Polarization and Correlation Phenomena in Atomic Collisions: A Practical Theory Course

250 Pages·2000·6.01 MB·English

by  Vsevolod V. Balashov

Polarization and Correlation
Phenomena in Atomic
Collisions
A Practical Theory Course
PHYSICS OF ATOMS AND MOLECULES
Series Editors
P. G. Burke, The Queen's University of Belfast, Northern Ireland
H. Kleinpoppen, Atomic Physics Laboratory, University of Stirling, Scotland
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o.
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Polarization and Correlation
Phenomena in Atomic
Collisions
A Practical Theory Course
V sevolod V. Balashov
Alexei N. Grum-Grzhimailo
and
Nikolai M. Kabachnik
Moscow State University
Moscow, Russia
Springer Science+Business Media, LLC
Library of Congress Cataloging-in-Publication Data
Balashov, V. V. (Vsevolod Viacheslavovich)
Polarization and correlation phenomena in atomic collisions : a practical theory course I
Vsevolod V. Balashov, Alexei N. Grum-Grzhimailo, Nikolai M. Kabachnik.
p. cm. -- (physics of atoms and molecules)
Includes bibliographical references and index.
ISBN 978-1-4419-3328-7 ISBN 978-1-4757-3228-3 (eBook)
DOI 10.1007/978-1-4757-3228-3
I. Collisions (Nuclear physics) 2. Pairing correlations (Nuclear physics) 3. Polarization
(Nuclear physics) I. Grum-Grzhimailo, Alexei N. II. Kabachnik, N. M. III. Title. IV.
Series.
QC794.6.C6 B35 2000
539.7'57--dc21
99-049492
© 2000 Springer Science+Business Media New York
Originally published by Kluwer Academic / Plenum Publishers, New York in 2000.
Softcover reprint of the hardcover 1st edition 2000
http://www.wkap.nl
10987654321
A C.I.P. record for this book is available from the Library of Congress.
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
Preface
This book is based on our training course with the general title "Theoretical
Practicum in Atomic and Nuclear Physics," which is used in Moscow State Uni
versity (MSU) as well as in other universities of our country as one of the com
ponents in the education program for students specializing in theoretical atomic,
nuclear, and particle physics. The concept of the theoretical practicum and the
main aspects of its content are also well known in a number of universities in other
countries as a result of special lessons given there either by ourselves or by those
who had learned the theoretical practicum during their studies at MSU. Among
them are the universities of Rome and Athens, Freiburg and Bielefeld (Germany),
Adelaide (Australia), Missouri-Rolla (United States), Cairo, and Ulan Bator.
The course was organized in the academic year 1962-63 at the Faculty of
Physics of the Moscow State University [1]. Since that time its content has been
regularly modernized taking into account current progress in research in this area
[2-4]. The principal goal of the course is to serve as a guide for students who are
being trained to perform theoretical calculations, including those which need a
long time of intensive individual work, based on the most important methods used
in practice. On the other hand, we know that the books in this series are widely
used by researchers, theoreticians and experimentalists, in their everyday work as
a collection of useful formulas and tables.
The complete program of the theoretical practicum contains such subjects
as modern methods in quantum theory of collisions, group theory, solid-state
physics, atomic spectroscopy, physics of nuclear reactions, computer simulation
of particle propagation through matter, and others. The concepts of density matrix
and statistical tensors as well as their application in polarization and correlation
studies has always been central to the program. To this end, we started from the
famous review by Devons and Goldfarb [5], accepted its style of presentation of
the subject, and intensified its pedagogical aspect.
In contrast to all our earlier publications, this book is devoted exclusively to
atomic physics. It is known that polarization and correlation studies playa grow-
v
vi Preface
ing role in this branch of modem physics. The main tendencies in atomic physics
along this line are the subject of excellent review papers and monographs such as
those by Fano and Macek [6], Kessler [7], Blum [8], Andersen et al. [9-12], Danos
and Fano [13], and many others. Several recent conferences and workshops have
been devoted to polarization and correlation phenomena in atomic collisions, and
many particular contributions are briefly described, for example, in Refs. 14-19.
These studies show that the density matrix and tensor operator method is
one of the most important and effective instruments in theoretical investigations
in the field. This method has been widely used and partly developed in our own
work on excitation of atomic autoionizing states in electron-atom collisions, the
coincidence (e, 2e) method in autoionization studies, polarization and angular
anisotropy of Auger and photoelectrons, inelastic and superelastic scattering in
electron-atom collisions, the coincidence (e, e' r) method, photon- and electron
impact excitation of atomic autoionizing and Auger states from laser-excited tar
gets, the theory of cascade processes, polarization effects in antiproton-atom in
teractions, and other studies. The reader will notice the obvious influence of our
research work on the general approach to the problem as well as on the examples
chosen to illustrate the basic theoretical methods.
We address this book to a reader with a background in elementary quantum
mechanics and general atomic physics at the undergraduate level. All necessary
information about the physical meaning of special polarization and correlation
phenomena considered in the book is given in the text. It is not our aim to present
a wide variety of approaches to describing the dynamics of the processes con
sidered. We see our duty as demonstrating the universality of the density matrix
and tensor operator methods. For this reason, throughout the book, we do not
link any of them to any special choice among the models and approximations of
the dynamic aspects of the processes under consideration as well as the choice of
the atomic wave functions. To make the book a real practicum, we supplied it
with a set of tables that summarize the most important information on the alge
bra of angular momentum. They overlap considerably with those collected in the
well-known book by Varshalovich, Moskalev, and Khersonskii [20].
This book is the first in the theoretical practicum series which is published
in English. We are very glad to meet our new readers. Let this book be our
small contribution to the international exchange of ideas and experience in the
education field.
ACKNOWLEDGMENTS
There is a long list of people whose kind advice and active support have
been of great importance for us on the long path of the theoretical practicum to
this special edition. First of all, we are grateful to our colleagues at Moscow
State University and the Joint Institute of Nuclear Research, Dubna: Grigory Ko-
Preface vii
renman, Yuri Smirnov, Nikolai Yudin, and Vladimir Belyaev, who were the first
authors of the series. We thank V. Korotkikh, V. Mileev, V. Popov, V. Senashenko,
S. Strakhova, V. Dolinov, and Yu. Krementsova, who continued their work.
We are also indebted to those who are not among the authors of the The
oretical Practicum, but with whom we conducted our research in atomic physics
and had numerous discussions on the problems considered in this book. They
include S. Baier, E. Berezhko, I. Bodrenko, V. Bubelev, S. Burkov, A. Dorn,
I. Golokhov, M. Gorelenkova, U. Hergenhahn, O. Klochkova, V. Kondratiev,
I. Kozhevnikov, O. Lee, O. Lhagwa, S. Lipovetski, B. Lohmann, A. Magunov,
S. Martin, I. Sazhina, V. Shakirov, V. Sizov, M. Wedowski, K. Ueda, and
B. Zhadamba.
We always met a deep interest in our theoretical practicum and in our re
search work on polarization and correlations in many leading institutes in the
physics of electronic and atomic collisions and atomic spectroscopy. We thank
Profs. V. Afrosimov, K. Bartschat, K. Becker, U. Becker, D. Berenyi, K. Blum,
A. Crowe, S. Datz, H. Ehrhardt, U. Heinzmann, H. Klar, V. Lengyel, H. Lutz,
J. Macek, N. Martin, I. McCarthy, W. Mehlhorn, V. Schmidt, B. Sonntag,
E. Weigold, and P. Zimmermann for very interesting discussions, their support
for our visits to their institutes, and their fruitful collaboration.
Our special thanks go to Prof. Hans Kleinpoppen for his very important
support of this edition.
V. Balashov
A. Grum-Grzhimailo
N. Kabachnik
Contents
1. Density Matrix and Statistical Tensors ...................... 1
1.1. Description of Mixed States by Density Matrix ............. 1
1.2. Spin Density Matrix and Statistical Tensors ............... 7
1.3. Spin Density Matrix and Statistical Tensors for Simple
Systems ......................................... 15
1.4. Symmetry Restrictions on the Density Matrix and Statistical
Tensors ......................................... 27
1.5. Efficiency Matrix and Efficiency Tensors ................. 31
2. Production of Polarized States ............................ 45
2.1. Polarization of Compound Systems ..................... 45
2.2. Polarization and Angular Distribution of Scattering Products ... 55
2.3. Direct Atomic Photoeffect ............................ 76
2.4. Direct Ionization by Particle Impact ..................... 97
3. Decay of Polarized States ................................ 107
3.1. Nonradiative Decay ................................ 107
3.2. Radiative Decay ................................... 115
3.3. Polarization State of the Residual Atomic System after
Decay .......................................... 121
3.4. Polarization in Cascade Decay ......................... 126
4. Resonant and Two-Step Processes ......................... 133
4.1. Two-Step Reactions of Excitation and Radiation Decay of
Discrete Atomic Levels .............................. 133
4.2. Two-Step Reactions of Ionization and Decay .............. 155
4.3. Polarization and Correlations in Autoionization Processes 170
ix
x Contents
5. Complements ......................................... 187
5.1. Polarization of Atoms by Laser Optical Pumping ........... 187
5.2. Time Evolution of Statistical Tensors and Depolarization of
Atomic Angular Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . .. 190
5.3. Influence of Time Evolution of Statistical Tensors on Angular
Distribution and Polarization of Decay Products ............ 193
Appendix ............................................... 197
A.l. Pauli Matrices .................................... 197
A.2. Legendre Polynomials and Associated Legendre Polynomials .. 198
A.3. Spherical Harmonics ................................ 200
A.4. Bipolar spherical harmonics .......................... 203
A.5. Solid Spherical Harmonics ........................... 205
A.6. Rotations and Wigner D-Functions . . . . . . . . . . . . . . . . . . . . .. 206
A. 7. Irreducible Tensor Operators .......................... 209
A.8. Clebsch-Gordan Coefficients .......................... 215
A.9. 6j-Symbo1s ...................................... 221
A. 10. 9 j-Symbols ...................................... 224
A.11. Sums of Products of the 3n j-Symbols ................... 229
Bibliography ............................................ 237
Index .................................................. 239
1
Density Matrix and Statistical
Tensors
1.1. Description of Mixed States by Density Matrix
1.1.1. Pure and Mixed States
The states of physical systems in quantum mechanics can be classified in
two categories: pure states and mixed states. A pure state is characterized by a
certain wave function or a state vector 10/) in an abstract Hilbert space. No vector
10/) can be related to the mixed state. The necessity of introducing mixed states
arises for quantum systems that are not closed. For example, a subsystem that is
part of a larger closed system, cannot be characterized by its own wave function,
which depends on coordinates of the subsystem, and therefore it is in a mixed
state. The system cannot be considered closed even when it is isolated at present
if it was not isolated in the past due to interaction with another system. This is
a typical situation in the analysis of different characteristics of reaction products,
when only part of the products (or part of the characteristics of the product) are
observed. Mixed states are described by a statistical or density operator p, which
operates in the Hilbert space and in general has the form:
L
p = Wn Io/n) ( o/n 1 (Ll)
n
where 1%) is a complete set of state vectors and the weight coefficients, Wn, are
real positive numbers which, as we will see later, characterize the probability of
finding the system in a particular pure state. The density operator is a general
ization of the concept of the state vector. Every mixed state is characterized by
its density operator.
1
V. V. Balashov et al., Polarization and Correlation Phenomena in Atomic Collisions
© Springer Science+Business Media New York 2000