Theory of Electron-Atom Collisions. Part 1: Potential Scattering

Transcription

1 Theory of Electron-Atom Collisions Part 1: Potential Scattering

2 PHYSICS OF ATOMS AND MOLECULES Series Editors P. G. Burke, The Queen's University of Belfast, Northern Ireland H. KJeinpoppen, Atomic Physics Laboratory. University of Stirling. Scotland Editorial Advisory Board R. B. Bernstein (New York. U.S.A.) J. C. Cohen-Tannoudji (Paris. France) R. W. Crompton (Canberra. Australia) Y. N. Demkov (St. Petersburg. Russia) C. J. Joachain (Brussels. Belgium) W. E. Lamb, Jr. (Tucson. U.S.A.) P.-O. Liiwdin (Gainesville. U.S.A.) H. O. Lutz (Bielefeld. Germany) M. C. Standage (Brisbane. Australia) K. Takayanagi (Tokyo. Japan) Recent volumes in this series: ATOMIC PHOTOEFFECT M. Ya. Amusia ATOMIC SPECTRA AND COLLISIONS IN EXTERNAL FIELDS Edited by K. T. Taylor, M. H. Nayfeh, and C. W. Clark ATOMS AND LIGHT: INTERACTIONS John N. Dodd COHERENCE IN ATOMIC COLLISION PHYSICS Edited by H. J. Beyer, K. Blum, and R. Hippler ELECTRON COLLISIONS WITH MOLECULES, CLUSTERS, AND SURFACES Edited by H. Ehrhardt and L. A. Morgan ELECTRON-MOLECULE SCATTERING AND PHOTOIONIZATION Edited by P. G. Burke and J. B. West THE HANLE EFFECT AND LEVEL-CROSSING SPECTROSCOPY Edited by Giovanni MOTUZzi and Franco Strumia INTRODUCTION TO THE THEORY OF LASER-ATOM INTERACTIONS, Second Edition Marvin H. Mittleman INTRODUCTION TO THE THEORY OF X-RAY AND ELECTRONIC SPECTRA OF FREE ATOMS Romas Karazija MOLECULAR PROCESSES IN SPACE Edited by Tsutomu Watanabe, Isao Shimamura, Mikio Shimizu, and Yukikazu Itikawa POLARIZATION BREMSSTRAHLUNG Edited by V. N. Tsytovich and I. M. Ojringel THEORY OF ELECTRON-ATOM COLLISIONS, Part 1: Potential Scattering Philip G. Burke and Charles J. Joachain ZERO-RANGE POTENTIALS AND THEIR APPLICATIONS IN ATOMIC PHYSICS Yu. N. Demkov and V. N. Ostrovskii A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher.

3 Theory of Electron-Atom Collisions Part 1: Potential Scattering Philip G. Burke The Queen's University of Belfast Belfast, Northern Ireland Charles J. Joachain Universite Libre de Bruxelles Brussels, Belgium Springer Science+Business Media, LLC

4 Library of Congress Cataloging in Publication Data Burke, P. G. Theory of electron-atom collisions / Philip G. Burke, Charles J. Joachain. p. cm. (Physics of atoms and molecules) Includes bibliographical references and index. Contents: Pt, 1. Potential scattering. 1. Electron-atom collisions. I. Joachain, C. J. (Charles Jean). II. Title. QC793.5.E628B dc20 CIP ISBN DOI / ISBN (ebook) Springer Science+Business Media New York 1995 Originally published by Plenum Press, New York in 1995 Softcover reprint of the hardcover 1 st edition All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher

5 PREFACE Electron collisions with atoms and atomic ions have attracted considerable attention since the earliest days of this century. This is partly because these processes provide a means of studying the dynamics of many-particle quantum systems with known interactions at a fundamental level and partly because a detailed understanding of these processes is required in many other fields such as astrophysics, atmospheric physics, plasma physics, laser physics and electrical discharges in gases. In particular, the modelling of non-equilibrium plasmas requires a detailed knowledge of cross sections for elastic scattering, excitation, ionization and recombination for neutral and for ionized atomic species. In recent years a number of important advances have been made both in experiment and in theory. On the experimental side these advances include absolute measurements of cross sections, experiments using coincidence techniques, polarized beams and targets, incident positrons and the use of synchrotron radiation sources and lasers. Many of these measurements provide stringent tests of the theory by enabling the magnitude of individual scattering amplitudes and their relative phases to be determined. They have stimulated the development of new theoretical methods as well as sophisticated computer programs to solve the resultant equations. In this monograph we develop the theory of electron--atom and electron-ion collisions and we illustrate the various theoretical approaches by comparing their results with experimental data. We begin in Part I, which is the subject of the present volume, by treating potential scattering. This serves as an introduction to many of the basic concepts which will be required in Parts II and III of the monograph where we discuss the general theory of electron collisions with atoms and ions at low and at high energies, respectively. In Chapter 1 we present a general introduction to the non-relativistic scattering of a spinless particle by a potential. In this way we introduce the partial wave expansion and the scattering phase shifts which are central to the theory of low energy collisions, and the Lippmann--Schwinger equation which provides a basis for the description of high energy collisions. In Chapter 2 we tum to approximation methods. This enables us to introduce the Born series as well as semi-classical, variational and R-matrix methods. In Chapter 3 we review the analytic properties satisfied by the scattering amplitudes. We introduce the concepts of poles, zeros and branch cuts of the S -matrix as well as effective range expansions which play an v

6 vi PREFACE important role in our understanding of electron-atom and electron-ion collisions. We also discuss in this Chapter dispersion relations. Finally, in Chapter 4 we conclude our discussion of potential scattering by extending the theory to take into account spin and relativistic effects. After describing some basic results concerning electron spin we present a review of the Dirac equation and its non-relativistic limit. We conclude this Chapter by presenting an analysis of polarization phenomena using the density matrix formalism. This volume also contains appendices giving a number of basic mathematical results needed in electron-atom collision theory. In conclusion, we wish to take this opportunity to thank our colleagues at Belfast and at Brussels for innumerable discussions on the subject matter of this monograph. We are also deeply grateful to our wives Val Burke and Halina Joachain for their continued and enthusiastic support during the lengthy period when this monograph was conceived and written. We also wish to thank Miss C. Vidal and Mrs L. Monaghan for their patient and careful typing of the manuscript. P. G. Burke, Belfast C. J. Joachain, Brussels December 1994

7 UNITS Atomic units (a. u.) will be used throughout this monograph. They are such that h = m = e = 1, where h is Planck's constant divided by 211", m is the mass of the electron and (-e) is its charge. Thus the atomic unit of length is ao = h 2 /me 2 ~ x 10-9 cm, which is the radius of the frrst Bohr orbit of the hydrogen atom with infinite nuclear mass. Using this unit of length, scattering cross sections, which have the dimensions of an area, are then expressed in units of a ~ ~ x cm2. Total cross sections are also often expressed in units o f 1 l " ~ a ~ x cm 2. The atomic unit of time is given by h 3 /me 4 = x s, while the unit of velocity is e 2 /h = x 108 cm S-l. The atomic unit of energy is e2 / ao ~ ev, which is twice the ionization energy of the hydrogen atom in its ground state (i.e. twice the Rydberg unit of energy). The fine structure constant a = e 2 /hc ~ 1/137 is of course dimensionless. vii

8 CONTENTS CHAPTER 1. GENERAL THEORy The Wave Equation Cross Sections Partial Wave Analysis Scattering by a Coulomb PotentiaL The Lippmann-Schwinger Equation Scattering by a Complex PotentiaL CHAPTER 2. APPROXIMATION METHODS The Born Series Defmitions and General Properties The First Born Approximation Higher Order Tenns of the Born Series Semi-classical Approximations The Eikonal Approximation Comparison Between the Born and Eikonal Series. The Eikonal-Bom Series Approximation Improved Eikonal Approximations The Eikonal Approximation in the Strong Coupling Case The JWKB Approximation Variational Methods Hulthen-Kohn Variational Method Anomalous Singularities in the Hulthen-Kohn Variational Method Hulthen-Kohn Variational Method for the Full Scattering Amplitude Schwinger Variational Method Extremum Principles The R-Matrix Method Variational Methods for the R-Matrix Homogeneous Boundary Condition Method ix

9 x CONTENTS Arbitrary Boundary Condition Method R-Matrix Propagator Methods Eigenchannel Method Other Generalizations Extremum Principles CHAPTER 3. ANALYTIC PROPERTIES OF THE SCATTERING AMPLITUDE The Jost Functions and S-Matrix Analytic Properties of the Jost Functions and S-Matrix Bound States and Resonances Time Delay Levinson's Theorem Effective Range Theory Finite Range Potentials Long-Range Potentials Behaving as r- S with s ;::: The Coulomb PotentiaL Dispersion Relations Mathematical Background Application to Non-relativistic Potential Scattering CHAPTER 4. SPIN AND RELATIVISTIC EFFECTS The Spin of the Electron Electron Spin Polarization and the Density Matrix Relativistic Scattering of Electrons by a Potential The Dirac Equation Free Particle Solutions Integral Equation Formalism The First Born Approximation Description of the Scattering in Terms of Two-Component Wave Functions. The M-Matrix The Dirac Equation for an Electron in a Central Potential. Separation in Spherical Polar Coordinates The Non-relativistic Limit of the Dirac Equation. Relativistic Corrections Through Order v 2 jc Partial Wave Analysis

10 CONTENTS xi 4.6. Polarization Phenomena Polarization Dependence of the Differential Cross Section Polarization of the Scattered Electrons Double Scattering Experiments Triple Scattering Experiments APPENDIX A. LEGENDRE POLYNOMIALS, ASSOCIATED LEGENDRE FUNCTIONS AND SPHERICAL HARMONICS A.1. Legendre Polynomials A.2. Associated Legendre Functions A.3. Orbital Angular Momentum and Spherical Harmonics A.4. Useftll Formulae APPENDIX B. BESSEL FUNCTIONS, MODIFIED BESSEL FUNCTIONS, SPHERICAL BESSEL FUNCTIONS AND RELATED FUNCTIONS B.I. Bessel Functions B.2. Modified Bessel Functions B.3. Spherical Bessel Functions APPENDIX C. DALITZ INTEGRALS APPENDIX D. THE DENSITY MATRIX APPENDIX E. CLEBSCH-GORDAN AND RACAH COEFFICIENTS E.l. Clebsch-Gordan Coefficients... '" E.2. Racah Coefficients... '" E j Sytnbols E.4. Higher Order 3n - j Sytnbols REFERENCES

11 Theory of Electron-Atom Collisions Part 1: Potential Scattering