Table Of ContentRELAXATION PHENOMENA
IN CONDENSED MATTER PHYSICS
Sushanta Dattagupta
School of Physical Sciences
Jawaharlal Nehru University
New Delhi, India
1987
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Library of Congress Cataloging in Publication Data
Dattagupta, S. (Sushanta), Date
Relaxation phenomena in condensed matter physics.
Includes index.
1. Condensed matter—Optical properties.
2. Relaxation phenomena (Physics) 3. Spectrum
analysis. 4. Stochastic processes. I. Title.
QC173.4.C65D38 1987 530.4Ί 86-3525
ISBN 0-12-203610-7 (hardcover) (alk. paper)
PRINTED IN THE UNITED STATES OF AMERICA
87 88 89 90 9 8 7 6 5 4 3 2 1
To my late parents
PREFACE
This book is divided into two parts. In Part A, which occupies about a
fourth of the volume, I discuss how various spectroscopy experiments can
be analyzed in terms of correlation funtions. In Part B, I deal with the
stochastic theory calculation of these correlation functions. Each stochastic
model is set up in the context of a physical process. The result of the calcula-
tion is then tied up with one of the experiments discussed in Part A.
There exist several treatises devoted to each spectroscopy technique
(analyzed in Part A) and its relation to correlation functions. However, I
felt the need of presenting in one place a collection of different methods.
This helps in appreciating the similarities and the dissimilarities between
various studies of relaxation phenomena and in formulating a unified
theoretical approach. The theory covered in Part B is based entirely on
stochastic methods. They provide a simple and physically motivated
mathematical framework for analyzing relaxation phenomena that can be
linked to one kind or another of diffusion process. Diffusion, though nor-
mally viewed to be connected with Brownian motion, is a paradigm of
dissipative or irreversible behavior of a variety of systems. Again, there are
numerous books on stochastic processes; the present contribution can,
however, be regarded as one in which applications to several problems in
condensed matter physics are described.
The background assumed is that of a graduate student who has had quan-
tum and statistical physics. As such, the book can be employed in a two-
semester course on special topics in nonequilibrium statistical mechanics. I
hope the book will be found useful to the graduate student of condensed
matter physics who is looking for a possible area of research. It ought to be
xi
xii PREFACE
helpful also to the experimentalist who would like to utilize stochastic
methods for interpreting data or wants to learn more about how similiar
relaxation phenomena can be investigated by other very different tech-
niques.
The study of relaxation phenomena pervades many areas of atomic and
molecular, liquid state, chemical, and condensed matter physics. The sub-
ject is so wide that it is impossible to cover it in just one book. What I have
attempted therefore is to highlight a few topics that have been selected on
the basis of my own prejudice and familiarity. I hope, however, that the
selection is representative enough to project a coherent picture of relaxation
effects in matter.
Because of the need to keep the book within a reasonable size, I have not
been able to discuss all of the techniques that can be employed for relaxa-
tion studies, but I hope the reader will be able to see the connection between
what is treated here and what is not. Similar considerations have compelled
me to give theoretical analyses, which a more mathematical minded reader
might find somewhat ad hoc, in certain parts o fthe book and convey an at-
titude of "let's get right down to business!" There has also been no attempt
to provide an exhaustive bibliography in this vast area o fresearch. The last
two chapters (Chapters XIV and XV) are concerned with relaxation effects
in cooperative and disordered systems. These are topics of great current in-
terest. My presentation is admittedly rather sketchy but is nevertheless in-
cluded here in order to indicate areas of future research in relaxation
phenomena.
ACKNOWLEDGMENTS
This book has resulted from the very enjoyable collaboration I have had in
the area of relaxation effects with Girish Agarwal, V. Balakrishnan, Marty
Blume, Deepak Kumar, Radha Ranganathan, Ajay Sood, Lukasz Turski,
and G. Venkataraman. In particular, I would like to thank Marty Blume for
introducing me to this topic and Girish Agarwal for constantly encouraging
me to complete the project. Discussions with Deepak Dhar and Subodh
Shenoy have been useful in presenting the material in Chapter XV. Lukasz
Turski has patiently read the manuscript. His criticisms have led to some im-
provement of the original material. Defects that remain are, of course, entire-
ly my responsibility.
The book would not have been written had Richard Cohen not persistently
pressed me to undertake it and initiated contact with the publishers.
I thank the Alexander von Humboldt-Stiftung for a fellowship that gave
me a wonderful opportunity for carrying out the project. I am grateful to the
University of Hyderabad for granting me leave and the Institut für
Festkörperforschung der Kernforschungsanlage Jülich for providing an ex-
tremely congenial working atmosphere. I am very thankful to Frau Her ff for
undertaking the arduous task of typing the manuscript with unflagging cheer-
fulness.
Finally, I should record my gratitude to my wife Ranu and daughters
Shahana and Sharmishtha for their unflinching support and understanding
during the time this book was written.
xiii
GLOSSARY OF ABBREVIATIONS USED
bcc Body-centered cubic
CTRW Continuous-time random walk
EFG Electric field gradient
EPR Electron paramagnetic resonance
ESR Electron spin resonance
fee Face-centered cubic
FPE Fokker-Planck equation
IR Infrared
KAP Kubo-Anderson process
KP Kangaroo process
LRT Linear response theory
/iSR Muon spin rotation
MJP Multilevel jump process
NMR Nuclear magnetic resonance
NQR Nuclear quadrupole resonance
PAC Perturbed angular correlation
RPA Random phase approximation
SCK Smoluchowski-Chapman-Kolmogorov
SCM Strong collision model
SLE Stochastic Liouville equation
SMP Stationary Markov process
TJP Two-level jump process
WCM Weak collision model
xv
INTRODUCTION TO PART A
The term "relaxation," as it is applied in condensed matter physics, means
the time-dependent approach of a system from one stationary state to
another. The systems that we have in mind here are thermodynamic systems,
and hence the "stationary" state referred to is to be understood a sa thermal
equilibrium state. A system can be made to "relax" either by subjecting it
to a "force" or by removing the force after having kept it on for a long
time. The central question posed in the study of relaxation phenomena is:
how is the infusion of energy or withdrawal of it shared by the various
degrees of freedom of an interacting many-body system? This question can
be conveniently addressed within the framework of nonequilibrium statis-
tical mechanics, and it is this approach that we shall adopt in the present
book.
The study of relaxation phenomena has largely benefited from the
development of numerous experimental techniques. One of our principal
aims is to discuss some of these techniques very briefly, with the limited
purpose of focusing attention on the type of relaxation phenomena one
investigates in a given experiment. This task is rather important, as the
development of the subject depends crucially on a very close relationship
between experiment and theory.
The experimental methods discussed here can be broadly classified as
electromagnetic, mechanical, and nuclear spectroscopy. Electromagnetic
spectroscopy includes the studies of magnetic and dielectric susceptibilities;
microwave, radiofrequency, infrared, and ultraviolet absorption; and
3
4 PART A
Raman scattering. Mechanical spectroscopy deals with the elastic interaction
of matter with small stress fields, as can be investigated by low- and
medium-frequency internal friction, and high-frequency ultrasonic devices.
Finally, nuclear spectroscopy involves the study of a subatomic
phenomenon, e.g., the recoilless emission or absorption of gamma rays by
nuclei, as in the Mössbauer effect, angular correlation of successive radi-
ations from a nucleus, or the decay of positive muons in matter. The topic
of the scattering of neutrons is also briefly mentioned, mainly to indicate
its relationship with Mössbauer spectroscopy in the investigation of certain
diffusion phenomena.
Each of these methods has myriad aspects that have been covered in
numerous books. Our objective is not to discuss them in any great detail
but merely to point out certain interrelationships between the various
measurements in the context of relaxation effects, especially when these
effects can be traced to one kind or another of diffusion phenomena. With
this limited aim in mind, we have left out the topics o fmagnetic scattering
of neutrons, Raman scattering due to excitations other than molecular
vibrations, and several electron and photon spectroscopic techniques.
The various spectroscopic methods mentioned are all linked by a common
principle that makes use of a weak coupling between the laboratory per-
turbations (e.g., an oscillatory electric/magnetic/stress field, light from a
laser, or neutrons from a reactor) and the sample. The weakness of the
coupling between the input "disturbance" and the system at hand allows
us to employ a first-order perturbation theory in the treatment of all the
methods mentioned here. This theory is popularly known as the linear
response theory (LRT); it is intimately connected with the golden rule of
perturbation theory in quantum mechanics. A detailed discussion is presen-
ted in Chapter I, which also contains an LRT derivation o fthe generalized
susceptibility that would be required for analyzing magnetic, dielectric, or
anelastic relaxation. In Chapter II we use the golden rule to calculate the
power absorbed by a system from an incident electromagnetic field. The
resulting analysis yields expressions for various spectroscopic line shapes.
In Chapter III we consider the theory of neutron and Raman scattering
based on the golden rule. Finally, the perturbed angular correlation (PAC)
of gamma rays and the muon spin rotation (μ-SR) experiments are described
in Chapter IV, again on the basis of the first-order perturbation theory.
We ought to emphasize here that the LRT or its equivalent does not really
yield a solution of the problem; it merely postpones the agony! The point
is, the eigenfunctions and eigenvalues, which occur in the first-order per-
turbation treatment of the LRT, refer actually to the entire many-body
system under study and are therefore not known for any nontrivial system.
However, the value of the LRT lies in the fact that it provides a convenient
PART A 5
language that connects experiment with theory—the language of correlation
functions. We shall see in Chapters I through IV that the measured quantities
in all the spectroscopic studies are expressed directly as a correlation
function or its Laplace transform. The typical correlation function is a
self-correlation or an autocorrelation,
CAA(0 = <A(0)A(0>o,
between a dynamical variable A (a quantum operator, in general) at time
zero and the same variable at a later time t. Physically, it measures the time
(often called the "correlation time" or simply the "relaxation time") over
which the variable A retains its own memory until this memory is averaged
out by statistical randomness. The angular bracket with the subscript zero
implies that the correlation function is the average over an equilibrium
ensemble of the system, i.e., the ensemble that pertained even before we
started the experiment (by switching the laser on, say) ! That is, the correla-
tion function ÇAA(Î) relates to certain intrinsic or; spontaneous statistical
fluctuations of the system. As we very well know from statistical mechanics,
such spontaneous fluctuations can never be "switched off," even for a system
in thermal equilibrium. The result of the LRT, which connects the response
(to an external perturbation) with spontaneous fluctuations, is essentially
a statement of inner consistency of statistical mechanics. It says that the
manner in which the input energy is shared or dissipated among the various
constituents of the system is determined by the time-dependent properties
of the intrinsic fluctuations themselves! Hence the studies of relaxation
behavior and the spontaneous fluctuations go hand in hand—one implies
the other. Therefore, we shall employ the terminology "relaxation
phenomena" not merely in connotation with the external disturbance that
makes the system evolve from one equilibrium to another but also to refer
to "relaxation" or "readjustment" of certain internal variables of the system
(e.g., the position of an atom as it jumps from one lattice site to another
due to spontaneous thermal fluctuations).
With the machinery of the LRT behind us, the task of model building
begins with the correlation function. If we can evaluate it, approximately
of course, we can link it with a given experiment in order to give a successful
interpretation to the observed data. It is also possible to analyze a variety
of experimental methods, as discussed in Chapters I to IV, from a fairly
uniform theoretical point of view. This helps to elucidate the similarities
and differences between various techniques, which are elaborated upon in
Chapter V.
It is evident that, by combining information from different experiments
on the same system, considerably more insights can be obtained. To cite
an example, consider the diffusion of an interstitial like H, which has many
