wamousomsomm amxflcms

Thesis for the Degree of Ph. 0. ¥
MIGHIGAN STATE unwmsm M f ‘ '
RONALD D. PAINTER *
1973

 

    
 

LIBRARY

Mill-a gen State
University

 

~.._- w-II. -‘-—-. ‘-

This is to certify that the

thesis entitled

VIBRATIONS OF DISORDERED BINARY CHAINS

presented by

RONALD DEAN PAINTER

has been accepted towards fulfillment
of the requirements for

BILLE— degree in Jhxsigi

69 1366/

Major professor

Date «3/44? /?]3

0-7 639

          

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ABSTRACT
VIBRATIONS 0F DISORDEREIS BINARY CHAINS
BY
Ronald D. Painter

We have examined the vibrations of harmonic

   
  
  
   
   
   
  
  
  
  
  
  

,linear disordered atomic chains which include an
parbitrary concentration of defects differing only in
mass from the host atoms. This study included computer

experiments on long chains and configuration average

.theories for infinite chains.

u
0

V ‘ - A : .. V
ffiww ' ». ~'n
,4. w.,, 1_.

pp n. ,

r . . .

.. I
'V' .
' l

Using the theory of ergodic Markov chains,we

. a

V

generated disordered binary linear chains with short-

' range-order among the constituents. Nearest-neighbors
'and second-nearest-neighbor correlations were explicitly
t_1ntroduced. For comparative purposes we also generated
{jfirrandom chains. The relationship between the Markov

: correlation and the Warren-Cowley short-range order
:‘parameters was explored. Although a simple analytic

' relation exists for the first-order Markov chain,

:correlations generated by the second-order Markov chain

Athese chains, in the harmonic approximation with all

 

i
l

 

 

Ronald D. Painter

   
 

force censtants equal. The spectra of chains of as many
as 100,000 atoms were computed and the effect of short-_
range order on the spectra was determined. We explicitly
computed the eigenvalues and eigenvectors for 1000 atom
chains. The localization of normal modes (A) was studied
as a function of energy by calculating Z Ug(l), where Uz
is the normalized displacement on site i, and also by
calculating an exponential decay parameter. We were,
therefore, able to describe the region of appreciable
amplitude of the eigenvectors as well as the decay rate

; away from this region. We have studied the vibrational
a. _ ' density of states theoretically. .Clusters of up to six
‘5. atoms were firstly embedded in a uniform chain of host

,; 30 atoms, and, secondly, periodically extended to form a

periodic chain. In each case the spectra of these chains,

    
    
  
 
 
    
     
 

averaged over all configurations with short-range order
yincluded, were compared to those found experimentally.
For defect concentration :.5 the embedded cluster gave
good agreement with experiment in the impurity band
especially in random systems, though it did not give
} information about the spectrum in the host band. The
periodically extended cluster gave qualitative agreement
' with experiment for the whole spectrum for all concentra-
tions of defects and all conditions of short-range order.
idfnghe periodic cluster theory, however, introduced many

’gpurious singularities in the density of states. Finally

ӎggmpdified self consistent cluster theory (cluster CPA)

 

 

   

Ronald D.

Painter

Voyed to calculate the density of states. For the

 

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it. ) .-. .

b...

_ .Aaz-“ m‘h "

 

 

BY

a“. '
Ronald D. Painter ,

.yvm,“ i .. a”, .—4

.~—,-,

A THESIS ' ' . *

t4
Submitted to M.
Michigan State University T'

for the degree of
DOCTOR or PHILOSOPEY
Department Of Physics , »;_;r' ',:if

1973

 

   

 

DEDICATION

     

If I dedicate this thesis to my wife, Gloria, for
t"
Isriienee and understanding of my virtual absence

. family for the last two years.

 

 

l
t.

1“

- ’5- A

 

ACKNOWLEDGEMENTS

    
    

Sincere thanks go to Professor William M.

v.15 for his guidance and friendship without which

iyress gratitude to the following members of my thesis
i-ttee for consenting to critize this work:

I Dr. William M. Hartmann

Dr. Thomas A. Kaplan

Dr. Jack Bass V

.Dr. Frank J. Blatt

'3.Dr. George F. Bertsch

 

9
l.
II
’1
'.

“"— wfi. ”gs—flrwl

l

 

   
  
  
  
  

'f,§EST OF
:7fLIST or
: Q Chapter

I.

TABLE OF CONTENTS

gmmTIoN I I I I I I I I I I I I I

,.’ ACKNOWLEDGEMENTS . . . . . . . . . . .

TABLES o c I I I o I u o I o o

FIGURES O I I o 0 0 o e o o 0

INTRODUCTION. . . . . . . . . .

Historical Review . . . . . . .
Classical Lattice Dynamics . . . .

NUMERICAL CALCULATIONS FOR RANDOM BINARY
CHAINS I I I I I I I I I I I I

Numerical Methods . . . . . . .
Numerical Spectra . . . . . . .

SHORT-RANGE ORDERED CHAINS AND THEIR
FREQUENCY SPECTRA . . . . . . . .

Short Range Order in Scattering Theory
Short Range Order Work. . . . . .
Markov Chain . . . . . . . . .
Frequency Spectra of Linear Chains with
Short Range Order. . .
First Order Markov Spectra Chains .
Second Order Markov Chains. . . .

LOCALIZATION OF EIGENSTATES OF DISORDERED

‘ CHAINS o o o o o o o o o o o o

THEORETICAL DEVELOPMENT . . . . . .

Green' 5 Function. . . . . . .
Defect Clusters . .

n Site Periodically Extended Model.
.Self-Consistent Cluster Theory . .

I O I I

~=p9Conclusions . . . . . . . . . .

iv

Page

NH

10
10
13
27
27
32
70
85

141

143
153
163
172

208

 

 

  
  
  
  
 
  
  
  
 
 
 
  
  

NUMERICAL COMPUTATION OF EIGENVALUES 0F
CHAINS I I I I I I I I I I I I I

SUPPLEMENTAL MARKOV.CHAIN THEORY . . . .
saonw RANGE ORDER PARAMETERS . . . . .

- NUMERICAL COMPUTATION OF EIGENVECTORS OF
CRIN S I I I I I I I I I I I I I

GREEN'S FUNCTION FOR A MONATOMIC CHAIN . .

ISOLATED DEFECT CLUSTERS EMBEDDED IN A HOST
CHAIN I I I I I I I I I I I I I

DENSITY OF STATES OF PERIODIC LINEAR CHAINS
WITH ALL FORCE CONSTANTS EQUAL. . . . .

i'r SELF4CONSISTENT SINGLE SITE APPROXIMATION

ISOLATED DEFECT IN A MONATOMIC HOST CHAIN

 

Page
212
218

219
226
267

283
294

306

313
328
333

 

 

 

 

LI ST OF TABLES

Page
The classification of Markov processes . . 35

Characteristics of two 100 unit Markov
chains I C O I I I O I I O I I I 6 8

Comparison of numerical spectra and
Equation (3.113) for C =.5, P =.5 . . . 73
d d,d
Comparison of Equation (3.113) to numerical
spectra for Cd='2 . . . . . . . . . 78

Comparison of Equation (3.113) to numerical
spectra for Cd='5 . . . . . . . . . 89

Probability of cluster of light atoms of

' size n for Equation (3.115). . . . . . 95

Probability of having n-host-defect clusters
for Equation (3.115) . . . . . . . . 99

Impurity mode frequencies of defect clusters
embedded in a host chain, III—h =2 . . . . 155
d

The bases for all possible 4 and 6 site
periodic chains. . . . . . . . . . 165

Statistical error analysis of a chain with
confidence limit error of Equation (3.105). 263

Statistical error analysis of a chain with
N=10000, Cd=.5, P d=0.1 compared to the
99% confidence limit error of Equation
(8.105) 0 o a u I o o v I o o o 264

Statistical error analysis of a chain with
N=100,000, C =.5, Pd,d=°-1 compared to
the 99% confidence limit error of Equation
(3.105) 0 I o o ‘I n u o o o o I 264

vi

 

 

   

.Lsd " I Page
c") I
’f,L£‘Statistical error analysis of a chain
..‘ . With N=1’000'000’ Cd=o5' Pd d=001
’: compared to the 99% confidence limit
error of Equation (8.105) . . . . . . 265

 

:‘
t
J

1’“
I"
j

_.._ 5.—
A u-

 

 

 

LIST OF FIGURES

‘s Figure
‘ 2.1 Numerical density of states for a 50%
random binary chain of length 1000 and
mass ratio 2/1, Case 1 . . . . . . .
‘L 2.2. Numerical density of states for 50% random
binary chain of length 1000 and mass
ratio 2/1, Case 2 . . . . . . . . .
2.3 Numerical density of states for a 50% random
binary chain of length 10000 atoms and
mass ratio 2/1 . . . . . . . . . .
_ 2.4 Numerical density of states for a 50% random
' binary chain of length 50000 atoms and mass
ratio 2/1I I I I I I I I I I I I
2.5} Numerical density of states for a 50% random
1 binary chain of length 100000 atoms and
3 mass ratio 2/1 . . . . . . . . . .
Numerical density of states for a monatomic
heavy chain of length 10000 atoms. . . .
Numerical density of states for a 1% random
light mass binary chain of mass ratio 2/1
and 100,000 atoms . . . . . . . . .
Numerical density of states for a 20% random
light mass chain of mass ratio 2/1 and
10000 atoms . . . . . . . . . . .
Numerical density of states for a 50%
ordered diatomic chain of mass ratio 2/1
and 100 00 atoms I I I I I I I I I I
Numerical density of states for a 80%
random light mass chain of mass ratio 2/1
and 10000 atoms. . . . . . . . . .
Alloweg values of Pdd d'Pdh d and Phd d for
Cd I I I I I I I

viii

 

Page

15

16

17

18

19

21

22

24

25

26

60

 

 

  
 

Figure

3.2

3.11

 

Allowed values of Pdd d’Pdh d and

for C .4 .

d=

Allowed values of Pdd d,Pdh d and

for Cd= . . .

Allowed values of Pdd d’Pdh d and

for C I 6 I I

d=

Allowed values of P ,P and
'for C _ 7 dd, d dh, d

d

Allowed values of Pdd d,Pdh d and

for Cd=. .8

Allowed values of P

for Cd=.9 . .

dd,d’Pdh,d

and

Numerical density of states with Cd=

mass ratio 2/1 and Pd d=0. O for a 10000

unit chain .

Numerical density of states with Cd=.2, mass
d d=.4 for a 10000 unit

ratio 2/1 andP

chain . .

Numerical density of states with Cda. 2, mass
Pd d=. .6 for a 10000 unit

ratio 2/1 andP
chain . .

Numerical density of_ states with C

ratio 2/1 andP
chain . .

Structure factor a(k) versus k for Cd=’2'
.6 and .8

Numerical density of states with Cd=-5r mass

ratio 2/1, and Pd d=.l for a 10000 unit

chain . . .

Numerical density of states with C

ratio 2/1, and Pd d” =.25 for a 10

chain . .

ratio 2/1 andP
chain .

I I

ix

Numerical density of states with C

I

83

8'

~2,

.2, mass
unit

.5, mass
0 unit

=.5, mass
d d” =.75 for a 100 0 unit

Page

61

62

63

64

65

66

74

75

76

77

79

81

82

83

 

   
 

. :55figure

3.16 Numerical density of states with C =.5,
‘ ' mass ratio 2/1 and Pd d='9 for a 10000
unit chain I I I z I I I I I I

3.17 Structure factor a(k) versus k for Cd='5'
"r Pd d=ol' .25, .75 and .9. o o o o I
‘.‘ I

3.18 Numerical density of states with Cd=.2,
Pdd,d=o' Pdh,d='4’ Phd,d='8 and a mass
ratio 2/1 for a 10000 unit chain . . .
-. 3.19 Structure factor a(k) versus (k) for Cd=.2,
LL ‘Pdd,d=o’ Phd,d='8 and Pdh,d='4' . . .
{ f 3.20 Numerical density of states with Cd=.5,
Pdd,d=Pdh,d='25’ Phh,d=Phd,d='75 and mass
ratio 2/1 for a 10000 unit chain . . .

Numerical density of states with Cd=.5,
Pdd,d=Pdh,d=‘75’ Phh,d=Phd,d=‘25 and mass
ratio 2/1 for a 10000 unit chain . . .

Structure factor a(k) versus k for Cd=.5
Pdd,d=Pdh,d=‘25’ .75 and Phh,d=Phd,d='25’
I 5' I I I I I I I I I I I I

Numerical density of states with Cd=.5,
Pdd,d=Phh,d='25' Pdh,d=Phd,d=‘75 and mass
ratio 2/1 for a 10000 unit chain . . .

Numerical density of states with Cd=.5,
Pdd,d=Phh,d='75’ Pdh,d=Phd,d=‘25 and mass

ratio 2/1 for a 10000 unit chain . . .

Structure factor a(k) versus k for Cd='5'
Phh,d=Pdd,d=‘25' .75 and Pdh,d=Phd,d='75'
Exponential localization length, L (m2), for

Cd='5 random and mass ratios of 1.5, 2.0,
and 3.0 O I I I I I I I I I I

 

Page

84

86

87

88

92

93

94

96

97

98

110

 

 

Page

Exponential localization length, LE(w2)
for Cd=.5 random and mass ratios of 4.0
and 8.0 I I I I I I I I I I I I 111

Exponential localization length, LE(w2)
for mass ratio of 2 and Cd=.l and .3
random. . . . . . . . . . . . . 114

Exponential localization length, LE(w2)
for mass ratio of 2 and Cd='5 random. . . 115

Exponential localization length, LE(m2) for
mass ratio of 2 and Cd=-7 and .9 random. . 116

Exponential localization length, LE(w2) for
mass ratio of 2, Cd=.5 and P6 d='1 and .3 . 118
I

Exponential localization length, LE(w2) for
mass ratio of 2, Cd=.5 and Pd d='7 and .9 . 119
I
Exponential localization length, LE(w2) for
mass ratio of 2, Cd=-5 and Pdd,d=Pdh,d='1
and P =I9 I I I I I I I I I
hd,d

Localization a=L;1(w2), for mass ratio of
2, Cd=.5 random and a 100 unit chain . . 123

Localization a=L;1(w2) for mass ratio of 2,
Cd=‘5 random and a 100 unit chain. . . . 124

\

Localization a=L;1(m2) for mass ratio of 2
Cd=.5, Pd d=.l and a 100 unit chain . .
I

Localization c=L-1(m2) for mass ratio of 2
Cd='5 and Pd d=.9 and a 100 unit chain . . 126
I

‘

Localization a=L;1(w2) for a mass ratio of
2, Cd='5 random and a 1000 unit chain . . 127

Localization a=L-l(w2) for a mass ratio of
2, cd=.s randofi and a 1000 unit chain . . 128

Localization a=L;1(m2) for a mass ratio of
2, Cd='5' Pd d='1 and a 1000 unit chain. . 129
I

_Loca1ization a=L;1(w2) for a mass ratio of
2, Cd='5' Pd d='1 and a 1000 unit chain. . 130
I

xi

 

 

Page

Density of states for a 1000 unit chain
with Cd=‘5' Pd d=‘1 and a mass ratio of
2/1I I I I ’I I I I I I I I I I 131

Eigenvectors n=100, 250, 353, and 450 for
a 1000 unit chain with Cd=.5, Pd d=.l and a
2/1 mass ratio . . . . . . ’. . . . 137

 

4.19 Eigenvectors n=700 and 775 for a 1000 unit
fio chain with Cd=.5, Pd d=‘1 and a 2/1 mass
.' ratio I I I I I ’I I I I I I I I 139

.?41 4.20 Eigenvectors n=776 and 950 for a 1000 unit
-1 chain with Cd=.5, Pd d=.l and a 2/1 mass
’ ‘ ratio I I I I I 'I I I I I I I I 140
, 5.1 Density of states for Cd=-5 random generated
T; by a 4 site embedded cluster . . . . . 156
I 5.2 Density of states for Cd=.5 random generated
4 by a 5 site embedded cluster . . . . . 157

Density of states for Cd=-5 random generated
by a 6 site embedded cluster . . . . . 158

Density of states for Cd=.2 random generated
by a 6 site embedded cluster . . . . . 160

Integrated density of states for Cd=.5
random generated by a 6 site embedded
cluster . . . . . . . . . . . . 161

Integrated density of states for C =.2
random generated by a 6 site embedded
cluster . . . . . . . . . . . . 162

Density of states for Cd=.5 random generated
by a 4 site periodically extended cluster . 167

Density of states for Cd=.5 random generated
by a 6 site periodically extended cluster . 168

Density of states for Cd=.2 random generated
by a 6 site periodically extended cluster . 170

Density of states for Cd=.5, P

=.l
generated by a 6 site periodically extended
cluster . . . . . . . . . . . . 171

 

 

Integrated density of
random generated by
extended cluster .

Integrated density of
random generated by
extended cluster .

Integrated density of
Pd d=.1generated by
extended cluster .

Density of states for

states for C =.2
a 6 site periodically

states for C =.5
a 6 site periodically

0 o c O o o 0

states for Cd=.5,
a 6 site periodically

Cd=.1 random gener—

ated by self—consistent l, 3 and 7 site

clusters . . . .

Density of states for

Cd=.5 random gener-

ated by self-consistent 1, 3 and 7 site

clusters . . . .

Density of states for

Cd=-9 random gener—

ated by self-consistent l, and 7 site

clusters . . . .

Density of states for

Cd=.5, P =31

d
generated by a 7 site self-consistent

cluster . . . .

Density of states for

Cd='5’ Pd d=‘9’

generated by a 7 site self-consistent

cluter . . . .
Integrated density of
random generated by
sistent cluster. .

Integrated density of

states for C =.5
a 7 site self-con—

states for C =.5,

Pd d=.1 generated by a 7 site se f-con-
I

sistent cluster .

Density of states for

3 site-periodic

system, h—h-d, and mass ratio 2/1. . .

Density of states for

a 3 site periodic

system, d-d-h, and mass ratio 2/1. . .

xiii

Page

173

174

175

199

201

202

203

204

206

207

319

320

 

 

CHAPTER I

INTRODUCTION‘

Historical Review

The modern theory of lattice dynamics began in

   
    
   
   
  
 
 
    
  

1912 with the work of Debyel and Born and von Karman,2
although Newton in Principia (C. 1686) began the study

:AI‘ of lattices by using a chain of masses connected by

L3;i{ harmonic springs to calculate the velocity of sound in

155‘“Zair. The first qualitative properties of lattices

‘0‘

_;;:}appeared around 1840 in response to the work of Cauchy
‘ 3

a .1
r‘t-
‘ -Jfi~,

.,_ a

‘On the theory of optical dispersion. Baden-Powell

aziandHamilton4 showed that there is a maximum frequency,
-41,W_ . .

 

7:- jgignificance of Hamiltods mathematical results. He, also,
y 7“: I} '

n§;;howed that the diatomic lattice has a forbidden gap in
Eng 2:;frequency spectrum.

7“ Lord Rayleigh7 in the Theory of Sound derived
FheOrems for a single defect in a monatomic lattice

A‘have a direct relation to defect modes in crystal

 

1. "If in a dynamical system composed of an array
of masses coupled to each other by Hookeian
Springs, a single mass is reduced (increased)
by AM, all frequencies are unchanged or .
increased (decreased), but not more than the
distance to the next unperturbed frequency."

2. Modes at the band edge may split off and

enter the forbidden frequency regions.

  
   
    
 
  
 
  

_ These theorems are clearly demonstrated in Appendix I.
Although these results were obtained in the 1880's, it
“Vwas not until 1957 that Bjork8 gave analytic expressions
‘for the location of the frequency of the mode in the
L‘nforbidden.gap-as a function of mass defect and force
'; constant defect for monatomic and diatomic chains. For
'9; detailed history and two and three dimensional applica-

ftions see Reference (9).

‘--~.¢

Classical Lattice Dynamics

Classical lattice dynamics formulated by Born and
anarman employed a Hamiltonian with a kinetic energy

Endancentral pair potential. For the one dimensional

2
P (£,t)
l (t)
gla2ma(£) 2 {’2’ 2,2 2,2

 

 

l’g'a = Rz-Rzl = (£‘2: )a

U(2.)'—U(9.‘)

_where a is the lattice constant and t is time.

     
 

Thigh atomic positions. Pa(2,t) is the momentum of the

th

thm of mass, m , at the ath atom in the 2 unit Cell.

Cl.

' . = ' 2 av ,
wry/”2,93? V(R9.,2’)+s WIIR’L VUBU“ )

+7 uEBUa”)a {INTI}; BU’ )+...

‘first term_is an arbitrary constant and the second term

:9 for a lattice in equilibrium. To first approximation

I -§;TIT- +7 22’ Ua(lut)¢aB(E,l')UB(£’,t) (1.3)

 

  
 
  
 
   
  
  
  
  
  
  

2
a 3 V
4” ”'2 ’ “‘TTfiTI
GB Bu“ 2 QUB 2 R2,2’
VTHamilton's equations of motion we get

. = Z ’ ’.
mh(2)Uq(2,t) ETB ¢uB(£,2 )UB(2,t) A. (1.4)

diet transforming Equation (1.4), we have

2 -__ .-, .
ma(2)w Ua(2) - £§B ¢a8(2,2 )UB(£ ) (1.5)

' U (z)= Hf: Ua (z, t)eimtdt

Hayes.
(k) ik-R
Ha“) == 2 -j———-o jug) e 9' (1.6)
k.) «fim ma

1
l
I
I
i
I
1

‘=“there the oj(k)h are the normal coordinates of
it the lattice and the oaj(k)'s are the expansion
. .coefficients, j is the branch index.

ng Equation (1.6) into Equation (1.5), we have

   

1k°R

L' . .-' 2 ‘ Z Z 0 (k)oB (k) ik~R£.
a; E.'Q-(k)oa.(k)e =~- ¢a8(2,z’) .—i—————i———e
'. - J 3 £78 k.j Vfim
1‘13 B
- -k‘-R
‘1ying by e and summing over 2 where
7 ik'R

e £=N6(k)

    
    
 

; than the LHS over k, we have

¢ (21%,) ‘ i(kR a-k’R )
fizz —1§-———-Qj(k)osj(k) e 2 2
, Mm N
e.jk B
- ' ' ¢ (21,24‘) ik.(R J-R )
iwge (k) = i.—J¥i-———-e z 2 (1.7)
. . a B

.2.2 ‘ _ _
w j Qj(k)0aj(k) -

2 z . = _ 1 2. i(k-k’)R2
j °j(kk%d(k) N z’BDaB(k)Qj(k)ojB(k)e
jk

‘Tquver 2 and k and rearranging we get

fl.,,‘ -2 =
H~»_§oj(k)[mj(k)ouj(k)+§ DaB(k)ij(k)] o

f: a2' ' =_ a '1
{“1,uj(k)oaj(k) g Da8(k)08j(k) (1.31

£1§Gj(k)08j(k) 3 6GB (109‘)‘

 

    
    
   
  

*
‘2 caj(k) qaj’(k) = ij, _ (1.9b)

£,monatomic linear chain, we have only one atom per
3§911 and one branch; therefore

:‘

'- ..
—} a ‘L‘
.' _ .xy
.r 04'

053(k) = -D(k)

D(k) = i m
:11'¢(2,2’) g igfiézil’
. 2,2
. D(k) = 1%(eika+e-ika-2) = - :TY- sin2(’§—a-)
(n: (k) = “is. s11;2 (’59-) fork = gin n=0,l,2, N-l (1.10)
(‘sity.Of states is
m, = 291; :2. 1
n w n ?;Z:;7;§
2» eta—2r; °if°=- “n? . . - 3'
I 0 w>m

 

no» ) = %-Re(———-—) w>o (1.11m

   
  
 
 
   
 
   
   
  
 
  

' Re(...) is the real part of (...). For the diatomic

11;;of masses ma and mb with all force constants equal

 

 

Hi7 + 1—)-w2
D(w2) = l. ( a mb \ (1.12)
"1” Re 2 I
’w - Iii—Y (0)2- 2_Y. /w2-2Y(11T— +l‘._)l
a mb a mb

where we define /:T = +i

7- vibe diatomic spectrum clearly has a gap between ferY" and 21.
,;F 4 , a

34" . The frequency spectrum of the monatomic chain was
t.£§£ given by Born and von Karman.2 Unfortunately, their
7'tgiémi8tic approach to lattice dynamics was neglected for
groximately thirty years because calculations in three
..IE;;nsions are difficult. Two approximations were used.

‘ 1907 Einsteinlo proposed treating the lattice as N

pupled harmonic oscillators of frequency wE. The density

tates normalized to one is

63(w) = 6(w-wE) (1.13)

:fved approximation was proposed by Debye,1 where

 

_ 3 2 _
vD(w) - ;§-— w 6(wmax w) (1-14)

max

   
  
  
   
 
 
   
  
   
   
 
   
   

2. where 6. is the unit step function. The l-dimensional

analogy of this density of states is

'1 6(w 'm) (1.15)

vD(w) _ ”max max

Both of these approximations allow simple calculation of

thermodynamic properties but the models are too simplified
for accurate calculatiOns, even for thermodynamic calcula-

itions..

The spectra of disordered chains were not considered

until the 1950's. The first significant work on disordered

11

lattices was by Lifshitz and, independently, by Montroll

12 Their work involved the use of classical

and-Potts.
‘Green's functions to compute the properties of isolated
defects in crystal lattices. Dysonl3 in 1953 had already
’;-presented a detailed theoretical approach to finding the
: density of states of a one-dimensional linear chain with
equal nearest neighbor force constants. Unfortunately,
I Dyson's analytic expressions have not been numerically

1

' solved. In 1957, Schmidt 4 derived a functional equation

{fdr the-random, linear chain which was solved numerically

 

   
 
 
 
 
  
    
   
  
  
  
 
 
 
   

: that the expansion of the density of states in terms

its moments involved only even moments. For the perfect

 

«:

_fi' atomic or diatomic chain 14 moments gave reasonable

.flhflmlts except near the spectra singularities. The moments

.1;

Z
J N

.k w§n(k) = Tr(DN) = {wznv(w)aw (1.16)

where tr is the trace

D is the dynamical matrix defined in Equation (1.7)
v(m) is the density of states

7 reasoned that the disordered system should
fi::%&3519°§sesg the high frequency singularities and that the

"Z Egfifient-trace method would be better for the disordered case
;Awit was for the ordered chain. ‘Their resulting spectra

‘fculated from 20 momenta were relatively smooth with some

ters of light (impurity) atoms. However, the arguments

 

CHAPTER II

NUMERICAL CALCULATIONS FOR RANDOM

BINARY CHAINS

Numerical Methods

   
  
  
   
   

To calculate the spectrum of a disordered chain

I requires some numerical effort. In one-dimension a single

:" defect (or disordered atom) destroys the translational

‘ ZSymmetry of the lattice. Plane waves will no longer

. g diagonalize the eigenvalue Equation-(1.5) and the solution

L;tanquation (1.5)‘must be found directly. ~

A Three types of boundary conditions are applicable

£§~a chain of length, N. These are:

V 1. Fixed boundary conditions where Uo=UN+l=°

2. Cyclic (Born and von Karman) boundary
conditions wherenU1=UN+1

3. Free boundary conditons

Equation (1.5) can be rewritten as
“ 2
“‘1‘” U1 ‘ 'Yi,i+1‘Ui+1‘Ui"Yi,i-1‘U1-1 U) (2'1)

'for equal force constants reduces to

10

 

11

2 _ - —

   
    
     
 

Q fiubject to the boundary conditions listed above. In

..:1.matrix form Equation (2. 2) become.s

A g =‘w2u (2.3)

when boundary conditions are applied,A is specifically

2*: given by

-m = 21 — L
.7 ‘ 7 1c Ai’j m. 6i’j m. 6iil’j (2.4)
- 1 1
for fixed boundary conditions.
. = _l _l_ ._l_ , ,
2. ‘A1,j i 611,j mi 6i %1 3 mi 6N,361'1
_ IL . . .
mN 6N,161,j‘ (2 5)

for cyclic boundary conditions.

for free boundary conditions.
‘9 matrix A.is tridiagonal for free and fixed boundary

.‘itions. The only difference in the matrix for the two

9; ary conditions. Clearly, one could easily mix the

bd fixed boundary conditions. For cyclic boundary

 

12

  
    
   
   
 
   
 
  
   
  

:gFCdnditions A is identical to the matrix for fixed boundary
1 hf Conditions except for the addition of non-zero (l,N) and

p _
9., ‘(N,l) elements. The question of which boundary condition

’3’. to impose on the problem is moot when one considers the

following theorem proved by Ledermann:22

If the elements of r rows and their corresponding
columns of a Hermitian matrix are modified in any
TWay whatever, as long as the matrix remains
Hermitian, the number of eigenvalues in any interval
' cannot increase or decrease by more than 2r.

The three types of boundary conditions differ by
only two elements from each other, and, therefore, no
,- frequency interval may contain more than four additional

3 A eigenvalues or four fewer eigenvalues. As long as a

-'%77 frequency interval contains many eigenvalues compared to
itfié-gjfour, the frequency spectrum will be independent of
: {' lboundary conditions. The eigenvectors, however, are
another matter which we will discuss later. Therefore,
fer ease of numerical computation, we use fixed boundary
_g‘conditions. The matrix A in Equation (2.4) can be made

A‘symmetric by the transformation

xi =71}? Ui . . (2.6)
when A’ 5 = (12 x (2.7)
2 = 2_Y _ ____v
where Ai'j mi (8in m m 1 5111,3‘ (2.8)
' i ii

,eigenvalues of A and A’ are identical although the

pending eigenvectors are different.* A’ is now a

 

 

 

13

 
    
  
  
  
  
  
  
  
  
  
    
   
   
 
 
   

,

_ 3 tridiagonal symmetric matrix. The Given's method for
6‘ 1
finding eigenvalues of a tridiagonal symmetric matrix

- can be employed.42 Appendix A gives a description of the

method and application to this problem. P. Dean18 in

1959 first applied this technique to a disordered linear

chain. Besides Dean and his coworkers,19 Payton and

0

Visscher2 have performed similar numerical calculations

for the spectra of disordered systems, including small
two and three dimensional lattices. Also, Dean has
performed these calculations on glass—like chains where
the force constants vary in a probabilistic fashion.

An excellent review article is Reference (21).

Numerical Spectra

‘2 The spectra of disordered chains presented in
4
8

.-’f, this section serve asaniintroduction to spectra with
short range order. These spectra have been recalculated

" ‘ from Dean's work to conform to the rest of the spectra

presented in the thesis. The spectra for the random

‘1

tzhains are for a mass ratio of heavy to light mass of 2.

s
‘
fu-

‘;_ The concentration of light mass will be called, Cd' To

{fkrfgenerate a chain of length, N, with a concentration
i Ca of light masses, we employ a random number generator
(which generates random numbers from zero to one. For

’“écn atom, the generator is sampled and if the number is

 

l4

   
  
  
  
  
  
   
   
  
   
   
   
   
 
  
    
  

other wise, it is a heavy mass (host). A statistical
analysis shows that 32% of the 100 unit chains generated
for Cd =.5 will have Cd<.45 or >.55. By contrast, for

N=1000, about 5% of the chains will have C <.49 or >.Sl.

d
One of the problems with random chains is a determination
of the length of chain necessary to insure a reasonably
accurate spectrum. To examine this variability, we look
at chains of 1000, 10000, 50000, and 100000 atoms.

Whereas a spectrum for a 10000 unit chain requires less

than thirty seconds on the UNIVAC 1108, the 100000 unit

'4' y...

i

N t
('-

chain requires over five minutes of CPU. Figures (2.1)

‘Ia

I

and (2.2) show Spectra of two 1000 unit chains with con-

.“ L—

’7

centrations of .43 and .48 light masses respectively. In
2

.4
Lo

. wgl

2"

the region 25w :4, the two graphs are nearly identical

,qu

$~g
WU

with the only serious descrepancy occuring near w2=4. The
region .3 to 2, however, shows a large degree of variability.
Figure (2.3) shows the fifty percent random 10000 unit

chain with an actual concentration of Cd=.4956. The m2
region 2 to 4 is much the same as in the 1000 unit chains;
however, there is a definite refinement in the spectrum

,in this region especially in the magnitudes of the maximums
<3 and minimums. The wz region from .3 to 2 is markedly
'Sgfiggther around D(w2)=.25 than the 1000 unit spectra.

igure (2.4) for the 50000 unit chain with C =.50093,

d

er smoothing in the .3 to 2 m2 region. Figure (2.5)

 

 

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20

shows the 100000 unit spectrum with Cd=.49809. There
are only minute differences between this spectrum and
the 50000 unit spectrum. The w2 region .3 to 2 does
possess some structure even for a chain of this length.

Dean21 recently published a 250,000 unit chain confirming

this structure. It is clear from looking at the figures

 

that a 10000 unit chain provides the significant spectral
structure seen in longer Chains and is much quicker to
evaluate; therefore with few exceptions as noted the spectra
presented in the rest of this thesis will be for 10000 unit
chains.

Dean previously found, and we clearly demonstrate
in the Defect Cluster theoretical section, that the peaks
in the disordered spectra from 2. to 4. can be associated
with clusters of defect atoms.

The next set of figures will show the variation
in random spectra as a function of defect concentration.
Figure (2.6) shows the spectrum of a monatomic heavy chain
corresponding to Equation (1.11). The first and last
bars extend to D(m2) = 2.2575 but have to be truncated to
give a more readable scale. Figure (2.7) is the spectrum
for one percent defect concentration. Using a 100000 unit
chain, the concentration obtained was .00975. The error
analysis at the end of Appendix B shows the Cd=.5 random
chain gives the most accurate calculations for any fixed

length chain. The mode at 2.66 corresponds to an isolated

 

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23

defectwhereas the mode at 3.22 corresponds to a nearest-

neighbor defect pair. There was only one such pair in the
chain. The bar at w2=l.98 was reduced from D(wz) = 2.26

i to 2.12 with the rest of the bar heights remaining nearly

the same. Figure (2.8) for Cd = .2(.2019) clearly shows
the three—defect cluster at “2 = 3.54 and the four-defect
cluster at w2 = .366 as well as the pair and single-defect.

The other peaks are more subtle and are discussed later.

As Cd is increased to 0.2 the heavy mass band becomes severely
depleted at the high frequency end with a small but
Significant decrease everywhere in the region w2<2. Where-

as D(2.66) = 1.45 in Figure (2.8),Figure (2.5) shows this

local modeseverehrdepleted, corresponding to the reduction

 

in the probability of having only one defect surrounded
by host masses. Figure (2.9) shows in contrast to

Figure (2.5), the perfect diatomic chain corresponding to

  
  
  
  
  
  

Equation (1.12). The interesting thing to note is that

 

the random chain spectra resembles defects in-a heavy

chain instead of a somewhat disordered diatomic chain.

5" Figure (2.10) gives the spectrum for Cd = .8(.8025). The

7 structure in the w2 region 2. to 4. is clearly disappearing

as the spectrum begins to resemble the perfect light chain.

 

 

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CHAPTER III

SHORT-RANGE ORDERED CHAINS AND

THEIR FREQUENCY SPECTRA

Short Range Order in Scattering Theory

   
  
  
  
   
  
  
   
    

The first suggestion that alloys may possess at
least some degree of order instead of being random was

made by Tammann.23 He observed that the copper in copper-
gold alloys dissolved in nitric acid only when the copper

. concentration exceeded 50 percent suggesting that CuAu is
ordered. The most conclusive evidence of order in alloys
is_presented by the superlattice lines on x-ray diffraction
patterns. Owing to the order in CuAu alloys first observed
by Johansson and Linde,24 early work on ordered alloys
.defined parameters in terms of sublattices, i.e. a copper
lattice superimposed on a gold lattice. A long range

'- order parameter S was introduced by Bragg and Williams25

hand a short-range order parameter a was introduced by Bethe.26

The Bragg-Williams order parameter is given by

ra-FA r -FB
S = W: "F (3.1)
A B

. where r“(8) is the fraction of atoms on sublattice
-;‘.v «(5) that are supposed to be on the sublattioe
'7; a(B). .

‘.f FA(B) is the concentration of constituent
:'~'__“_-. I. A (B) o i
‘ '3.. 27

 

 

28

For a perfectly ordered system, ra = rB = l and 8:1, and
for a disordered system (i.e. a system with no long range
order) ra = FA and r8 = FE and 8:0. The Bethe short-range
order parameter is concerned only with the configuration

of nearest neighbors.
o = (q-qr)/(qm-qr) . (3.2)

where q is the fraction of unlike pairs among
nearest neighbors q is the fraction of
unlike nearest neighbors for an ordered
system (usually one) and q is the fraction
of unlike nearest neighbors for a completely
random system.
Therefore o=l represents a perfectly ordered system and
0:0 a random system. Only in the limit o=l does the
short-range order parameter imply long-range order;
generally, a system can have short-range order without
long-range order. These order parameters represent the
two extremes of the definition of order. The Bragg-Williams
deals with the order present in the whole system whereas
the Bethe parameter is concerned with only nearest neighbors.
The concept of short range order must be generalized to
understand the results of x-ray scattering experiments.
27

Warren, Cowley and M05528 introduced short-range order

parameters in terms of conditional pair correlation functions

for binary allbys as pi’dl , pg’h£ , pg’dg and pg’hg and
1' 2 1' 2 l' 2 1' 2
d,d _ _
0,1’22 — Cd + (1 Cd)a£ 2 (3.3)

1' 2

29

d . . . . .
where 03' 2 is the conditional pair correlation
l' 2

function between defects at sites 21 and

£2, and mg ’2 is the short-range order

parameter.

pi'dz
1’ 2

a defect at site £1, then there is a defect at site £2.

Specifically, is the probability that if there is

From this definition we can deduce the following

og’dg + og'hg = l (3.4)
1' 2 ”1' 2

pz'hz + pg'dfi = 1 (3.5)
l, 2 1' 2

since given that site 21 is occupied by a certain atom,

£2 must be a host or defect for a binary system. For an

isotroph25ystem, the probability that site £1 is a host

and 12 is defect must be equal to the probability that

site 21 is a defect and site 22 is a host, i.e.

dph

h,d
d 02

C = ChDR (3.6)

2

1' 2 2

1' 2

From Equation (3.3), we see a can assume any

21,12
value between +1 and -l with a. q = 6 . being the random
ly‘z il'”2
limit. For any case c2 2 is necessarily one.

1’ 1
Using these short-range order parameters, we show

in Appendix C that for waves incident on a short-range
ordered system the scattering intensity in the Born

approximation is

30

m 'k
Id 2 el na

nzdoo

[(fh(l-Cd)+fdc d) 2+Cd (l— C dn)a (f— (3.7)

d fh) 2]

where a is the lattice constant,
fh is the structure factor for the host atom

fd is the structure factor for the defect atom

k is the incident wave vector

Therefore, for structure factors independent of k,

2

Id N 26(ka- 2nm)[fh (l- C dd)+fC d]2 +Cd (l- C d)0L(k)(fd- fh)

(3.8)

where d(k) = §_ eikana (3.9)

The first term of (3.8) gives the Bragg reflection
from an average host-defect chain where all sites are
randomly occupied by host or defects in proportion to their
respective concentrations. The second term is a diffuse
scattering.

There are several limiting cases in which the
short-range order becomes long-range order.'

Case 1: cl = (-li

In this case, the lattice is an ordered diatomic

chain and

a(k) = %[6[ka-(2m+l)n]] (3.10)

The unit cell in real space is now twice as long as that
for the monatomic case and Bragg peaks occur at both

even and odd multiples of w/a.

31

Case 2: d2

In this case, it is impossible to have any mixed
chain, a possible interpretation of this case is to super-
impose two perfect chains. The Bragg peaks occur at the

superlattice reflection points with strength

IdN%(C f2+(l-C

d d f:)6(ka-2nm) (3.11)

d)

and there is no diffuse scattering.

The experimentalist measures d(k) using various
energy x-rays and neutrons and inverts the results to find
29

C1

n In generating chains with short range order (SRO) we

can find an and can consequently explore the implications

of the order in terms of diffuse scattering.

Short Range Order Work

Using the long-range order parameters of Bragg and
Williams with the Warren-Cowley short range order para-
meters, numerous contributions have been made to the theory
of order-disorder transitions in alloys. A good summary
of the work is given in Reference (30). The first attempt
at introducing general short-range order into the theory
of thermodynamic properties and frequency spectra was made
in 1968 by Hartmann.31 Using Green's function techniques,
he was able to calculate a variety of physical properties

in the low concentration of defects limit for three

32

dimensional crystals. These calculations were for
arbitrary short-range order describable by pair correla-
tions. Subsequently Taylor32 was able to introduce the
short-range parameters into his self consistent Green's
function formalism (coherent potential approximation).
Taylors work will be discussed in more detail in the theory
section (V). Taylor's formalism with short-range order
was valid, however, only near the random limits.

The short-range order discussed above is in
terms of pair correlation functions, which are in fact
adequate for scattering theory in the Born approximation.
However, for generating a binary chain and for exact
calculations of physical quantities which can depend on
scattering to higher order than the second in the defect
perturbation, we need all orders of correlation functions.
Also, the extension of pair correlation function formalism
to n(>2) constituent chains is not readily apparent. There-

fore, we consider the Markov process.

Markov Chain

 

To examine the eigenvalues and eigenvectors of a
linear chain by numerical methods, a linear chain must be
constructed. An ordered chain can be simply generated by
placing unit cell clusters one after another. A random
chain with n constitutents can be numerically generated

as discussed in Chapter II. To generate a chain with

33

short—range order with the proper concentration of each
constituent and the correct stochastic relationships
between atoms, the theory of Markov processes is employed.35’36
Stochastic processes where the probability that a
physical system will be in a given state at time t2 may be
deduced from a knowledge of its state at an earlier time t1,
aind does not depend on the history of the system before time
tl are called Markov processes. More formally, a discrete
parameter stochastic process EX(t), t=0,1,2 ....J or a
continuous parameter stochastic process [X(t,), tZOJ is
said to be a Markov process if for any set of n time points
tl<t2<t3<....<tn in the index set of the process, the
conditional distribution of X(tn), for given values of

X(t1),X(t2),....,X(tn_l) depends only on X(tn_1) the most

recent value; more precisely, for any real numbers xl,x2

§[X(tn):xnlx(tl)=xl,...

§[X(tn):xan(t )=x (3.12)

n-1 n-11

This equation should be read as follows, the probability
that the variable X at tn is greater than or equal to

the state xn subject to the conditions that the variable X
at tl,has the value x1, the variable X at t2 has the value
x2, and so on up to the condition that X at tn-l has the

value xn_1 is equal to the probability that the variable

34

X at tn is greater than or equal to the state xn subject
only to the condition that the variable X at tn—l has

the value xn_1. The order among the states can be included
in any self consistent arbitrary way.

Markov processes are classified according to the
nature of the index set of the process (parameter) and the
nature of the state space of the process. The index set
of the process can be continuous or discrete. The index
set is a special type of ordered set. When the parameter
is time, an intuitive interpretation of a Markov process is
a process where the future depends only on the present and
not on the past. For the linear chain, the parameter is
distance down the chain. This distance, however, must be
directional in the sense of proceeding down the chain in
only one direction. For a position along the chain 2.,
the position RO—Ql,£l>0 must be considered as a previously
occurring state and 20+£2,£2>0 as a future state. For
atoms on lattice sites the index set is discrete. The
state space of the linear chain is the set of the types of
atoms in the chain. The state space is called discrete
if it contains a finite or countable infinite number of
states as does the linear chain. A state space which is
not discrete is continuous. If the state space is discrete
the Markov process is called a Markov chain.

Table 3.1 shows the four basic types of Markov

processes. The one used to generate the linear atomic chain

35

is theidiscrete parameter Markov chain. Theoretical work
is still very limited for the continuous state space
Markov processes. A little more is known about continuous
parameter Markov chains with the discrete parameter
Markov chain being the best understood and most widely

applied.

TABLE 3.l.--The classification of Markov processes.

 

State Space

 

 

 

Discrete Continuous
Discrete Discrete
Discrete Parameter Parameter
Markov Markov
Chain Process
Parameter
Continuous Continuous
Space . Parameter Parameter
Continuous
Markov Markov
Chain Process

 

The remainder of our discussion will be concerned
only with discrete parameter Markov chains.

Equation 3.12 can be rewritten for the discrete
parameter Markov chain with lattice sites as the parameter
as follows: .

Definition:. Let X£ be a random variable where the value
of X1 represents the atom at position (lattice site) 2.

The sequence [X2] is the linear chain. The sequence [X1]

is a Markov chain, if the set of possible X2 is discrete

36

and if for any integer m>2 and any set of m points

£1<£2<....<£m, the conditional distribution of X2 for
m

given values of xi ,...,X2 depends only on X2 , the
1 m-l m-l

closest atom; i.e. for any real numbers x1,x2,...,xm,

=x (3.13)

PEX2m=Xm|X£ =xl,...,X2 =x 1] = PEXQ =x IX m-lJ

l m-l m- m m z

A Markov chain is described by a transition
probability function, Pmk(£0,21), which represents the
conditional probability that the state of the system Willlxg
at point 11 in the state k, given that at point
£0(<21) the system is in state j. The Markov process is
said to be homogeneous in space or to have stationary
transition probabilities, if Pj'k(£0,£1) depends on 21
and 20 only through the difference (ll-£0). The transition

probability function is also called the conditional

probability mass function and_is

>10 (3.14)

Pj’k(£o,ll)=P[X£1=k|X£0=j] for £1

In order to specify the probability law of a
discrete parameter Markov chain, the probability mass

function (not conditional)

Pj(£) = PEX£=jJ (3.15)

37

must also be specified. The linear atomic chain should
have homogeneous or stationary transition probabilities.
For such a homogeneous chain it is physically realistic
to expect the same stochastic relationships between atoms
in different regions in the chain. For the homogeneous
discrete parameter Markov chain, Equation (3.14) can be

rewritten as

Pj’k(n)=P[X2+n=k|X2=j] for any integer £10 (3.16)

Equation (3.16) is called the n step transition probability
function. In words, Pj k(n) is the conditional probability
I
of making a transition to state k, n steps after being
in state j. P. (l) is usually rewritten as P. .
Jrk Jrk

The transition probability function of a Markov

chain [Xn] satisfies the Chapman-Kolmogorov equation:

for any lattice sites 23>£2>£lzp and states j and k

P

j,k(£l'£3) = in’i(ll,22)Pi'k(22,l3) (3.17)

This is a necessary but not sufficient condition for a
Markov chain.

The transition probabilities are best exhibited in
the form of a matrix called the transition probability

matrix

38

P(11,22) = [{Pi j £1,22)}] with rows and

columns i and j. (3.18)
The matrix elements also satisfy the following conditions
Pi’j(£l,22):0 for all 1,] (3.19)
and

E Pj,k(£l'£2) = l for all 3 (3.20)

The Chapman-Kolmogorov equation can be rewritten in matrix

form as

P(£1,23) = P(21,£2 )P(£2,£3) (3.21)

From the Chapman-Kolmogorov equation some funda-
mental recursive relations can be derived for the discrete
parameter Markov chain. For the homogeneous chain the
.transition probability matrix P(Rl,£2) depends only on the
difference n=(£2-£l) and can be rewritten as P(n). Equation

(3.21) can be rewritten as
P(n) = P(m)P(n-m) where m<n (3.22)

P(l) is usually rewritten as P. Using Equation (3.22)

O
recursively, we see

n

P(n) = P(n-l)P=P(n-2)PP= ... =P (3.23)

39

Next, from the probability mass function defined
in Equation (3.15), we can define the unconditional

probability vector as the row vector

p(n) = [{pj(n)}] with columns j (3.24)

Given an initial unconditional probability vector p(O),

it follows that
p(n) % p<0)9(n>=p(0)P“ (3.25)

As a consequence, the probability law of a homogeneous
Markov chain [xn] is completely determined once one knows
the one step probability transition matrix P and the
unconditional probability vector p(O).

A Markov chain [Xn] is said to be a finite Markov
chain with k states if the number of possible values of
the random variables [Xn] is finite and equal to k. The
transition probabilities pj,k are non—zero for only a
finite number of values of j and k and the transition
probability matrix P is then a k x k matrix.

An example of a discrete parameter finite homo-
geneous Markov chain would be a linear atomic chain 1
consisting of two states a host state (atom) h and a
defect state (atom) d. The unconditional probability

vector would be

40

p(O) = [ph’pd] (3.26)

and the transition probability matrix is

ph,h ph,d
p = (3.27)
pd,h pd,d

Although we have not examined how we determine

P as yet, it does have some interesting properties.

2+ (+)
ph,h ph,dpd,h ph,d ph,h pd,d

2
ph,d(pd,d+ph,h) pd,d+ph,dpd,h

(3.28)

and for lpd,d+ph,h-l |<1, the n - step transition probability

matrix is

l‘pd,d 1‘Ph,h
P(n) = l/(Z-ph,h-pd,d) 1- l-
Pa,a ph,h
(3.29)
n
(Ph,h+pa,a 1) 1 ph,h ‘(1’9h,h)

+
Ti'ph,h'pd,d) —(1

 

-pd,d) l-pd’d

The proof of Equation (3.29) is given in Appendix B. The
asymptotic expressions for the n - step transition

probabilities are

41

 

 

l-p
' — = d,d
l~p
' ° h h
1 = I

To determine the transition probabilities, the
evolution in parameter space (time or distance) of a
discrete parameter homogeneous Markov chain [Xn] must be
studied. First, the states of a chain can be classified
according to whether it is possible to go from a given
state to another state.

Definition: A state k is said to be accessible from a
state j (j+k) if, for some integer nil, pj'k(n)>0. Two
states j and k are said to communicate fitvk) if j is
accessible from k and k is accessible from j.

For a fixed concentration linear chain all states
must communicate; otherwise, some pj'k(n) = 0 for all n
implies that once the state j is entered state k can
never be reentered modifying the concentration of the
- state k.

Given a state j of a Markov chain, its communica-
ting class C(j) is defined to be the set of all k states

in the chain which communicate with j, i.e.,

k e C(j) if and only if k++j

42

For the linear atomic chain, we require all states of the
chain to communicate with each other. Since the communica-
ting class C contains all the states of the system, there
are no states outside the set and C is defined as a

closed set. More formally,

Definition: A non-empty set C of states is said to be

 

closed if no state outside the set is accessible from any
state inside the set.

Next, we can define the occupation number Nk(n)
of the state k in the first n transitions. More precisely,
Nk(n) is equal to the number of integers v satisfying

lgyin and Xv=k. The total occupation time of k is

(3.31)

(3.32)

Nkm = 53,59 Nkm)
The occupation times can be represented as the sum of
random variables. Define for any state k and n=1,2,....
Zk(n) = 1 if Xn=k
= 0 1f ank
Then, we can write
n
.Nk(n) = Z Zk(m) (3.33)
=1
and
Nk(°°) = 2 2km) (3.34)

=1

43

With these relationships we can define the following

probabilities

fj,k = PENk(m)>0[xO=j] (3.35)
and

gj,k = PENk(w)=w|xO=j] ‘ (3.36)

In words, fj,k is the conditional probability of ever

visiting the state k given that the chain is initially in state
j, and gj,k is the conditional probability of an infinite
number of visits to the state k given the chain is at

some initial time in state j. For the linear chain with

fixed concentration of constituents, the requirement that

every state communicates implies

fj k = l (3.37)

and in an infinite chain every state occurs an infinite

number of times to maintain fixed concentrations implying
gj'k = l (3.38)

Definition; A state is said to be recurrent if f l

 

k,k:

or a state k is recurrent if the probability is one that

the Markov chain will return to state k.

Definition; A recurrent Markov chain is said to be

 

irreducible if all pairs of states of the chain communicate

(f.

3 k>0 for all j,k).

44

Therefore, the linear atomic chain has a closed
recurrent irreducible set of states. The chain must also
have a fixed concentration of constituents as the length
of the chain approaches infinity.

Definition: A Markov chain with state space C possesses

 

a long run distribution if there exists a probability
distribution {Wk’ keC}, having the property that for every

j and k in C
fig Pj,k(n) = wk (3.39)
summing over k

i; lpg pj'k(n) = lig i_pjlk(n)=l=ifik (3.40)

which gives a useful relationship between the concentrations
flk' The interchange of the summation and limit in Equation
(3.40) is not rigorous but Appendix B has a rigorous

proof- No matter what the initial unconditional probability

distribution {pk(0), ksC}, the unconditional probability

pk(n) tends to “k as n tends to infinity

r133; pkm) gig ;pj(0)pj,k(n>

J

gpjm) gig! pj,k(n)

gpjmmkmk (3.41)

45

Definition: A Markov chain with a state space C is said

 

to have a stationary distribution if there exists a
probability distribution ({"k' keC} having the property

that for every k in C

(3.42)

In order to state conditions under which the
irreducible Markov chain possesses a long run distribution,
we need to introduce the concept of the period of the state.

Definition:' The period d(k) of a return state k of

 

a Markov chain is defined to be the greatest common divisor
of the set of integers n for which pk’k(n)>0. A state is
aperiodic if it has a period of 1.

For an irreducible Markov chain, if pk,k>0 for any
k in c, then the state is aperiodic. Also, if an integer
n can be found such that pj’k(n)>0 for all j and k in C,
the chain is aperiodic. In fact, for the linear chain
only ordered (periodic) chains are not aperiodic.

If a chain is irreducible, aperiodic, and recurrent
it is called an ergodic chain. For our purposes we want
a homogeneous chain, for which the stochastic relationships
.are the same throughout the entire length of the chain.
Therefore werequire a Markov chain which is ergodic. Such

a chain has a unique long-run distribution.

46

Theorem 1: An ergodic Markov chain has a unique
long-run distribution, {nk,k€C} with 0<wk<w. The long-run
distribution of an ergodic Markov chain is the unique
solution of the system of Equation (3.42) satisfying
Equation (3.40).

Theorem 1 is proved in Appendix B.

The converse of Theorem 1 is also true.

Converse of Theorem 1: If a Markov chain has a

unique long run distribution in keC} with 0<Wk<m ,

k'
then the chain is ergodic.

Proof: The converse is most easily proved by
showing that any Markov chain which is not ergodic does
not possess a unique long run distribution with 0<nk<w.
A Markov chain is not ergodic if it is periodic and/or

reducible and/or non-recurrent. The definition of

periodic is the contrapositive of the definition of

 

aperiodic.
Definition: If a chain is periodic, then an integer n
cannot be found such that P. (n)>0 for all j and k in C.

31k

The definition implies some P.

j k(n)=0 for any n and

therefore %;g Pj k(n) does not exist in the ordinary sense
I
or is equal to zero. Therefore ”k is not unique or is

equal to zero in violation of the conjecture.

47

A reducible Markov chain is defined by the contra-
positive of the definition of an irreducible Markov chain.

Definition: If. a chain is a reducible Markov chain, then

 

not all pairs of states communicate. That is to say some
fj,k=0° This implies Pj,k(n) = 0 and therefore lig Pj,k(n)=
0=nk again in violation of the inequality 0<n<m.

Finally, the contrapositive of,a recurrent Markov
chain, gives the definition of the non-recurrent Markov
chain.

Definition: If a Markov chain state is non-recurrent,

 

then the probability that the Markov chain will return to
state k from a state k is not one (fk k7‘1). This implies
I

fk k<l' In Appendix B, we show the f <1 implies
I

k,k

=0 and 2 Pk’k(n)<w. Therefore; éi§ Pk’k(n)=0='nk

gk’k n=0
in violation of the inequality 0<nk<w. This completes the
proof of the converse. Therefore, the solution to
Equations (3.42) satisfying Equation (3.40) will give a
ergodic chain for all wk#0.

{For the two component linear chain with concentra-

tions ch and 0d Equation (3.42) gives

c (3.43)

h = Chph,h+cdpd,h

cd = Chph,d+cdpd,d (3.44)

Equation (3.40) gives

48
ch+cd = l (3.45)
and Equation (3.20) gives
ph,h+ph,d = l (3.46)
pd,d+Pd,h = l (3.47)

From this set of equations, it is obvious the combining of
Equations (3.44), (3.45), (3.46), and (3.47) gives
Equation (3.43) which is a redundant equation. Given cd

and pd d’ the rest of the variables can be specified.
I

ch = l-Cd (3.48)
pd,h = l—pd,d ‘ (3.49)
ph’d = Cd(1_pd,d)/(l-cd) (3.50)
ph,h = l‘Ph’d (3.51)

The first atom in the chain is selected as in the
random case. A uniformily distributed (0+1) random number
is picked and compared with the concentration of defects

c If the number is less than Cd, the atom is a defect;

d'
otherwise, it is a host. After the first atom has been
selected, a random number is picked and compared with either
ph d or pd d depending on whether the previous atom was a

I I .
host or a defect respectively. If the random number is less

than this value the atom is defect, and otherwise it is a

host. This procedure generates the correct Markov chain

49

with the proper concentration in the limit of long chains.
A statistical analysis of these chains is included in
Appendix B.

Another example is a three (state) constituent
linear chain with concentrations Cl, C and c for atoms

2 3
of types 1, 2 and 3. Equation (3.42) gives

C1 = c1P1,i+°292,1+°3p3,1 (3°52)

C2 = c1Pi,2+czpz,2+°3p3,2 . (3'53)

03 = c1P1,3+°292,3+°3p3,3 (3'54)
Equation (3.40) becomes

cl+c2+c3 = 1 (3.55)
and Equation (3.20) gives

p1'1+p1'2+pl,3 = l (3.56)

p2,1+p2,2+p2,3 = 1 (3.57)

p3.1+P3,2+Pz,3 = 1 ' (3°53)

It is easily shown that one equation is redundant;
therefore, eliminating Equation (3.54), six variables must
be specified to construct an ergodic Markov chain. Picking
cl,c2,pl'1,pl’2,p2’l, and p2,2 as spec1fied, Equations (3.52) -
(3.58) can be solved.

50

p1,3 = l"P1,2’pi,i

p2,3 = 1‘p2,i‘pz,2

c3 = l-cl-c2

P3’l [C1(l-pl,l)-C2p2
p3,2 [C2(l'pz,2)‘cipi
p3,3 = 1"‘-:’3,i’p3,2

For the two and three state linea
constraints which limit the range
tions or the input probabilities

between zero and one.

specifying arbitrary concentration for the defect c

transition probability is constra
0<ph,d<l
Os[cd(l-pd’d)1/(l-ca)<l
0<l-pd’d<(1-cd)/cd

(2_l/cd)<pd,d<l

For the three constituent chain,

complicated.

(3.59)

(3.60)
(3.61)
Ill/(l-cl-cz) (3.62)
’23/(l-cl-c2) (3.63)

(3.64)

r chains, there are
of the input concentra-

to a range less than that

'For the two constituent chain when

d' the

ined for cd<.5.

(3.65)

the constraints are more

51
' The restriction from the concentration of constituents is
0<c3<l
which leads to
0<(cl+c2)<l (3.66)
The restrictions from the probabilities are: first,
0<p1’3<l'
which implies
0<(p1,1+p1,2)<1 (3.67)
Next,
0<p2’3<l
which implies
O<(p2'l+p2’2)<l (3.68)
The third restriction is
O<p1,3<1
' which gives'
0<[cl(l-p1,1)~c2p2’lJ/(l-cl-c2)<1

P2,1<Cl(l-pl'l)/C2<[p2’l+(l-cl-c2)/c2] (3.69)

52

The final restriction is

0<p2,3<l

which gives
0<[C2(l_p2’2)-Clpl,2]/(l-Cl-C2)<1
Pl,2<[C2(1‘P2’2)/Cl]<[Pl’2+(l‘Cl—C2)/Cl] (3.70)

So far we have discussed strictly Markov chains.
In constructing such a chain the choice for the state
(occupancy) at site 2 is affected only by the state at site
2-1. 'We can generalize our method by bending somewhat the
definition of the Markov chain. In the two state system
for example an obvious extension would be to relate the'
atom at 2 to the atoms at 2-1 and 2-2 instead of just the
atom at 2-1. In fact it is desireous to relate the atom at
2 to the atoms at 2-1, 2—2, l-3,....,2-n; By analogy to
the ergodic Markov chain we will define the following:

Def: the sequence [Xg] is an nth order Markov
chain if each X2 is discrete and for any integer m>n+l and

any set of m points 2162 <...<2m, the conditional distri-

2
bution of X2 for given values of X2 , X2 ,...,X£ ,
m l 2 m-l
depends only on X2 ,...,X2 , the n closest atoms. For
m-n ~ m-l

any real numbers x11x2,...,xm

53

P[X£m=xmlx£l=xl,xlz=x2,...,X£m_l=xm_1
(3.71)

Ptxg =Xmlx2 =x ,xg- +xm_n+l,...,x£ =Xm-1]
m m—n m-n+l m-l

The one step probability transition function would be

pi'j'-°-,s;t=P[x2=tlX2-n=l'X2-n+1=3v--~Xi_l=S] (3.72)

These transition probabilities satisfy the Chapman-

th

Kolomogorov equation, but for n>l, the n order Markov

chain as defined is not a Markov process. A transformation,38

however, can be made to make it a Markov process. For an

ergodic Markov chain with m constituents, an nth order

n

Markov chain will have mg states with an mn by m transition

probability matrix with mp(mn-m) identically zero transition
matrix elements. The one step probability transition

function is redefined as

. . =P. . . =
pl,J,...,S;t l'J’oooIS;J,oooS,t

PEXQI=tIX =S,..,X i,X (3.73)

2-1 2-n+1=3lX2-n= 2-n+1=3""X2-1=s]

. For the two state linear chain, the second order

Markov chain would consist of 22=4 states; namely,

hh
hd (3.74)
dh
dd

54

with the transition matrix being 4 x 4 with 4(4-2) = 8

zero transition matrix elements.

phh,dh = phh,dd = 0
phd,hd = phd,hh = 0
(3.75)
pdh,dh = pdh,dd = O
pdd,hd = pdd,hh = 0
and 8 non-zero matrix elements
pdd,d = pdd,dd
pdd,h = pdd,dh
Pdh,d = Pdh,hd
pdh,h = pdh,hh
Phd’d = phd,dd (3°76)
phd,h = phd,dh
phh,d = phh,hd
phh,h = phh,hh
Equation (3.40) then gives
chh+chd+cdh+cdd = l (3.77)
which we need to relate to ch and Cd' The relation is
ch = chh+(cdh+chd)/2 (3.78)

55

c +(C

d = Cdd thCha)/2

so that

+c = l

C h

d

Equation (3.20) gives

phh,hd+phh,hh+phh,dh+phh,dd =
phd,hh+phd,hd+phd,dh+phd,dd =
pdh,hh+pdh,hd+pdh,dh+pdh,dd =

pdd,hh+pdd,hd+pdd,dh+pdd,dd =

which because of the zero elements reduce to

phh,d+phh,h

phd,d+phd,h = 1 (phd,dd+phd,dh =
pdh,d+pdh,h = 1 (pdh,hd+pdh,hh

pdd,d+pdd,h = 1 (pdd,dd+pdd,dh =

Equation (3.42) gives

Chh = Chhphh,hh+chdphd,hh+cdhpdh,hh+cddpdd,hh

cdh = chhphh,dh+chdphd,dh+cdhpdh,dh+cddpdd,dh

l (phh,hd+phh,hh =

1)

l)

-l)

l)

hd = Chhphh,hd+chdphd,hd+cdhpdh,hd+Cddpdd,hd

(3.79)

(3.80)

(3.81)
(3.82)
(3.83)

(3.84)

(3.81')
(3.82')
(3.83')

(3.84')

(3.85)
(3.86)

(3.87)

56

Cad = Chhphh,dd+chdphd,dd+cdhpdh,dd+cddpdd,dd (3°88)

These equations can also be simplified to

Chh = Chhphh,h+cdhpdh,h (3'85')

Chd = Chhphh,d+cdhpdh,d (3'86')

th = Chdphd,h+cddpdd,h (3'87')

Cdd = Chdphd,d+cddpdd,d (3°88')
Using Equations (3.75) - (3.88), one equation is

redundant. We can arbitrarily eliminate Equation (3.85').
From theremaining equations, we have ten equations with
fourteen unknowns requiring us to specify four variables.
. If we take these four speCifications to be Cd’pdd,d’pdh,d

and phd d' the equations can be solved in terms of these
I

variables.

Ch = 1‘Ca (3.89)
pdd,h = 1'Paa,a ' . . (3.90)
pdh,h = l‘Pdh,d (3.91)
phd,h = 1‘Phd,d ' (3.92)

from Equation (3.88') we have

Chd = Cdd(l-pdd,d)/phd,d (3°93)

57

from Equation (3.87') using Equations (3.93), (3.92), and

(3.90)
th = Cdd(l-pdd,d)(l-phd,d)/phd,d+cdd(l_pdd,d)
= Cda(l’pad,d)/phd,a
th = Chd

Using Equation (3.80) with (3.94), we get

Substituting Equation (3.95) into (3.93) gives

C-C

d dd = Cdd(1-pdd,d)/phd,d

Cdd = cdphd,d/(l+phd,d-pdd,d)

Using Equation (3.96) in (3.95), gives
c

dh = Chd = cd(l_pdd,d)/(l+phd,d-pdd,d)

Equation (3.81) gives

l-c

Chh aa‘zcah

Chh = l-CdEPhd,d+2(l-pdd,d)J/(l+phd,d-pdd,d)

and from Equation (3.86'), we get

phh,d = Chd(1'Pdh,d)/Chh

(3.94)

(3.95)

(3.96)

(3.97)

(3.98)

58

Cd(l-pdd ’d) (l-pdh’d)
l+phd,d-pdd,d-Cd[phd,d+2(l-pdd,d)J

(3.99)

 

phh,d

Finally,

phh,h = l—phh,d (3.100)

The procedure for generating the chain is identical
to that for the first order Markov chain except the states
are two atoms long. The first state is hh, hd, dh or dd
depending on where the random number falls in the unit
interval. The unit interval is divided into Chh' Chd’
cdh and cdd respectively. From there on the probability
the next atom will be a host or a defect depends on the
preceding two atoms.

As in the case of the first order chain, the pro-

babilities are constrained to the unit interval.

' . ‘ = O O <
—' 1:) IJk( p13 Ik)—1

Unlike the first order chain where pi j = l or 0 for a two
I

state system gives an ordered chain, some pij k may equal
I
zero or one and not produce an ordered chain. However,

for a given c are not always

d’ pdd,d’ pdh,d and phd,d
allowed any value from zero to one. Since nghh ail, we

have

Oicd(1-pdh’d)/[l—2cd+(l-cd)phd’d/(l-pdd,d)Jfil (3.102)

59

One question we can easily answer is at what concentration
if any does Equation (3.102) constrain the probabilities.
Examining Equation (3.102) carefully for different values
of cd reveals the :1 is always the more constraining limit.

Therefore, setting phh,d equal to one we get
1 = Cd(l-pdh,d)/[l-2Cd+(l-Cd)phd,d/(l-pdd,d)J (3.103)
phd’d = (l_pdd,d)Cd[3'l/Cd'pdh,d]/(l"cd) (3.104)
since phd,d:0

Gail/3 (3.105)

For Gail/3 all pdd,d' phd,d and pdh,d from zero to one are
allowed. Figures 3.1 to 3.7 show the allowable values for
pdd,d’ phd,d and pdh,d for cd=.3,.4,.5,.6,.7,.8 and .9.
Only the volume in front and above the surface is allowed
for a ergodic chain. Though our choice of independent
parameters is a convenient one it is arbitrary and there-
fore these figures, for the three parameters which we have
chosen,may be somewhat misleading. One should not infer
that the parameter space of the second order chain is more
restricted for higher concentration. In fact the replace-
ment of cd by l-chch and of all d subscripts by h subscripts
leaves the figures still true.

A two state third order Markov chain and three
state second order Markov chain are considered in Appendix

B.

60

   

   

- Q
- .
o”..
o .

  

.
.

     

9:
...

 

BM +1.0

         

o

 

. .
0'
a:
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v
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dd,d’Pdh,d ”d Phd,d f“

=.3.

FIGURE 3.1.-—Allowed values of P
C
d

 

61

 

’05,;0,
IIIIII’IIIIIIZIII’ ”will,”
III” IIIIIIIIIII
;’ I)/ III,” III/”II”!
’ I... III III/l [/III’III ”III III [III/[III ”llzII/IIIIII
II" II'IIII III/I’ll ‘
fiiggggfiffifizfié I??? 0'9"": IIIII 6993/0”; (III/II

fl" I'IIIII” "(III III
III)!
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I III
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1.9.!9’“ I

 
  

gt-Dtsd:

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FIGURE 3. 2. --Allowed values of Pdd d’ Pdh d and Phd,d for

d

62

 

     
   

 

 

   

‘0
I”
I:
' I
4333;953:532???"
O 0 '
I01.0,$0,’0,:0:I?:z1:’?:1
gzgzozczm ..g/g m} III/III ”II/III! ”III,”
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FIGURE 3.3.--Allowed values of P

= dd,d'Pdh,d and Phd,d for

d .5.

63

    

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FIGURE 3.5-“gliowed values °f Pdd.d'Pdh,c1 and PM!“ for

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66

  
     
    

 
  

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FIGURE 3.7.-rAllowed values of P

Cd .9.

for

67

One of the major problems when generating a finite
. length linear atomic chain via the ergodic Markov chain
theory is to generate a chain with the desired stochastic
relationships between atoms. As the number of atoms in
the chain becomes infinite the ergodic Markov chain theory
guarantees the correct relationships, but when one looks
at a 100 or 1000 unit chain the relationships can be

quite different from those desired. The question we need
to answer is: For a given length chain N and a given
confidence level C what magnitude of error between the
obtained and desired stochastic relationships can we
tolerate before we discard the chain? A 99% confidence
level says that 99% of the chains of length N, generated
by a given transition probability matrix will fall within
certain error limits. 'Since the actual value of pd'd(n)
in the first order two constituent Markov chain is a

binomial statistic; it has a mean number of Occurrences
= N .
u pd'dm) _ (3 106)
with a standard deviation of

02 = N Pa,a(n)[1‘Pa,d(n)3 (3.107)

68

For the binary random chain with cd=.5, pd’d(n)=.5
independent of n. First, for the 100 unit chain, the
mean number of defects is u =100(.5)=50 with a standard
deviation of o=[100(.5)(l-.5)]5=5cm768% of the time the
number of defects will range from 45 to 55 and 95% of
the time the defects will be between 40 and 60 out of
100. In other words to a 95% confidence limit a random
chain of length 100 could have 40 to 60 defect atoms and
still be representative of a random chain. For the 100
unit chain Table 3.1 gives pd’d(n) for two computer

generated chains.

TABLE 3.l.--Characteristics of two 100 unit Markov chains.

 

 

cd=.43 cd=.49

n pd,d(n) pd,d(n)
1 .357 .510
2 .452 .388
3 .463 .490
4 .450 .479
5 .385 .468
6 .385 .511
7 .461 .362
8 .333 .522
9 .513 .578
10 .289 .432

 

69

By a similar analysis, for N=l0,000, the 68%
confidence limits are 4950 to 5050 defect atoms out of
10,000 atoms. The concentration of defect atoms varies
from .495 to .505 versus .45 to .55 for the 100 unit
chain. Therefore, any calculations involving 100 unit
chains should be averaged over many chains.

In Appendix C, we have related the Markov
chain to the short-range order parameters. For the first
order Markov chain, the pair correlation function is
related to the conditional transition probability by

i,j

p£ = P. .(l2 -2

1 2I) (3.108)

1112 1:3

The reason for the absolute value Ii -2 is that the

lzl
Markov chain has direction; one can examine a chain in
only one direction at a time. Since the Markov chain we
generate is isotopic, it does not matter which direction
we consider. This directionality is important if one
considers the relationship of Markov probabilities to
triplet correlation functions.

The short-range order parameter for the first order

Markov chains is

4 _ n .. _ _
an — (a1) where al_(pdd,d Cd)/(1 cd)

The Fourier transform of an can be computed in closed

form

0
_ n
—l+2n;l alcos(nka)

(l-alcos ka)

 

=1+2[ - 1] (1.353.3)39

l-Zalcos ka+af

l—ai

2
l-2alcos ka+al

 

a(k)= (3.109)

a(k) is characterized by a single broad peak centered

at k=0 or k=§ depending on whether a1>0 or al<0 respectively.
The relationship of the second order Markov conditional
transition probabilities to the pair correlation function

is not as simple as that for the first order chain; it is
examined in Appendix C.

Frequency Spectra of Linear Chains
With Short—Range Order

 

Before examining the spectra of chains with short-
range order, we will mention other work using nearest
neighbor short-range order. Payton33 calculated a limited
number of spectra using Dean's technique with short-range
order for Cd=.5. Payton used nearest-neighbor pair

correlation functions to generate his short-range order

which for Cd=.5 are particularly simple, i.e.

(3.110)

71

In general it is conceptually incorrect to generate a
chain using pair correlation functions. However, for
Payton's particular case of nearest neighbor correlations
only,the logical fallacy does not result in an incorrect
chain because the first order Markov transition proba-
bilities are numerically identical to the nearest neighbor
pair correlation functions. More recently, Papatriantafillou,34
has introduced short-range order into the one dimensional
electron problem. His order, although not stated in the
paper, is a first order Markov chain. He does not generate
afrequency spectra, however. Neither, Payton or Papatriantafillou
justified their method of generating short—range order or
studied its implications in terms of correlation functions.
We, on the other hand, have mathematically justified our
method of generating short-range order and have examined
the pair correlation functions for this order.

Matsuda and Teramoto41 used first order Markov
chain theory to calculate a formula for the integrated
density of states of the harmonic mass defect linear chain
to certain special frequencies. One point to note here
is their use of Markov chain theory was in a much differ-
ent context than presented in this thesis and was not
easily reformulated into generation of linear chains. The
special frequencies correspond to zeros in the density

40

of states of random chain and are given by

72

2 _ 2 2 ns
w (s,t) — wmax cos (2t (3.111)
where “max is the maximum allowable frequency of a
perfect chain of light atoms (in our case
wiax=4.) and s and t are integers prime to each
other.

5 and t are determined by the condition

m
52‘: l+cot(n/2t)tan(sn/2t) (3.112)

L
mh
for a mass ratio of ——=2, 5 must be equal to 1. In this
case the Matsuda and Teramoto formula for integrated

density of states is

N<w2<s,t>)=x-[cd<1-pd,d)p§j§/<1-pg,d>1 (3.113)

where N is the integrated density of statesand pd d is the
Markov transition probability. '

The justification for introduction of short-range
order into the this formula is not clear. It seems to
depend on the assumption that the Special frequencies do
not change with introduction of short-range order.
Unfortunately numerical studies cannot adequately examine
the special frequencies since we can use only finite length
chains.

Table 3.3 gives some of the values of t,w2(s,t)

and N(w2(s,t)) for the 50 percent random chain.

73

TABLE 3.3.--Comparison of numerical spectra and Equation

 

 

(3.113) for c=.5,pd’d=.5.
2 Eq. (3.113) Approximate
t w (s,t) N[w2(s,t)] Numerical
2 2. .66667 .667
3 3. I p .85714 .858
4 3.414 .93333 .934
5 3.618 .96774 .968
6 3.732 .98413 .984
7 3.802 .99213 .992

 

Looking at Figure (2.5) the first five zeros are visible.

The agreement is excellent in this case.

First Order Markov
Chain Spectra

 

 

First we will examine, the effect of short-range
order generated by a first order Markov chain. Figures
=.2 and P

(3.8), (3.9),(3.10) and (3.11) are for c =0.0,

d d,d
0.4, 0.6, 0.8 respectively. These can be compared to
'Figure (2.8) for the random case. For Pd,d=0' no defect
atoms can be next to each other, therefore, we get an
increase of the isolated defect peak to D(wi)=l.62 vs 1.45
for the random case. The rest of the structure above
w2=2. is due to defect pairs, triples which are not
nearest neighbors. The host mass spectrum is consider-
ably more depleted at the high energy end of the spectrum
than in the random case. For P

dd=.4, .6, .8, the opposite

is true, with the w2=2.66 peak decreasing in magnitude.

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78

For Pd,d='8 the impurity band spectrum is approximately
evenly Spread over the whole band. For this case the
probability of long defect subchains and long host
subchains is high. Figure (3.12) gives d(k) for these

four chains. Table (3.4) compares the numerical integrated

density of states to Equation (3.113).

TABLE 3.4.--Comparison of Equation (3.113) to numerical
spectra for cd=.2.

 

 

Pd,d t N(t) Numerical
0.0 2 .8' .7966
3 1.0 1.0
0.2 2 .8333 .8288
(random) 3 .9677 .9706
4 .9936 .9936
0.4 2 .8571 ‘ .8538
3 .9487 , .9467
4 .9803 .9792
5 .9927 ‘ .9922
0.6 2 .8750 .8704
3 .9388 .9371
4 .9669 .9654
5 .9813 .9796
6 .9891 .9880
7 .9936 .9929
0.8 2 .8889 .8963
3 .9344 .9401
4 .9566 .9601 ’
5 .9695 .9720
6 .9778 .9785
7 .9834 .9849

 

The agreement between the numerical results

and Equation (3.113) is remarkably good-considering that

79

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80

the concentration for Pd,d=0'8 came out to be .1852
instead of 0.2. Also, Equation (3.113) is exact only
for infinite chains. The results in Table (3.4) tend
to confirm the correctness of Equation (3.113) when
short-range order is included.

Figures (3.13), (3.14), (3.15) and (3.16) are
for cd=.5 with Pd,d='l' .25, .75 and .9 respectively.
Figure (3.13) shows a slight deviation from the binary
ordered chain in Figure (2.9) which would be obtained
at cd=.5, Pdd=0' The peaks between w2=1 and 2 come from
2 heavy mass clusters (w2=1.5) and 3 heavy mass clusters
at w2=1.22. The peaks above w2=3 come from the light
atom clusters. The band edge singularities at w2=l,

and 2 disappear. For P .25, Figure (3.14) shows that

d,d-
much of the ordered diatomic structure has been lost with
the spectrum filling in the gap from w2=l to 2. .This

can be compared to the cd=.5 random case (Figure 2.5)
where the spectrum looks like a depleted heavy mass
spectrum with many modes in the forbidden region.

Figures (3.15) and (3.16) show progressive clustering of
light atoms which also gives clusters of heavy atoms.
Clusters of heavy atoms produce the peak at the upper

edge of the host band. For P 75, the host band

d,d='
structure is reappearing and the impurity modes
although highly structured are equally distributed between

m2=2 and 4. For Pd d=.9, the structure in the impurity
I

81

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band is diminishing as the spectrum approaches

the superposition of a light chain spectrum and a heavy
chain spectrum with the integrated spectrum normalized

to 1.0. Figure (3.17) shows the a(k)'s for Figures (3.13)-
(3.16). For cd=.5, the a(k) corresponding to pd,d=u is

a mirror image of the a(k) corresponding to pd d=l.-u.

The case pd,d=0'l approaches Case 1 and the case

pd,d=0‘9 approaches Case 2 of Section II on x-ray
scattering. Table (3.5) compares Equation (3.113)

to the numerical integrated density of states at the

special frequencies.

Second Order Markov Chains

 

Although numerous spectra were generated by
second order Markov chain theory, we present only a few
of the most interesting spectra.herein, Figure (3.18)
shows the only cd=.2 spectra generated by second order
Markov theory to be presented. Figure (3.18) is for
pdd,d=0' pdh,d=‘4 and phd,d='8° Or, in words, the chain
will not contain any defect clusters greater than two
long (pdd,d=o)' In general, the defects will come in
pairs (phd,d=’8) or separated by one host (pdh,d=°4)
with few isolated defects. The spectrum shows a large
nearest-neighbor defect pair peak, w=3.26 with only a

small single defect peak w=2.66. Figure (3.19) shows

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89

TABLE 3.S.--Comparison of Equation (3.113) to numerical
spectra for c =.5.

 

 

 

d
Pd,d t N(t) Numerical
0.1 2 .5455 .5466
3 .9550 .9539
4 .9955 .9960
0.25 2 .6000 .5970
3 .9048 .9014
4 .9765 . .9767
. 5 .9941 .9938
0.75 2 .7143 .7225
3 .8378 .8433
4 .8971 .9013
5 .9309 _ .9330
6 .9519 .9544
7 .9658 .9670
0.9 2 .7368 .7363
3 .8340 .8345
4 .8822 .8818
5 .9110 .9098
6 .9300 .9298
7 .9434 .9441
a(k) for this chain. It is quite different from that

of the first order chain showing a local maximum between
k=0 and n.

dd,d=Phd,d;Pdh,d= l-Cd -Phh,d 1“
second order Markov space is equivalent to the first

)
The line P dd

 

order Markov process. Figure (3.3) shows that for C =.5,

d
this line is one of the diagonals in the unit cube. Two

other diagonals for Cd=0.5 result in simple forms for a(k).

These diagonals in Figure (3.3) are for Cd=.5:

90'

 

 

 

Pdd,d=Pdh,d7Phd,d=Phh,d=1-Pdd,d (3'114)
and
Pdd,d=Phh,d;Pdh,d=Phd,d=l_Pdd,d (3°115)
Appendix B has a derivation of the short-range order
parameters for each case.
l-ai
a(k) = 2 (3.116)
l—2a cos(2ka)+a
2 2
where a2 = (Pdd,d-Cd)/(l-Cd)=2Pdd,d-l
corresponds to Equation (3.114) and
1-dg _
a(k) = 27 (3.117)
l-2a cos(3ka)+d
3 3
(C ) P -C
_ d dd,d d 2_ _ 2
where a3-1_Cd ( l-Cd ) _(zpdd,d l) (3.117a)

corresponds to Equation (3.115).

The period of Equation (3.116) is one-half the period of the
short-range order function, a(k), for the first order

Markov process; whereas the period of Equation (3.117) is
only l/3 of that for the first order Markov process. In
addition from Equation (3.117a) we note that Pdd,d can have

two values for any a3, therefore the identical short-range-

order function can result from two different P This

dd,d'
ambiguity illustrates an important point which will be even
more dramatically made at the end of this section. Whereas

there is a one to one correspondance between chains which

91

can be generated by a first order Markov process and a
set of short-range order parameters, namely the very
simple set given by Equation (3.108), there is no such
correspondance for the second-order Markov chain. In fact,
quite different second order Markov chains may produce
the same set of pair correlation functions. Therefore
in order to study the statistical properties of a Second-
order Markov chain one needs triple (and perhaps higher)
order correlation functions.

Figures (3.20) and (3.21) show vibrational spectra
for second order Markov chains satisfying Equation (3.114)
.25 and P

with P .75, respectively.

dd,d: dd,d=

With Pdd d=.25, the spectrum is not radically
I

different from the random case with only a reduction in

the single defect mode (because Phd h=.25) and a
I

corresponding increaSe in the nearest neighbor pair

hd,d:
Pdd d=.75 is, however, quite different. The band edge
I

defect mode (because P .75). The spectrum for

at w2=2 is not visible. Since P =0.75 the chain

dh,d=Phd,h

has a structure rather like the binary ordered chain (with

Pdh,d=Phd,h=l°O) but because =0.75 this

Pdd,d=Phh,h
structure includes some long clusters of similar atoms.

The short—range order functions for these two chains differ
by n/Z phase shift as shown in Figure (3.22) for Pdd,d='25
and .75 respectively.

92

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95

Figures (3.23) and (3.24) show spectra for chains
with shortrrange order satisfying Equation (3.115) with
Pdd,d='25 and Pdd,d='75 respectively. The single curve
in Figure (3.25) shows the short-range order function for
both cases. Though the pair correlation functions are all
identical, two spectra are never the less quite different!
The probability of having defect clusters of size n

surrounded by single host masses is given in Table 3.6.

TABLE 3.6.--Probability of cluster of light atoms of size
n for Equation (3.115).

 

 

 

Cluster size Probability of occurrance
h-nd-h Cd='5'Pdd,d=Phh,d=l-Pnd,d
n random Pdd,d='25 Pdd,d='75
1 .125 .0625 < .1875
2 .0625 .140625 > .015625
3 .03125 .035156 > .011719
4 .015625 .008789 E .008789
5 .007812 .002197 < .006592

 

Table 3.7 gives the probability of having n host-defect

clusters for these two chains.

965

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99

TABLE 3.7.--Probability of having n-host defect clusters
for Equation (3.115).

 

 

 

Cluster size Probability
n-(hd) Pdd,d='25 Pdd,d='75 random
.25 .25 .25
.046875 .046875 .0625
.008789 .008789 .015625

 

From these two tables it is clear that the major difference

between the two chains is that isolated defects are more

probable for ghid=0.75 and nearest neighbor defect pairs
I
are considerably more probable for P =0.25.

dd,d

CHAPTER IV

LOCALIZATION OF EIGENSTATES OF

DISORDERED CHAINS

There is considerable physical interest in
whether eigenstates of disordered systems are localized
or nonlocalized. In terms of thermodynamic quantities,
thermal conductivity for phonons and electronic
conductivity for electrons depend on the localization
of the eigenstates of a system. The degree of local-
ization of each eigenstate, also, gives information
on the basic quantum mechanical mechanisms working
in the system.

A precise general definition of a localized
mode versus a nonlocalized mode is not available.
However, an acceptable working definition is available.
An eigenstate is localized if the eigenfunction is
appreciable over some region of space and decays
exponentially away from this region. In infinite
systems, this is most likely as precise a definition
as one needs. However, in finite systems, this definition,
while catagorizing some of the eigenstates as localized,
is insufficient to adequately describe the character of

other modes.
100

101

For one dimensional systems with nearest neighbor
interactions considerable localization work has been
done. Mott and Twose45 conjectured and Borland46 proved
that all one electron eigenstates of the disordered
chain are localized. This proof is valid for eigenstates
of arbitrarily large energy for an infinite chain with a
finite fraction of disorder, and therefore applies to the
exact solutions of a Schroedinger equation. Demonstration
of localization in the lattice dynamics problem does not
require such a dramatic result. The lattice dynamics

47 where the

problem is more akin to the Anderson model,
one-electron wave functions are expanded in functions which
are eigenstates of a single atomic energy level (Tight
binding model) or which correspond to a single band
(Wannier picture). Dean48 presented an analogous proof
to Borland's theorem for the phonons in a disordered chain.
Dean showed that all eigenfunctions of an infinite linear
chain are localized. More recently, Economou and Cohen49
presented a more general proof of the localization
character of eigenstates of the Anderson Hamiltonian
and of the lattice dynamics problem.

In the electron problem, the degree of randomness
corresponds to the difference in energy levels and
hopping terms between the two constituents in a binary

Chain. One might think that the mass ratio would be a

Similar measure of randomness in the phonon problem. It

102

is but only one of two measures in the phonon case, the
other being the frequency of the eigenstate. As w+0

50 and coworkers

the modes must become delocalized. Hori
have argued that Borland's definition of localization
may be too loose for practical use, since according to
it any one dimensional system with disorder (finite
concentration of impurities) would have all eigenstates
localized and no conductivity could occur in contradiction
to physical intuition.

The problem with Borland and Dean's proofs
involves the definition of infinite systems. In con-
trast for infinite systems with boundary conditions,
P. Taylor51 has argued that no eigenstate is strictly
localized. (His argument is based on the fact that changing
the boundary conditions will change all eigenvalues and
eigenvectors and a localized state would not be subject
to "distant" boundary conditions. Hori has suggested
that this definition of localization may be too strict.

Two note worthy attempts have been made in the
one dimensional harmonic phonon problem to find eigen-
frequencies in random systems beyond which all states
are localized. Matsuda and Ishii52 give an expression
for the approximate number of vibrational modes in the
finite mass-disordered system that are not "well localized".
' Using the assumption that the localization of eigen-

functions in "large" finite systems is equivalent to

103

the localization of eigenfunctions in infinite systems,

the number of non-localized modes is

 

n = % /Nkm>/J:(m-<m>)2> (4.1)

where N is the chain length
and <m> =Cdmd+Chmh-is the average mass of the chain.

They admit that this formula is valid only to an order of
magnitude since "well localized" is not a precisely
defined quantity. Visscher,53 using Matsuda and Ishii's
ideas, performed some numerical Studies on thermal energy
transport in chains up to 1000 atoms. He arbitrarily
defined the eigenfrequency above which all states are
"localized" as the eigenfrequency above which the sum of
the remaining eigenfrequencies gives a total contribution
to the thermal conductivity of only 10%.. Visscher obtains

an emperical formula for the demarcation mode of

15 . ‘
nC = 5.5(N) (4.2)

for a two to one mass ratio random system. Both methods
suffer in that a few modes of high frequency in nonrandom
disordered systems could carry a significant amount of
the thermal transport energy and be quite nonlocalized.
These modes would not be of interest under Visscher's
criteria and for finite system could violate Matsuda and

Ishii's assumptions.

104

From the preceding discussion, it is apparent
we need to catagorize the degree of localization as a
function of frequency, mass ratio, and Short-range order.
Thouless54 has given a number of localization criteria
for the electron problem, some of which we can adopt to
the phonon problem. The eigenfunctions of the electron
exist continuously throughout Space whereas the components
of the eigenvector in lattice vibrations are defined only
with respect to lattice points. Since Thouless considers
an infinite system his criteria are binary in nature,
either an electron is localized or it is delocalized. We
propose that some of his criteria may be applied to the
lattice dynamics of finite systems with the localization
parameter taking on a continuum of values bounded by the
localized and delocalized values for the infinite system.

Thouless feels the following criteria should
give equivalent indications of localization. Using
wx(r) as the electron wave function and U£(A) as the

displacement of the 2th

atom for eigenfrequency A, we have
1. Non-zero value of f|¢x(r)l4d3r' which

corresponds to

o. = EIUKWM/ZIUgnnz = iluimll; (4.3)
2

105

for a eigenvector normalized to one;a is an inverse
localization length. For the perfect crystal with

' . . 1 ikARa l
periodic boundary conditions U£(A)= — e and a = E
.where N is the number of atoms in the chains. A Similar
derivation for fixed boundary conditionsis given in
Appendix I. We can calculate the eigenvector of linear
chains as given in Appendix D and easily apply this
criteria.

2. Discrete (but dense) spectral density
ilwx(r)I26(E-A) which corresponds to-iIU£(A)|26(w2-wi)
=J£(w2). For finite chains, the Spectral density is
necessarily discrete, therefore, the proper application
of this criteria is difficult. Although we have
looked at this criteria for finite chains, the interpre-
tation of the results is inconclusive.

3. Non-infinite value of f Ir-RAIZIwA(r)|2d3r
for some value of RA corresponding to Q=i(£-L)2IU£(A)|2.
This criterion only applies to the infinite system
because it does not give a unique value of
localization because of the ambiguity of RA’

4. Vanishing of d.c.conductivity (for a static
lattice) and an A.C. conductivity of the order wz. This

essentially corresponds to the work of Visscher and

our comments on this work apply.

106

5. A change of boundary condition shifts energy

A?

levels by an amount of order e- rather than order

N-l. This method holds some promise in the phonon problem
although we have not attempted to eXploit it. To properly
apply this criteria we would have to use cyclic
boundary conditions so that the boundaries themselves
do not influence the eigenvectors. Then, we could switch
from periodic to antiperiodic boundary conditions and
note the change in eigenvalues. Unfortunately, periodic
boundary conditions make the dynamical matrix non-tridiagonal
requiring more complicated methods than we have presented
for finding the eigenvalues and eigenvectors.

Therefore, when we lookanzspecific eigenstates
of finite chains, we will use the criteria (4.3). We can

55

interpret a, in terms of ideas developed by Bush and '

expanded by Papatriantafillou.34 They report that there
are two quantities of interest when we talk of localiza-
tion, especially in one dimension.

1. The length over which the eigenfunction is
appreciable, Le(A).

2. The eXponential decay rate of the eigenfunction
away from this region LE(A) where LE(A) is given by

R

EEO.)
Il£(l)a e (4.4)

107

The quantity 0 =EIU£(X)(4 is inversely proportional
to 03(1) since the sum over A of lUg(A)l4 will pick out
only the U£(A)'S which are appreciable. Bush has given
a Monte Carlo technique for calculating LE(A) which we
have adopted to the phonon problem. The procedure is much
like that outlined in the first page of Appendix D for
eigenvectors. First we assume we have a semi-infinite
chain, i.e. a starting point but no end. Since Bush found,
as expected, the value of LE(A) is not subject to the
initial boundary conditions, we can find successive dis-
placements of atoms from equilibrium starting with fixed
boundary conditions

m.

___1_2 _
u _(2 Y m )Ui Ui-l (4.5)

i+l

where U0=0 and Ul=l.
For the semi-infinite system as for infinite systems
all values of w2 are eigenvalues except for a very limited
number of values of w2 (special frequencies). Therefore,
we can sample Equation (4.5) for a uniform distribution
of wz. Given initial conditions for U0 and U1 we generate
a chain using the appropriate Markov transition probabili-
ties. As Dean proved the Ui's will always Show an

'exponential increase as we proceed along the chain. We

start sampling the chain after some Ui satiSfieS

108

IUi(w2)I>e4. We then Store the number of sites we must
proceed before another Uj satisfies |Uj(w2)|>e5. We
collect 30 data points for a given computer generated
chain, i.e., Ui's vary from e4 to e35. The number of
sites for an e increase in Ui is LE(A). We then find

the mean and probable error of LE(w2). We repeat this
Monte Carlo experiment for 20 chains for each w2 finding
20 means and probable errors. We finally combine the
results using a weighted mean and probable error. Roberts
and..Makinson56 have Shown that even though the use of
Equation (4.5) often will not generate the correct
eigenvector given an accurate eigenvalue (see Appendix D),

the eXponential increase displayed in the successive Ui's

has the same slope as that of the true eigenvector

away from the region of where the eigenfunction is
appreciable. Therefore, the Monte Carlo procedure should
work and display small errors.

Papatriantafillou has in fact shown from
probabilistic arguments that for infinite systems LE(A)
is Sharply distributed (zero deviation about the mean)
and Le(A) has a finite distribution. We find that the
probable error associated with an LE(A) is usually less
than 5% and the variability between chains is of the order

of 10%.

109

We will first examine the exponential localization
length, LE(A) and , then, the length over which the
eigenvector is appreciable. a =[Le(A)]-l. Since the
computation of the eigenvalues and eigenvectors of a
single 1000 unit chain requires nearly 45 minutes CPU on
the UNIVAC 1103:and a complete sampling of the Monte
Carlo exponent decay run usually requires ~2-3 minutes,
we study LE(A) in much greater detail than we can a
which depends on the explicit eigenvectors. We will
present the data as generated with no attempts made
to smooth the data or to draw a smooth curve through it.
It is clear that there is some actual structure in the
data but we cannot at this time conclusively say whether
some of the smaller variations in the data are due to the
Monte Carlo technique or to actual structure. The circles
around the data points give an indication of the probable
error of LE(X).

Figures (4.1) and (4.2) Show LE(w2) versus w2
for binary random chains with Cd='5 and mass ratios of
1.5, 2, 3, 4, and 8 respectively. In every case, the

maximum allowable frequency for these chains is normalized

such that l—-.=1, or w2 = 4. Therefore the perfect
mL max
heavy chain spectrum would have a maximum allowable

2

' 8 4 1
frequency of mm = 3, 2, 3' 1 and 2 for m = 1.5, 2, 3,

H
4, 8 respectively. A number of things are immediately

apparent: (l) in all cases localization length tends

3

LC 5370
LOC OL‘Efl‘l'i'on

 

(wz), for
Cd='5 random and mass ratios of l. , 2.0, and

3.0.

FIGURE 4.1.-~Exponentia1 localization length,

.1 ”H”?— .
. l.__.b -..---

4 3
(Phoenix: C EPQN

 

(02) for
and 8.0.

Cd=.5 random and mass ratios of 4.

FIGURE 4.2.--Exponentia1 localization length,

112

to infinity as w2+0; (2) the variability in the data.

is small from point to point following a definite
functional trend; (3) the data Show an almost uniform
decrease in localization length with increasing frequency
and (4) increasing mass ratio decreases localization
length at a given wZ. A few other trends are not as
apparent: (l) the increase in mass ratio increases
localization even within the host band. This can be seen
by picking a point equivalent point in each host band,
i.e. the band edge; (2) near the host band edge a marked
drOp in localization length is observed (and increase in
the negative Slope of the function LE(w2) and (3) the
localization length is not monotonically decreasing with
increasing wz. If we repeat the given Monte Carlo run
for any given value of the mass ratio, we see a definite
persistence of peaking in the function LE(w2). In

“1.

Figure (4.1) the ——-= 1.5 curve shows a definite peak

ML
2 mh

at w =2.94. The fi—-=2 curve shows a peak at w2=2.66
l
and possibly other peaks at w2=.98 and a number of places

above w2=3. The fi—=3 curve has a peak a 82=2.38 among

L m
others. In Figure (4.2) the -§=4 curve gives a peak
around w2=2.30 and the ——=8 curve gives a peak around
w2=2.06. A much finer grid and a much more detailed

analysis of the data would almost Surely reveal other

structure. Even with the data we have presented, the

113

localization length increases although only slightly

at the isolated light defect impurity frequency in

each case. Equation (F.6) gives the local mode frequencies

as w2=3, 2%, 2%, 2% and 2%g-for mass ratios of 1.5, 2, 3,

4, and 8, respectively. We would expect similar peaks

at all the local mode frequencies of isolated defect

clusters in host mass chains. Again, we must emphasize

that Figures (4.1) and (4.2) are for 50% random systems.
Figures (4.3), (4.4) and (4.5) are random chains

with defects of mass % mH in concentrations of 0.1, 0.3,

0.5, 0.7 and 0.9, respectively. Figure (4.4) for

Cd='5 is an independent Monte Carlo run compared to

the data in Figure (4.1) for 2§=2, and is included to‘

Show the variability from run to run. One can see that

some structure in LE(w2) is present and some of variability

in the data is due to the Monte Carlo technique. .Figures

(4.3), (4.4) and (4.5) Show some common features: (1)

the localization length in the impurity band increases

with increasing impurity concentration although not in a

regular or uniform fashion; (2) there is a marked decrease

in the change in localization length at the hOSt band

edge (w2=2), varing from a change of an order of magnitude

at Cd=0.l to a change of less than 10% for C =0.9 and (3)

d
localization length in the host band is smallest for

Cd=.5 and increases as Cd+0 or Cd+l.

114

  

 

.:
p
6‘
S
3
c
,0
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p
C
o)
J
¢.
U

2:3

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J

_b

2"":-

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5:

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(.5

Q

.\,.

. 110‘
FIGURE 4.3.--Bxponential localization length, I¢(mz) for
mass ratio of 2 and Cd-.1 and .3 random.

AG 5370
on: -r .. . l 1.

“:1 '2'..

'12:.“ LCGARITN‘J. .
. .. _ 4 .... ..~
..u.u. -.

FIGURE

 

2.

I4.)

4.4.--Exponential localization length, LIL-(012) for
mass ratio of 2 and Cd-.5 random.

.. I.“ . \.

AIUI‘VCL a (51:11 (‘0.

an

I V-‘f SIM! LOGARITHMIC
_ _ ‘ . . -. .M.

5"“.1.

46 5370

Lang Liv“

LocflLl Qfinon

 

be“

FIGURE 4.5.-—Bxponential localization length, L;(u2) to!
mass ratio of 2 and Cd-J and .9 random.

117

As before Figures (4.3), (4.4) and (4.5) Show a
general decrease in localization length with increasing

frequency. In all cases, the localization length is less than

100 atoms for wzil. In all cases the function LE(w2) has
definite structure, i.e., peaks.

Figures (4.6) and (4.7) are for C =.5,MH/ML=2,

d
and first order Markov chain transition probabilities

of Pd,d=0'1 and 0.3, and 0.7 and 0.9 respectively. Short-
range order radically modifies the structure of the
localization length as a function of frequency. These
figures can be compared with the Cd=.5 random case given

in Figures(4.1) and (4.4). We have previously presented

the frequency spectra for P =0.l, 0.25, .5, 0.75 and 0.9

d,d

which provide insights into localization. Figure (4.6)

with Pd d=0.l is close to the ordered binary chain P 0;
I

d,d:
therefore, we expect as shown in Figure (3.13) a degraded

ordered binary frequency spectrum. The localization

2

length is "large" in the 01w :1 band, dropping sharply

at the band edge. Also, the length increases in the
optical band 2:w2:3, with a maximum occurring near the
middle of the band. A sharp peak in localization
length at the local mode frequency (w2=2.66) is observed.
In the region of the ordered binary chain band gaps
(1<w2<2 and w2>3), we see a marked decrease in LE(w2).

Figure (4.6) also Shows a Similar trend for P .3 but

d,d:
the band gap decrease and in band increasing in localiza-

tion length are not as pronounced. Figure (4.7) with

46 5370

mu. .. . |.I

.«rcm;
mun-i a -«..I|'I v.4

S:Ml L’JGARITHM LC

I 1::
I‘ve- a 1 \t.

Loan... IZATIOn LENfiTHJ Lng‘)

 

N1-

FIGURE 4.6.--Exponontial localization length, 1.30.12) for
mass ratio of. 2, Cd-.5 and Pd d-.l and .3.
I

" LC:

:3 5370

‘I'HYDI'! C

TM ,LKV~U

LCC fit| 2 fi‘IIv-l LEI’IG

to

. 1

W

7.

FIGURE 4.7.--Exponentia1 localization length,
mass ratio of 2.

Cd=.5 and Pi.d=

H

 

(wz) for
and .9.

120

Pd,d='9 can be compared with frequency spectrum Figure
(3.16). For Pd,d='9 we have large clusters of like
masses, and the structure of the host band is recovered
over the random case. The localization length in the host
band is greatly enhanced with a marked decrease outside
the host band. Comparing Figure (4.7) for Pd,d=°9 with
Figure (4.3) for Cd=°l we see that the short-range order
makes the localization length in the host band for

Cd=.5 greater than that of Cd=0.l random an. unexpected
result. Near w2=0, this however is not the case. For

=0.9 and Cd=.5 it looks as if LE(w2) approaches 600

Pd,d
atoms as w2 approaches zero and only the lowest wz point
'indicates that actually LE(w2)+w as w2 goes to zero.
Figure (4.7) with Pd d=0.7, Shows behavior similar to

I

that for P 0.9, though the delocalization of the

d,d—
host band is less pronounced as expected.

5 We finish the analysis of the exponential
localization length with a single second order Markov
chain to Show the effect which second nearest neighbor
correlations can have on the localization length. Figure
(4.8) is for Cd=0.5, MH/ML=2, and Pdd,d=Pdh,d='l and
Phd,d=Phh,d=0'9' This is one of the special Cases we
discussed before. Figure (4.8) shows three definite humps
in the localization length. This chain resembles an

ordered binary chain of the form (AA-BB-AA-BB)

(Pdd,d=Pdh,d=0 and Phd,d=Phh,d=1)' The peaks in localization

. ~ 2| .- I511" ('4.

121

  

LOC

FIGURE 4.8.--Exponontia1 localization length, 1.36.3) for
mass ratio of 2, C -.5 and P -P -.l
a dd,d dh,d

and P -.9.

hd,d

122

length occur in the four bands (0+.313, .5+1.22, 1.5+2
and 3.l9+3.28) (see Appendix G, Equation (G.12)), and
the decrease in localization length occurs in the band
gaps. There is a decrease in the maximum LE(w2) in each
successive band as w2 increases.

Next, we can study the length over which the
eigenfunction is appreciable. Comparing the following
results with the exponential localization length we will
be able to make some interesting statements about the
localization problem. ‘In the remaining figures in this
section we will be plotting, a=[Le(w2)]-l versus wz
instead of Le(w2), this makes direct comparison with
LE(w2) difficult although general trends are apparent.

Due to the large cost in generating eigenvectors
of large chains (N>50), we will restrict the discusSion
to Cd=0°5’ mH/mL=2 chains introducing short-range order
via the first order Markov chain theory. Using fixed
boundary conditions, Figures (4.9) and (4.11) show
localization as a function of w2 for the perfect host
chain and perfect diatomic chain with (N=100).. (The
lower curve in each figure). a=.01485 for N=100 for the
perfect.host chain. The zone boundary values in Figure
(4.11) are (a=.01485, w2=0), (a=.0297 w2=l,2), and
(a=.0202, w2=3). These values are also found analytically

in Appendix I.

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(MOMIO

132

Figures (4.9) and (4.10) shows the localization
plots for two different random Cd=0.5, N=100 chains.
Since the chains are generated using a random number
generator, one expects different stochastic relationships
between atoms in the two chains. In Table (2.2) we gave
a detailed analysis of these relationships with Cd=.43
corresponding to Figure (4.9). We can easily see that
a does not follow a definite functional relationship with
w?. The spread between individual points as well as
differences in the two figures might seem to make the
data inconclusive. This however is not the case. One
problem is that a 100 unit chain does not contain a
representative set of eigenmodes. The data are not there-
fore contradictory but complementary. We would need
to run many 100 unit runs to get a decent data set.
Looking at Figure (4.1) with mH/mL=2, we can expect that
this collection of data would be quite good for wzzl.0
where the exponential decay length is less than 30 atoms,
and nearly exact for 02>2 where LE(w2)<5 atoms. The
data below w2=0.5 is almost certainly affected by the
chain length, which means that these data do not have
the generality which we ascribe to the other data.

The eigenvectors in this region are extended over the

whole one hundred unit chain. Near the local mode a shows

a great variability. Never the less one can distinguish

133

lELe(w2)

a peak in a[a(w2=2.666)=.633] or a dip in a—
(Le(w2=2.666)=l.58) at the local mode frequency. A
comparison between this plot and Figure 4.1 for LE(02)
shows that whereas the local mode extends over a smaller
number of atoms than do modes nearby it decays less
rapidly with distance away from the region of appreciable
strength, than do modes nearby. Therefore the two
measures of localization vary in opposite directions as
one approaches the local mode. However, as w2+0 -
Le(w2)+N and LE(w)+w, or the two parameters follow each
other. This comparison shows that although our two
measures of localization actually measure different
things the general trend of both parameters are the same.
In Figures (4.9) and (4.10) the region of appreciable
amplitude of the eigenfunction is less than 10 atoms for
w2>2 for the random system with an exponential decay
length of less than 5 atoms. TheSe modes are clearly
localized even in chains of length 100 atoms.

Figure (4.11) is for Cd='5 and.Pd’d=.l and
N=100. Unfortunately for N=lOO we get very few modes
in the ordered binary chain gaps. Comparing this figure
~ toFigure (4.6) we see that they follow the same general
trend, i.e. as LE(w2) increases Le(w2) generally increases.
For w2<0.9<1can be limited by the chain length since.
LE(w2)>50 atoms. Figure (4.11) shows that a is near

its minimum possible value for w2<.5. The general increase

134

in Le(w2) in the 21w253 band over the random case is
clearly significant.

Figure (4.12) is for Cd=.5 N=lOO and Pd,d='9 or
the case of Clustering of atoms. Looking at LE(w2) in
Figure (4.7) we expect the results below w2=l.9 to be
severly chain-length limited and not truly representative
of the values of longer chains. With clustering of alike
atoms, N=lOO gives especially poor statistical relation-
ships in the sense of seeing representative impurity
clusters. Looking at the chain composition we can easily
identify the modes in the Ziwzi4 region that occur in this
chain, all of which show rapid exponential decrease away
from the region of appreciable displacements (LE(w2)=0(l)).

The chain can be shown as follows:
I- llmL - 9mH - 6mL - 7mH - 21mL - 3mH - 3mL - 6mH -

BmL - 3mH - 1mL - 10mH - 10mL - lmH - 1mL -|
for a Cd=.61. The isolated impurity at site 78 gives
rise to the eigenfrequencies at w2=2.667 with a =.644.
The isolated defect cluster of 3 atoms gives a=.33 and
eigenfrequencies w2=3.52 and 2.414. The points for a=.20
comes from the 6 defect cluster; those at a=.16 arise
from the 8 defect mode; a=.l4 corresponds to the 10 defect
cluster; a=.12 corresponds to the 11 defect cluster, and

a=.066 corresponds to the 21 defect cluster.

135

Figures (4.13) and (4.14) show 0 versus w2 for
a random 1000 unit chain plotted in two ways. Figure
(4.13) gives the reader a feeling of the variability of
a from point to point, although Figure (4.14) is the more
useful. For N=1000, the minimum a is .00145 for the
perfect monatomic chain. Figure (1.2) shows the frequency
spectrum of this chain. Figure (4.1) indicates that some
Iralues of a could be chain—length limited for 0210.4
but other values may be valid. A negligibly small
number of higher frequencies can be chain-length limited
because they happen to have their appreciable displace-
ments near one end of the chain. The data in Figures
(4.9) and (4.10) for wz>l,Cd=J5random fall within the
'boundaries of the N=1000 unit chain spread supporting
the use of the exponential localization as a test for
the accuracy of a(w2) for a given chain length. To
get accurate values of a for the region .02102:.4
would require approximately N=10,000. Since a dot in
Figure (4.14) represents an eigenfrequency, the zeros
in the frequency spectrum at 02:2,3,3.414, and 3.618
are clearly visible. The clustering of points around
defect cluster frequencies is also clearly seen. Also,
by arguments given for the 0 versus 02 in Figure (4.12),
we can explain the tendency to get lines with a constant
in the impurity band. To give the reader a more funda-

mental understanding of the mechanisms involved we would

136

need to look at each eigenvector. Although this is
obviously not possible herein we will look at a few of
the eigenvectors of the next chain to be discussed.
Figures (4.15) and (4.16) give localization for
Cd=.5 and Pd,d=0'l' Figure (4.17) gives the correspond-
ing frequency spectrum for this chain. From Figure (4.16)
the following can be said: (1) here, as in the random

case, Le(w2)=a-l

has the same general structure as
LE(w2) but has a much greater spread about a mean; (2)
the data in the band 2<w2<3 show a marked increase in
Le(w2). In fact some modes are longer than 100 atoms
so that the N=100 chain, Figure (4.11» can never show
completely the localization in this region; and (3) the
values of a may be chain-length limited for w2<.5 because
of the long exponential decay given by Figure (4.6) below
this frequency.

Figure (4.18) gives the n=100, n=250, n=353, and
n=450 eigenvectors (n=1 labels the eigenvector of lowest
energy) for the above chain, (N=1000, C

d d,d=
The figure shows that at w2=.06 (n=100) the eigenvector

=05'P 01).

is completely delocalized over the whole chain. Therefore
for this frequency we know Le(w2)>>1000. The mode n=250,
w2=.36 is clearly also nonlocalized although structure

is appearing in the eigenvector. The n=353 mode is one

of the lowest energy modes that does not have amplitude

137

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at one boundary or the' other, and it displays a clearly
local character although extending well over 600 atoms.
The n=450 mode is near the ordered binary chain band gap,
the localization is quite distinct extending over about
100 atoms. The modes in the gap 1<w2<2, usually extend
over 30 to 40 atoms. In the band 2<w2<3, the extent

of appreciable magnitude of the eigenvector can often
increase to over 100 atom as shown in Figure (4.19) for
n=700 and n=775. The variation in the localization can
be seen in Figures (4.19) and (4.20) for n=775 and

n=776 respectively. Figure (4.20) also shows for n=950
as we get near the band edge w2=3, the localization

increases; Le(w2) extends over ~20 atoms.

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£30133!“leq I 3

CHAPTER V

THEORETICAL DEVELOPMENT

To explore the theory of vibrations of the
linear chain, we will develop, in review, the quantum
mechanics of the chain and the classical and quantum-
mechanical Green's functions necessary for the theory.
Given the classical Hamiltonian of the system (Equation
(1.3), we can rewrite it in terms of the displacement
and momentum operators (Ua(£,t) and Pa(£,t))

*
Pa(£,t)Pa(£,t)

 

*
a a ,
2,2 -
(5.1)
where the quantization condition is
[Pa(£,t); UB(£,t)] = -Ifi5a35£,£ (5.2)
Ua(£,t) and PB(2,t) are Heisenberg operators; Heisenberg's
equation of motion for a general Operator X(t) is
int) = -1-—[X(t)H(t)] ‘ (5 3)
. in ’ '

141

142

The time derivative of the displacement operator is
, Pa<z,t>
Ua(£,t) = W . (5.4)

CI

and the second time derivative is

u 1 Pa(£'t)
Ua(l:t) = ffi [EETET——v H]

1 » - ’
_ _ figTEY AB§£, ¢a8(1,2 )UB(£.t) (5.5)

This is identical in form to the classical equation of
motion (1.4). The Fourier transform of Equation (5.5)
gives Equation (1.5) in operator form.

For the perfect (periodic) chain ma(£) = ma,
and we can expand the displacement operator in terms of

a normal coordinate operator, Qj(k)

Q-(k)

00(2) = 2 —l———- 0a.(k) eik'Rg -(5.6)
k,j VNma 3

where Oaj(k) is an expansion coefficient.

The expansion coefficients obey Equations (1.9)
governing closure and completeness. Substituting
Equation (5.6) into the Fourier transform of Equation

(5.5) will again give Equation (1.8) for the eigenfrequencies..

143

Green's Function
. . ll ‘ 12 .
Lifshitz and Montroll and Potts independently
formulated the lattice dynamics problem in terms of the
classical Green's function, defined to be the solution

to the matrix equation.

ILQ
II
"H

(w g-g) (5.7)

where M is the mass matrix
and 2 Is the potential matrix

In component form we have

2 0 M ’I I _
$2”[w ma(~)5a’Y a£,£,,¢ay(2,2 )ngB(2 ,2 )—5m8522. (5.8)

where ¢ is defined in Equation (1.3).

The formal solution to Equation (5.7) is
M-Q) (5.9)

In Appendices E, F and G, we make use of this solution
to solve for specific elements of g for some special systems.

' g is therefore the inverse of the matrix A in Equation

(1.5).

"3’
I C‘.
ll

0 where §=(w2§-g) (5.10)

144

First for the perfect periodic chain, Q can be

diagonalized by the normal coordinate transformation,

 

1.8.

++

(2 2’) 1 z 1 -k.R£ *(k)' (k k’)
g I =— e 0' g“’ I
a8 N j,j’ yfigfig' 31 J]
k,k’
iK’-§f.
e . Oj’8(k ) (5.11)

From Equation (1.7) and (1.8) it is clear that the matrix
transformation Ti: E og(k)elk.R£ diagonalizes the

periodic matrix (wZM-¢) in coordinate space (£,a)

-1 jk 2 _ 2. -12’8_ 2_ 2
M TJLOLW M (Mg,B T'j’k” (w wj(k»5k,k’6j,j"

the same matrices also diagonalize the inverse of (wZM-¢)
or

, l
g.0’(k’k ) = g. (k) = 5 £6.0’ (5.12)
33 3 LUZ-w? (k) kvk 3:3

 

Substituting back into Equation (5.11) we have

* ik(R ’-R )
Oja(k) Oj8(k)e 2 2

z (5.13)

. 2 2
j,k (w -wj(k))/mamB

 

Ell--|

ga8(£’£’) =

where N is number of unit cells in the crystal.

In Appendix E, we use this equation to solve for the
monatomic chain Green's function. The diagonal element

of Equation (5.13) is

145
* (k) (k)
C. O.
m g (2,2) = l 2 ~3“ 3“
a do N . 2 2
j,k w -wj(k)

 

(5.14)

We can now find a relation between the Green's function

and the density of states

v(w) = % z 6(w-wj(k)) (5.15)
jk

where we normalized the density of states to one

U)

m
Io V(U.)) (100:1 (5.1.6)

Summing Equation (5.14) over a we have

_ 1 1
Z magaa(2,£) — fi 2

a2 . jk wZ-w§(k)

ZIH

which we can convert to an integral (w+w+ie)

 

w
$2 ————2 % =sPIm————"§°°”§“’
jk w -wj(k) ° w -w’
ins
+ 2w f (6(w-wj(k))+5(w+wj(k)))

3k

where

P is the principle part
and s is the number of atoms per unit basis

In terms of the density of states, this becomes

146

mm v(w’)dw’ nis
imagaa(2,£) = SPIO m + W(V(w)+v (“U”)

real + imaginary.

Since v(-m) = O for w>0 we have
v(w) = - 39- Im z m g (2 2) (5 17)
SN a a do ' ‘

where Im( ) is the imaginary part of ( )-

Also
v(w) = - £2—-Im 2 m g (2 2)
an 2 a aa ’
I
v(w) = — 2&- Im tr (M9) (5 18)
an =— '

where tr( ) is the trace of the matrix ( )

and since D(wz) = v(w)/2w
D( 2 _ l
w ) — - NSn Im tr (gg) (5.19)

 

for a chain with N lattice points and s atoms per basis

and w2

+(w2+ie).
Although we have derived the density of states
for the perfect lattice, Equation (5.19) is valid for

the imperfect lattice where Ns=N the total number of

T
atoms. To show this, we look at the solution to the
eigenvalue equation for the disordered system. For the
imperfect crystal, we must start with Equation (1.4). We

define A¢aB(2,2f) as the difference between the force

147

constant matrix of the disordered system and that of the
perfect system (¢’-&3) (=0 for the mass defect problem)
and mg as the mass of site, a, of the perfect chain. We

can define two quantities

‘ma-ma(2)

m (5.20)

ea(£) =
_ a

and

2

CaB(2,2’) = A¢a8(£,2’)+€a(£)maw 6&86£,l’ (5.21)

Then, we can rewrite Equation (1.4) in terms of the

perfect lattice where

_ w2¥3+2,

IIO

-w2gp+§o+

We can expand the displacements in terms of the normal

coordinates of the imperfect lattice

Ua(£) = i Xa(f.£)Q(f) (5.22)

where Q(f) are the normal coordinates and
xa(f,2) are the eXpansion coefficients

and we get

_ 2 O ’ ’ ’ :-_-
22’[ w (f)ma(2)6qfi6£fi, + ¢a8(2,2 ) +ca8(2,2 )JXB(f,2 ) o

(5.23)

148

for the eigenvalue equations. The Green's function for

the system is formally

G = ( OwZ-go-g‘l (5.24)

Clearly, it is also diagonalized by the normal coordinate
transformation of Equation (5.22). If we normalize the
eigenvectors such that

2-
3g ma(2)|Xa(f,2)| — 1 (5.25)

we again can derive Equation (5.19). For a chain of
length, N, we would in practice have to solve an Nx
N matrix for the eigenvalues and eigenvectors.

We can also write the Green's function of the
disordered lattice, G, in terms of the Green's function

for the periodic lattice, 35g. From Equation (5.24)

which gives

IIC)

ll
“'0

+
""0
HO
IIO

(5.26a)

IIO
ll
"'27
+
HO
HO
II'U

(5.26b)

149

These two equations are often called the Dyson equations.
The classical Green's function is a generating
function which allows us to solve complicated interaction
problems by solving much simplier force free equations.
However, the classical Green's function has no quantum
mechanical foundation. Following the work of Elliott and
Taylor,57 we restructure the problem in terms of the
Zubarev58 double-time single-particle Green's functions.
These functions are generalizations of correlation
functions and, therefore, have a definite physical inter-

pretation. We use the retarded and advanced Green's

functions
Gr(t,t’) = - 24,12 e(t-t’)<[A(t),B (t’)]> 1(5.27a)
Ga(t,t’) = %%i e(t’-t)<[A(t),B(t’)]> ‘ (5.27b)

where A(t) and B(t) are two operators in the

Heisenberg representation

th/fiA e-th/fi

A(t) = e (5.28)
B(t—t’) is the unit step function
_ 1 t>0
B(t) — 0 t<0 (5.29)

The commutator is defined as

[A(t) .B(t’)] = A(t)B(t’)-n' B(t’)A(t) (5.30)

150

where n = l for bosons
- -1 for fermions

The phonon problem is a boson problem, therefore, n = 1.
'Also, the average is

tr(p...)

<....> = ' 5.31
E's—(35‘— ( )
where p = e-H/T. (TEkT)

H is the Grand canonical Hamiltonian and is related to the

canonical Hamiltonian by
H = H—uN (5.32)

where u is the chemical potential and
N is the number of particles

Using tr(AB) = tr(BA), and Equation (5.28) we

can Show
G(t,t’) = G(t-t’) (5.33)

Next, we define the correlation functions

FAB(t,t’) <A(t) B(t’)> (5.34a)

~FBA(t,t’) <B(t’)A(t)> (5.34b)

The correlation functions also depend on the time variables

through their differences, and one related by

ifi _
FBA(t + ?—) — FAB(t) , (5.35)

151

The relation between the Green's function and correlation

function can be shown to be

 

. . iwt
FBA(t) = in lim f (G(w+i§&'G(w'l€))e dw (5.36)
€++O E—'
e - n

The correlation functions of interest in the phonon
problem include those of the operators Ua(2,t),
Pa(2,t), a;(k), aj(k). We will look at the displacement-

displacement correlation function
FAB(t) = <Ua(2,0) UB(2 ,t)> (5.37)

For d=B, 2:2’ and t=0, F gives the mean square displace-
ment used to calculate the Debye-Waller factor in
scattering theory and in the Mossbauer effect, and to
calculate the frequency spectrum. The mean squared
momentum correlation function is used for calculating
the doppler energy shift in the Mossbauer effect which
is velocity dependentwhereas the probability of emission
is dependent on the mean squared displacement. The creation-
destruction operator correlation functions are used in
phonon scattering calculations of lifetimes of modes

and transport processes. The displacement operator

Green's function is

r ’ 2 _ _ZTTi _ ’ ; ’
GaB(2,2 ,t,t ) — -:fi—-6(t t )<[Ua(2.t).UB(2 .t )]>

(5.38)

152

First, we want to find the equations of motion of this
quantity. Taking the first time derivative with reSpect

to t, we have

 

 

O
°r .0 ’ _ “'ZTTi _ I »
Ga8(2,2 ,t,t ) — (h 6(t t )<[Ua(2 ).UB(2.t)]>
-2Tri ’ . 2 ’
‘h B(t-t )<[Ua(£1t)rUB(£lt )1)

where fia(2,t) = Pa(2,t)/ma(2). Taking the second time

derivative, we have

-2ni 6(t-t’)(
'h ma(2) \

 

..r ’ ’_ ’
Gae"" ,t,t ) — [Pa("t)'UB("t)]>

.43ng )9(t‘t'><[5a(2.t).UB(2:t’)]>

0.

Using Equations (5.2) and (5.5), we have

..r

I ’ _ -2“ _ ’
GQB(2,2 ,t,t ) — EETET 6(t t )6m861,1’

+

 

2ni 6(t-t’) ” ” ’ ’
h. ma(g) Z£”¢GY (2,2 )<[Uy(2,t)Ud2,t )]>

6:8(2,2’,t,t’)

5372) 6(t-t’)

1 II r II I I
WyiӢay (119' )GYBUZ' IQ' Itlt)

Taking‘UnaFourier transform we have

2 II r II I .-
ig”(w md(£)6wyéfi,£” ¢ay(2,2 ))GYB(2 ,2,w)—6q86g£, (5.39)

' .6 Cd(1-C

153

.This is identical to Equation (5.8) showing that the
Zubarev double-time Green's function and the classical
Green's function satisfy the same equations with the
corresponding quantum operators replacing the classical
variables.

With the theoretical background we can now look
at various theoretical approaches to the density

of states.

Defect Clusters

 

Our first model for the density of states for
the phonon spectra of disordered chains will concern
itself only with the light impurity-band modes. Dean18
associated the structure in the impurity band
(2<w2<4 for TE): 2 and L— = l) with isolated clusters

, mL “5

for random chains. We will attempt to reconstruct the
impurity band density of states by configuration averag-
ing all the impurity modes in an n site defect cluster
embedded in a host chain. For example, for a 6 site
cluster one possible configuration would be one defect

and 5 hosts (h-h-h-d-h-h), etc.). For the random system,

the probability of occurrance of their configuration is

5. This configuration gives rise to only one

d)
impurity mode as we shortly show. For this mode we

5

have weight 6 Cd(1-Cd) . Using the monatomic chain Green's

function derived in Appendix E, we use the Green's

154

function techniques to enumerate all the impurity modes
arising in defect cluster to size 3 in Appendix F. We,
also, have generated the impurity mode frequencies for
defect clusters up to size 6, by embedding the clusters
in a 1000 unit host chain, then finding the eigenvalues
and eigenvectors for w2>2 by the methods previously
described. Table 5.1 lists the eigenfrequencies in the
impurity band of all possible clusters of size less than
or equal to 6, where the defect mass is half the host
atom mass.

Once we have the relative weights of each eigen-
frequency, we can form a bar graph like those in Sections
II and III. To get the correct density of states in
the impurity band we can normalize this bar graph to
correspond to the Matsuda-Teramoto formula, Equation
(3.113). First, for sake of comparison with the
.numerical plots we normalize the bar graph instead so that
the peak at w2=2.66 is equal to that of the corresponding
numerical calculation. This normalization gives a cumula-
tive density of state well under that given by Equation
(3.113). .Figures (5.1), (5.2) and 5.3) are the density
of states plots in the region 2<w2<4 for 4, 5, and 6 site
embedded cluster approximations for Cd='5 random systems.
The corresponding numerical density of states is super-
imposed. The four site cluster displayes 6 of the peaks

of the numerical density of states structure. The 5 site

155

TABLE 5.l.--Impurity mode frequencies of defect clusters
embedded in a host chain, mh/md=2.

 

 

 

Cluster eigenmodes in
size type region 2iw214
l d 2.6667
2' 2 dd 3.23607
3 ddd 2.4142,3.5214
dhd 2.4142,2.8393
4 dddd 3.675,2.839
dhdd 2.623,3.258
dhhd 2.592,2.727
5 ddddd 2.306,3.l32,3.7659
ddhdd 3.132,3.332
dhddd 2.293,2.707,3.528
dhhdd 2.6618,3.237
dhdhd 2.274,2.643,2.906
dhhhd 2.643,2.6876
6 , dddddd 2.636,3.332,3.8236
ddhddd 2.3929,3.224,3.536
ddddhd 2.573,2.8995,3.678
ddhhdd 3.212,3.259l
dddhhd 2.3928,2.6724,3.5217
dhddhd - 2.552,2.6876,3.278
dhdhdd 2.3921,2.815,3.2599S
ddhhhd 2.666l,3.236l4
dhdhhd 2.3920,2.6622,2.847

dhhhhd 2.659,2.6738

 

156

F. I'Hsin qumr
p /v *Numuficnu

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

86)
N

FIGURE 5.1.—-Density of states for C -.5 random generated
by a 4 site embedded cl ater.

 

157

H— Sim-Te, Clusnv‘
N - Humor-(cm.

r P“

 

 

 

 

 

 

 

 

 

 

/ I
C—
l
_j
I j

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

M)

.J;_

 

 

 

 

 

.OllY
w

W3

FIGURE 5.2.--Denaity of states for C -.5 random generated
by a 5 site embedded elector.

0(4)”)

158

37- . r'—-Q.SH1. chu5rmx*

A, ~ A merlckp

'F

in
1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

FIGURE 5.3.--Density of states for C -.5 random generated
by e 6 site embedded clfister.

159

cluster displays 10 of these peaks. The 6 site cluster
gives no major new information as to the defect cluster
modes responsible for the peaks. The 6 site cluster
does, however, weight the peaks much more realistically
than does either the 4 or 5 site cluster.- It also

gives a clue to the reason for the non-zero density of
states between the 1, 2 and 3 site cluster modes. This
peak broadening is due to an effective cluster-cluster
energy splitting (dd-h-h-dd). A ten site cluster would
give even a better frequency spectrum. As one might
expect, as we go to lower concentrations of defects

this approach works even better since the defect clusters
will be more isolated from each other by host masses and
the probability of many defect strings is greatly
diminished. Figure (5.4) gives the 6 site embedded
cluster spectrum for Cd='2 random along with the
corresponding numerical plot. The agreement is remark-
ably good. In each case presented above, we will look

at the integrated density of states of the 6 site cluster
"properly" normalized compared to the numerical results.

Figure (5.5) is for Cd='5 and Figure (5.6) is for C =.2.

d

The numerical results for Cd=.2 random are for a 10,000
unit chain and give an integrated density of states in
‘ the impurity band of .1712 versus .1667 theoretically. If

we adjust for this error, we can see that except for a

160

1.6)

1 22' ' f‘ (a s'rre. dMsTer

‘/v - NumemifiL.

1.0“

D/wz)
5?

I .4
1:3

 

 

 

 

 

 

 

 

 

 

1 U1 I“ Li fl A. l .
204 L02 3 4

 

 

FIGURE 5.4.--Density of states for C -.2 random generated
by a 6 site embedded cl ster. '

161

_JJ— (os'n-z. (lunar
30- ' fl‘NqumcfiL

 

 

 

2

FIGURE 5.5.--Integrated density of states for C =.5 random.
generated by a 6 site embedded cluster.

 

162

 

[Jr—(0' Ewe, CLUSTET‘
/— Numerical. '

 

 

1

2 3

FIGURE 5.6.--Integrated density of states

for C =.2 random

generated by a 6 site embedded clugter.

163

region 2.66<w2<2.80 the fit is almost perfect. The Cd='5
plot shows serious discrepancies throughout the region.
This is one indication of larger clusters than we con-
sidered being important. Any short—range order only
makes this problem more serious. The correlation between
the correctness of this model and the localization
length is apparent. If the localization lengths Le(w2)
and LE(w2) are short (~l—5 atoms) in the impurity region,
the model works well.

Not only does this model fail at high defect
concentrations, it also gives no information on the
density of states in the host band (Dim212) and on how

it is depleted to form the impurity band. Therefore,

we look at another simple model.

n Site Periodically Extended Model

 

Butler and Kohn59 first suggested reconstructing
the electronic denSity of states by taking clusters
as we did in the last section and periodically continuing
a given cluster throughout the system. Then they
configuration averaged the density of states over all
possible clusters. We, in a similar manner, take the
possible spectra resulting from a periodic chain with
an n site basis, and obtain an "average" spectrum by
properly weighting each individual n site periodic

spectrum. The weighted average is based on the probability

164

of obtaining the given sequence of atoms that make up
the basis. Essentially, the average is performed as if
the basis were in fact a cluster in the medium. This
procedure has sOme definite advantages over the embedded
cluster approach. These are: (l) we obtain the total
frequency spectrum not just the impurity band; (2) the
errors are not as concentration and short-range order
dependent as in the embedded cluster; (3) the frequency
spectrum is normalized to one as constructed without

the introduction of some ad—hoc renormalization criteria.
It however also has some bad features. These are: (l)
the averaged n site periodic system does not display the
theoretical zeros in the density of states often tending
to infinity instead of zero at the theoretical zeros;

mH mL

(2) the host band (0<w2<2 for 5; =2; 7+ = 1) has many
infinite singularities in it. This is a very undesirable
feature since theoretically most of the singularities
should not occur and numerically most of them do not

show up.

In Appendix G, we derived the theoretical density
of states of these n site periodic systems for the '
binary chain with n=l,2,3,4 and 6. In Table 5.2 »we
list a complete set of bases which give all the unique
frequency spectra for 4 and 6 site periodic systems.

We note some of the 4 and 6 site bases are not primitive

bases but are composed of multiple 1,2, and 3 site periodic

bases.

165

TABLE 5.2.--The bases for all possible 4 and 6 site
periodic chains.

 

 

Basis Size Complete Basis Set

4 (l) hhhh
(l) dddd
(2) hdhd
(4) hhdd
(4) dddh
(4) hhhd

6 ' (1) _ hhhhhh
(l) , dddddd
(2) hdhdhd
(3) hhdhhd
(3) ddhddh
(6) hhhhdd
(6) ddddhh
(6) V hdhhhd
(6) dhdddh
(6) dddhhh
(6) hdhhdd‘
(6) dddddh

(6) hhhhhd

 

166

The four site periodic system has only 6 unique
bases out of 24=l6 and the six site periodic system has
only 13 unique basis out of 26=64 possibilities.

By using the analytic expressions for density of
states given in Appendix G we can generate the average
density of states to any desired accuracy. We have,
however, found difficulties in getting a reasonable
integrated density of states with the numerous singulari-
ties in each spectrum (about 12 singularities per 6 site
configuration). We have in fact 104 unique singularities
in an average 6 site density of states spectrum. There-
fore, we average the numerical spectra generated by
10,000 atom chains with an n site basis, instead of the
analytical expressions to avoid the infinite singularity
difficulties in the analytic expressions. Figure (5.7)
shows the Cd='5 random spectrum produced by an averaged
4 site cluster. Comparing with the superimposed numerical
results we conclude that the 4 site periodic system gives
a quite poor reproduction of the frequency Spectrum.
Figure (5.8) for Cd=.5 random produced by an averaged 6
site periodic system looks much more like the experiment.
The structure in the region 0<w2<2, however, does not
appear in the numerical results. The density of states
is in good agreement with numerical results in the region

2<w2<4. The density of states in the impurity band is

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vousuosom Sounds n... o How nous». no unassuming...» Hana:

 

167

 

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168

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169

in fact overall in much better agreement with experiment
than is the embedded cluster density of states. We see
very good agreement in the structure of the density of
states except at w2=3.14 and 3.34 where the 6 site periodic
system does not reproduce the expected peaks which the
embedded cluster approach did. Figure (5.9) is the six
site periodic averaged spectrum for Cd=.2 random. The
host band structure is much more vivid than in the Cd='5
random case. This is just opposite to the experimental
behavior where the in-band structure for Cd='2 is less
than that in the Cd=.5 random spectrum. The calculational
reason for the increased structure in the periodic
cluster calculation is clear. Of the possible 6 site
periodic clusters, the clusters containing 0 or 1 defects
are given much more weight than the bases containing)
5 or 6 defects. For Cd=.5, all clusters have equal
weight. In the impurity band, the embedded clusters
spectrum is of equal accuracy if not superior to the
averaged periodic structure.

Figure (5.10) for Cd=.5, Pd,d=0‘l was generated
by the 6 site periodic system. Comparing this figure
with the superimposed numerical results we see that the
6 site periodic system gives the general overall structure
quite reasonably. However, the actual detail is quite

2 2

poor. The in-band structure (0<w <1, 2<w <3) is, for

'the most part, not present in the numerical results and

170

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172

the peaks in the rest of the density of states do not
carry the prOper weights.-

Figures (5.11), (5.12) and (5.13) show the
integrated density of states for Cd=.2 random, Cd=.5
random and Cd=.5, Pd,d='l' respectively. The 6 site
periodic system even with the spurious structure in the
host band region does a remarkable job of recreating
the integrated density of states. The errors are less
for the random systems but the relative error is in no_
case over 5%. We would expect a 10 site periodic cluster
to do even better.

Using the basic concepts develOped in these two
simple theories for the density of states, we can now

look at a more complicated theory which will eliminate

the deficiencies of these theories.

Self—Consistent Cluster Theory

With clusters imbedded in a.host chain we found
that the basic impurity modes could be described, but we
had no handle on the hdst band structure. From the
periodically extended cluster model, we were able to
reasonably reconstruct the whole density of states.
The periodicity introduced in this model had the effect
ofproducing regions of finite spectral density versus
isolated peaks in the embedded cluster approach. The
difficulty with the periodically extended cluster method

is that in each configuration a given cluster only

 

_-~,
I

 

,:_

Ow

 

173
_,
1.“
xx “(0S\T€. 'Peyfodlg
-/\—— ”NMWWlC'PSL-
x
i. I. I
1 L231 Z 3

FIGURE 5.ll.——Integrated density of states for Cd=.3 random
generated by a 6 site periodically extended

cluster.

174 2"

 

07L 7 . ' . ’
' XX “-43 SITL'. pence-he,
o" fl - ngemcsu ' ~

0.1 [am

«'5‘

i.
T

 

Cf ; 1
‘ Lot 2 3
5.12.-—Integrated density of states for Cd=.5 random

FIGURE
generated by a 6 site periodically extended

cluster.

 

 

175 'W
xx ‘
’1... - ."' - . x
" XX'b SITE, )D’JXH'OLMC.
A, (\(MMQJNCAL.
r
OC-
X
X
x
.6» x
x
2‘
3 ~gxltll
<3 2 '
5
.“ ‘6‘
“X
X
.‘f.- x
X
.5
:Z-
'U
Of
("1_ 1 l L I
vi) 1 LL)1 2:. :3 ‘1
FIGURE 5.13.—-Integrated density of states for Cd='5' Pd d=.l
I

generated by a 6 site periodically extended

n'l- . 5---
l‘uka-‘t L a

176

interacts with identical clusters, clearly an unphysical
~assumption. An ideal cluster approach would allow a
given cluster to interact with all other possible clusters
throughout the entire chain. In practice however, such
a calculation is impossible and therefore we now consider
a theory in which the cluster interacts with an effective
medium of identical average clusters. By including, in
an average way, the interactions among different types
of clusters we remove the unphysical singularities of the
periodically extended cluster method, while retaining the
short-range correlations among atomic vibrations, which
is the virtue of the above cluster methods. Clearly there
must be some optimum way to chose the effective medium;
in the work which follows we shall show how the effective
medium can be chosen in a self-consistent way. First we
shall review some of the history of self-consistent
effective medium approaches to scattering theory. Then
we shall introduce the so called single-site Coherent
Potential Approximation (CPA) and finally we shall consider
the cluster CPA and apply it to our problem.

Lax60 examined the many body problem in terms
of multiple scattering theory. Lax used the method of
self consistent fields. This method assumes that a wave
is emitted by each scatterer of an amount and directionality

determined by the field incident on the scatterer. The

177

incident field is the effective field which includes the
effects of all other scatterers. In other words, the
effective field is composed of the waves emitted by all
other scatterers. Although a self consistent solution is
often difficult to obtain, the solution will contain all
orders of scattering. Ten years later in 1961, Langer,61
recognizing that a phonon in a disordered system can be
treated asra wave scattered at lattice sites, attempted
to calculate the density of states of a one-dimensional
harmonic random chain With no force constant changes.
In the scattering wave formalism, the Green's function
has the interpretation of the phonon propagator. Langer
reasoned that in a random system any site is as likely
to be occupied by a host or defect as any other site

and that for the density of states or other quantities
directly calculable from the prOpagator a configuration
averaged phonon propagator is needed instead of the
propagator for one possible configuration of atoms. The
averaged crystal will possess the same symmetry as the
monatomic chain where we have placed a self consistent
mass on each lattice site. Langer reiterated the Dyson
equation (5.40a) in a k space representation where the
.perfect-chain Green's function is site diagonal. He,
then, took the configuration average of each term of

the series. By diagramatic arguments he extracted from

each term all single particle scattering. He was able to

178

solve the self consistent problem where only single
particle or two particle multiple scattering occurred.
In each case, the singularity at the top of the host
band was reduced from a square root singularity to a
fourth root singularity. The density of states was
[not an analytically continuous function as one would
eXpect.

Davies and Langer61 were able to write a self
consistent equation to all orders in scattering, simply
by replacing certain unperturbed propagators in their
expansion for the self energy with the final propagator
for which they were solving. The physical reasoning
for this replacement however was unclear. Davies and
Langer were not.able to solve their equation for all
concentrations of defects. They, however, found a solution
for small Cd. The solution had the bad feature that r
as concentrationcd increased, one can obtain a finite
density of states in regions forbidden even to the light
mass chain, a completely unphysical result.

‘Taylor63 and Soven64

, on a suggestion from Lax,
reformulated the self-consistent scattering problem
for the phonon and electron problems, respectively. We
will present the basic theoretical structure of the
theory in text with the specific details of Taylor's
single site self-consistent method left to Appendix H.

First, we define a number of different Green's

functions.’

179

(l) g = (gw2-2)-l, the Green's function of the

perfect monatomic chain.
(2) g = (ng—g-g)-l, the Green's function for a
specific configuration of a disordered chain.
(3) <§?E§ = (ng-g—g)-l, the configuration
averaged Green's function for the disordered
'lattice. (We use both a bar and <> to
indicate configuration averages.) G has the
symmetry of the perfect lattice. X is often
(called the self energy although for the

phonon problem it is really a self consistent

mass.

(4) g = (ng-g-g)-l a reference Green's function

displaying the periodicity of the perfect
chain.

We define the scattering matrix, 2 by

g =1; + 225 <5-4o>
Using the definitions of G and R we have
5‘1 + g = 5w2-2=9'1+g
(5.41)
M g=§+yssg

Substituting Equation (5.40) in (5.41), we get .

-1

fl?
no

u
"a

($213

III-3

) (5.42)

180

For the mass defect problem g is site diagonal and so is
g by definition. We expand T and $1; in terms of a site

representation

(5.43)

III-3
ll
20M
“#3
go

(5.44)

IIO
"<3
Ill
won
ll<‘.

It is important to understand Equations (5.43) and
(5.44). Where as g and and (g-g) are diagonal matrices
with all diagonal elements non-zero, the matrices 22

and 22 are zero except at diagonal site ‘2. For example,

Ilr-B
II
o

(5.45)

 

 

 

 

whereas

£+1 (5.46)

 

181

Substituting Equations (5.43) and (5.44) into (5.42)

we get

Equating terms in R, we get

(5.47)

"a
u
H<
30
W3
+
um
20M

Removing the term £’=£ from the summation and solving

for T1 , we get

§)-l¥g(;+§ Z 21’) (5.48)

2 #2

"*3

= (4:2

1 2

We now define the single site scattering matrix

(5.49)

llrf
II

(;‘¥

Even though g has all elements possibly non zero, g2
has only one matrix element, that at (2,2). Substituting

(5.49) into (5.48) we get

(1+5 2 T 4) (5.50)

Finally, taking the configuration average we get

<T>=<

Ilfl'

g(;+§ Z T ’)> (5.51)

£¢2’=
The theory is exact to this point. The configuration
average of the RHS of Equation (5.51) requires the

solution of the full NxN matrix for a chain of length, N.

182

The single site approximation decouples the
configuration average into

><1+5 2 <2

<
2 £,#£

"a

> = (2 ,>) (5.52)

R R

where we have in effect made an error of

{figR §£(g£,—<T2.>)>. (5.53)

The first Equation (5.52) describes the average
effective wave seen by the 1th ion while the second
Equation (5.53) describes fluctuations in the effective
wave.65 Neglecting the terms in (5.53) means that we
neglect all correlations between scattering on different
sites and consequently cannot see the effects of short
range order between sites.

If we add the £’=£ term back into the RHS of

Equation (5.52) we get

<2£> = ($1) (;+§<g>)-<E£>§<gl>
<22) = (;+<EQ>§)-1<Eg>(;+§<T>) ' (5.54)_
Since from Equation (5.43),
(g) = X (22'), (5.55)
R

A
"*6
V
II
p54
Ill--l
+
A
llrf

£>§) .<E£>(;+§<g>) (5.56)

183

Using the definitions of the reference Green's function

and the configuration averaged Green's function, we have

no:
u
um
+
um
fig
ud
IE:

or

<g> = g = R + <I>

"w

Substituting Equation (5.58) into (5.57) yields

(é-g) = <2><4+5<2>>’1

Using Equation (5.56), this reduces to

z - g = Z (1+<:->B)’1 <: >
— - —£ — —2
k
In site representation, we have
2 = o + (1+<t >R)-l<; >
:2, :2] :2, = _2

(5.57)

(5.58)

(5.59)

(5.60)

(5.61)

The single site approximation has resulted in a scattering

expression in terms of the single site t matrices t

It

includes in an approximate way all scatterings apart from

that at site 2. Self-consistency is achieved by asserting

that on the average the remaining scattering, by t2, be

zero. Therefore t£=0.

184

For this case, we see by Equation (5.56) that
<3>=0 and g = g and g = g. This approximation is also
called the coherent potential approximation (CPA). Setting
the configuration average Green's function equal to the
reference Green's function requires a self consistent
solution to the configuration average of Equation (5.49),

l-g)-1. We show this calculation in Appendix

where §=(g_
H. For this single site CPA we can also obtain CPA self-
consistency by reiteration. The procedure is as follows.
(1) initially take g = 0
(2) calculate g = (g

(3) calculate <; > using Equation (5.49)

l
(4) calculate ; from Equation (5.61)
(5) set g = g and start over at step (2).
We reiterate until <§z>becomes as small
as desired.
The only real complication in the procedure is
finding §,which is non-trivial because g-1 is finite
over the entire matrix. We use the method of matrix
inversion described in Appendix E. The explicit calcula-
tion is done in Appendix H.
In the corresponding electron problem the configur-
ation averaged density of states is-simply given by the
fimaginary part of the trace of the configuration averaged

Green's function. In the phonon case, the configuration

averaged density of states is given by

185

_ 2 _ 1 .__
D(w ) - m Im Tr (gg) (5062)
where we used Equation (5.19 ).

Therefore we will have to get an expression for the

configuration average of gg. First we have, in site

representation,

5
z (5.63)

IIO

6
2

IIZI
IIO
IIZ

(mg) = = 2 ){(S
— 5,2

where x6 's the oncentration of constituent 6,
of mass M and G is the conditionally configura-

tion averaged Green's function when we require
that there be an atom of type 6 on site .

The Green's function for the disordered system satisfies

the Dyson equation

HO
NC)

:24-

""0

g

which we configuration average to yield

is = Egg—g? = 2+1; 2 x5925: (5.64)
5,2
5 6 2

where £2 = (M-g )w

Therefore

, we can solve for fig in Equation (5.63)

getting

2 X5M595 = MG - (P_1G+I)w 2
-2-) == = — =
62
.since 5 = <E’l-§>’lv g'lé-g = 55
and
—- 2

5G = (fi'é/w )§ (5.65)

where g is the mass matrix of the perfect chain and
;=g. For the single site self consistent theory, the
site representation of most of these quantities are
scalars.‘ The reason we have kept the full matrix notation
will become apparent in the next few pages. From Equation ‘
(5.65) we see in the phonon case, the parameter 2/w2=o/w2

plays the part of a frequency dependent complex effective

mass i.e.

2

~where = (g-g/w )

“5|
ll
"3 2
IIQI
Hz 2

By using the single site approximation we are
neglecting all correlations between scatterings on
different sites no matter how close together those differ-
ent sites may be. All scatterings are either multiple
scattering from a single site or else they are scatterings
from an effective medium. Such an approximation might be
alright if all modes were so localized that there was no
overlap between them, but as we have seen from the plots
of localization of the eigenstates, there is appreciable

overlap of the modes even at rather high frequencies. The

187

advantage of a cluster calculation is that local correla-
tions between scatterings from different but close sites
can be exactly included.

An obvious generalization of the single site
approximation is to consider each site in the above
formalism as a cluster. ‘The site representation becomes
a cluster representation. Again we neglect correlations
in multiple scattering between clusters. We can however~
introduce short-range order in the cluster and treat all
scattering in the cluster correctly. In most general
form, the self consistent parameter Q becomes block

diagonal with each block the size of‘a cluster.

 

 

 

 

 

II

 

 

 

 

II

 

 

 

OIV , (5.66)

 

 

ID
H

 

 

 

 

Cl 02 03
c2 = O4 ('5 CI6 ‘ (5.67)
or7 08 O9

188

for a three site cluster.

=0 _0

8 2‘ 6

there are only 5 independent parameters instead

Butler67 has noted that by symmetry 01:09, 04:0
and 03=07;
of 9. The reference Green's function R has the same
periodicity as the self energy a. A characteristic block
2 of R can be found by using the methods of Appendix E

for the inversion of an infinite tridiagonal matrix.

Introducing the symbol

8 = mwz-ZY (5.68)
we have
_ -l -l B-A-o1 y-oz -03 -1
R2 :(P ‘0) 2 = )‘02 5‘05 Y’Oz (5.69)
-U

3 Y-OZ B‘43-‘51

where A and B are the boundary diagonal elements of

matrices as follows

 

-1
2 Y"0'2 ’03 .
A/Y = 8-05 y-oz (5.70)
-l
2 8-01 y-oz -o .
B/Y = Y-02 8-05 -02 . (5'71)
-03 Y_02 B-Ul-B 1'1

from which we see A=B.

189
Now generally Equation (5.69) may be rewritten
5'1 = g‘1 -X QR. (5.72)

As before, g1 takes on the interpretation of a matrix

as big as the entire system but with all elements, apart
from those in the cluster i, equal to zero whenever g1
appears as a term in an equation for matrices with the
dimension of the whole system. The Green's function
describing a system with a cluster of configuration 6
embedded in an average medium with the periodicity of

the cluster has the following form within the space of

that cluster, 2.

(Gi)-l = (RQ)-l-(Ci-O£) (5.73)

Performing the same algebra as above for inversion we

V find
8 -A Y 0
(G6)-1 = l B (5 74)
2’ ‘Y 2 Y O O
O y B3-A
_ 2 _ 2
where Bi: miw -2y:m(1—ei)w -27 .(5.75)

The coherent potential self-consistency now takes the form

II C)!
II
ll 31

(5.76)

or, because the set of configurations over which we average

gives G the translational periodicity of R,

190

cS:
8

(5.77)

um

6 6
26" SR 8'

HQ

We can show that this equation for the self consistency

can be obtained by setting the average cluster t matrix

to zero.
PrOOf‘. 0 = <22) = Z6 X6£i"§i‘22)§2]-1(§2’gz' (5'78)
0 = is x5{l-E5;l-<gi>"135)}’1E8;1-<85>'11
(5.79)
or, multiplying by 52 on right and left
Xaxéfgi-ggl (5.80)
or, since g x‘3 = l, (5.81)
5Q = $2 Q.E.D.

which is the same as Equation (5.77).
A possible self-consistent procedure now becomes clear.
'One must numerically set the configuration average of
the inverse of the matrix in Equation (5.74) equal to
the matrix for RR in Equation (5.69).
We have attempted to use the reiterative solution

of the CPA given above for the three site cluster. How-

ever, we have never achieved convergence beyond w2=1.52.
AS long as the off-diagonal elements of g remain small,

Convergence is rapid; however as the values of c and 02

3

and 0 , the number of steps to

become as large as 01

5

191

convergence increases. In fact as we approach w2=l.50
rapid changes begin to occur in the off diagonal elements
of 0.

Because this method of solution failed we use the
ideas recently developed by Butler68 for a cluster treat-
ment of the electron problem. In fact, Equations (5.69)
through (5.74) are in.But1er's form and make the following
argument perspicuous.

Because matrix A is a function only of the
external medium it is independent of configuration.
Therefore once we have determined A we have essentially
determined G or R. The statement 6L:RL is, of course,
the statement that corresponding matrix elements are
equal. Therefore if we can find an equation including A
for one of the elements of this matrix equality and if
we can solve that equation for A then we have found the.
desired configuration averaged Green's function. We have
such an equation in the equality of the boundary elements,
say the (1,1) elements.

From Equation (5.69) we find

R£(ll,w2) = [8-6 ~gyg

T ‘1
l 1

(5.82)

where, for the 3-site cluster
9 = (y-cz, -c3); UT is the transpose of U (5.83)

and

B-G y-o
_ 5 2
2 ‘ Y-o 8-0 —A (5.84)
Similarly from Equation (5.70)
2 . -
A = Y [5-01-92gT] l (5.85)

Inserting Equation (5.85) into Equation (5.82) we find

an equation for R(1,l) involving A only.

Rg(l,l,w2) = AEYz-A2]-l (5.86)

Next we can write the (1,1) element of GE as

a continued fraction, and finally set

. 2 2
. ’2 . . u" = u) .

818283
where the probability P of various configurations may

include short-range order within the cluster, and Bi

may take on two values.

2
{mdw -2Y=m(l-€)w2-2Y
(5.88)

 

8i : mth-Zy=mw2-2y=8
we obtain
2A 2 = Z 8(818283){81-A-y2[82-y2(83—A)’l]’1)'1 (5.89)
Y —A B.

l

for our final self-consistent equation. The result may
be easily extended to clusters larger than the 3x3 which

(we have used here for illustrating; on the right hand side

193

of Equation (5.89) one simply extends the continued
fraction to as many sites as desired.

Before discussing the numerical solution of
Equation (5.89) we will make one more point. The cluster
CPA which we have solved above is akin to the periodically
extended cluster method discussed previously. The
difference, of course, is that in the cluster CPA a given
cluster can interact, though in an average way, with all
other possible clusters, whereas the periodically extended
model includes only configurations in which a given cluster
can interact with other identical clusters. This point
raises the question of whether there is a CPA analogy to
the embedded cluster method. Indeed such an analogy
motivated Butler's recent work on the cluster problem.

We show below the CPA analogy to the embedded cluster
method and then, following Butler, we show that if one
Inakes the so-called Self-Consistent Boundary Site (SCBS)
approximation one obtains the same result for G: as

we obtained above in the cluster CPA, namely Equation
(5.89). i

For this analogy we embed a n(=3)xn cluster with
a particular configuration 6, with probability
P(Ble...Bn), in an average chain. The average chain

has the periodicity 2f the host lattice With atoms of

 

average self-consistent complex mass m(l-E). Using the

symbol

194
- _ - 2
B : m(l-e)w -2Y (5.90)
and as before in Equation (5.75)
I 2-
B. _ miw 2Y

The inverse Green's function for a 3-site system is

 

 

 

 

 

Y
5 —l _
(G ) — B Y (5.91)
Y Bl Y
Y 82 Y
0 Y 83 _
Y Y
Y E Y
Within the cluster 2 then
Bl-A’ Y 0 -1
<3 _
G2 ‘ Y 82 Y (5.92)
I 2 - I -l
where A /Y = (B-A ) (5.93)

We determine the self consistent mass E as Butler did

by requiring the configuration averaged boundary element

of the Green's function (here 11) to be equal to the

195

diagonal element of the reference Green's function, which
is

' -o)‘ = (B-2A’)‘1 (5.94)

where the constant A’ on taking the inverse is the same
as in Equation (5.93). From Equations (5.92) and (5.94)

we have

2 -l

R = A’(y2-A’ ) . (5.95)

11

Writing Gil in continued fractions the self consistency

condition is

-——§———-= 2 P(B B B ){B -A’-Y2[B ‘Y2(B -A’)'1]’l}‘l
2 ,2 l 2 3 1 2 3
(Y -A ) 818283
(5.96)
The important point is that because G for the

11

embedded cluster analogy is the same function of its.

argument (A’) as is G for the cluster CPA, the self

11
consistency equations for A’ and for A in the two methods
(5.89) and (5.96) are the same. Therefore the two methods
are identical! 9

We now return to a solution of the cluster CPA.
We can reiterate Equation (5.87) or the n site equivalent
to the correct solution. We have found in practice that

we always get convergence to the correct solution by

'initially picking A to be the value of the perfect

196

monatomic heavy chain plus a small imaginary part.

A = B-JB:-4Y2

(5.97)

We used the Newton-Raphson method69 to speed the convergence
of the reiteration to A. The procedure takes 3 to 6
reiterations in general to give A to .01% error. For

the 3-site cluster, for example, we have the error F

on any reiteration

P(A) = A/(YZ-AZ)~2x5{81-A-Y2[82-Y2(83-A)’1]’1}'1 (5.98)
6
and F’(A) = dF/dA
= -—53113— —X x6{B -A-Y2[B -Y2(B -A)-l]_l}"2
(YZ'A2)2 6 l 2 3
X{1+Y4[BZ‘Y2(83-A)-l]-2[B3-AJ-2} (5.99)
Therefore the value of A chosen for the (J+l)-th
iteration is
AJ+l = AJ-F(AJ)/F’(AJ) (5.100)

There is one difficulty to overcome in calculating
the density of states for the phonon problem which does
not arise in the electron problem where the density is
the trace of the configuration averaged Green's function.

For the phonon problem

_ _ £_ -———— (5.101)
tr Im MG ‘ nn Im tr Mzcz ’

IH

‘ D(wz) = -

Z

”(T

197

where the last eXpression is that for an n site cluster.
In fact we calculated several candidates for the density
of states. In order of increasing agreement with experi-

mental spectra we tried

1. 0(02) = - % Im tr m(1-E)R (5.102)

11

an expression involving only the effective medium in
_the embedded cluster analogy. We solved for E by using

Equation (5.93) in the form
- 2 2
(l-€)mw = A+2y+y /A (5.103)

The result for D(wz) was a series of broad flat peaks
rather similar to the results of the single site CPA as

one might have expected.

2. D(wz) = - — Im Exam G (5.104)

a configuration average of the boundary site expression.
Because the configuration average of the boundary site Green's
function is Rll this result for the density of states
closely resembles the result (1).

3. We calculated the full cluster trace indicated

by Equation (5.101)

Im tr) x‘SM‘SG‘S

D(wz) '%fi-

0')

1 2
EH.1m2 Z? .(w ) (5.105)
i 0

198

This spectrum showed some of the peaks of the

' experimental spectrum but the peaks are rather broad and
do not have their centers at the right frequencies.
Actually our test of this expression was only for the
3-site cluster. This method has the advantage that

as the number of sites in the cluster gets larger the
method is guaranteed to improve, a statement which cannot.

be made for our final and best expression for D._

4. D(wz) = - -1- 1m) xdma 05 (5.106)
TT 5 CC CC

where cc indicates central site. For the 3-site cluster,

for example

-1

D(wz> = - % Im2x5m<1-ez>{82—Y2[81-A1'l-YZEB3-A1’l} (5.107)
5

We now show computations of the density of states
using the central site expression (5.106).

Figure (5.14) shows the density of states for
Cd=.l random for the l, 3 and 7 site self-consistent
clusters. All three cluster sizes reproduce the host
band structure reasonably well. The seven site cluster
is able to display some of the fine structure in this
region. In the impurity band, the improvement with
increasing cluster size is remarkable. The seven site
cluster reproduces the total density of states with great‘
precision. The superimposed 10,000 unit numerical chain
probably shows some small spurious structure in the host

band due to its short length.

199

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counuonom Havana H.000 Mom magnum mo unaccoaul.ca.m mmDUHm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. 3.1
1. 1
I); 1 . - Y-
mm.
\/
.11 m
G
M 1
1
111
11 1
1 1 _
.. b If
.1 .. 11 : .
1 fl 1 1 Wm)
\wt m NI )\ _1 1
9.1.5... M In“ 1 1L .1
Awk1mu .0; fi(r1mw 11.1.. m “w 1
. M 1 ,
Piutnazoo o: 0m 1 "1 1
11 1
1 ..
Jn1u1L®<15Z 11H. _1 u _
1i. 11 .1 .1 U .\

200

Figure (5.15) shows the density of states for
Cd=.5 random for the l, 3 and 7 site self-consistent
clusters. The single site CPA completely fails to
reproduce any structure. With the three site cluster,
we pick up some of the major structure. The seven site
cluster again does a remarkable job recreating both
host and impurity band structure.

Figure (5.16) shows the single and 7 site self-

d

numerical plot is for 10,000 atoms which probably includes

consistent cluster for C =.9. The superimposed

some small spurious structure. The seven site cluster

. . . . 2
progress1vely shows more structure With 1ncreas1ng w .
.1

Figures (5.17) and (5.18) are for C =.5,

d Pd,d=
and .9 respectively. For these two cases with very high
degrees of order, the seven site self-consistent cluster
does not give as good a frequency spectrum as in the

random cases. Both figures however, display all the
correct peaks in the frequency spectrum. They also seem

to display structure apparently absent from the experi-
mental spectra. We say apparently because the experi-
mental grid is A=.04 whereas the cluster grid is .01333.
-The spectra show that a cluster size of seven is not

large enough to adequately reproduce the respective spectra
with high degrees of short-range order.

As a final comparison of the 7 site self consistent

cluster with experimental results we compare the

201

 

 

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mumsflo u: umwmzoo mama ouwm n m an

m .m.u 0 you moumuu uo auamcmonu.na.m mmaon

1

 

 

 

...
.\ ,1

\

 

 

 

——_..1.._._...

 

 

 

 

 

pram. wacoo
(Sam. 0.2m \) Y )\

Y1 «1019a25c111

2“:

 

 

 

 

 

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s
cwumumcmm .m.nu cm .m.ucu now nounua no madmaoo-u.ma.m mmoon

‘

 

 

 

 

 

 

 

 

 

3
1m. m. N a H - mo 0
K
F
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as (gag: NT
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11 111
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205

integrated density of states for C =.5, random and

d
Pd,d='l' Figure (5.19) gives the comparison between

the experimental and analytic cluster results for Cd=.5
random. From this figure we see that at each sharp peak

in the spectrum we lose some contribution to the density
of state for the seven-site self-consistent cluster.

The total integrated density of states is .975 instead

of one.‘ When we compensate for this, the curves of the
numerical and 7 site cluster integrated density of states
coincidence until w2>1.9, whereupon the deviations between

the two are about 1% at most. Figure (5.20) shows the

comparitiveintegrated density of states for Cd=‘5

Pd,d=°l' Although significant deviations between the
,two curves are evident, the maximum relative errors'

are of the order of 5-6%.

2(36

.Hmumaao ucmumamaom mama muam h n ma vmumumcmm
socamu m.nuo mom mmuuum mo spa-cue uououmoucunn.ma.m mmoon

m“ “w .d55 a-mfi;|!):|:-:

J .

\1r&.u11rywu/>149.71 \1#\\.

1.13me1913 from. mkmk. ......)

    
  

 

 

1Q.\

207

.umumaao

s
unmumamco uamm muam h m an Umumumcmm a.uw 6m
.m.u u how mmumum mo xuamcmo Umumummucanl.on.m mmsuam

1. m N «3.1 a

1 1 1 «

  

1331,5552 \/\
LthQmCoouiWw 01.1w K

 

CHAPTER VI

CONCLUSIONS

In the preceding chapters, we have examined the
vibrations of disordered binary chains in the harmonic
approximation with all force constants equal. First,
we numerically examined the phonon density of states
of binary chains with short-range order generated by
ergodic Markov theory. The theory was used to introduce
nearesteneighbor and next-nearest-neighbor correlations.
When these spectra were compared to the spectra of random
binary chains with the same concentrations of constituents,
we found marked differences between them.

For the first order Markov chain we found that
the pair correlation functions were particularly simple
and that all higher order correlation functions can be
expressed in terms of the pair correlation functions.

The second order Markov chain theory does not give as

simple pair correlation functions and higher order
correlation functions cannot be expressed in terms of the
pair correlation functions. With the second order Markov
czhain, we demonstrated that a pair of quite different
frequency spectra can be obtained from chains with identical

pair correlation functions.

208

209

In addition to numerically examining the frequency
spectra (eigenvalues) we also numerically examined the
eigenvectors of the chains. We were able to characterize
both the extent of appreciable amplitude of the eigen-
vectors and the exponential decay rate away from this
region of appreciable amplitude. We found that both
localization parameters displayed the same general functional
relationship versus wz; however, they were found to have
quite different functional trends at certain points. Short-
range order was found to radically change localization
values and their functional relationship to m2.

Theoretically, we constructed the phonon frequency
spectrum in three ways. First, we embedded impurity
clusters of size n in‘a host chain. We found the impurity
modes arising from each possible cluster configuration.

We were then able to reconstruct the impurityband spectrum
by properly weighting all possible n-site configurations.
This reconstruction conclusively demonstrated the origins
of the peaks and peak broadening in the impurity band.

It did not give any information on the host band region

of the frequency spectrum. The embedded cluster theory

was found to be highly dependent on defect concentration,

working much more satisfactorily below C =0.5 than above

d
this value.

210

The second approach for reconstructing the phonon
density of states involved periodically extending the
n-site cluster throughout the chain. We then obtained both
analytically and numerically the spectrum of each possible
periodic chain with an n-site basis. The density of states
was reconstructed by averaging over all possible spectra.
Unlike the embedded cluster calculation, this method
reconstructed the whole frequency spectrum. For a six-

. site periodic system, the impurity band was reasonably
accurate independent of concentration and short-range
order. The host band region, although generally correctly
reproduced, possessed many discontinuities which are not
physically realistic.

The third method for reconstructing the frequency
spectrum was a self-consistent Green's function method.

An n—site cluster was embedded in a self-consistent host
,medium. The self-COHSistenCY was determined by requiring
the phonon scattering from the configuration average of

all clusters to be zero. We were able to simplify

greatly the problem by using the self-consistent boundary
site theory developed by Butler for the one-dimenSional
electron problem. We were able to prove that Butler's
approach gives zero scattering from the configuration
averaged cluster. For the seven-site self-consistent
cluster we obtained excellent agreement with the numerically

generated density of states for all concentrations in

211

random systems. When high degrees of short-range order
were introduced,general agreement was obtained. By
correlating the cluster size, localization lengths, and
goodness of frequency spectrum, we were able to predict
the approximate size of the cluster required to give

agreement with numerical results.

REFERENCES

212

10.

11.

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C. Domb, A. A. Maradudin, E. W. Montroll, G. H. Weiss,
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H. Bethe, Proc. Roy. Soc. A150,552(1935).

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S. C. Moss, P. C. Clapp, Phys. Rev. 132,418(1966).

S. C. Moss, Journal of Applied Phys. 35,3547(l964).

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ed. F. Seitz, D. Turnbill, Academic Press, Inc., New
York, 1955.

W. M. Hartmann, Phys. Rev. 112,677(1968).

D. W. Taylor, Private communication to W. M. Hartmann,
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Dynamics of Disordered Harmonic Lattices (1966).

C. T. Papatriantafillou, Preprint.

E. Parzen, Stochastic Processes, Holden-Day, Inc.,

 

1964. This was the primary reference for this work.

J. G. Kemeny, J. L. Snell, Finite Markov Chains,

 

D. Van Nostrand Co., Inc., Princeton, New Jersey,
1969. _

A. T. Bharucha-Reid, Elements of the Theory of Markov
Processes and Their Applications, McGraw Hill Book
Co., New York, 1960.

P. Billingsley, Statistical Methods in Markov Chains,
Address presented on August 23, 1960 at Stanford
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Series and Products, Academic Press, New York and

London, 1965.

216

40. J. Hori, Prog. Theor. Phys. 31,940(l964).

41. H. Matsuda, E. Teramoto, Prog. Theor. Physics 34,314(l965).

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Clarendon Press, Oxford, 1965.
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Mechanics, Addison-Wesley Publishing Co., Inc.,

 

Reading, Mass. (1961).

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Dimension, Academic Press, New York and London, 1966.

 

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text.

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extensive publications in this field, a good

bibliography is given in the Hori reference.

53.

54.

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56.

57.

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59.

60.
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217

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461(1972).

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630(1962).

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189(1964).

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Ehrenreich, Phys. Rev. §£,3383(1971).

W. H. Butler, Preprint.

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and Fortran Programming, John Wiley and Sons, New

 

York, 1964..

APPENDICES

218

APPENDIX A

NUMERICAL COMPUTATION OF EIGENVALUES

OF CHAINS

219

To calculate the density of states (spectrum) of
a linear chain of length N, we need to be able to cal-
culate the eigenvalues of the eigenvalue Equation (2.3)

in text. Rewriting Equation (2.3) as

2

(g-gw )5 = 0 (A.l)
where
i.= r261 (Si.
3 i I]
ZL—cs..
xa‘ia—‘“ l'Jtl
i iil

g is clearly a symmetric tridiagonal matrix. Now, we
denote the principal minor of order r of (§_;w2) by
Pr(w2). Letting P0(w2)=l, we can write a recursion

relationship between these principal minors.

2 — _
Pl(w ) — H— w (A.2)
l
2
2 _ 2Y _ 2 2 _ Y 2
Pin» ) — (fin-.- w ’Pi-l‘“ ) (In—.1“. “”14“” ) (A.3)
1 1 1-1
ZiiiN

The zeros of Pr(w2) are the eigenvalues of the leading
principal submatrix of order r of Q. We denote this
submatrix by gr. Since g and 2r are Hermitian their
eigenvalues are all real. Now, we will show that eigen—

values of gr are separated in the strict sense from the

220

221

eigenvalues of B

—r+1' The proof is by contradiction.

Assume u as a zero of Pr(u) and Pr_l(u); then, using
Equation (A.3), we have u is also a zero of Pr_2(u) since
(wZ/mimi_l)#0. Continuing this argument, we have u is

a zero of P0(u) which is a contradiction of the definition
of P0(u).

Because of the strict separation of zeros of the
Br's, Given's was able to devise one of the most effective
ways of determining the eigenvalues of a symmetric tri-
diagonal N by N matrix. The chain with periodic boundary
conditions is not tridiagonal requiring a transformation
to reduce it to tridiagonal form, whereas the chain with

fixed boundary conditions is already tridiagonal.

Theorem: Let the quantities P0(w2), Pl(w2)....

PN(w2) be the principal minors of a symmetric

tridiagonal matrix, evaluated for some value of

wz; then S(w2), the number of agreements in sign

of consecutive members of this sequence, is the

number of eigenvalues of g which are strictly greater

than wz.
To apply this theorem we must find a way to deter-

mine a sign if Pr(w2) is exactly zero. If Pr(w2)=0,

then Pr(w2) is taken to have the opposite sign of

Pr_l(w2).

222

The proof of the eigenvalue theorem is by induction.
Assume that the number of agreements in sign Sr(u), in

the sequence P0(u),Pl(u)...Pr(u) is the number of eigen-

values of gr which are greater than u. If we denote the
eigenvalues of gr by xl,x2...xr, then
xl>x2>x3>...>xs>u:xs+l>xs+2>...>xr (A.4)

Next, the eigenvalues of §r+l’ we denote by yr,y2...yr.

Since the eigenvalues of gr and gr 1 are separate in

+

the strict sense, we have
yl>xl>y2>x2>'°'>Y3>Xs>ys+l>xs+l>'"Yr>xr>yr+l'(A‘s)

B

=r+l' therefore, has 5 or s+l eigenvalues greater than u.

In terms of these eigenvalues the principle minors can

easily be evaluated.

r
Pr(u) = i21(xi-u) (A.6)
and
r+l
Pr+1(u) = i£l(yi-u) (A.7)

If no xj=u and no yj=u, there are two cases to
conSider. If ys+1>u, then Pr(u) and Pr+l(u) have the same

sign and S (u)=Sr(u)+l. If y u, then Pr(u) and

<
5+1

Pr+l(u) have oppOSite Signs and Sr(u)=Sr+1

definition of Sign of Pr(u) when xs=u and Pr+1(u) when

r+1

(u). Using the

223

ys+l=u completes the proof.‘ A sequence of principal

minors, P0(w2), P1(w2)...PN(w2) satisfying the above theorem
is said to possess the Sturm sequence property.

For the linear chain, Rosenstock and McGill43

have shown that the atomic displacements from equilibrium,

U1,U2,U3...UN form a Sturm sequence with initial conditions

U0=O and Ul=l where Equation(2.2) is rewritten as
mi 2
Ui+l = (2 - '7— ’.L) )Ui-Ui’l (A.8)

Since the Sturm sequence generated by Equation (A.8)
requires either approximately one half the computer
storage or one half the computing time of Equation (A.3),
we have used Equation (A.8) for most of the spectra
presented.

To determine the range of permissible eigenvalues

of the matrix B in Equation (A.l), we use the fact that

the absolute value of all the eigenvalues of g must be
less than or equal to the infinite norm of 2,42 or
2
lw |:|lBl|m (A.9)
N
where IIBIIoo = Mix §=llbij|

For the linear chain

IlBl|m= I-Yfi—Hmflw -,;Y— =,‘3—} mm

L L L i L

224

where mL is the lightest mass in the chain. For

M
computational convenience we take 7£'=l; therefore,

Iw2|14 sets bounds on the frequency spectra. We, also,

know that physically w must be real or wzio. Therefore,

the spectra of all chains must be in the interval 0:w2:4.

If we divide the interval into 100 equal parts each .04

wide,a reasonable histogram frequency spectrum is produced.
To calculate the eigenvector of a chain given in

Appendix D, we will need to know individual eigenvalues

to a considerable degree of precision. Any eigenvalue can

be determined to any degree of precision by the method

of bisection. First, we know that all eigenvalues must

lie between 0.0 and 4.0. If we take a =0, and b =4.,

0 0

then for the 5 step in the bisection we have

CS = %(as_l+bs_l) (A.ll)

s-l’bs-l)'

. 2
Next, we compute the Sturm sequence Wlth w =Cs and

where Cs is the midpoint of the interval (a

determine S(CS). Suppose we are looking for mi, then

if S(Cs):H, then we take aS=CS and bs=b and if S(CS)<K,

s-l'
we take a =a and b =C . By this method we can locate
s s-1 5 s
an eigenvalue mi in p steps to be in an interval (ap,bp)
of width (bO—a0)/2p = 4/2p = 22‘9.
For implimentation of this procedure on a digital

computer, the precision to which the computation is carried

is important because of computer rounding errors. The

225

error in the bisection described above is %(4/2p) or Zl-p.

In addition, Wilkinson42

has shown the computer error
is 18.08/2t where t is the binary precision to which the
computation is carried. One significant fact is that
the error is independent of the chain length. The total

error is, therefore,

ET = (18.08)2-t + 21'p (A.12)

where p is the number of steps in the reiteration
and t is the conputational binary precision.

For example, for computations on the UNIVAC 1108 single
precision computations (27 binary bits) give a minimum

error of 1.35x10-7 and double precision computations
-17

(60 binary bits) give a minimum error of 1.57x10 . To
obtain errors of less than 10-8, (~2-27) we must use
double precision computations requiring 2—2732]‘-p or

p128.

For each computation of the Sturm sequence there
are of the order of 2N multiplications and subtractions.
For each eigenvalue there are 2Np computations. To find
all the eigenvalues requires 2N2p computations by the
method of bisection versus N3 for most other methods.

For N>2p, the method of bisection is favored.

The method of bisection must be carefully examined
for the case of eigenvalues near zero and close eigen-
values. For eigenvalues near zero, the accuracy of the

eigenvalue computed by bisection may be small. For example,

226

'if mizlo—4, single precision computation with errors of
at least 1.35x10“7 will give less than three digit
accuracy for mi. In double precision with p=28, the
error is ~10.8 or m: is good to about four digit accuracy.
For close eigenvalues, more than one eigenvalue may lie
in the interval (ap,bp) of error. The eigenvector

computations give indications of this trouble and are

discussed in Appendix D.

APPENDIX B

MARKOV THEORY SUPPLEMENT

226-A

PROOF OF ERGODIC MARKOV CHAIN THEOREM

Theorem 1: An irreducible aperiodic recurrent Markov chain
possesses a unique long run distribution. The long run
distribution {ny, keC} of an irreducible aperiodic recur-

rent Markov chain is the unique solution of

1T1: = g 7Tj Pjyk (8.1)
satisfying
2 mk = 1. (B.2)

k
Recall that Pj}:(n) is the n step transition probability

to go fnmnstate j to k and P. ,=P. (1).

‘ JI}‘ Jlk

To prove this theorem, a number of new quantities
and additional theorems will have to be introduced. By
analogy to Equation 3.35 for the conditional probability,
fj P’ of even visiting the state k, given the chain was

’5
in state j, we define the conditional probability fj k(n)
I

of the first passage from j to_k occurring in exactly

n steps.

f = P[Vk(n)IXo=j] (3.3)

j,k‘“’

where for any state k and integer n=l,2,... we define

chain segment, or set, V

Vk(n) = [Xn=k,Xm#k for m=l,2,3...n-l]

227

228

Theorem 2: For any states j and k,
(D

f. = X

3,k r1 fj,k(n) (8.4)

1

Proof: Define
Vk = [Xn=k for some n>0], then
Vk is the union of the sequence odeisjoint sets

{Vk(n),n=l,2,...}

f.
J.k

IIMS

PEVk(n)IXO=j] = : f.'k(n)

t= PEV Ix =j] =
k 0 1 n o 3

n

Q.E.D.

The n-step transition probabilities Pj k(n) and the first
I

passage probabilities f. (n) can now be related.

J,k

Theorem 3: For any states j and k and integer n11,

n ‘ .
Pj,k(n) = mil fj,k(m)Pk,k(n'm) (B.5)

where we must logically have

P = 6 (the Kronnecker delta)

. 0 .
J.k( ) J,k
Proof:

n

z ptxn=k,xm=k,xq¢k for

P[x =k|x =j] =
n j o m=l

q=l,2,...m-llxo=j]

= z P[Xn=klxm=k]P[Xm=k,Xq#k for q=l,2,...m-l|Xo=j]

229

Pj,k(n) = mil Pk'k(n-m) fj,k(m) Q.E.D.

In the step before the last one we multiplied the conditional
probability that if m is in state k then n is state k by

time probability that m is the first passage to state k to
obtain the probability that both occur. Such a multipli-
cation of probabilities is correct only if the pro-
babilities are independent, which is not true for arbitrary
chains but is true for Markov chains. Next, we can relate
the conditional probability of ever visiting a state fj,k

to the conditional probability of visiting the state an

infinite number of times, gj k'
I

Theorem 4: INN: any states j and k

— ' n
gk'k — 11m (fk'k) (B.6)
n—mo
and
. = . 3.7
gi,k f3.k gk,k ( )

Proof of B.7: From Equation 3.36

gj,k = PENk(m)=m|XO=J]

ll
"M8

PENk(w) - Nk(n)=w, Xn=k' Xq#k for

n 1

q=l,2,.. .n-llxo=j]

230

= X f n = X f. n
n=l gk,k j,k( ) gk,k n=1 j,k( )
by Theorem 2
v = I ‘ OEODO
931k 919k f3.k Q

Proof of (B.6):

For nil look at

PENk(m) : n|XO=kJ =

co

mil PENk(m)-Nk(m)in’lr Xm=kr quk, for q=1,2,...m-l|x0=k]

mil P[Nk(w)-Nk(m):n-1|Xm=k] x

PEXm=k, Xq¢k for q=1,2,...m-11X0=k]

Z PENk(w):n-1|X0=k] fk,k(m)

II
T]
|'—'I
Z

k(m)_>__n-1|XO=kJ fk,k
repeating this procedure n times

n

PENk(w):nIXO=kJ = fk,k n11

therefore

pEnk(w)=w|xO=kJ=iim PENk(w):nIXO=k]

n+co

_ . n
_ :12 fk,k Q.E.D.

231
From Theorem 4’since Oifkkil,
gk,k=l or 0
and
gk,k=l if and only if f = l
gk,k=l if and only if f < l
TheorennS: For any state k,
fk kgl if and only if 2 P (n)<m
' =

and

IIMB

fk,k=1 if and only if

Pk’k (n)=oo
n

1

Proof: To prove theorem 5, we need to perform a

transformation on the transition probability and the

conditional probability of first passage

00 n m n
P. z ZZP. n=6.+ZZP. n
J:k( ) n=0 J:k( ) 3k n=l J,k( )
f. (2) = 2 2n f. (n)
31k - n=l 31k
for |Z|<l

(B.8)

(8.9)

(8.10)

(B.11)

(B.lZ)

(3.13)

(B.l4)

Next, take Equation 3.5, multiply by Zn and sum over n

from one to infinity; we get

00 n g -
i Z Pj,k(n) —

||M8
N
M

n l 1 m=1 ’

fj k(m) P

232

interchanging the sumation on the RHS

m m w n-m
P. Z -6. = Z Z f. 2 Z P n-m
jlk( ) jlk m=1 j,k(m) n=m k,k( )
Pj,k(Z)—6jk = fj,k(z) Pk,k(z) (B.15)
From this
Pk,k(Z)-l = fk,k(z) Pk,k(z)
or fk k(Z) _ 1 - P 1 Z) (B.16)
’ k,k
Next Since fk,k = E fk,k(n) (B.17)
n—1
and lim_ f (Z) = X f (n) (8.18)
Zel k’k n=l k'k
Lim_ P (2) - 2 P (n) ’ (3.19)
Z+l k’k n=0 k’k ,

where l- is approaching one from less than one we have

fk,k<l lff E fk,k(n)<l from Equation 8.1?
n—l
X f (n)<l iff Alim_ fk k(Z)<1 from Equation B.18
n=l ‘Kk Z+l ’

lim_ fk k(Z)<1 iff lim; Pk k(Z)<0° from Equation B.16
Z+1 ’ Z+l ’

co

Lim P, (Z)<oo iff Z P (n)<0° from Equation 8.19
Z+l n-O
Similiarly
f =1 iff 2 f (n)=l iff Lim_ f (Z)=l
iff iiT— Pk,k(z)= m 1ff n: Pk,k(n)= w Q.E.D.

O

233

Next, we define the expected first passage length

m = i n fk,k(n)

Theorem 6: ILet k.be a state such that

. _ ~ . 1 m 1
fk,k-l and mk,k< , then Lim H X Pk,k(m) m

n+w m=1
and for any state j

. 1 m 1
Lim ~ 2 P. m = .
n+m n m=1 j,k( ) j,k m

 

Proof: Assume that

Lim_ ((l-Z) A(Z))=L exists where
Z—>l
a zn

A(Z) = 0 n

n

IIM 8

Expand A(Z) in a Laurent series about Z=l

A(Z)

II
+

n
bn(Z-l)

= L( x Z ) +
m=0

:3
IIM8llM8
o

:1
O

(8.20)

(8.21)

(8.22)

(8.23)

(8.24)

interchanging summations in the second term on the RHS

A(Z) = L( z zm) + 2 2m 2 (“H-1)”m b
m=0 m=0 n=m m
= 2 2m (L+ z (“H-i)“-m b )
m n
m=0 n=m

Therefore, equating powers of Z to Equation 3,24

_ m n _ n-m
an — L+ 2 (m)( 1) bn
n=m

 

234

Now, look at

1 N N=m w
lim — X am=L+lim
N+w m=0 N-*0° m=0 n=m

m n
. 1 n _ n-m _ n_
“.23: N brig <..>< 1’ - <m> “moi

 

= L+lim —— = L
N+m N
1 n 1
Therefore to Show lim H X Pkk(m) =
n+m m=1 mk,k

we need to show that

 

 

. l
Lim_ ((l-Z) P (Z)) - ————
2+1 k'k mk,k
or equivalently
Lim 1 = m
2+1- (l_Z)TPk,k(Z)3 klk
using Equation B.16 , we get
l-f (Z)
. k k
Lim ’ = m
2+1' ( 1-z ) k'k
’ _ m n m _ n
fk,k(z) E fk,k(n) Z 3 fk,k(n)((z l)+l)
n-1 n—1
- 2 f (n) z (n) <z—1>m
n=1 k’k m=0 m
= X f, ,(n) +- 22 f (n) X ( )(Z-l)
n=1 L'L n=1 }"k m=1 m
_ m n _ m
- fk,k + E fk,k(n) X (m)(Z 1)
n—l =1

235

Since fk,k=l is given

n

 

m n m-l
_ - 2 f (n) 2 ()(2-1)
1 Z n=1 k,k m=1 m
l—fkk(Z) w ( )
Therefore, lim_ (————:———) = X n f n = m
2+1 1 Z n=1 k’k k,k
l n
Finally for Lim — Z P. (m) we can look at
n _ j,k
n+m m—l
L" 1-z P. z
m_ H ) 3,1,} ))

Z+l

using Equation 8.15 we have

Lim_ [(l-Z) fj,k(z) Pk,k(z)]=

 

Z+1

. (1-2) 1
Lim_ f. (2) _ = f. —— Q.E.D.
2+1 3'k 1 fk,k(27 3'k mk,k

For an irreducible recurrent Markov chain

fj k=l for all j and k, implies

1 n 1 “
Lim K E P.,k(m) = le.H E Pk,k(m)
n+m m—l n+w m—l

1
mk,k

 

(8.25)

The sum is independent of the starting state. This implies

the limit exists in the ordinary sense so that

. 1 .
Lim P (n) = = lim P. (n) = 1r
n+m k,k mk,k n+w j,k

 

k (8.26)

where Equation 3.39 is employed. This shows the ergodic
chain possesses a unique long run diStribution. Now,

we can complete the proof of Theorem 1.

236

First, we define

n

clearly,

= ' *
Wk lim P j,k(n)

n+oo
Summing over k, we get
X n = 2 lim P? (n) < 1im Z P? (n) = l
k k k n 3'k “'n+w k 3'k

or Z "k <1 (8.28)

Next, using the Chapman-Kolmogorov equation, we have

Pj'k(n+1) = i Pj'i(n) Pi’k (3.29)
Then,
1 n+1
Pj,k(n+l) = H:I mil Pj,k(m)
n+1 n+1
(n+l)P§’k(n+l) = mil Pj,k(m) = Pj,k + :2 Pj,k(m)
n
= Pj,k + mil P.'k(m+l)

Substituting Equation 8.29 into the above

. * — . =
(n+l)Pj'k(n+l) Pj Z Z Pj i(m)Pi

m=1 i ' 'k

' Dividing by n and interchanging the: summations of the RHS,

we get

l l l n
(1+n)Pj,](n+l) P. P ( X Pj,i(m))Pi,k

= *
Z Pj i(m)Pi

i I Ik

Taking the limit as nrm, we get

. 1 * 1 _ '_
Lim [(l+fi)P ,k(n+1) fi-P. J _ w _

n.. j

Lim Z P*..(n)P.

> (1im P*. . n P.
n+e 1 31 .,k - i n+m 3:1‘ )’ 1k
= ; Tri Pi,k
i
or Wk 1 2 Ni Pi,k (8.30)
1
Now, we sum over k
E w > Z in P. = Z n. X P. , = Z n.
k k “'k i k i,k i i i i,k i 1
Therefore, the equality is proved and
“k = 2 Hi Pi,k (8.31)
i
and the inequality in Equation 8.28 is actually an
equality giving
2 ”k = 1 (8.32)
k Q.E.D.

Finally, the following definition is often employed
Definition: 11 recurrent state i is called positive
if mk k<oo or null if mk k=9. A recurrent
I I

state which is neither null nor periodic is
called ergodic.

238

Third Order Two Constituent Markov Chain

 

For a two state, third order Markov chain, the

3

transition probability matrix is 23 by 2 or has 64 elements

3+1

of which only 2 =16 are nonzero. These elements are:

Pddd,d = Pddd,ddd

Pddd,h = Pddd,ddh
Pddh,d = Pddh,dhd
Pddh,h = Pddh,dhh
Pdhd,d = Pdhd,hdd
Pdhd,h = Pdhd,hdh (3'33)
Pdhh,d = Pdhh,hhd
Pdhh,h = Pdhh,hhh
Phdd,d = Phdd,ddd
Phdd,h = Phdd,ddh
Phdh,d = Phdh,dhd
Phdh,h = Phdh,dhh
Phhd,d = Phhd,hdd
Phhd,h = Phhd,hdh
Phhh,d = Phhh,hhd
Phhh,h = Phhh,hhh

Using the reduced notation and eliminating identically

zero elements, Equation 3.42 gives

239

Cdddpddd,d + Chddphdd,d = Cddd (3'34)
Cdthddh,d + Chthhdh,d = thd (3.35)
Cdddpddd,h + Chddphdd,h =9Cddh (3°36)
Chhdphhd,d + thdehd,d = Chdd (3'37)
thhpdhh,d + Chthhhh,d = Chhd (3'38)
thdpdhd,h + Chhdphhd,h = Chdh (3°39)
Cdthddh,h + Chthhdh,h = thh (3'40)
Chhhphhh,h + thhpdhh,h = Chhh (3°41)
Equation 3.20 becomes
Pddd,d + Pddd,h = 1
Pddh,d + Pddh,h = 1
Pdhd,d +Ipdhd,h = 1
Phdd,d + Phdd,h = 1 (3°42)
Pdhh,d + Pdhh,h = 1
Phhd,d + Phhd,h = 1
Phdh,d + Phdh,h = 1
Phhh,d + Phhh,h = 1
Finally, Equation 3.40 gives
Cddd + thd + Cddh + thh + Chdh + Chhd + Chdd + Chhh = 1
(3.43)

One of the above equations is redundant. We can arbitrarily

eliminate Equation 8.41. Also, we must relate the cluster

240

concentrations to the host and defect concentrations.

This relationship is

l

2
ddd 3 (Chdd+cdhd+cddh) + §(C

hhd+chdh+cdhh) (3'44)

and - c + c = 1. (B.45)

Therefore, we have 16 equations with 24 unknowns

, requiring the specification of 8 variables. If we specify

Cd'Pddd,d'Pddh,d'Pdhd,d’Phdd,d'Pdhh,d'Phhd,d and Phdh,d'

we'get

 

 

Chdd = Cddd(l-Pddd,d)/Phdd,d ' (3°46)

Cddh = Chdd (8.47)

c = C l-Pddd,d Phhd,dphdh,d-Phdh,d-Pddh,dPhhd,d
dhd ddd Phdd,d Phhd,dPhdh,d—Pdhd,dphdh,d-Phhd,d
(3.48)

Chhd = (Cddh-thdpdhd,d)/Phhd,d (3°491

Chdh = (thd-Cddhpddh,d1/Phdh,d (3°50)

chhd = cdhh (3.51)

Chhh = l-(Cddd+cdhd+zcdhh+2Cddh+chdh1 (3°52)

where

241

C

 

 

d
C :
ddd. 1+ 1'Pddd,d 2Phhd,dphdh,d-2Pdhd,dphdh,d-Phhd,d(l+Pddh,d)+Pdhd,dedh,d‘)
Phdd,d Phdh,d(Phhd,d-Pdhd,d)-Phhd,d
(3.53)
and

Phhh,d = Chhd(l_Pdhh,d)/Chhh

The Specification of all eight inputs in a coherent
fashion is clearly difficult. The constraints on these
inputs are not clearly visible requiring trial and error
specification. The complications involved in the third

order Markov chain will inherently limit its application.

Second Order, Three Constituent Markov Chain

 

We will briefly examine the three constituent
second order Markov chain because it can be applied to
generating "salt" like chains. 'For'the 3 state, second
order chain the transition probability matrix has

32-32=81 elements of which 32+1

=27 are nonzero.
Equation 3.20 in text when we eliminate the zero

matrix elements becomes

11,1 11,2+P11,3

P +P

12,1 12,21”P

12,3

P13,1+P13,1+P13,3

P21,1+P21,2+P21,3 =

+P

P22,1+P22,2 22,3

P23,1+P23,2+P23,3

+P

P31,1+P31,2 31,3

P32,1+P32,2+P

P33,1+P33,2+P33,3

32,3

II
H F4 ta

242

F4 )= H

H

where we have constituents l,

'U
I I

33,1

C11P11,1+C21P21,1+

C

C13P13,1+C23P23,1+

C11P11,2+C21P21,2+

C

C

c11P11,3+C21P21,3+C31P31,3
C12P12,3+022P22,3+C32P32,3

C13P13,3+C23P23,3+

12P12,1+

12P12,2

’ P33,31'

Equation 3.42 in text

C

C

C

C

+C P

22

13P13,2+C23P23,2+C33P33,2

C

31P31,1
22P22,1+C32P32,1
33P33,1

31P31,2

33P33,3

22,2+C32P32,2-

(8.55)

2, and 3 and, for example,

becomes

11
21

31

0 0 0 0

12
22
32
13
23

00000

33

(3.56)
(8.57)
(B.58)
(3.59)
(8.60)
(8.61)
(3.62)
(8.63)
(8.64)

243

Finally Equation 3.40 in text gives

C11+C12+C +c +C +C +C +C +C = 1 p (8.65)

13 21 22 23 31 32 33

These cluster concentrations can be related to the in-

dividual concentration of constituents by

_ 1
Cl — Cll + 21C12+C21+C31+C131 (B.66)
c = c + 1(c +c +c +c ) (3 67)
2 22 . 2 12 21 32 23 '
and
C +C +c = 1 . (8.68)

1 2 3

Here once again one equation can be shown to be redundant
leaving 21 equations with 39 unknowns or 18 Specifications
being required. In itself 18 Specifications makes the use
of the three constituent second order Markov chain almost
impossible.

However, with the solution of the 2 state, first
and second order Markov chains, we can make the following
hypothesis:1) n constituent, tth order Markov chain will
require (n-l)(nt) specifications.

Next, we will look at the three state, second

order chain where adjacent alike constituents (11,22,33)

are disallowed. Therefore, we have

C = C := C = 0 (B069)

We now rewrite

P12,2 = P

C1+C2+C3 = l

C1=C12+C13

12,1

P

P23,1+P23,2

P31,2+P31,3=
+P =1

P

21,2

32,1

+P =1

21,3

1

32,3

244

C12P12,1+C32P32,1=C21

C

C21P21,2+C31P31,2=

13P13,1+C23P23,1=

C

C

31

12

Equations 8.55 - 8.68 as

=0 (8.70)

(8.71)
(8.72)
(8.73)

(8.74)

(8.75)

(8.76)

(8.77)
(8.78)
(8.79)
(B.80)
(8.81)

(8.82)

(8.83)
(8.84)

(8.85)

This equation set includes 15 equations with

21 unknowns requiring only 6 Specifications.

first four specifications as

C

l,

C

2' P12,1'

and P

32,1

We take the

245

The solution of'C3,P12’3 and P32,3 are trival. In

addition, we get

C _2P12,1(C1+C2)-P
21 l-P

12,1+P32,1‘P32,1(2C1+C2)

32,1+P12,1

 

(8.86)

Using 8.86 we can easily solve for C C

23' C13' C12' 32'

and C31 in that order. Since all the concentrations
have been found with these four Specifications Equations

8.84 and 8.85 give
Pl3,1 a P23,l (B.87)

P21’2 a P31,2 (8.88)

Apriori, we would not have expected this dependence among

probabilities. -For the last two Specifications we can.

take P13,l and P21,2 completely speCifying the problem.
For the special case of an 1 21-X 3X salt in which

the substitutions are made on only one sublattice of an

otherwise perfect chain we must have Cl=0.5 and

C11=C22=C33=C23=C32=0° We then have x=2c3 and,
c2+c3=o.5
P21,2+P21,3=1
P31,2+P31,3=1
C2P21,2+C3P31,2=C2

four equations in Six unknown leaving only two parameters

to be specified.

246

Examination of Structure of Pn

 

The 2 transformation defined by Equation 8.24
gives uS a powerful technique for finding the n step
transition probabilities Pj,k(n)' If we define, the
z transform of the transition probability matrix as the
z transform of each element and Simiarly for the un-
conditional probability vector, we can take the transform

of the Equation

p(n+1) = p(n) P I (8.89)

— ’

where p(n) is defined by Equation 3.24 and P by Equation
3.18 P=P(l) has no n dependence and as such is

constant under a z transformation, p(n) goes to

p(z)l=- Z p(n)zn and p(n+l) goes to
n=0 .
co on n
2 p(n+1)zn = Z p(n)§_
n=0 n=1 2
1 _ °° 1
=35; Z p(n)z - 12(0)! =—(p(2)-p(0))
_ z
_n-O
Therefore, Equation 8.89 becomes
1 _
5(p(2)-g(0)) — 13(2) :1: (B.90)

This equation can be rearranged to give

or g<z>=g<0><;—zg)‘

247

If we take the inverse z tranformation we get

p(n)=p(0k9[(£fz£)_l 3 (3.91)

-’

Comparing Equation 8.91 with Equation.3,25 we see the

important result

1] (8.92)

—_.

En =~9[(£-z§)—

where the inverse z transform is performed on each

element of (I-ZP)_1. For_the 2 state first order Markov

chain we can now easily derive Equation 3.29 from

Equation 3 . 2 7 .

l—ZPh,h - ZPh,d l-zPh’h - z(l-P

(I—zP) = =

-2Pd,h - l-de’d -z(1-P

h,h)

d,d) 1'zpd,d

The determinant of (I-zP) is

2

' z Pd,hPh,d

det( ) = (l-ZPh,h)(l-zpd,d)

1—z(P ) + 22(P

h,h+Pd,d h,th,d_Pd,hPh,d)

2

l-z(ph,h+Pd,d) + (Pdld+PhIh-l)z

(l-Z)(l-(P -l)Z)

+
h,h Pd,d
The inverse of the matrix is

l l-de’d ZPh,d

(I-zP)- = 2P 1
d,h l

) (l-z)[l-(Ph’h+Pd,d-l)z]

 

_ZPd,d

248

We can expand each element by partial fractions. For

example the (1,1) element is

l-zP

 

 

 

 

 

 

d,d ' =A + B
(1-2111—(Ph,h+Pd,d—l)z1 l-z l-<Ph,h+Pd,d—l)z
giving A+B=1
=AP +P -1+B

pd,d 1 h,h d,d )

A = (l-Pd,d)/(2_Ph,h+Pd,d)
and 8 = l-A = (l-Ph’h)/(2-Ph’h-Pd’d)
Therefore, we have
(I_ZP)-1 = 1 1'Pd,d 1'Ph,h 1

‘ 2_Ph,h_Pd,d l-Pd,d l—Ph,h l-z

+ 1 1’Ph,h’(1‘Ph,h) 1

2-Ph’h--Pdld -11-Pd,d)l-Pd,d 17(Ph’h+Pd'd-l)z

The inverse z transformation is now easily performed Since

. _ n _ 1
if f(n) - a , then f(z) — l-az'

 

N _ 1
- (2

 

-Ph,h-Pd,d) 1’Pd,d 1‘Ph,h

N

l) -(1-P

(l-P

(l-PhIh)
- (l—Pd,d)

h,h)
d,d)

(Ph,h+Pd,d—

+
(Z'Ph,h’Pd,d)

 

which is Equation 3.29 in text.

249

Since we know from Equation.3.50 and 3.51

ph,h = 1-Cd(l-Pd d)/(l-Cd)

I

 

 

l-P
d d
- ’ _ = C (3.93)
(1 Pd’d)+(l Ph’h) d
1-P
2 P §éh = 1-cd (3.94)
d,d h,h
and
- P -C
_ d,d d
(Ph,h+Pd,d—l) l‘Cd , (8.95)
we obtain
l-C C P -C N C C
pN = < ‘1 d)+(_€1_rii__§.> < d d) (B 96)
l-Cd Cd l—Cd -(l-Cd) (l—Cd)
l-C C
lim p” = (l_cd Cd) (3.97)
N»w d d

For the two constituent second order Markov chain, the

matrix inversion becomes more difficult. For this case
\
phh,h phh,d 0 0
0 O P P
P = hd,h hd,d (8.98)
Pdh,h Pdh,d O O
0 0 P P

where p(n) = [Phh(n) Phd(n) Pdh(n) Pdd(n)]

250

l_z(l-Phh,d) ”ZPhh,d O 0

(I-zP) = 0 1 ”2(1’Phd,d) -ZPhd,d
—z(l-Pdh,d) -2Pdh,d 1 O

0 0 _z(l-Pdd,d) l-ded’d

Taking the determinant, after much Simplification we get

D = det(I-zP) = l-z(P l)

dd,d—Phh,d-

+ z (Pdd,d—Pdh,d-Pdd,dPhh,d+Pdh,dPhd,d)

3

+ z ( 2

Pdh,d_Phh,d+Phh,dPhd,d+Pdd,deh,d- Pdh,dPhd,d)

4 , -
+ Z (Phd,dpdh,d+Phh,ded,d-Pdh,ded,d Phh,dPhd,d)

)+22(P

U

= (l-z)[1+z(

Phh,d—Pdd,d hh,d—Pdh,d-Pdd,dPhh,d+Pdh,dPhd,d)

3
+ z (Phh,dphd,d+Pdd,deh,d—Pdd,dPhh,d—Pdh,dPhd,d)J (3'99)

. D must be factorable to use the expansion by partial
fraction. A Simple factorization of D does not seem
pOSSlble. For speCific numerical values of Pdd,d’ Pdh,d’
Phd,d’ and Cd where

Cd(1-P )(l-P

P : dd,d dh,d)
hh,d 1 +2(1-P

+Phd,d_Pdd,d-Cd(Phd,d dd,d11

one can factor D.

251

12 13 14

22 24

(I—zP)‘

ll
Cl)"’

32 34

42 a43 a44
where

_ _ _ 2 _ 3 _
a11‘1 zpdd,d z (1 Phd,d)Pdh,d+z (Pdd,deh,d Phd,deh,d)

2 3

a =z (l—Pdh’d)(1-Phd’d)+z

21 (Phd,d-Pdd,d+Pdh,d(Pdd,d_Phd,d)

_ 2
a3l-z(l - z (l-P

“Pdh,d) dh,d)Pdd,d

=zz(l-P )(l

a

41 dh,d _Pdd,d)

=zP - 22

a12 hh,d

Phh,ded,d

2

a22=1'z(Pdd,d'Phh,d+1) + z Pdd,d(l-Phh,d)

=zP‘ 2

a dh,d-z (Pdh,d-Phh,d+Pdh,ded,d)

32

+ z3[P (

dd,d )]

Pdh,d"Phh,d

_ 2 I 3 _ _
’2 Pdh,d(l 1+2 1Phh,d Pdh,d+Pdd,d(Pdh,d Phh,d)J

a42 -Pdd,d

. _ 2 _ 3
a —z (1 P + z Phh,d

13 (P

hd,d)Phh,d hd,d-Pdd,d)

=z(l-P d)-22(1-P 2

a23 hh,d+Pdd,d- Phd,d+Phd,dPhh,d)

3
+ z (1_Phh,d)(Pdd,d-Phd,d)

252

2

=l-z(P 1 )+2 P d(l-P )

a33 dd,d+ "Phh,d dd, hh,d

= a22

2

a43=z(l-Pdd’d) - z (l'Pdd,d)(1'Phh,d)

2

a =z P

14 hh,dPhd,d

=zP -22P d(l-P

a24 hd,d hd, hh,d)

2 3

3 =2 Pdh,dphd,d + z Phd,d(Phh,d’Pdh,d)

34

2

=l-z(1-P (1

a44 hh,d) ’ z “Phd,d)(Pdh,d)

3
+ z (1’Phd,d)(Pdh,d‘Phh,d)

We want to examine this matrix for three cases.

I. First order Markov chain equivalent

Pdd,d=Phd,d’ Phh,d=Pdh,d

and Phh,d=cd(l—Pdd,d)/(l-Cd)

11' Pdd,d=Pdh,d‘ Phh,d=Phd,d and

Phh,d=cd(1’Pdd,d)/(1'Cd)

III. for Cd='5' Pdd,d=Phh,d; Pdh,d=Phd:d

and Pdh,d=l-Pdd,d

Unlike cases I and II, the relationships in case III apply

for only a single value of the concentration.

253

Case I:

D=(l-z)\l+z(Phh,d-Pdd,d)

=(l—z)(l—Z(Pdd,d_Phh,d))

and

a =l-zP -22(1

11 dd,d "Pdd,d)Phh,d

=22(1-P )(l-P

a21 hh,d dd,d)

a =z(l-P 2(1

31 hh,d)’Z ‘Phh,d)Pdd,d

_2_ _ _
"141’2 (1 Phh,d)(l Pdd,d)‘azi

2

=ZP ’2 Phh,ded,d

a12 hh,d

2

=l-z(P +1) + 2 P (1

a22 dd,d'Phh,d dd,d ’Phh,d)

2

a32=zphh,d‘z Phh,dpdd,d=a

12

=22P d(l—P

a42 hh, dd,d)

2

a =2 (l

13 ‘Pdd,d)Phh,d=a42

2

a =z(1-P )-z (1

23 dd,d -Phh,d_Pdd,d+Pdd,dPhh,d)

2

a33=l-z(l-Phh,d+Pdd,d)+z Pdd,d(l’Phh,d)=a33

. _ - _ 2 _ _ -
a43‘2” Pdd,d z (1 Pdd,d)(l Phh,d)‘az3

254

_. _2 —
a24‘2Pdd,d z Pdd,d(l Phh,d)
a =22P P =a
34 hh,d dd,d 14
a =l-z(l-P. )-22(l-P )P
44 hh,d dd,d hh,d

The expansion by partial fractions is tedious. For
all z Phh,d(1‘Pdd,d)+(1‘Pdd,d)(1'Phh,d) 1

 

 

 

 

example _ _ _

l D Phh,d Pdd,d .1 Pdd,d+Phh,d 1 z

+ Phh,d(1’Phh,d) 1 1
(1‘Pdd,d+Phh,d) 1’(Pdd,d'Phh,d1z (Pdd,d'Phh,d)

the inverse of this term is

(l (l-P )(l-P )

 

~9(a11) = Phh,d 'Pdd,d) 5 + dd,d hh,d
D Phh,d—Pdd,d “'0 1’Pdd,d+Phh,d

Phh
l-P

,d(1’Phh,d)

)n-l
dd,d+Phh,d

+

 

(Pdd,d'Phh,d

USing the relation between Phh,d and Pdd,d for case I

we have

(l_Pdd,d+Phh,d) = (1’Pdd,d)/(1‘Cd)

(l-Phh,d) = (l-2Cd+Cded,d)/(l-Cd)

(Pdd,d‘Phh,d) = (Pdd,d‘cd)/(1‘Cd)

therefore,
2

 

 

all Cd11-Pdd,d) l-2Cd+Cded,d
‘9‘ D 1 P -c 5 o + +
dd,d d n'
n-l
+ Cd11"2Cd+Cded,d) (Pdd,d Cd)
1-cd 1-cd

255

The inverse has prOpertieS

(all) _ 1 n=0
‘8‘ _

= (1‘Phh,d)

as required, since for no transition (n=0) a state must
transition into itself (éii) and n=1 give 3 again. We
can now write, the total inverse in matrix form

~ Cd (1- 833,4): " Cd Pdd,dL1-Pddr3) Cd ( rad”)? Cd?oa,d(\‘fia,d)

(‘ 'PMAX Hem-m) FAIR 1‘2Cd +6. 8,4,6) 41-8344) (FILM. 214,4) fimu- 2614418421)

 

‘1 0,0
P‘ Pddfii'Q ”'Pddfi (i-icdtcdflw) ‘Cd Pod“) U‘Pdd,d) (Odd/d (l-lCdi'QaJA) 931344804344)

. _ . - 4) 2
(I-Bn,4>(l-i<4+C3 844,4) C'r4(l"?<ld,cl)z '("EJ,<1)("Z‘1+C‘11>‘1‘1"9 Q“ “’0

1‘2Cd+Cded,d Cd(1‘Pdd,d) Cd(1'Pdd,d) Cded,d
+ 1—2cd+chdd’d Cd(1'Pdd,d) Cd(1'Pdd,d) Cded,d
1’2Cd+Cded,d Cd(l-Pdd,d) Cd(1‘Pdd,d) Cded,d
1‘2Cd+Cded,d Cd(1-Pdd’d) Cd(l-Pdd,d) Cded’d

Cd ( than?” a) C} ( ram (I) - C401 Pdd d) ‘Cd Pddml
W -——-—-——Kl _Cd I

 

€33.10 ”(P2984848 «am—3.1,» (hour-W) ("€018.48
+ . , ,

(‘g((-1C4+dedtd) CiU’Pddm1 —Cd(l~Pdd,¢|) 'Cd PAM
(“Q1

l-Cd

 

\ .. “-2ch Pam) 44018.48) Max:813) ((‘Cdfl’cidd

(3.100)

'256

Case II:

2

3
)+2 (Pdd,d’Phh,d)

— — - - -2 -
D-(l z)(l z(Pdd’d Phh,d) z (Pdd,d Phh,d

=(1-z)(l-z(P ))(1—22(P

dd,d-Phh,d dd,d—Phh,d)

 

 

=(l-z)(l-z(P ))(l+z/P

) x (l—z/P

dd,d—Phh,d dd,d'Phh,d dd,d-Phh,d)

and

=l-z(P )-22(l-P +z3(P

a11 dd,d hh,d)Pdd,d dd,d)(Pdd,d‘Phh,d)

=zz(l-P 1)

3 .
d)+z (Pdd,d Phh,d)(Pdd,d'

a21 dd,d)(1‘Phh,

_ _ 2 _-
a31‘2” Pdd,d)"z (1 Pdd,d)Pdd,d

=z2(l-P

a )2
41 dd,d

_ _ _ 2
a12—2(Phh,d) z (Phh,d)(Pdd,d)

=l-z(1+P ) + 22P

a22 dd,d-Phh,d dd,d(1'Phh,d)

=zP -22

3
a32 dd,d (Pdd,d Phh,d+Pdd,d)+z (Pdd,d)(Pdd,d Phh,d)

2

3
Pdd,d )+z (P

=2 (l—Pdd,d dd,d-Phh,d)(Pdd,d-l)

_2 -»3 _
5113‘2 (1 Phh,d)Phh,d+z (Phh,d)(Phh,d Pdd,d)

hh,d

2

2 3 '
1‘2 (l-3Phh,d+Phh,d+Pdd,d)+z (1‘Phh,d)(Pdd,d‘Phh,d)

3=z(l-P

a2

a33=a22

257

2
1‘2 (1‘Pdd,d)(1“Phh,d)

l

Phh,d( "Phh,d)

_2 3 _
(Pdd,d)Phh,d+z Phh,d(Phh,d Pdd,d)

=l-z(l-P )-22(1-P +23(1-P

a44 hh,d hh,d)Pdd,d hh,d)(Pdd,d"Phh,d)

For the term

a.. = q+rz+szz+tz3

1]

the partial fraction expansion is

 

 

 

where d = P - P

(r+s+t+q)/(1-d)2

 

A =
B =(qd3+rd2+sd+t)/d(l—d)2
C = _(t+rd)+/d(s+dg)
2d(1-/d")2
D' = (/d'(s+dq)-(t+rd))/2d(1+/d‘)2

Therefore, for case II, the inverse transform gives

258

. 2 2
(1—cd) cd(1-cd) cd(1—cd) Cd
2 2
P“ _ (1-cd) cd(1—cd) cd(1—cd) Cd
” 2. 2
(1—cd) cd(1-cd) cd(1—cd) cd
. 2 2
(1-cd) cd(1—cd) cd(1-cd) cd
2 2 2 2
Cd 'Cd “Cd Cd

n
+(Pdd,d-cd> -Cd(l—Cd) cd(1-cd) cd(1—cd) -cd(1-cd)
-c

 

 

 

 

 

 

 

 

d -cd(1-cd) cd(1-cd) cd(1-cd) -cd(1-cd)
2 2 2 2
(1-cd) -(1-cd) -(1-cd) (1-cd)
_ n/2
+ 1 pdd,d Cd 1 + (-1)n x
2 1-cd (l-(Pdd'd-Cd)%)2 (1+(Pdd'd-Cd)%)2
I‘Cd I-Ed
o Cd(1’Pdd,d’ Cd(1'Pdd,d) 0
I-cd 1-cd
l-P l-ZCd+Pdde —2(l-2Cd+Cded'd) -Cd(1-Pdd,d)
dd,d 1-cd i-cd 1—cd
(1_P ) _2P l_zcd+Pdd,d Cd(l-Pdd,d)
dd,d dd,d 1-c 1-c
d d
o l-P -(1 o

“Pdd,d)

259

 

 

 

 

 

 

 

 

 

n- _.
P .. -C .. 2, 2 . n
1 dd d d '7‘" l (-l)
+ 2‘ T-CL__ P “‘-‘.C‘ 157 " ‘8 P w —c 15 T
d (1—( dd d d) ) (1+( dd,d d 1
~ - T1———c - I-C )
. .2. d d J
‘ . _ _ 2. _P )2
C (...? I): C-dedéU-Pd”) -L_d(‘ Pddfi)“ JQT'CdPQJ) -C-6 (‘ dd,d
dI-de )‘Cd (‘Cd 1 L1 “411$
. . 1 2 3 - P )
-(I-P ,Xi-KNQPM) _ Qi“?dd,d) -:~29+ch..«,4 41.11.; ext-'8) Mum a
z + Pan d'Cd> '2? AI CdU‘Pdfhcl) -C_d 84,3 U‘?d4)d)
814,3 04344.4) , P443! l-Cd 11' I"Cd Pit-c.)
q? ( 8 (07¢...
_ t—qu dd )‘ ad
-.-...)2 .. 3.1m.» w sot/~42... =.2—.—
(8.101)
For Case III, we have
D = (1-z)‘(1-z3(1-2Pdd d)2)
I

= (l—z)(l-a2/3z)(l¥a2/3ein/3z)(1+a2/39-ifl/3z)

where a = l-Zpdd,d

2

3
all=l-2Pdd,d_z Pdd,d(1"1’dd,d’~+2 (zpdd,d"1’(1‘Pdd,d’

260

_ 2 3 _
“‘21‘z Pdd,d+2 Pdd,d(l 2Pdd,d)
_ 2 2
a31‘zpdd,d z Pdd,d
a =22P (l-P )
41 dd,d dd,d
_ _ _ 2 2
a12‘a31‘zpdd,d z Pdd,d
a =l-z+22P (l-P )
22 dd,d dd,d
a =z(l-P )-22(1—P -P2 )+23P (l-2P )
.32 dd,d — dd,d dd,d dd,d dd,d
_ 2 _ 2 3 _ _
"142‘2 (1 Pdd,d> +2 (1.2Pdd,d)(Pdd,d 1)
_ 2 2 3 _ _
"113‘z Pdd,d+z (1 2Pdd,d)Pdd,d‘a21
a -zP +22(l-3P +P2 )+z3(1-P )(2P -1)
23 dd,d dd,d dd,d dd,d dd,d
a33=a22
_ _ _ 2 _ 2
a43‘2” Pdd,d) z (1 Pdd,d)
a =22P (l-P )=a
14 dd,d dd,d 41
_ 2 _ _ 2 _ 2_
5124“2 (1 Pdd,d) z (1 Pdd,d) ‘a43
_ 2 2 3 _ _
5134‘z (1‘Pdd,d) z (1 Pdd,d)(l 2Pdd,d)
a =l-z(l-P )—22P (l—P )+z3P (l—ZP )
44 dd,d dd,d dd,d dd,d

dd,d

for the term aij=q+rz+szz+tz3 the partial fraction ex-

a.
pansion for the term -%;-is

261

 

 

 

 

A+IB+ C +—D-
l-z l-dz l+d iw/B l+de—in/3z
/3
h d = -
w ere (l 2Pdd,d) and
A = q+r+s+t
4Pdd,d(I_Fdd,d1
B = t+ds+d2r+d3q
‘2
-3d (l-d)
c = [}t+d3q)((i-d)(d+2)-(1-d)(1+2d)e1“/3)+rd2(-(1-d)2+(1-d)
(d+2)e1“/3)
+ sdt-(i- d)2e 11/3- (1-d)(2d+1)]/-3d2(Ld3)(e11/3-e‘11/3)

D = +C* (since d is real)

The inverse transform in rather symbolic form

is
1 1 1 1
n _ 1 l 1 1 1
P“:
1 1 1 1
1 1 1 1
+(l—2P )3§'(3 )+(-(1-2P )2/3e 11/3) n (c )
dd,d ij dd, d ij
%/ -ifi/3) n
+(—(1-2Pdd d) (c*j ) (3.102)

262

where we note that the last two terms can be combined as

, 2n ‘ inn —iwn
(—1)11(1-2Pdd’d)—3— (CijeT + C3113 e—r)
2n .
=(-l)n(l-2Pdd'd)-—§ ZRe (cije11m/3) (3.103)

Clearly, the long run distributions of constitutents,
C is completely random, C =(.5)(l-.5)=.25. Also Pn
gm £,m
is always real as we require since imaginary probabilities

are meaningless.

Statistical Analysis of Generated Markov Chains

In this section, we will statistically examine the
Markov chains we generate to insure the computational ac-
curacy of the computer programs which employ a random
number generator. For Simplicity, we present the error
analysis only for first order Markov chains since Pn is
quite complicated for the second order Markov chain. In
addition, we will only examine Pd,d(n) since the other
probabilities give similar results. Using Equations 3.106
and 3.107in text, we will use a 99% confidence level C,

expressed as a certain number of standard deviations,

C = a0 (8.104)

263

the maximum relative error we will tolerate is

(n+C)-u) C 0
u u (11)

From Equations 3.106 and 3.107

[(N P (mu—P (nm.15
E = a dd,d d,d

 

 

NPd,d(n1
_ 1'
or E — a/%(FE—ETHT - 1) _ (8.105)

The 99% confidence limit gives a =2.58. Tables 8.1, 8.2,
8.3, and 8.4 are for Pd,d=‘l' Cd=‘5' and N=1000, 10000,
100000, and 1000000 atom chains respectively. For each
case, the n=1 case provides a test of the statistical
accuracy of the random number generator.

Table B.1.--Statistic error analysiscmfa.chain With N=1000,
Cd=.5, Pd d=0.1 compared to 99% confidence
I

limit error of Equation 8. 105

 

 

 

Pd,d(n) Relative Error
n
Calculated Experimental Acceptable Experimental
(Eq. 8.105)

1 0.1 0.09419 .245 .058

2 0.82 0.82129 .0382 .0016
3 0.244 0.24346 .144 .0022
4 0.7048 0.70423 .0528 .0008
5 0.33616 0.32661 .115 .0284
6 0.63107 0.63306 .0624 .00315
7 0.39514 0.40404 .101 .0225
8 0.58389 0.57374 .0689 .0173
9 0.43289 0.44332 .0934 .0241
10 . 0-55369 . 0.55061 .0732 .0056

 

264

Table B.2.--Statistic error analysis offia chain with N=100000,
Cd='5' Pd d=0'1 compared to the 99% confidence
I

limit error of Equation 81105

 

 

 

 

 

 

 

Pd,d(n) ‘ Relative Error
n
Calculated Experimental Acceptable Experimental
1 0.1 ' 0.10112 .0774 .0112
2 0.82 .81418 .0121 .0071
3 0.244 .24955 .0454 .0227
4 0.7048 .69678 .0167 .0114
5 0.33616 .34415 .0363 .0238
6 0.63107 .62159 .0197 .0150
7 0.39514 ‘ .40313 .0319 .0202
8 0.58389 .57836 .0218 .0095
9 0.43289 .43507 .0295 .0050
10 0.55369 .55322 .0232 .0008
Table B.3.--Statistical error analysis of a chain with
N=100,000, Cd=.5, Pd d=0.l compared to the
99% confidence limit error of Equation 3,105
Pd,d(n) Relative Error
n
Calculated Experimental Acceptable Experimental
1 0.1 0.09818 .0245 .0182
2 0.82 .82278 .00382 .00339
3 0.244 .24144 .0144 .0105
4 0.7048 .7078? .00528 .00436
5 0.33616 .33337 .0115 .0083
6 0.63107 .63433‘ .00624 .00517
7 0.39514 .39190 .0101 .0082
8- 0.58389 .58810 .00689 .00721
9 0.43289 .42916 .00934 .00862
10 0.55369 .55811 .00732 .00798

 

265

Table 8.4.--Statistical error analysis of a chain with
N=l,000,000, Cd='5’ Pd d=0'1 compared to the
I

99% confidence limit error of Equation FL105

 

 

 

Pd,d(n) Relative Error
n
Calculated Experimental Acceptable Experimental

1 0.1 0.09950 .00774 .005

2 0.82 0.81999 .00121 .00001

3 0.244 .24392 .00454 .00033

4 0.7048 .70457 .00167 .00033

5 0.33616 .33562 .00363 .00161
10 0.55369 .55481 .00232 .00202

 

The table Show that the computer routines are clearly
generating first order Markov chains. The only two vio-
lations which occur at n=8,10 for N¥100,000, are out of
acceptable error limit by less than .1%, which we do not
Consider significant. With confidence that the chain
generation is correct, we can rearrange Equation 8.105
to give another important statistic tool. We might like

to generate a chain with C =.5 and P -0.1; with a 99%

d d,d—
confidence that the relative errors will be no greater

than 1%. Solving Equation 8.105 for N we have

2
_ a l
E d,d
P =0.l is the most severe constraint of any P (n) with
d,d d,d

=.01 and a=2.58, we have

266

N 1 599,076 2 600,000 atoms

We must generate quite long chains to insure a high degree
of statistic accuracy in the chains.

From this analysis, we note that, for a given
length chain, the Cd='5’ random chain will contain the
smallest overall statistical errors of any binary chain.
In this case Pgm(n)=°5 for all n, whereas any other con-
centration and Short-range order will give some
P2,m(n)<.5 (£,m=h or d) and therefore insures higher
errors. Very low or high concentration of a given con-
stitutent as well as a high degree of correlation will
greatly increase overall statistical errors.

Although we have not presented a statistical
analysis for other Cd and Pd,d and for the second order

.Markov chain, computer studies have been performed to

verify the validity of these computer programs.

 

APPENDIX C

SHORT-RANGE ORDER PARAMETERS

267

X-ray and neutron scattering have provided solid
state experimentalists with a powerful technique for
examining the structure of solids. The differential
scattering cross-section per unit solid-angle %% in the

Born approximation is given by

Eff _ E— 211 [81091 )'r V(r)drI2 ((2.1)

where k is the incident wave vector
+ ’ I
k is the reflected wave vector

u is the reduced mass in the center of mass
system

and V(r) is the interaction between the incident
wave and the scattering center.

The interaction V(r) is the electron density for x-ray
scattering, and V(r) is the nuclear density for neutron
scattering. In the case of elastic scattering, we have
k=k’. We recall that the Born approximation is valid only
when the scattering center is quite localized and has a
sufficiently small scattering strength.44

For the perfect crystal, the scattering potential

V(r) will be periodic or translationally invariant.

V('£+E) = w?) (c.2)
—> -+ —> -+ .
where 2 = £1al+£2a2+23a3, 21,22,13 are integers,

and 31,32,33 are the basis vectors of the

primitive lattice cell.

268

269

.+
Rewritting V(r) as

.+->
+ o
V(r) = ) V+elG 1

E G

then
V(r+i) = V(r) =
or

+—>

6'2 = 23n, n=integer

(C.3)

(c.4)

6 is the reciprocal lattice vector of the crystal defined

by Equation (C.4).

Using Equation (C.3) in Equation (C.l), we get

CL
O
W

_ ’ I1 2

1f i(E-E’+E)-§d+lz
47? 69 r

Cl.
20
”I

_B.2V
2“)
’fi G

’ 3 -> +
E— |i%%l— 3% 2 V+0(k-k’+G)|2
.n E G

or the cross section vanishes unless

, + ->
= k+G

3'4

(C.5)

(C.6)

Equation (C.6) is the Bragg reflection law for the perfect

crystal.

For the disordered lattice, we must return to

Equation (C.l). The integral can be usefully broken into

three parts

1. a sum over the lattice sites I

o + i
2. a sum over the baSis r3.

2

270

3. an integral over the rest of space 2”

+ + + +
-i Ak ° r +rT+r”)
( ) ( Q 3

fe1(k‘k 1'rV(r)d§ = ) [dr”e V(r”)
i,j
+,+
where 1k = k -k
-1Zk-E£ -1Zk-§. ~1Zk-r”
= E e 2e fe V(r”)dr”
1 3
'1k I
-i °r
= Z f(Kk)e R (C°7)
2
where we define the structure factor by
y -1Xk-Ef -1Xk-r” +
f(Ak) = Ze Jfdr”e V(r”)
j
The differential scattering cross section is therefore
proportional to
iZkoE
2|2 (c.8)

Isl) f£(Zk)e
2

This is true for the perfect lattice as well as
the disordered one, the only difference is that all
f2(Ak) are equal for the perfect lattice and can be
different for the disordered lattice. For the binary
atomic system a lattice point can have a structure factor
fd(q) or fh(q) depending on whether the Site is occupied
by a defect or a host mass. Rewritting Equation (c.8)

we have

271

134} .-; )
Id 2 f£(q)f£’(q)e 2 R

12’ + y +
iq-(r ,-r)
= 2 ffi(q)e i 2
11’ 1*-(} —E )
+ X [f (q)f (q)-f2(q)]eq 9" ‘1
1=d h d h
£’=h

+ + + ia(;Q’—EQ)
+ Z [rh<q)fd<q)-rfi<q)1a

i3. (E ’--E )
[f§(q)-fi(q)]e 2 2 (C.9)
R

+
H H54

d
2 d

We can further simplify this equation by the introduction

of some new quantities
Nd
C = ——- The concentration of defeccswhere Nd is

d N
the number of defects in the system and N
is the total numbir of particles.
1 if atom at Site is a defect

0 if atom at site I is a host

ed<1) =

h + 1 if atom at site E is a host
8 (2) = 0 if atom at site I is a defect

Lets examine the function

fr“

o1'j(f) = 261(I)ej(E+I) (c.10)
2

i
in words, pl'3(i) is the number of times there is an atom
of kind 1, separated from an atom of kind j by a distance

i in the whole crystal divided by the number of

272

atoms of type i in the crystal. Clearly, pl'J(L) is
defined only for lattice points.

Since 0h(l) + 8d(i) =1 (C.ll)

we have the following relationships

ph'd(E) + oh'h(L) = 1 (c.12)
od'd(i) + od'h(L) = 1 (c.13)
Next we look at the following
03'1(E) = $7 2 ej(l)ei(i+l)
j l
= $3.; 91(E)ej(l-E) (C.l4)

Comparing Equation (C.l4) with Equation (C.lO), we get

the important result

Nipl'j(L)=ijj'l(-L) .(C.15)

If we have an isotropic crystal, then

i,j _ j,i
Nip (L) — Njo (L)

which, if we divide each Side N, gives for the binary

chain
Cdpd’h(L) = c (L) (C.lG)

pl’J(L) is also called the conditional pair correlation

function between atoms i and j separated by a distance L.

273

By using the 8 function we can eliminate the

restricted sums in Equation (C.9) and we get

h +,
16 ) [r§(q)+ed(2>e (2 )(fh(q)fd(q)—ffi(q))
n,
+eh(i)ed(i’)(fh(q)fd(q)-ffi(q))

+ + iq°(r ~r »)
+ed<2>ed(2’)(f§(q)-f§(q)le 1 1

_ + + + —+ ,
if we take r -r ,=L the sum now goes over L and I .

1 2
I=(I’+L)pwe get

Ia Z [Z f§(q) + X ed(24£)eh(i’)(fh(q)£d(q)-rfi(q))
+ +, 2’
L R

+ 2 eh(E’+E)ed(2’)(fh(q)fd(q)—ffi(q))
2,

...}

+ 2 ed(2’+i)ed(l’)(f§<q)-ffi(q))1e1q°1
2’

or

re ){Nf§(q)+N oh'd(L)[rh(q)fd(q)-ffi(q)J

L h

d,
+Ndo h<L><fh(q)fd(q)-ffi(q))

+

d,d 2 '+~
+Ndo (L)<f§(q)-fh(q))}elq 1

With

where Equation (C.lO) has been employed. Using Equation

(C.l6) with Equation (C.l3) and dividing by N, we get

In zeiq'1(ffi(q)+zcd(l-0d'd(R))(fh(q)fd(q)-ffi(q))
2

+Cd(f§(q)-ffi(q))od'd(£))

(C.l7)

274

These pair correlations functions can be expressed in terms

of the Warren-Cowley Short-range order parameters a as

d,d _ _
o (1) — Cd+(1 Cd)a (c.18)

0,2

Equation (C.l7) becomes with rearrangement

.+.E 1' 2 2
Id Ee1q [(fh(q)(l-Cd)+fd(q)Cd) +Cd(l-Cd)ao'l[fd(q)-fh(q)J

(C.l9)

Relationship of the Pair Correlation
Function to the Markov Probabilities

 

 

Examining Equation (C.lO) in the limit as Ni goes
to infinity with Ni/N held constant, pi’j(L) is the
conditional probability that starting at state i,L sites
away the site will be occupied by a state j. Specifically,
pd'd(n) is the conditional probability that starting at
a defect, n Sites away the site will be occupied by a defect.
The relationship between the pair correlation functions
and the first order Markov chain probabilities can now

be simply expressed. First, to start at a defect, the

initial unconditional probability vector is
p(O) = [ph(0)=o. pd(0)=1] ' (0.20)
Then,

od'd(n) = pa(|n|) (c.21)

275

where p(n) = p(O)Pn-

using Equation (8.96) for Pn, we get

l-C C

[ph(n>.pd(n)l = [0,11 ‘1 ‘1
1-cd Cd
' c
+ [0,1] d
—(l-Cd)
and
pd(ln|) = od’d(n)=Cd+(l-Cd)( 1_C
or
d,d
o (n) = Pd,d(|nl)

Comparing this equation with Equation (C.

P -C
’ l-Cd 01
P -C
where d =( d’d d)
0,1
l-Cd

Pd,d'

"Cd (Pd,d’cd)n
(l-Cd) 1-cd
c )
d )1“. (c.22)
d
(C.23)

18) we see

(0.24)

For the second order Markov chain, the pair correla-

tion functions are not as easily computed.

pd'd(n), the initial state must be taken

To calculate

as a linear

combination of the stateSphdandpdd normalized to unity

and is given as

‘ 276
1°) = [phh=0’phd=chd/Cd’ pdh=0’pdd=cdd/Cd1 1C°251
[OI (l-P

)/(1+P P

dd,d hd,d"Pdd,d)'°' hd,d/(1+Phd,d‘1’dd,d)J

where we employ Equations (3.96 ) and (3.97 ).

Then,

od’dm) = pdd(lnl)+pnd(lnl)

where again p(n) = p(0)Pn

Next, we will examine pd’d(n) for the three special cases

of the second order Markov chain described in Appendix 8.

Case 1: First order Markov chain equivalent

Pdd,d=Phd,d’ Phh,d=Pdh,d and
Phh,d=cd(1'Pdd,d)/1'Cd)

For this case, the initial unconditional probability

vector is

P(O) = [0, (l 0, (C.26)

”Pdd,d)' Pdd,d1

Using Equation (8.100) for Pn

pdd (n): -6n,0(1-Pdd,d) (Pdd,d(l-2Cd+cdpdd,d)

+5 (l—P 2

n,0(‘Pdd,d)‘Cd) dd,d)

+Cd(P )(l-P

dd,d dd,d+Pdd,d)

P -C

dd,d d)n-1
l-C

d

+1 (1‘Cd) (Pdd,d)(l-Pdd,d+Pdd,d)

 

277

Pdd1n1 = -6n,0 (1‘Pdd,d1(1'cd1pdd,d

+cd Pdd’d+(Pdi_g;Cd)n-l(l-Cd)Pdd’d (c.27)
Phd(n) = 0n'o(l-Pdd'd)(Pdd'd)(l—Cd)

+Cd(l-Pdd’d)+(39%f%ésg)n 11-Cd)(l_Pdd,d1 (C.28)
Therefore
od'dm) = pdddnl) + phddnl) = cd+(P—8‘1i};—§—;—C91“"1(Pddid-cd)
od'd(n)==c2d+(1-c&(43}£L—3-)1“1 (c.29)

Cd

For the first-order-equivalent second order chain

the Markov transition probabilities P were chosen to

lj,k
be independent of the first atom 1. They therefore

Simulate the first order chain; thus Pdd,d=Phd,d=Pd,d in

the notation of the first order chain. Using this fact we

note that the equivalence between this equation and

Equation (C.22) demonstrates that the calculational method
d d

of finding 0 ’ (n) for the Second order Markov chain is

correct.

Case 2. Pdd d=Pdh d:Plh'd=Phd d and

Phh, d= Cd11 Pdd,d,1/(1 Cd 1

278

For this case the initial unconditional probability

vector is

 

 

 

p(O) = [0, l-Cd,0,cd] (C.30)
Using Equation (8.101)for P“, we can write
Pdd(n)=cd2‘cd(l—Pdd d)[3 2111/2 (—-—-1 2 + ————(‘11n2)
’ (l-/§) (l+/§)
aux-1V2 _ (-1)n )1
2
(1-/§)2 (1+/E)
_ (Pdd’d-Cd)
where a — l-C
d
and
phd(n) = cd(1-cd)+(l-cd)(1+Pdd d)[3a“/2( 1 2 +"11n 2)]
' (l-/§) (1+/E)
(n-D/Z 1 (-l)n
+(C -2P +C P )[%a ( - ]
d dd,d d dd,d (1_/§f2 (1+/3)2

Therefore, after some simplification

pd, |n|/2 1 (- -l)1n1)
d(n) =C d+(l- 2C )[ka ( + )]
d 1'1 (1-(3) (1+(3)1

a(|n|-1)/2 1 (-1)1“1
+(C -P E______._ _______]
d dd, d1 (1_/§)2 (1+/3)2

pd, Pdd, d Cd)|n|/2

Cd

d(n) =c d+(1- c d)( s[l+(-1)|“1] (c.31)

Comparing this equation with (c.18), we see

279

O‘o,22+1 = 0

do = (-—§§%——9)l£| (c.32)

The Fourier transform of Equation (C.32) is

eiqlaa‘ = E eiq22a( Pdd, d Cd I2]

0,2 1— —Cd

a(q)

£=~m

CD

1+2 E cos(2q£a)(a2)£

where a2 = (Pdd,d-Cd)/(l-Cd)

Therefore,

2
. l-a2
a(q) = " 7? (C.33)
1-2azcos(2qa)+a2

 

=l-P and C =.5

case 3‘ Pdd,d‘Phh,d’Phd,d=Pdh,d dd,d d

For this case, the initial uncondition probability

' vector is

9(0) = [OI l5: 0: 15] (C.34)

Since in Equation (3.102) we have not explicitly written
out Pn, we will solve explicitly for the elements we need.
Since p(n) = p (O) Pn and we require only pdd(n) and phd(n),
we need to solve for only four elements of the matrix
(2,2), (2,4), (4,2) and (4,4).

If for convenience we let an element of Pn be

given by Qi 3, then

280

Pdd(n) = 8 (042+Q44)

The pair correlation function is

d,d _
D (n) — %(022+QZ4+Q42+Q44)

Now, instead of writing out each Qij needed, an

examination of the Bij and Cij coefficients in Equation

(8.102) shows a running sum of the four Qij will greatly

simplify the results. In fact, we get

Z(Bij) = l/3
and
X(Cij) = 1/3 ei”“/3
Finally
od'd<n) = %+%[l/3(l-ZPdd,d)2n/3(l+2(-l)ncos(%£))J

We next examine three cases for Equation (C.35)

Case 1: od'd(3n—2)=s+a(1/3(1-2pdd'dfl6n"®/3

but

cos(nn-%1) cos(nn)cos(%1) since sin(nn)=0

<-1)“<-x)=k(-1)“*1

therefore

(1+2(-1)3n

(C.35)

)

2 fi(3n-2)
cos(-—§————

281

2<-1)3“‘24(-1>“*1 = (—1>4“(-1>=-1
and
pd’d(3n-2) = 3 (C.36a)
Case 2: pd'd(3n-1)=g+g[1/3(1-2pdd’df6n’m/3(1+2(-1)3“‘lcos(313%:11)
again
cos(nn— %) = cosnn cos-g = 35(-l)n
and
pd'd(3n-l) = % (C.36b)
Case 3: pd'd(3n) = %+%(l/3(l-2Pdd’d)2n(l+2(-l)3ncos(nn))
since (—l)3ncos(nn) = (-l)4n =1
pd'd(3n) = 5+3(l-2Pdd'd)2n (C.36c)
Comparing these equations with Equation (c.18) for Cd=%,
we see
O‘o,3;z-2 = 0
00,32-1 = 0 (C.37)
O‘o,352. = (l-ZPdd,d)2|£|

If we take the Fourier transformation, we have

282

2lfil

CD . '9‘ CD 3. 2‘
a(q) Z elq adopfi = 12 e lq a(l-Pddpd)
2=-m =‘m

1+2 2 cos(3q2a)ag
2:1 3

l-ag

0t(q)= (C.38)

2
l-2a3cos3qa+a3

 

where a = (l

2
3 ‘Pdd,d)

APPENDIX D

NUMERICAL COMPUTATION OF

EIGENVECTORS OF CHAINS

283

Given an eigenvalue of a matrix one can cal-
culate the corresponding eigenvector. By the method of
bisection described in Appendix A, we can compute the
eigenvalue of the linear chain to any desired accuracy.
For a computed eigenvalue, A, the computation of the
corresponding eigenvector seems at first almost trival.

Using Equation A.l, we can write

_ 1
X. — [(A ai) Xi bixi

] (D.l)
1+1 bi+l

 

-l

where Xi = Vmi Ui (Ui is the displacement of the ith atom)

b'+l = -y/Vm.m.

1 1 1+1

and a. = 21
mi

Equation 0.1 requires initial conditions to be complete.
For fixed boundary conditions, Xo=0. Arbitrarily, we take
Xl=l, since X1=0 would give all Xi=0' Then, Equation D.1
allows us to compute all Xi's and correspondingly the Ui's.

In terms of the leading principle minors

_ _ r-l _
xr — ( 1) Pr_l(A)/b2b3...br (r—2,...,n) (0.2)

Unfortunately eigenvectors computed in this manner can be

completely wrong. The procedure will in fact work when

the exact eigenvalue is given and exact computations are

performed. However, although we compute on eigenvalue

284

285

to 8 significant digits, it is not exact. In fact, the
computation of XN+1 will not usually give 0 as specified

by the fixed boundary condition but

bN+lXN+l = 5 = ‘bNXn—l + (A-aN)XN#O (D'3)

In the matrix form of Equation A.l, we have

(B-A_I_) 33: 6e (D.4)

th

where en is the n column of the identity matrix. For

simplicity, we can renormalize X to give

(13.-u.) E = em

or

1

en (D.5)

5 = (a-Ay’

Next for X1>A2>A3>>AN, the approximate eigenvalue is

almost Ak or (A-Ak) "small" and (A-Ai)(i#k) "not small".
Let 21' 22""’ZN be the exact set of eigenvectors of E

corresponding to Al, A2, ..., A We can expand the

N'

vector in in terms of these eigenvectors

N
En = .Z Yiyi (D.6)
i=1
N 2
where X y. = l.
. 1
l=l

286

Rewriting Equation D.5, we have

N Yiyi YR! z A A
x = X r—J = x—J + Y.v./( .-) (v.7)
" i=1 i k 151k 1’1 1

For g to be a good approximation to 2k we require

Yk Y1

>> ' k . ,
fl}; Wi- (15¢ ) (D 3)
Often, Ak-A>>yk and Equation D.8 is not fulfilled.

Solving Equation D.6 for Yk we have
= V e (D.9)

Therefore, any time, the last (Nth) component of the
eigenvector is >>l,the eigenvector g calculated by this
procedure will usually be incorrect. The procedure would
work if we computed the eigenvalue and carried all com-
putations to greater accuracy by five or six digits than

the value of the Nth component of V
-20

. T
k' Given ykyk=l' and
(yk)N=lO , then computations to 26-30 digits would give
accurate results. In practice, this method fails on

disordered linear chains working to 18 digit accuracy for

chains of length N>30.

If we knew apriori that the rth component of 2k

was not small, we could take Xr=l instead of Xl=l, and

reiterate the Equation D.l to a correct solution.

287

A more satisfactory method of computing eigen-
vectors is that of inverse reiteration. First, consider

the set of equations

(33:13) = g (D.10)

llx

where C is an arbitrary vector normalized to one. We can,
as before, write 9 in terms of the exact eigenvectors y

of g,
N
E = Z y.V. (D.ll)

For A close to Ak' Equation D.7 shows that g is much richer

in the yk than is 2, namely by a factor Xéxf >> 1, since

Equation D.ll gives
= V C (D.12)
Next, we solve the equation
(E'X£)X.= g (D.13)

y can similarly be expanded in terms of the eigenvectors

of

Ilw

y}: 2 diyi/(A-Ai) (v.14)
l

where

5 _ VT x - Yk _ (D 15)
k‘-k-‘x—-T];'r:r; -

288

Therefore,

1 = (3:9) vk/(X-Ak)2 + X

i¥k aiyi/(A-Ai) (B.16)

This process can be reiterated to any desired ac-

8

curacy for V with

_kO
12, the first reiteration gives Yk/(>\->\k)=10"4 or

If, for example, we take (X—Ak)=10-
Yk=1o’
E is very deficient in Vk but the second reiteration
gives Ok/(X-Xk)2=104 or y is quite a good approximation
to Yk' As long as g is not orthogonal to 2k the method
will always work given (X-Ak) "small" compared to
(K-Xi)(i#k).

Applying this approach requires the inversion of
the matrix (E—AI). Since this matrix is tridiagonal,
the method of tfiangularization, also called Gaussian

elimination with partial pivoting, is the most efficient.

We look at the elements in Equation D.10 as follows

C1
a -A b2 C
b a -A b O C
2 2 3 3 (0.17)
b3 a3-A b4 9

 

 

289

We start by comparing the elments in the first

column,

1. if Ibzl > ai-A, we exchange the elements of
rows one and two; otherwise, we do nothing. Denoting
the nonzero elements in row one by U1, V1, wl, and d1

and those in row two by X2, y2, 22, and dé, we have

C'.
I

1 ’ bz' V1 = O‘2"A' w1 = b3' d1 = C2

9!
:3
Q.
X
ll

2 al-A' y2 = b2' 22 = 0' d2 = C1

for Ibzl > (al-A), or

ll
0‘

1 al-A, V

- = — = I =
and X2 - b2, y2 (a A, 22 b3, d2 C2,
otherwise,

2. we compute

2 = Xz/Ul and replace X2 by zero.

3. compute

"U
II

2 Yz‘mzvl
q2 = zz’mzwl

I
C2 ‘ dz‘mzdl

290

then, replace
Y2 by P2:

22 by q2

I I
and d2 by C2

Matrix D.l7 looks like

 

/ ”1 V1 w1 d1
m2 ‘ 0 .P2 q2 0 C5
b3 a3_A b4 , C3 (D.18)
:
0 bn ag-A Cn

 

For the rth step, we proceed as follows

1. If lbr+ll>Pr’ we interchange the r and r+1

rows. If lbr+ll>Pr' we have Ur=br+l' Vr=ar+l-x'
wr=br+2' dr=Cr+l' and Xr+l=Pr' yr+l=qr' zr+l=0'
, _ I
dr+1-Cr
otherwise
= = = = '
Ur Pr' Ur qr, wr 0, dr C
_ -- ... = ' 2:
and r+1"br+l' yr+1—ar+l A' zr+1 br+2' dr+1 Cr+1

291

2. We compute mr+l=xr+1/Ur and replace Xr+1

by zero.

3. Compute

Pr+1 “ yr+1'mr+1vr

=2

qr+1 r+1-mr+1wr

I I _
Cr+1 dr+1 mr+ldr

and replace

by P

Yr+l r+1

zr+1 by qr+1

l !
dr+1 by Cr+1

th

The r principle minor of (EfAI) is, (also given

by Equation A.3)

k.
1

Pin) = (-l) U U U ”'U.

1 2 3 1-1Pi (D.19)

where ki is the total number of row interchanges occurring

to the end of (i-l) steps. Therefore, the process of tri-

angularization could also be used to determine eigenvalues.
We can now solve for the eigenvector g given by

Equation D.1O

 

292

xN = Cfi/PN
xN—1 = (dN—l’XNVN-1)/UN-1 (0'20)
Xi = (di-Xi+lVi-Xi+2wi)/Ui

(13191-2)

If A is the exact eigenvalue PN would be zero. Usually,

it is not zero, however, if it is we need only to take

<<
PN 1.

On the first reiteration, since C, is an arbitrary

initial vector, we can pick d saving a calculation of d

from C. We take all di=_£' This choice is not necessarily

the best for all cases but it works in every case we have

encountered. Once we have (EfAI) in triangular form, it

does not have to be recalculated on successive reiterations.
We therefore only have to compute d from the initial

vector E. From step 1 in the reiteration we notice that

wr=0 if no exchange took place in the rth step; otherwise,

wr¢0. Therefore, we have

1. If wr=0, we compute dr=xr and dr+l=xr+l-mr+ldr'

2. If wrfo, we interchange the elements Xr and

X then calculate d =X and
r r+1

r+l'

-X -m d

dr+1_ r ‘r+1 r’

Then, we can reapply Equation D.20.

293

Once, we get the eigenvector of (EfAI), we must

finally return to the displacements

ui = xi/Jr‘n‘i' (13.21)

and renormalize U to unity.

 

APPENDIX E

GREEN'S FUNCTION FOR THE MONATOMIC CHAIN

294

We will examine the Green's function of a monatomic
linear chain in the harmonic approximation in two ways.
First, we will explicitly calculate the general Green's
function g(£,£’;w) from the k space transformation given
by Equation (5.12). Next, since diagonal elements or
near diagonal elements of g(£,2’,w2) are all that are
usually required in calculation of thermodynamic quantities,
we will examine a simple method of calculating these from
the inverse Green function. Since this method does not
require a k space transformation, it will be useful for
calculating the impurity mode frequencies of defect
clusters in a host medium and for calculating the density
of states of an n site periodic system. These applica—
tions will be described in the next two appendices.

Using Equation (1.10) for the monatomic lattice,

we can rewrite Equation (5.12) as

 

6 ,
,. _ k,k
g(klk ,(U) " 2 2 . 2 ka (E01)
w -w Sln (——)
m 2
where w2 = 31
m m
The transformation into real space gives
ikLa
, 2 _ l e
9W“ N“ ’ mm 1% 2 2 Tka (3'2)

w -wmsin (§—)

295

296

where L 2-2’ and

all (n), n=0,1,2.. ;N-l

For N , the sum over k can be converted to an

integral,

. Na 2% .
2* 7; / dk . (E.3)
k 0

Therefore,

 

 

 

 

21
'kLa
2 _ a a el dk
9‘L'M‘mf 7 .21.. “3"”
0 w -meln 7—
Let 2 = eilka for L(z)0 (E.5)
LI
2 2 2l dz
g(L;w ) =1 . ¢ +;- (E.6)
inm w222+(4w2-2w2)z+w2
.m m m
where the integral is around the unit circle
(clockwise (-), counterclock wise (+))
The only contributions to the integral will be from
poles of the integrand inside the unit circle. These
poles are
2+ = -2<§——)2+1:2,/(g—)2\/(%—)2-1 (3.7)
‘ m m m
= -2y+1:2/§ /§:l where y= (ELL)2

297

For yil, we have

-1

2(1)

w . . _ _ t _ l l
and z( ) $gg ( 2y 1 2y(l % §+0(;I))

0 (+)
z(m) = m (_) - (B.8)

Clearly only the + root is inside the contour and

 

 

 

:2 |L|
g(L;w2)=( ’ (hunky—(£77)
iwmw 2 2:2
m l
where
21 = -2y+1iz/§ /§Ti
2
.. /—-. ILI
9(L;w2) = 42 (lyflfl/iy 1’ (y21)
mwm 4/§ Vy-l
or
._ |L| - _ 21 l
g(L;w2)=( %) W17 “1 1) (11:1) (E.8)
mwm J; Vy-l
For y<l
z = -2y+lti 247/Ty
and [z|= (z-z*)$5 = l , (E-9)

Both poles are on the contour and the integral has no

. . +. 2
prec1se value. However, Since we need g(L,(w-1e) ), we
can calculate these directly. First w+wtie implies

y+yti5 as long as e<<l. Therefore,

298
zi(yt16)=-2(yiié)+112 yiia /(yiiG)-1

=-2y+1¥215:2E/§ /§:Tti5(2y_l)]
/§ Vy-l

 

to first order in 6. For y<l, we must show the imaginary

parts explicitly, i.e. /§:I=i/T:§

2+(ytid)=-2y+1:(i) élZXlll+2i(:/§ /1-yld)
_ /§ /I:§ .

and

26(2y-l)2 +6

 

 

 

Iz+(yti5)I2=(4y2-4y+l+4y-4y2)t(;( /§ /1—y)
’ J; /I:§
Izi(ytid)lz=1:(126)(1/(/§ «I:§)>
For w+ie, 2+ is inside the contour and
for w-ie, z_ is inside the contour.
_ /—:— ILI
g<L,(w+ie)2)= 42 ‘ 3Y+ 1+21§11_1>
mwm 4/§ Vy-l
_ (‘1)L (_6' vy-l)2|L| (B.lO)
mwi /§ Vy-l
or
_ L _._—_ 2|L|
g(L,(w+ie)2) = ( 1) (5 l 13) (y<1) (E'll)

 

mw; i/y Vl-y

showing the explicit imaginary part.

Similarly,

299

(-1)L<gg§+i/i:§)2'L‘

2 (E.12)
mwm -i/§ Vl-y

 

g<L,(w-ie)2)=

_ w 2
y—(—w ) <1
m

In the band g(0,(wii€)2) is purely imaginary and
outside the band purely real. We can calculate the density
of states using Equation (5.18) with 5:1 and compare

the results with Equation (l.llb).

D(wz) —l Im tr (mg(0,(w+i€)2)

 

 

1 1
Im
" d2 . r w 2 w 2
m 1 (5-) ‘fl-(E—)
m m

1
)
" f 2 J”2“2
U.) LU'U)
m

Inside the band we can rewrite the Green's function in an

(E.13)

 

alternate form. First, define
_ . -l
6 — Sin /§ (B.l4)

Then cose = Vl-y

and (/§:i/l-y)2|L|= :i(i sinetcose)2|L|
(ii)2ILIe;2i(L)6=(_l)|L|e;2iILI
Therefore
2 e:Zi|L|6
g(L,(U):i€) )=2(_‘;i) ——2———-—' (B.15)

mm sin26

 

300

Using this form the spectral density is

2 i<g<L.(w+ie)2)-9(L,(w—ie>2)>
JL(w ) éfiw/T-l

= 4 cos(2|LJe)

(B.16)
mm: sin(2e)(éKw/T-1

and the pair displacement correlation function is

 

_ 45 iwt cos<21Lle) dw
F (t) — [e (B.17)
L mu; sin(2e)(é““/T-1)

with the autocorrelation being

_ 25 I eiwt dw

 

(E.18)
(éfiw/T-l)

2 2
w w -w
m

Alternative Green's Function Derivation

From Equation (5.8) we can define an inverse
Green's function for the perfect monatomic crystal in

the harmonic approximation as
G (2 2'°w2) = (me-ZY)5 ,+y5 , (2.19)
’ ’ 2,2 2,2 :1

In matrix form, we have an infinite tridiagonal matrix, i.e.

(E.20)

 

 

301

If we wish to know G(2,2;w2) for all 2,2; we have to

invert this infinite matrix, an impossible job. However,

we can find matrix elements of G(2,2;w2) for I2-2’l of

the order of unity relatively easily. In fact for

L=I2-2’l, we have to invert an (L+l)x(L+l) matrix. We

will first calculate g(0,w2).

as follows,

 

G(2,2:w2) = ;

 

 

G‘1(2.2: 02)
’(a) (b) (C)
'Y O O
2
_1 mm -2Y Y
2
o 7 mm -2y Y o
(d) (e) (f)
2
Y mw -2Y Y
o { 0 Y ‘
I
(g) (h) (i)
where G-1
E=G(0,w2). We can write the
2 2 + Q E 0 =>§= ‘
§§+2§ £§=1
2 g + E Q 0 i’H =
Therefore
(2 - ge‘le - 21’12>§

 

We partition the matrix

 

 

 

 

A B c
D E F = 1
G H I

 

(3.21)

and G are similarly partitioned. Notice that

matrix equations

IIQJ

l
IIP
IIU‘
lltTJ

= 1 (E.22)

302

If we denote the (m,m) element of a.1 as the last element

-1 -l _ 2 -l .-l _ 2 .-l
of g , then Q; g-Y (am,m)62,od2:o and g; Q — Y (gill)
62,n62:n where 2,2 are the element descriptors e2,2’

and n,n the last element of g. For n=1 we have

(e-Y2(a;fm)-Y2(i;{l))3 = l (2.23)

Next, we look at 9-1, using

 

 

 

 

 

 

 

 

g 3-1 = A we have
(b)
(a) o
. o (W) (X)
Y - .
2 — o 1 0
Y mw -2Y Y 1
(C)o y me-zy (Y) (2’ ° 1

 

 

 

 

 

since g is semi-infinite the partitioning of g as shown
gives 3 as one of its submatrices and we have
_ _ _ -l
g g + 2 z - 0 — x ->-§ Dz

ll
H

+ (mwz-ZY)z

IIO
IIX

(-g g-lg + (mwz-ZY)Z = l
where g g-1 b = Y a-1
- - - m,m
Since 2 = a-1 we have

303

 

 

 

 

 

 

2 -l 2 -l _
(-Y am,m + (mm -2Y))am,m - l_
or
2 l 2 2 2
m,m 2
2Y
Next, if we similarly partition ; g-1 = l
we find
.-1 _ -l
11; - am,m (E.25)
and from Equation (E.24) we have,
g(o,w2)=E= 1
me-Zy-(mwz-ZY- (mwz-ZY)2-4Y2)
2
9(Olw ) = (E.26)

1
4Y

where wi=fi— and we took the negative square root.

The only difficulty lies in the sign of the root of aéi
we take, As Equation (E.26) is definedJit is g(qxw+ie)2).
If we wanted g(l,w2) the matrix 3 in Equation (E.22) will

need to be a 2 by 2 and the matrix g is given by

m 2-2 - 2a.1 -1
w Y Y mm

Ill?!
II

2 2.-l
Y mw -2Y-Y 1ll

(me-ZY)+ mw2(mw2-4Y) Y -1
2
= mwz-Zy+ mw2(mw2-4y)

Y 2

 

304

where we take the negative square root.

The determinant of g is

2
2 w
det(g) = {g—sz (wz-wi)£m+ (UP-313)]

2

 

2

0..)
m
I22_27 /2 2_2/2 2_2 2_“m
m w (w mm) 2m w (w mm) m (w wm+(w 5—9

 

 

-w 1

2
/22 /222°’m /222
meVw wm(w w -wm+w -f— m w (w -wm)

 

 

 

(E.27)

Clearly, E11 and E22 are g(o,w2) as before. First, using

Equation (E.8) with 2=l we have

w2 __
2 2 “NATE” “’2
9(1103 )z +—

mwz 2 2 2
m (/w Vw -w
m

if we rationalize E12, we have

2 ___
w
2 -(w2-§m)+7w2(w2-wi)

2
mwm ([1:12(w2-wri)

 

(wz-wi)

 

 

E12 =

 

 

9(11032) Q.E.D.

305

Comparing E12 with Equation (E.10), we see we again generate
g(l,(w+i€)2). For g(2,w2), we need to invert a 3x3.

In this case
<3(0) 9(1) 9(2)
E= 9(1) 9(0) 9(1) :

9(2) 9(1) g(0)

   

This procedure can get quite messy for large L.

APPENDIX F

ISOLATED DEFECT CLUSTERS EMBEDDED

IN A HOST CHAIN

306

 

If a light mass or cluster of light masses is
embedded in a heavy mass chain, vibrational modes often
appear in region forbidden to the pure host lattice.
In Appendix I, we will look in detail at the single defect 1
in a host chain. In this appendix, we will look at the
impurity modes. 2

For the light defect mass, we define

 

 

e = m (F.l)
where m is the mass of host chain atoms
and m’ is the impurity mass.
For the single defect at the origin we can easily solve

the Dyson equation
G = P + PCG (F.2)

, _ 2
where C(2,2 ) - emw 62'06£;0

and P is the monatomic chain Green function. Upon
reiteration we have

2 .
6(221w2) = P(2,2fw2)+ 5“ ”P(“11w)P(0r2'w)

 

(F.3)
l-emw2P(0,0,w2)
We can find the local mode frequency from the pole of
G(00,w2).
G(o,gw2) = P(O,gw2)/(l-€mw2P(0,0,w2)) (F.4)

307

308

An impurity mode occurs at frequencies satisfying
l-emw2P(O,0;w2) = 0 (F.5)

In Appendix E, we found

1

“firm;

 

P(O,0;(w+i€)2) =

and we have an impurity mode at

 

w2
(.12 = m 2 (F.6)
l-e
for a mass ratio E, = 2, e=3 and
m!
2 _ 4 2 _ 2 2_ .
(.0 - 301m - 2 3' for (Um—2 (F.7)

Alternately, we could have found G(0,0,w2), by
the inversion of P-1(2,2,w2) with the element at the

origin replaced by mw2(l-e)-2yin this case,

 

2 _ 2 _ _ _ 2 -l_ 2.-l -l
g(0,w) - (mm (1 E) 2Y Y amm Y 111) (F-B)
by using Equation (E.21) and
mg(o,w2> = [fiwz-wfi-wzej'l (9.9)

again, the impurity mode occurs at

w2(w2-wfi)- wze = 0

1J

 

309

 

The utility of the inversion method becomes evident for
the two defect cluster. zThe Dyson equation method becomes
cumbersome at best whereas we can write the diagonal
Green's functions for the cluster by matrix inversion

almost by inspection. For the 2 defect cluster we have

2 2 -l -1
mm (1-€)-2Y-Y amm Y

(F.10)

2 1

y mw2(l-€)-2Y -y ill

For convenience, since agi = iii for the host chain,

we will take

 

 

 

 

 

2 2 2
-1 2.-1 m“ ’ZY‘/Q;(mw ‘4Y) (F.11)
A — Y am,m-Y ll'l— 2
. 2 . 2
G(OIOI(U)+1€) ) = G(llll(w+1€) )
w2
2 w2-ifl-+ w (w -w.
-w E +
= 2 (F.12)
w2
2 -m 2 2 2 2
w —2+Jw<w comm w 2

 

 

310

The denominator vanishes for

 

w2 = wi/[4e (l-€)] (F.13)
and for
2
w
2 m
w = 8(€)(1-€Y ((4e-l)+/1+8€) (F.14)

for €=% Equation (F.13) gives w2=w; and is not of much
interest since the perfect chain has a zero denominator
in the Green's function at w2=w;. However, for larger
mass ratios (€58), a second mode emerges into the
impurity band. Equation (F.14) gives for 6:8

2
(A)
— 55(1+/§) = 1+/§ for wi=2 (F.15)

E
I

3.236068

For a three defect cluster two possible cluster

Q

configurations are present namely, d-d-d and d-h-d. For

these cases

-1
2
m(l-€l)w -2y-A Y 0
_ 2
E _ y mw (l-€2)-2Y Y
2
0 y m(l-el)w -2y-A

(F.16)

 

311
-wnere £1 = e and £2 = 0 or e
and the determinant of g gives the poles of the Green's

function.

0 = det(E) = (mw2(l-€)-2Y-A)((m(l-€)w2-2y-A)(mw2(l-€2)-2y)-2y2)

The poles are at

w2 = w; ‘%§%%:E)’ (F.17)

land

4w6(l-€l)28(€—l))+2w4wi(l-el)(3-(€1+25))€

4 2 6
w w w

+ :2 [(51+25)2-(1+8€)]+z§l= o (v.18)

 

Equation (F.18) is cubic in oz and we will solve it only

for specific values of 5,61 and mi. First, Equation (F.14)

will introduce a mode into the impurity band for e>t or

“n

a mass ratio M— >4/3. For e=k and wiéz Equation (F.17)
L

gives

w2 = 1+/§ = 2.414214 (F.19)

independent of the values of 6 indicating that the middle

1
atom does not participate in this mode. For el=0(d-h-d),

2
€=%, wm=2, we get

 

312

86-4w4+4wZ-2 = 0

or 82 = 2.8392868 (E.20)
for 8 =5 €=k and w2=2 we get
1 ' m '
w6-6w4+llw2-8 = o (E.21)
2
or w = 3.5213797

This method with some algebra can be extended to many
site defect clusters. For sake of brevity we will stop
at three site defect clusters since n site defect clusters

requires us to solve an 11th order polynomial for its roots.

 

APPENDIX G

DENSITY OF STATES OF PERIODIC LINEAR

CHAINS WITH ALL FORCE CONSTANTS EQUAL

313

 

 

For a periodic linear chain, we.can calculate

the density of states using Equation (1.8) and

dk
a— (G.l)

D(wz) = w.

wra
el
:Im

For the monatomic chain, we get Equation (l.ll) derived
in text. For a two—site periodic system only the h-d

binary chain is possible and Equation (1.8) gives

—£I_ _
1 - +ml cl Zycos(ka)02//mlm2
(G.2)
___2_1 _
w 02 — m 02 2ycos(ka)ol//mlm2

2

The determinant of coefficients must vanish for a solution

to exist, or

 

 

 

 

(wz- ii 2y coska.
l Vm m
1 2
=0
ZY coska wZ-él
mlm2 2
or

42 2 1 1 2 4 A
nTYTn‘ sin ka=(2y(I-fi— +548 -w) (6.3)
1 2 . 1 2

2'd4_fi

 

315

 

 

 

 

 

Then
Y(m-l-**%1—) -w2
dw a Y m—m
(2y(}n-+%—)-wz)[1-—123w2(2y<;—+1—-)—wz)J
m
1 2 4y 1 2

upon simple factoring

(Y(1—+-1-—)-w2)

m
D(w2 )= —1Re( m1 2

1 1
fl Thad—+3?) 21-w21fl2-fillfifu2-fi-z- i

For ml=m2, Equation (6.4) reduces to Equation (1.11).

-.M!‘ ..n.‘ N?
I .

(6.4)

 

 

 

 

For ml=2m and 1— = 1,
m2

(1.5-w2)

/;§-(3-w2 /;2-1 JEZ-Z

D(w ) = — Re( (6.5)

 

 

m1
and

Figure 2digives the comparative numerical frequency
spectrum.

For a three-site periodic system, Equation (1.8)

 

 

gives
ika -ika
8281 = + 2.1 81 i.e.—.82 - Ye 83
ml /m m /m m
1 2 1 3
(6.6)
-1ka ika
“202 = gl’oz ' e ' O1 ' ye O3
2 /m m /m m
2 1 2 3
ika -ika
w 03 = $1 0 - ye 01 - lg———— 02
3 3 Vm1m3 szm3

316

Again the determinant of coefficients must vanish for

a non-zero solution to exist.

 

 

 

 

 

 

w2_ £1 Yeika Ye-ika
m
1 lem2 ,/mlm3
e-ika w2_£l eika = 0 F1
r———' m
mlmZ 2 vm2m3 .
Yeika Ye-ika w2-£1 ,
m
lem3 szm3 3 i
or
1 1 l 2 2 l l
w-w2y(—+—+—)+w 3Y( gt + )
m1 m2 m3 mlmz m1’“3 m2m3
+ ——2-y—3— (-2)51n2(3ka) - 0
1m2m3 2
Therefore

 

 

 

1 1 2 1¥_y 1 y 1
W?) +3Y ( I l

3 m1““2 m1m3 m2m3

 

[Jw4-w22y(l—+
m1

317

 

 

m m m
x (1-w2.1 § 3(w4-2vwz<%—+l—+§—>+3y2 (m mnm1—* 1 )1
4Y 1 2 3 ml 2 1m 3 2 3
QE has zeros at
dw

 

N
N
.<

_ 111.44
w ‘3‘1+(fi‘+fi‘+fi‘11 -2*-2*
1 2 3 m1 m2

which we designate w ,w2

11o
'7

and poles at

 

 

 

 

 

 

 

 

 

2 1 1 1 1 1 l 1
w =Y(——+——+——-i/——+ -( + ~+ )
“‘1 m2 m3 m2 n71? mimz m11113 I“2‘“3
1 2 3
. 2
which we deSignate w p+ andw p-
(wZ-w +><w -w:_>
l
D(w ) =— Re(
8 .1. 3
400sz -w2 JwZ-wz [rim m -3Y2w21m m +mlm 1m m
+ p- 1 2 3 2 1 3 2 3
2Yw4(%—+:’T-+%—)-w6 ((3.7)
1 2 3

For m1=m2=m3 we again get Equation (1.11) as a check to

our calculations. There are two other unique combination

both With ml=m3,

Case 1: m1 = 2m2 and Case 2: m2 = 2ml

For Case 1, l— =
m

1 and the density of states has zeros at
2

318

 

)= .60685, 2.05986
and poles at O, %, 2:8, 8(71/17)
= O, .5, .719224, 1.5, 2.5, 2.78078.

and

(w2-.60685)(w2-2.05986)
(82(82-1.5(wZ-2.5 «Ks-82 JwZ-.71922 J83-2.78078

 

 

 

D(w2)=%-Re(

 

 

(G.8)
Figure (G.l) gives a numerical spectrum for
Case 1 (h-d-h).
For Case 2; %— = 1 and we have zeros in D(wz) at
1
82 = %(5:/7) = .78475, 2.54858
and poles at
w2 = o, 1, 8 (5:1). (21/2)
= 0, 1, 2, 3, .585786, 3.41421
and
2 2
2 _ 1 (w -.7847S)(w -2.54858)
D(w ) - ; Re( f___ /_ 2__ )
(wzuw2-1(w2-2(w2-3 .585786-w21w2-3.41421
(C.9)

Figure (6.2) gives the corresponding numerical spectrum.

“:1

 

 

U
a
.. .lw'

. .wa 033 name was 6.17:
.Emumhm ofléauomlouda m How nounun no hufluconll..n.u mmouHh

 

 

319

 

 

 

 

 

 

 

 

 

 

 

 

2—(20 >O¥¢<t muo¢o Ozouum 00.0 00.0 00.0 MM.0
. 3.3
.. Dara Dram Doom 0
.: m.n 0.m m.m 0.m m.~ , o.~ 0.0 0.0
r. 00.0
mm.0
L _ 0m 0
:
mb.0
L
A Sh.

 

 

 

 

 

 

 

 

 

 

 

 

00.—

(HoM)O

1320

.m

.qu 038 33 65.. 5.6.8

.amumhm cacofluom wufiu n a you ucuana mo had-coonu.m.u mmsuHh

z_<:u >ox¢<z zuomo ozouum
3.3
o.m m.m o.m m..

00._ 00.
on 000m
.0

001m 01
m

00.0

00.

u

0

0.0

 

 

 

 

 

 

 

 

 

 

 

()5

 

 

 

 

 

 

 

 

 

 

 

 

00.

mm.

om.

mh.

00.

.(MoH)0

321

Alternative Approach

 

A very useful alternative approach to the calcula-
tion of density of states for an n site periodic system
is available. By the method of matrix inversion as
described in Appendix B, we can generate the diagonal
Green's functions for the System. The density of
states is
n

2 — .
D(w )=E% Im g=lmagaa(0,0,(w2+1e)2 (6.9a)

where 8 refers to atoms in the basis of an

n site periodic system.
This approach is much simplier than at first it may
seem for several basic reasons. First, the diagonal
elements can be written in terms of continued fractions
of the inverSe matrix; also, the procedure is easily
adaptable to computer calculation.

An additional benefit is we calculate the Green's
function itself and can relatively easily calculate near
off diagonal elements. The inverse Green's function for

a two site periodic system is

2
‘ Y mlw -2Y Y
-1 ; 2 y 2-
Ga8(£,£,w L. Y mzw 2y Y (G.10)
m w2-2
Y 1 Y Y
2
Y mzw ‘ZY Y

Y

 

322

Partitioning G-lG=1 as we did in Equation (E.19) where

e is now a 2x2 matrix, we find

 

 

 

 

 

 

-1
2 2 -1
m w —2Y-Y a Y
1 mm (6.11)
E = 2 2.-1
Y mzw -2Y-Y 111
where, following the arguments in Appendix E,
m w -2 - Za-l -1
-l 1 Y Y mm Y
amm = _ (6.12)
Y m2w2-2Y 2,2
and 2 -1
1w '2Y Y
-1 (C.l3)
i =
11 Y msz-ZY Yzill 1,1
I
Therefore: a.1 = l ‘——
mm 2
mzw -2Y-m w -2 - a-1
1 Y Y mm
1
.-1
and i = 2
1,1 2_ ‘1
m1“ ZY 2_2 21-1
2 ”2“ Y Y 11
m wZ-ZY
a-1 = —l7(mlw2-2Y- -l—7——-4w2)(m1m2w2-2Y(ml+m2))
mm 2Y mzw -2Y
2
“-1 =—l—(m 2 Mm-Zy(w)[(m Luz-2 (m+m)J
l1,1 2y2 2“ Y“;2:§;’ 1 1m“’2 Y 1 2

. -1 .-1
Notice am,m and 11,1 are no longer equal.

 

323

 

 

 

 

 

 

 

 

2 1
G l(0,0 ,(w +ie) ) = _1 Y2
1“ ”ZY'Y amm ' mw2_2 2i-1
m2 Y Y 11
- £1
(w m2 )
-1.
m1
”V w 2-2y(——+% —) Z-gl. 2-%1
1 2
1
(0 OI(U)+16) ) = -2 2 _1 12
m2“ Y Y i11mw2_2 _ 2a-1
ml Y Y mm
(m 2&1)
= l_
m ‘:—f
2 (wZJFZ-2y($—+%—) (w2_%l w2_%l
1 2 1 2

The density of states from Equation (G.9a) is

(Y(-—+a—)‘w2 )

f_“J;_:_—_I—I—'27‘2 __1/V2_21

which is identical to Equation (C.4).

D(wz)

 

 

sly

We can set up the three site (or the n site)
periodic system Green's function by inspection! Being

more explicit than necessary we have

-1
2 2 -1
mlw -2Y- Y am m Y 0
E = Y mzw -2Y y
0 Y 3w ~2y-Y il 1

(G.14)

 

 

(G.15)

(C.l6)

324

 

 

 

 

 

 

 

where
-l
2 .2 -l
mlw -2Y-Y amm Y 0
-1 _ Y 2_ Y
amm — mzw 2Y (6.17)
0 Y. m3w —2Y 3,3 r
and
-1
m w2-2Y Y 0 . -
l i 1
1-1 — Y m w -2Y Y (G 18)
1,1 2 °
2 2.-l
0 Y m3w -2Y-Yill 1,1
In continued fractions, we would write
1
. 2 _
611(0,0,(w+i€) )--In 2-2 - 2 -1 - Y2
1” Y Y amm 2’ 2
mzw -2Y- Y
m w2_2 _ 2.-1
3 Y Y l1,1
(6.19)
1
. 2 _
622(O,0,(w+ie) )— 2 y2 Y2
m w -2Y- -
2 m w -2 - 2a.1 m w2-2 - 2i-1
1 Y Y m 3 Y Y 1.1

(6.20)

325

 

 

 

 

 

 

 

 

 

l
633(0,0,(w+i€)2)= if
2 2.-1 Y
m w -2Y-Y i - I2
3 11 2 Y
m w -2Y-
2 m w2-2 - 2a-1
1 Y Y m,m
(6.21)
where
2
2 -1 = Y
Y ,m 2 (6.22)
m w -2Y- Y
3
2 2
mzw -2Y- Y
2_2 _ -
mlw Y Y am!m
and
2
'2.-l _ Y
Y
mlw -2Y- 2
2 Y
mzw -2Y-

2 2.-l
m3w -2Y-Y 11,1

We can solve these equations relatively easily by hand

and almost trivally by numerical computer techniques.

For the three site periodic system, we have shown that
ml=m374m2 give all possible spectra that can result from a
three site periodic system except for the trival case,
m1=m2=m3, the monatomic lattice which is really a one site
periodic system. For ml=m3, Yzaifm = Yzi-J;1 (only true

for this special condition), and 611(O,O,w2)=633(0,0,w2).

 

326

6 1(0,0,(w+i€)2) = G 3(O,0(w+i€)2)

3 2
(-(w4 -2Y(——+m2 —)w 2+m1m2 —)
(6.24)

ml *1 wa “W1 2_ ;I.Z 2 f 2 2 l 2 2
w -w w -w w —w
r3 r4 r5

 

,1221

 

 

 

 

 

 

 

where

2 = 1.
r1 ml b
2 .. 2 1. l.__1__2
w r2 Y(m1 m2 iJle m2) )

r3

2 _ I 3 l2_ (9 4 _ 4
wr4 — 21ml m2i (;2+ 2 mlmz)

r5 1 2

_ 4 4 2 BY
(w "11 + 2)
. . l
G22(ODXw+l€)2)= 53’ 2 2 ti
Jw {w- w2 -w: M“ @JJifi w2 -w2 r5
((3.25)
The density of states is
D(w2)=--}- Im(2m 6 (O 0 (w+ie)2)+m 6 (O 0 (w+ie)2))
3H 1 11 ’ ’ 2 22 ’ ’
_ 1 (wZ-w+)(w2-w2 )
fl.R
")(2Y[;1:;—J;2w2wr2 w2 -w: 3Yw2 -wr 4Vw2 -wr
(6.26)

327

Where

 

2 I 4 2 4 _
u) = (-—-+—-— i: + — - ,
m m mg mlm2

312

|+

é::::]

This is identical with Equation (G.7) with m

l

=m3.

m. LJ'A“. '

 

 

APPENDIX H

SELF-CONSISTENT SINGLE SITE APPROXIMATION

328

D. Taylor in 1968 first derived, the results
we will now present. We present this derviation for
two reasons. First, it is necessary for the sake of
completeness of this thesis and, second, the published
papers all contain misprints. From Equation (5.49 )

in text, we can write the self consistent condition

_ _ d _ h
(.155) _ 0 - CdEs + (l c3d):'s (H'l)
where
5 _ _ 5_ -1 5_ '
gs — (1 (gs gs)B) (gs gs) (H.2)

The subscripts indicate a site representation where the
matrix gs is zero except at the site (5,5). For the single

site,

U104

mw2(1Le) (H.3)

IIO

(023‘

= 0 . (H.4)

"O

For convenience, we take

gs = mw2(l-E) (H.5)

For a single site, the matrices can be treated as scalars;

whereupon Equation (H.l) becomes

329

 

330

c (e—E) (l-c )(-E)
o = d 2 _ .+ d?
(l-mw (e-€)R00)

 

 

ER)

(1+mw 00

We can rearrange and solve for R00, getting

E—Cde
R00 = _ _ 2 (H.6)
(e-€)emw

 

Now, we must solve for R0 and substitute into Equation

0
(H.6). Since

= E -g (H.7)

where g is the perfect chain Green's function and g is
diagonal. We can take the k space transformation since

g possesses the symmetry of the monatomic crystal.

 

-1
R(k) = (P(k)-wZE)-l (B.8)
. _ 1
Since P(k) - —§-—2—— p
w -w.(k)
3
RM = 1 01.9)

w2(l—E)-w§(k)

The inverse transform is now obvious when we look at the
perfect monatomic chain Green's function derivation in

Appendix G.
’ 2 ’ 2 ' -
R(£,£ ,w ) = P(£,£ ,w (1-€)) (H.10)

and

l

_ _ 4- (H.11)
(w2<1-e)fnw2<1-e)-w§fi

 

R(o.o,w+i<s)2> =,%

 

331

Alternatively, we could have derived R(0,O(w+i6)2) by
the real space inversion of an Equation (H.7). Using

the methods of Appendix B, we have

R(0,0;(w+i5)2) = 1 (H.12)

mwz-Zy-o-ZyzA

 

where

A = 1 (H.13)

mwz-Zy-o-YZA

 

 

fi

 

or A = —l§(mw2-2y-o- (me-Zy-o)2-4Y2) (H.14)
2Y
and R(0,0(w+i6)2)= 1

( mw2-0)15(mw2-o-4y) l5
. . _ 2- 4y _ 2 .
substituting o-mw e and 5"” mm, we get Equation (H.11).

Substituting R(0,0(w+i6)2) into Equation (E.6) and

squaring to remove the radical, we get

 

- 2 —2 2
(5‘5) 8 w = (l-E)(w2(l-E)-m2)
(E-C e)2 m
d
2 2
First dividing each side by mm and defining x = E: ,
w
m

we, then, collect terms in.powers of E.

E3[2x(l-e(l-Cd))-l]
+E2[x(ez(1-cd?) -(l+4eCd))+l+2€Cd]

+E[cde(ecd-2(1+Cd e)(l-x))]

 

 

332

2

+cde2(1-x) = o ' (H.15)

This equation has three roots, all three will be real or
one will be real and the others complex conjugate numbers.
When all three roots are real, the Green's function will
always be real, i.e., the density of states will be zero.
The single real root, we can discard as unphysical. To
get R(w+i6)2 we have to take the complex root with the

negative imaginary part. Then, the density of states is

'D(w2) = - % Im R(O,0;w2) (B.16)
Finally, we wish to examine two special cases of
Equation (B.15). First, for Cd=0 or equivalently, 6:0,
we get E=O which gives §=gpthe perfect chain Green
function. Second, for Cd=l, we get E=€ which gives

1
(w2(l-€)fflw2(l-€)-wi)

w
u
SIP

8 (H.17)

This is the Green's function of the chain with mass
m(1-e) where m;;= wi/(l-e). Therefore, the self-con-
sistent method presented in this thesis is exact in both

limits Cd+0(e+0) and C +1.

d

 

APPENDIX I

ISOLATED DEFECT IN A MONATOMIC HOST CHAIN

333

In this appendix, we will briefly examine the
eigenvectors and localization parameter for a monatomic
chain and a diatomic ordered binary chain of length, N.
Next, we will explore, the effect a single defect, either
lighter or heavier than the host mass, has on the eigen-
frequencies and eigenvectors of a monatomic chain. For

the monatomic chain the equatiomsof motion are

_ m 2
Un+1 —(2—7w )Un Un_l (1.1)

In text, we showed that we could diagonalize this coupled

set of equations by expanding in running waves

Un = u eilkna (1.2)

This solution for the eigenvector is valid for an infinite
chain and for chains with periodic boundary conditions. For
fixed boundary conditions, we can expand in a linear combina-

tion of outgoing and incoming waves.

. *_.
Un = U elkna +U elkna (1.3)
where U E U(1)+ iU(2)

subject to the boundary condition Un = 0 and UN+1 = 0.

334

 

335

— O we have'

For U0 —
U = -U(2)
and
Un = U sin(kna) (1.4)
For UN+1 = O
U sin(k(N+l)a) = 0
or
(1.5)

TTI‘

k = W , r-integer

Substituting Equation (1.4) into Equation (1.1), we get

2 _ 2 . 2 ka
w - mm 51D (2 (1.6)
Where
(”2..-£1
m m
nd k= "r -1 2 N (I7)
a M r— ' ..., .
Equation 1.6 is identical to the equation in text. The
eigenvector for a given value of r is
(1.8)

= . TTrn
Un U Sln (m)

The localization parameter a is

336

 

 

 

 

1?
4

_ IUI

OL = n—l n (1.9)

()I In |2>:2
n=1 n

First, since

N 2 N ' [(N+1) J ' (N ) (1 351 1)39

Z sin (qx) ___ __ cos $.81n x . . n

_ 2 2 Sin(x) .

q-l )
we will want to evaluate

2 N 2 )

ngllUnl :n£1 Un (Since UN+1:0)

(N+2)rn .
_ N+l 2 2 nrn _ N+l. cos(——fi:T——)51n(nr) 2
- 2 U s (——)-(—- )U
n=1 N+l 2 zsin(1£—)
N+l
= 5%; 02 (1.10)

Since sin(nr) 0 for all r.

Next since

{3N+cos[2(N+l)x]sin(2Nx)cosec(2x)

(DIP

Z sin4(qx)=
q=l

-4cos[(N+l)x]sin(Nx)cosec(X)}

we again add the N+l term to the sum

N+l N+l
4 _ 4 . 4 nrn _ 3(N+l) 4

n=1 n=1

 

337

since sin(an) = sin(nr) = 0

_ 3 1
Therefore 0. - '2- m (1.12)

For N=lOO, a = .01485

and FOr N=1000, a =.0014985

For the ordered binary chain, the equations of

motion are

“‘1 2
(1.13)
“‘2 2
U2n+2 = (2'7“ ‘” )U2n+1‘U2n

where for a fixed boundary conditions n takes on the values

1 to N/Z. We take the eigenvectors to have the form

U2n = U Sin(ka2n) ‘ (1.14)

U = U sin(ka(2n+l)) (1.15)

2n+l

From the boundary conditions, we have

U0=O satisfied by Equation (1.15) with (n=0) and

UNI1=0 giv1ng
k - NT (r ' t r) (I 16)

as before. Using Equations (1.14) and (1.15) in Equations

(1.14), we have

 

 

 

2U cos(ka) = (2 -7—w )U

2U’ cos(hfl= (2 -V£w2)U

And from (G.3)

 

2--Y+Y+Y---(-I—24Y2 '2 117
w - E_ —— - (m m ) - ( m )51n ka ( . )

1 2 1 2 m1 2

 

B

For each value of k, we get two eigenvalues and eigenvectors;

therefore, for

 

_ nr _ 11
k - m , r—1, 2, ..., 2 (1.18)

will give all eigenvalues.
' m

 

 

 

 

For our case _3_= 2, 1— = l, and l—-= l and
m m m 2
1 l 2
we have
m: = g 3/% - 251n2(;:1) (1.19)
(-1: W<——§>
Uzn = U 2 COS("r ) . Sln(-fi:-l-) ’ (1.20)
N+l
_ . nr(2n+l)
U2n+l U $1n( N+l ) (1.21)

We can find approximate localization values near the band
edges fairly easily. First, for r=l and N large, we take
. Tr ~
5111871371?) — 0

‘TT
and COSW - l

339

The two eigenvalues are
m 20,3 (1.22)

For w2=0, the eigenvector is

. TT
U211 —U Sin(m2n)
_ - w
U2n+l — U Sin(N+l(2n+l))
_ . fin
or Un -' U Sln(NTl—) (1.23)

Near w2=0 the localization a is nearly the same value

as in the monatomic chain.

fl2n

2 __ .
For 0) -3, U - 2U Sln(m)

2n

n(2n+1)

n+1 ) (1.24)

(U2n+1 = U Sln(

or the light masses are vibrating in opposite direction
of the heavy masses with twice the amplitude of the

heavy masses. Then we have

‘n 2n 2n+l
n=1 n=1 n=1
N/2
N/2
_ 2 2 n2n 2 . 2 n(2n+1)

n
C‘.

N
A
m

:42
V

 

340

N
and Z 0:: £%'E;) U4
n=1
_ 51 l
and O. - -2-—5- fi' (1.25)

For N = 100, a=.0294

and for N=1000, 02.00204

Whether we should take N or N+1 in the above equation
is questionable but doesn't matter within the error of the

approximation.

For r=§, we take

. n N
Sln (E-fi:—)— l
and cos (— ( N ))~ 0
2 N+l
The eigenvalues are w2 = 1,2 (1.26)
For w2 = 1 u ~o
’ 2n -
_ - N
and U2n+1 — U 51n(n(2n+1)(fi:f) (1.27)
for w — 2, U2n+1 — 0
U = U’sin(n2n(—E—))' (I 28)
Zn ' N+l °

Within our approximation, these two eigenvectors will have

the same localization, a.

 

341

 

~ 3
_ fi _ (1.29)

For N=lOO, a =.03
and for N=1000, a = .003.,
Before looking at the single defect, we will

examine the eigenvector of the monatomic chain in greater

detail. For small eigenvalues, r=l, 2 .. the eigenvector
is a sine wave with period 2(Egl). For large eigenvalues

(N+l)-p, p = 1,2,3...

 

_ - (N+l)-p fin _ _ . nn
Un —U 51n( N+l ) — cos(nn)51n(fi:§)
--n+1 - r22 _
Un — ( 1) Sln(N+1) p-l,2,3 ..

This is the product of a wave where all the atoms are

vibrating in opposite directions and an envelope function

212)
N+l '

Increasing p, increases the number of nodes in

sin(

the envelope function and therefore decreases the number
of nodes of the alternating chain giving lower frequency.
If N is odd, then N+l is even and atom at (N+1)/2 will

be at a node for all r or p odd. Other atoms in the chain
can also be at.nodes. If N is even, then N+l is odd.
First if N+l is a prime number, then no atom can ever be
at a node. If (N+1) is odd but not prime, then no atom

can be at a node at p or r = 23, j=0,l,2,...,1n(N)/ln(2).

 

342

If we place a single heavy defect in a light chain,
Rayleigh's theorem says that all eigenfrequencies may)
decrease but not more than to the next lowest eigenvalue
of the perfect chain. For an almost infinite heavy mass
placed at or as close as possible to the center of the
chain, we can easily demonstrate this theorem. First, if
N is even the heavy defect is placed at N/2, dividing the
chain into one even chain of length N/2 and one odd chain
of length N/2-1. The heavy mass almost acts like a fixed

boundary. The eigenvalues of the short chain are

2~ 2 . 2 nr’
ws- wm Sin ( N ) (1.30)

r’ = 1,2 ...N/2-l

And the eigenvalues of the longer chain are

2 _ 2 . 2 nr”
(11L - (Um $111 (w) (1-31)
r” = l, 2, 3 ... N/2

compared to Equation (1.7) for the monatomic chain. First,
we see we lose the lowest frequency mode wzzo, r=l for the
infinite defect. For a very heavy defect, this mode will
be the only mode to propagate throughout the chain. The
r=even modes become the modes of the longer chain (r=2r”)

and the energy shift is

 

343

I

2 _ 2 . 2 wr _ .
Awr -wm[51n (N+l ) Sin 2(N+2 —)]

 

2N+3 l

 

 

 

— 2 I o I!
—wm Sin[(nr )(N+1Y?N+2)]Sln[(flr )(N+l)(N+2))]
(1.32)
r = l, 2, ..N/2
The r=odd.modes become the modes of the shorter chain
with frequency shift (r=2r’+l)
2 _ (2r +l)n _ . 2 nr’
Awr, - mm 2[sin2 (m) 8.111 (__N )1
r’ = l, 2, 3, ...N/2-l
2 _ 2 . 4r N+N+2r
Aw ’—wm $1n(w 2NTN+1) )sin(n—§TN%TT) (1.33)

r

The long chain frequency shift increases with
increasing r” and the short chain frequency shift decreases
for increasing r’. Therefore, we expect the modes to
cluster in groups of 2 near w2=0 and a); and be well

separated from one another for wzzwfi/Z.

N+l

If N is odd, the heavy defect is placed at 2

dividing the chain into two equal chains of length
(N-1)/2 separated by the heavy defect. If the heavy defect
is infinite, we have two identical short chains with two

fold degenerate eigenvalues at

I 2 = 2 sin2 (—
w wm

N+1) ' (1.34)

 

344

For this Case, the even modes of the perfect chain remain
unchanged with the odd modes shifting as far as Rayleigh's
theorem allowsithat is, the r odd modes have an energy
decrease of exactly the spacing between the eigenfrequencies
of the perfect chain. The non-shift in r even modes is

not surprizing when we recall that for N odd (N+l, even)

the atom at 5%; is stationary for the r even modes; there-
fore, the heavy defect cannot affect these modes giving

the same eigenvector as the perfect chain. For the r odd

modes, where the atom at le is at an antinode for the

perfect chain, the heavy atom chokes this displacement giving

a mode symmetric about (Egl) instead of the antisymmetric
r-even mode.
= ' £1.11
Un U Sin (N+l)

for r-even modes

_ . r’nn , _ N+l
_ » . r’nn _ N+l
- U 51!) (WT) n— T, .. ., N

for distorted r-odd modes.
These modes can also be found by symmetry considerations
as the modes of two identical length chains placed side
by_side.

To consider the case of one light impurity in

a heavy chain, we look at an impurity of almost zero mass.

 

345

The isolated frequency in the impurity band occurs near
w2=w. The almost zero light mass creates an unusual
boundary between the two light chains. We are therefore
restricted to a qualitative analysis versus a quantitative
approach. For N odd (N+l even) the light mass divides

the chain into two identical small chain of length (Egl).

The r even modes (eigenvalues and eigenvectors) of the

perfect heavy chain will remain unchanged since the atom

at Egl is at a node for r even. For r odd, the atom at
+ . . .
E7; is at an antinode for the perfect chain. For r small,
we have
Uri-_lgufltf UN_+3;
2 2 2

and the light mass at UN+1 will not greatly effect this

mode. For r large (=N),:we have

for the perfect chain. The effect of the light mass

(=0) at U will be to make

N+l
2

N+3 ‘

causing a frequency shift to the next highest frequency

of the perfect lattice. For r large, the zero mass

 

346

defect acts much like the infinite mass defect but for
r small, the zero mass defect hardly disturbs the perfect

system.

 

142 6764

303