Table Of ContentIMPURITY SPECTRA
OF SOLIDS
Elementary Theory of Vibrational Structure
IMPURITY SPECTRA
OF SOLIDS
Elementary Theory of Vibrational Structure
KARL K. REBANE
Institute of Physics and Astronomy
Academy of Sciences of the Estonian SSR
Tartu, USSR
Translated from Russian by
JOHN S. SHIER
NASA Electronics Research Center
Cambridge, Massachusetts
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PLENUM PRESS -NEW YORK-LONDON -1970
Karl Karlovich Rebane was born in 1926 in Piirnu in the Estonian SSR. He was
graduated from Leningrad State University in 1952, and attained the degree
of Doctor of Physico mathematical Science in 1964. His scientific work has been
largely devoted to the theory of absorption, luminescence, and scattering of
light in activated crystals. His dissertation was devoted to the theory of vibrational
structure in impurity spectra of crystals and the quantum-mechanical method
of moments in spectrum problems. He has been a Corresponding Member of the
Academy of Sciences of the Estonian SSR since 1961, and an Academician
since 1967.
Since 1968 Rebane has been vice president of the Academy of Sciences of the
Estonian SSR. During the preceding four years he was academic secretary of the
division of physicotechnical and mathematical sciences of the Academy,
and a member of its presidium. At the Institute of Physics and Astronomy he has
led the solid state group since 1956 and also the crystal spectroscopy group since
1963. Since 1955 he has taught at Tartu State University, where he has given a
number of courses on theoretical physics and solid state physics.
The original Russian text, published by Nauka Press in Moscow in 1968 for the
Institute of Physics and Astronomy of the Academy of Sciences of the
Estonian SSR, has been revised and corrected for this edition by the author.
The translation is published under an agreement with Mezhdunarodnaya Kniga,
the Soviet book export agency.
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ELEMENTARNAYA TEORIYA KOLEBATEL'NOI STRUKTURY SPEKTROV
PRIMESNYKH TSENTROV KRISTALLOV
Library of Congress Catalog Card Number 69-12540
ISBN-13: 978-1-4684-1778-4 e-ISBN-13: 978-1-4684-1776-0
001: 10.1007/978-1-4684-1776-0
© 1970 Plenum Press, New York
Softcover reprint of the hardcover 1s t edition 1970
A Division of Plenum Publishing Corporation
227 West 17th Street, New York, N. Y. 10011
United Kingdom edition published by Plenum Press, London
A Division of Plenum Publishing Company, Ltd.
Donington House, 30 Norfolk Street, London W.C.2, England
All rights reserved
No part of this publication may be reproduced in any form
without written permission from the publisher
Preface to the American Edition
It is very rewarding for an author to know that his book is to be
translated into another language and become available to a new
circle of readers.
The study of the optics and spectroscopy of activated crys
stals has continued to grow. The development and first remarkable
successes of light scattering by impurities in crystals have occurred
in the comparatively short time since my original book was sent to
press. After experimental observation of the sidebands (wings) in
impurity infrared absorption spectra, interest in these spectra as
a source of information on the vibrations of a crystal in the neigh
borhood of an impurity has increased significantly.
Therefore, in addition to making minor corrections, r have
supplemented the section on the effect of anharmonicity (section25)
and written two new sections and another Appendix on infrared ab
sorption, scattering of light by an impurity center in a crystal,
and the adiabatic approximation, respectively.
The bibliography has received several dozen new entries,
but it nevertheless does not pretend to be complete.
r hope that the American edition is useful and in some de
gree corresponds to the general deepening of our physical under
standing of solids.
K. Rebane
v
Preface to the Russian Edition
Interactions with lattice vibrations are an important problem in
contemporary solid-state theory and crystal spectroscopy. These
interactions are clearly seen in the vibrational structure of the
absorption and luminescence spectra of activated crystals which
are, in essence, always vibronic spectra. The properties which
determine whether a crystal can be used in a laser depend largely
on them.
The vibronic transition problem first arose in its present
formulation in the theory of molecular spectra. It is comparative
ly simple for diatomic molecules. Therefore it was here, on the
basis of the Franck-Condon principle, that a correct understand
ing of the physical situation was achieved, the principle was given
a quantum-mechanical basis, and 'the first quantum-mechanical
calculations of the distribution of intensity over the vibrational
components of the electronic transition were made [1-4].
The interaction between an optical transition and the vibra
tions of a polyatomic molecule is quite similar to the problem of
an impurity center in a crystal. The need to explain the nature of
broad structureless spectral bands whose shapes could be repre
sented by a certain more-or-Iess universal function applicable to
a large number of complex molecules first arose long ago in the
theory of the luminescence of complex molecules (Vavilov [5]). The
current status of the theory of the shapes of vibronic absorption
and luminescence bands in complex molecules, whose development
has received major contributions from the work of Neporent [6]
and especially of Stepanov and his co-workers, has been developed
in monographs by Stepanov [7], and Stepanov and Gribkovskii [8].
vii
viii PREFACE TO THE RUSSIAN EDITION
The present quantum-mechanical theory of the shapes of ab
sorption and luminescence bands arising from impurities in crys
tals originated with the work of Pekar [9, 10], Kun Huang and Rhys
[11], Davydov [12], and Krivoglaz and Pekar [13]. In these papers
the nature of the broad vibronic bands was elucidated, proceeding
from the idea, first put forth by Frenkel' [14], that changes of the
normal modes with a change of the electronic state could be the
reason for the appearance of multiphonon transitions. Effective
methods of calculation were developed and formulas were obtained
which give a good description of such basic properties of these
bands as the Stokes shift, the temperature dependence of the half
width, and Levshin's mirror symmetry law [15, 16].
In connection with the main objective ofthis book, the theory
of spectra with clearly-defined vibrational structure, we take par
ticular note of the work of Krivoglaz and Pekar [13], in which a
theory was developed which takes account of the dispersion of the
normal mode frequencies in the crystal. In the light of our present
understanding, it is especially noteworthy that they showed that a
narrow resonant purely electronic line (ideally with the radiation
width) exists even with dispersion of the vibration frequencies. *
In essence this line is the same kind of analog and "precursor" of
the Mossbauer line as the zero-phonon Lamb line in the spectrum
of neutrons absorbed by nuclei in a crystal [17].
It cannot be said that the impurity spectra of crystals have
been neglected by experimenters. Since the discovery of line spec
tra in organic crystals by Obreimov [18] and in uranyl crystals by
Becquerel [19], many investigations have been made of the absorp
tion and luminescence spectra of a wide variety of inorganic and
organic crystals. In our opinion there are three series of papers
which are the most interesting in connection with vibrational
structure: the spectra of rare-earth ions in ionic crystals [20,
21], investigations of molecular crystals with impurity molecules
[22], and investigations of uranyl ion spectra [20, 23].
In recent years interest in the vibronic spectra of crystals
has increased substantially. This is chiefly due to the develop
ment of crystal lasers.
• Unfortunately, Pekar and Krivoglaz did not pay due attention to this line; in es
sence they only remarked in [13] that there is a transformation of one of the "parti
al spectra" into a line which corresponds to the purely electronic line.
PREFACE TO THE RUSSIAN EDITION ix
In a laser which uses a crystal containing impurities as the
active medium, the level scheme is the scheme of vibronic levels.
A large class of problems which have previously been thought of
as more-or-Iess "purely academic" details of the theory of lumin
escence have turned out to be part of such realistic areas of study
as laser theory. The theory of the shape of a spectral line on
which a crystal laser works is the theory of the shape of a vibronic
line, in particular a purely electronic line.
The most attention has been given to spectra with a sharp
purely electronic line, but the vibrational replicas of the purely
electronic line and the broad structureless band are of great inter
est also. It suffices to mention the development of a laser which
operates on the vibrational replicas [24] and the theoretical possi
bility of making a laser using the F band in colored alkali halide
crystals [25].
There is a close analogy between the Mossbauer effect and
vibronic transitions [26-28]. The purely electronic line has even
received a new name - "the optical analog of the MOssbauer line"
[29, 30].
The great interest in the Mossbauer effect has led to a sub
stantial revival of theoretical studies on the role of lattice vibra
tions in this interesting phenomenon. In a short time significant
results were achieved on local lattice dynamics in the vicinity of
an impurity atom [31-37] and the effect of various subtle effects of
the interaction with vibrations on the properties of the Mossbauer
line [37-39]. In the theory of luminescence it has long been under
stood that the local lattice dynamics about an impurity center
play an important role in determining the structure of the spec
trum. In recent years the development of theories for both pheno
mena has proceeded in parallel, and often even simultaneously. A
number of investigations have developed a unified theory of zero
phonon lines (MOssbauer lines and purely electronic lines) and
their vibrational backgrounds [38-41].
The mathematical methods developed for vibronic transitions
and many formulas obtained for them are almost directly applica
ble to the theory of the Mossbauer effect. In essence, the analogy
of these two phenomena gives a deeper understanding of the phys
ics of interactions between the vibrations and an impurity center.
x PREFACE TO THE RUSSIAN EDITION
In connection with lasers and the Mossbauer effect, vibronic
transitions in crystals containing impurities have become of inter
est for real problems. In our view, this is revealed by rapid
growth of interest on the part of experimenters. This has been
shown by the large number of publications and the organization of
conferences especially devoted to the spectra of crystals which
serve as active media for lasers and which have richly detailed
vibrational structure in their spectra [41]. Of the experimental
investigations already specially set up to make a detailed check of
the conclusion that the purely electronic line is an optical analog
of the Mossbauer line and should basically have its characteristic
properties, we note the work of Gross, Permogorov, and Razbirin
[29. 30], and that of Hopfield [42], Fitchen, Silsbee, Fulton, and
Wolff [43], and Fitchen et al. [44-47].
* * *
This book is an attempt to give an elementary treatment of
the present concepts and basic results of the theory of vibrational
structure in the vibronic absorption and luminescense spectra of
impurity centers in crystals. In this case lIelementary treatment"
does not mean that we use only classical and the simplest semi
classical ideas about the physics of the process, or that higher
mathematics is not used. On the contrary, the basic conclusions
of the theory are derived from a proper quantum-mechanical
treatment, since all of the features of interest in the vibrational
structure of the spectrum are consequences of quantum effects of
the lattice vibrations. These effects cannot be understood on the
basis of classical ideas; they are tractable semiclassically, but,
as usual, good results can only be had from a quantum-mechanical
treatment.
The author believes that elementary" means a quite thor
II
ough development of both the basic quantum-mechanical problem
and its solution. It is not necessary for this purpose that the
reader's knowledge go beyond elementary ideas of quantum mech
anics. Therefore the book is easily accessible to readers who
have had elementary quantum mechanics in the usual experimental
physics or chemistry curricula. The author also feels that the
book may be of use to well-prepared readers who are interested
in the theory of crystal spectra.
PREFACE TO THE RUSSIAN EDITION xi
The book includes several dozen problems and exercises of
varying degrees of difficulty.
The author thanks 0.1. SHld for his great assistance in pre
paring the manuscript for the press, and 1. G. Virko for work on
the final editing. The author sincerely thanks V. V. Khizhnyakov,
G. S. Zavt, N. N. Kristofel', R. A. Preem, A. P. Purga, and L. A. Re
bane for valuable remarks.
Contents
C HAPT ER 1. The Ad i a bat i cAp pro x i mat ion • . • . . 1
1. The Adiabatic Approximation [3, 4, 48] . . • • • . . . . 1
2. Molecular Energy Levels and Potential Curve
Diagrams. The Harmonic Approximation 8
3. Lattice Vibration Theory for a Crystal
Containing Impurities. Band and
Localized Vibrations. Pseudolocalized
Vibrations . • • • • . • • • • • . . . . • . • . . . . . •• 14
4. Vibronic Transition Probabilities. The
Franck-Condon Principle. • • . . . . • • • • . • •• 23
C HAPT ER 2. The 0 r y 0 f Qua s i Ii n e V i b ron i c
Spectra of Impurity Centers... 35
5. The Impurity Center Model. . • . • • • • . • . • . . • •. 35
6. The Energy of a Purely Electronic Transition,
Zero-Phonon Transitions, and Quasilines • • •• 36
7. The Integrated Intensity of the Purely Electronic
Line and Its Temperature Dependence in the
Simplest Case . • . • . • • • . • . • • • • • • • • • .• 39
8. Position, Width, and Peak Intensity of the
Purely Electronic Line in the Simplest Case .• 50
9. The Integrated Intensity of a Vibronic Band. . . • .• 53
10. One-Phonon Transitions ••••.•.••••...•••• 55
11. Two-Phonon and Multiphonon Transitions. • • • • •. 62
12. The Purely Electronic Line and the Vibrational
Background . • • . • . • • • . • • • . . . • • . . . • .. 64
13. The Effect of Localized Vibrations .•••.•..... 68
14. Several Localized Vibrations 78
xiii
