Theoretical Foundations of Electron Spin Resonance

404 Pages·1978·5.29 MB·English

by  John E. Harriman and Ernest M. Loebl (Auth.)

This is Volume 37 of
PHYSICAL CHEMISTRY
A Series of Monographs
Editor: ERNEST M. LOEBL, Polytechnic Institute of New York
A complete list of titles in this series appears at the end of this volume.
THEORETICAL FOUNDATIONS
OF ELECTRON SPIN RESONANCE
JOHN E. HARRIMAN
DEPARTMENT OF CHEMISTRY AND
THEORETICAL CHEMISTRY INSTITUTE
UNIVERSITY OF WISCONSIN
MADISON, WISCONSIN
ACADEMIC PRESS New York San Francisco London 1978
A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHT © 1978, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED.
NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR
TRANSMITTED IN ANY FORM OR BY ANY MEANS. ELECTRONIC
OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY
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PERMISSION IN WRITING FROM THE PUBLISHER.
ACADEMIC PRESS, INC.
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United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
24/28 Oval Road, London NW1 7DX
Library of Congress Cataloging in Publication Data
Harriman, John E.
Theoretical foundations of electron spin
resonance.
(Physical chemistry, a series of monographs ; )
Bibliography: p.
Includes index.
1. Electron paramagnetic resonance spectro-
scopy. I. Title. II. Series.
QC763.H37 538'.3 77-75573
ISBN 0-12-326350-6
PRINTED IN THE UNITED STATES OF AMERICA
PREFACE
When I was in high school and college, one activity I enjoyed was formal
competitive debate. The first speaker in such a debate always began by
defining terms so everyone would be in agreement as to just what was the
subject. I will begin this introduction by giving my "definitions" of the terms
which appear in the title. In doing so I hope to establish part of the context
in which the book was written.
Theoretical. It is clear that this book is theoretical in its approach. Only
a few references to experimental' results are made, although the nature of
experimental work in the field has influenced the selection of material to
be treated. More important, perhaps, is that the organization of the book is
on a theoretical basis. I have tried to exhibit some of the unity present in
this branch of theoretical chemistry; it is a unity that is sometimes obscured
when results are developed on the basis of need in a particular experimental
context.
Foundations. This book is not elementary. I have in fact assumed reason-
able prior knowledge on the part of the reader of both the fundamentals
of ESR and ordinary quantum chemistry. Foundations are referred to by
way of distinction from applications. I have tried to establish the foundations
firmly; I have not attempted to build upon them applications to other
branches of chemistry or physics.
Electron Spin Resonance. If the assumption that the reader already has
some knowledge of the subject is correct, the only word requiring comment is
"spin." Objection might be made that the term electron spin resonance is
inappropriate in certain cases and thus that the more general term electron
paramagnetic resonance is preferable. It is precisely because of this implied
limitation that I have chosen "spin." In selecting the topics to be discussed
I had in mind applications to polyatomic, probably organic, free radicals in
condensed phases. Questions of orbital angular momentum in first order, ap-
plicable to transition metal complexes, and of rotational angular momentum,
IX
X PREFACE
applicable to small radicals in the gas phase, have not been addressed.1
Electron paramagnetic resonance in nonmolecular solids has also been
excluded from explicit consideration. Many of the results obtained here are,
of course, applicable to these systems, but problems, methods, and approxi-
mations specifically addressed to them have been avoided.
Another major restriction on the subjects addressed is not inherent in
the title. This book confines itself to essentially static phenomena: the
description and determination of stationary-state energy levels. Dynamic
aspects other than the ESR transition itself have not been considered. When
the book was first planned, I expected that topics such as relaxation, hin-
dered rotation, inter- and intramolecular electron transfers, etc. would be
discussed. I have since abandoned these topics for a number of reasons.
One is just the fact that I have done little work in these areas and thus
feel less comfortable with them. A related consideration is the time that
would have been required to develop a careful treatment. Another reason
is the recent appearance of several books in this area [145, 159]. The decisive
factor, however, is my feeling that the theory of relaxation and related
processes is still in a state of rapid development. While present treatments
are entirely satisfactory in some cases and at least useful in others, they
do not as yet present an essentially unified, firmly grounded, a priori theory
such as is possible (at least very nearly) for the static phenomena. I came
to suspect that I would be unsatisfied with anything I did in this area, so I
did nothing. Besides, the book was getting rather long anyway.
As 1 have indicated, I have assumed the reader to have some familiarity
with elementary ESR. To provide some background and possibly an in-
dication of motivation to those readers with very limited previous experi-
ence, a brief review has been provided in Chapter 0. I have also assumed
some familiarity with elementary quantum chemistry, I hope comparable
to that provided in a typical graduate course. The necessary aspects of
relativistic quantum mechanics are developed as needed. More advanced
techniques such as second quantization and diagrammatic perturbation
theory have been avoided. Relevant features of classical mechanics and
electromagnetic theory, as well as of the quantum theory of angular momen-
tum and rotations, are reviewed in appendices.
Where appropriate, atomic units have been used. In the early sections
I have chosen not to use atomic or natural units because constants with
dimensions make it easier to keep track of terms. Gaussian rather than SI
units have been chosen. I believe they will be more familiar to more readers
f These topics are treated by, for example, Abgragam and Bleany [2] and by Carrington, [31],
respectively.
PREFACE xi
and, in addition, despite the clear advantages offered by the International
System, I prefer Coulomb's law to have the simplest possible form for this
work, where Coulomb forces dominate. A comparison of systems of units is
presented at the end of Appendix A.
A few words next about the organization of the book. The fundamental
unit is the section, although sections vary considerably in length. These
sections are grouped into chapters, the titles of which I believe to be self-
explanatory. The progression is one of logical development rather than
historical or in terms of practical interest. This is one aspect of the assump-
tion of some prior knowledge of ESR. Motivation—why something is of
interest—is in the early stages to be supplied by the reader if required. When
significance is not apparent, I request a suspension of judgment. I hope it
will eventually be recognizable.
References have been presented primarily on the basis of two criteria:
when my treatment was explicitly based on another work, appropriate
credit is given; when another point of view or further development is likely
to be useful, references are provided. In both cases my primary concern
was utility rather than historical precedence. I have therefore cited review
papers or books in preference to the original literature in many cases. With
the exception of a very few topics, I have not attempted to completely survey
the literature in the field. To the many workers who might have been cited,
and to the hopefully few who clearly should have been but were not, I offer
my apologies.
A CKNO WLEDGMENTS
I would like to express my appreciation to the friends and colleagues,
past and present, who contributed to this work. I was introduced to ESR
by A. H. Maki when we were both at Harvard, and to modern theoretical
chemistry by P. O. Lowdin in Uppsala. J. O. Hirschfelder is responsible,
more than anyone else, for the existence of and favorable environment
within the Theoretical Chemistry Institute where this book was written. I
thank them especially.
Among those who read portions of the manuscript and provided helpful
comments were C. F. Curtiss, J. M. Dietrich, S. T. Epstein, E. M. Loebl, J. M.
Norbeck, B. T. Sutcliffe, and F. A. Weinhold. Suggestions and corrections
were also received from graduate students with whom the manuscript was
reviewed. They were Melodye Block, David Fish, Nancy Piltch, Charles
Szmanda, and James Tortorelli. A truly superb job of technical typing, which
I greatly appreciate, was done by Patty Spires.
Financial support that contributed directly or indirectly to the develop-
ment of the book was received from the National Science Foundation and
the Alfred P. Sloan Foundation.
I would like to express my sincere thanks to my wife and son for putting
up with me during this project and for not asking too frequently, "How is
the book coming?" To my colleagues on faculty committees where I spent
many hours that might have been used to hasten completion of this work,
my forgiveness; it was my choice.
xiii
CHAPTER 0
REVIEW OF ELEMENTARY
ELECTRON SPIN RESONANCE
Systems Studied by Electron Spin Resonance
In magnetic resonance spectroscopy, transitions are observed between
energy levels which depend on the strength of a magnetic field.* Electron
spin resonance (ESR) is a branch of magnetic resonance spectroscopy
dealing with molecules (or occasionally atoms, or "centers" in nonmolecular
solids) in which the total spin quantum number S is different from zero. This
statement implies that the quantum mechanical states of the system are, at
least to a good approximation, eigenfunctions of ff1. When this is not the
case a somewhat more involved description is necessary. For now, we will
assume each molecular state can be characterized by, among other things,
spin quantum numbers S and M.
s
Many of the spin-dependent properties of such a molecule are determined
by its spin density distribution. The spin density at any point in the molecule
is the probability of finding an electron there with spin "up" minus the
probability of finding one there with spin "down." The entire spin density
distribution is proportional to M. When M = 0, including the case of a
s s
singlet state with S = M = 0, the spin density is everywhere zero. Most of
s
the nondynamic information obtainable from an ESR observation is
information about the spin density distribution. This in turn gives infor-
mation about the structure of the molecule being observed.
The overwhelming majority of molecules exist in singlet states and are,
therefore, unobservable by ESR. We can use the term "molecule" in a very
general sense in considering systems which are of interest to ESR. Many
atoms have ground states or accessible excited states in which S # 0. They
can be observed by ESR in the gas phase or trapped in relatively inert
matrices. Equivalent information may also be available from high-resolution
optical spectroscopy. Transition metal and lanthanide ions form many
1 There are very many books on magnetic resonance, among them Refs. [32, 192], which
include both nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR).
1
2 0. REVIEW OF ELEMENTARY ELECTRON SPIN RESONANCE
complexes in which S # 0. It is often necessary in these cases to consider
electronic orbital angular momentum as well as spin, and the term "electron
paramagnetic resonance'1 (EPR) is probably preferable to ESR for such
cases. This is also true for a variety of paramagnetic, localized defects or
impurities which can be observed in nonmolecular crystals.
Some molecular systems also have stable nonsinglet ground states. The
most important now are probably the nitroxides used as spin labels. Less
stable molecular radicals may, under appropriate circumstances, have
lifetimes long enough that they can be observed by ESR. Small, highly
reactive radicals are of interest in interstellar and upper atmosphere chem-
istry. Irradiation of solid materials often produces radical species via bond
rupture or ionization. They are potentially very reactive but are unable to
move rapidly enough through their environment to find a reaction partner.
Oxidation or reduction reactions in solution under carefully controlled
circumstances may produce radical ions with lifetimes ranging from milli-
seconds to days. It is clear that many samples suitable for study by ESR
can be obtained, and that information about them is of interest in a variety
of applications.
The Basic Electron Spin Resonance Experiment
We examine now the basic ESR experiment. Consider a molecule with a
net electronic spin different from zero. Associated with this spin will be a
magnetic moment. If a magnetic field is applied, the moment will interact
with it. If the molecule is, for example, in a state with S = \ but spatially
nondegenerate, there are two energy levels which are degenerate in zero
field but are split in the field by an amount proportional to the field strength.
Any other interaction of the net spin will for now be assumed to be absent
or negligible, so we have a two-level system. The basic ESR experiment
consists of observing a transition between these two levels.
Transitions between the two levels can be induced by an appropriately
oriented, oscillating magnetic field if the resonance condition hv = AE = g[W
is satisfied. In cgs-Gaussian units the Bohr magneton fi is 9.27 x 10~21 erg/G
and Planck's constant h is 6.63 x 10~27 erg sec. The dimensionless constant
g is very nearly 2 for organic free radicals. A typical x-band microwave
frequency of 9.5 GHz thus corresponds to a field strength of 3400 G. The
oscillating field has the same probability of causing upward and downward
transitions, but at equilibrium in the static field the lower level will be more
highly populated than the upper so there will be more upward than down-
ward transitions and a net transfer of energy to the sample from the oscillating
field. It is this absorption which is observed.
REVIEW OF ELEMENTARY ELECTRON SPIN RESONANCE 3
Let n+ be the number of molecules in the sample with energy gfiB/2 and
n_ be the number with energy —gfiB/2. At equilibrium in the static field
j_ = ~AE/kr = ~gPB/kT.
n n e e
Boltzmann's constant k is 1.38 x 10 16 erg/K so at room temperature of
298 K, gPB/kT= 1.5 x 1(T3 for B = 3400 G. Then njn = 1.0015, and
+
the population difference An = n_ — n as a fraction of the total population
+
n = n- -h H+ is
tanh - 0.003.
Let P be the probability per unit time that a transition in either direction
will be induced. Clearly P depends on the strength of the microwave field
as well as other factors. The rate of absorption of energy by the sample from
the microwave field is W = hvP An, and unless there is some mechanism
tending to reestablish equilibrium, An will approach zero.
We assume that there is a relaxation mechanism tending to reestablish
equilibrium, and further assume it to be first order:
The first-order rate constant is written as \/T and T is known as the spin-
{ x
lattice or longitudinal relaxation time. This combines with the changes
produced by the microwave field to give
2PAn 4
total
For steady-state conditions we assume this rate of change to be zero and get
An,
The steady-state absorption of power by the system due to ESR transitions is
v steady state = hv 1 + 2PT, '
Before treating other interactions of the electron spin which give structure
to the spectrum, we will briefly consider other aspects of relaxation, obtaining
a simple description of the resonance lineshape. A detailed treatment is
beyond the scope of this chapter, and will in fact not be considered at all in
this book. At this point we continue with a macroscopic, phenomenological
point of view.