ABSTRACT

PARAMAGNETIC RELAXATION OF SUBSTITUTIONAL

IMPURITIES IN A1C13~6H20

by Robert F. Vieth

Spin-lattice relaxation times of substitutional

+++

+++ .
Fe and Cr in AlCl3-6H20 have been measured. The data

was taken over a wide range of temperatures, 1.10K. to

o +++ o o , +++ .
48 K.for Fe and 1.1 K.to 80 K. for Cr . Both spin-
echo and saturation recovery techniques were employed pro-
viding a cross—check between the two methods. The two
methods yielded data which agreed within the limits of
experimental error («’lO%). Departures from the usual

, , +++ ' ,

Raman behaViour were observed in Fe . In the Raman region:

the spin—lattice relaxation time had a TSJ (e/T) dependence°

4

+++
The Cr obeyed a T7J (G/T) Raman dependence. The Debye

6
9 was found to be 1600 in both cases. The full dependence

for Fe+++ wa3'% = 7lT + 7.58x10.4T5 and for Cir-F++ was
1

l = 67T + 1.2x10-8T7.

”‘1

A full discussion is accorded those relaxation

theories which could explain the observed effects. The

Robert Vieth

mathematical section also includes a resume of Group and
Representation Theory for use in the field of paramagnetic

resonance. Applications are made to the A1C13-6H20 system

studied.

PARAMAGNETIC RELAXATION OF SUBSTITUTIONAL

IMPURITIES IN AlCl3°6H20

BY

‘/
{\L

h’ .
Robert Ft Vieth

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy

1963

PREFACE

This thesis was written with two objectives: to
provide a clear presentation of the experiment and the data,
and to provide an outline of the theoretical integrants of
paramagnetic resonance and relaxation. In cases where
methods or results were used but not proven, an effort has
been made to provide references which do not require addition-
al reading to understand. The bibliography is arranged
so that it can be a useful entity.

Acknowledgement and thanks are due to all my
professors. Special recognition is certainly due to
Professor Jerry A. Cowen for his suggestion of the problem
and continued assistance, to Professor Robert D. Spence
for his many useful suggestions, and to Professor Truman
Woodruff and Dr. Peter Schroeder for their helpful comments
during the preparation of the dissertation. I should not
forget the machine and electronics shops whose assistance
and skill were always welcomed. The support for the
research by the U. S.-Army Signal Corps has been deeply

appreciated.

ii

This thesis is dedicated to my parents and wife
who have provided me a constant source of encouragement,

assistance, and inspiration.

iii

TABLE OF CONTENTS

Chapter Page
1. INTRODUCTION . . . . . . . . . . . . . . . . 1
History 1

Summary of Theory 2

Outline of Mathematical Theory 6

2. MATHEMATICAL THEORY . . . . . . . . . . . . 8
Introduction to Crystal Symmetry 8

Group and Representation Theory 11

Groups 11
Representations 13

Hilbert Space 18

The Character Table 19

Projection Operators 21

Direct Product of Representation 22

Direct Product of Groups 23

Double Valued Representations 25

Application 26

Coupling Coefficients 32

General Hamiltonians 38
Introduction 38

The Crystal Field 39

Interaction Terms 41

Coulomb Terms 42

Zeeman Energy 42

Calculation of the Hamiltonian 42

Koster-Statz Hamiltonian 44

Spin Hamiltonian 47

iv

Chapter
Lattice Vibration Theory

Introduction

Lattice Modes

Creation and Annihilation Operators
The Dispersion Law

Relaxation Theory

Introduction

The Direct Process

The Two-Phonon Process
Phonon Bottleneck
Localized Modes

Cross Relaxation

Relation to Line Shape
Relation to Relaxation Time

3. EXPERIMENT AND TECHNIQUES . .

General Description
Spin Echo
Saturation Recovery

4. RESULTS AND CONCLUSIONS
The Experiment

Analysis
Discussion

Final Discussion

BIBLIOGRAPHY

Page
52

52
53
59
62

71
71
78
84
92
93
95

95
97

99
99
-103
‘108
111

111

112,‘

117

120
121

121

134

LIST OF FIGURES

Figure Page
1. Local Symmetry of Al+++ Site . . . . . . . . 9
2. Fe+++ Energy Diagram . . . . . . . . . . . . ‘28
3. Cr+++-Energy Diagram . . . . . . . . . . . . 31
4. Phonon Distribution . . . . . . . . . . . . . 73
5. Typical Energy Diagram . . . . . . . . . . . “85’33'
6. Saturation Equipment . . . . . . . . . . . . 100
7. Spin Echo Equipment . . . . . . . . . . . . . 101
8. Dewar Cap and Cavity . . . . . . . . . . . . 102
9. Spin Echo Sequence . . . . . . . . . . . . . 1105
10. Spin Echo Receiver . . . . . . . . . . . . . 109
ll. Spin Echo Data . . . . . . . . . . . . . . .‘ 113/19’
12. Saturation RecOvery Data . . . . . . . . . . 114
13. Cr+++ Temperature Dependence of Relaxation . 119
14. Fe+++ Temperature Dependence of Relaxation . 121

vi

LIST OF APPENDICES

Appendix Page
A. KRAMER'S DEGENERACY . . . . . . . . . . . . 126
B. DEFINITION OF A REAL ROTATION OPERATOR . . . 129
C. JAHN-TELLER.EFFECT . . . . . . . . . . . . . 130

D. FURTHER DISCUSSION OF CRYSTAL FIELD
CALCULATIONS . . . . . . . . . . . . . . 131

Appendix Tables

1. Cross reference for group notations . . . . 132
2. Cross reference for irreducible
representations . . . . . . . . . . . . 133

vii

CHAPTER I

INTRODUCTION

History

Spin-lattice relaxation phenomena in crystal salts
has been a subject of both experimental and theoretical
interest for three decades. Gorter89 in the 1930's studied
the relaxation phenomena using susceptibility techniques
and Waller.106 Van Vleck,101 and Kronig57 established a
theoretical explanation of the results. With the advent of
electron paramagnetic resonance the interest in relaxation
phenomena was renewed, since resonance provided the
possibility of measuring relaxation times in a more direct
fashion. Experimental resolution limited the results to
4.20K. and below until the late 1950's. During this period
emphasis on spin-spin interactions played the major role.

104

Work in K Co(CN)6 and K Fe(CN)6 coupled with advances in

3 3

resonance technology has reopened interest in Van Vleck's
22 18 . . .
theory. Orbach and Klemens, Within the past two years,
. . . . 1
have made modifications in the theory. Castle 9 has re-
ported results up to about 900K. in MgO using a saturation-

inversion technique. To date. this is the most comprehensive

high temperature relaxation study in crystal salts. This
is the situation at the present time: theories predicting
higher temperature results have been formulated but still

await direct experimental evidence before they can be expanded

and/or corrected.104

Our experiment contrasts the relaxation time of an

+++ 6 +++

S state ion, Fe (

(4F

S ). and a non-S state ion, Cr

5/2

), in the same host lattice AlCl »6H 0 over a temperature

3/2 3 2

range of 1.10 to 800K. In order to enhance the value of the
data two dissimilar methods were used to check the validity

of the results.

Summary of Theory

 

In order to understand relaxation time measurements;

. . , . .. 111, 48
one must consider the theory of lattice Vibrations and

the theory of paramagnetic resohance.43’ 88’ 97 The over—
all aim is to calculate the characteristic time (relaxation
time) for electron spins that have absorbed energy to
relinquish the energy to the thermal bath.

The absorption of energy by a paramagnetic electron
is governed by the resonance equation,

hv = El - Ezrov gBH (1.1)

where\) is the frequency of the incoming r.f.. h is Planck's

constant, E1 is the final energy of the electron, E2 is the

initial energy, 9 is a parameter for the paramagnetic system

called the spectroscopic splitting factor, B is the Bohr

magneton, and H is the applied magnetic field. Assume a

certain group of spins with g = 91 have their resonance

condition met (1.1). If the r.f. field is turned off, then

the spins will be in a non-equilibrium state. There are two

processes that may occur. The first process is called cross-

relaxation and has two aspects. The spins with g = 91 can

exchange energy among themselves in a characteristic time

T2: also if there exists another set of Spins with 92 # 91.

the spins with 91 may undergo mutual spin flips with them

in a characteristic time denoted 112. The second process

is termed the spin-lattice relaxation. The spins dispose

of their energy to the lattice heat sink by means of the

coupling between their orbital magnetic moment and the time

~varying electric field. The relaxation time for this

process is labeled Tl.
It is the latter process that shall be of concern

to us. In order to calculate T1, the crystal field must

be expanded in terms of the strain. Knowledge of the

"exact" wavefunctions of the paramagnetic system is

necessary in order to calculate the required matrix elements.

These wavefunctions must reflect the symmetry of the crystal

field. The "exact" wavefunctions would be calculated by

expansion in terms of d electron wavefunctions (and if need
be admixtures of wavefunctions of the bonding electrons)
which have been corrected for spin-orbit coupling by means of
perturbation theory. The expansion coefficients are the
Clebsh-Gordon coefficients (or their equivalents) which are
arrived at by group theoretical arguments. However the
coefficients and the correct wavefunctions for the expansions
require a detailed knowledge of magnetic energy levels of
the system under study, which, for our case, unfortunately
is not available. One can write down the Spin Hamiltonian,
which is a highly specialized form of the Hamiltonian used
to describe only the states that participate in the resonance
phenomena.

In spite of our difficulties in writing an exact

expression for T it is still possible to extract the

l I

theoretical temperature dependence of T and compare it with

1
our measured values. By writing down the lattice waves of

a crystal in a simple classical form, one can obtain several
quantum mechanical operators. These operators have the
effect of creating and destroying a phonon, a process which
takes place when an electron spin absorbs energy from or
relinquishes energy to the lattice. Two systems of ions

must be considered:

1) those which have an even number of electrons

(non-Kramers' systems), and

2) those with an odd number of electrons (Kramers'
systems). Two types of relaxation can occur for each system:
one which involves one phonon (the direct process) and one
which involves two phonons (the Raman process). The direct
process occurs at lower temperatures while the Raman is
dominant at higher temperatures. For non-Kramers' systems

we find %. u.T for the direct process and % d.T7 for the

1 1
Raman process. The situation for Kramers' systems is not
so simple. If zero order wavefunctions are used in the
computation of the necessary matrix elements, the direct
process will vanish and the Raman process will have a T
dependence. If the zero order wavefunctions, however, are
admixed with higher levels a direct process that again is
proportional to T and a Raman process proportional to T
are obtained. However, if there are energy levels that lie
low enough, the Raman process may go as T5.

One can also consider the situation in which the
thermal bath does not remain in equilibrium during the
absorption of energy from the electrons. This is found
to give an additive term proportional to T2. The concen-
tration of the impurity ions (< .03%) is low enough so

that this effect is negligible in our experiment.

Terns proportional to T3 and T11, for Kramers'
case, and T5 and T13, for non-Kramers“ case, arise from the
effect of the impurity on the lattice modes. The results are
obtained from the consideration of damped vibrational modes
which are localized around the site of an impurity. These
modes produce a spike in the frequency spectrum centered
around a frequency which is termed the local mode frequency.
For lattice frequencies above the local mode frequency, the
impurity mass will not follow the lattice vibration and the
resulting strain will be proportional to wave amplitude

(in the normal situation the strain is proportional to the

derivative of wave amplitude).

Outline of Mathematical Theory

 

In Chapter 2 we hav indicated the calculations that
are pertinent to our problem. In the first section, the
group theory concepts applicable to paramagnetic resonance
are set forth. The groups used in the examples are those
which are directly applicable to our problem. In the second
section we derive the local electric field at the Al+++ site
and the Spin Hamiltonian. The construction of the "exact"
wavefunctions and Hamiltonians is outlined using the Koster—
Statz formalism. In the third section the normal mode

relations are obtained and related to the creation and

annihilation operators. A discussion of the approximations
that are made in the diSpersion law is also included.

Finally the relation of the Debye 9 to other measured
parameters is examined. In the fourth section relaxation
times of the direct and Ranan processes are determined in
terms of transport integrals. In addition a brief discussion

of cross-relaxation has been included.

CHAPTER 2

MATHEMATICAL THEORY

Introduction to Crystal Symmetry

 

In any discussion of the properties of a crystalline
solid a definition of the geometric structure must be estab-
lished. A perfect single crystal consists of a regular
array of lattice sites (atom positions) with some periodic
arrangement. Real crystals possess impurity atoms, vacant
sites, and localized departures from the regular array
extending over many sites.

The periodicity of the array of sites is commonly
termed symmetry. If only the symmetry about each lattice
site is considered, there are just 32 different point
symmetries found in nature. A cross reference for notation
of point-groups is given in the appendix. The addition of
translational operations gives 230 space groups,

A resonance experiment, however, can only sample the

- . +++
pOint symmetry. The local pOint symmetry about the Al

site in AlCl ~6H O is C .110

c . . . .
3 2 3i (see Figure 1.) This implies

+++ . . . . .
that the A1 Site is a center of inverSiOn and lies on a

three-fold axis (the z-axis). Hence the neighboring atoms

 

 

 

P
~p_—-—_o_

 

 

 

 

0 H20

+++

. +++ .
Figure 1. Local Symmetry of Al Site.

10

lie at vertices of pairs of equilateral triangles whose
planes are perpendicular to the three—fold axis. If one
of the triangles lies in a plane at -z, the other is re-
flected through the inversion center to +2. The nearest
neighbors are two sets of three water molecules located
1.8 A from the A1+++ site. The two triangular planes are
nearly coincident, being separated by less than .2 A. The
next nearest neighbors are two sets of three Cl- atoms
which lie at 3 A and whose planes are separated by only
.4 3. Another group of three C1- also lies 3 A from the
Al+++ site but in a plane midway (2.1 A) between two
Al-H-+ sites.

It is useful to think of C3i symmetry as a "distortion"
of one of the body diagonals of a cube, since it is this
"distortion" which changes upon substitution of an impurity
for the A1+++ such as Fe+++ or Cr+++. Thus the local symmetry
can still be described as €31,109 but with a different
distortion.

We shall now develop the basic ideas of group theory

and use them to investigate the C distorted cubic, type

3i'

of symmetry and apply the results to our resonance problem.

11

Group and Representation Theory

 

Groups. Group theory is an extensive topic in modern
algebra. Most of the literature is highly mathematical in
nature, the physical application aspect having been somewhat
. 107 , .68 .

neglected. Wigner and Lamont are two of the best standard
references combining a good mathematical introduction with

. . . . . 2
phySical applications. Several other texts5 33' 34 6
and articles6 referenced in the bibliography were invaluable
aids in the following discussions.

A group G consists of a set of elements, A,B,C.....
which may be real or complex scalar functions, vectors.
matrices, or any type of operator. The elements may be
combined according to certain rules of "multiplication."

(l) Multiplication is closed (i.e., the product of any
two elements is always in the group} and single valued.

(2) Multiplication is associative.

(3) Every group contains the identity element E (e.g.
EA = AB = A).

. , —l
(4) Every element has an inverse (e.g. AA = E where
-1 _ .
A is the inverse of A).
The elements of a group do not necessarily commute.

If the elements do, the group is termed Abelian.

The group multiplication properties can be exhibited

12

in a table form. Consider the group of six elements, E.A,

B.C,J.F with the multiplication table:

 

 

 

..._EIB B 0.1-?

EEABCJF

A A E J F B C Example: BA = F and

B B F E J C A (2.1)

C C J F E A B AB = J. The group

J J C A B F E

F F B C A E J is not Abelian.

The table may be rewritten so that E's appear on the diagonal.

E A B C J F

E E A B C J F

A A E J F B C

B B F E J C A (2.2)

C C J F E A B

F F B C A E J

J J C A B F E

 

Some basic terms must be defined:

1) g = the order of the group (i.e. the number of elements
in the group).

2) n = the order of the element (i.e. An = E where no
number less than n satisfies the equation. The order of an
element is always a divisor of the order of the group.).

3) A subgroup is a subset of G which itself forms a
group.

4) A' is conjugate to A if SAS—l = A' and A and S
belong to G.

5) The set of all elements conjugate to an element of G

forms a subset of G known as a class. It follows that all

the elements of a class have the same order. An invariant

l3

subgroup is one that is made out of self—conjugate elements.

“Representations. Given a group G, a matrix representation
of the elements of G may be constructed. For each group
element one matrix is chosen such that the multiplication
table is the same as that of the group elements. Such a
one to one correspondence is called an isomorphism. Denote
the matrix corresponding to the group element A by D(A),

B by D(B), etc. For the group described by (2.1), a matrix

representation then is (it is not unique):

0 l

1 o -l/2 V372 ’-1/2 4/372

D(A) = 0 -1 D(B) = V372 1/2 D(C) = 4/572 1/2 (2.3)
-1/2 V37? -1/2 413/2 1 o

D(J) = -W§72 -1/2 D(F) = V372 -1/2 D(E) =

Once a representation has been found, it is possible
to generate "new" representations that obey the multipli-
cation tables by using matrix transforms known as similarity
transformations. These "new" matrices, since they are not
unique, are termed equivalent matrices. If R is a general
element of a group and D(R) is its matrix representation.

the transformation is

(5):.(12) (s) '1 = D' (R) (2.4)
and is carried out for all the elements of the group. (S)

does not have to be a matrix in the representation. It is

l4

always possible to transform a matrix representation so that

all the representation matrices of the group elements are

simultaneously unitary, that is the complex conjugate trans-

pose of the matrix is equal to the inverse of the matrix.
0* = 0* 0*
The sum of the diagonal elements of a matrix is

called the trace, spur, or. in group theory, the character

 

of the matrix. A similarity transformation leaves the
character of a matrix invariant: the determinant is also
left invariant.

There is a method other than inspection for writing
a matrix representation of a group by just using the multi-
plication table. This representation is constructed by first
rewriting the multiplication so that E's appear on the
diagonal. Then the representation matrix of the element
is obtained by replacing that element in the table by l and
every other element by zero. From (2.2), the representation

for A is:

010000
100000
(A) = 000001 (2.6)
000010
000100
001000

This is called the regular representation of A. The other

 

elements can be written down in a similar fashion. By the

15

proper similarity transformation, it might be possible to
completely diagonalize this matrix. Failing in this we try
to form blocks along the diagonal, then try to diagonalize
each block. In (2.6) there is already a small block of 2x2
elements in the left hand upper corner and a large block of
4x4 elements in the lower corner. Let us diagonalize the

2x2 block:

Construction of the similarity transformation can be done

immediately. The secular equation23 is

(O-k)g' + n' = O
(2.8)
g" + (0-1)n' = 0
and we are free to pick a normalizing condition
£2 -+ ‘n2 = l. (2.9)
For i = 1 (g) = (1N2) s Ix'> (2.10)
T)" l/‘VY
For >\ = "l S" _ l/W 2 n

(2.10) and (2.11) are called the basis vectors. Notice they
are orthogonal and hence Span a two-dimensional subspace.

The diagonalization just performed can be interpreted as

16

a rotation of the basis vectors in two-dimensional subspace.
If S is the unitary matrix (that is a rotation matrix which

keeps an orthogonal system orthogonal) that transforms(%)to

n)
(g:) I then
H

0 l -l _ 1 O
isiil 0) (s) ’ (o -)
. . . —l 24
W111 transform A to a diagonalized form. S can be shown

to be a matrix made from the transformed column basis vectors.

3'1 = (lx'> iy°>) = (if); -53) (2'12)

Since S is unitary. S.1 = S* («rindicates transpose) and thus

S __= IN? IN?)
-1/v5 1/1/17/

By permuting the elements B.C.J.F. in the multiplication

(2.13)

table. still keeping the E's on the diagonal. the 4x4 matrix

can be reduced into smaller blocks.

0001 ’0100
0010 1000
0100 ‘> 0001 (2'14)
1000 0010

This allows us to work with two two—dimensional parts of the
'4' dimensional subspace. Then the similarity transformation

for A in block form of (2.7) and (2.14) is. if

l7

1N2 1/412“ 0 0 0
-i/v2' 1/«I2‘ 0 0 0
0

000

0 0 1/V2 1/V2’ _
0 0 -1/v2 1N2 0 0 ' (S) (2'15)
0 0 0 0 1/V2‘ 1/V'2‘
0 0 0 0 —1/V2‘ 1/V2’.
then
0 1 0 0 0 0 .1 0 0 0 0 0
1 0 0 0 0 0 _1 ’ 0 -1 0 0 0 0
(s) 0 0 0 1 0 0 (s) = 0 0 1 0 0 0 (2.16)
0 0 1 0 0 0 0 0 0 -1 0 0
0 0 0 0 0 1 0 0 0 0 1 0
0 0 0 0 1 0 0 0 0 0 0 -1

This similarity transformation is then applied to the regular
representation of each of the other elements. The direct

sum rule may be illustrated for (B)reg:

-l/2 V372 '\
N372 1/2

-1/2 V§/2
’vI/z 1 /2

-1

-1/2 «1‘3/2 a, {-1/2 “(i/2 a. (1) e (-1) (2.17)
«1372 1/2 M372 1/2

Inspecting the results of (2.15) and (2.17) shows that the
same two-dimensional matrix occurs twice; two different one-
dimensional matrices occur once. The reason for not
interpreting (2.15) as six one-dimensional matrices is that

it is impossible to completely diagonalize all the regular

18

representations simultaneously as (2.17) clearly shows.

If we were to write the regular representations for all the
elements and then perfbrm the similarity transformation as
shown above, each matrix would have at most two two-dimensional
matrices. the remainder being one-dimensional. These matrices
then are said to be irreducible matrix representations, that
is their dimension cannot be reduced. It can be shown33 that
the regular representation contains each possible irreducible
representation a number of times equal to the dimension of

the irreducible representation. Our example group has two
one-dimensional irreducible representations and one two-
dimensional irreducible representation. A by—product of

this result is

l

2
A - g (2.18)

yhi

th

where fix is the dimension of the I irreducible representation

 

and g is the order of the group. Schur's lemma33 states
that if D(R) is an irreducible representation, then if and only

if for all R. AD(R) = D(R)A, then A = constant - l.

Hilbert Space. A complete unitary space is called a Hilbert
space. The unitary Spaces of a finite dimension are always
complete. In our example the basis vectors of the entire

group span a four—dimensional Hilbert Space. By finding the

irreducible matrices and the correSponding basis vectors

19

the space was subdivided into one two—dimensional subspace
and two one-dimensional subspaces. (This is the equivalent
of subdividing relativistic four—dimensional space
[Minkowski space] into an x-y plane. the z-axis. and the time
axis.) Each of these spaces is invariant under the action
of the group.

Among the irreducible representations there is always
a trivial one, the unit representation, given by a single
base function invariant under transformations of the group.
In the unit representation all the elements of the group
are represented by the one—dimensional unit matrix and hence

all the characters equal one.

The Character Table. The fact that the character of a matrix
is invariant under a similarity transformation suggests it
might be seen that the character of the irreducible repre-
sentations is significant; in fact a great deal of information
may be gleaned in this way. The table of the characters of

our example, which is commonly called Group 32. is now

 

 

listed.
E ABC JF
r1 1 1 1
F2 1 ’1 l (2.19)
(*3 2 0 -1

Since elements in the same class have the same character.

20

only the characters for each class are listed. (Elements
in the same class are related by a similarity transformation.)
[Fl refers to one of our one-dimensional representations.
P2 to the other. f3 is the notation of the two-dimensional
representation. (The notation is that of Bethe. There is
another common notation, the Mulliken notation. The two
notations are cross referenced in Table l of the appendix.)
Several important relations between group characters
can now be written.108 Remembering k is the number of
classes. 2 is the dimension of the irreducible representations.
g is the order of the group, hi is the number of elements in
th

the class Ci' and Xa(Ci) is the character of the d

. .th
representation, 1 class. we have

5

ll
l><
IN

k a
gOGB ) = i hi X (Ci) X (Ci) (2.20)

*
O.

_ t a
(zlci).§(cj)) - : h. x (ci) X (cj) (2.21)

gfiij

These relations are known as the orthogonality relations.
Relation (2.20) implies that if each element in a row of a
character table is considered as a vector component, then
the rows are mutually orthogonal with respect to one another.
These vectors are normalized to one by the appearance of the
g's and h's in the formula. Relation (2.21) is a similar

orthogonality condition on the columns. Any general

21

representation of the group with the vector character I of
the group can be resolved into its component irreducible
representations by the same device as used in vector algebra.

the scalar product:

CY.

_ a 1 k
d — (§ ,_§) =-§ 2 hi X (Ci) X (Ci) (2.22)

CI

One other orthogonality relation is important. For

irreducible representations. G. of the same group

(2.23)

a =.9
: G' G. h“ 6ilakmadfi

Prpjection Operators.42' 35 If there exists a particular
vector in the Hilbert space spanned by the basis vectors

of the irreducible representations. then it is reasonable

to ask for the components of that vector along each of the
basis vectors. If the ath irreducible matrix representation
is (Da(A» for an element A, then the component of ¢ along the

ith basis vector in the subspace of the dth representation is

9 *
0? = z Da(A).,A¢ (2.24)
i A 13

where j is fixed at any value. If. as is usually the case.
only a character table is available, we must settle for the
total component of¢ in the ath SubSpace.

0a = XQ(A)*A¢ (2.25)

22

Direct Product of Representations. Consider the vectors

 

spanning the subspaces of two irreducible representations

B
J

can serve as basis functions of a new representation equal

' o
of the same group. l¢a> and (ya >. The products (1 w

in dimension to the product of the dimensions of the
irreducible representations. The characters of the new
representation are equal to the products of the characters
of the two component representations. that is XaVZXB(G) =
Xa(G)XB(G). The product of a representation with itself
can immediately be decomposed into two representations, one

with symmetric. the other with antisymmetric functions.

The character for the symmetric product is

[X2](G) E -:- [X(G)]2 + X<G2)i (2.26)
. )
and that for the antisymmetric product is
f 2 - l 2 2
ix (G) = -2- [X(G)] - X(G) . (2.27)

J C’
No antisymmetric product can be formed if the |¢.> and
the ($6) are the same functions.
As an example of representation products consider
. o a. . th
the integral ~[widq where (1 is the baSis vector of the a
representation of G. The integral is taken over all space

and hence is invariant to any symmetry transformations.

Then .
a 1 a a a
‘iwi dq -‘iGwi dq —\[E Gik wk dq

23

and if we sum over all members of the group

gfwiadq = i: J g Gikc (l/kadq, (2.28)
but by (2.23) this is zero is a is not the unit representation.
Hence the product of an integral of three functions must
also transform as the unit representation to be different
from zero. Since the product of a representation with it-
self contains a unit representation. then for the integral
jyiij, the product representations according to which

vi and w

transform must contain a representation under

J

which V transforms.

 

. , (1 t
Direct Product of Groups. If functions 1w > span the a h
. _ [.tfls‘ th
representation of the group G and ~ ) span the B
-2 C1,
representation of group H. the products of the form ¢iwj

are the basis functions of a 1Q° 28 dimensional irreducible
representation of the group G 3 H. The characters of this
representation are obtained by multiplying together the
characters of the original representations so that if
element C belongs to G 8 H, 6' belongs to G, and H' belongs
to H and C = G'H) then ,

Bur) (2.29)

We) = when)

where 0 will be an irreducible representation of G® H.

An example of a product group is

24
C = C ® C.. (2.30)

The corresponding character tables54 for Ci and C are:

 

 

 

3

C E I C E C C2

i 3 3 3
A 1 l A l l l

9
A l -l E 1 52 -e (2'31)

u

1 -e 62

 

e = exp #1/3

where I is the inversion operator. C3 is a rotation of 1200

around the three-fold axis. C2 is the rotation of 2400 around

3
the same three-fold axis. All the representations are one-
dimensional since the representations of E are all one.

From g = 2922. the classes all contain only one element: the

 

two groups are thus Abelian. The product group is:
2
C I C I
w? L3.)
C3i E C3 C2y I S6 S6
A l l l l 1 1
19
E 1 e2 -e 1 e2 -e (2.32)
9
l -e e2 l -e 62
A 1 l 1 -1 -1 -1
lu
Eu 1 £2 —6 -1 _62 e
1 -e 62 -1 6 -€2

 

25

Double Valued Representations. Since the group elements of

the axial rotational group are Abelian the representations

are one-dimensional and thus X(¢1)X(¢2) must equal X(¢1+¢2).34

. ' ¢
We also must have X(2v) = X(O): thus X(¢) = e1m where m
is an integer corresponding to the mth representation. The

rotations by an angle ¢ about different axes belong to the

 

same class of the full rotation group. Thus36
j _ im¢ _ sin (j + l/2)¢
X (¢) - E-Je - sin(l/2)¢ . (2.33)

Frequently the total angular momentum j takes on
nonintegral values, for example j = 1/2. Using equation

(2.33) for the character of the full rotation group.

sin (j + 1/2)¢

J - _ 21
X (¢ + 2W) - ( 1) sin (1/2)¢

(2.34)

Xl/2(¢ + n) = -x1/2(¢). (2.35)
This does not correspond to the effect that is expected by a
rotation of 360°. The introduction of spin has made the
characters double valued. If a spin system corresponding to
half integral values is to be used, double groups must be
employed. In order to do this a new element R = -E is
introduced to the old groups. Koster54 has tabulated the
double groups for all the crystal pointegroups. Several
rules have been written_down for calculating the double

groups.38 These rules and the character orthogonality

26

relations allow us to construct the double-point group of

C31.

Application. The group and representation theory can be
applied to determine the energy level degeneracies of a

paramagnetic ion in an electric field of a specified symmetry.

 

Consider Fe+++. 685/2. ion as a substitutional ion for
A1+++ in AlCl3-6HZO. The local symmetry in such a site is
c31.1°9' 5 Using (2.33) to find the characters of the
rotation group correSponding to the group elements of c3i'
it is seen that
E C C2 I S5 S R RC RC RI R85 .R8
| 3 3 6 6 3 3 6 6
(2.36)
I 6 O 0 6 0 O -6 0 0 -6 0 0 .

D5/2
The identity character is 6 because of the dimensionality of
the 5/2 manifold. We have used the fact that the electrons
of interest are d electrons. and the inversion operation
shows no effect on the characters of the rotation group
whose corresponding basis functions display even parity.

The decomposition (2.22) of the reducible representation
yields

D = flag e 2Qg 3 2V . (L3H

5/2 69

This is shown in Figure 2.

. . ++_.
The case for Cr+++, 4F substituting for Al +.

3/2'

27

CHARACTER TABLE FOR ‘3

 

 

 

 

 

 

2 5 2 5

C3 C3 I 86 86 R RC3 RC3 R RS6 RS6
["19 1 1 l l l l l 1 l l l
[‘29 62 -e l e2 -e l 62 —e l 62 -e
1“‘39 -e 62 1 -e 62 l -e 62 1 —e e2
[“49 e e2 1 e 62 -1 -e —e2 -l -e _62
r59 -e2 -e l -e2 -e -1 £2 e -1 62 e
[‘69 -l l l —l l -1 l -l -l 1 -1
Flu 1 1 —l -l -l 1 l l -l -l -l
.7211 62 -€ -1 -62 e l 62 -e —l -ez e
[“3“ -e 62 -l e -e2 l -e 62 -l e -e2
F4u e 62 -l -<—: -<—:2 -l -e -<—:2 l e 62
r5u -62 -e -l 62 e -1 62 e l -e2 -e
r6u -l 1 -l l -1 -l 1 -1 1 -1 1

Note here that e = er/3.

 

139 + [Lg <‘IIIIII""I

 

a 2Psg
S
T59 +119
Figure 2. Fe+++ Energy Diagram.

5/2

3/2

I/2

-l/2

-3/2

—5/2

29

can be handled in a similar manner. One must remember that
the values of the characters of the rotation group for half

integral angular momentum (58.1) are double valued.

 

2 5 5
In c3 c3 I 56 36 R RC3 RC3 RI R86 R86
(2.38)
133/2'4 -1 1 4 -1 1 -4 1 -1 -4 1 -1
Then
+++

The same results may be obtained for either Cr
+++ . . . .

or Fe by assuming an aXially distorted cubic symmetry. We

, . +++ , . , .
shall illustrate this for Cr . First a cubic field is
considered: then an axial distortion along one of the body
diagonals of the cube is carried out. The complete
character table can be found for the cubic group in Bethe's

. 6 . . . 54

article or Koster s compilation. From Table 12 of
Bethe's article (or by computing in a similar manner as we

, +++ +++
did for Fe and Cr )

D3 =r2 cal—'4 efs D =r (2.40)

3/2 8
where I; is one-dimensional, r; is three-dimensional, and
f; is three-dimensional. (We note that D3 is seven—dimension—
al which serves as a check to see if the reduction is

correct.) The ground state. which must be identified by

some means other than group theory. is f}. The product

30

representation r; x f; is the representation of an LS
coupled state for the ground multiplet. However, the product
may be reduced into irreducible representation, I; x I; = f&.
From Hund's rule we note that the lowest j multiplet is

j = 3/2 which indicates directly that the ground multiplet
transforms as 1;. These results are indicated in Figure 3.

For reference we note also that

[gxrll =16 4317 e 2TB (2.41)
1.8 xii; = J—g e _f; 6 2f; (2.42)

which gives the splitting of the higher multiplets.

New the cube is distorted along one of its body diagonals.
This gives the symmetry of double group 32, which has the
following character: (Note the group 32 multiplication
tables and irreducible representation have been derived

(2.1). (2.3) and (2.19).)

 

E RE 2c3 2Rc3 3c2 3RC2
ill. 1 1 1 1 1 1

—A.2 1 1 1 1 —1 -1

Ji.3 2 2 —1 -1 0 0 (2°43)
Jl 1 —1 1 -1 1 -1

4

.A_5 1 -1 1 -1 -1 1

/ _ _

_\.6 2 2 1 1 0 0

 

Figure 3 .

I‘.

 

' 8 [)g

 

+++
Cr

 

 

 

 

 

Energy Diagram.

 

 

 

1".
(3(39
4 F
1‘.
I()l)g
3/2
1/2
12 _
-112
-3/2

j

32

Those of the corresponding cubic operators are:

(2.44)

11
F2
["3
f; 3 3 0 0 1 1
F5
F6
F7

[‘1
‘8

 

C2 is the two-fold rotation operator along a cube face:
C3 is the three-fold rotation operator along a cube diagonal.

Thus

1; =A4 +15 +146.

since again the character of the rotations forming a reducible

representation are double valued. Note thatj\6->>fgg.

\ - a .
y5—+>.fgg, and/\4 I49 on the diagram.

90. 107. 31

Coupling Coefficients. The product of two repre-

sentations in general is reducible. Consider a general

representation product

‘1‘:

X<a x 5*) 6

(C1) X“(ci) x (ci) (2.45)

then using (2.20)

33

k *
a = 2 h X(a X B )

.1 0*
G 9 i=1 i (ci)x (01). (2.46)

The number of times the identity representation appears then

is just

k
a = 2 h. xa(c.)x
i=1 1 1

8*
(Ci) (2.47)

alt:

= 565 (2.48)

The last statement is just (2.20). The do of (2.46) are
. . . a ‘ .
the expanSion coeffiCients of D (Ci) x DB<Ci)' that 18

Da x D6 = Z a D6 =‘ Z (dfiJ)DO. (2.49)
0 0 o

(The matrix multiplication here is the same as that used in

multiplying the character tables of C1 and C together (2.30).)

3
Equation (2.49) is called the Clebsch-Gordon series. As
previously indicated the products of the vectors spanning
th th .

the two subsets of the a and S representation can serve
as a basis of space spanned by the product representations.
The question is how to pick out the orthonormal set which
spans each irreducible subspace by taking the proper linear
combination of the product functions. Calling the correct

set ¢, then we have (summing over repeated indices)

(911(1):
5

wjawkfi <Gj.BklkT(i) s> (2.50)

as the correct linear combination. The coefficients are

Clebsch-Gordon coefficients. A refers to the Ath representation

34

spanned by Da x D6; s is the sth vector necessary to span
the subspace of DA. If the product representation contains
the Ath representation m times, then there will be m 'correct
linear combinations‘ which we distinguish by T(%) labeling.
(The total number of 'correct combinations' must equal the

th

product of the dimensionality of the a and Bth represen-

tations.) The inverse relation.

(jawka = ¢:T‘))< )T())laj.ak> (2.51)

implies that

(1'1” (N )s' )aj.Bk><aj.Bk|>\T(MS.\/= 6 (2.52)

ii'5T(i)1'()')5ss'
and

(01'.kaI)T())S><ii(i)slcj.6k> = bjj. fikk. . .(2-53)

The transformations are again a rotation in Hilbert space and

the Clebsch-Gordon coefficients are hence chosen to be unitary.

that is
(17(1) ldj,Bk> = (aj.BklM(i)>*. (2.54)
Letting a member of the group G act on ¢sx%(%). we have
on M”) = 4 .MWD ,MMNG). (2.55)
s s s S
. AT(A) .
We can pick Ds's to be independent of T(%). Then
G0 )vr(S) = <b 'AT(7\) D I')\'t'()\)
s s s s s
a . , A A
= (P). ¢E<O£J.Ekl)vr(?\)s > Ds'sT( )(G) (2.56)

35

also
G¢:T(S) = G[¢i v51<ai,fillir(i)é>
= w u D G(G) 4 fl 0 B(a) <41 BQJ)T(1) > (2 57)
j ji k k2 ' 3 ° '

.- up,
Since the wj wk are independent. from (2.56) and (2.57)

Dj:(G)Dk:(G)<ai,Fibrous)-= <c‘zj,Bklm-(7\)s> 03):”)(6). (2.58)

We can then write the irreducible representation in terms of

the product representation by using (2.52).

.» B I
<i'r' (i')s"laj,5k> Dj:(c)nk'g(c)<'..81.Bilm(7\)s> =

AT(A)
6 6
Ds's AA“ T(A)T'(A') s's" (2.59)

The inversion of (2.59) can also be written using (2.58)
with (2.53).

The more familiar Clebsch-Gordon coefficient is a
specialized case. The characters are assumed to be real.
This implies that each representation is equivalent to its
complex conjugate and that the Clebsch-Gordon coefficients
will be real. We have also pointed out that there are 'several
correct linear combinations' if the product representation
contains any certain irreducible representation more than
once. As a result, such coefficients will vary by an arbi-

trary phase factor. Equation (2.50) then can be rewritten

36

in a more standard form.
(1112JM> = |11m1>l12m 2><1112m 1 m2lsJM> (2.60)
= lili2m lm2><jlj2m lm m2|sJM> (2.61)

(s = jlj2 and is sometimes not written on the right side.)

where the phase is fixed by the relations:

<111211(jl-J)IJJ> is real

 

and JiisJMD = ’VFJ(J+l)-M(M:1) lsJ,n:1>. (2.62)
Clebsch-Gordon coefficients are tabulated in a publication of
the Atomic Energy Commission.20

Wigner coefficients, or the '3j' symbols, are related

directly to the Clebsch-Gordon coefficients by:

J

_ —j2 +M
= i 1)) l (jljzmlmzlJND (2.63)

1 m2 —M ’V2J+l

 

By consideration of the possible dimensionalities of a product

representation. it is seen that
lJ1’3213.JIS_31+32
and by the orthogonality of the Hilbert space (or of the

spherical harmonics) that

m1 +1112 = M

The Wigner coefficients are:

37

a) invariant under a circular permutation of the columns.
b) multiplied by (-1)jljzJ by a permutation of two
columns or under a sign change of all the m's.
Racah coefficients V(abc,dBy) are defined in the

following manner:

-b-c a b c (-l)c-Y
V(abc,aB)) = (-1)a , ) - (abafiic.-)>
a B V 2c+l

 

Coupling of the product of more than two representa-
tions follows in a similar fashion. A table of recursion
relations can be found in Messiah77 for Racah W coefficients
and 6j symbols (for the product of three irreducible
representations) and 9j symbols (for the product of four

irreducible representations).

38

General Hamiltonian

Introduction. For an exact solution of the Schroedinger
equation, one must know the Hamiltonian which expresses the
interaction of all the electrons with themselves and their
environment. At the present time such a Hamiltonian cannot
be written in closed form. Instead the Hamiltonian is
separated into component parts expressing various inter-
actions; the magnitude of each term is estimated and per-
turbation theory is applied accordingly.

One problem arises in such a separation: Is the
paramagnetic ion to be considered 'free'. residing in a
crystalline field laid down by its neighbors (crystal
field theory) or must the local bonding be considered,
that is, must we treat in detail the interaction between
the paramagnetic center and its neighbors (ligand field
theory)? The latter approach is usually too difficult for
exact solution. The problem reduces to the question whether

5'26'43 Usually it is

the bonding is ionic or co—valent.
a mixture of both.

Using the former approach, in the absence of magnetic

field, where the summation is over the paramagnetic Sites

39

3<i 2 2 2
H = z [(Pi/Zm) — (Ze /r) + (e /rij)]
i1

+ [1. 1, s .bijgi) + c. s.-s] (2.64)

11-1°-1 -1 11-1 -1

+ Kai's

3 , . 5
/rij) - 3(§i.rij)(§j.£ij)/rij] - [ei¢(ri)]-

J

Inside the first set of brackets is the expression for the
kinetic energy, the coulomb interaction between the nucleus
and the electrons, and the coulomb interaction between the
electrons. It is called the free ion term. The second
bracketed term contains the interaction between the
electrons' orbital and Spin magnetic moment, the orbit—
orbit interaction, and the exchange effect for the electrons.
The term contained in the third bracket is the magnetic
dipole interaction between the spins. The term in the fourth
bracket is the crystal field interaction where V2¢= 0
(Laplace's homogeneous equation) for the region around

the paramagnetic ion.

The Crystal Field. A crystalline electric field of a definite
symmetry will cause Splitting of the terms of the unperturbed
ion. The number of components into which a term of a free
ion is split will increase with decreasing local symmetry

of the electric field.

We shall first expand the crystalline field in a

40

series of spherical harmonics. The solution to Laplace's

equation.V2¢ = 0 is
9. -£-1
¢ = . Y n .

Aim and Bfm are expansion coefficients; r is the radius about
the origin of the paramagnetic ion. For solutions finite
at the origin, Bim = 0. YTm are spherical harmonics, with
phase factors chosen in the Slater manner, that is YQm =
(-l)mY12_m. Only the effect of the crystalline field on d
electrons is of interest. Since the wavefunctions of these
electrons can be expanded in a series of d wavefunctions and
the product of two d representations is four-dimensional.
then only terms up to.£= 4 need be included. The reason

. . . . 2 I
for this is that matrix elements of the form <IJ|A1mr YflmLZJ\
will vanish for12>4 because of (2.28). All terms of odd ,6
must vanish since in order for the integral of the product
of three spherical harmonics not to vanish, the sum of the
.2's must be even. (If the crystal has no inversion center.
other configurations may be admixed allowing oddfifs.) The
series must also transform according to the symmetry of the
site under consideration, C . in our problem. The group

31

operator C (1200 rotation) implies that

3
. eimZTT/3.
.Lni (,m

but the group element C2 (-1200 rotation) implies that

3

41

_ —im2v/3
Ygum - YLme °

Therefore m=3t. where t is an integer.

. m m
¢ 0 = _ ’ = - * 0
Since is real and 22m ( 1) Y2,-m then.AZm ( 1) Aim

Thus, neglecting the constant energy term, we have

2 4 4 e 4
¢ — Azor YZO + A4Or Y4O + A43r Y43 - A43r Y4’_3 (2.66)

The coefficients must be determined by calculating the electro-
static potential produced by the nearest neighbors (in our

case, the dipoles of the 6H 0 molecules) expressing this

2
potential in terms of Spherical harmonics. To do this, one
uses the expansion for the dipole (the dipole being the

origin and the z—axis being directed along the direction

of the dipole)

 

¢ = , .
e leO(9 ¢) (2 67)

e
4weh
where p is the dipole moment. Each of the six expansions

can be transformed to the origin corresponding to the para-
magnetic site such that the z-axis is as Shown in figure 1.
Further calculation of ¢ is difficult since the orientation
of the dipole is unknown. If an impurity is located at the

C3i symmetry center. actual distortion must also be known.

Interaction Terms. The second and third bracketed terms of

 

equation (2.64) represent interactions of the spin ensemble

with itself. They are typically about 102 and 1 cm-1

42

respectively for the transition elements. These terms are

more fully discussed under the consideration of T processes

2
in which their action is of large importance. In general
only the spin—orbit interaction term will dominate. It

should be noted that all interactions between our electron

spin ensemble and any nuclear ensemble have been ignored.

Coulomb Term. The first bracketed term is of the order of

105cm-1. Besides the coulomb interaction it contains the

 

kinetic energy of the electrons. Its effect is to shift
all energy levels by a constant energy and hence it will

be ignored.

Zeeman Energy. Upon application of a magnetic field. the

 

interaction of the field and the magnetic moment of electrons

produces what is called the Zeeman term of the Hamiltonian:

' + ' . .
95 £35]; -S-i B 12 (-L-i Eli) (2 68)

Here 9 is the spectroscopic splitting factor and is ~42.

H is the applied magnetic field, and B is the Bohr magneton.
The Bohr magneton is defined in the m.k.s. system as

eh/ch. Here again it has been assumed that only one 'type'
of spin is present. and hence no nuclear spin term is in-

cluded.

Calculation of the Hamiltonian. The problem now is to

43

construct a plan for perturbing the free ion Hamiltonian.
One must know the order of magnitude of the various terms of
(2.64). Since they cannot be calculated. as yet, from first
principles. one must rely on experimental observations.

Such observations reveal that the problem usually falls

into one of three categories.

1) The crystal fields are very large (""10"4 cm‘l). The
coupling between the orbit and electron is overcome. The
spin-orbit term then is ignored.

2) The crystal fields are of the order of magnitude of
the spin-orbit coupling. The field and the spin-orbit
coupling are then considered as simultaneous perturbations.

3) The crystal field is small compared to spin-orbit
coupling. J is a good quantum number in this case. The
crystalline field is applied as the final perturbation.

Cr+++, like most iron transition elements, falls
between cases 1) and 2). Fe+++, being an S state ion, is
an exceptional case. Both Cr+++ and Fe+++ can be handled
in one of two general approaches that are commonly used
in.constructing transition ion Hamiltonians. The first
method, the Koster—Statz Hamiltonian,ssis a more fundamental
approach. Here one must consider various possible models

of crystalline environments and then check the calculations

against empirical results. Unfortunately, the experimental

44

results compiled at the present are usually insufficient to
complete the calculation in full. One must settle for some
approximation.

An alternate method is the Spin Hamiltonian approxi-
mation. This method represents a phenomenological approach
to the problem. Most resonance data can be described with
Spin Hamiltonians in a relatively simple way without re-
quiring a detailed knowledge of complicated effects such as
crystalline field and spin-orbit coupling. since the parameters
involved are directly measurable. The danger in the last
approach is that the picture may be grossly oversimplified.54
We shall outline the two methods using our paramagnetic

systems.

‘Eppper-Statz Hamiltonian. The Koster—Statz Hamiltonian

‘being a more 'exact' form in return requires more information.
In this form of the Hamiltonian. the exact perturbations
appear (e.g., g, A, and A appear as scalars in the pertur—
bations gfiggg, ijg, and AIfiS, where the Algg term repre—
sents an interaction between the nuclear and electron spins.).
The actual wavefunctions, or in lieu of these the best
approximation to them, are used. The wavefunctions are
associated with the irreducible representations under which

they transform. This approach to the Hamiltonian involves

45

the use of the projection operator of equation (2.25).
, +++ , ,
ConSider Fe which has a spin of 5/2. We shall need to

know how Hz’Sz and H+,S+ transform to evaluate the terms

gfig-g = get - (H+S_ + S_H+) + H2821. (2.69)

Using the projection operator (2.25) and the character table

for C we find the projections of Hz to within a multi—

3i'

plicative constant are

¢1(H ) = g xl(A)* A¢ = g 1(a)*H
9 1
= 2 X (A)H = gH (2.70)
A=1 z z
9
¢2(H ) = 2 4(1 + e2 -e) H = 0 (2.71)
z A=1 z

In this manner it is seen that Hz transforms asi 1 only.
These projections can also be seen by inspecting character
64 . . . .
tables that contain the baSis functions for the various
representations. Thus S2 and H2 transform aS'FIg; H ,S

transform as[1 7 and H .S transform asi—1 .
2g + + 3g

The wavefunctions transform (2.37) as (rig'f‘5g)'
2 2 l 2
(f4gnfwsg). (F6gif6g) where the l and 2 are used to separate

the levels since each representation occurs twice. The
Kramers' degeneracies are easily identified Since the characters

are complex conjugates of one another, which is the necessary

46

condition for time conjugate states.39 and they are paired
in the parentheses. Applying the time conjugate operations

[(A.9) and (A.10)] to a Kramers' conjugate pair. we see

4‘3.

_ 2 _ l 2
J‘sg KI‘ig 4759 KI‘l -F

g 69 - 6g

and (2.72)

2 2 _ .2 __ 2 2 1 _ 1 __ 1 2 1 _ 2 __
K F4g-Kf'sg— 1’49 K F4g—Kfsg— {‘49 K Fég—Kfég- F69.

and since (gfififig) under time inversion changes sign. then

typically

*
Gfiglgasoelfég = - ffiglgfia-slfép (2.73)

Following on with the use of (2.72) to establish relations
between as many matrix elements as possible and remembering
that any matrix element which transforms like <f;lf;lf;>
will be zero if f: xnf; does not containwr; [equation
(2.28)]. we write down the matrix Hamiltonian in as exact

a form as possible without the knowledge of the actual eigen-
function of the problem. It should also be noted that the
matrix is Hermetian. The remaining matrix elements are
then evaluated with wavefunctions that best approximate the
given problem. They. of course, must transform according
to the correct irreducible representations. The method can
be adapted to the use of molecular orbitals as is done in

ligand field theory.5

47

Spin Hamiltonian. The second approach is the one which has
been used most extensively for fitting experimental results.
Its success justifies its use although the approximation
might seem severe. Even though higher lying energy levels
which do not participate in paramagnetic resonance may
influence those that do, those levels which participate

in magnetic resonance are treated as if they were isolated.
An effective spin S is assigned to the 'isolated' levels
such that 23+l equals the number of levels participating

in the resonance. For our system the 2S+l levels arise
from an orbital singlet, that is they have a common orbital
quantum number, (L) = 0. This is true for most transition
ions, and it is the usual case treated by the Spin-Hamiltonian
formalism. (Several methods for handling the exceptions
have been proposed.70)

Instead of the 'exact“ wavefunctions which are called
for in the Koster—Statz formalism, a set of free ion wave-
functions which transforms according to the irreducible
~representations of the ZS+1 levels, namely the ground status
in Figures 2 and 3 are chosen. Since, in fact the ion is
not free; the various perturbations must be written in the
most general form possible to account for the effects of its

actual local surroundings. Thus no longer will A and g

48

(nor A in Aggg) appear as scalars. and tensor forms take
their place. The whole Hamiltonian must transform according
to group symmetry. In fact for C3i symmetry the approxi-

mation is made that the local symmetry is actually axial.7

The Spin-Hamiltonian then will consist of a Zeeman term.

SE

1 .
g” 8H2 + -2- (H+S_ + H_S+). (2.74)

where H = H +iH and S = S + is
iX‘y :X—y

and the contribution from terms involving L. We must select
terms similar to those in (2.66)which will be the leading
terms in the contribution of the perturbation involving L.
This can be done by perturbation theory: the leading terms.
as in the usual case, come from the lower order perturbation
theory. Since <0|9J0> = 0, the first contribution comes
from second order perturbation theory. Here we eXpect no
terms in 8 higher than 82. Higher order perturbation terms
must be found if the Hamiltonian does not agree with experi-
mental data as is the case with Fe+++. Cr+++ however can

be fit with only terms up to quadratic. (The actual pertur-
bation calculation is carried out in full in Slichter98 for
the second order. It should be noted that terms like

S(S+l) shift all the levels by a constant amount.)

The crystal field Hamiltonian for C symmetry is

3i

taken into the operator equivalent form as outlined in the

49

method for writing equation (2.66). This calculation can

be facilitated by the use of the irreducible tensor operators

 

of Racah. Tlm' which transform in the same way as Ylm'97
The T establish the relationship to the J . J . J or
1m x y z
Sx' Sy' 82. Using the following commutation results:
[Sx' Sy] = iSz [SY, Sx] = lSY [Sx' $2] = in
2.
[S . Si] - 0 [82. Si] -.1 Si [S+. S_] - 282 (2.75)
[321 5+] = 5+ + S+SZ
and the necessary definitions:
(Si, Tim] =’\/I(£+1)-m(mi1) T}, ml
(2.76)
[82' Tim] = MTIm
n
S+ °c Tnn°
Thus. equation (2.66) becomes
2 2 4 2 2 2 2 4
e¢ — B10(3Sz-S ) + 840(3582-305 82 + 2582-68 + 38 )
(2.77)

3 3
+ B43[s+(zsz+3) + s_(2sz~3)].

This result can be obtained in another way which by
invoking the Wigner-Eckart theorem that operators which
transform in the same way have proportional matrix elements.78
The proportionality constants are called reduced matrix

elements and depend upon the operator and the initial and

50

and final total angular momentum and are indicated by

IIOperatorlI J > These are tabulated69

<Jfinal ,initial °

for some operators and J manifolds. One then proceeds to
replace the cartesian co—ordinates in the potential function
by equivalent operators. that is x by Jx' y by Jy' and 2 by

Jz' Care must be taken to proceed with proper regard for

»the commutation rules of J operators, such as replacing xy
1
— + .

by 2 (JXJY Jny)

Wong's109 results for spin resonance in A1C13-6H20

. . . +++ +++ . .
With the substitution Cr for A1 at C3].- Site fit the

axial spin Hamiltonian

wIH

as + DIS: - s(s+1)] + All _1_:.§ . (2.78)

H

At liquid air temperatures. D = hc(.043) cm”1 and A =

3 ‘1 +++

hc(l.7x10- )cm and g = 1.977. The results for Fe

. +++ , . . . .
in the same Al Site fit the spin Hamiltonian

a 4 4 4 2 35
H - Bg” H + 6(sx + Sy + $2) + D(Sz'iz)
(2.79)
F 4 95 2 .81
. _ -2 —l _ -1
With (a) - hc(l.6x10 )cm , D - hc(.15)cm , and
F = (3.1x102)cm-1. Note for the S state ion it is necessary

to carry almost all the crystal field terms.

One should note that D = 0 for cubic symmetry. Thus

51

D is related to the axial part of the crystalline field.

Recently an expression has been presented for D in terms of

the actual crystalline parameters and wavefunctions.27

52

Lattice Vibration Theory

Introduction. Our difficulty with the Hamiltonian prevents
us from theoretically obtaining numerical expressions for
the spin-lattice relaxation times for our particular experi-
ment: however, it is possible to extract both the temperature
and field dependence from our final expressions. Before
making these calculations, we must discuss lattice vibration
theory in order to fully display the concepts and approxi-
mations which must be used.

We begin by defining a set of three fundamental

translational vectors a_ such that the atomic arrange-

1’3

2’3

3
ment appears identical whether viewed from an arbitrary

origin in the crystal or viewed from any point related to

this origin by

i = £1§1 + l2a2 + 13:13 (2.80)

where 91. 92, and}.3 are integers and the 'af are not
necessarily orthogonal. The choice of these vectors is
somewhat arbitrary. The lattice vectors are termed primitive
if all points, whose atomic arrangement appears the same.
satisfy equation (2.80) so that £1.22. and 03 are integers.
The unit cell is defined as the polyhedra bounded by the

fundamental translational vectors. We shall use primitive

translational vectors and the unit cell they define in

53

order to avoid any ambiguity. If the unit cell contains
more than one atom. an origin must be taken in the unit
cell and a vector is prescribed from the origin to each

atom in the cell.

Lattice Modes. Consider the general 3-dimensional lattice.
Arbitrarily choose an origin. Let &_be the vector to the
1th . . th

unit cell and p be the baSis vector to the b atom
in the cell. The displacement of this atom from equilibrium
is just

étb‘ (_l+_)g) (2.81)

92.1.
whereggfl’b is the actual position of the atom. Now the
potential V in a Taylor series in terms of the displacement
is

28v

2
a v
V((1.0.b) = VO + Rb 0g

’ ° 9.. .
b lb dgfibdgnrb. 9 b

(2.82)

The derivatives are evaluated at the equilibrium position of

the atom, for by the definition of equilibrium position.

——- = o v can be arbitrarily chosen
Ba 0
£13 xflb = £ + 9.

equal to zero since we are free to choose the zero of the

54

potential energy. This equation corresponds to Hooke's

1
Law. V =‘E’kx2-e> §_= -k§. and thus it can be said that

only the harmonic terms are being carried in this calculation.

Following the Hooke's Law analogy further:

22.);

k—)- c 2 (2.83)
5-IL1:>59-l'b'

1.13!

The Hamiltonian can now be written

H -'£ 2 'l P - P 2 G a
‘ 2 flb rub—L'b' —2b 0b 9216' =11: '—L'b"

lib! Alba Lib!

1 .
+ 2 (2.84)

The motion is subject to the Born-vonKarmén boundary condition
which reflects the fact that the motion, as in an infinite
crystal, is subject to no limiting surface condition. This
mathematical artifice can be handled easily by considering

a one-dimensional linear chain of N atoms. The translation
group is Abelian (of. 2.33) and, we therefore have N one-
dimensional representations whose N basis functions differ
only by a phase factor. Noting the wave functions for phonons

are symmetric if

= iq \
Iola2 ..... aN—laN’> e I o2o3 ..... oNdl/ (2.85)
then
_ i2q .
[ala2.....aN_loNC> — e I c304 ..... oNalaz (2.86)

and so on until

55

_ i(N—1)q
l 2 N-l N — e I oNal ..... GN_2aN_lGN_>. (2.87)

The trick is to go one step further as if the chain formed

a large circle with no boundaries. Thus

I01 01 ..... OLN > = eiqu a on ..... OLN > (2.88)
which implies q = 2vn/N. Since eiq = eiZTm/N
= ei2w(n+N)/N the distinct values of q are said to be those
for which 0 S.n S_N; Note q, called the wave vector. is

often written as k and equals ZF/A. A linear function which

transforms as (2.85). (2.86), and (2.87) is

..._1._ iqfl _ _l__ iq 12q iNq
A —~fN E dee - {N (Tie + aze + ... + aNe )(2.89)

where N is the total number of atoms.

These functions and their conjugates are the normal
coordinates of the one-dimensional problem. The analog
in three dimensions will give a normal coordinate for each
unit cell.

Rather than find the P conjugate to A in classical
mechanics, it is advantageous to go directly to the operator
equivalents for quantum mechanics since quantum mechanical
results are desired. If A13 a 'spatial position' operator,
the P is the proper momentum operator conjugate to A if and
only if

[A’P] =ih I

56

where‘fi = h/Zn.

Then Pq where [5tp1]= 5 i h (2.90)

for

q q
_ :1; z e]. (ql-q I) l h 6 ' .
N 2_ qq (2.91)
Also
.. ’ 1 ~ +qu2 1 1 ~ —iq.0_
P = —-——— Z = s s
Q VN— q Pq e and d1 'Ffi' Zq Aq e (2 92)

In the above equations implicit use is made of a Fourier

inverse theorem. namely

 

23 eiq(.Q-Q') Ngl eiZWn(Q_£.)/N = e[l_ei27r(Q-Q')]
q “=0 1 _ eiZ‘IT(9.-Q.')/N
z ”N 551-1)) n <N (2.93)
and
2181(q—q.w = E: eizTth—n' ”N = 1/N 5 _q. . (2.94)

Utilizing only the ideas already employed. formulas
(2.90). (2.91), and (2.92) can be carried into three dimensions.
However, even with the use of a compact vector notation. the

symbolism is cumbersome. The three dimensional results are

___l_ - is-JL 7 -' _l_ 2‘ -i9-..9.
.12 " WV 2191.12 8 33-3,). " WV 21 21.11 e ”‘95)

14””

~ __1 2 -_q-9. . __.1 ~ 191-9.
= Z = '
-a2.b NV q—Ag,b 9 Lb NV Zq—Pg,b e (2 96)
and
[A ’b, P '.b'] = 1 h = 6 . Sbb' . (2.97)

where N is the number of unit cellsgxnrunit volume V. All
that remains now is the substitution into the Hamiltonian

(2.84) and the summation over both lattice cells and sites.

1 1 1 -1g.g_ -i_g_'. 9.
H = — (——-—)z — z . P e -
2 NV 1.12an 9.9 9.32 9.2
+1(_1___)21 2. A- .G .1 e-i<q.1+ 4122')
2 NV 1.2 991" ‘91-; =L2 ‘9'49.’
13.19} g: .__' (2.98)
Yet 2 e1)3 +.g )£'= NV5 , from Fourier theory [cf. (2.94)].
L g.-g
therefore
1 1 l
H _ 2 29.10. mb Paula P-QIP + ZNV Ems-Ame
__ 2'9:
ig.(_,_Q-_£') -i(_q+g').9
Z , G e e — - A , ,. (2.99)
M; 1.2 19.19.
222'
Since g (b depends only on the separation of the two
1'71»:

lattice cells rather than the absolute position of the cells.

it is only a function of 2_- Q} E‘h and of p.p'. Thus the

sum over 2] is the sane as a sum over h and each i contributes

Using

2 el<3 +3 )i
(2.99) reduces to

1
2 Z Z

H=sle

Inspection of
choice of coordinates

inside of unit cell.

usual normal mode method must be applied.29

1P .
"Lb—’9

58

= Egg-’2, (2.100)
0
sttza' (2.101)
*
23,10 + 22.1-35302
' *
£99, 5910' (2.102)

the above equation reveals that our
is only normal up to the interaction
In order to complete the problem, the

Thus the

Hamiltonian can be reduced to a form which has no cross

terms .

th

If p is the correspondence label connecting the p -

type branch mode to the pth—root of the resulting secular

equation, and if we assume
. -l/2 iwt
= .10
'AQLQ (m2) 'EQLQ e (2 3)
then
1 . *. . . *.
H = — 2 C C + w B B (2.104)
2 p slp‘zqtp QIP’QJP'ZCIIP
where
. _ —l/2 * 2
‘9ng — 222 ( _) ..QIEIP '23:}; (2.105)

59

B B - A (2.106)
11:9 _ ml; 92:9 ‘92

a“

with the normalization

2 B -B , = 0 ,. (2.107)
.19. sup 9,949 PIP

The creation and annihilation operators can now be expressed

in terms of the normal coordinates:

a ‘ a (2m) ' )‘l/2 c - 1(0) /2vn)1/2§* (2.108)
_9.IP HIP —9.Ip g,p 3:13 .

3* = (2m) )‘1/29" + i(co /25)1/2§ . (2.109)
9:13 sip sip 9:9 9p

Creation and Annihilation Operators. At this point some of
the important properties of a and 5* should be discussed.
To do this. consider once again, for simplicity, the case of

one atom per unit cell. The mass will appear explicitly

and [A,§] = ifi. Then for each mode and polarization
3 = (Zmfim)-l/2 [E - imwA] (2.110)
5* = (thw)-l/2[ E + immA ] (2.111)

Some of the more important relationships are derived

in the following paragraphs.

A) a*a = N the number of phonons (2.112)

This follows from a rather interesting line of reasoning.

60

l 2

Since H = (2m)‘ (P2 +_gw A2) = E classically, and the
corresponding operator equivalents obey the quantum mechanical
commutation brackets [A,§] = in, then the operator equivalent
of the classical Hamiltonian is the correct quantum

mechanical Hamiltonian which has the energy eigen values

E = (N+ @1110).

 

Then N=%:)—E—%
=d‘1—i— ("712—6 ]P2 +m200 2A2]) -%
= Zmfiw (5+imw A)(§—imw A)
= 5*5 (2.113)
B) [a,a*1 = 1 (2.114)

This follows from the relation [A,§] = in

*n -*(n—l)
a

 

C) [3,5 ] = n (2.115)
A A*

D) 9 = (thm)l/2 (a g a ) (2.116)

. 1/2 A ~*

A 1(2mfim) (a - a )
A = (2.117)

NH) 2
Assume that there exists a state such that

510) = o (2.118)

Interpreting a as the annihilation operator, equation (2.118)

means physically that Id> contains no phonons and the state

61

with fewer phonons is non-existent.

We then define the property of 5* by the following

equation:
|n> = ——1——— a*lo>
Vni
E) N|n> = nln > (2.119)
Since
Nln > = a*5ln > = FT: 5*5 5*n|0> = 712-111 5*(n5*(n‘1)
+ 5*“a|o>) = ——-. 7—111. 5*na*(n‘1’|o> = Ti.— na*“|o>
1 1
F) Hln> = (N + §)hw = (n + §)hw (2.120)

Since n is the same as the number of phonons present, the

In) correspond to the Hermite polynomial eigen functions

which are the solutions for the quantum mechanical oscillator.59

G) <m|a|n> =Vfi5m,n—1 (2.121)
H) <m|§*|n> = ‘Vn+1 6m n+1 (2.122)

(Note the matrix element corresponding to an increase in
phonons exceeds that for a decrease.) From (2.121) and
(2.122) it is easy to deduce that the matrix elements con-

necting two phonon states are zero everywhere except adjacent

to the diagonal.

62

The Dispersion Law. In order to be able to understand

some of the problems encountered in the calculations pertinent
to this thesis, it will be necessary to discuss the dispersion
law. Inspection of the equations derived so far seems to
imply that the basis vectors are orthogonal. This is not
necessarily so. Mathematically all that has been said

will hold for a non-orthogonal system if the meaning of the
scalar product can be preserved. In order to do this, the
reciprocal lattice must be introduced. If-él' 32, and a3

are unitary basis vectors in the regular lattice (or direct
lattice), then basis vectors of the reciprocal space are
constructed such that the vector reciprocal t°.élr that is

*

.QI, is perpendicular to g_ and g3 and :3 21.31 = l. Reciprocal

lattice vectors with these properties are defined by:

2

 

 

 

* 'QZ X'QB

31 . a ( X a) (2.123)
—1'32 3

e X 2

52: = a .‘Z’a x1 ) (2.124)
—2 3 a1
.9 X .9

e: = a {(a :a) (2.125)
—3 —1 2

(Note the reciprocal and direct lattices coincide for the
orthogonal case.) In the scalar products qnfl (cf. (2.95)
and (2.96)), q is assumed to be written in reciprocal space.

It also should be noted that all the physically distinguishable

63

values of.q lie inside the region in reciprocal space defined

* * *
1 , ZKQZ , 2ma3.

by the vectors 2mg This region is called
a reciprocal lattice unit cell or the unit cell in q—space.

It should be mentioned in passing that if non-
orthogonal coordinate systems are utilized, tensors must be
expressed in two forms (co-variant and contra-variant)
corresponding to the two-coordinate systems reciprocal to
one another. The reciprocal and direct lattice vectors
are physical examples of co-variant and contra-variant
tensors. The same ideas also apply when writing bras and
kets in non-orthogonal systems.

The next step is to write down the distribution of
modes p(w). Consider for simplicity a primitive lattice,
although the argument is easily extended to the general

case. If the number of modes in a volume qu in q-space

is m, then

N N N
l
.___ = .___{%_l_ * *' (2.126)
q (2N) (a* x a. X g_)
‘1 2 3

where the ratio on the right is just the number of modes per

unit volume in q-space. Hence
dT V
m ___ ___q__3 (2.127)
(2?)

where V is the volume in direct space. The volume dT

64

between two constant w surfaces in q—space is

1
d1 = .( ———— dA d 2.128
9 (lzqwl 9)“) ( )

where Z¢>is the gradient in q—space and the integral is
over a constant w surface. It is evident that both the
constant w surface and the group velocity, lqul = .39,
are required to evaluate this expression. Both can be deter-
mined from the dispersion law, which is the relationship be-
tween w and q. Such a relationship is usually semiempirical.

It is not known for the crystal being studied. For a one-

dimensional chain, the relationship can be arrived at

 

easily:
w =.: (40/14)”2 sin (qa/z)48 (2.129)
.a v9 = vO(V1-(w/wmax) ) for w ————>— 0 (2.130)
where 'a' is the lattice spacing and v0 5 'w/q.

Since the problem of calculating the density of states
.for an anisotropic distribution is sometimes confused, two
different derivations will be presented.

More often than not, it is necessary to introduce a
very simple dispersion relation

w = qu (2.131)

In this type of relation v is an 'average' velocity

0

65

characteristic of the crystal. This relation produces no
distinction between the group velocity (v9 = dw/dq) and
the phase velocity (vp = w/q). This dispersion law will
be used in the following discussion.

The first method for obtaining the density of states
uses the basic idea of the relation of phase space to the

number of quantum mechanical states, AP.

Ar'='éEég- s 'where s is the number of degrees of
(2H ’15)
freedom of a given subsystem. (2.132)
1
2 Af‘. ‘4’ -—-—-——- ‘fd d d d d d = total number
i 1 (Zw’fi)8 px py pz qx qy qz
of states 5 Id[' (2.133)

The probability of occupancy in a particular volume of

phase space then is

w = Ide where p is the distribution or density function.

(2.134)
Assuming p = p(q), then
JP(q)dP= "Jl—jf' Sp(g)dpxdp dpz (2.135)
(2? h) y
If
P = h/K and w 5 qv
q 3 2W/% or q = p/h (2.136)

66

and p (g) has spherical symmetry,

 

 

w = __LS. (pm 4,,p2dp (2.137)
(20 h)
where
2
p dp = fi3q2dq. (2.138)
Thus
2V - 2
w = 2 jp(q)q dq. (2.139)
2?
Therefore
AP = V 3 47rqqu (2.140)
(27r)

for one mode.

A more physical way of formulating the same result
is to consider running waves in a crystal. Choose a crystal
with a cubic lattice of one type of atom with the dimensions
Qx"Qy' 22. The assumed Born-von Karman boundary condition
on which polarization of the wave then implies qx = 2Wn/JX,
etc. For each point in the real lattice there corresponds a
point in a reciprocal q lattice such that each lattice cell
has sides ZW/flx, 27/2y, ZW/Qz, and each such point corres-

ponds to a mode. Then each volume (2w)3(l/flxflyg;) corresponds to

 

one mode. Then in the volume 4wq2dq there are 3
(27r)

4wq2dq modes per polarization, that is

67

v
(2W)

AF” -—? 4Wq2dq (2.141)

3

the total number of states in the interval dq.

The total number of states is always equal to three
times the number of atoms. Thus the integral that totals
up the number of states must be cut off at 3N regardless

of the dispersion law used. From (2.141) for three

 

polarizations
3N = 191-3— )qmax3qqu (2.142)
(20) 0
3N: YET-L— g; . (2.143)
3 ax
(2W)
Assume for the dispersion law [eq. (2.131)]
m = voq
the“ :13 (2w) 33mg 3
3 3 = = (k6) (2.144)
fiwmax 47rV
3 ”V 2 1
___ .___. +___ .
where v3 V3 V3 (2 145)
0 t E
vt = transverse velocity
vi = longitudinal velocity
k = Boltzman constant
6 = a characteristic constant of the crystal, termed

the Debye temperature.

68

The velocity is related to the elastic constants

(X)

of Hooke's law.

 

(C) (e)

 

(Stress) = (C) (Strain) (2.146)
For a crystal with tirgonal symmetry
11 C12 C13 C14 0 0
C12 c11 c13 'C14 0 0
(c) = c13 C13 C33 0 0 0 (2.147)
c14 'C14 0 C14 0 O
0 O O 0 C44 2cl4
.0 0 0 0 2c14 2(cll - C12)

and 1

V

Combining this with the experimental guide that v

 

IV

1
+ 3 M
V

<l
cone»

.2.
V3
t

c 3/2
,0 33

or v3
0

Solving for 93

C33

‘VcBB/p along the c axis where p is the density.30

3

in terms of c :

~..Y...
1 2 t '
._1_ = .ééc_e___)3/2
V3 16 C33
(2.148)
33
\3/2 (2.149)

 

3 3
63 1a (2w) = 3N (
4w 3

Vk P

The Debye 9 can also be related to the specific

 

69
heat (CV) since the ensemble average of a function is just
[cf. (2.132) to (2.139)]

‘A’ =Lpp Adr' (2.150)

where the distribution function for phonons is the Einstein

distribution.65

 

P = ——;-——— (2.151)
efiw/kT-l
Therefore
E = 8%(m) (60)) ( V2 3 2 dq) (2.152)
0 e -1. Zn q

Let the energy per unit volume E/V E U. Then

”8 2
U = 3g 2 ‘3 Engfii (2.153)
2” v0 0 e -1

 

Extracting the temperature dependence by letting fim/kT =
~x = G/T
4 4 x 3
U =._EE;£___ “‘ fiLQE (2.154)
2 3 3

ZV‘fi v

0

Differentiating with respect to T (note the limits of the

integral depend upon T):

BU _ 3 Xm exx4dx
(5-) = C = 9Nk(T/e) S ---- (2.155)
cV = 9Nk(T/e)3J4 (2.156)

The integral is one of a class of integrals known as

7O

transport integrals. The integrals are defined as follows:

xm xnex xm xn
Jn(x)5 g ___— dx = g ___— dx (2.157)
0

(ex-1)2 0 ex+e-x-2

The tranSport integrals cannot be evaluated in a closed

form: however, they are well tabulated.91

71

Relaxation Theory

Introduction. A discussion of the phonon distribution as
a function of frequency or energy will provide a basis for
the organization of the discussion of relaxation theory.
It is assumed that the energy sink for the spin energy is
the phonon bath.

Equation (2.134) gives the probability of finding
a particle in a particular section of phase space. By
rewriting (2.134) using w = voq, an expression for the

distribution as a function of energy may be obtained.

 

w = deP
= (V/Zvr) (qu dq
= ([9819] ( ‘2’3 wzdw) (2.158)
- 2w v

where p(w) is the occupation number and the remaining terms

represent the density of states. For phonons

p(w) = 1
éfiw/kT_l

 

(2.159)

is the Bose-Einstein distribution with the chemical potential
(Fermi energy) equal to 0. A factor of three should also
‘be included to account for the two transverse and one longi-

tudinal modes of vibration.

72

The functional dependence of (2.158) is depicted
graphically in Figure 4. In general, the maximum for such
distributions occurs at Esva, that is, at about the energy
associated with the temperature of the crystal. The resonant
energy of an electron in a field of 3,000 gauss corresponds
to about l/ZOK., the maximum for the distribution shifts
to the right, and thus, as the temperature of the crystal
increases there are decreasingly fewer phonons at the resonant
energy. In Figure 4, the shaded area represents those
phonons capable of interacting in a direct exchange of
energy with the electrons.

It becomes evident then for high temperatures that
perhaps some other process than the direct interaction will
be important. Such a process is the Raman interaction. In
a Raman interaction an incoming phonon is inelastically
scattered.by an electron and an outgoing phonon departs with
an energy different by just the resonant electron energy.
Since the Raman effect involves the creation and annihilation
of a phonon in one.process,the prbfiabifity is more
unlikely than one involving only one phonon: however, at
elevated temperatures the number of likely phonon candidates
for such a process is large enough to make the process become

important. Although all the phonons in the spectrum participate

n (w)

 

 

nun)

 

 

    

?

/

 

Figure 4.

w (H)

h maxgk d2
9.39.2

Phonon Distribution. This figure illustrates
the effect of lowering the Debye 9d for a
distribution. The occupation number contri-
bution will remain the same; however, the
maximum in the density of states is shifted
to the left for a lower Debye Gd.

74

in the Raman process, the most important contribution comes
from phonons at the energy corresponding to the temperature
of the crystal.

Since the electrons that are of interest are non-
interacting, they obey Boltzman statistics. If all the spins
are either in state |a> or 16>, then nb/na = exp (”Bab/RT)
When the distribution varies from equilibrium we speak of

spin temperature, Ts, defined by nb/na = exp (-Eab/kTs)

{ .
The time it takes for a spin system to return from
(some non-equilibrium.state to a state of thermal equilibrium
with the surrounding heat bath is referred to as the spin

lattice relaxation time T , defined as

l
d(.nb-na) -1 '
'-—aE——— = ';; [(nb -na) - (nb -na)co ] (2.160)
or

(nb-na) — (nb -na)co =(nb -na)0 - (nb -na)co e_t/Tl (2.161)
It is necessary to consider the rate equations
under the condition that our ensemble is in contact with a
phonon bath. Consider a displacement a whose time dependence
we shall, for the moment, disregard. From the solution of
the lattice mode problem
.01 =@/ W)eig"2-°

The matrix element for annihilation of a phonon is

75

(cf. (2.121) and (2.122)).

 

 

12/
(2mm . _ -* _. 2
<ng-1l_A(_g_)|n; =<nfl-ll‘ffim0: ___a__2__a elg_| n;
(2.162)
(21:66)) / a -i 2
= Gig-1| v... 3 e 3‘- In; (2.163)
N mm

— i(g) (2.164)

II
(D
I

 

2Mw
q

and for creation

_ fin +1 -iqh£
(n3 + 1l1_\ In; - ‘l _q e _h_(_q) (2.165)
' 'Zqu

and g is the unit vector in the direction of the polarization

 

and m = VF, the total mass of the crystal.

Rather than displacements, elements of the strain
tensor will be used [cf. equations (2.146) and (2.147)].
A brief review of the facts in reference30 is in order.
The fractional changes per unit length, are Bxx’[;y’ and.Zzz
where x,y, and z are the three orthogonal axes. The off-
diagonal terms represent shear type forces and represent
the angular change between the principal axes indicated by
the subscript.

Let g = ui + vi + wk_(i,i,k_are cartesian basis vectors) be

76

the displacement of a point at.r = xi+yl+zk before the

 

 

strain
’1
i
J
K
before strain after strain
and r' = xi'+yj'+zh' (i'hj'hh' are not necessarily ortho—

gonal or unit vectors) after the strain. If

i'-_i_52xx_i_+xy_1+£zx}£ (2.166)
.'1' - .1 E flyxi + flyx—j— +9”); (2.167)
11' "35'2sz +1”; +2221; (2.168)
and _ _
lxy - fiyx, sz -— sz' lyz "sz’ (2.169)
then
-_1_ 9.2 .911.
exy _ 2 ( x + By) (2.170)
or, in general notation
Bu Bu .
__l_ __B'. _E
655 _ 2 ( X5 + axK). (2.171)

It should be noted that the off diagonal elements are some-
times written so the 1/2 will not appear in equation (2.170).
Instead of using the strain components, an average

strain will be used, 6. Then (2.164) and (2.165) become

77

iq.2

(With the use of Bax/8x = quxe —')
I<n -l|€|n > IZ’V qzhn /’ ZMw (2.172)
.9 .9 .9 .9
I<n +1|€ I hi) laou qzfi(n +l)/ Zflw (2.173)
.9 .9 9

In 1932, Waller106 originally pointed out the
possibility of two relaxation phenomena: the diffusion

of energy among the spins themselves, termed T and a

2!

spin-lattice coupling, termed T1. For the latter mechanism

.he proposed that as the interionic distances are modulated
by the lattice motions, the dipoles of the neighboring spins
set up an oscillatory magnetic field. The frequency components
of the motion at the Larmor frequency, hm = gfiH, provided
the coupling of spins and the lattice. Waller also
introduced the idea of the Raman process.
J. H. Van Vleck101 showed that the order of
magnitude of the Waller coupling is too weak by a factor

of 102 to 104 and.that the T will be too long when

1

compared with the observed values. In 1939, Van Vleck100

and Kronig 57' 58

established the now accepted theory;
the thermal vibrations modulate the crystalline electric
field which effects the orbital motion of the electrons and

then the spins by way of spin-orbit coupling.

The crystalline electric field arises in ionic salts

78

from the charges surrounding the paramagnetic ion. A
distortion of the surrounding ions due to a strain will
result in a change in the eleCtric field. We expand the

crystalline field in powers of the strain.

V = VO + V16 + V2€€' + Vaee'e" + ... (2.174)

The first term in the series is the static term
producing a stationary state. The second produces the
perturbation on the stationary state which we shall assume
-is responsible for the spin-lattice relaxation. In this

approximation second and higher order terms are temporarily

neglected.

The Direct Process. This process involves the creation of
one phonon equal in energy to that given up when the
electron relaxes. The probability for a spin to relax

.in such a combination of events can be calculated from first

order time dependent perturbation theory.6o'94’75
20 2
dwif 'fi lFifl 6(Ef-Ei‘fiw)df (2.175)

Here dwif is the probability of transition per unit time to

a range of final states df due to a perturbation F and Ei

and Ef are the initial and final energies of the system as

u

a whole.

79

 

 

E. l E.
E; EL '1

imam fume.)

 

 

The figure depicts a spin in the initial state 'a
relaxing to state 'b' with the emission of one phonon.
E3 is the difference in the spin energy levels and hwq is
the phonon energy. The collective state function 0 of
the system is separable, that is w = fi(phonons)a(para-
magnetic ions).

We would like the probability for all phonon.states
and all lattice sites. The former involves use of the

relation (2.132)

 

ZAP -——+ 3‘2, S 002cm) (2.176)
20 v
0

which allows one to evade counting the quantum mechanical
states and to substitute an integral in the approximation
.that the states form a continuum. The latter operation is
accomplished by summing over those sites with a spin in the
desired state, in this case na of them in the 'a' state.

It should be noted that these are randomly situated; thus

the phase factor in (2 164) and (2.165) may be neglected.

80

Then

nb S dwb _, a (2.177)

Using (2.173) for the creation of a phonon and

W
b-e»a

 

 

 

27r 3v wmax 2 2 2
W = n'—— w dwl<a|V |b>l °I<n +1I€ln I
b->a b‘h 27T2V3 )0 l q 9)
6(Es 416)) (2.178)
then 3
3(E8/h) pnb - 2
wb _J’ a = '5 K alvllb> InAE (2.179)
fiv
where p = V/m. Therefore
“Es/m3 K I I>I2
-dN/dt=W -w _= aVb
b b-epa a-—9b Znfipvs 1
_ 2
KnAE +l)nb nAEnaI (2.180)

Since we have assumed a two level system (i.e., only two

levels are occupied),

dnb/dt = — dna/dt.

3
d(nb -na) 3(Es/h) 2
————= |<a|V l10>) [(n +1>n -
dt Whp V5 1' AB b

 

nAEna]’ (2.181)

nb and na are the spin populations at the temperature Ts'

At equilibrium d(nb-na)/dt = 0 implies [(nAE+l)nb-nAEna] 0.

T =T=
B

It is assumed that the thermal bath remains at the thermal

81

equilibrium temperature during the process, therefore

1
nAE — eAE/kT-1 . (2.182)

Thus
nb +nAEan ‘nAEnaT = nAE[(an 'naT )-(an-naT)]
3 S 3 8
AI 1
+ans'an = )2 + nA8)[(ans’naT)’(an'naT)]

~, 1 eAE/kT+1
= (2°(;KE7EE:;)[(ans'naT)'(an’naT)] (2°183)

Combining the above expressions with equation (2.160),

3<33 3 2 eEs/kT
= _Z—___§ l<alVi|b>| (
1 6 Zva

+1
s/kT_l

) (2.184)

Aha

E
e

or assuming Es < < kT

.l -.i£E§1:__ l< IV lb>|2 __121__. (2 135)
w 4 5 a 1 '

1 ‘fi Zva (Es/RT)

2
1 3(Es) RT 2 (2.186)
g' = 4 5 |<a|V1|h>|
1 'fi va

 

If Ia) and [6) are time conjugate states, then the matrix
element in (2.186) will be zero by Kramers' theorem. Kramers'
theorem holds for the case of an odd number of electrons.
This also can be seen by applying the idea of time reversal

(see Appendix, Section A). Thus

 

82

<aIV16(b> = (Kblvllb) (2.187)
= K<bIKOV1|Kb>
= <Kblxoxovllm>
.__ -<a|Vllb> (2.188)
since KK = -l for an odd number of electrons. The only way

for (2.188) to be true is for the matrix element to vanish.
Non-time conjugate states, however, can be admixed by
the magnetic dipole-magnetic field interaction to the wave-
functions Ia> and lb) by use of perturbation theory to
produce a non-zero result. The admixed wavefunctions correct

to first order, then are63

 

 

‘ <m18-iilb>

Ibu) = |b> +511 Eb-Em I m) (2.189)
ml .

la'> = la) + g<Eaifla> (2.190)

vvhere Elia the magnetic dipole and H is the magnetic field.
The only terms that contribute are those which lie near to
'the ground state (see Figure 5). If c and d are the nearest

lying time conjugate states, then

<c lash» <dI11-slb>
lb‘) = |b> + E_E lc> + _E |d>. (2.191)
a c Eb d

Since IAml is at most equal to one, both bra-kets cannot be

 

 

different from zero. Thus if

 

 

j’ I+-,':_-t>

7\,

 

 

 

 

 

 

 

 

 

 

I--'2-I>
I
I
I
AMI ~ 3 ling“ k)
I f "‘2 k)
I
l
I Afe d l+-'.;,- r>
' < .
l c I--2- n
I
I
I
I ‘Acd
I
l
I
I
I
l
I ‘ t I I
1 +— )
l ’ T b 2 9
Es
Jr 0 I—-'é- p)

 

Figure 5. Typical Energy Diagram.

 

84

 

 

<CIEoE|b>
lb'> = |b> + A I C>
cd
then
(am-Fla)
la'> = (a) + A la) (2.192)
cd
where Acd is the energy difference between the unsplit

a c A“
(b) and (d) levels and Ec-Eb~ Acd Ea E (2.193)

d

Following the notation of Figure 5, then

2 2
_ 3(ES) HkT \ 2

_ 4 5 2 |<Clulb><alVIIC/ +<aluld><dlvllb>l (2.194)
1 h vpv Acd

 

Aha

This can be put into a slightly more compact form by a time

83
reversal argument.

12(Es)2H2kT

= 4+5 2 |<c|ula> <dlvllb>lz (2.195)
1 fi wpv Acd

 

ape

where u is the magnetic moment in the direction of H.

The Two Phonon Process. There are many mathematical approaches
to the Raman process. The two different methods uSed here
follow.the two basically dissimilar physical approaches.

The first method consiSts of using the second order
term in equation (2.174), that is V 62, and applying first

2

order perturbation theory.

 

85

 

-21: _ 2.
wv—9a — ”h Sl<a,nq +l,nq lIVZelezlb,nq,nq> I
l 2 1 2
3V 2 3V 2
2 3 wldwl 2 3 wzdwz 6[fia§-wl) - Es] (2.196)
20 v 20 v ,

where dE8 = fi(w2-wl) and hwl is the energy of the incident

phonon and hwz is the energy of the scattered phonon.

 

 

 

 

Then
2
9|<aIV2|b>| 3 .3
wb__>a = 2 3 10 7 (“(6031) d(fiwl)(h(.02) d(hw2)6[h(wl-w2)-Es]
8p W v fi .
(n +l)n (2.197)
q1 q2
and
2 3
9|<aIV2Ib>I (hwl)
n = -fi = [n n (n +1)
b a 8p21r3vl°h7 g a q2 q1 eXP(fiw1/I<T) 1
(642)3’
_ exp(hw /kT)-l nbnq (nq +1)]
2 l 2
dwldw2[6(hw2-wl-Es)] (2.198)
The nq, the phonon occupation numbers, are assumed to have

thermal equilibrium values. If ES<<kT and ES<<hwl, then
fiw ‘fiw

 

. l 2 .
‘fiwlaxhwz. Equating kT_ and kT— to x, then (2.198) yields
[cf. (2.181)]
9|<aIV lb>l2 G/T 6 x
l _ 7 2 x e dx
? " (RT) 2 3 10 7 x 2 (2'199)
1 4p W v ‘6 0 (e -1)

 

86

Since the transport integral is nearly constant for e/T > 30

i 04 T7. (2.200)
T1

The bra-ket will vanish if a and b are time conjugate
states [cf. (2.187) and (2.188)]: however, as in the direct
case, higher order states may be admixed. By comparing
(2.194), (2.195), and (2.186) to our present case, we find

7 2
9T H J6

_ 2
T - 2 3 10 7 2 I<clu|a><le2|b>l (2.201)
1 p F v h Acd

._l

 

The second method used first order strain terms in
second order perturbation theory. Mathematically, the only
major difference from the direct process is in calculating

the new ‘IF. [2. Here an appeal is made to second order

1f

time dependent perturbation theory.60

 

F. = z <fLAh><11AL9 jaéi + f (2.202)
if j E.-E,
l J
j = all the intermediate states (and is not equal to either
i or f). We use equation (2.175) in the same fashion as

before, letting the initial state consist of a phonon and
the electron in state b, the intermediate state consists
of only an electron in a virtual level j. Denoting A as
the difference in initial and intermediate magnetic states

(see Fig. 5), we find

 

87

(a, n +1IV1(2)€2|j’ nql><jl nqz-l

 

 

..| _A
JJ 11002 J

IVl(l)ellb, nq2>

£-

 

 

 

.2
I I _
(a,nq +1|Vl(l)ellj’ nq ><j’ nq 1lvl(2)ezlb,nq >
1 1 2 2
+ -h<1> - A
1 3"
3V 3v
,- 2
'-§—%—'widwl g 3 wzdwz 5[E8-fi(w2-wl)]. (2.203)
2? v 2V V

If we were dealing with a non-Kramers' system there
would be only one term inside the absolute value signs. The
phonon contribution can be separated from the spin terms
giving an expression multiplying the spin contributions as

in the direct case, as in(2.l96) and (2.197), namely

2 2
q ‘fi(n +1) q ‘fi n

'U
II

h 2km) ZMm

l 2 4? v

MES-sz-wl) 1. (2.204)

Subtracting W and remembering ha=-hb. l/Tl can be found

a-A>b

as in the previous calculations.

aha

2
=2 (I, I2 @lvul mijlvmh» (2.205)
1 1 ha” j

o~¢ w ,'hw <<A, and noting that the

Assuming as before that ml 2

 

88

same form appears as in (2.198) and (2.199), we have

 

 

. 2
_1_ = 90.107 EgalquiXflvml» J

(2.206)
Tl 403pvlqfi7 J j 6
In the special case where AJ.corresponds to an energy
difference between a non-Kramers' ground state and a non- r27

Kramers' first excited state, the denominator of (2.205)

will display a resonance. For this resonance to be operative, (

 

J

A must be less than the Debye cut-off energy. If this g}
. ’ !;
condition is met, then the relaxation time may display an

(exponential‘dependence.22

T1 = Ae‘Aj/kT (2.207)

This is known as the Finn-Orbach—Wolf process.

Our discussion will now be limited to systems which
display a Kramers' degeneracy. We shall temporarily fix
our attention on the term inside the absolute value signs in
(2.203). Let us expand V [eq. (2.174)] in a series of

spherical harmonics.

V =,§m §£m¥1m

5B
where '22m = r£g2m(a0) + ea( Sim)l O+ gee aa'(§aaT—-I 0 (2.208)

 

Upon rewriting we can express V in terms of efim'

v = 2 v1 (2.209)

1m Inelm

89

The factor in the absolute value signs [eq. (2.203)]
becomes, when considering only a particular term in the
expansion of‘%

2
(alvz.m.|j><1|vzmlb> (alvzmlj'><j'|v .m.lb>
+

-Aj+hQ£m -AthQ£.m.

 

 

(2.210)

Consider the second term. The matrix elements must be

invariant under a time inversion. Let

p 1" -
and

'1

(Figure 5),

__l.
2
l

+— ' +
2P J

Ans: th
'1

then from the orthogonality of the spherical harmonics

m' = '% (p-r) and m = '% (r+p).

Hence m + m' is odd since, by assumption, p and r are Kramers'
states and therefore odd. Then, since this term is invariant

under time reversal

(_l)m+m'

<a|vgmlj'><j'lvg.m.lb> <blvfi.m.lj><jlv;.mld> (2.211)

(-1) <a|v .m.lj><jlv2mlb> (2.212)

since the term is also invariant under Hermetian conjugation .

Note also, since

 

 

2
. . 1 1 ~
KaIV 'm' |J><J |V£m|b> (m "’ -A.-f1(l) I ')|
J .fim J .1 m
2
moojmwpm.) (2 213)
A2 <a|V .m. IJ><JIV£m|b>

J'

 

90

where the left hand side is in the approximation that

Aj>>fiw1m

 

 

 

 

that
1 + 1 _ _ Ajmwl'm'-Aj+fiw£m
(-A+hc0 ) (A+f1c0, i,) 2 2 _
j 1m j .1 m Aj +(fi mgmw£,m,)+Ajh(mzm w£,m,)
”F16” . . .)
,V, .2 m lrn . (2.214)
2
A)

This is the Van Vleck10 cancellation and is based upon the
assumption that the splitting of j and j' levels by the
magnetic field is small compared to Aj' The effect of the
cancellation is to raise the power of fiw in the integrand
of equations (2.203) and (2.205) by a factor of two. Using

the definition of transport integrals (2.157), then

I

2
1 9h k'I‘ 9
— = ——————. (-—> J (e/T) (2~215)
T
1 4v3p2v10 h 8
9. .1. 9
andforT> 35 Tour.

Cancellation of the Van Vleck type may be inhibited by

application of a magnetic field. Admixture of adjacent

a
Kramers' levels to the ground states and/or the excited
' b
1
states can destroy the time conjugate nature of these

 

91

states. The admixture enters again as in (2.194) and (2.195)
giving an equation comparable to (2.201).

Orbach and Blume86 have considered the Van Vleck
cancellation term further. In the above arguments we assumed
that j and j' were split apart from the ground state by a
large energy, but there is no reason to demand this a priori.
Now cancellation [(2.210) and (2.213)] gives for Aj<<fiw£m

and fimgm =‘fiw 'm'

<a|V£.m. |j><j lvflmlb Mai; (2.216)

The absolute value squared of (2.216) replaces the
absolute value squared in equation (2.206). The effect is
to multiply the previously determined absolute value of the

relaxation time in equation (2.206) by
1/2

/2
2 2 2 'kT 2 l 2 2
(("—*—) 13.5 or (4(—") (—) A.) . Thus
'hmwm J ‘fiwrm kT J
E 15
% N 93krf0 7 J4(e/T) (2'21?)
1 pw v h
l 5
and for e/T >25 ; ocT .
l

A rough order of magnitude criterion for the above process

to dominate in the Raman region is given by Orbach and Blume as

)(—-§—) > kT (2.21s)

 

92

where A is the appropriate crystal field splitting and A is

the spin—orbit coupling constant.

Phonon Bottleneck. In all the preceding derivations the
phonon bath was assumed to remain in equilibrium during the
relaxation process. Our measurements actually measure spin—

which is identical to-T if our

bath relaxation time, T 1

b’
equilibrium assumption remains valid. The non-equilibrium
case has been debated by many authors.102' 28' 10' 17

The most recent theoretical and experimental effort of note
has been carried out by Scott and Jeffries.99 They envision
a strong 'localized heating' in the phonon spectrum at the
resonant electron energy. In the direct region this

process can be dominant. The solution of the rate equations
for this process yields two time constants. One of these
‘would.be of the order of one microsecond for a ten gauss
xnide line. The other should be larger and have the pre-

dicted temperature dependence of

3

l = ___.AET (2.219)
b DT +AT

)Mhere D is related to (density of spins) (average linear
Ciimension of crystal).1 (line width), and A.is an empirical

canstant. Then for DT2>>AT (no bottleneck)

 

93

.l
T

b l

= AT (2.220)

dIH

and for DT2<<AT (bottleneck)

% = DT2. (2.221)

b

Localized Modes. It has been assumed that the phonon spectrum
with which the spins are on speaking terms is the phonon

spectrum characteristic of the crystal. The effects on the

 

phonon spectrum introduced by an impurity or defect site

have been studied theoretically by I. M. Lifshitz,67

E. Montroll and R. Potts,81 P. Klemensso' 51' 52’ 53

' 56 . 14
B. I. Kocheleav, and R. Brout and W. Vissher. Castle,
Feldman, and Klemens have applied their theory to para-
magnetic relaxation experiments.l7' 18 A brief summary of

their theory is given below. The contribution of the strain,

e, to the J transport integral in a normal T7 Raman process

6

is €4a(hw)4. The strain introduced by the impurity is:

 

2
w
e' =-é ' (2.222)-
a0 2
w. —w
1
‘where mi is the local mode frequency. Then for w<mi
e' = (-42- )2 -—§— elmt (2.223)
m. a
1 O

and for <0>c0i

94

 

e' = a e1“)t (2.224)

This leads immediately to the relaxation law for a T7 Raman

process.

l- = AT7J + B(T3J -T3J'%%) + C(TllJ

g;
1 6 2 2 ( )) (2.225)

10 T

The second term arises from the contribution of the modes
from 61(wi) to 6D(wD). The third term arises from all

modes up to 01(wi). The equivalent law for T9 then is

J + B' (TSJ ~T5J4(%—;i-))+c'(rrl3a 33)). (2.226)

9 (
8 4 12 T

1.- .
T A T

 

95

Cross Relaxation

Relation to Line Shape. The general theories of line shapes
and various cross relaxation processes are intertwined. In
theory if the correct interaction Hamiltonian for a spin
system can be written down, then all possible cross relaxa-
tion interactions will be present and the relaxation time
line shapes can then be calculated. A general form of such
an interaction Hamiltonian for two spin ensembles is

.1. +_s_. ._S_ (2.227)

H =§.g._§ +1. ._1_ +1,

int

"'0
u?
II>J

which represents the internal interaction between spins in
ensemble S and those in ensemble I and the interaction between
the two ensembles. The first term can correspond to two
physical phenomena, the contribution of the_exchange integral8
and that of dipole-dipole interactions4 for the ensemble of
spins S. The second term has the same interpretation for the
ensemble of spins I. The third term is the interaction be-
tween the ensembles. The strength of this interaction is
determined by the amount of frequency overlap in the Fourier
spectrum of individual members of ensemble S and I. The
fourth term is the spin orbit coupling. Bloembergen, et al,
have written down the dipole interaction plus a pseudo-

dipole interaction in terms of raising and lowering operators

0

which is an equivalent of (2.227).1 (The dipole expansion

96

is done in detail in reference 4. The discussion in
reference 10 provides the basis for much of the work in
.spin-spin interaction theory at the present time.) The
operator form carries a physical significance with each term.
S+I_ or S_I+ indicates a mutual spin 'flip'. SzI_ or _
SzI+ denotes an interaction in which spin I, being coupled fin:

to spin S, finds components in the frequencyspectrum of S which

 

correspond to frequency components in the Fourier transform J":
of the time dependence of its relaxation. S+I+ or S_I_ ’1
corresponds to a simultaneous spin transition of spins S

and I. Using a hybrid of perturbation theory and Van

Vleck's103 method for calculating the line moments using

traces of the Hamiltonian and spin components, it is possible

to evade calculating the repeated perturbation action of Hint
which would account for the reshuffling of the dipole fields

caused by Hin In making the calculations certain terms

t'

normally are dropped as being negligible, but these terms

have recently been investigated for special systems in which

they are important. For example, A. Kiel considers in detail
46, 47 .

the effect of the exchange-terms. An important

result of Bloembergen's calculations was to show that even

if resonance absorption lines are clearly resolved, there

still may be considerable overlap between the lines as

calculated from the Hin terms since the absorption line

t

97

does not represent the effect of all the terms in Hint'

Relation to Relaxation Time. The typical resonance line can

 

be thought of as made up of a group of Lorentzian ensembles
or lines randomly distributed in a Gaussian-like manner.
Each Lorentz packet is said to be a homogeneous line, that
is a line made up of an ensemble of spins seeing essentially
the same local field. 72 is defined as the characteristic
time it takes for the transverse components of the spin in

a homogeneous line to lose phase coherence, that is, if at

of the spins are precessing about

-t/':2

time equal to t = 0,NO

the 'z-axis' in a coherent fashion, N = Noe

will still be coherent at a time t and NO-N of the spins

will have had a mutual spin transition with another member

of the same ensemble. 72 is related to the ensemble width,

Af, through the Heisenberg uncertainty principle.

—1- = 2Af (2.228)
T2

(NOte that we have tacitly assumed that none of the N spins

0

process.) T is defined as the

will undergo the T 12

l
characteristic time for a Spin in one particular ensemble to
exchange energy by a mutual Spin 'flip' with a member of

another ensemble. The whole line consisting of all the

ensembles seeing slightly different local fields is termed

98

a non-homogeneous line. 7: is defined as the inverse of
twice the non-homogeneous line width.

The diffusion time, T, for a packet of energy to
diffuse through a non-homogeneous line can be calculated in
the first approximation using a random walk method. T:/72

is the chance that a neighbor is on 'speaking terms'.

1%
Therefore the probable time to cover a step is T2(T2/T2).

 

The number of steps to go from boundary to boundary, however,

 

is (Tz/T;)2. Therefore

4 *3
T 12 /T2 (2.229)
A more accurate expression is derived by Bloembergen:
2 1/2 4 -3
TN(<vc> 12) T12 (2.230)

where <v§>l/2 is the mean square cross relaxation line

width.
One other effect has been observed. Upon the
application of an r.f. field there will be a reshuffling of
. . 79, 12 .
the local dipole fields. Depending upon the system

and the magnitude of the r.f. field, the system will take a

finite amount of time to come to a quasi-equilibrium.

CHAPTER 3
EQUIPMENT AND TECHNIQUES

General Description

The equipment for this experiment may be divided
into two categories, the saturation recovery apparatus (see
Figure 6) and the spin echo apparatus (see Figure 7).

Both pieces of apparatus are connected to a standard X-band
microwave waveguide passing through a vacuum sealed head
into a double Helium dewar (Figure 8). The waveguide-termi-
nates at a cyclindrical reflection microwave cavity (Figure
8) excited in the TE-Oll mode.80 The frequency of the
cavity can be varied with a pair of Teflon rods 1/4" in
diameter which enter the cavity at 1/4 and 3/4 the
diameter of the cavity. This positioning of the rods
allows maximum coupling to the electric field and permits
tuning over a range of 150 megacycles. A small bifilar
Manganin wire coil was wound around the waveguide just
above the cavity. This coil was used to regulate the
temperature between 4.20K. and 800K. The samples were-

mounted at the center of the cavity on a polystyrene rod

99

 

 

)

 

 

I CAVITYIf

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\
’\ IATTENUATORI
IATTENUATOR ‘
WAVE PARIABLE I I ISOLATORJ
METER SHORT ‘
IATTENUATORI
‘ IATTENUATORI
IATTENUATORI _ 0 ,
IATTENUATORI ' | 11
l IISIOLATOR I IISOLATORI
IISOl LATORI . I
PULSED -
IKLYSTRON I LOCAL I
MOINITOR _ OSCILLATOR
KLYSTRON
I PULSE
‘ IGENERATOR _
WAVEFORM ~ .
GENERATOR PULSE I—T RECEIVER SCOPEI
- _ if _
.L _________ J

 

I * GENERATOR

6 CRYSTAL 05750103

; 20 db DIRECTIONAL

COUPLER

Figure 6. Saturation Equipment.

 

 

I TERMINATIONI

 

CRYSTAL
DETECTOR

    

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I LOCAL I
OSCILLATOR , 2
- - I TERMINATIONJ
IISOLATORI
IATTENUATOR I I ATTENUATORI
IATTENUATOFJ I ISOLATOR I I ISOLATORI
IMIXER IMAGNETRONI ‘ I MAGNETRONI
I RECEIVER I MODULATORI I MODULATOR I
ITIME DELAY I
GENERATOR
t' 21 1 » 7
r, _ ‘l ,

 

 

 

 

' s PULSE WAVEFORM ‘
COPE GENERATOR GENERATOR

 

Figure 7. Spin Echo Equipment.

 

PLUNGER KNOB

PUMPING PORT

DEWAR CAP ASSEMBLY
SHOWING POSITION OF CAVITY

 

 

PLUNGER ROD

 

WAVEGUIDE

 

 

 

‘qu PLUNGER

I r CAVITY

 

 

 

 

 

I WAVEGUIDE

m I ' TEFLON PLUNGERS
GOLD - COBALT THERMOCOUPLE
a...)

I SAMPLE

 

 

 

 

 

 

 

HOLE FOR PLUNGER _‘~
-— ——-——5TYROFOAM
LL

 

 

Figure 8. Dewar Cap and Cavity.

Thu-J I. '_

 

103

embedded in styrofoam which filled the bottom of the

cavity. A thermocouple made from 3 mil silver 32 atomic %
gold and gold 2.1 atomic % (cobalt wire was attached to the
crystal with Glyptol. The other junction of the thermo-
couple was in contact with the liquid Helium. The temper-
ature was monitored with a K-3 potentiometer and a null
indicator. This method of measuring temperature was checked

by observing the temperature dependence of the Splittings

 

of the nuclear resonance lines of the protons in NiSO4.6H20.
The splittings in this salt are known to obey l/T Curie law
dependence down to 4.20K. Temperatures were stabilized at
boiling Helium (4.20K.) and Nitrogen (77.20K.) temperatures
as a further check. Temperatures below 4.20K. were obtained
by pumping on liquid Helium with a Kinney vacuum pump at

rates up to 230 cu.ft./min. Temperatures as low as l.l°K.

were obtained in this fashion.

Spin Echo

The spin—echo techniques and apparatus used are those
45,15 .
developed by D. E. Kaplan. A block diagram of the
apparatus is shown in Figure 7. The idea of spin echoes,
although relatively new in the area of paramagnetic resonance,

was originally introduced in nuclear magnetic resonance work

by Hahn.32 The effect can be explained readily on the basis

104

of a classical model (Figure 9). Consider the spins initially
precessing about the z-axis under the influence of a static
magnetic field applied in the z direction. At time 0,

a linearly polarized magnetic field in the x direction is
applied at the resonance frequency of the Spins for an
extremely short period of time ('\/50 x 10—9 seconds).

The linearly polarized field can be decomposed into two
circularly, oppositely polarized components. The component
rotating with the spins pulls the Spins down toward the xy-
plane. If the power is correct, the pulse will pull the
spins 900 into the xy-plane. The spins then start to
precess coherently in phase in the x-y plane at w =

TH E gBH/fi, where H is the combined externally applied

and local magnetic crystal fields. Due to differences in
the local field, the spins will become more and more out of
phase as time progresses. This loss of phase coherence
because of local field differences is not an irreversible
process. By applying a second pulse (Of twice as much
power) at time T, it is possible to have the spins essentially
reverse their angular velocity and retrace their paths back
to the phase coherent condition. Actually the second pulse
flips the y component of spins a full 180°, but the effect

is the same. At a time 2T a pulse may be observed as spins

IL

 

 

¢—-‘T --++—-"T"'*

90° 180° ECHO
PULSE PULSE

 
 

 

 

 

 

Z
Y
X
. /\
‘l, T ~$¢-3r-——++-'T--O
90° 90° |80°

PULSE PULSE PULSE

Figure 9. Spin Echo Pulse Sequence.

106

pass through the phase coherent condition. This process of
bringing the spins into coherence can be repeated. However,
as time passes it will be observed that the size of the pulse
decreases due to various relaxation processes. Normally at
low temperatures T2 <<Tl, so the decreases in pulse Size are
due to cross relaxation processes. Although energy is
conserved, such a process will destroy the phase coherence.
*

2: nOt T12 0r “[2 I

If one wishes to actually measure T
(see cross relaxation section for definitions) it is essential
that the whole resonance line be spanned by the Fourier
components of the pulse.

Let NO be the initial number of spins tipped by the
900 pulse, and N be the number that have not undergone a

relaxation process. Then

.. 'c
e t/.

N = N 2. (3.1)

0
If the system is initially pulsed at time equal to

zero, and then at some time t>> T the system is inspected

2,
for echoes with a 900 and then an 1800 pulse. The echo then
formed will depend upon the number of spins that have relaxed
in the time t, since the relaxed Spins are again eligible

for a spin echo process (Figure 9).

Thus

N = N (1 -e ”1). (3.2)

107

Both equation (3.1) and (3.2) must be written in an
operational form. If V is the voltage monitored on the
oscilloscope, then Vk = P, the power emitted by the echo
'where k is a characteristic parameter of the equipment. It
can be shown15 directly from Bloch's equations that

p = N2. (3.3)

Hence, (3.1) becomes

v = v0 e"kt/272 (3.4)
and (3.2) becomes
1 - (V/V0)k/2 = e-t/Tl. (3.5)

In making such a measurement it is essential that the initial
pulse tip the spins 90°. The Criterion on the-line is that
it be sufficiently narrow so that it can be spanned by the
first 900 pulse. Times down to .lps can be measured with
this technique.

The spin echo equipment is shown in blockform,
Figure 7. The repetition of the pulse sequence is control—
led by a waveform and pulse generators (Tektronix 161, 162, 163).
These are used to trigger a General Radio 1392 delay gener-
ator and to gate the receiver. The delay generator triggers
the magnetron modulators and the oscilloscope. Two sets
of magnetrons (Litton 3028B) and modulators are used to allow

for the magnetrons to recover in short-time pulse work.

108

{mnemagnetronlines join at a matched-T. Pulses are distri-
buted through a circulator with 40db isolation to the
cavity. Pulses from the cavity are returned through the
circulator, mixed with the local oscillator frequency, and
detected by a superhetrodyne receiver (Figure 10). The
receiver has a bandwidth of 50 megacycles at a center

frequency of 350 megacycles.

Saturation Recovery

 

The saturation recovery technique is similar to that
used by Jeffries and Scott.99 A microwave pulse of 10..5
seconds is applied to the microwave system. The pulse is

of sufficient power to saturate the signal. The recovery
of the cavity and the sample is then monitored at low power
level. The monitor is modulated to compensate for any
drifts from the cavity frequency. The monitor power must
be low enough so that its effect is negligible. The
sensitivity of our system is around 80 dbm. The shortest
measureable time is about .1 ms. The operational equation
for this system is

-kt/Tl

V = V e (3.6)

0

where Vk represents the response of the saturation apparatus

to a power level P, since the power absorbed by the sample

. ‘ 71
is proportional to N, the number of centers.

 

Spin Echo Receiver.

Figure 10.

110

The equipment is shown in block form, Figure 6.
The pulse klystron is connected to standard magnetic resonance
spectroscopy bridge with reflection cavity. A side arm
with 40 db isolation connects the monitor klystron to the
pulse klystron arm. The detection arm contains a balanced
mixer and local oscillator. The output of the balanced
mixer is fed to a superhetrodyne receiver which operates at

150 megacycles with a bandwidth of 10 megacycles.

CHAPTER 4
RESULTS AND CONCLUSIONS

‘Thquxperiment

Single crystals of A1C13.6H20 were grown from an
aqueous solution. Extremely slow growth produced crystals
which were optically clear. The c axis was clearly defined
by the intersection of the large crystal faces. When either
the chlorides of Fe+++ or Cr+++ were added to the solution,
the impurity ions readily substituted for the Al+++. The
actual percentage by weight of the substituted ion was
estimated by E.P.R. absorption and checked by chemical
analysis. The results showed the percentage of iron to be
.009 and the percentage of chromium to be .028.

For the experiment, the crystals were mounted as
described in Chapter 3. T1 data was taken in the 4.20K.
to 1.10K. temperature region using the saturation technique.
The spin-echo technique was normally used from 4.29K. to
800K. since it is more sensitive and can measure shorter

times. Temperatures over 4.20K. were obtained by heating

the cavity as indicated in Chapter 3. However, as a check of

111

I]
,
l l 2 /,/ 3) .3

the reliability of the data, both sets of apparatus were

used interchangeably in the region between 4.20K. and 150K.

Within the limits of error (10%) both gave the same results.

The upper temperature for obtaining data was limited by

the combined effects of T2, the ultimate time resolution of

the spin-echo apparatus, and the Boltzman population factor.

This upper limit was at 470K. for Fe+++ and 800K. for Cr+
For each run the equipment was checked to find its
response law, namely input power was plotted against output
voltage. The response law then was determined in the power
region used for obtaining data; that is k in P = Vk was

obtained.

Analysis

The saturation recovery data is read directly from
the photographs of the oscilloscope traces (Figure 12).
The modulation clearly shows when the apparatus is on the
cavity frequency. In the photograph shown, we would read
the bottoms of the recovery trace. The voltage is then
plotted against time on semi-log paper. The spin-echo
data must be handled differently (Figure 11). Here we do
not see the continuous recovery, but only the pulse
corresponding to a particular delay between the first and

second pulses. Since the experiment is not continuous, but

++

 

 

Figure 11. Spin Echo Data. The pictures correspond
to elapsed times of V0, .1, .25, .5, .75 m.s.; and
V0, 1, 2, 3,1) m-S-

 

Figure 12. Saturation Recovery Data. The top trace
is with modulation. The sweep is l m.s. per large
division.

115

actually takes place over a time >>Tl, one must be careful
to check V0, the pulse in a spin-echo after the system has
had a time >>Tl in which to recover. Thus the top trace

in each picture is V For the Spin—echo technique, 1 -

0'
(V/Vo)k/2 is plotted against time on semi-log paper (3.4).
Using this technique, the best straight line through the

data will be a good statistical average for the determination
of the relaxation time from the data. Using (3.4) and

(3.5) T is readily interpreted. The initial points

1
(especially in the Cr+++ data) probably show a Bowers and
Mims effect.12
Resonance data was taken on all three lines in the
Cr+++ spectra between 1.10 and 20K. No appreciable difference
in relaxation times was observed. The relaxation times
were also found to be angle independent. Since the 1/2,
-l/2 transition was the strongest, it was used for
determining the temperature dependence of the relaxation
time. All five transitions for iron gave the same relaxation
time between 1.10 and 20K. when the crystal was oriented
parallel to the magnetic field. The line at 3800 gauss was
strongest and therefore was used in obtaining the temperature
dependence.

We have seen in Chapter 2 that two distinct types

of temperature dependence are expected. At low temperatures,

116

equation (2.195) is expected to dominate since it is a
single phonon process. At higher temperatures we expect
the two-phonon (Raman) process to prevail. The actual
temperature dependence of the Raman process depended upon
the splitting A of the energy levels near to the ground
state levels (Figures 2, 3, and 5 and equation (2.218)).

Summarizing, we have two processes which occur for transition

ions:
1 7 pg A2
:r-l = our + 8T J6( T) for Z— (kT (4.1)
or
1 5 g; A2
-T—1= our + 6T J4( T) for K >kT (4.2)
where 9 a, 5, are experimental parameters. A more

d,
specialized process called the Finn-Orbach process which is

highly unlikely for transition ions would contribute an

additional term (2.207)

Ue-A/RT (4.3)
to equations (4.1) and (4.2), whereq/is a constant to be
determined.

The relaxation time dependence was plotted against
temperature on a log-log plot. The curves were fitted trying
various theories as outlined in Chapter 2. A Finn~0rbach

process gave a very poor fit. A bottleneck term if present

118

. +++ .
Figure 13. Cr Temperature Dependence of Relaxation.

Io"

I0"

   
    

I0"

I I IIIIII

TI (5 cc.)

Io"

lllllll

I0"

IIIIIII

 

lo” I I IllIIJI I I IIIIIJJ
l 5 I0 50 I00

T (°K)

120

. +++ .
Figure 14. Fe Temperature Dependence of Relaxation.

Io"

 
    

IO"

 

I IIIIIII

I0"

IIITIIII

T, (596)

I

I0"

I IIIIII

Io"

TIIIIII

 

I0-7 1 1 [141.1% I IIIIIIJJ
I 5 Io '50 I00

T(°K)

117

would have to have a negative coefficient if it were to

improve the fit. Both of these effects were not expected

for our systems. Therefore, they were discarded. The Cr+++

data fit well using a T7 dependence and much more poorly for
a T5 dependence and very badly using a T9 dependence. The
Fe+++ data fit extremely well to a T5 dependence and quite

poorly to a T7 dependence. The results as shown in

Figure 13 for Cr+++ are

0
-1- == 67T + 1.20x10’3T7J I—150 I (4.4)
T1 6 T

In Figure 14 the results for Fe+++ are

0
= 71T + 7.58x10-4T5J4 (;§9—) (4.5)

l
T T

1
It should be noted that in both cases the same Debye 0,
160°, gave the best fit. We should now like to compare these

results with the theoretical predictions for the T7 and

T5 Raman processes.

Discussion

In general for T5 to be a probable process, Az/A

16 O

32 Since kT = hc/A, then 1.38x10- x 1 =

> RT.

(6.6xlO-27)(3x1010)/A, or each degree Kelvin equals .7 cm-l.

Since the roll off of T occurs at about 4.2°K., the

l

criterion is Az/A > 4. We now consider our two cases,

118

. +++ .
Figure 13. Cr Temperature Dependence of Relaxation.

Io"

Io"

Io"

'l'I (sec)

I0"

I0"

ID”
I

I IIIIII

     

ITI IIIII .

I

IIIIIII

I

IIIIIII |

NIH” I

 

LIIII

Il_l_l

 

IO
T (°K)

50

I00

’2')

 

120

. +++ .
Figure 14. Fe Temperature Dependence of Relaxation.

IO"

     

IO“

 

I IIIIIII

I

I0"

I IIIIIII

T, (see)

I

I0"

IIIIIII

I

I

I0"

I'IIIIIII

 

 

 

W” I llI_L‘L_I.’_LL I IIIIIILI
I 50 I00

122

+++ +++
Cr and Fe .

g;:::. The local symmetry of the Al+++ is basically C3i.
Since the ratio of the ionic radii of the Cr+++/Al+++ =
.064/.056 and the coordination is in direct correspondence
with (radius of metal/radius of ion) = l.28/.64 = 2 -—9
C.N. = 3, then the Cr+++ has the tendency to lend itself
to a situation in which it is coordinated by atoms forming a
triangle about the Cr+++. This strain will tend to produce
an extra axial distortion. However Wong109 has fit the
E.P.R. data to an axial form of the Hamiltonian with a 9
factor of 1.977. The 9 factor for Cr+++ in an octahedral
field 1372
g = 2.0023 - 8A/A.

(This expression will vary slightly with axial distortion.)
Hence 8(A/A) = .025 or A/A“’.003 where A is the splitting
between the F2 and T; levels.

We assume a A of 55 cm.1 which is the value given
by Low,73 or that of the free ion (A is certainly less for
the non-free ion) given by Dunn,21 C = 275 or A = 275/ZS
= 92. Thus A2/A< (92)(.003) = .276 or, in other words T5
is important as a Raman process somewhere below .SOK.

At these temperatures the direct process will be dominant.

We have available one other scheme of attack,

 

123

namely using the direct optical observation of A-é I; -/;

= 17,400 cm-l.44 Using the A of 92, we see A2/A =
8.1x103/1.7x104 ~.5‘.’K.

+++ . . _ ., +++ +++
Fe . The ratio of the ionic radii of Fe /A1 =

.067/.056 and the coordination is given by l.26/.67

1.88 P]

and this also leads to a coordination of 3, but to a greater

tendency than does the Cr+++. Thus we would expect a larger

 

axial term, D, than for the Cr+++ and indeed the resonance a
work of Wong109 bears this out. Therefore, one should be

' . . +++ .
more careful in handling the Fe data. This plus the
‘ . ' . +++
fact that no other spectral data is available for Fe
in a similar site makes the situation nebulous. However,
one would expect that the A should be small since the admix-
ture of excited states needed to account for the large

++ is proportional to

1

splittings seen at zero field in Fe+
(A/A)2.11 Using Dunn's extrapolated values ”V’lOO cm-
(and this is but a very crude estimate), A would have to be
of the order of 200 cm“1 if T5 were to be a dominant process

at 50°. This implies a value for the admixture which may be

reasonable, (A/A)2 hv'.25.

Final Discussion

The possibility of a local mode dependence should

124

not be overlooked. The theory for one local mode, however,

will not give a very good fit. A priori, there is no reason
why more than one mode cannot occur. This could lead to an

alternate explanation of the data.

The possibility of cross relaxation has been checked
carefully by making relaxation time vs. concentration measure-
ments. These measurements showed no concentration dependence
in systems as dilute as we have reported here.

The data are reproducible by two techniques, the
temperature measuring techniques have been proven reliable,
and the possibility of T2 effects have been ruled out. In

conclusion, the effects reported are real Tl effects in the

AlCl .6H 0 and do represent departure from the normal Raman

3 2

process.

APPENDIX

 

 

APPENDIX

A. Kramer's Degeneracy

In addition to the usual types of symmetry such as
space groups, it is also necessary to consider the symmetry
of the time variable. Consider the Schroedinger equation

and its complex conjugate.

 

 

‘5 2 -1119. _
" 2m V + “TI ¢ “ i at ’ m ”3"“
' *
_ 12 V2 + V(r) <I>* = - 123% = 30* (A.2)
2m

 

(It is assumed ¢ is non-spin dependent.) Two conclusions
can be drawn. ¢ and ¢* form a degenerate pair of eigen-
functions and hence (A.2) implies that time inversion
(replacing t by -t) is the same as taking the complex
conjugate. We will denote the complex conjugate operator
as K0 (K0 is a non-linear operator.). By taking the
proper linear combination of ¢ and ¢* it is always
possible to construct a real function. Hence, real oper-
ators have real eigenvalues and imaginary operators have
imaginary eigenvalues. Since any operator with an odd

power of t is imaginary, the expectation value of such

operators (e.g. 5, f) must vanish over the entire set of

126

 

127

degenerate non-spin states.

The addition of spin to the picture necessitates the

introduction of a new operator, K,

for time reversal. The

following properties will enumerate some of the more

important features of K.

A) K = e(-iWSy/mxO

where S is the spin of the system.)

K = -ioyKO

for one electron where 0y is a Pauli spin matrix.

-1
B) K K - (KOKO Y y

C) [KI A] = 0

where A is a real operator.

D) KB = -BK (anticommutation)

where B is an imaginary operator.

E) KxK-l¢ = x¢ KpK-l¢
F) K = inoyloy2 ... Oyn
G) K2¢ = (-1)“¢

H) <¢|Iv> = <K¢IK¢>

-l)(-0 o -

(A.3)

34,p.232

o (A.4)

(A.5)

(A.6)

(A.7)

(for n spins) (A.8)

(for n spins) (A.9)

(A.10)

An inspection of the above equations shows that most of the

features of complex conjugation remain.

Using the properties

of K several important conclusions can be draw.

 

128

A) K9 l-¢ for an odd number of electrons.

(M I <9) = <K¢ (1009) = .(m | q.) = 0

.3 <K¢I¢> = .o (A.ll)
B) <K¢IA¢> = o (A.12)

where A is a real Hermetian operator.

The important theOrem of Kramers can now be easily
derived: for n odd, every state is at least two fold
degenerate in the absence of a magnetic field. In the
absence of a magnetic field and even with an electric field
present, H is real. If H9 = E9, then from (A.5) KH¢ = EKO
shows that K¢ is also an eigenfunction. Kramers‘ theorem

says that it is independent. Assume that this is false.

Then
K¢ = a¢ (where a is a constant)
K2¢ = a* K¢ = a*a¢ = IaI2¢
But n is odd and K2¢ = Q9, thus
IaI2 = -1

Therefore our assumption leads to a false conclusion leaving
only the possibility of at least two-fold degeneracy. In
the presence of a magnetic field, the Hamiltonian contains
terms linear in the angular momentum95 and is not invariant

under time reversal.

129

B. Definition of a Real Rotation Operator
. ' a. . I '
The rotational operator R = el)- ;) Here 18 the
infinitesimal rotation operator and a is the angle through

which the rotation is carried. The angular momentum

operator_g can be defined in terms of I.

.g E? (B.l)

Consider a rotation about the y-axis by T radians.

- (in /h)
Ky — RyKO e y KO (13.2)
The effect of KY on the spherical harmonic Vim when
J = )1 is
K Y = e-iwaY* = e-iwfy(_1)mY .
Y 2m m 2,-m
_ 2
- (-1) 21m (B.3)
since
(Y )* - ( 1)‘Y d
Am - .Qt‘m an
SI-m 76
Rymm> = (-l) l2,-m>. (6.4)

V2” m is termed real if £_is even since then it commutes
,-

with K .
Y

130
C. .gahn-Teller Effect

In fields of high symmetry, the ground and excited
states of an ion may often be degenerate. The Jahn-Teller
effect37 shows that a molecule having an electronic energy
level which is degenerate can undergo a nuclear displacement
such that the degeneracy is removed. This gives rise to an
axial distortion in a cubic lattice which will remove the
symmetry causing the degeneracy. Physically this corresponds
to the paramagnetic complex seeking the lowest possible
energy: for given a small distortion perturbation the energy
levels shift so that their center of gravity remains the same
which in turn means that there is a higher and a lower
energy level possible. There will therefore be a distortion
in the molecule to remove the symmetry degeneracy. Several
configuration distortions may be stable, giving rise to a
new degeneracy. There are two exceptional cases. The
Kramers' degeneracy of the lowest level cannot be removed
and, in the case of strong spin-orbit.coupling, the effect

may be overcome.

131
D. Further Discussion of Crystal Fieldpgglculations

It should be noted that the approach established
by Bleaney and Stevens7 will yield the same results for the
expansion of the crystal field (2.65). The local symmetry
is axially distorted cubic symmetry. The distortion is along
one of the three-fold axes of the cube, which is labeled -1

as the z-axis [cf. (2.43) and (2.44)]. The cubic field is

 

_ee 191/2 6 3
¢—A4Y4-(7) A4[Y4+Y

‘3
4 ] (0.1) '

and with axial distortion it is just equation (2.66).

Matrix elements of the potential between two states
are needed when one applies the crystal field as a perturbation.
These elements take the form<LMI¢I£lm1 . Two common
techniques are needed. If the potential is expressed in
spherical harmonics, ¢ reduces immediately to a sum of Wigner
coefficients. The ones of interest may be found69 tabulated.

Hence

<LMI¢|£1m1> = a20[3M2—J(J+l)]+(a43- )

*
c‘43
(LTM) '. (LiM+2) '. 1/2
(le) '. (LT-M-Z)‘.

 

(214:3) (0.2)

where M = ml + m, L =II', and 2L 212 O.

132

Table 1. Cross reference for group notations

 

 

 

Schoenflies International
Crystal Type Notation Notation
Rhombohedral-trigonal C 3
. 3
(one 3-fold aXis) -—
C . 3
3i
C3v 32
D3 3m
D3d 3m
Hexagonal C 6
. 3h
(one 6-fold axis) D '6
3h
C6 6/m
C6h 622
C6v 6mm
D6 6m2
06h 6/mmm
Triclinic Cl 1
(no rotational symmetry) _
Ci-S2 T
Monoclinic Cs 2
(one 2-fold axis)
C2 m
C2h 2/m
Orthohombic 2v 222
(three mutually V= mm2
2-fold axis) 2
Vh= 2h "um"
Tetragonal S4 4
(one 4-fold axis) _ '-
Vd-D2d 4
C4 4/m
C4h 422
C4v 4mm
D4 'ZZm
D 4/mmm

133

Table l.--Continued

 

- A-

Schoenflies International

 

Crystal Type Notation Notation
Cubic T 23
(four 3-fold axis) T m3
h
Td 432
0 '33m
0h m3m

 

Table 2. Cross reference for irreducible representations.

 

 

 

 

Dimension 1 l 2 2 2 3 3 4
Bethe I‘1 r2 r.3 F6 1—7 I74 I”5 F8
Mulliken Al A2 E E1/2 E5/2 T1 T2 G
Variations B1 B2
Subscripts:
g = gerade (even with respect to inversion)
u = ungerade (uneven with respect to inversion)
Superscripts:

prime indicates double group representation

10.

11.

12.

13.

14.

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‘30

31

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