n} A. . 512'. ’

VIBRATIONAL PROPERTIES OF ‘
IMPURE QUANTUM CRYSTALS

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
RICHARD D. NELSON
1972

 

.H, F:
and

    
 
 

L10:

 
 
 

RY

Michigar Scam
Lhnverflyr

This is to certify that the

thesis entitled

Vibrational Preperties of Impure
Quantum Crystals

presented by

Richard D. Nels on

has been accepted towards fulfillment
of the requirements for

Ph.D. Jegreein Physics

 

 

cw; \JAA M

 

Majorptot‘essor
W .14.. Hartmann
Date 11 Feb. 1972
0-7639

       
       

‘-
_.

2? amomc av
IIUAG & SONY ‘
BIIIIK HINDU" INC.

LIBRARY SINGERS

ABSTRACT

VIBRATIONAL PROPERTIES OF IMPURE
QUANTUM CRYSTALS

BY

Richard D. Nelson

Quantum crystals lack well localized atomic motion
making classical treatment (Born-von Karman) inapplicable.
Beginning with an equation of motion for the double-time
thermal Green's function for atomic displacements, we derive
a Dyson equation for the imperfect quantum crystal. The
self-consistent force constants are found to be determined
by both the bare two particle interactions and the particle
dynamics. An isotopic substitutional defect will have
vibrational properties which differ from those of the host
atoms and will therefore induce force constant changes.
Quantum crystals are the only systems where a mass defect
induces a force constant defect. The equation of motion
was solved in several approximations for the induced force
constant changes. The results are applied to a calculation
of defect induced spin-lattice relaxation time, displace-
ment correlation functions, specific heat, and thermal

conductivity.

Richard D. Nelson

To complete this work theorems concerning phonon
lifetimes and static distortion fields are derived in the
Appendices. In calculating phonon lifetimes using displace-
ment propagators (descriptor of lattice displacement waves)
it is necessary to ascertain the relationship between the
phonon prOpagator and the displacement propagator. This
relationship is found as well as the relationship between
the associated T—matrices, which are a simple way of
expressing phonon lifetimes. Several derivations of phonon
lifetimes are given. Lattice distortion near a defect has
an important effect on the phonon scattering rate and a

simple general method for calculating it is found.

VIBRATIONAL PROPERTIES OF IMPURE

QUANTUM CRYSTALS

BY

Richard D. Nelson

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Physics

1972

DEDICATION

I dedicate this thesis to my wife, Leigh, for her

patience with my mumblings and this physics in the first

two years of our marriage.

ii

ACKNOWLEDGMENTS

Sincere thanks go to Professor William M. Hartmann
whose guidance and friendship cannot escape a warm citation.
I wish to thank my wife, Leigh, for her fortitude in helping
prepare the rough draft of the original manuscript. To
Dr. S. D. Mahanti, I owe appreciation for the physics he has
taught me in the last two years. Finally I wish to express
gratitude to the following members of my thesis committee
for consenting to criticize this work and in several cases
for their frequent encouragement:

Dr. William M. Hartmann
Dr. Gerald L. Pollack
Dr. Wayne W. Repko

Dr. Thomas A. Kaplan

*****

iii

TABLE OF CONTENTS

CHAPTER
I 0 INTRODUCTION 0 O O 0 O O O O O O O O O 0

Historical Review . . . . . . . . .
Quantum Crystals . . . . . . . . . .

II. THEORETICAL DEVELOPMENT . . . . . . . .
Derivation of Equation of Motion . .
Commutator Theorems . . . . . . . .
Force Constants . . . . . . . . . .

III 0 APPLICATIONS 0 O C O O O I O O O O O O 0

Perfect Crystals . . . . . . . . . .

Debye Temperatures . . . . . . .

Phonon Dispersion Curves . . . .

Perfect Crystal Green's Function

Defect Calculations . . . . . . . .

Approximate Inversion Scheme . .

Thermal Conductivity . . . . . .

Defect Specific Heat . . . . . .

Phonon Phase Shifts . . . . . .

IV. CONCLUSIONS 0 O O O 0 O O O O O O O O 0
APPENDIX

A. COMMUTATOR THEOREMS . . . . . . . . . .

B. STATIC DISPLACEMENTS . . . . . . . . . .

C. HARMONIC OSCILLATORS AND STATISTICS . .

D O DENSITY OF STATES O O O O O O O O O O O

E O SCATTERING O O O O O O O O O O O O O O 0

REFERENCES 0 O O O O O O O O O O O O O O O O O 0

iv

20
20
23
28
34
34
34
35
35
40
51
54
60
61

65

68
79
86
90
96

114

TABLE

LIST OF TABLES

Potential Parameters for Solids

Perfect Crystal Debye Temperatures for

BCC 3 He 0 O O O O O O O O O O 0

Thermal Conductivity Experiments

PAGE

15

36

55

LIST OF FIGURES

FIGURE PAGE
1. Helium-Three Phase Diagram . . . . . . . . . . vii
2. Helium Lattice Potential . . . . . . . . . . . l6
3. Phonon Dispersion for Helium—Three . . . . . . 37
4. Phonon Dispersion for Helium-Three . . . . . . 38
5. Perfect Crystal Green's Function . . . . . . . 39
6. Helium Force Constants . . . . . . . . . . . . 42
7. Helium Force Constants . . . . . . . . . . . . 43
8. Phonon Phase Shifts . . . . . . . . . . . . . . 62

vi

om

om

ZfimwdHQ mmdmm mmmmBIZDqum

OH

.H HMDOHM

 

oom

mo:

1

l

 

 

000m

 

OOH

0cm

Vii

CHAPTER I

INTRODUCTION

Historical Review

 

Quantum crystal theory is a transfiguration of
classical lattice dynamics in an attempt to overcome the
failures of the quasi-harmonic expansion of interparticle
interactions and account for the quantum mechanical behav-
ior of the atomic motion in some crystals. For historical
perspective we begin with a cursory review of the infancy
of lattice dynamics followed by a proleptic discussion of
more recent developments.

In Principia (c. 1686) Newton began the study of

 

lattices by using a one dimensional lattice of harmonic
springs to calculate the velocity of sound in air. He
considered only one dimension because the three dimensional
problem was insoluble; partial differential equation methods
had not yet been discovered. In 1753, John and Daniel
Bernoulli1 created the idea of proper vibrations. Lagrange2
(1759) supplied mathematical connection between the con-
tinuous and the discrete vibration problems. These last

two developments were both heavily criticized at the time

because the principle of superposition was disputed until
Fourier's proof in 1807. That the velocity of sound
depended on wavelength was discovered by Baden-Powell3
(1841) and Cauchy (1830). They both understood that the
velocity at long wavelengths was a constant but their
velocities at short wavelengths did not agree with experi-
ment. The concept of group velocity was yet to be discov-
ered. Lord Kelvin4’5 understood the significance of phase
velocity and a maximum lattice frequency in 1881.

The first mechanical and electrical filters were
conceived as a result of response properties of discrete
lattices. The first mechanical filter was built by Vincent6
in 1898 to test the ideas of Lord Kelvin. The first elec—
trical filter was built in 1906 by Campbell.7 Born (1912)
derived the existence of several branches of the sound
dispersion curve due to several masses in a solid, and with
von Karman8 related the microscopic and macroscopic proper-
ties of a solid. The shortcomings of the Einstein9 theory
of specific heat were partially rectified by Debyelo when
he related the spectrum of allowed frequencies to elastic
properties. Of importance to the study of defects in solids
are two theorems of Lord Rayleigh.ll These are applicable

to the case of a single defect which differs from the host

atoms in mass and/or harmonic interactions.

1. Except for frequencies at a band edge the frequency
of no mode is shifted by more than the distance to
the next unperturbed frequency.

2. Modes at band edges may split off and enter the

interband forbidden gap.

Lord Rayleigh's12 discovery in 1885 of elastic
surface waves explained at that time the violent surface
waves which follow the volume waves of earthquakes and has
recently had a very important practical manifestation
(acoustic surface wave devices) in microelectronics.13

Classical lattice dynamics as formulated by Born and
von Karman considers a Hamiltonian with a kinetic energy and
a pairwise potential which is expanded in a Taylor series in
powers of atomic displacements. The symbol R will denote

unit cell positions and u will denote atomic displacements

from mean atomic positions.

P: 1
H=2——°‘-+—XV(R..+u..) (1)
g 2m 2 ij 1 13
P2 3V(R--) 2
= 2: Ji+ %_- >;,(V(Ri.)+ ——1—J—u + zumaaz m u 8
2 2ma 13 3 auja 3“ a i. 38 3

Latin indices label unit cells in the lattice and
Greek indices refer to cartesian components and different
atoms in the unit cell. Double subscripts on a Latin
variable denote a vector between two unit cells. As is
usual in small oscillation theory only the third term in
the potential is kept. The first term in the potential is
an arbitrary constant, the second term is zero for a lattice
in equilibrium and the third term represents the lowest
order term of importance. The coupling second rank tensor
between the displacements in the term quadratic in displace-
ments is known as a force constant (¢). With the Hamiltonian
the equations of motion for the atomic displacements can be

found from Newton's second law.

 

1d 1
H = 2 + — 2 u. ¢.. u. (3)
£a2ma 2 ij 1a 1% 38
a8 a
mula = — a? ¢££'u2'8 (4)
8
08

Equation 4 is a set of coupled linear differential equations.
The order of this set is the same as the number of unit
cells in the crystal to be considered. The solution to the
above equations is made possible by the translational sym-
metry of the lattice which implies that the equations of

motion can be diagonalized by running waves.

i(k-Rx-wt) (5)

Insertion of the above into the equations of motion,
Fourier transforming the time variable, and summing on unit
cells leaves a simple equation with dimensions equal to

three times the number of atoms in the unit cell.

2 _ (
mam ua(k) — 2.9a8(§) ”8(5) (6)
_ -ik-(£-£')
Da8(k) — Z¢££,e ~ ~ ~ (7)
1 a8

D is known as the dynamical matrix. The sum in

equation 7 is independent of 2 prime because of translational

symmetry. Equation 6 is solved by equating the determinant of

coefficients to zero. The dynamical matrix has eigenvectors

o which obey orthonomality and closure relations.

_ 2 —
lDaB (5) mm 6a8| — o . (8)
j _ 2 3

EDGE (§)Ua (k) — mwjk 0a (E) (9)
o ‘ j -

2033(1‘) COL (.15) _ 6jj' (10a)
*3 j _

I 0a (5)08 (k) — Gas (10b)

3

In order to effect a transformation of the
Hamiltonian to quantum mechanical form, a set of normal
coordinates will be found and conjugate variables will be
found using Lagrange's equations.

Although the kinetic and potential energy in the
harmonic Hamiltonian do not commute it is still possible to
find a transformation which simultaneously diagonalizes both
provided the kinetic energy is a positive definite form and
the potential energy is an arbitrary quadratic form in the
same number of variables.16 Moreover if both terms are in

addition symmetric the eigenvalues of

|¢ - mwzl = o (11)

are all positive.17

The principle axis transformation is given by

expanding the displacements in terms of plane waves

1 ° .
Ra r—NMa 15] )5] a

Wave vectors are labeled by k and branch indices by j.
If it were not for the mass term in the expansion
the principle axis transformation would be unitary. In

terms of the normal coordinates ij the Hamiltonian becomes

- i .*' O 2 *
H - 2 Z ‘ng 91;: + “’12: °1sj 012:"

_ l t 2 t
- 2 2 (ij PEj + ij Q]:j QEj) (13)

using Lagrange's equation to find the conjugate momentum
to the normal mode coordinate Q.
The Hamiltonian is now quantized by requiring the

conjugate P and Q to satisfy commutation relations.

[ijI Pkljl] = i“ 6"I6k’kl+G (14)

G is an arbitrary reciprocal lattice vector.
In order to obtain the second quantized form the
coordinates Q are expanded directly in annihilation and

. 18
creation operators.

= +
ij Fakj + a—kj] (15)

~ 3k
Mm .
= _' _;£l _ +
ij 1 2 [akj a_§j] (16)

The results of using the annihilation and creation
Operators are listed below. Details may be found in quantum

mechanics books such as the texts by Dirac or Messiah.

l l
H = Z w . +. + — - Z w . N . + — 17
[a af ] = 6 6 (18)
kj' kj jj' k, k'+G

 

[(a1)nl (a§)n2,--- (ag)9k ...]

[/Ql‘ /£2! AQET— ...1

~

Inln2n3...nk...> I0 0.. 0.. >

 

(19)

The ket on the LHS of the last equation above represents a

many phonon state with occupation numbers n1, n etc.

2' nk'
and the ket on the RHS represents the many particle~vacuum
State.

While Lord Rayleighll considered the problem of a
single defect differing from the host particle in mass
and/or harmonic interactions (point defect) the beginnings
of the current analytic structure of the defect problem were
contributed by Lifshitz19 in Russia and slightly later in

time by Montrollzo’21

at the University of Maryland. The
method in the papers of Montroll and Lifshitz was a Green's
function method for the classical equation of motion

(equation 4).

2 (mm2 5 5 + ¢£2.) G . u = a a (20)

82' av lfi'

This Green's function is labeled G and the same symbol will
be used throughout this thesis. If the atomic displacements
are expanded in terms of normal mode coordinates as in
equation 13, then an equation for the normal mode coordi-

nates of the defect lattice can be found. The normal mode

coordinates of the defect lattice can be used to determine
the eigenfrequencies and the mean squared atomic displace-
ments of the defect lattice. In the following set of
equations chi represents the eigenvectors of the force
constant matrix. Perfect crystal normal mode coordinates

are labeled (k,j) and defect crystal normal modes by (f).

ufia = :j ijxza(53) perfect crystal (21)
X£a(§j) = fi%— 0; (E) elE°R£

a
ufia = E Qf X£a(f) defect crystal (22)

2’!le
BY a8 BY
_ _ 2
C££. — AMa(£) w 5a8511' + A¢££, (24)
a8 a8
det Il-GCI = o (25)

Equation 23 is the general result for the eigen-
vector of a differential or matrix operator having Green's
function G and possessed by a perturbation C. Equation 25
follows directly from equation 23 and determines the new

eigenvectors and eigenvalues.

10

As representative of the type of calculations
performed with this classical (Newtonian equations)
Green's function method we reference the work of Dawber

and Elliott22'23

where the vibrational and optical prop-
erties of an impure crystal are studied. The difficulties
with the classical Green's functions were threefold. The
equations lacked a quantum mechanical foundation. The
calculation of the eigenvectors was a serious numerical
difficulty. And finally, there was lacking a physical
interpretation of the Green's function. The first and

second difficulties were resolved in the case of a harmonic

Hamiltonian by Elliott and Taylor24 in 1963. They use a

retarded Green's function of the type described by Zubarev.25
_ _ 2ni _

G12). (1:) — T 0(t)< [u2l8(t) Iu£|8(0)]> "'

a8
2n
__ << u£a(t)l u£.B(0)>> (26)
M

<Q> = tr(e'BHQ)/(tr e'BH) . B = M/kT (27)

Beta is the inverse temperature. Taking two time deriva-
tives, equation 20 is rederived. The quantum mechanical
foundation is buttressed by taking the time derivatives
using Heisenberg's equation of motion. Needing the defect

lattice eigenvectors (see equation 23) is circumvented by

ll

calculating the defect crystal Green's function by a Dyson

equation

G = G0 + GOCG (28)

where matrix multiplication is implied. The calculation of
mean squared displacements is simplified by a general rela-

tion between a retarded Green's function and its related

 

correlation functions:25
G(t-t') = -i®(t-t')‘é[A(t),B(t')]> (29)
<A(t) B(t')> = 1: J(w) e+8we-iw(t-t')dw (30a)
<B(t') A(t)> = i: J(w) e‘i“(t’t')dw (30b)
J(w) = limit _ 1 G(w+i§L-G(w-ie) (31)
6+0 e 21

The minus sign is for bosons; plus is for fermions.
Since we are concerned with phonons the minus sign will
always be used.

The contribution of this thesis to the general
framework of lattice dynamics is to derive a Green's func-
tion equation of motion for a lattice having general two
body potential interactions (no harmonic approximation), to
derive a set of useful formulas for lattice properties, and
finally to probe the physical interpretation of the Green's

function.

12

Quantum Crystals

 

Compared to the history of lattice dynamics, the
active history of quantum crystals is still in its youth.
The subject has been actively pursued now for about six

years. In 1965 deWette and Nijboer26

calculated the eigen-
frequencies of solid helium and found them to be pure
imaginary throughout the first Brillouin zone. Previous

to this explosive catastrophe it had been known that certain
crystals could not be accurately described by classical
lattice dynamics but the deviations from classical behavior
were considered innocuous. Quantum crystals are crystals
where the zero point energy of the basis particles is com-
parable to the binding energy of the particle to a partic-

ular lattice site. This can be expressed using the

uncertainty principle.

(A)2_ 3‘
—.§)fi—‘§%”Eb (32)

The square root of D is a measure of the localiza-
tion of the particle.
— 8 . .
/D = 10 cm. atomic solids

= 10-13cm. nuclear solids

As a consequence of large zero point energy the particles

are not well spatially localized.

13

One of the few evidences of a nucleon solid is the
star quakes of neutron stars. These quakes or sudden
changes in rotational velocity have been interpreted as
the cracking of a rigid surface of the star.

The degree of nonclassical behavior of the atomic
motion in a solid has been historically labelled by a
parameter lambda. Lambda had its genesis in attempts to
write reduced equations of state for solids which would
allow all solids to be described by a single universal
function.27 In terms of the Schroedinger equation for a
solid written in reduced variables, lambda is the coeffi-
cient of the atomic kinetic energy.28 Because lambda is
usually small the atomic kinetic energy is neglectable.
Lambda is defined in terms of the atomic mass and inter-

action parameters.

A = —1"—— V(r) = e f(r/o) (33)
«HE—c
Epsilon is the interaction strength (well depth) and sigma
is the value of the interatomic distance for which the two
particle potential is zero. For the rare gas solids a
Leonard-Jones potential is often used.

12

_ o 5 o
VLJ - '4€[(;) —(E) ]

Note that lambda can be approximately interpreted as the

ratio of de Broglie wavelength to diameter of a basis atom.

14

A measure of localization is implied. Some values of lambda
are displayed in Table 1. For values of lambda less than
one-half, classical dynamics has proved applicable with good
results. In the case of neon quantum corrections amount to
several percent.29 Neon is then a quantum crystal or at
least quasi-quantum. The value for a neutron solid was
calculated by averaging the Hamada-Johnson singlet and
triplet s=0 central potential for two nucleons. The
values for the rare gas solids were taken from Cook.30

To have an idea of what the potential seen by a
helium atom is in the proximity of its lattice site, we have
calculated this potential due to two shells of nearest neigh-
bors interacting via a Leonard-Jones potential. The result
is in Figure 2. The abscissa is the displacement from the
lattice site in the (100) direction of a BCC lattice. The
curvature at the origin indicates the cause of the imaginary
eigenfrequencies of de Wette and Nijboer. Clearly the clas-
sical quasiharmonic expansion makes no sense here. The use
of the harmonic expansion should also be questioned for
several additional reasons.

1. The root mean square displacement of the atoms can

be as much as one-third of the interparticle dis-

tance. The harmonic expansion is essentially a

power series in u/R. In quantum crystals this is

not a small expansion parameter.

2. A Taylor series for an inverse power series con-

verges slowly.

15

TABLE 1

POTENTIAL PARAMETERS FOR SOLIDS

 

 

 

Solid E/kB(K) 0(1) A*
Xe 221 4.10 .063
Kr 171 3.60 .102
02 118 3.58 .198
A 120 3.41 .186
N2 95 3.70 .230
Ne 35.5 2.75 .593
H2 37.0 2.93 1.73
“He 10.2 2.56 2.68
3He 10.2 2.56 3.08
NUCLEON 750 MeV .5 .950

 

V [10.16 x ergs]

16

-.82 - V = I VLJ(ROj+uO)
J
-.83 - 0 5 0 ’2
VLJ(r) = -4€[(;') - (?) 1
-.84 P
8 = 14.11 ergs

—.8 I- 0

5 O = 2.56 A

 

 

I I J l l I 1
-.95 ' . ‘ 1

0 .l .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1

O
u0 = Displacement in [100] Direction from Lattice Site (A)

FIGURE 2. HELIUM LATTICE POTENTIAL (BCC PHASE)

l7

 

 

_ 0 m _ w u n
V(R+u) - (R15) - :0 Vn(o)
vn+1 _ _ m+n (g)
V n+1 R
n
1im Vn+l = - g
V R
n

Both of these difficulties are overcome by not
making the Taylor expansion at all but by averaging the
potential over the two particle distribution function.

The average of the Taylor expansion keeping all even order
derivatives will be shown to be equivalent to using a
gaussian distribution function.

Nosanow31 has done a variational treatment of solid
helium. He finds values of the solid energy which are too
large (in fact greater than zero). However, his values of
pressure and compressibility agree with experiment to within
ten percent. More will be said about Nosanow's procedures
when we calculate force constants.

There are now a number of reasons motivating the
study of isotopic defects in solid helium.

Experimental motivation:

1. There is a large enhancement of the longitudinal
spin-lattice relaxation time in 3He provided by only

a small addition of “He defects. The experiments of

32

Gifford and Hatton find at T==.425 K and specific

18

volume = 20 cc/mole 1-1: .004 + 103x = .004

(l-+ 2.5 X 105x) sec-1. Tau is the relaxation

time constant and x the defect concentration.
Thermal conductivity remains unexplained by a mass
defect theory. This topic is further complicated by
lack of agreement between different experimentalists.
A chart of experimental results is included in the
discussion of thermal conductivity.

Mixtures of helium isotopes undergo phase separation
below a critical temperature. Phase separation has
been studied experimentally by Edwards, McWilliams,

33"34 and theoretically by Mullins.35

and Daunt
Phase separation in solid helium mixtures may be of
further interest because it is perhaps the only

solid to undergo phase separation in laboratory

times.

Theoretical motivation:

Phonons exist. The elementary excitations of a
harmonic crystal with translational invariance are
phonons. Solid helium lacks both of these criteria
because of isotOpic disorder and anharmonicity.
However, phonons are still observed experimentally.
Harmonicity is salvaged by use of an effective two
particle interaction in the solid which is related
to the free two particle potential. This was first

demonstrated by Nosanow.31 In the case of defects

2.

19

the phonon lifetime can be related to the imaginary
part of the phonon self energy.

A quantum crystal is the only system where an
isotOpic defect induces a force constant defect.

The interparticle interactions will be determined

by the particle dynamics. Changing the dynamics
with a mass defect will change the interparticle
interactions, to wit, the interparticle force
constants.

In a classical crystal a mass defect changes the
density of vibrational states and consequently the
vibrational specific heat. In a quantum crystal the
aforementioned force constant change will compensate
for the mass change. The degree of this compensa-

tion is discussed later.

CHAPTER II

THEORETICAL DEVELOPMENT

Derivation of Equation of Motion

 

This thesis is an attempt to deduce the collective
behavior of quantum crystals by a Green's function tech-

nique. As is well known,36

the poles of the Green's func-
tion yield the excitation spectrum (viz. phonons) and the
Green's function itself is a linear response function. It
will be later shown that the density of states is prOpor-
tional to the trace of the product of a mass matrix and the
Green's function and also that the defect crystal Green's
function contains the phonon scattering rates. From the
density of states all of the thermodynamic properties may

be readily calculated. The displacement double-time Green's

function is defined as:

Zni _
G££.(t) = --H_ 0(t)<[u£a(t),u2,8(0)]>:

08

<<u£a(t)lu (0)>> (34)

2'8

20

21

_ 1 t>0
0(t) — {0

t<0
[A,B] = AB-BA

<Q> = tr (e-BHQ)/(tr e-BH) (35)

B = M/kT

Beta is the inverse temperature. The retarded
Green's function has the previously mentioned prOperties
(equation 29) which allows the calculation of displacement
auto and cross correlations. We now specialize to the case
of one particle per unit cell. The Hamiltonian to be used
assumes pair-wise interactions and we shall remove into an

Operator form the particle dynamics by using Taylor's

expansion.
H = z :23 + l 2 V(R +u ) (36)
_ PEG 1 11' '.V0 0

The gradient operator is defined to operate on
lattice coordinates only. Using Heisenberg's equation of

motion we take two time derivatives of the Green's function.

G22. = %fi'<<[u£a(t)' H] u£,8(0)>> (38)
08
P (t)
£0 I
<< m ul'B(0)>> (39)

22

 

m G 201

9.22' "' M (0)]>+

5(t) <[P£a(t) I 1127.8

Ilfi<< ””26“" H] “938(0)”

P — 1 iv [P egijwij] V(R )
52,0: ‘ 17 2 ij 80' ij

l-‘

 

II
I
th
~M
A
09
20
I...»
I
09

m£G££,(t) = '2fl5(t) 622'6a8 -
08

u .(t)°V
ZI<< e~13

lj a
j Iu£,8(0)>>V£jV(R£j)

(40)

j)

(41)

(42)

The RHS of the Last equation contains a higher order

Green's function. Shortly this will be reduced to a new

quasi-harmonic series, but first some preliminary theorems

are needed.

23

Commutator Theorems

 

The proofs can be found in Appendix A. A, B, and x
will be general operators (which are distributive, associa—

tive, but not commutative) and we define:

 

 

 

A = A0; A1: (X.A]; A2= [X. IX.A]]; An+1 = [X,An]
(n) 2 n!,
k ' k! (n-k)!
Theorem 1
n
An = z (2) (-1)k xn‘k A xk
k=0
Theorem 2
n
A xn = 2 (i) (-'l)k xn-k Ak
k=0
Theorem 3
m _ P
[ex,A] = -ex 2 ( }) A
p=1 P- P
Theorem 4 If [H,A] = 0, H(t) = eIXt He.lXt then
m - P
[H(t),A] = —4H(t) 2 (It) A
P=1 P! 2p

24

Theorem 5 If ex ey = e2 then
2 = x + y + f a y E (y )
p=l P P p,q=l Pq P q
33° a ((y)) +...,

where ap and aij ... are defined in Appendix A.

Theorem 6
If we define a symmetry ordering Operator 8 such that

n n
S(yx ) = Z x yx

then

n

n- n
+ x8 x = S x
ypx (yp ) ( yp)

Theorem 7

The difference in the ordering of the product of two

Operators is given by the symmetry ordering operator.

y x - xny = - S(y xn-l)
p 9 9+1

25

Theorem 8

where

The higher order Green's function in equation 42

can now be written as a series of lower order functions

using Theorem 3.

u .(t)-V . u .(t)°V - m _ P
[e~23 2’] I u n] = _e~£j £3 2 L—{i—— (ug') (43)
2. P=1 p: “' P
(E£')l = [Efij.v' 132"] (92')n = [B j.v' (92,.)1’1-1]

The term p=l corresponds to the original Green's function.

The average of the above commutators is now done using a

cumulant expansion37’38 which is defined by
<eXt>x = exp {2 fig Kn}
Kl = <x>
K2 = <x2>-<x>2

K3 = <x3>-3<x><x2> + 2<x>3

26

For simplicity we redefine variables in equation 43.

u .-V
-<[eӣJ , 91']> E + <exy>

 

3 j l j ~j
= (e >|
57' i=0
Z w _ p
= <e > f 4 + Z ( :1.) a (ugl) + (45)
p=2 p- p=1 p q ~ 9 q
q=0

The coefficients a are defined in theorem 8
(Appendix A). Note because of the evaluation at 1:0 only
terms linear in y survive. Zn contains the terms of the

th

n-— cumulant function which are linear in y.

The equation of motion becomes:

 

u u .-V
_. _ _ I “'R'J 0°
m£G££,(t) - Zn 6(t) 621'6a8 I <e >{ E' <<Z >> +
08 p—2
f (‘1)p a <<(u ) >>}v v (46)

p=l p! 2' qp+q 8 ji
q=0

<<Z >> = -iO(t)<z>
n n

The term p+q=2 represents cubic anharmonicity. The term

p+q=n-l represents nEE order anharmonicity.

27

We now keep only the term p=l, q=0, the first two cumulants

of the exponential average, and Fourier transform the

result.
A .-v+0 .ovv
- 2 (w) _ _ _ . 23 £3' a Y
m m£G££' ‘ 512'568 $0 9 szvxj an sz'
08 3 Y8

n J a Y
+ g2 e ngvzjvzj Gfll'

yB

(47)

Besides being an exact approach the calculational
methods derived here have the advantage that they are good
for finite temperatures and also include long-range order.

Nosanow's variational theory suffers in all these respects.

-> +
Azj - <u£j> (48)

a u£B>-+<ujd ujB>'-<u£a ujB>m<uj01 u28> (49)

-<u£a><u£B>-<uja><ujB>-+<uja><u£B>-+<ula><uj8>}
The vector A represents the mean displacement of an atom
from its lattice site. In the pure crystal this should be
zero. If not it can be made zero, because of translation
symmetry, by a translation or uniform dilation of the
lattice coordinates. A strain field corresponding to non-
zero A can accompany a substitutional defect. This is

considered in more detail in Appendix B.

28

The bilinear terms in D are determined from the

Green's function.

— . . u m l— . _ _ .
(ulauj8> - lgmgt 2fl_£ eBw-12[G£j(w+10) G£j(w 16)] dw
08 GB
= limit 35 Im I” coth (EB) G .(w+i6) dw (50)
Zn -m 2 2)
0+0 08

In matrix notation the equation of motion becomes

+mw2G = 1+0G (51)

G = (m2 1 - <9)"1 (52)

Force Constants

The Green's function matrix G, unit matrix 1, and
force constant matrix phi have dimensions 3N, where N is the
number of lattice sites. If the exponential Operator in the
force constant expression (equation 47) is neglected the
Classical harmonic force constant remains. The present

result is more clearly understood by Fourier transforming

to an integral operator.

29

B

 

 

 

 

o + :
¢££' = e5 Y 2 YY gay V(R2£,) (53)
08
co iq'R , 00 -iqu -q.g.q -A '-q
= 3 f d3q e ~ ~££ fwea ~ d3R e ~ ~e ~2£ ~VGVBV(R)
(Zn) -w - ~ ~ ~
m m -q.(R -R-A) -q.g 0q
= 1 3 f d3R VaVBV(R)fm daq e ~ 71" ~ 7 e ~ '1' 7
(28) -w ~ ~ ~
522' " 135‘}
1 m —(u-A)-2;% ~(u—A)/4 a B
= 1 1m d3u e ~ ” ' ” ” V££,V££,V(R££,+u) (54)
/(2w)3l.D.T .. 1 ~ ~

The new force constants are just a spatial average
of the Old ones. If the mean square displacements of the
particles go to zero, D goes to zero and the exponential
becomes a delta function. In this limit of stationary
particles the classical result is again obtained. The
introduction of a force constant change by a mass defect

Can now be understood as a change in gaussian width D.

3O

 

 

69(R££,) = 0-00 = vvv - e vvv

= (ev'éD 'V -1)eV°D°V vvv
= (eV°6D -V -1) ¢ (R )

o ~££'

/ —1 I
1 w “'5D22"u/4
= { __ [m d3u e” ~ 90(RM.+u)}-<I>O(Ru.) (55)
/(2n)‘|6DT “ ~ ~
u' = u-0A

If the change in width 6D tends to zero, the
gaussian becomes a delta function and the change in force
constant disappears.

While the spatial density function for a harmonic
oscillator is a gaussian (see Appendix C) the two particle
density function for quantum crystals cannot be quite gaus-
sian because of short-range correlations which might arise
as a result of a hard core interaction or of fermi statis-
tics. For solid helium the hard-core interaction is respon—
sible for short-range correlations (SRC). For the short-
range correlation function we may either develOp the above
formalism further (SRC should exist in the neglected terms
of the equation of motion) or assume a phenomenological form
used by Nosanow. Our objective is to study defect properties;
we have therefore done the latter. Nosanow's idea is to use

two particle wavefunctions of the form

31

*6
'JU

:1
II

ij ¢i (Ri +ui ) 0. J(Rj+uj ) fi j(Rij+ uij) (56)

fij = e-KV<Bij+Bij>
where 0 is a single particle wavefunction and fij is the
short-range correlation. The product of the single particle
gaussians is a result similar to ours (equation 54) except
our gaussian width contains long-range correlation effects
in addition to single particle effects. Nosanow determined
parameter K and the gaussian width by minimizing the total
crystal energy. He found K to be the same for many molar
volumes and for both isotopes. We therefore accept K as a
constant and determine the gaussian width (DOB) by self-
consistency in equations. Jackson and Feenberg39 have shown
how to absorb the SRC effects into an effective potential so
that the many-body wavefunction can be approximated by a
product of single particle gaussians. This product of
gaussians then reduces to our earlier result for the two
particle density function in equation 54. The effective

interatomic potential becomes31

2 2
V(r) = f2(r) [v (r) - 2215 v ln f(r)] (57)

The second term is a correlation to the kinetic energy.
From equation 52 it is evident that diagonalizing the

Green's function is equivalent to diagonalizing the force

32

constant matrix because a unitary transformation which
diagonalizes a matrix also diagonalizes its inverse. In
the perfect crystal case where translation symmetry exists
the force constant matrix can be inverted by the plane wave
transformation

1
s = (Nm) 2 0;(q) e18 5

where sigma is an eigenvector of the dynamical matrix.

Different branches of the phonon spectrum are distinguished

by index j.
2 l iq'(£-£') (58)
268(3) ' fifi' 2. ¢22' e ~ ~ ~
22
~~ dB
=15: 2 4“. 9,1848% ’ (59)
2-2'
~ ~ 08
j = 2 3'
22018“? 08 (g) wig 001(3) (60)
2d 2'8 2
Z S . 0 S , , - 5 , 6..,w
22' 53 22' 5 3 55 33 5
dB 08

Applying the transformation S to the pure crystal Green's

function we find

33

1..
_ 9.01 0 1'8
688' ‘ gi. 555 sz'sk'j'
jj' qp GB

The equation of motion can be comp
case of mass and force constant de
Epsilon will denote the negative 0

change.

[m+(m'-m)] sz 1+(0+50)G

(m-m')/m

(mw2-¢)G l + [04+ m6w2]G

1 + CG

The matrix C contains all defect i
on the left by the perfect crystal

a Dyson form and a T-matrix form 0

G + G CG
0 o

T

G + G TG
o o o

If the above can be solved all of

= Ojj' (Skkl

~~

(61)

2

(L) --(.L)2

kj
leted by considering the

fects in the lattice.

f the fractional mass

(62)

nformation. Multiplying
Green's function gives

f the above equation

(63)

l

C(1-GOC) (64)

the phonon properties Of

the imperfect crystal will be known within our self-

consistent approximation.

CHAPTER III

APPLICATIONS

Pure Crystal Properties

 

Before beginning the calculation of defect
properties our formalism should be tested by a calculation
Of perfect crystal properties. We calculate some Debye
temperatures and later some phonon dispersion curves.

The Debye temperatures are determined by equating
the mean square lattice frequency calculated in a force
constant model with the same quantity in the Debye model.
A calculation of Debye temperature has also been done by

40 Their calculation used

de Wette, Nosanow, and Werthamer.
Nosanow's variational parameters which were found by mini-
mizing the ground state energy. Our self-consistent calcu-

lation is compared for reasonableness with experiment and

Nosanow's results in Table 2.

2__l_
<‘*’>‘3N

(WM

=‘% w Debye model

34

35

$0
2
00

m

2 (l-elg % force constant model

)3 )
850'

_1_
3N

We use the following definitions:
1. N(1) is the number of lattice sites in shell 1.

2. The volume per mole of a crystal is V.

_ _ 2 1/3
3. XA-qmale— (60 /V) R)

4. The nEE spherical Bessel function is jn'

5. The average of the force constants linking the

AER shell of atoms to the origin is ¢aa(1).

_ 5 1 j1(5))
0) — 3?“- )2 (Daa(}\)N(}\) (3' " T) (65)
a 1

 

On the following pages are some phonon dispersion
curves calculated using equation 60. The upper curves are
longitudinal modes and the lower curves (often degenerate
in these symmetry directions) are the two transverse modes
for B.C.C. helium three. The purpose of their presentation
is to show that the self-consistent harmonic approximation
produces normal innocuous phonons, in contrast to the
unstable modes with imaginary energies found by Nijboer
and de Wette. Figure 5 shows the real and imaginary parts

Of the perfect crystal displacement Green's function.

 

 

vom. mom. m.om H.mm u- mv.m no.mH
nu mam. nu m.am «.mm om.m Hm.mH
6mm. Hmm. m.mm m.om m.km mm.m nn.om
mmm. Hem. m.mm ~.mm m.H~ mo.m mm.mm
6
23 Na mmm. Na mom. x o.mm x n.m~ M m.na a mn.m waoe\oo mv.vm
O 0 O
ABOOMmozv AOOmamzv Azocmmozv AcOmHOZV Amxmv OOOMUmHQ mEOHO>
ca 86 800:6 800:6 600:9 22 oAMAommm
Hoo Hoo

 

 

(1

mmm 00m mom mmmbedmmmZMB mummn Afifimwmv BUmmmmm

N mqmdfi

01/1012 (radians/sec)

 

37

.l .2 .3 .4 .5 .6 .7 .8 .9 1.0
111 Direction

(k/k zone boundary)

I?IGCHU3 3. PEKNNONIIDISEWKRSIIHQ FCHQJHELJIHH-TTHIEE
(Molar Volume 24.5 (cma)

38

1.601 m.v~ msaao> umaozv mmmmanzqumm mom onmmmmmHo

Amumocson OOON x\xv
cowuowuflo ooa

2020mm

.v MMDUHM

 

(ass/pal) ZIOI/m

39

Ansov m.vm u masao> umHozv
oneozom m.zmmmo amamwmo aommmmm

Axmsa\3v mocmsvoum

o.H m. m. h. m. m. v.

.m mmDOHm

m. N. H.

 

 

_ J! _

1 q _

0 Hmmm

[—

 

o mmsH

L_l_|.

 

2mm

X9111

33
O

40

Defect Calculation

The second derivative of the potential can be
conveniently written in terms of derivatives with respect
to the interparticle distance. For completeness higher

derivatives are also listed.

 

 

 

VO‘V(r) = 3:01 v”)
r r (1) (l)
vavsvu): “8(v‘2’-V )+6 V
r r r2 r GB r
r r8
= 0‘ (R-T) + 8 r
r 08
VO‘VBVAWr) = rarer)‘ [v(3) --3- v”) +-3-2 v(3)]
r3 r r
_1:_ E. I. (2) __1_ (1)
+ (aaBrz-55Bkr2-Féalr2) (V r V )
VGVBVAVBVM) = ——r0‘r8r"r3 [v(4)— 9 v(3) +-1—5 v‘z) -£5— v”']
r“ r r2 r3

+(rar86X8+rBrA63a+rarxésé+rar30Bx+r8raéax+rxr36a8)X
l (3)._§_ (2) ;i_ (l)

[Jr—3 v r v +r2 v 1+ (5885)8+5815aa+561588)x

l (2) _ l (l)

?- (V r V )

The superscript indicates the order of the deriva-
tive which is evaluated at the interparticle distance. The

ratio of the first derivative to the interparticle spacing

41

is the tangential (I) force constant. The second derivative
with respect to interparticle distance is known as the
radial force constant (R). The radial force constant as

a function of gaussian width is shown in Figure 6, which has
several physically interesting interpretations. The ordi-
nate intercept is the bare force constant, that is the
second derivative of the interatomic potential appropriate
for two free atoms with no spatial averaging. Note that the
bare force constant is negative implying lattice instability.
The maximum divides the curve into two parts. To the left
of the maximum short range correlations are not very impor-
tant; increasing the gaussian width allows a particle to
sample more of the region of positive curvature of the bare
interatomic potential yielding a larger force constant. To
the right of the maximum however the short range correlation
function prevents the gaussian from any further sampling of
the positive curvature portion of the bare potential. In
fact since the wavefunction is spatially expanding with
increasing width more of the negative portion is now sampled
and the force constant decreases. The decreasing portion is
roughly linear, a fact which is exploitable for numerical
calculations. The effect of a mass defect on the force
constant depends upon which portion of figure represents the
lattice involved. The helium solids all lie to the right of
the maximum. A helium four substitutional defect in a
helium three lattice will decrease the amplitude Of atomic

vibration and cause a force constant increase.

42

 

Magoo Hm.ma u masao> unaozo mezaemzoo momom onqmm .m mmoon
203:
66
HO
lav 60643 68466666 a
N O ywmv
mm 6m mm om hm am am an ma “a m m gm
m n .1 I L. x 4 o u I m V q o
x
\
ATm
z. + m nnnnnn
m 1.0H
I 3: 05328 0886 H3631 \
1...:
:om
).mm
.0m
4
-.mm
.466

 

 

(Zmo/sfiza) queqsuog aoxog

43

mm.

mxaov m.vm u mssao> nmaozv mezaemzoo momom onqmm .5 mmoon

dd

0
NAav gueflz cmflmmsmo a o
0

mm. mm. om. hm. vm. Hm. ma. ma. NH. mo. 00. mo. 0

 

 

4 . a _ . . _ 1 (1 4( 4 a

   
 

5+9" uuuuuuu 1

 

Amy ucmumcou mouom Hafiomm

 

man

(3
CV

(zmo/sbxa) :ueqsuoa eoxog

44

The calculation of the defect lattice Green's
function involves the inversion of some large matrices.
We divide the crystal into a defect subspace and a comple-
mentary subspace. The defect subspace is modified due to
a crystal defect. The dimension of the matrix requiring
inversion is three times the number of atoms in the defect
subspace. This can be demonstrated by writing the equation

.of motion in T-matix form.

G = G0 + G0 TGO
O O O O O O
G = G11 G12 _+ G11 G12 I'11 T12 G11 G12 (66)
- O O O O O O
521 G22 G21 G22 T21 T22 G21 G22

Elements ll refer to the defect subspace, 22 identifies the
complementary region of the lattice, 12 and 21 denote the

coupling between the defect subspace and its complement.

T = C(1-GOC)‘l
o o o -l
(1_GoC)-1 = l G11C11 G21C11 = (1'G11C11) 0
o O -1
0 l “G21C11(1'G11C11' 1
o -l
c (1-G c ) 0 T 0
T = 11 11 11 = 11 (67)
0 0 0 0

The T—matrix has elements in the defect subspace
only and the matrix to be inverted involves only elements

in the defect subspace. One should not construe this to

45

mean the remainder of the lattice is unaffected. Equation 66
shows that the perturbation introduced by the defect is cou-
pled to the whole crystal. Note that the perturbation of
each element of the matrix G is quadratic in G.

Because the off-diagonal elements (Gi£.) of the

perfect crystal Green's function fall off as I I for

 

r22'

large the perturbations of the diagonal elements of

6”.)

the Green's function far from the defect subspace fall off

as«JL and the off—diagonal perturbations as i.

r2 r
O O O O
O G11T11611 G11T11612
g = Q + o o o o (68)
G211111611 G21T11G12

The necessary matrix inversions have been done by
three methods all of which give qualitatively similar
results for the perturbed force constant. First we used an
Einstein approximation for the Green's function. This
reduces the behemoth to a 3X3 matrix. The second method
used an approximation to the off-diagonal elements which
allowed the inversion to be done by hand, and the third
method used a computer to do an exact inversion. Since the
matrix must be inverted many times in seeking a self-
consistent result the third method is computationally too
expensive to be practical, especially when many shells of

neighbors are considered.

46

As a simple example of the effects resulting from
an isotopic defect we consider an Einstein oscillator
approximation to the equation of motion where the particles
are taken to be independent oscillators. Independent oscil-
lators imply that the Green's function is diagonal on site
indices. The change in two particle force constant will be
written in two parts, the first resulting from the vibra-
tional properties of the new mass and the second from static
relaxation of the cage of neighbors around the defect. It
was noted earlier that the interatomic force constants were
roughly linear in the gaussian width (D). This will be
exploited by using the change in D as an expansion parameter.
In this Einstein model all perturbed quantities will be
expanded to first order in the change in the width parameter
and the resulting equations solved for the perturbed gaus-
sian widths.

The following symbols are used:

8
ll

unperturbed mass
m = unperturbed Einstein frequency

52 = (mz'mi)/m2

¢£2 = force constant for site 1
¢££, = interatomic force constant
n = static relaxation of nearest neighbors of defect
atom

R = radial force constant, second derivative of
intersite potential

47

T = tangential force constant, first derivative
intersite potential divided by the intersite
distance

D = width of two particle gaussian

D0 = width in unperturbed crystal

V(n) = n211 derivative of intersite potential

In general we have that

 

 

 

 

— 2 :_—_
(1 6£)mw Ggg, 0££,+-§ (¢££;+0¢££,)G2£,
But in the Einstein model,
_ 2 =
(l e£)mw G£2 l + (¢£2+<S<I>££)Gu
or
1 l l
G = _ ( +' _ ) (69)
2% 2m(1 €£)w w+w£ w mi
where
w =/¢22+5¢22 = / (Du (1-l- MM)
2 m(l-6£) m(l-6£) 2 ¢££

The last expression is the Einstein frequency for site 2.
The expansion makes less than one-half of one percent error

in frequency if %? is less than twenty percent.

48

 

 

_ - Z w . -iwt
<u£a(t)u£a(0)> —-§:§ fl £m[n(w)+1] Im G2£(w+16) e dw
dd
_ . Z -iwt 8g .1 l 1
——%33 0 Im [we COth(22)2m(l-e£)w (w-w£+iS-+w+w£+16)dw
M cos(wt) sz
= n COth (T) (70)

m(l-6£)w£

We can consider the change in force constants due to

relaxation. We do a cubic average of force constants.
= = EX. _
‘1’) NTTA 2:) (1’02 22”?) '1) + 5xy

For the first shell of a BCC lattice the force constant

becomes
41 = (Rl+2T1)/3 (71)

(3)4_g
r

(R—T)1/3 (72)

Eta is the relaxation of the nearest neighbors of the defect.
This effect is considered in detail in Appendix B and can be

written here as

1 1 3 3
v( )-vc() )+2(DOVC(, )-DV( ))

 

 

R+Ro
(1) (3)
3V (3) av
80 + 2{“79 '13— D9]

= 5D

2?

 

49

6¢relax E 06D

The effect of the change in mass on the intersite potential

is taken as the slope of the effective interatomic potential

versus D.

 

 

69 = 00D
mass
00 = (p+a)6DEEA6D (73)
D=£(<uu>+<uu>)=D+0D
2 O o l l o
D +0D = l M coth (Bwo/Z) + M coth<8wl/2)
O 2 2m(1-e)wb 2m(1)l

 

 

 

 

 

 

_ 0-(6 _ _ l 5
“'1 ' TE ‘ “E (l 1671?)
1 6
D +5D _ M coth (BwEZZ) (1“? 7?)
4me /'1-_e—
. 1 -15 - /_T- l5
x 1-8wE‘71rz‘1’2‘g' ( l-e 1) 2 ‘95
8w BM
—".3. ' h (J)
L cosh ( 2 ) Sln 2 J
+ (1__£§_?;) l+§mE (5(1) / COSh(B—u3) Sinh(-B':”£)
16 q) “in" 2 2

50

Using 0¢==AOD the above can be solved for 0D

 

 

 

l
(1‘. BwE[l" If? ] )_ 1
SD = 22')( 1:? posh (BwE/Z) Sinh (BwE/z)
2
l 2 l
1+ADo 1 + 1+ 888 ['8‘_1-e+ Te
46 1-6 8

cosh (BwE/Z) sinh (BwE/Z

In the low temperature limit (BwE>>l) the temper-
ature dependence is exponentially small. Taking the zero

temperature limit

 

D T—1 -1
(SD = —9- A65 (74)
2 l+__o _]_'_ + l
400 l-e 8

 

The numerator is the result for a mass defect theory
and the denominator is the self—consistent renormalization
due to quantum mechanical effects (change in shape of the
two particle wavefunction). Evaluating the derivatives in
equation 73 for a 3He host lattice with molar volume 24.6
cm3 we find a large self-consistency effect. The denomi-
nator of equation 74 gives a self-consistent enhancement

of sixty-seven percent.

51

ADO _ r _
-——'- .406 E— — -1.9%
4¢O 0
molar volume = 24.45 (cm)3/mole
5D
—— = -ll.2%
Do
6¢ AOD
-—-= ———-= 18.1%
¢0 90
A = p + a

Approximate Inversion Scheme

If we consider a model where only force constants
which connect an atom to the defect are perturbed and the
force constants connecting neighbors of the defect among

themselves are unperturbed, GOC can be written as

O _ O _ O o _ o _ _
O
(iGikaOij (75)
_ o _ o o _ o _ o
O O O O 0
Cl: 0
1 10
O O O 0
(G01 G00'C1 (G02 G00'C2 ""

 

 

52

where the defect has been placed at the origin and Ci is an

abbreviation for Ci For a cubic crystal Gii is diagonal

0.
on cartesian coordinates. Taking all matrices to be diag-
onal on cartesian coordinates the model in equation 80
becomes three one-dimensional models. Now Gij depends only
on Ii-jl and within any shell of neighbors about the defect,
diagonal elements in GOC are equal. To complete the approx-
imation note that the off diagonal elements (GIj-GIO)Cj are
the same except for Gij which is a function connecting dif-

ferent sites within a shell of atoms surrounding the defect.

We replace this Gij by

—3 l
G) '7'

O

O 0
G02

(3 03+G04’

__ l o
— N) z Gij -—> +3G (81)

3 (1.36))
for the first shell of a BCC lattice.

Using one shell of neighbors a force constant

 

defect (C), and a mass defect mewz, (l-G80C-m6w2G80)
becomes.
(1-1) -a -a -a .... 1
-0. (l-A) -0. ‘0. o o o o A
-a -a (1-1) -a .... A
l-GOC = -a -a -a (1-1) . . . . 1
A A 1 1 .... A
O

 

53

For a BCC lattice and one shell:

0 O

5 = (Goo'Glo)C
_ _b_ 0
a — (G GlO)C
Y = A-+7a-m€w2 G0

01

YO 1 81 G00(m€w)

...v

(1-GOc)‘l = 5%? B Det = Det(l-GOC)
Det(l-GOC) = (1-)+a)7(1-1-7a)y0-8(1-)+a)7y)
B11 = (1-)+a)6[(1—1-6a)yO-7y)] = Bii (i#0)
B12 = (1-)+a)6(ay0+)y) = Bij (i#j, i#0, j#0)
B01 = -1(1-)+a)7 = Boi (i#0)

B10 = -y(1-)+a)7 = Bio (i#0)

B00 + (1-)+a)7(1-)-7 )

The final defect calculation performed was one using a
computer to do the necessary matrix inversions. The defect
equation of motion was first rewritten as two equivalent
equations because the mass defect equation can be solved
exactly and easily. A force constant defect equation of
motion was then solved self-consistently using the mass

defect Green's functions and equations 84 and 85.

54

G = G0 + GO(6¢+m€w2)G (83)
_ 0 0 2
Gm — G + G mew Gm (84)
G = G + G 69G (85)
m m

The results of this model are listed below.

N33525:: 5% HS) {137% (SB) 1%;- (5%) Coordinate
1 -9.1 14.0 -1.2 § (111) 8 sites
2 -7.5 3.1 -0.5 (200) 6 sites
3 -6.95 -1.4 -0.2 (220)12 sites

Thermal Conductivity
A number of calculations and experiments on solid
helium have been reported. Aside from questions of quantum
crystals, solid helium has the historical significance of
having been the first solid to exhibit phonon umklapp
scattering in 1951, twenty-two years after Peierls' pre-
diction in 1929.41 A summary of published results of
analyzing conductivity experiments is given in Table 3
(in references 42-47 and 52-54). Several difficulties
beset the interpretation of experimental results:
1. All calculations assume an elastically isotropic
and dispersionless solid.
2. The necessity for inclusion of three-phonon

42,54

processes seems understood, however, the

various authors differ on the amplitude and

55

.mumu mcflumuumom uomwwc mmmE\wumu mcfiumuuwom powmmo««

.mumu mcfluwuumom cocosm mousse

 

 

80H onma mom uu = o.~a uu o.H =
m.on who mom nu = m.vH nu H.H =z
m.mv omm mom uu = o.ea un m.H me~3zm
m.em mea mom uu 0mm m.mH uu m.a ~a~3 m Lone cmaumm
Goa mmea mom muH = mo.HH uu o.H =
moom. 00m mum II. : moMH II. NoH .-
mNomV OWN mum II mm: mm .GH II m oH .-
o.mv oma mom I) mm: m.na nn m.H = Awov coaumm
uu un uu m.a+m. 0m: m.ma omum omuom :2
un un mom m.H+m. 0mm m.mH omuH H.0H mews m Ammo :msuumm
uu mva mom mua 0mm oanmmo. m.H =
H.mm oea mom muH mm: H.ma Hana n.H =
m.~m om mom muH 0m: m.mH Hana m.m ez
o.mm om mom mua 0m: v.om Hana H.m ~e~3 m Amov cmaumm
z
uu nu uu H.N+H.H mm. nu m.mum. o.m maea m lame smzmaamo
u. z
Axove AEumv Ousuosuum Axove umom AOHOE\MEOV vao 24M<\¢ . an H cofiumasoamo
wusmmmum Ho> «

 

 

mezmSHmmmNm wBH>HBUDQZOU

m mqmdB

A¢2mmm9

56

. 50
frequency dependence of these processes. Herring
has shown that the three phonon scattering rate

for long wavelengths is given by

T-l(k) = ks T5_s

where s is usually taken as 2, the value character-
istic of longitudinal modes in cubic crystals.

3. The experiments are done principally by two groups.
Berman at Oxford University does experiments at
constant pressure, while Bertman at Duke University
does experiments at constant volume. These differ-
ent techniques seem to yield different results for

the defect scattering rate as shown in Table 3.

Rather than to do the full calculation for the
thermal conductivity we shall calculate a scattering ampli-
tude from the scattering T-matrix in Appendix E which can
then be compared to experimental values.

The mass defect scattering can be done exactly in
the low concentration limit where now noninteracting defects
are imagined to exist, while the force constant defect case
is a little more complicated. The model used is one where
the change in potential between any atom and the defect is

proportional to the unperturbed potential.

0 _ o
¢id ‘ ®id - 3 Y1 91d (86)

08 08 0 up 08

57

If Aa8(£) represents the change in force constants between

site 2 and the defect

A22' {-Gid AdB(£)'-6£'d AaB(£')-+62£,Aa8(£)}{l-6£d6£,d}
dB
+zidAaB(£)) éfldéfi'd , (87)
= lq°l J -lq"i *3'
A99. 15' e ~ ~0a(q) A££.e ~ ~0B (g)
33' 08 GB
_ ’ u i(q-q')°d
— A... +A... - -A... - ~ ~ ~ 88
[ J] (g) j] ( g) j] (g g )] e ( )
= iq (2-2') *3'
AJJu(g) _ Ri'e ~ ~ A££.0 (q) 08 (3)
dB us

These equations are perfectly general in the case of no
perturbation of force constants among the defect neighbors.
We now take Y to be a constant independent of site coordi-
nates. This model has been previously considered by Elliott

and Taylor.51 In this scaled force constant model

_ 2

3 31'

Using a multiband Debye model and the total perturbation C,

the scattering T-matrix can be evaluated.

_ 2 i(q-q')-d
A _ 6 o o 2 V I . ' e ~ N “I 90
qq. J]. Y 3 g ( )

Vj = velocity of sound for branch j

 

 

... 2
C22' ‘ A12"*m€w 620520588 (91)
GB 08
o —l
T = C(l-G C) (92)
em? 2yV? q°q
qu' = 3 o + 2 32; ~2 o (93)
jj' l'mew G00 (1+37)‘3 m” G00
08 GB

The T-matrix can be written as a sum of partial waves pro-
vided we take Go(k) to have an isotrOpic dependence on wave
vector.

It is shown by several methods in Appendix E that
the lifetime of a phonon with wavevector k is given by
equation 94. Va is the atomic volume. x is the concen-

tration of defects.

 

 

 

g. : __X_la_. IdaK- ITKK'l2 600-0012) (94)
K (210240)2
K
= 93; 2 Va ITKK, (2 (95)
' 4 V ..
3 n J 33'
2 u 472w“
. e w .__3__ ...
=mVa (—2§ —1§— KO 24" 25 2 0‘2 (96)
V V l-mssz l 1+———-m6w2G
t 2 00 .3 3 00

 

59

Equation 95 was obtained by considering spherical bands and
averaging over these bands.

In thermal conductivity one is really interested in
the rate of loss Of forward momentum rather than the total
rate of momentum loss. This shortcoming is rectified by the
insertion of a l-cos(O) factor which occurs naturally if one

begins with a Boltzmann equation for thermal conductivity.55

x Va

 

55': x dk' (l-cosO) IT (26(w-wk) (97)

I
K (21f)2 4w; KK

This angular factor couples 3 and p partial waves and adds

to equation 96 a term

-2 w)
Va 3 Y
Zn

X

1+1
3.
Vt V2

 

 

*+ C.C. (98)

H

 

(l-msszgo)(l+%g—%myw2680)
The denominators of the scattering rate expressions
were checked for possible scattering resonances (vanishing
of the real part of the denominator). The resonance con-
dition for the mass defect is shown in Figure 5. A reso-
nance occurs if the curve labeled l/(ewz) crosses the real
part of the Green's function. The resonance curve for the
force constant denominators lie in the upper right some
distance off of the graph in Figure 5. The failure of a
resonance to occur eliminates these resonances as a possible

source of enhanced scattering rate.

60

The total relaxation rate appropriate for thermal

conductivity to order m“ is then

 

 

 

1 X Vauuu" 2 1 2 4y2 46y
= —— — + —5- 8+ + (99)
T(w) lZfl Vt Vfi 3(1+%})2 3+2Y

In our calculation of a 1'He defect in a 3He host we found a
value of y which was approximately minus one-half epsilon.
The above equation then implies the scattering rate should
be twice the mass defect scattering rate. This result
agrees more closely with the experimental analyses of
Callaway and Berman than the analysis of Bertman (see

Table 3). There are still several scattering mechanisms
which we might consider. The helium atoms are only very
loosely bound to their lattice sites in the ground state.
It may be that higher excited states are not bound and the
helium atoms actually become nomadic, allowing inelastic as
well as elastic phonon scattering. If the helium four
impurity changes the local exchange frequency a change in

scattering rate might result.

Defect Specific Heat

 

A heavy mass defect will increase the density of
states at low frequencies and will therefore give an
enhanced lattice specific heat at low temperatures. In
quantum crystals a mass defect is accompanied by an induced
force constant change and we would like to assess the impact

of this force constant change on the specific heat. If

61

there is an increase in force constants for an isotopic
defect as in helium there will be a diminution of low
temperature specific heat. The force constant change will
mask the effect of the mass defect. If a decrease in force
constants accompanies a heavy mass defect the opposite
effect would be expected. The phonon specific heat is the

derivative of the total vibrational energy will respect tO

temperature. The change in specific heat per mole is then56
3N kBBZ w
AC(T) =-——z———-fm w2A(w) csch2(Bw/2) dw (100)
B = M/kT

The change in specific heat is expressed in terms of the

phonon phase shift (see equation 147) as

Ag(w) = 523% (101)

where x is the concentration of defects. The phonon phase
shift for a mass defect and for the scaled force constant

model (defined in section on thermal conductivity) is

 

 

_ 2 0 .2 2 O
mew Im G££ 3 mew Im ng \

0 = tan"1 2 a: + 3tan'l A: a“ (102)
1'm5“ Re 622 l+%%w-%%-mw2Re Gig,

C10
(10.

This expression was evaluated for €==-l/3 and y==.15. The
result is displayed in Figure 8. In the very low frequency

limit where the density of states is quadratic in frequency

Phase Shift (radians)

62

 

 

...].1

 

 

 

: 1 1 1" 1 1

 

J 4 l
.l 3 .4 .5 .6 .7 .8 .9 1.0
Frequency
FIGURE 8. PHONON PHASE SHIFTS

(Molar Volume = 24.5 (cm3))

63

the change in density of states can be found from equation
101 and there is then a simple form for the corresponding

very low temperature specific heat.

Ag((0) = §3£3 (103)
1 J (”D
3
0cm = -x(-3-§E + -in—) C(T) 03- << 1 (105)
1+—3Y- D

Substitution of the phonon phase shift into equation

100 and an integration by parts gives

3kaN

 

0cm = - [:0 (100(0) @22 csch2(Bw/2){1—§§coth%‘3} (106)

This expression was evaluated using the phase shift
of Figure 8 and at .2 K eighty-one percent of the enhance-
ment due to the mass defect was cancelled. At 1 K seventy-
five percent cancellation was found.

The change in the ground state energy was also
evaluated and ninety percent of the diminution caused by
the mass defect was restored by the induced force constant

change.

 

1 .1.
2

AE(T) = I” 019(0) dw ( +
O eBw—l

) (107)

64
integration by parts,

x <n Bw Bw/Z
3? a) 0(w) [cothjf-+ -f-§EJ dw
Slnhjr

AE(T) =

AE(T=0) = -— fo°° 0(0) do

(108)

(109)

CHAPTER IV

CONCLUSIONS

In the preceding develOpments we have derived a
general equation of motion for lattice vibrations which is
applicable to all crystals and temperatures but is espe-
cially useful in the case of quantum crystals where clas-
sical methods fail. The formalism developed is exact and
has several advantages over the existing variational theory.
The variational theory minimizes the ground state energy and
is therefore good at zero temperature and only as good as
the trial wavefunctions. This formalism should also be use-
ful in cases where classical calculations are extended by
perturbation theory to include anharmonicity. This possi-
bility has recently been prOposed by Gillis and Koehler57
as a method of treating paraelectric crystals. Spatially
averaging the interatomic potential would reduce anharmo-
nicity and provide an alternative to summing many orders of
phonon interactions. It has been shown how a force constant
change is induced by an isotOpic defect and quantitative
results were obtained by these methods.

The three methods of calculation consisted of an

Einstein model for a solid, an approximate matrix inversion

65

66

scheme for calculating the defect crystal Green's function,
and an exact computer inversion of the necessary matrices.
Although the calculations differed in sophistication the
quantitative results were very similar. This can be under-
stood because the results depended on integrals of the
phonon spectrum and not on a detailed description of the
spectrum.

We can now comment on the experimental effects noted
earlier. It was the original expectation that the enhanced
spin-lattice relaxation rate which occurs in 3He with only
a small addition of isotopic defects could be accounted for
by a resonance in the density of states due to a heavy mass
defect. While the defect is heavier than the host atoms the
fractional mass change is insufficient to cause a resonance;
furthermore, the increase which does occur in the density of
states is cancelled by the induced force constant change.
The effects cancel because an increase in mass enhances the
amplitude of low frequency vibrations while diminishing the
weight of high frequencies. A force constant increase has
an opposite effect. The conclusion is that phonons are
unable to explain the spin-lattice relaxation experiments.
Other inelastic processes which might account for the exper-
imental spin-lattice relaxation results are enhanced vacancy
diffusion and nomadic particles.

The anomaly in thermal conductivity experiments is
a phonon scattering rate which exceeds the value given by

mass defect theories. By considering the scattering due to

67

the induced force constant changes, we find a scattering
rate which is twice the mass defect prediction. This
result is roughly in agreement with most experiments while
considerably smaller than the experimental predictions of
the Duke University group.

An enhancement of the phonon specific heat of
helium because of a heavy mass defect has not been reported.
Experiments measuring specific heat have been done in
temperature ranges (above isotopic phase separation tem-
peratures but below vacancy generation temperatures) where
this effect should be observable. The absence of this
observation is explained by the masking of the mass defect
by the induced force constant changes.

To complete this work we derive theorems concerning
phonon lifetimes and static distortion fields in the appen—
dices. In calculating phonon lifetimes using displacement
prOpagators (descriptor of lattice displacement waves) it is
necessary to ascertain the relationship between the phonon
prOpagator and the displacement prOpagator. This relation-
ship is found as well as the relationship between the asso-
ciated T—matrices, which are a simple way of expressing
phonon lifetimes. Several derivations of phonon lifetimes
are given. Lattice distortion near a defect has an impor-
tant effect on the phonon scattering rates and a simple

general method for calculating it is found.

APPENDICES

APPENDIX A

COMMUTATOR THEOREMS

APPENDIX A

COMMUTATOR THEOREMS

Presented here are some theorems used in the
derivation of the equation of motion for the displacement
Green's function. For two arbitrary operators x and A
(which are associative, distributive, but not commutative),
'we define

A0 = A A1 = [x,A] A2 = [x, All An = [x, An-l]

(n) __ n!
k '- (n-k)! k!

 

Theorem 1

(EH-l)k xn-kA xk

0

An =
k

IIMU

The proof is by induction. For n = 1 the relation is true.

For n = 2, 3, and n + 1:

A2 = [x,[x,A]] = sz -2xAx +Ax2
A3 = [x,Az] = x3A - 3x2Ax + 3xAx2 - Ax3
n n k n-k k]
An+1 = [x, An] = Z (k)(-l) [x, X Ax
k=0
n+1
n n k n-k+lek + Z xn-k+lek( n )(-l)k
= Z (k)(-l) x k-l k-l
k=0 -

68

69

n
= 2 {[(E) + (kfl)] (—l)k xn k+lA xk} +(_1)n+len+l +
k=l
xn+1A
n+1
= Z (n;1)(_l)k xn kAxk
k=0

Theorem 2

(fin-1)k xn‘kA
0

Axn =

k

"015

k
The proof proceeds again by induction.
AX = XA-Al

sz = sz - 2xA + A

l 2
n+1 n n k n-k
Ax = 2 ( )(-l) x A x
k k
k=0
n n+1 k
= 2 (:)(-1)k xn+1-kAk +. Z (knl) xn+1 k Ak(—l)
k=0 k=l
n
= 2 {(n) + ( n )}(-l)k xn+1—kA +xn+1A +_(_l)n+l An+1
k k-l k
k=l
n+1
= Z (n;1)(_l)k xn+1-kAk
k=0

Q.E.D.

70

Theorem 3

A A

_ _ (-1)p
[e ,B] - ep —p—l-—Bp

IIMS

1
This relation is derived by expanding the commutators,

using theorem 2, and explicit resummation.

 

[eA,B] = )3 l (AnB -BAn)
n!
n=0
°° n
= - >3 Ill! 2 (fi) («1)k An‘kBk
n=0 k=l
w n k
A (-l) n-k
[e B] = - Z 2 _ A B
' n=l k=1 (n k)! k! k
A _ (-l) o
'[e '31 " 01—171) El

(-1)1 (-1)2 o
+1—1T—A11B +0121AB2

 

(-l)1(-1)2 (- 1)3

+1sTn—A31 +1T2T—A32”0T§T—A°B

3

('1)1 ('1)2 2 (-l)3
+3TIT"A Bl +2121 A 32 +IT§T"A B3 +

 

(- l)“A
"Df—_A

-L—%L:.A1B4+éT%%:-AOBS

71

 

 

_ (-l) A
sum of column 1 — —IT+ e Bl
_ (-1)2 A
sum of column 2 — -§T——-e B2
_ 3
sum of column 3 = (3}) eAB3
“ _ P
[eA,B] = -eA 2 ( }) B
p=1 P- P
Q.E.D.
Theorem 4
If [A,B] = O and A(t)=elHtAe-lHt then

[A(t),B]=-4A(t)p£1%%2§rsz where Bl=[H,B] BN=[H,BN_1]

The proof uses several applications of theorem 3.

 

 

 

 

. _. w . p _ w _.
[eifltAe 1Ht,B] = _A(t) 2 (1t) B _e th z (- -if)P Ae th
p=l P P p=1 P BP
” ' P _ P m - P _°
= A(t) 2 (if) B -A(t) z ( 1?) lHtA 2 (if) {B ,e lHt}
P=1 P P=1 P P=1 P P
_ “ (it)2p th _ -th ” (1t)pit)q
= 2(t) PE __1B—§-P_f—— + e A( e )pil qu qu! Bp+q
- - P+q _
Coeff1c1ents of t Bp+q for p+q—N
Ngl (-1)p = Ngl (-1)P = 1, g (_1)p N1 + _1_,_(-1)N
p+q=§ plqlq p=l pl(N-p)l N p=0 p!(N-p)! N N!
p:q=
N 2
0 + l-éIl) N' N even
0 N odd

72

[A(t) :B] = '4A(t) Z W B

2
If e ey = e and y1 = [x,y] y [x,yn_l] then

n—

00 00 CD

z=x+y+ 2 a y +

 

p=1P P pq_1an P g pqr-l qu P g
=x++2a +Za()+ (15*-
y p=l pyp q=1 pq yq p pqr_1apqr (yp ) q)

This is demonstrated by writing
etxety=ez 2= 2 z tp

p=l p m

z ptp n
(2 )
m n m+n

2 x t = 2 p=l
mn m n! n=0 n!

Equating powers of t the above form is found. The

[y .y 1 + Z a [y [y yr] + -

coefficients a.. . . . are sums of the coefficients ak.

13

Q.E.D.

Theorem 6

If we define a symmetry ordering operator S such

that

n
S(yxn) z 2 XY Y xn"Y

Then

Y xn + XS(Y xn‘1

_ n
P P )—S(XYp).

Y xn + XS(YP xn'l

n
P ) = Y xn + z x“'7 Y xY

n‘Y Y _
X YP X YP X

II

|-<

X

+
"P15

n
S (X YP).
Q.E.D.

Corollary

n n—l _ n
x YP + S(YP x ) x — S(X YP)

Theorem 7
The difference in ordering of the product of two
operators is given by the symmetry ordering operator.

Y - S(Y n’1)

P P X

P+1

The proof is an induction proof.

n=l YPX = XYP + (YPX - XYP)
= XYP - YP+1
n-2 Y x2 - XY x‘ Y x
-42— P ‘ P P+1
= sz + x(Y x - XY ) ’Y x
P P P P+1
= x2Y — (XY + Y X)
P P+1 P+1
2

X Y - S(XY

P P+1)

 

 

74
Show-n+1
Y xn = xn Y - S(Y xn'l)
P P P+1
n+1 _ n _ n-l
YP X — X YP S(YP+1 X ) X
_ n+1 n _ _ n
_ x YP + x (YPX XYP) S(X YP+1) +
n
X YP+1
_ n+1 _ n _ n n
" X YP X YP+1 S(X YP+1) + x YP+1
_ n+1 _ n
— X YP S(X YP+1)
Q.E.D.
Theorem 8
%X ex eAYI 5 ex Y _‘%_ eX+AZ l
A=0 A=0

where Z = Y+

j=0 3 3
We prove this by expanding the exponential (ex) and
using the symmetry ordering operator before resumming.

First we need the following Lemma.

 

 

Lemma 1
XnY n n-1 n-2
—r-1T=S(Yx.) +alS(le )+a28(Y2x )+ anYn
(n+1)! n1 (n-l)!
n .
= 2 a. S(Y. Xn-J)
j=0 J l

 

(n+1-j)!

75

The proof is by induction.

 

 

 

 

 

_ _ XY + Yx + XY - Yx = S(XY) 1
n-l XY — 2 -—§——-+ 2 Y1
2
_ x Y _ x S(XY) 1
n—2 -—2- — —T—— + Z- XYl
2 2
_ s(x Y) _ Y x 1 _
— '—I_—_ _I_“'+ I’ XYl + le + XYl le
(Using Theorem 6)
S(X2Y X2Y S(Y X) 1 1
= - + l + _ S(XYl) + Y2
4 4 4 8 8
3 2 S(XZY) 3 1
4 X Y + S(XYl) §|+ §' Y2
X2Y S(XZY) s(le)
T=T+a1 —71—+ a2Y1
.. 1 _ 1 z
where a1 - 3 a2 — I2 ao _ l
n+1
n n n-1 n-2
X Y S(X Y) S(X Y1) S(X Y2)
‘HT" TH:ITT’+ a1 n! + a2 (n-l)! "‘ anYn
Xn+lY XS(XnY) XS(xn'lY1)
n! = (n+1)! + a1 n! + "‘ an x Yn
now use theorem 6
Xn+lY S(Xn+lY) S(anl) S(xn'le)

 

“ET"= “TH¥TYT’+ a1 “‘ET‘" + a2 (n-l)! + "'

1 l

n+ n n-
Y X al Y1 X a Y X

2
’W'T' Z‘TrFTST’“'

 

76

now use theorem 7

 

 

 

 

 

 

1 1 S(Xn+lY) S(an )
XnH'Y [— + ] + a ———l— +
n! (n+1)! (n+1)! l n!
n-l
a2 S(X Y2) + ...
(n—1)Tv
n n n-l
+ s(Yl x ) alYl x _ a Y2 x ____
(n+1)! n! 2 n- !
+l a 1
n+1 n+2 S(Xn Y) n l
x Y (n+1)! (n+1)! + S (x Y1) [ET + (n+I) I] +
a 1
n-l 2
S(X Y2) [THZITT + HT ] + ...
alan1 xn'le xn’zY3
’ n! a2 TH=TTT ‘ a3 n-2 1 ' °
= S(Xn+lY) + S(an ) [ a1 + 1 _ a1 ]
(n+1) 1 1 FY (n+1) 1 (n+1) I
a a a 2 a;
n-l 2 l l 2
+ S(X Y2) [ n- 1 +‘HT" ‘3? ""T 1
a a a a a
n-2 3 . l 2 3
+ S (X Y3) [(n-Z)! + n- 1 - (n—I)! - n-
+ O O O
S(Xn+1Y) n-l EK S(an’K'1 YK)
= ‘THIIYT + K31 (fi+2-K)l
El-= (n+1) + a0 - a1
— 2 —
E2 + n a2 + a1 a1 a2
E3 = (n-l) a3 + a2 - ala2 - aza1 - a3

77

 

K-l
EK = (n-K+l) aK + aK-l - .E a. AK—j
j—l
= (n+2) aK - (K+l) aK + aK_1 - g ajAK-j
= (n+2) aK
where the recursion relation for the coefficient aK is
K-l
(K+l) aK = aK-l - Z_ ajaK__3
j—l
_ _ l l _ _ l _
ao ‘ l'a1 ‘ i'az + 17'a3 ‘ °'a4 ‘ 730" a2n+1 ’ °
(ns‘O)
Xn+1Y S(Xn+lY S(xn-K-l YK)
(n+1)! = n+2 1 + Z aK (n+2-K)!
Q.E.D.

The theorem now follows easily. We rewrite the expansion
of-exY in symmetrized form.
Y = Y

3 (XY)

21 Y

+a

 

1 l

2
X Y S(X Y) alS(XY) a

T=T+T+ 2Y

2

x3Y S(x3Y) S(XZY) S(XY)

“T =““IT‘ + a1 ‘—‘§T"+ a2 “§T"+ a3Y3
In resumming, the RHS is summed by diagonals and we use the

definition

78

 

ex Y _ Z + S(ZX) + s(zxz) + s(zx3) +
- ‘71— ‘7‘." T
ex(l+Y) - 1 + (x+Z) + £31§l§§1 + x3 + S(ZSZ’ +
‘ 21 ‘1‘:—
eX Y _ 8 ex (IL-FAY) l = L ex XYI
’ a) 1=0 a e A=0
a (x+12) (x+12)2 (x+12)3
=§xll+T+T+T—+m1 l1==o
_ a x+Az

Q.E.D.

 

 

APPENDIX B

STATIC DISPLACEMENTS

APPENDIX B
STATIC DISPLACEMENTS
Static displacements of atoms from the perfect ‘1

crystal lattice sites resulting from a substitutional

impurity are calculated here by minimizing the total Gibbs ;‘

 

Free Energy with respect to the atomic positions. Other 5’
methods of calculating static displacements will be

critized at the end of this appendix. We divide the

Gibbs Free Energy into a phonon energy, phonon entropy,

configurational entropy and an applied pressure term.
G=V +U -TS -TS +PV (110)
s p p c
=V +F+PV—TS
s c

where

VS = static lattice energy

F = Helmholtz Free Energy of phonons

Sc = configurational entropy

If the substitutional defect is not allowed any
mobility the configurational entropy may be neglected.

Static displacement vectors will be denoted by eta.

79

80

G=VS+F+PV

3? = 53— + 80 (F—FO) (defect lattice) (111)

We consider first the potential term and later the free
energy.

If only the potential term is considered for the
present and every atom is required to be in equilibrium we
may expand the potential about the perfect crystal lattice

sites and determine the static displacement vectors.

3V(ri.)
Z—E—J— =0 for all i
j 11'
o 2 o
0:23V(r1 ) _ 2(3V(ri ) + g v(rij) .n )Ez{v' +¢13'”13}
J r13 rij 3r?. 13 3

A-P-n (112)

II
- M
<
I
M
'9'
J
m

Eta may now be found by inverting the force constant
matrix (P). This is difficult.since P has dimensions

3N X 3N where N is the number of atoms with which any one
atom interacts. Assuming the defect is not displaced and
does not interact with other defects, it is a center of

inversion symmetry and only a spherical distortion

 

81

is expected. In this case the number of coordinates
involved in the relaxation is just the number of shells of
atoms considered. If a vector of direction cosines for

atom j of shell A is defined by I; we find a

 

Amax X xmax matrix equation.

“ijnf 1 l“

1
m 'm m

z r¥-¢§§f§ ri-nk — z M-AA ,2

kijA' l 13 3 i 1 L}
I I

z 0*“ fix = AA (113)

AI

Many shells of atoms are included here because it if often
not possible to consider the relaxation of the nearest
neighbors of a defect decoupled from the remainder of the
lattice. As a trivial example we consider the case of a
single defect in a one—dimensional lattice with nearest
neighbor interactions where the first derivative of the
potential connecting the defect to its nearest neighbor is
changed and no force constants are changed. The effect on
an infinite chain is easily visualized to be the displace-
ment of every particle by an amount n1 towards the defect.

For a 2N+l member chain the equation becomes

82

 

 

 

 

 

 

 

1 ~ 1
_ _ '
-¢ 2P -¢ ... 0 n 0
-¢ 2Q -¢ no. 0 o 0
0 -¢ 2¢ ... 0 . = . (114)
O O O O O 0 .-¢ 0 O
b 0 O 0 0 .- 2¢J nN~ - 0 -
which is easily inverted to give
5v' ‘
_ _ .m_+1;m_>. 10
nm “ N+1 0 (m > O) (115)

which goes to the correct result in the limit N-—*w. The
result of keeping only nearest neighbors is in error by a
factor of 2.

Rather than putting all atoms in a shell in equi-
librium as in equation 113 it is easier (and equivalent) to
put one atom in each shell in equilibrium. The direction
cosines for this chosen atom then can be expanded in terms
of eta for purposes of iterating equation 112.

In the following Vij denotes V(Rij) where Rij is the
vector between sites i and j in the unperturbed lattice and

cos(A,B) is the cosine of the angle between vectors A and B.

+

o = 2 6r V(R..+n..) = Yr {v.; + 7r v..-n..
j -ij 11 13 ij 13 ij 11 11
+ avi.
= Z Vr.;V..4-$r.. Se—l {cos(ij,i)n.-cos(ij,j)n.} (116)
j 13 13 13 rij . l J

 

83

If the defect is a center of inversion symmetry the
sum on j is a radial vector in the direction of site 1. We
therefore take the projection of each term along the radial

vector and leave vector signs implicit.

 

 

A -+ .. = 0
Z11n1
avi.
A. = 2 ———l cos(ij,0i) (117)
1 . 8r..
1 11
l
V 2v..
§——i1-cos(ij,01) cos(ij,0j) 1¢j (118a)
8r?.
11
-z §——il cosz(ij,0i) i=j (ll8b)
L 3r?.
11
We again have a A X A system of equations. One can
max max

include effects of displacements from shells beyond the Amax
without changing the order of equation by using the elastic

limit for these displacements

(n l dimension

nk(k>Am ) = i nz fi; 2 dimen51ons 5L<Amax (119)

 

R2
an (——)2 3 dimension

84
Phonon Free Energy

If the normal modes of the defect lattice are
harmonic phonons the change in the free energy can be
written in terms of a density of states g(w) or the phonon

phase shift 0(w).
6F = kT 4: 5g(w) 1n[2sinh(%§)] dw (120)

It will be shown in equation 147 that

6g(w) = - % 5% 1n Il-GQI (121)

where 0 is the phonon phase shift, G the defect lattice with
no static relaxation, and Q is the change in force constants

due to the static relaxations.

 

 

_ 2 _ 2
Qij — VXyVij(R+n) nyvij(R)
xy '
(2)_ A (3) (2)_ A
=51{(v‘3’-‘-’——-—T-)(r-n)+(v““-V +Y—'-—-T-)(r°n)2}
r2 r ~ r r2 ~
(2) A (3). (2)_ A
+ 5x {V ‘T r-n+ (V V T)(r-nm
Y I' ~ r ~
5 0(1) f-g +1% 0(2) (£°Q)2 (122)
V(n) is the nEh derivative of V. Differentiating and

performing the integrals the free energy contribution to

equation can be found.

 

 

85

 

__ (1) .. .0 _ JP”
IfifijéF — ZXQij Djicos(13,01)ri+2{ 2DijQji 1TfOIm(Giiij
3 3 as Ba 3 08 Ba 08 ya
(1) (l) ‘83 2 .. .
+GijGij)QBY Q50 coth (:2) dw} (cos (13,01)ni
dB yé
-cos(ij,0i)cos(ij,0j)nj) ri (123) IL!
There exist other methods for calculating static é

relaxations. The method of lattice statics by Hardy58 Lg
yields displacements in the elastic limit which in general 3

will not be applicable near the defect. The method of
lattice statics uses the first two derivatives of the poten-
tial and is not easily iterated. Two previous calculations
applied to isotopically impure helium used a continuum model.
Maradudin and Klemens59 neglected the potential energy con-
tribution to the free energy, while Klemens et a1.52

neglected the possibility of an isotopically dependent

potential. Other calculations consider only the potential
60,61

part of the free energy.

APPENDIX C

HARMONIC OSCILLATORS AND STATISTICS

 

.4 nfrnnv
I I fir .

 

APPENDIX C

HARMONIC OSCILLATORS AND STATISTICS

The purpose of this section is to further examine
the relation of gaussians and harmonic oscillators and then
for completeness to gather some statistical relations. The

ensemble average of an operator A is given by

 

 

<A> = trpA=-:- tr(e-BHA) (124)

a e-Bw(n+122) -a2§—2— ..aZZ‘:

= —Z n (H (ax)e 2,A(x)H e 2)
Zr) 2 nl/F_ n n
a ’Bw(n+%) _ 2 2

=f{;2:e H2(ax)e a x } A(x) dx (125)

n 2nn1/1r—- n
E f p(x)A(x)dx (126)

provided the Hamiltonian is harmonic. Hn are Hermite poly-

nomials. p(x) is an average of field Operators
4: +
0(X) = <‘P (x,t ) ‘1’(X,t)> (127)
The braced quantity can be evaluated using62

H 0011 (Y) _ 2 2
z n n“ tn = (1-(;2)'1/2exp{zxyt (x +1 ”2} (128)
=0 2 n! 1-t2

 

n

86

 

 

87
32 ‘ 2 "Bw 2
p(x) = 77 tanh(Bw/2) eXp(—a tanh IT'X ) (129)
l 2 2
= /— exp(-x /2<x >)
20<x2>
<x2> = f p(x) xzdx = coth(Bw/2)

 

2a2 I

The density function is a gaussian. Note it is also (

the square of the ground state wavefunction with a tempera-

 

ture dependent width. The density function (or number L]
operator) is the diagonal term of the field theoretic

Green's function
G = <W+(x,t)W(x',t')>0(t-t') (130)

Using equation 129 we can evaluate G.

G = writ; z e[-B+i.(t-t')] [n+§10
Z n
2 2 2 .2
Hn(ax)Hn(ax')e-Ta x e-Ta x

 

X

2n n! /0

 

88

 

. . __ _2_a_ sinh(Bw/2)
G(Xt'x t ’ ‘(3 / 1T sinh(ew-i(t-t1)w)

- 2 2 I
exp _%_'(x2+x'2)COth[Bw-i(t-t')w]+sinh[B:-IwIt-t')]

23 sinh(Bw/2) 0(t-t')

G(Xt'x't') n sinhmw-iE-t'm)

 

exp -a2x2 tanh[Q—2w- - EFL] (131)

 

Useful statistical functions are the characteristic
(x), moment generating (M), and cumulant functions (K).
Because the density function is a gaussian these properties

are easy to calculate and are listed below.

. , _ 2
x(b) = (elbx> = f elbxp(x)dx = e b /4a (132)
_ 2
M(b) = <e bx> _ b /4a (133)
-_ -1
Kr — log M(S)IS 0 2a 5r'2 (134)

For completeness we include the classical analogs of
the above. For an oscillator with fixed energy and ampli-

tude A:

Pc(x) = — —— (135)

89

n n n—l !!
n1! n,even

. n
x<q) = 2 ii%%- <xn> = JO(Aq)

_ . -r’ 3r _ _A2 _27A“
Kr - (1) 3;;'109X(q)|q=0 Kodd_0 KZ-jr' K4-_7r—

The quantum and classical statistics are inosculated

by considering the classical ensemble average.

/5' e-b2x2/2 (136)

_ 2 2
e mm Bx /2 fl

 

 

_ msz
PC(X) - n

This is similar to the quantum result; the difference is in

the inverse width of the distributions.

2

 

‘ mw .

b — ICE cla551cal (137a)

a = / —§21(tanhg%%)+l/2 quantum (137b)
. . m0)2

llm1t a = W (138)

kT>zhm

Equation 138 is a demonstration of the correspondence

principle.

 

APPENDIX D

DENSITY OF STATES

APPENDIX D

DENSITY OF STATES

We derive here some formulas for the density of

phonon states. Q will be a unitary transformation which

diagonalizes the product of the diagonal mass matrix and

the displacement Green's functions. The eigenstates of the

defect crystal will be labeled 5.

2

mew Ggg. = 6a8612' +- Z ¢££2G£2£,
08 Y 2 my Y B
@222
2a 2 - ax 12 y 2 Y 12'3 _
2 Q5 (62260Yw m ) Q5 2 Q53 m2 G2 2' s' - 655'
22223 2 2 l 1 3 3B
dB'y Y
S1
id id _
2 Q Q n - (S
Rd s 5 ss'
20 2'8 _
: Q3 Q5 ‘ 622'6018
q’u'
la 08 Ti 8 — 2
2 _____ _
22' 3 m2 Qs' - wséss'
GB

90

(139)

(140)

(141a)

(l4lb)

(l4lc)

 

91

Using equations 141 in equation 140 we find

1 + l l
Efitx[QWQ ]=‘§§Z'jr_:
Stu-w
5
lim Im 3le tr[QmG(UH~i5)Q] = - 3% 2 Sufi-(D? = -1Tg(w2) (142)
6+0 5
2 ___ _£_ . +
g(w ) - WBN 5; mRGQ£(w+iO ) (143)
ad

The last step follows by invariance of the trace to repre-
sentation.

If we consider a mass defect

2 ____i_ ’ _ 0
A9“) ’ ’ N3N f {szst TM}
a aa aa
-.._l_ _ 0 _ o o
‘ mu 5 { miezsummu 82’ “3 TG ’22}
a ad dd

This result is cumbersome. A more eloquent general result

can be found in terms of the phonon phase shift.

 

 

 

g(w?) = -3L Im tr 1 w =w+io+ (144)
NN mg _¢ +
1 2
= -F§ Im tr 8 2 1n(w+-¢)
w
- 1 3 2
- -Ffi Im awz ln Det(w+-®)

where we have used a relation true for a nonsingular A.

tr In A = 1n Det A

 

 

92
Then since
2 _ 2_ _ 2_ -1
w+ - Q - (w+ @gfll (w $0) C},

2 _ _ .1. .11. 2- _ 0 2
g(w ) — “N Im sz ln Det {(w+ ¢O)(1 G (w+)C)}

The determinant of the product of two square

matrices is the product of the determinants.

Ag(w2) = - i Im __8__ 1n Det {1-G°(w2)C}
TTN 3032 +
2 _ _ 3;.Ji_ 2 2 z _ 0 2
Ag(w ) — “N 3w2 O(w+) O(w+)_ Im 1n Det l1 G (w+)C|

 

(145)
L)
(146) 1. .
ii
(147)

This can be rewritten in a form which is sometimes

more useful by expanding the determinant and expressing the

force constant perturbation in an orthogonal basis.

 

 

Ag(w2) = Im -—2—-ln Il-GOCI
am?
0 11
Ag(w2) = - %—3 2 tr (G C)
awzrl n
using, 1n Det A = Z 1n A. A.(eigenvalues)E l-x.
1 1 i 1
co (xl)n
= Z ln(l-x.) = - Z X
i 1 1 n=1 n
n
= -tr 2 E—-
n n

After differentiation equation 149 can be resummed as a

geometric series.

(148)

(149)

(150)

93

-Im %- tr 020 (GOC) ' (GOC)n (15].)
n=0

Ag(w2)

1

-Im %-tr (GOC)'(1-GOC)- (152)

In equation 151 the derivative could be commuted to
the left in each term because of the trace operation. The
prime denotes differentiation with respect to wz. We now
expand the perturbation in an orthogonal basis. Band

indices will be labeled j and V is a folded matrix.

* I
qu, — .2 Vik ‘l’i(q)‘l’k(q) (153)
... 1k ...
33 33
g(w2) = -Im-];tr{(GoC)' +(G°C)' (3°. c +...}
" 93 331 q131 31?
33 331 313
l 0 U
— -Im E tr{(Akivik) +(Akivim ) Amn vnk + ...}
jj jj jj jjl j1j2 jzj
1 -1
= -Im E-tr{(AV)'(l-AV) } (154)
: O *
Aik _ ij, Z_Wi(q)qu Wk(q) (155)
3'3" 3

Mass Defect Example

» _ 2
egg, _ mew 518 52'8 Gas (156)

as

94

 

 

iQ'2 -iq"£ j *j'
C _ ~ ~ ~ ~
Qq' 5:. e e 00(g) 08 (g) C21.
jj' a8 d8
= mewz 6--I
33
o
A ‘ 651' G21
jj
2 O I
[mew G22]
_ - 221 (AV)') - _ £9. jj
139(9)) ' Im 17 2. (l-AV)j - 77 Im 2' 1-mt:szo
] 3 El
ii
2 O I
[mew G£ll
= - 3ng 1m 2 2a:
a 1-mew G££
(1a

(157)

The following is a p wave example for a partial wave expan-

sion of the perturbation.

C .= V”. "
qq 13 93
13'
= 'v 31y ()Y*(')+Y ()Y*(')+Y ()Y
qq jj' 3 11 3 11 3 10 3 1o 3 1-1 3
j z * o
A-lO - Z q Yl_l(g) Ylo(g) qu
3 ~
3 = 2 *
A1—1 2 g Yl-1(g) Yll<g) qu

(158)

(qW

~

and V are now 9 X 9 matrices. By imbedding the band indices

j into azimuthal matrix elements, we find the following form.

95

”'- __1

A11 V11 A1-1V11 A10 v11
= *
AV A1-1V11 41-1V11 A-10V11
1* v A*

 

 

L- 10 11 -1-1V11 A-10V11

In order to find a simple result we consider the case of

spherical bands.

j- 20_ _20
Agm — 6£m : q qu — 6£m(1 mw GOO) (159)
Ill

GOD is the diagonal element of the real space perfect

33 crystal Green's function.

 

 

1 o 0
AV = o 1 o (l-meGO )V..,
00 j]
o o 1 33
- _ 22 _ 2 0 . _i_ _ 2 '1
A g(w) - Im 1T ;g'[(l mm Goo)vjj'] [ ‘3(1 mw GOO)V]jlj (160)

ii

If the perturbation does not couple bands and is the same

for all bands,

v... = a..,v
31 JJ

[(l-meGgO)V]'

 

119(6)) = um???- 22 1 33 o (161)
j 1—§(1-mw2c; )v
00

ii

 

 

APPENDIX E

SCATTERING

 

 

APPENDIX E

SCATTERING

In transport problems the phonon lifetime is
required. It will be shown that the phonon lifetime can
be calculated from the T-matrix for the displacement Green's
function. For structureless scattering centers the lifetime
is found to be temperature independent. This appendix
begins with the calculation of the phonon Green's functions
by several methods, comparison of the various T-matrices,
lifetime calculation and finally the connection with phase

shifts is given.

Retarded Green's Functions

The displacement Green's function (G) is a linear
combination of a phonon Green's function (g_) and a "hole"
Green's function (g+), the propagation of the state with

one phonon removed.

__ 1

 

1 l
—— +
(0+0) (1)—(L)

1

)

k

In deriving equations for these Green's functions several

definitions are necessary:

96

 

 

97

g = -2fl<<a+(t)a (0)>> E (-2n)[-i@(t)<[a+(t) a (O)]>] (163a)
+kk' k k' k ' k'

g = 2n<<a (t) a+ (0)>> (1631»
-kk' k k'

_ + +

fkk' — 2n<<ak(t) ak,(0)>> C163c)
hkk' = -2n<<ak(t)ak,(0)>> C163d)

The name of the phonon Green's function is justified

by the interpretation of the related correlation function

I . 1 0° -iwt
< > -— __
a] (t) a ' (O) - 11m 2 f e

6+0

g_kk,(w+i(3) [n(w)+1] dw (164)
as the probability amplitude for creating a phonon of wave-
vector k' at time zero and destroying a phonon of wavevector
k at time t. In this sense then g_ allows one to follow the
evolution of the n+1 phonon state.

The Hamiltonian in momentum states

_ + 1 1 + +

H - X €k(akak+-§) + -2- Z ka, (ak+ak) (ak,+ak,) (165)

k kk'
is found by transforming the real space Hamiltonian

1’3.

l 1
H = m + E- Z'¢££,umu£,8 + 3 2' Gig. uzau£,8 (166)
£2 £1
a8 08

a8

 

 

 

 

98
where
/I . 'mwkM j + ik°£
PQG. = fi 2 1 0a (k) (ak-ak) e ~ ~ (167a)
kj /2 - -
/T f M + ik°£ j
1110. — SN 1:. 2mm (ak+ak) e ~ ~Ua(§) (1671))
Isa k
We identify ka,=Ckk, -———E———- . For a mass defect ka, =
4m/w w .
k k
-M€/wkwk,
4N

The equation of motion for g_ is found by taking the

time derivative using Heisenberg's equation of motion:

f ).

in g-kk' = 6(t)6kk, (Znh) + e + E Vki (g-ik'+ ik'

kg—kk'

1“ fik' = —€ifik' ' Vik(fkk'+g-kk')°

WM

Fourier transforming and solving for g_

l —l 0
90V} 9

g- = {(1-ng) + gSV(1+goV)- + _

Superscript zeros denote unperturbed functions.

9+ is found similarly:
ng+kj = 2"”6(t)ij'ekg+kj-vk£(g+2j+h2j'

lhhfij = €£h£j+vfik(g+kj+hkj)

 

 

99

Fourier transforming and solving for g+

_ o o _ o -l o -l o
g+ — [(l+g+V) + g+V(1 g_V) g_V] g+

Having derived the Green's functions we display the results

in closed, Dyson, and T—matrix form.

_ o o _ o -l o -l 0
9+ — [(l+g+V) + g+V(l g_V) g_V] g+
g_ = [<1—gEV> + gSV<1+gEV>'1 givf1 9°
G = (l-GOC)'1GO
9+ = g: + 9:3V(1+[l-gc_DV]-l 93V) 9+
9_ = g? + gfvu-mgfw'l gEV) g_
G = G0 + GOCG
O O O
9+ = g+ + g+t+g+
O O O
9. = g- + g_t-9_
G = G0 + GOTGO
T-matrices
t+ = -V(1-[1-gfv1'1gSV) {1+g$V(1-[1—g‘_’v1'1g‘_’V)}‘l
_ o -l o o o -1 o -l
t_ - -V(l-[l-g+V] g+V) {l+g_V(1-[1-g+V] g+V)}

T = C(1-GOC)'l

 

100

C is the force constant perturbation

Note that g+(w) = -g_(—w)

t+(w) -t_(-w)

A relation between t_ and T can be found by equating two

T—matrix forms for G.

o c) o 1 1 o c) o C) o o
G — G +G TG — 2(1) (g++g_) —§E(g++g+t+g++g_+g_t_g_)

o o 1. o o o o
G TG - 35-(g+t+g++g_t_g_)

 

 

 

 

T = ‘w‘wk't+kk"w"w'wfi’ + (m+wk)t_kk.(w) (w+w£) (168)
kk' 2m 2w
= _ (w—wk)t_kk,(-w)(w-wg) + (w+wk)t_kk,(w) (w+wk.)
2m 2w

In the case of elastic scattering (wk=wk,), T and t_ are

simply related by a factor 2w.

Time Dependent Scattering

 

The phonon transition probability will be found to

be linear in time. The inverse lifetime is then given by

T 2
(akf(t) ak(0)> (169)

 

 

i: 2 limit l
T t

k k' t+0

 

 

/(nk+1)(nk,+l)

The thermal factors in the denominator normalize the states

1.

ak,

|o>, a:|0> .

101

 

T i w -iwt
Akk' - <ak, (t) ak(0)> — 3; [co [l-n(w)] e g_k, ((1)) t-kk'g-km)
[H(w )+l] t (0) ) e"'"""kt
5k -iw t -kk' k
= W— t-k'k(wk) e k +[n(wk)+l] w _w
k k' k' k

Once Ak'k is divided by the nomalizing thermal factors it
is independent of temperature. The poles of t have been
neglected since these give short transient effects except

in the case of a localized mode where its residue must also

be included. Performing the square and limiting operation:

 

l 2
—= 211 2 It ,(w) | (5(a) ~16.) (170)
'& Id -kk k }< k
2
T ,(w )
=2nz'kkkl (S(w—w) (171)
k' «62 1‘ k.

k

Where equation 168 has been used. If the T—matrix
depends on the final wavevector only via the final state
energy the delta function may be replaced by the density of

final states.

9(a)) 2
i=m " IT .(w)| (172)
T 2 kk k
k 2w
k
--EHLI G0 (T ( ,|2 (173)
’ wk m 21 k'k wk

102

Perturbation Scattering Theory

 

Perturbation techniques make use of the time ordered

Green's function
15 (t) = <T a (t) a+(0)> (174)
k k k

which can be related to the advanced and retarded functions

 

 

via the spectral function.36
gR(kw) = i: g: wffé'f“: )in (175a)
gA(kw) = _f_: :9“ 686(1)“)— )in (17512))
E = [1+e-Bw]-l gR(ku)) + [1+e8w]ul gA(kw) (175C)
For real frequencies these reduce to
R 1
Im 9 (km) = — 3mm) (176a)
A 1
1m 9 (km) = + 3906») (176b)
Im E = - -12- coth (Eglmmw) (176C)
Re G = Re gR = Re gA (176d)

These relations permit the calculation of the spectral func-
tion from either the time ordered or the retarded Green's
functions previously considered. Note that for zero
temperature the real and imaginary parts of the time ordered

and retarded functions are equal.

 

103

The imperfect crystal propagator can be expanded in

the usual way.36

m ._ n
Ekk'(t) = -i Z l—l%—-IB..IB..dR..dT <TV(T1)...V(T )ak(t)a:,(0)> (177)
n=0 IL 0 O n n connected
I I
V - k2}; ka' (ak+ak) (ak'+ak,)

This potential allows only the following inter-

actions between two phonons (order of operation plotted

WT??? _____ .1 1---- k1 ..... 1 '
k) k) w) )4

Using Wick's expansion of the ordered product

-1
1k 1

= M +11: +11"; H T'- + .......

Dyson's equation can be employed to simplify the summation

by extracting the irreducible self-energy by grouping graphs.

6° = 60 GO 2 G’ = (1-602)"1 GO (178)

+

104

X = ka'+vkklg:>kl Vklk'+kangleklkzgok2Vk2k'+ (179'
= V(1-g:V)-l (180)
G’ = -i<T a (t) a+(0)> 5° = 9) (181)
k k 0
g<(kt) = i@(t)<a:(t) ak(0)> g:(kw)=-w+w:_i6 (182)
g>(kt) = iO(t)< ak(t) a:(0)> g:(kw)= w-;o+i6 (183)
5': G°+G°ZG = g>+ g>25
= (l-g>Z)g>
= {1-g:V(1-g:V)-1}-lg> (184)

Time Independent Scattering

 

Beginning an equation of motion for displacement
waves, we now derive the time independent scattering

equations
(mwz-Q) u> = O,¢=®O+c (185)

¢ are the atomic force constants. The above can be written
as a Lippmann-Schwinger equation by use of the displacement

Green's function.

 

 

105

(mw2-€)I u> = c|u> (186)

|u>==lu >+G Clu = In >+G T u >
O O O O 0

where T=C(l-GO C)-l

. . . ik°r ' .
If we con51der an 1nc1dent wave <r|uo>=e ~ ~ 03(k) With

polarization 0, branch j and wave vector k, the scattering

equation in real space becomes

0
<r£a|u> — <r£aluO>-+ Z GQQ TR 2 <r£ Iuo> (187)
2122 881 B1 2 2
BY Y"
The scattered wave is
o ik'22 j
W(r£) B$£ G£21 Tklgz e OY(k) (188)
2 1 88 B Y
2

and can be written as a spherical wave by expanding the
Green's function for large distances from the scattering

center.

oxj (k) oj (k)
a .. 8 ~ 'k'R

 

 

Using a two band isotropic model the completeness condition

may be used to sum the eigenvectors.

106

 

 

 

 

 

05 Ci otlot +ot20t2+olo£
a B a B B a 8 £ 2 1 1
Z 2 2 = 2 + Ca 08 2 2 2
j w -wjk w -wtk w -w£k w -wtk
k k
= 608 + a B l _ 1- (190)
2_ 2 2 2_ 2 2_ 2
w wtk k w wlk w wtk
Equation 189 becomes
. 5 k k
G:B(R,w) = ——!——g f d3k e15 5 2882 -+ “28 (191)
m(2fl) w -wtk k
)< 1 _ 1
2_ 2 2_ 2
w wlk w wtk

In the above equations 2 labels the longitudinal branch and
t labels the degenerate transverse branches.

There are two types of integrals here:

 

 

3 ik°R q q iq'R
I=f-—-——dke “vase ~ d3q
wZ-vzkz qz wZ-vzqz
I = 4flf:k2dk SliRkR l
wz-vzk2
_ _. 2 m 4“ sin 3R 2
J — ( 1) VaVBfO 2 qR q dq

 

 

 

 

 

 

 

 

 

107
. 5 2R R R R q
vaVB §$2§35-= —g§-cos qR - a 8 cos qR - a sin qR —
q R2 R“ R
sinR 3RR RR
-———Jl—-+ sin qR - cos qR
Raq qu Ru
RR 2.
= - a 8 q' Sln qR to order (é)
R2 qR ,
‘rr-
RR
J = a B I to order (i)
R2 R
Both integrals reduce to the same form for large R. i
o RxR V
G (Rw)={6 I+—X-(I-I)}———
d8 d8 t R2 i t 6W2m

in terms of a function I which depends on the branch index

and which can be evaluated analytically in a Debye model.

 

H
H

6 qmax SlgRqR 2 12 q2 dq d9
w -vjq

3E-{[Cid(w+w3
a max

)-Cio¢(wj -w)] sin dw+[Si 01(8)j +w)
max max
+ Si 01(8)j -w)-n] cos am} (192)
max

Si and Ci are sine and cosine integral functions and a is

R/v.. In the large R limit or more exactly when £L(w3 iw)>>l
j Vj Inax
nv. .
I = _ J e1(R/vj)w

J 2Rv?
J

 

 

108
0 Va Sasei(R/Vt'w RXR' ei(R/vgyu ei(R/vtnu
G (OR) = - —— + y( ——-‘-———
GB 4 v2 R2 V2 v2
t i t
Using the equalities
. . “k...R A
1. lim el(RR-R21)(Q/Vj) = elR(w/vj) e ~J ~£1 (k!=$1-2 )
R1 3 j
-——->>1
R2
1
_ j j
2. 6GB — I Ga 08

J

The scattered wave may be written in terms of a scattering

 

amplitude.
W (r,2) = 2 G OJ OJ T Oj(k) eik°£2
a J86 2 221 B 5 £122 Y
fiY1.dB 6 7
v oz: 1%
= ' 4 2,32- 'I.'k'k R (193)
3 J J 3
eikJR
= 2 fJ(k,k') R (194)
3
Where the scattering amplitude is
OJ
_ _ _L_0‘
fJ(®kk.) - 4 2 Tk'k (195)
V
J J j

 

109

In finding the total cross section we neglect the

rapidly oscillating interference terms between different

 

 

 

 

 

bands.
do 2 2 2
61—9" |.z fial — Ifgl +2|ftl
38
a 2 2 Va 2 2
= 2( ) lTk,k | + ( ) |Tk, I (196)
4va2 4'1va2
t t j R 1 j
As an example we may find the phonon lifetime in the case of I I
a mass defect where the T-matrix is easily calculable:
mew2 ij,
T = ———————- (197)
kk: _ 2 o
33 1 mew GOO
ad
V V
1 s s 2
Tu») " VQOT ‘ vq'fkk" d9
V 2 4
=4“: em 112.11) (198)
l-meszo v3 v3
00 t 1
ad

where an average over initial polarizations has been

performed,

 

110

Phase Shifts

We write down the expression for phase shifts by
analogy with the relation between the T-matrix and phase
shifts for particles. So that we may have noninterfering
phase shifts, the T—matrix will be considered in an

irreducible representation labeled by y and degeneracy g.

10 .
_ e Sine .
T — “TE'G" or equ1valently, (199)
G = tan”1 (1E_£) (200)
Re T

The T-matrix was originally defined in equation 64. It can
be rewritten for one irreducible representation and the

total phase shift found by summing all representations.

 

T = C(1-GOC)_1

-1 -Im (GOC)

0 = tan ( 0 Y) (201)
Y l-Re(G C)

Y
= Im Det Il-(GOC)YI (202)
0 = 2 g 0 = Im Det |1—G°c| (203)
Y Y Y

One way to find the T-matrix and phase shifts is to
decompose the perturbation on an orthogonal basis and use

folded matrices.

 

111

qu, = Z c (1-GOC):I q, (204)
... q j .. .l..
J] l 311 113
_ jj' * .
qq'-.Z Vik Wiuv Wk(q ) (205)
... 1k ...
JJ 33
_ * . ‘jj '* 0 j j' * * .
qu, _ .2 vik Wi(q)wk(q ) + .2 Vikl‘Pi(q)'i’k(q)Gq j v£11m W£(ql)'¥m(q) +.
... 1k ... 1k 1 l
3] 3] 1m
_ * , jjl jl jlj' * ,
" 12k Vikwimfllkm ) + “Em Vik Akfl. Vim “11"!) Wm“; '
* I
+ .2 (VAVAV).1m Wi(q) ‘Pm(q)
1m
-1 * ,
‘=.Z {V(l-AV) }im' wi(q) wm(q ) (206)
1m 3]
where

jj' _ * o
Akz ‘ 6jj' z fk(q) qu f2(q' (207)
In a mass defect example the function indices i,m do not

occur 0

c , = mewz 6... = 6... v (208)
qq 33 33 0°
jj'
A = 2 G32) = G139, ‘(Real Space Green's function)
q H

33

 

112
mew2 6. . ,
qu, = 33 o (209)
_ 2
jj' l mew X G211
jj
mm2 6... o
, = 3 or 0 = arg (l-mewzcu’) (210)
qq , l-mrioozGaz
J] «j 00
p wave example:
c E ”I - ' (211)
qq' V33 q q
jj'
--41 v ' {Y* ( ) Y ( ') + 2* ( )Y ( ') + 2* ( )Y ( ')}
" 3 jj'qq 1—1 q 1—1 q 11 q 11 q 10 q 10 q
4w
V10 _3— vjj'
j __ 2 *
A1-1 ' i q Y11'3' Y1-1'3' qu

A and V are now written in folded form; each matrix element
below represents a 3X3 matrix of band coordinates. The

indices below are azimuthal labels.

-

I) = j j j
A A11 A1-1 A10 1 0 0
j j j = gn' 0 1 o
A1-1 A-1-1 A-10 ij' 3 ij' o 0 1

j j 1
A10 A~10 A00

 

 

F“

 

*T‘

 

113

 

 

 

 

___ _1 1-
5 j
q Yll (q) :10 0
4n
= y ——-v.. 0 1 0
T q 1-1 (q) 3 33 '
0 0 1
q Ylo (q) _ _j
1- _
— -Ir )— fl
_ ' I
A11 V11 A1-1 V11 A10 v11- 1 q Y11 (q '
— ' I
)( l A1-1 V11 A-1—1 v11 A-10 V11 q Y1-1 (q '
' '
A10 v11 A~10 V11 A 00 V11) q Y10 (q '
_ _. J... ...)

 

 

 

 

If we now consider spherical bands, A is diagonal on

azimuthal indices.

1
— 4_TT _ -1 I I * I
T , — 3 .2 v3.3. {1 Allvll}j j qq g-Z Yl£(q)Yl£(q )
jj, 31 1 1 —-1

..l '
T FZijfl-Allvll}jlj' q q
13" 31

If the perturbation is diagonal on band indices

 

q'q' Vj'j
T ,= 6.” ‘41 v
33. 33 (1 11 11'
33
G = arg(1-vllvll) (212)

 

REFERENCES

 

10.

11.

12.
13.
14.

15.

16.

REFERENCES

J. Bernoulli, Pertop. Comm. 3, 13 (1728).

J. Baden-Powell, View of the Undulatory Theory as
Applied to the Dispersion of Light. London,
1841.

 

J. L. Lagrange, Mecanique Analytique. Gauthier-
Villans, Paris, 1888.

W. Thomson, Baltimore Lectures on Molecular Dynamics
and the Wave Theory_of Light. Clay, London, 1904.

W. Thomson, Popular Lectures and Addresses, 2nd edition,

MacMillan, London, 1891.
J. Vincent, Phil. Mag. 26, 537 (1898).

G. A. Campbell, Bell System Tech. Journal 1, l
(1922).

Born and von Karman, Physik Z. 13, 297 (1912).
A. Einstein, Ann. Physik 22, 180 (1907).
P. Debye, Ann Physik 39, 789 (1912).

Lord Rayleigh, Theory of Sound, pub. 1877, second ed.
1894, reprint Dover, New York, 1945.

 

Lord Rayleigh, Proc. London Math Soc. 11, 4 (1885).
Kino Matthews, I.E.E.E. Spectrum 8, 22 (1971).
Born and von Karman, Physik z. 14, 15 (1913).

Dicke and Wittke, Introduction to Quantum Mechanics
Addison Wesley, Reading, Mass., 1960. ’

Franz Hohn, Elementary Matrix Algebra, MacMillan Co.,
NOYOI 1966’ p. 349.

114

 

 

17.
18.

19.

20.

21.

22.

23.

24.
25.

26.

27.

28.

29.

30.

31.
32.

33.

34.

35.

36.

115

Ibid., p. 348.

Messiah, Quantum Mechanics.

 

I. M. Lifshitz, J. Phys. (U.S.S.R.) l, 211, 249
(1943).

W. E. Montroll, J. Chem. Phys. 15, 575 (1947).

E. W. Montroll and R. B. Potts, Phys. Rev. 102, 72
(1956).

P. G. Dawber and R. J. Elliott, Prec. Roy. Soc. 273,
222 (1963).

P. G. Dawber and R. J. Elliott, Proc. Phys. Soc. 8_,
453 (1963).

Elliott and Taylor, Proc. Phys. Soc. 83, 183 (1964).
D. N. Zubarev, Soviet Physics Uspekhi 3, 320 (1960).

F. W. de Wette and B. R. A. Nijboer, Phys. Lett. 18,
19 (1965).

G. L. Pollack, Rev. Mod. Phys. 26, 748 (1964).

G. L. Pollack, Unpublished Colloquium at Michigan State
University, 1970.

Fennichel, Phys. Rev. B 3, 439 (1971).

Cook, Argon He and the Rare Gases. Interscience
Publishers, N.Y., 1961, p. 317.

 

L. Nosanow, Phys. Rev. 146, 120 (1966).
Grifford and Hatton, Phys. Rev. Lett. 18, 1106 (1967).

Edwards, McWilliams, and Daunt, Phys. Rev. Lett. 9,
195 (1962).

Edwards, McWilliams, and Daunt, Phys. Lett. 1, 218
(1962).

Mullins, Phys. Rev. Lett. 20, 254 (1968).

Fetter and Walecka, Quantum Theory of Many Particle
Systems, McGraw-Hill, N.Y., 1971.

 

 

 

37.

38.

39.

40.

41.
42.
43.

44.

45.

46.

47.

48.

49.
50.

51.

52.

53.

54.

55.

116

R. Kubo, Journal of Phys. Soc. of Japan 11, 100 (1967).

Kendall and Stuart, Advanced Theory of Statistics,
second ed., C. Griffin, London, 1945.

Jackson and Feenberg, Ann. Phys. (N.Y.), 15, 266
(1961).

de Wette, Nosanow, and Werthamer, Phys. Rev. 162,
824 (1969).

R. Peierls, Ann. Physik 3, 1055 (1929).
J. Callaway, Phys. Rev. 122, 787 (1961).
Bal Krishna Agrawal, Phys. Rev. 162, 731 (1967).

R. Berman, C. L. Bounds, and S. J. Rogers, Proc. Roy.
Soc. A289, 66 (1965).

R. Berman, C. L. Bounds, C. R. Day, and H. H. Sample,
Phys. Lett. 26A, 185 (1968).

R. Berman and C. R. Day, Phys. Lett 33A, 329 (1970).

B. Bertman, H. A. Fairbank, R. A. Guyer, and C. W.
White, Phys. Rev. 142, 79 (1966).

E. J. Walker and H. A. Fairbank, Phys. Rev. 118,
913 (1960).

R. Berman, Proc. Roy. Soc. (London), A208, 90 (1951).
C. Herring, Phys. Rev. 95, 954 (1954).

R. J. Elliott and D. W. Taylor, Proc. Roy. Soc. (London)
296, 161 (1967).

P. G. Klemens, R. De Bruyn Ouboter, and C. Le Pair,
Physica 30, 1863 (1964).

H. R. Glyde, Phys. Rev. 177, 262 (1969).

P. G. Klemens, Encyclopedia of Physics, edited by
S. Flugge (Springer-Verlay, Berlin, 1956), Vol. 14,
p. 198.

J. M. Ziman, Theory of Solids, Cambridge University
Press, Cambridge, 1965, p. 187.

 

117
56. W. M. Hartmann, H. V. Culbert, and R. P. Huebener,
Phys. REV. B. l, 1486 (1970).
57. Gillis and Koehler, Phys. Rev. B. 4, 3971 (1971).
58. J. R. Hardy, J. Phys. Chem. Solids 15, 39 (1960).

59. Maradudin and Klemens, Phys. Rev. 123, 804
(1961).

60. C. M. Varma, Phys. Rev. Lett. 23, 778 (1969).

61. H. R. Glyde, Phys. Rev. 177, 262 (1969).

62. W. W. Bell, Special Functions, D. van Nostrand, 1968.

 

 

 

\Ilfl’tjnlywuu'

(I!

v
1

4
3
0
3
9
2
1