~ r n sec-'. - ~
Banks ( I 966b) also calculates approximate energy transfer rates for ions scattered by the molecular gases N, and 0, , again based upon the long range polarization forces, which he describes as elastic energy loss. Some care is needed in interpreting his formulas. Unlike the case of electrons scattered by homonuclear molecular gases, the rotational levels of the molecule are strongly coupled during an ion-molecule collision even at thermal velocities. A substantial fraction of the scattering is inelastic. It can be argued that the total scattering is satisfactorily described by an elastic scattering model in which inelastic scattering is not possible (Bernstein et al., 1963), so that the formulas given by Banks (1966b) may yet be satisfactory representations of the energy loss associated with the transfer of momentum during a collision, but the argument implies that energy loss by rotational excitation is large. A similar argument applies to the scattering by atomic oxygen for which energy loss by excitation of the fine structure levels through say He+
+ O('P,) +He+ + O(3P1)
(21)
396
A . Dalgarno
may also be large. A simple formula for processes like (21) has been derived by Dalgarno and Rudge (1964). Detailed calculations will be necessary. An energy loss rate larger than that associated with momentum transfer appears possible. Banks (1966b) has presented also a collection of formulas describing the energy loss rates associated with resonance charge transfer processes such as (20a) and(20b). Some of the formulas are reproduced in Table 111. For application to the thermal balance of the atmospheric ions, it is desirable to supplement this list by formulas appropriate to the possible inelastic processes. TABLE 111 RESONANCE CHARGE TRANSFER RATES".~ COOLING
i
M
R x 1015
o+
0 He H N
2.1 4.0 14 2.1
He+
H+ N+
From Banks (1966b). dU,/dr = - n,n(M)(T, T,,)l/z(T,- T.)ReV~rn-~sec-'. a
+
The role of the accidentally near-resonant charge transfer processes
+ O+ H +O+
+ O(3P,)+ 0.019 eV H + + O(3P1)- 0.001 eV H f O f + H++ O(3P0) - 0.009 eV
H
+
+
H+
(22)
on the ionospheric thermal balance has been studied by Banks (1966~)on the assumption that the resonance is exact. An attempt has been made by Dalgarno and Walker (1967) to include the effect of the finite energy discrepancies. They point out that (22) is a source of cold protons which are then heated by collisions with the ambient thermal proton gas. Substantial differences between the temperatures of the different ion gases would occur in the topside ionosphere were it not for the close coupling of different ions with each other, which occurs because of the long range Coulomb interaction. The energy transfer rate from an ion gas X + at a temperature T ( X + )and with mass M(X +)to an ion gas Y + at a temperature T( Y +)and
397
COLLISIONS IN THE IONOSPHERE
with mass M( Y +) is given by Spitzer (1956) as dU -( x + Y , + )= 3.3 x 10-4 dt
T ( X +)- T( Y +)
n ( x + ) n ( M~ (+X )+ )M ( Y +)
eV ~ r n sec-'. - ~
(23)
It has been used by Banks (1966~)and by Dalgarno and Walker (1967) to investigate the ion temperature differences that may occur in the ionosphere. It appears that temperature differences exceeding 200°K are unlikely unless the electron temperature is usually high. The conclusion is reinforced when the effect of ionic thermal conductivity is included in the analysis (Banks, 1966~). The atomic nitrogen ion may be an exception. Nicolet and Swider (1963) have noted that dissociative photoionization of N, is an important source of N + ions, and McElroy (1967) has pointed out that the process produces N + ions with substantial kinetic energy so that the effective temperature of the N + ions may exceed the temperature of the other ions.
IONAND ELECTRON TEMPERATURES
A variety of observational techniques has established that the ion and electron temperatures substantially exceed the neutral particle temperatures in the F region of the ionosphere during the daytime. Some recent studies are those by Farley (1966), Brace et a/. (1967), Evans (1967), Knudsen and Sharp (1967), Carru et a/. (1967a), and Mahajan (1967). Early theoretical studies (Hanson and Johnson, 1961; Hanson, 1963; Dalgarno et a/., 1963) assumed a local equilibrium, cooling being due only to collisions. Hanson (1963) pointed out that thermal conduction in the electron gas would modify the profile at great heights, and Geissler and Bowhill (1965) confirmed the importance of conduction. Geissler and Bowhill adopted a conductivity appropriate to a full ionized plasma. Formulas which include the effect of collisions with the neutral particles have been given by Banks (1967a, b) and by Dalgarno et a/. (1967a). Calculations by Da Rosa (1966) show that the assumption of a steady-state time-independent equilibrium is valid. The theoretical studies all emphasize the control over the electron temperature that is exerted by the electron density, and direct experimental verification is provided by the satellite observations of Brace e t a / . (1965). In circumstances where the electron density is low, large electron temperatures may occur. Dalgarno and McElroy (1965) have considered the situation near dawn and have shown that the electron temperature will rise rapidly. Their study ignores the effect of conduction which serves to limit the increase (Dalgarno et a/.,
398
A . Dalgarno
1967a) so that their analysis is appropriate only at equatorial latitudes. Predawn rises in electron temperature have been observed and it is proposed that the heat source is provided by photoelectrons streaming from the sunlit magnetic conjugate ionosphere (Carlson, 1966). Hanson (1963) had earlier pointed out that a proportion of the photoelectrons produced in the topside ionosphere would escape upwards along the magnetic lines of force. The conjugate point photoelectrons may also explain the predawn enhancement of the red line by thermal excitation (Cole, 1965) and by direct electron impact. An enhancement of the twilight glow of 3914 1$ radiation has been interpreted as due to conjugate point photoelectrons by Broadfoot and Hunten (1966). The theoretical studies all show that for solar minimum conditions, electron conductivity produces an isothermal region above about 300 km if it is assumed that there is no conductive flux from above. Because of the increased electron densities, local cooling is more important at sunspot maximum and the altitude profile may contain a maximum (Geissler and Bowhill, 1965). The backscatter data appear to establish the existence of a small positive temperature gradient at high altitudes at sunspot minimum (Evans, 1967), and Geissler and Bowhill (1965) suggest that heat, deposited in the protonosphere by the escaping photoelectrons, is conducted downwards. According to Evans (1967), the flux is probably sufficient to explain the night time observations (see also Nagy and Walker, 1967), but it may be necessary to invoke an additional source of heat during the daytime. In an earlier attempt to explain the night time satellite observations of Bowen el al. (1964), Willmore (1 964) suggested that the heat source might be the low energy particle fluxes observed by Savenko et al. (1963), and Nathan (1966) has shown that the particles remain trapped sufficiently long so that there is no violation of the limit placed on precipitated fluxes by the observations of 3914 A radiation. The electron temperature at night is very sensitive to the incidence of soft corpuscular radiation, and Dalgarno (1964a) has derived an upper limit of 0.03 erg cm-2 sec-' for the possible flux. A detailed study of the diurnal variation of electron and ion temperatures during solar minimum has been carried out by Dalgarno et al. (1967a), who computed the solar ultraviolet heat source as a function of time during the day and took account of the varying neutral particle temperatures and densities. Bearing in mind the possibility of a heat flux from the protonosphere, the agreement with the backscatter measurement of Evans (1965) is satisfactory above 300 km, but there are significant qualitative differences in the electron temperatures at lower altitudes though the predicted and observed ion temperatures remain in harmony. The observations in the E region (Spencer et al., 1965; Smith et af., 1965; Knudsen and Sharp, 1965; Hirao, 1966; Smith, 1966) cannot be explained on
COLLISIONS IN THE IONOSPHERE
399
the conventional theory which is based upon heating by collisions with photoelectrons and cooling by collisions with neutral particles. Evidence of stratification is found in rocket experiments but not in backscatter observations (Carru et a/., 1967a,b) which suggests that the stratifications, if real, are short term fluctuations. The theory appears to require generalization. The influence of recombination heating on the electron and ion temperatures in sporadic E-layers has been studied by Gleeson and Axford (1967), and it may be relevant to the normal E region. Joule heating, possibly associated with a n energetic charged particle flux, may also be a contributing source (Cole, 1967).4 Thermal equilibrium probably prevails in the D region (Dalgarno and Henry, 1965). Electron and ion temperatures during auroras have also been the subject of theoretical study. According to Rees et al. (1967), in one particular auroral arc the electron temperature attained a value of about 4500°K at high altitudes and much of the atomic oxygen red line emission was due to thermal excitation. High ion temperatures in auroral zones have been observed by Knudsen and Sharp (1967).
V. Ion-Molecule Reactions The positive ions of the major constituents produced by photoionization and by corpuscular ionization take part in a complex sequence of ionmolecule reactions. The rates of most of the reactions that are of importance have now been measured at least at room temperature (cf. Ferguson, 1967). There have been many investigations of ionic content using rocket-borne mass spectrometers and ion traps [for recent work see Hoffman (1967, Young et a/.(1967), Smith et a/.(1967a)], radar backscatter observations [for recent work see Farley (1966), Carlson and Gordon (1966)l and using whistler data, Shawhan and Gurnett (1966). Several reviews of the reaction rate scheme have appeared (Nicolet and Swider, 1963; Dalgarno, 1964b; Nicolet, 1965; Donahue, 1966). The interpretation of the He' content in the high ionosphere encounters a difficulty. The He' ions diffuse downwards in the atmosphere and are there destroyed by the ion-molecule reactions He+ + N z + H e + N z + He+ + N z + H e + N + N + He+ + O z + H e + O , + He+ + O z - , H e + O + O + . Walker (1968) has suggested that (10) may be a significant heat source, N2(u) being produced by deactivation of O('D) atoms.
400
A . Dalgarno
According to Bauer (1966), the measured concentrations of He' imply that the rate coefficient of any of the reactions in (24) does not exceed lo-" cm3 sec-l, whereas the values measured in the laboratory are of the order of cm3 sec-' (cf. Ferguson, 1967). No satisfactory resolution of the discrepancy has yet been advanced. In contrast, it appears from the discussion by Donahue (1966) that the ionic composition in the daytime E and F regions can be satisfactorily interpreted in terms of known reactions, though some modifications may be necessary to reflect the fact that ions are produced in metastable states, which react differently. Thus Dalgarno and McElroy (1966) have suggested that O'('0)
+ Nz + 0 + Nz+
(25)
is an important source of N2+ in the F region. The persistence of the ionosphere at night appears to require a source of ionization in addition to diffusion from above. Prag et al. (1966) suggest a flux of low energy protons, and Ogawa and Tohmatsu (1966) suggest that ionization by geocoronal emission of hydrogen and helium lines is a sufficient source. Difficulties of interpretation emerge again at lower altitudes, and we present a more detailed discussion of the theory of the D region. Ionization is produced in the D region mainly by absorption of Lyman c( by nitric oxide and by absorption of x rays and cosmic rays by the atmospheric constituents (Nicolet and Aikin, 1960). Variations in intensity occur during the solar cycle, and the x ray intensity can undergo very large temporary enhancements following solar flares. Cosmic ray ionization depends strongly on latitude. Large increases in D region ionization occur at high latitudes during auroral absorption and polar cap absorption events when the atmosphere is bombarded by high energy electrons and by solar protons. A further variation can occur since the Lyman c( ionizing flux is attenuated by the atomic hydrogen in the atmosphere. The resulting chemistry is complicated. The NO' ions produced by Lyman ct photoionization of nitric oxide NO
+ h1216 = NO+ + e
(26)
can be removed by dissociative recombination NO+ + e + N + O ,
(27)
by mutual neutralization with negative ions X NO+
+ X- + N O + A',
(28)
by charge transfer to a minor constituent (such as a metal atom M ) which has a low ionization potential NO++M+NO+M+,
(29)
40 1
COLLISIONS IN THE IONOSPHERE
or by ion-molecule reactions with minor constituents such as NO+
+ 0,
NO,+
+
(30) The positive ion O,+ can be produced in the D region by absorption of Lyman a and of x rays and cosmic rays. It can be destroyed by dissociative recombination +
0 2 .
(31)
02+ +e+O+O,
by mutual neutralization
o,++ x-+ 0 2+ x, and by transformation into NO' by the reactions
+ N2 NO+ + NO O,+ + N + O + N O +
0 2 +
(33)
+
(34)
02' +NO+NO+ + 0 2 . (35) The positive ion N z + , produced by x ray and cosmic ray absorption, is rapidly converted into O,+ by the reaction
Nz+
+
0 2
+
Nz
+ Oz+
which has a rate coefficient 1 x lo-'' cm3 sec-' (Fite et a/., 1966; Warneck, 1967), whereas N2+
+0
+ NO+
+N
(36) et a/.,
1963, Goldan (37)
is more important at higher altitudes. The atomic ions 0' and N f are also produced by x ray and cosmic ray absorption. They participate in the ion-molecule reactions N + +O,+NO+ + O Nf+0,+02+ +N
o++o,+o,++ o 0' + N , + N O + +o. It follows from the measured reaction rates (cf. Ferguson, 1967) that the main positive ions in the D region should be NO+ and 0,'. The positive ion distribution has been measured by Narcisi and Bailey (1965) (cf. Narcisi, 1966). Above 83 km, masses 30 and 32 become predominant, though five metal ion peaks appear. There is, however, a serious quantitative discrepancy. At 85 km, the production of 0,' by X-ray absorption is about lo-' cm-3 sec-' during quiet conditions, whereas the removal rate through dissociative recombination (31) is about 10 cm-3 sec-' and through (35) about 100 cm-3 sec-', if the measured rate coefficients are accepted.
402
A . Dalgarno
Thus if the mass 32 peak consists only of O,', and the measured concentration is correct, there must be an unidentified ionization source.' The discussion may require modification since 0,' ions are also produced in the metastable a411 state, whereas the laboratory measurements refer to ground state ionic species. Below 81 km, the predominant ions have masses 37,30, and 19, and there is an important contribution from masses greater than 47. Masses 18 and 28 are also detected. Mass 28 is probably Si+ since (36) is too rapid to permit significant concentrations of N,+ to exist. Narcisi and Bailey identify mass 18 as H,O+, mass 19 as H 3 0 + , mass 37 as H 5 0 2 + ,and they suggest that the heavy ions may be higher hydrates of H 2 0 + and other cluster ions. Fe+ may occur in the heavier ions. Narcisi (1966) has advanced several arguments countering the obvious criticism that the measurements are contaminated by water vapor carried aloft by the rockets. His claim that the sharp decrease of H 3 0 + and H,O,+ near the mesopause is consistent with the expected decrease in the atmospheric water vapor content may not be convincing since the low temperature may also affect the rate of outgassing, but the fact that masses 19 and 37 reappear at the same altitudes on descent is more persuasive. The ions H 3 0 + and H 5 0 2 + can be produced from H,O+ by HzO+ + H z O + H , O + + O H
H,O+ + H z O + M + H , O z + + M
(42) (43)
The difficulty, recognized by Narcisi (1,966), is that the reaction HZO+ +O,+O
+ HzO
(44) proceeds much more rapidly than (42). The' rate coefficient for (44) quoted by Ferguson (1967) is comparable to that of (36) so that N,+ should be much more abundant than H 2 0 + (if each ion is produced by ionization of the parent molecule). The measured concentration of mass 18 is, in any case, much too large if (44) is as rapid as laboratory measurements suggest. Hunt (cf. Narcisi, 1966) has suggested tentatively that ionization of water vapor conglomerates, ranging in size from 0.0005 to 0.01 p, may account for the presence of ions H.(H20),+, many ions being produced by simultaneous ionization and fragmentation in a single collision with a cosmic ray. The metal ions detected by Narcisi and Bailey (1965), Narcisi (1966), Istomin (1963, 1966), and Young et al. (1967) are of considerable importance since they cannot participate in ion-molecule reactions with the major constituents and so recombine very slowly. They are presumably responsible I +
Hunten and McElroy (1968) have argued that photoionization of 02('Ag) is the additional source.
COLLISIONS IN THE IONOSPHERE
403
for the persistence of sporadic E layers (cf. Narcisi, 1966; Young et al., 1967). Little is known of the mechanisms responsible for their eventual removal. Donahue and Meier (1967) have proposed the reaction M+f
MO+ 4-0
(45) and Hunten and Wallace (1967) suggest that recombination occurs on aerosol particles. The contribution of mutual neutralization M+
0 34
2
+ x-+ M + x
(46) depends upon the abundance of X -. The electrons produced in the D region are removed by dissociative recombination processes, but also by attachment. The most efficient attachment process throughout most of the D region is e
+ 0,+ M - , 0,-+ M ,
(47)
but e+03-+O- +02 (48) (Branscomb, 1964) may also be important. Both 0-and 0 2 -can be destroyed by associative detachment
(Massey, 1950). Dalgarno (1961) noted that the rate coefficient might be as large as lo-'' cm3 sec-', and Fehsenfeld et al. (1967) have recently measured the rate coefficient of (49) to be 3.3 x lo-'' cm3 sec-' at 300°K and of (50) to be 1.9 x lo-'' cm3 sec-'. Processes (49) and (50) are more efficient than photodetachment above 50 km. The rate coefficient of 0- +O,-+O,-
+o
( 5 1)
may also be large, but it has not been measured. Depending upon the ratio of the concentrations of ozone to atomic oxygen, 0-may react alternatively to form 0 3 -through
+
0-
0 3 +0 3 -
f 0.
(52)
+Oz.
(53)
Another source of 0,-is 0 2 -
fO,+O3-
Fehsenfeld et al. (1967) have found that 0 3 -
+ coz
-+
co3-
f
0 2
(54)
404
A . Dalgarno
has a rate coefficient of 4 x lo-'' cm3 sec-' at 300°K so that 0 3 -is probably rapidly converted into C 0 3- . There are exothermic competing mechanisms
but only the rate coefficient of (57) has been measured. The C 0 3 - can be transformed back into 0,-by the reaction COB-
+0
+0 2 -
+ coz
(58)
but some of the C 0 3 - will react with nitric oxide to produce the very stable NO2- negative ion according to
coo- + NO + NO,-
+ COz .
(59)
(Felsenfeld et. al. (1 967). NO,- has frequently been advanced in explanation of the polar cap absorption twilight anomaly, which suggests that radiation longer than 2900 A is ineffective in destroying negative ions. The sunrise rocket measurements of Bowhill and Smith (1966) can be interpreted as supporting this requirement. It should be noted that there is a production of NO and NO, during PCA events, and that charge transfer of negative ions in collisions with NO, to form NO2- is usually a fast process. The problem of the nocturnal destruction of NO,- has not been satisfactorily resolved. The laboratory studies by Pack and Phelps (1966) show that C0,- forms readily in a mixture of O2 and C O , . It may be produced in the D region through 02-
+ coz + M+C04- + M.
(60)
Many other reactions can be written down involving minor constituents, such as H,, H 2 0 , H 0 2 , H 2 0 2 , OH, and H. H 0 2 - may have an electron affinity of nearly 3 eV (cf. Branscomb, 1964). Of special interest may be the metastable species O,(a A,) since it is produced by ultraviolet radiation . and Hasted (1965) have suggested that it may play an absorbed by 0 3 Megill important role since the detachment process
'
+
0,- 0 2 ( al A g ) + 0 2
+ O2 + e
(61)
is similar to a Penning ionization process and may be very fast. A discussion of the ionization balance in the D region which recognizes that there may be several major ion constituents has been presented recently by Adams and Megill (1967).
COLLISIONS IN THE IONOSPHERE
405
VI. The Slowing Down of Fast Protons The penetration of fast protons into the atmosphere at high altitudes is established by observations of Doppler-shifted hydrogen lines (Vegard, 1939, 1948; Meinel, 1957) and by rocket measurements (McIlwain, 1960), the hydrogen being produced in an excited state by the capture of an electron from a neutral constituent. The role of solar protons in polar cap absorption phenomena has been clearly identified (cf. Bailey, 1964). Detailed discussions of the absorption of protons in the atmosphere have been given by Chamberlain (1961), Davidson (1965), Prag et al. (1966), and Eather (1966, 1967) with special reference to bombardment in the auroral zone, and by Reid (1967) and Sandford (1967) with special reference to the bombardment over the polar cap. Some modifications may be necessary in these calculations because of the quenching of the excited hydrogen atoms through collisions (Bates and Walker, 1966). Penning ionization processes in collisions with O2 may also reduce the intensity of the hydrogen emissions.
ACKNOWLEDGMENT The research reported has been sponsored by the US Office of Naval Research for the Advanced Research Projects Agency, Department of Defense under Contract N62558-4297.
REFERENCES Adams, G. W., and Megill, L. R. (1967). Planetary Space Sci. 15, 1111. Andrick, D., and Ehrhardt, H. (1966). Z. Physik 192,99. Bailey, D. K. (1964). Planetary Space Sci. 12, 495. Banks, P. M. (1966a). Planetary Space Sci. 14, 1085. Banks, P. M. (1966b). Planetary Space Sci. 14, 1105. Banks, P. M. (1966~).Earth Planetary Sci. Letters 1, 270. Banks, P. M. (1967a). Ann. Geophys. 23, 577. Banks, P. M. (1967b) Planetary Space Sci. 15, 17. Barth, C. A., and Pearce, J. B. (1966). Space Res. 6 , 381. Bates, D. R. (1951). Proc. Phys. SOC.(London) B64,805. Bates, D. R., and Dalgarno, A. (1954). J. Atmospheric Terrest. Phys. 5 , 329. Bates, D. R., and Walker, J. C. G. (1966). Planetary Space Sci. 14, 1367. Bates, D. R., McDowell, M. R. C., and Omholt, A. (1957). J . Atmospheric Terrest. Phys. 10,51. Bauer, E., and Bartley, C. E. (1965). J. Chem. Phys. 43, 2466. Bauer, S. J. (1966). Ann. Geophys. 22, 247. Belon, A. E., Romick, G. J., and Rees, M. H. (1966). Planetary Space Sci. 14, 597.
406
A . Dalgarno
Bernstein, R., Dalgarno, A., Massey, H. S. W., and Percival, I. C. (1963). Proc. Roy. SOC. A274,427. Bethe, H. A. (1930). Ann. Physik 5, 325. Boness, M. J. W., and Hasted, J. B. (1966). Phys. Letters 21, 526. Bowen, P. J., Boyd, R. L. F., Raitt, W. J., and Willmore, A. P. (1964). Proc. Roy. SOC. A281, 526. Bowhill, S., and Smith, L. G. (1966). Space Res. 6, 511. Brace, L. H., Spencer, N. W., and Dalgarno, A. (1965). Planetary Space Sci. 13, 647. Brace, L. H., Reddy, B. M., and Mayr, H. G. (1967). J . Geophys. Res. 72,265. Branscomb, L. M. (1964). Ann. Geophys. 20,49. Bridge, N. J., and Buckingham, A. D. (1964). J. Chem. Phys. 40,2733. Broadfoot, A. L., and Hunten, D. M. (1966). Planefury Space Sci. 14, 1303. Burgess, A. (1960). Astrophys. J. 132, 503. Carlson, H. C. (1966). J. Geophys. Res. 71, 195. Carlson, H. C., and Gordon, W. E. (1966). J. Geophys. Res. 71, 5573. Carru, H., Petit, M., and Waldteufel, P. (1967a).J. Atmospheric Terrest. Phys. 29, 351. Carru, H., Petit, M., and Waldteufel, P. (1967b).Planetary Space Sci.15, 944. Chamberlain, J. W. (1961). “Physics of the Aurora and Airglow.” Academic Press, New York. Chan, Y. M., and Dalgarno, A. (1965). Proc. Roy. SOC.A285, 457. Chubb, T. A., Byram, E. T., Friedman, H., and Kupperian, J. E. (1958). Ann. Geophys. 14, 109. Cole, K. D. (1965). Ann. Ceophys. 21, 156. Cole, K. D. (1967). Planetary Space Sci. 15, 873. Craggs, J. D., Thorburn, R., and Tozer, B. A. (1957).Proc. Roy. SOC.A240,473. Daiber, J. W., and Waldron, H. F. (1966). Phys. Rev. 151, 56. Dalgarno, A. (1961). Ann. Geophys. 17, 16. Dalgarno, A. (1962). In “Atomic and Molecular Processes,” (D. R. Bates, ed.). Academic Press, New York. Dalgarno, A. (1963). NASA Contractor Rept. CR-8. Dalgarno, A. (1964a) Ann. Geophys. 20, 65. Dalgarno, A. (1964). Discussions Faraday SOC.37, 142. Dalgarno, A., and Degges, T. C. (1968). Planefary Space Sci. (in press). Dalgarno, A., and Henry, R. J. W. (1965). Proc. Roy. SOC.A288, 521. Dalgarno, A,, and Khare, S. P. (1967). Planetary Space Sci. 15, 938. Dalgarno, A., and Lane, N. F. (1966). Astrophys. J. 145, 623. Dalgarno, A., and McElroy, M. B. (1965). Planetary Space Sci. 13, 143. Dalgarno, A., and McElroy, M. B. (1966). Planetary Space Sci. 14, 1321. Dalgarno, A,, and Moffett, R. J. (1962). Planetary Space Sci. 11, 463. Dalgarno, A., and Moffett, R. J. (1963). Proc. Natl. Acad. Sci. India A33, 51 1. Dalgarno, A., and Rudge, M. H. (1964). Astrophys. J . 140, 800. Dalgarno, A., and Walker, J. C. G. (1964). J. Atmospheric Sci. 21, 463. Dalgarno, A., and Walker, J. C. G. (1967). Planetary Space Sci. 15, 200. Dalgarno, A., McDowell, M. R. C., and Williams, A. (1958). Phil. Trans. Roy. SOC.London, A250, 41 1 . Dalgarno, A., McElroy, M. B., and Moffett, R. J. (1963). Planetary Space Sci. 11,463. Dalgarno, A., Latimer, J. D., and McConkey, J. W. (1965). Planetary Space Sci. 13, 1008. Dalgarno, A., McElroy, M. B., and Walker, J. C. G. (1967a). Planetary Space Sci. 15, 331. Dalgarno, A., McElroy, M. B., and Stewart, I. A. (1968). In press. Da Rosa, A. V. (1966). J. Geophys. Res. 71,4107.
COLLISIONS IN THE IONOSPHERE
407
Davidson, G. T. (1965). J. Geophys. Res. 70, 1061. Davidson, G. (1966). Planetary Space Sci. 14, 651. Davidson, G., and O'Neil, R. (1965). Proc. Intern. Conf. Phys. of Electron. At. Collisions, 4th, Quebec, 1965. Degges, T., Dalgarno, A., and Williams, D. A. (1967). Proc. Phys. SOC.(London). 92, 291. Dickinson, A. S. (1968). Proc. Phys. SOC.(London) In press. Donahue, T. M. (1965). Planetary Space Sci. 13, 871. Donahue, T. M. (1966). Planetary Space Sci. 14, 33; J . Geophys. Res. 71, 2237. Donahue, T. M., and Fastie, W. G. (1964). Space Res. 4, 304. Donahue, T. M., and Meier, R. R. (1967). J. Geophys. Res. 72,2803. Eather, R. H. (1966). J. Geophys. Res. 71,4133, 5027. Eather, R. H. (1967). J. Geophys. Res. 72, 1481. Engelhardt, A. G., Phelps, A. V., and Risk, C. G. (1964). Phys. Rev. 135, A1566. Evans, J. V. (1965). J. Geophys. Res. 70, 1175. Evans, J. V. (1967). Planetary Space Sci. 15, 1387. Farley, D. T. (1966). Ann. Geophys. 22,448. Fastie, W. G., and Crosswhite, H. M. (1964). Planetary Space Sci. 2, 1021. Fastie, W. G., Crosswhite, H. M., and Heath, D. P. (1964). J. Geophys. Res. 69, 4129. Fehsenfeld, F. C., Schmeltekopf, A. L., Schiff, H. I., and Ferguson, E. E. (1967). Planetary Space Sci. 15, 373. Ferguson, E. E. (1967). Rev. Geophys. 5, 305. Ferguson, E. E., and Schluter, H. (1962). Planetary Space Sci. 9, 701. Ferguson, E. E., Fehsenfeld, F. C., and Schmeltekopf, A. L. (1965). Planetary Space Sci. 13, 925. Fite, W. L., and Brackman, R. T. (1959). Phys. Rev. 112, 815. Fite, W. L., Rutherford, J. A., Snow W. R., and Van Lint, 5. A. (1963). Discussions Faraday SOC.33, 518. Fournier, J. P., and Nagy, A. F. (1965). J. Atmospheric Sci. 22, 732. Fox, R. E., and Hickam, W. M. (1954). J. Chem. Phys. 22,2059. Garrett, W. R., and Jackson, H. J. (1967). Phys. Rev. 153, 28. Geiger, J., and Stickel, W. (1965). J . Chem. Phys. 43, 4535. Geissler, J. E., and Bowhill, S. A. (1965). J. Atmospheric Terrest. Phys. 27, 122, 457. Geltman, S., and Takayanagi, K. (1966). Phys. Rev. 143, 25. Gerjuoy, E., and Stein, S. (1955). Phys. Rev. 97, 1671; 98, 1848. Gleeson, L. J., and Axford, W. I. (1967). Planetary Space Sci. 15, 123. Goldan, P. D., Schmeltekopf, A. L., Fehsenfeld, F. C., Schiff, H. I., and Ferguson, E. E. (1966). J. Chem. Phys. 44,4095. Green, A. E. S., and Barth, C. A. (1965). J. Geophys. Res. 70, 1083. Green, A. E. S., and Barth, C. A. (1967). J. Geophys. Res. 72, 3975. Grun, A. E. (1957). Z. Naturforsch. 12A, 89. Gryzinski, M. (1959). Phys. Rev. 115, 374. Haas, R. (1957). 2. Physik. 148, 177. Hake, R. D., and Phelps, A. V. (1967). Phys. Rev. 158, 70. Hanson, W. B. (1963). Space Res. 3, 282. Hanson, W. B., and Johnson, F. S. (1961). Mem. SOC.Roy. Sci. Liege 3, 390. Hayakawa, S., and Nishimura, H. (1964). J. Geomag. Geoelect. 16, 72. Henry, R. J. W. (1967). Phys. Rev. 162, 56. Hermann, 0. (1936). Ann. Physik. 25, 166. Hirao, K. (1966). J. Geomag. Geoelect. 18, 333. Hoffman, J. H. (1967). J. Geophys. Res. 72, 1883.
408
A . Dalgarno
Holland, R. (1967). Private communication. Hunten, D. M., and McElroy, M. B. (1966). Rev. Geophys. 4, 303. Hunten, D. M., and McElroy, M. B. (1968). J . Geophys. Res. In press. Hunten, D. M., and Wallace, L. (1967). J. Geophys. Res. 72, 69. Istornin, V. G. (1963). Space Res. 3,209. Istomin, V. G. (1966). Ann. Geophys. 22, 255. Jobe, J. D., Sharpton, F. A., and St. John, R. M. (1967). J. Opt. SOC.Am. 57, 106. Kaplan, S. A., and Kurt, V. G. (1965). Kosmich. Issled Akad. Nauk SSSR 3, 256. Kaplan, S. A., Katyushina, V. V., and Kurt, V. G. (1965). Space Res. 5, 595. Katyushina, V. V. (1965). Kosmich. Issed. Akad. Nauk SSSR 3, 248. Kishko, S. M., and Kuchinka, M. Yu. (1959). Opt. Spectr. 6, 378. Knudsen, W. C., and Sharp, G . W. (1965). J. Geophys. Res. 70, 143. Knudsen, W. C., and Sharp, G. W. (1967). J. Geophys. Res. 72, 1061. Larnpe, F. W. (1957). J. Am. Chem. SOC.79,6129. Langstroth, G. 0. (1934). Proc. Roy. SOC.A146, 16
