Introduction to the Theory of Atomic Spectra

607 Pages·1972·10.038 MB·English

by  I.I. Sobel'man, G.K. Woodgate and D. ter Haar (Auth.)

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INTRODUCTION
TO THE THEORY
OF ATOMIC SPECTRA
I.I.SOBEL'MAN
PERGAMON PRESS
Oxford • New York
Toronto • Sydney • Braunschweig
Pergamon Press Ltd., Headington Hill Hall, Oxford
Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford,
New York 10523
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Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street,
Rushcutters Bay, N.S.W. 2011, Australia
Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig
Copyright © 1972 Pergamon Press Ltd.
All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photocopying,
recording or otherwise, without the prior permission of
Pergamon Press Ltd.
First edition 1972
Library of Congress Catalog Card No. 79-148054
This book is a translation of BBE/liEHHE B TEOPHK) ATOMHblX CnEKTPOB
by I.I. SobeFman, published by Fizmatgiz, Moscow, 1963,
and incorporates revisions supplied by the author during translation towards the end of 1969
Printed in Germany
08 016166 9
Dedicated to the memory of
Grigorii Samuilovich
LANDSBERG
Preface
MORE than 25 years have passed since publication of the widely known monograph
Theory of Atomic Spectra by E. Condon and G. Shortley. Naturally, many sections of
the book have, to a considerable extent, become out of date during this time. This also
particularly applies to those chapters where the fundamentals of the theory of angular
momentum and the methods of construction of antisymmetrized wave functions are stated.
The series of papers by Racah on the theory of complex spectra were published in
the period 1942-1949. Thanks to these papers, the theory of angular momentum was
enriched by new effective calculating methods. The method of fractional-parentage
coefficients, which proved to be very fruitful when considering electronic configurations
containing equivalent electrons, was developed in these papers.
It is difficult to overestimate the significance of Racah's work for the theory of atomic
spectra. Many calculations, which previously required lengthy and laborious calculations,
are carried out almost instantaneously by means of the Racah "technique", the results
being expressed in terms of tabulated coefficients—^-coefficients and fractional parentage
coefficients.
Racah's methods, which were further developed in the works of a large number of
other authors, are now widely used in many fields of theoretical physics, especially in
nuclear theory. At the same time there are no monographs or textbooks, at present, which
contain a systematic presentation of the theory of atomic spectra on the basis of these
new methods. One of the objects of the present book is to fill this gap to some extent.
Besides the traditional range of questions usually included in a handbook on atomic
spectroscopy and associated with the systematics of spectra, this book also examines
many questions which are of interest from the point of view of using spectroscopic methods
for investigating various physical phenomena. These include, for example, continuous
spectrum radiation, excitation of atoms amd spectral line broadening.
For the convenience of the reader, the main body of the book is prefaced by a summary
of elementary information on atomic spectra (Chapters I—III). In the other chapters of the
book, experimental data are discussed only for the purpose of illustrating theoretical con-
clusions or for justifying approximations being used. References to experimental work
are therefore of a selective character. The bibliography of theoretical works also does not
claim to be complete. As a rule, references are made only to monographs, reviews and
papers, whose results are used directly in the text.
Abbreviations are used for a number of papers and monographs which are particularly
frequently quoted; these are given on p. 603.
For reading the book, knowledge to the extent of the ordinary university course of
quantum mechanics is necessary (this does not apply to the first three chapters, for the
xiii
XIV PREFACE
reading of which very elementary knowledge on the quantum theory of the atom is ad-
equate). Knowledge of group theory is not required. Because of this limitation, caused by
the endeavour to make the book intelligible to a wider circle of readers, a number of
difficulties arose in presenting some sections of the second part of the book. For example,
it was very complicated to explain the physical meaning of the quantum number v (seniority
number), introduced by Racah. When using group theory, it is trivially simple to solve
this question. This limitation caused one to give up any detailed consideration of the
classification of levels of atoms with unfilled/-shells.
The course of lectures on atomic spectroscopy and the facultative course of lectures
on the theory of atomic spectra which the author gave in 1956-60 at the Moscow Physics
and Technology Institute were taken as the basis of this book. Notes of lectures on atomic
spectroscopy given by Prof. S. L. Mandel'shtam at the Moscow Physics and Technology
Institute were used in writing Chapters 1, 2 and 3. Section 33 and Chapter 11 were written
jointly with L. A. Vainshtein, and § 46 jointly with L. A. Vainshtein and L. P. Presnyakov.
In conclusion, I wish to express sincere thanks to Prof. S. L. Mandel'shtam, on whose
initiative this book was written, to Prof. M. G. Veselov, who read through the manuscript,
and also to L. A. Vainshtein, Yu. P. Dontsov, N. N. Sobolev and V. I. Kogan, who checked
individual chapters of the manuscript, for many valuable comments. I also thank T. I.
Sokolova for her help in the layout of the manuscript.
I. SOBEL'MAN
Preface to the English edition
The Russian edition of this book was prepared for publication in 1962, i.e. nearly
10 years ago. In a number of new fields in physics such a period would make a complete
rewriting of the book necessary. Atomic spectroscopy, however, is a discipline which was
built up and consolidated relatively long ago. In preparing the English edition it has there-
fore proved possible everywhere, except in Chapter 11, to make no more than relatively in-
significant corrections and changes, and also to add a number of references to new work.
Chapter 11, which is devoted to the problem of atomic excitation by collisions, occupies
a special place in the book. Questions of this type usually refer to the theory of atomic
collisions and are not included in a book on atomic spectroscopy. In the present author's
view this is not quite correct, since the interests and requirements of spectroscopists, and
of those working in many other branches of physics, are quite different from the interests
of specialists in the field of atomic collisions. In the theory of atomic collisions most atten-
tion at the present time is being paid to the development of special methods of approxi-
mation, and also to the study of a number of subtle effects such as, for example, the resonance
peaks in scattering cross-sections. At the same time, numerical calculations using the
simplest formulae of the Born approximation are often inconvenient in other branches of
physics, so that one is confined to estimates made with the help of quite crude semi-empirical
formulae.
In making the considerable revision of Chapter 11 compared with that of the Russian
edition, the aim has been to make its contents coincide as closely as possible with the inter-
ests of spectroscopists. In particular, tables of parameters have been added, so that calcula-
tions of Born excitation cross-sections can be made for a broad class of atomic transitions,
without resorting to numerical calculations.
Of the other chapters in the book the greatest difficulty in revision lay in determining
the nature of the necessary changes in Chapter 10, since a number of problems in the theory
of spectral line broadening are now being worked on intensively. This applies especially to
hydrogen-like spectra. Unfortunately, one cannot consider that a revision of the theory of
broadening in hydrogen-like spectra has yet been accomplished. It was therefore considered
advisable to retain the contents of Chapter 10 in the main without appreciable changes.
As in the Russian edition, the bibliography does not pretend to be complete. In general,
references are only given to work used directly in the text.
Dr Woodgate put in a lot of work in the preparation of the English edition. Besides
pointing out many shortcomings and misprints, he took upon himself the taks of replacing
the Racah W-coefficients by the now more commonly used 6/-symbols.
I am glad of this opportunity of expressing my sincere gratitude to Dr. Woodgate.
I. SOBEL'MAN
June 22, 1971
XV
Note
THIS book is devoted to systematically presenting the physical principles and theory of
atomic spectroscopy. The presentation is based on the modern system of the theory of
angular momentum. Questions of atomic excitation and radiation are also examined
systematically in the book. These questions are interesting from the point of view of using
spectroscopic methods for investigating different physical phenomena.
The book is intended for advanced-course university students, postgraduate students
and scientists working on spectroscopy and spectral analysis, and also in the field of
theoretical physics.
XVI
1.
The Hydrogen Spectrum
§ 1. Schrddinger's equation for the hydrogen atom
1. Energy levels
The problem of the relative motion of an electron (mass m, charge — e) and a nucleus
(mass M, charge Ze) reduces, as is well known, to the problem of the motion of a particle
with an effective mass ft = « m in a Coulomb field .
m + M 2 r
Ze
The Schrodinger equation for a particle in the field — has the form
fi2 Ze21
IL.V2 + £ + _U = 0. (1.1)
2fx r J
The wave function y) which is the solution of this equation, describes a stationary state
9
with a definite value of the energy E. With motion in a centrally symmetric field the angular
momentum of a particle is conserved; therefore among the stationary states there are
those which are also characterized by a definite value of the square of the angular momen-
tum and by the value of one of the components of the angular momentum. We shall select
the z-component as this component, i.e. we shall consider stationary states which are
characterized by definite values of the quantities E, the square of the angular momentum
and the z-component of the angular momentum. The wave functions ip of these stationary
states are eigenfunctions of the operators I2 and l and must therefore also satisfy the
z
equations
l2W = l{l+ 1)V, 0.2)
lip = mxp, (1.3)
z
where /(/ + 1) and m are eigenvalues of the operators I2 and / . We recall that, in quantum
z
mechanics, the square of the angular momentum can only take a discrete series of values
fi2l(l + 1), where fi = ; h is Planck's constant, and also / = 0, 1, 2, .... In exactly the
2n
same way, the momentum z-component can have the values fun, m = 0, +1, ±2, ... with
the additional condition \m\ ^ /.
In future, for brevity, we shall simply talk of the angular momentum / and the z-com-
ponent of the angular momentum m, meaning the angular momentum whose square is
equal to h2l{l + 1) and whose z-component equals fim.
3
4 INTRODUCTION TO THE THEORY OF THE ATOMIC SPECTRA
The components of angular momentum /are connected with the components of momen-
tum p by the relation
W* = yPz - Wy, My = ~XPz + ZPx, ^z = *Py ~ Wx- (L4)
rs
Replacing in these expressions p , p and p by the quantum mechanical operators — ifi —,
x y z
r d A -r d ^
—//z— and — in —, and introducing the spherical coordinates r, 6,cp, we obtain the follow-
ing equations instead of eqns. (1.2) and (1.3)
-L- ±(sin8^) -L-^X + 1(1 + 1) = 0, (1.5)
+ V
sine 36 \ 86 J sin2 6 dy2)
i^L + = 0. (1.6)
my)
dtp
We also write down eqn. (1.1) in spherical coordinates
dr J r2 [sin 6 36 \ d6 J sin2 6 3cp2) fx2
(1.7)
Comparing eqns. (1.5) and (1.7), we see that the angular part of the Laplace operator V2,
apart from the factor r~2, is the operator of the square of the angular momentum, and
therefore we obtain instead of eqn. (1.7)
1 d ( dyj\ 1(1 + 1) 2M [^ Ze2l
2
We shall seek the solution of this equation in the form
Y = R(r)YJp,<P)> (1-9)
l
where the angular part of the wave function, Y (d,(p), satisfies eqns. (1.5) and (1.6). Sub-
lm
stituting eqn. (1.9) in (1.8), we obtain the equation for the radial part of the wave function
±A( ^)J(L±Jl ^\ in o. (i.io)
r R + E + R =
r2 dr\ dr) r2 h2 \_ r \
The asymptotic behaviour of the radial function for r -> oo is defined by the equation
m ±?.E-R=0. (1.11)
+
dr2 h2
Thus, for r -» oo, we have
R^ e^r C e-^Er. (1.12)
Cl + 2
The constants C, C can be found from the condition for matching (1.12) with an accurate
t 2
solution of eqn. (1.10) and from the normalization condition. These constants are func-
tions of the energy E and angular momentum /. If E > 0, then V — 2juE = iy/2/u \E\ and
i ,
— y/-2ftEr
the function (1.12) is bounded. But if E > 0, the term eh increases without limit