triplet correlation and resistivity in liquid metals. -...
TRANSCRIPT
TRIPLET CORRELATION AND
RESISTIVITY IN LIQUID METALS
A. Lakshmi
B.Sc., University of Madras, India, 1968.
M.Sc., University of Madras, India, 1970.
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE IZEQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
in the Department
of
Physics
A. Lakshmi 1974
SIMON FRASER UNIVERSITY
August 1974.
All rights reserved. This thesis may not be reproduced in whole or in part, by photocopy or other means, without permission of the author.
APPROVAL
Name : A. ~aksPuni
Degree: Master o f Sciknce
T i t l e of Thesis: Triplet Correlations and R e s i s t i v i t y of Liquid Meta 1s
Examining Committee:
Chairman: A. E . Curzon
- -
L. E . Bal lent ine Senior Supervisor
V Q. D. Crozier
T. M. R i c e
Date ApproveA*
PARTIAL COPYRIGHT LICENSE
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o r p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l lowed
w i t h o u t my w r i t t e n pe rmiss ion .
T i t l e of T h e s i s / ~ i s s e r t a t i o n :
Author : I I - - -v---
( s i g n a t u r e )
A. LAKSHMI
(name)
( d a t e )
ABSTRACT
Express ions f o r long wavelength l i m i t s of t r i p l e t and
h i g h e r o r d e r d e n s i t y f l u c t u a t i o n c o r r e l a t i o n f u n c t i o n s
a r e d e r i v e d r i g o r o u s l y , u s ing thermodynamic f l u c t u a t i o n
theory . These r e s u l t s show t h a t i f t h e l i q u i d s t r u c t u r e
i s s u f f i c i e n t l y r e s i s t a n t t o compression, any p a r t i a l
long-wavelength l i m i t of any s t r u c t u r e f u n c t i o n i s s m d i l l .
W e have eva lua t ed t h e t r i p l e t s t r u c t u r e f u n c t i o n i n t h e
long-wavelength l i m i t f o r Rb, N a and K , by means of
e x i s t i n g exper imenta l d a t a f o r t h e p r e s s u r e d e r i v a t i v e
of t h e s t r u c t u r e f a c t o r . Using t h e hard-sphere model, we
have shown t h a t suceess ive2y h i g h e r o r d e r s t r u c t u r e
f u n c t i o n s a r e s u c c e s s i v e l y s m a l l e r i n t h e long-wavelength
l i m i t .
An approximate form f o r t h e t r i p l e t c o r r e l a t i o n f u n c t i o n
s a t i s f y i n g t h e long-wavelength l i m i t and o t h e r p h y s i c a l
c o n d i t i o n s , i s cons t ruc t ed . The s t a n d a r d approximations
v i o l a t e t h e long-wavelength l i m i t badly . A p e r t u r b a t i o n
theo ry c a l c u l a t i o n of p, t h e r e s i s t i v i t y , i nvo lves a
p roduc t of t h e s t r u c t u r e f u n c t i o n and t h e e l e c t r o n - i o n
pseudopo ten t i a l . The l a t t e r i s never s m a l l compared t o
t h e Fermi energy, i n t h e long-wavelength l i m i t . Hence,
f o r p e r t u r b a t i o n theo ry t o be v a l i d , t h i s l a r g e va lue
should be c a n c e l l e d by t h e smal l v a l u e of t h e s t r u c t u r e
f u n c t i o n i n t h e long-wavelength l i m i t . Using t h e approxi-
mation constructed, the third order contribution to the
resistivity of liquid Rb is calculated and is found
to be surprisingly large. While examining which regions
of q, the wave-vector of density fluctuation, contribute
dominantly to P , we find that the long-wavelength region,
"-1 f o r our mcdel, from q = O to q = 0,4A is important.
O-1 The intermediate wave-length region from 0.4A to 2kF
is found to make the major contribution. Our model is
certain in the small g and large q regions but uncertain
in the intermediate q region.
Ours is the only calculation that has examined the
long-wavelength region, in the calculation of the third
order contribution to the resistivity. Any further work
-..--- 7 2 L ---- &- - ...---.-...w- w u u ~ u LU i ~ r b v ~ y u ~ ~ t ~ OUT ~ c s U ~ ~ S i2?3 it.
- iv-
To my beloved p a r e n t s
ACKNOWLEDGEMENT
I wish to express my sincere gratitude to my
supervisor, Dr. L.E. Ballentine, for suggesting this
problem and for continual guidance through every
stage of research. E have benefited greatly from
discussions with many people, in particular Drs.
W.J. Heaney and E.D. Crozier. Thanks are also due to
Mrs. Georgina Carlson, Miss Margaret Linquist and
Mrs. Linda Yim for typing this thesis.
Finally, the financial support of the Physics
Department in the form of a teaching assistantship is
gratefully acknowledged.
TABLE OF CONTENTS
Abstract
Dedication
Acknowledgement
List of Tables
List cf Figures
CHAPTER 1.
1.1
1.2
CHAPTER 2
2.1
CHAPTER 3
CHAPTER 4
MOTIVATION ,AND 0,UTLINE
Introduction
Outline sf Thesis Contents
GmEN'S FUNCTION FORMALISM
General Introduction
Green k Function
Kubo-Greenwood Formalism
Diagrammatic Analysis
Deri.vation of the Ziman formula from the Kubo-Greenwood formalism
CORRELATION FUNCTIONS
TRIPLET STRUCTURE FUNCTION IN THE LONG WAVELENGTH LIMIT
Fluctuation Theory
A p p l i i ,a&xan of Fluctuation Theory the liquid structure functions
Relation to other work
Page
iii-iv
v
vi
ix
X
1
1
10
13
13
14
21
24
28
37
42
42
5 0
55
- vii -
E Page i i CHAPTER 5. LONG WAVELENGTH STRUCTURE FUNCTIONS: i HARD SPHERE MODEL AND COMPARISON WITH i EXPERIMENT 6 0
5.1 Hard Sphere Model 6 0
5.2 Experimental Data 67
5.3 Structure Functions when all q's approach zero
5.4 A thermodynamic method, using the pressure derivative of sound velccity 79
CHAPTER 6.
6.1
CHAPTER 7.
MODELS FOR THE TRIPLET STRUCTURE FUNCTION 85
Conditions to be satisfied by any model and data used in the calculation 85
Models for b3 (q1tq2 tq3) 89
Solving for unknown functions t(q) and U (q) 92
Least Squares Minimisation Method 96
CALCULATION OF RESISTIVITY
Expressions for W iii and w(3j
The pseudopotential and screening 112
7.3 Calculation of W ( 2 ' and W (3) and resistivity 118
1 CHAPTER 8. CONCLUSIONS i
- viii -
L I S T OF TABLES
The u-function for various approximations
with values of adjustable parameters and
minimum error, E...
Resistivity ...
Page
7 7
L I S T OF FIGURES
Figure -+ -+
I Hard Sphere Model: S2 (q) . S3 (ql -q. 0) -t -+
S*kI -q.O,O)vs q
-+ -+ II Xb: Bard Sphere an8 Experimental S 3 ( q , -4. o)
-+ -+ I11 Na: Hard Sphere and Experimental S3(qt -qt 0)
vs q
-+ -+ IV B: Ward. Sphere and Experimental S (q, -q, 0) 3
V Graphs of the u- and t- functions vs q. for
2 approximations, the 4 and 6 parameter u (q)
given by ( 5 ) and ( 8 ) in Table I1
Page
65
+ - b e VI S, (q. q1 q) for approximations (5) and (8)
J
in Table T I and for the Greenwood approximation 105 1
VI I g j (r. r. r) for approximation (8) in Table I1 and
comparison w i t h g3(rtr,r) from the Superposition
approximation 107
w ( ~ ) 3 ,(2) VIIf - q and - il q3 vs q for several approximations 1 2 4 0
CHAPTER I: MOTIVATION AND OUTLINE
" A m i g h t y maze, b u t n o t w i t h o u t a p lan" .
--A. Pope
Sect ion 1.1
This t h e s i s t r e a t s t h e problem of t h e c a l c u l a t i o n of
t h e e l e c t r i c a l r e s i s t i v i t y of l i q u i d metals . I n p a r t i c u l a r ,
it concerns i t s e l f wi th t h e t h i r d o rde r c o r r e c t i o n t o t h e
near ly- f ree-e lec t ron c a l c u l a t i o n of t h e r e s i s t i v i t y of
l i q u i d metals.
Ziman (1961) has der ived a simple formula f o r p, t h e
r e s i s t i v i t y of a l i q u i d metal . H i s well-known formula k s
-1
(I. 1.1)
where z i s t h e valence, kF t h e Fermi momentum, S 2 (q ) i s t h e
s t r u c t u r e f a c t o r , v ( q ) i s t h e screened e lec t ron- ion form
f a c t o r , x = -% and g i s t h e momentum t r a n s f e r r e d upon 2 k ~
s c a t t e r i n g .
Ziman assumes that a valence e l e c t r o n moves i n t h e
s e l f - c o n s i s t e n t p o t e n t i a l due t o t h e ion cores and t h e
o t h e r valence e l e c t r o n s . This t o t a l p o t e n t i a l i s usua l ly
taken t o be t h e sum of i d e n t i c a l s p h e r i c a l l y symmetric
p o t e n t i a l s cen t red on each atom,
t h e p o s i t i o n vec to r s of t h e i o n s 9 being c o r r e l a t e d , wi th j
each o t h e r i n a complicated manner. Ziman t r e a t s t h e t o t a l
p o t e n t i a l f i e l d a c t i n g on a conduction e l e c t r o n a s per-
t u r b a t i o n on p lane wave s t a t e s and he c a l c u l a t e s t h e r a t e
of t r a n s i t i o n by p e r t u r b a t i o n theory.
Although d i s o r d e r i s t h e prime c h a r a c t e r i s t i c of a
l i q u i d , an e s s e n t i a l f e a t u r e of a l i q u i d i s t h a t t h e i o n i c
d i s t r i b u t i o n i s n o t completely random, bu t t h e r e i s a s h o r t
range o rde r , exh ib i t ed i n a set of ion-ion c o r r e l a t i o n
funct ions . Ziman t a k e s i n t o account t h i s e f f e c t of cor re- -+
l a t i o n between t h e p o s i t i o n v e c t o r s of t h e ions R j "
Taking t h e matr ix element of v(;) between unperturbed
s t a t e s ,
(I. 1 .2)
where N i s t h e number of atoms, and L! i s t h e macroscopic .&$es
volume, and C e j = p+-. V+ 3 s e p a r a t e s i n t o two f a c t o r s , j q k , k l
a form f a c t o r desc r ib ing a s i n g l e atom r e l a t i v e t o i t s own
c e n t e r , and a s t r u c t u r e f a c t o r depending only on t h e p o s i t i o n s
of t h e atoms, which i s r e l a t e d t o t h e r a d i a l d i s t r i b u t i o n
func t ion of t h e atoms i n t h e l i q u i d .
The s t r u c t u r e f a c t o r i s given by
(I. 1" 3 )
where t h e b racke t s denote an ensemble average over configu-
r a t i o n s . g (r) i s t h e p r o b a b i l i t y of f ind ing another atom 2
a t a d i s t a n c e r from a given atom i n a f l u i d . I t i s c a l l e d
t h e p a i r d i s t r i b u t i o n funct ibn . The l i q u i d s t r u c t u r e f a c t o r
i s r e a d i l y obta ined from e i t h e r X-ray o r neutron d i f f r a c t i o n
experiments. (Furukawa 1 9 6 2 ) . From t h e Boltzrnann equat ion, t h e r e s i s t i v i t y i s given by
(I. 1 . 4 )
where n i s t h e number of e l e c t r o n s p e r u n i t volume and T e
i s a r e l a x a t i o n o r s c a t t e r i n g t i m e .
The s c a t t e r i n g t ime T may be def ined such t h a t
where 0 i s t h e angle between t h e i n i t i a l momentum d and
t h e f i n a l momentum 2 ' , and t h e i n t e g r a l i s over t h e ~ e r m i
Pk, k' i s t h e Born approximation t r a n s i t t i o n pro- su r f ace.
b a b i l i t y
(I. 1 . 4 )
given
Pk,k'
i s d e r
An i n i t i a l
ived, f o r example, by Mott and Jones.
s t a t e 1 % i s s e l e c t e d and a sum over f i n a l
s t a t e s 1'&+6> of t h e same energy i s taken. The d e n s i t y of
such f i n a l s t a t e s pe r u n i t volume of wavenumber space is
R (2T) 3
Thus t h e nclmoer of s t a t e s wi th in an energy 6E
of t h e Fermi su r face segment of a r e a dS' i s Qds '
o r t h e d e n s i t y of s t a t e s per u n i t energy a t t h i s
segment of t h e Fermi s u r f a c e i s
R dE ' dE' - 'h2 kF
n ( E P ) = (2.~1
dS ' ( , where 7 - - dk m
This i s s u b s t i t u t e d i n ( 1 . 1 . 6 ) and summed over t h e Fermi
s u r f a c e according t o (I. 1 .5) t o g ive
(I. 1.7)
I f t h e matr ix elements depend only upon t h e magnitude of 6 -t -+
( f o r 5 and k+q on t h e Fermi s u r f a c e ) , dS' may be taken t o
be a r i n g on t h e Fermi su r face wi th a r e a 2 ~ k i Sin0 dB. Thus,
S u b s t i t u t i n g (1.1.8) i n ( 1 . 1 . 4 ) and a simple change of
- 5 -
variable leads to the Ziman formula for resistivity.
The Ziman formula calculates the resistivity in the Born
approximation for elastic scattering from each configura-
tion of the ions. The physical reason for the appearance
of the factor (1-Cos0) in the scattering time T is the fact
that an electron scattering through an angle 0 loses a
fraction (1-Cos0) of its momentum in the initial direction
of motion. Thus each scattering event is weighted by the
loss in momentum due to that event.
An interesting fact about the Ziman formula is its
remarkably close agreement with experimental results. (See
for e.g., Sundstrom, 1965, Ashcroft and Lekner, 1966).
Sundstromss calculations of resistivity are based upon a
pseudopotential due to Heine and Abarenkov (1964) and
experimental structure factors from X-ray and neutron
diffraction measurements. (Gingrich and Heaton 1961;
for review sf experimental results, see Furukawa, 1962).
Her p calc for the alkali metals agree fairly with experi-
ment, and the agreement for A1 is particularly good. For
Sodium at 100•‹C, fo r example, there is 98% agreement between
theory and experiment. Aschroft and Lekner (1966) use a
theoretical S (y) curve for all metals, obtained by applying 2
the Percus-Yevick approximation to a hard sphere model.
They quote values for pcalc and 'expt corresponding to a
packing fraction of 0.45, differing by not more than 10%
in at Peast the weak scattering liquid metals. (See their
results quoted in Faber, 1972, p.326).
Very little is known about the nature of higher order
terms involving higher order atomic correlation functions,
which are not directly measurable. In this thesis, we
set out to examine the nature of third order corrections
to the resistivity. Experimental results are used both
to suggest major features which the theory must incorporate
as well as to test the theory whenever possible.
In the past, a few attempts have been made to calculate
higher order terms. In the study of higher order terms,
an appropriate correlation function has to be chosen, that
is tractable as well as accurate. Springer (1964) has used
a simplified form of the Kirkwood superposition approximation,
where the triplet distribution function is written as a
product of three pair distribution functions. H ~ S results
for Zn, even though evaluated with a pseudopotential, indicate
that the correction term destroys the good agreement with
I experiment, that had been obtained from the Ziman formula.
Greenwood's (1966) approximation for the three-particle
structure function consists of a sum of three terms, each
term being a product of two structure factors and is symmetric
in the three q's. His estimate of the third order correc-
I ' tion is much smaller than Springer's, not because of the
t-matrix formalism. Faber (1965) has obtained a similar
result. Toombs (1965) has approached the problem in an
entirely different manner, using a theory of collective
movement to describe the ionic structure. He has obtained
explicit expressions for the structure factor and three-
particle structure function involving the ion-ion potential.
His corrections to the Ziman formula are very small.
Ashcroft and Schaich (1970) used an approximation in which
higher order terms involve a non-symmetric product of
structure factors and obtained results surprisingly large.
Bringer and Wagner (1971) have calculated higher order cor-
rections, using a Thomas-Fermi screened potential for Rb.
In their approximation, only those terms in the third order
which involve only two scattering centers, are considered.
Their value for the resistivity when they included higher
order terms differed only slightly from their value of the
resistivity which they calculated using the Ziman formula.
Both p Z and the resistivity which they calculated including
higher order terms, differed vastly from the experimental
value. This may be attributed to the nature of the pseudo-
potential they used. It is not clear what their results
would be if they had used a different form for the pseudo-
potential.
The third order term in the perturbation expansion
involves a product of three matrix elements V+ -+ Vj++,g+, k,kt
V . This simplifies to a product of three form factors
and a triplet structure function
+ + + ' - P' P + > where = ij<Pq 1 2 3
' -4 -tp + + + + Here, ql = k-k , =2 - k"-k"f d 3
= k"-g. S (c ,; ,+ 1, the 3 1 2 3
triplet-structure function is similar to the structure
factor S? (q) given by (I. 1.3). The vector sum of the q' s - add up to zero, due to the translational invariance of the
structure functions, i.e., 6 +G +; = 0. 1 2 3
Ballentine (1966) conjectured that in a weak scattering
liquid metal, the electron scattering amplitude should be
small in a11 orders of perturbation theory. The reason for
this is as follows:
It is well-known that (see, eg, , Ziman 1964)
lv-t-t+= ~ ( q ) = - - 2 E lim q+O E k,k+q lim q+O 3 f
and
lim q -+ 0 S2 (q) = nk Tk B T*
N where n = - i s the number d e n s i t y of atoms, kg i s Boltzmann's
52
cons tan t , T is t h e abso lu te temperature, and kT i s t h e i so-
thermal compress ib i l i ty . A s (1.1.10) i s never small com-
pared t o t h e k i n e t i c energy EF (which i s con t ra ry t o t h e
c r i t e r i o n f o r t h e Born approximation t o be v a l i d ) , t h e n t h
o r d e r t e r m i n t h e s c a t t e r i n g amplitude can be small only i f
3 t h e n - p a r t i c l e s t r u c t u r e func t ion Sn (G1 ,Zj2 , . . . qn) becomes
small when any one of i t s arguments approaches zero. This
i s well-known t o occur f o r n = 2 , according t o (I. 1 . 1 0 ) and
(1-1-11). (1.1.11) is t y p i c a l l y about 0.03 f o r most l i q u i d
meta ls near t h e i r me l t i ng p o i n t , s o our weak pe r tu rba t ion
c r i t e r i o n i s s a t i s f i e d a t l e a s t f o r small q. So t h e s t r u c -
t u r e of t h e l i q u i d and t h e n a t u r e of t h e atom a r e important.
Near t h e c r i t i c a l p o i n t , where kT i s i n f i n i t e , t h e s e argu-
ments may no t be v a l i d . No s i m i l a r r e s u l t s f o r n > 2 have
been previous ly repor ted . I n Chapter I V of t h i s t h e s i s , w e
prove B a l l e n t i n e ' s con jec tu re t o be t r u e , t h a t i f t h e
l i q u i d s t r u c t u r e i s s u f f i c i e n t l y r e s i s t a n t t o compression,
any p a r t i a l long wavelength l i m i t of any o rde r s t r u c t u r e
funct ion w i l l be small , compared t o u n i t y , and it w i l l can- Y
c e l t h e l a r g e va lue of ' the screened p o t e n t i a l i n t h a t l i m i t .
Recently t h e work of Greenf ie ld and Wiser (1973) shows
t h a t f o r weak s c a t t e r e r s l i k e N a , K, A l , pcalc agree c l o s e l y
wi th pexpt, d i f f e r i n g from it by less than 4 % . However,
f o r Pb, Zn and Cd, where t h e r a t i o of p o t e n t i a l energy t o
to kinetic energy is larger than for Na, K, Al, the ratio
~calc'pexpt increases from 1.12 to 2.4 to 5.3. So we
expect the third order correction to be small compared to
the Zirnan formula, at least for weak scatterers.
It is thus interesting to see quantitatively the nature
of higher order terns even for weak scatterers, which this
thesis attempts to do. In an actual calculation, the subtle
cancellation between the small value of the structure func-
tion in the n-particle case, as one of its arguments tends
to zero, and (1.1.10) which is never small, is not automatic
and can easily be lost by approximations. Hence, we take
special precautions to ensure that the structure functions
beasme small in the long wavelength limit.
Recognizing that the structure tunctions represent car-
relations of density fluctuations, according to (1.1.3) and
(I.1.9), and that a long wavelength (q-tO) fluctuation may
be treated thermodynamically, we derive an exact expression
for long wavelength limit structure functions in any order.
In order to do this, we use Callen's (1960, Ch.15) fluc-
tuation theory of thermodynamic statistical mechanics, which
readily offers itself for this purpose.
Section 1.2. Outline of Thesis Contents.
A comprehensive review of the entire subject of liquid
metals will not be attempted here. That task has been
excellently performed by Wilson (f365), March (1968) and
more recently by Faber. Recent advances have been reported
in the Proceedings of the Tokyo Conference (1972). W e
shall only investigate the order of magnitude of the third
term in the Kubo formalism.
At the outset, we introduce the Green's function formalism
and the general method of obtaining P and we use this method
to derive Ziman's result. In Chapter 11, we introduce the
concept of the pseudo-potential and the Kubo Greenwood for-
malism and diagrammatic analysis. - In Chapter 111, we introduce the notation for correlation
functions and the properties of the triplet-correlation
function.
1n Chapter IV, we use fluctuation theory to derive an
exact expression for long wavelength limit third and higher
order structure functions and calculate it for the hard
sphere model and compare it with results we obtain using
existing experimental data in Chapter V.
In Chapter VI we invent a triplet structure function
incorporating the above-mentioned feature, in other words,
giving the correct long wavelength limit and also satis-
fying other physical conditions. We then test for its
correctness.
In Chapter VIL we calculate and compare the third order
terms with the second order terms in the Kubo formalism
for resistivity.
Chapter VIIIcontains some concluding remarks.
- 13 -
CHAPTER 11: GREEN'S FUNCTION FORMALISM
Section 2.1: General Introduction
Ziman's simple theory of resistivity, although agreeing
closely with experiment, cannot easily be extended to higher
orders. It is good only in the case of weak scattering, when
the simple perturbation theory works. In order to go to
higher orders, a more general method is necessary, for which
the Green's function method readily offers itself.
The study of disordered systems presents great theore-
tical difficulties. Liquid metals exhibit structural dis-
order and do not possess the translational periodicity that
their counterparts, metals in their solid state exhibit.
In the latter case the periodicity of the lattice allows one
to use Bloch's theorem to greatly simplify the problem.
No corresponding simplification exists for a liquid.
However, some of the techniques of solid state theory
can be taken over directly for the study of liquid metals.
As the ions are much heavier than the electrons, we may use
the adiabatic approximation and consider the motion of
electrons in the static field of the ions, which are regarded
as being at rest. The electrons will be regarded as inde-
pendent (subject only to the Pauli exclusion principle)
and the many body nature of the problem will be taken into
account only in the construction of self-consistently
screened potentials.
The specifically liquid features of the system enter
through the dependence of the potential energy of the electron
on positions of the ions. Detailed instantaneous spatial
arrangements of the ions are not important, but only certain
statistical correlations are necessary to calculate any
interesting physical quantity. The basis of this thesis rests
upon these statistical correlations and the need to average
over the ensemble of all configurations of the ions, thereby
introducing the pair, triplet, etc. correlation functions.
The formal theory of resistivity is described in the
next few sections.
Section 2.2 - The Green function method was first introduced lnto the
theory of liquid metals by Edwards (1962). The Green operator
is defined as
Here H is the one-electron Hamiltonian (Ho + V), and En and are its eigenvalues and eigenvectors. The spectral ope-
rator is defined as
p ( E ) = lim ) / G (Etiq) -C (E-in) 17-f 0
2ni t
Although any representation may be used for these operators,
it is convenient to use the momentum representation because
of the translational invariance of the ensemble after ave-
raging. We refer to the quantity
or its ensemble average <G($,E) > as the Green function. The
spectral function is the diagonal matrix element of (11.2.2)
in the momentum representation.
p (LE) = <$I p (E) l ib = ~l<tl$ n n >I26(~-En) (11.2.4)
which tells us the momentum distribution of electrons with
energy E. For a perfect crystal, the II)~>'s are eigenstates
of crystal momentum and the spectral function is just a sum
of delta functions.
From the completeness of the set of states {I),}, we ob-
tain the sum rule
The trace of the spectral operator gives the density of
states per unit energy (for one Spin orientation)
which holds for any representation, as the trace is an in-
variant.
To obtain the ensemble average Green's function, we
expand the Green operator
= Go + GoVGo + Go VGo VG + 0
where Go (E) = (E-Ho)-'
2 e - where we choose ( h = 2m 2 - 1). Now, we introduce the
self-energy
-+ -+ Expanding [ E - ~ ~ - c (k,E) 1 - 1 in a power series in C (k,E) and
substituting from (II.2.8), we get back the diagonal element
of (11.2.7).
.'. G($,E) = 1 (11.2.9)
E-k2-c (&,E)
The self-energy ogerqtor C can also be defined without
recourse to perturbation theory:
<G($,E)>c = <GV>
Averaging (112.11) and using (II.2.10), we obtain the result
(11.2.9). (For review, see Ballentine 1974).
The essential criterion for the validity of perturbation
theory is the weakness of the potential. The true potentials
due to the ions in a real liquid metal are certainly not weak.
In order to apply Edwards' formalism to perform quantitative
calculations, the concept of the pseudopotential (Phillips
and Kleinmann 1959, for review, see Harrison) is very useful.
It is ciear from Chapter i that i% ;V / g+$> canmt be snzll
for all larger values of q, other than for q + 0, because
v($) is strong enough to bind the core electrons. But the
core states are of little interest, being so tightly bound
that they are essentially the same in a solid or liquid metal
as in a free atom or ion. Therefore, it is convenient to
introduce a p s e u d o p o t e n t i a l whose lowest eigenvalues corre-
spond to the valence eiqenvalues of the true potential.
Instead of the usual Schredinger equation
Austin, Heine and Sham (1962) have shown that the above may
be transformed into
containing the pseudopotential W. W is defined to be equal
to the true potential V in the interstitial region outside
of the ion cores, but is much weaker than V inside the ion
cores. The valence pseudo-wavefunction qv (t) is equal to the true valence wavefunction qv (;) in the interstitial
region, but @ (2) has no nodes within the core, whereas v
$v(%) must have several nodes in order to be orthogonal
to the core wavefunctions. In other words, %he conduction
electron wavefunctions must oscillate rapidly inside the
core regions so they may be orthogonal to the core wavefunc-
tions. The large positive kinetic energy associated with
these oscillations almost completely cancels the deep
negative potential within the cores, resulting in a weak
net effective potential. So + is smooth inside the core and the valence energy eigenvalue E is preserved.
The total screened potential W, within the small core
approximation (see Harrison) can be written as a sum of
spherically symmetric terms w centered on each of the N ions.
-f + 3 W (I) = Ew (r-Ra) and the plane-wave matrix elenent can
a be factored as
the first term depending only on the properties of a single
ion and the second term being a function of the positions
of the ions. a
The nth order term in the diagonal element of the expan-
sion G(E) can be written as
3 + 3 3 3 3
3 3 -+ * R +q *R +. . .+qn-RJ where Cn(qltq2t =qn) = e i(qk 0: 2 B
a,8,. . 3
The ensemble average of G(k,E) is obtained by replacing 'n
with its average
- - - + - + -+ Cn(q1,tq2t * = qn) = < C > 1
avs
which is related to the n-particle correlation function. 3 3 3
NSn(ql,q 2....qn) is the continuous part of <Cn> ave.
It is convenient to introduce diagrams to represent the
various terms. Some examples are shown in fig. 1. A solid
line represents a propagator go($,^) = (~-k~)-', an inter-
section of two solid lines with a dashed line (a vertex)
-b -f
represents a factor < k l w l k l > and a node connecting n-dashed
lines is related to the n-particle correlation function
. ' a"' a? '. '*.*A
3 a) A typical reducible diagram in the expansion of G(k,E).
(b-d) - Some irreducible diagrams in the expansion of
3
The self-energy x(k,~) consists of the sum of all irre-
ducible diagrams, i.e., those that cannot be separated into
two disconnected parts by cutting a propagator line such as
those in fig. (b,c,d) . G($,E) is the sum of all reducible
diagrams like the one in fig. (a) . alle en tine's review
paper (1974) is excellent in its details about the diagram-
matic approach, which we do not wish to describe any further
in this thesis.
Section ( 2 . 3 ) : Kubo-Greenwood Formalism
In this sect ion and in section ( 2 . 4 ) , we follow closely
the treatment in T. Chants thesis (1971).
We first describe the general theory of resistivity and
the Kubo-Greenwood formalism and then reduce it to a convenient
form for the d.c. case.
In the independent-electron model, the Kubo-Greenwood
formula for the frequency-dependent conductivity of a metal
can be written (in units % = 1) , as (Kubo 1957, Greenwood 1958)
The notation is the one in conventional use: f(E) is the
Fermi-Dirac distribution function, 4J the pth component of the one-electron current operator, and
where $r and Er are the eigenvector and eigenvalue of the
one-electron Hamiltonian.
Only the absorptive part, which is equal to the real part
in the absence of a magnetic field, needs to be considered
as the imaginary part is related to the real part through the
Kramer's-Kronig relation. Thus
In the absence of a magnetic field, the current operator
is given by
-#
where p is the canonical momentum. Expanding in terms of
momentum eigenstates, 1 k> , we have
(11.3.4)
On substituting (11.3.4) in (11.3.2) and introducing the
operator p (E) given by (11.2.2) , we have
In t roduc ing t h e Green o p e r a t o r ,
can be expressed as
< C / ~ ( E ) = - ?T I ~ < $ ~ G + ( E )
t h e m a t r i x e lements o f p ( E )
Following t h e n o t a t i o n of Langer (1960) and Neal (1970)
w e i n t r o d u c e
Using (11.3.7) and (11.3.6) , (11.3.5) s i m p l i f i e s t o
For a liquid metal, the ensemble averaging ensures
isotropy, and so the conductivity tensor is diagonal,
++ -+ K~-(~,E,E-~) - K (k,E,E-w) (11.3.9)
Equation (11.3.9) is the frequency-dependent generalization
of the expression for the d.c. conductivity given by Langer
(1960) and Neal (1970).
Section ( 2 . 4 ) : Diaarammatic Analvsis
Using the expression for the Green operator, as given by
(II.2.1), the total electron-ion scattering potential being V,
a diagrammatic expansion for K" can be provided. For example, lJv K+- (k,E1Ef ) is equal to the sum of all diagrams of the form
--?b - - --- . .
An upper solid h e represents a free propagator -f
+ + -+ Go (k,E) = (~+io+-k~ I - ' , a lower solid line
- fhE) , represents Go (k (~+io"-k' ) -' , an intersection of two solid lines with a dashed line (or vertex) represents a
matrix element <$lv/$'7 of the scattering potential due to
a single ion, and a node connecting n dashed lines represents -?- -f +
a factor NSn (ql.q2.. . .gn) . The free propagator line is represented by
---+-I
but
the full propagator G 6 , ~ ) = [E-~'-c (k,E) 1" is represented
by . -+
The leading diagrams are
Analogous to the Dyson equation for the one-particle Green's
function, a Bcthe-Salpeter type of equation (see e,g., Nozieres,
1964, Sec. 6.1) may be written for the ensemble average guan-
tity,
where the upper tiorizontal line is associated with complex
energy z, and the lower with z'. Here we have introduced
the irreducible interaction part
(with external pro-
pagator lines removed)
The above is the sum of all irreducible
grams, which by definition cannot be divided
interaction dia-
into two by
cutting both propagator lines but without cutting any inter-
action line. Thus,
K ( 2 , z . z ' ) = kU > P V 4
For this thesis (11.4.2) is the most important dlagram used.
For other aspects of the diagrammatic approach, we refer the
reader to the review given in Chanb thesis (1971) and Ambe-
gaokar ( 1 9 6 2 ) . -+
An integral equation to determine K(k,z,z1) can be ob- 3 3
tained fram (11.4.1). Defining a quantity P(k,klz,z') by
writing
-+ where G ( 2 , z ) denotes the ensemble average of G (k, z) , we have
Hence, K ( k , z , z 8 ) = k 2 G ( k , z ) G ( k , z ' ) + ~ ( k , z ) ~ ( k r z ' )
-+ For an isotropic system, ~(k,z,z') is independent of the
-+ 3 -+ direction of k ' , , W ( k , k " z , z s ) depends only on the relative
-+ 3 angle of and <" and P (kt ' , k , z , z ' ) depends only on the
3 -+ relative angle of k w and k'.
Following the treatment given in Chan's thesis (1971)
again, choosing $' ' as a polar axis in doing the 2 ' sum, and using isotropy, (11.4.5) simplifies to
with 0 being the angle between 2 and ' , and W being given
by (11.4.2).
Having thus obtained an expression for K , the conductivity
a can be evaluated using ( 1 1 . 3 . 9 ) .
Section ( 2 . 4 ) : Derivation of the Ziman formula from the Kubo-
Greenwood formalism
We first obtain an expression for the- d.c. conductivity
( w o ) from ( 1 1 . 3 . 9 ) . O n taking w +o, in ( 1 1 . 3 . 9 ) ~
L i m f (E-o) -f ( E ) becomes u+o W
which behaves like a delta func- aE
tion, -6 (E-EF) , as T+O. But, for a liquid metal, T is certainly * - - -
not zero. However, if T is small compared to TF, the ~ e m i
degeneracy temperature, it is a good approximation allowing
us to perform the energy integration very trivially. Hence
the d.c. conductivity is given by
In order to solve (11.4.6) and evaluate the conducti-
vity, we must examine the propagator G(k,E) in some detail.
If we evaluate the proper self-energy part Z(k,EF) and
write it as the sum of its real and imaginary parts, we have
From (11.4.6) and (II.5.1), we note that we need to eva-
+ - + + luate G G and G G . + G and G- are given by
+ 1 thereby giving G G- = - r r (E~-~z-A) 2+r 2
If ris small compared to EF and constant in the region near
E ~ - ~ ~ - A = 0, (11.5.3)
as it should for perturbation theory to hold, the expression
in (11.5.2~) has the familiar Lorentz shape. In this case,
r is a measure of the width of the spectral function.
From (11.5.2) , making use of the properties of delta
functions, we have the spectral function
The second i i n e of (11.5.5) is arrived at, by the property
6(k-ki) where k is of a delta function that 6(f(k)) = i
a root of f (k) = 0 and the
derivative is evaluated at k = ki.
In our case, k being a root of (II.5.3), we have F
hence, giving g = 1 in (11.5.5). 1+ ------ - 2k1 F it 1 k=kF
which we have obtained by expanding (11.4.6).
Now, substituting (11.5.5) in (11.5,. 6) , we obtain
Lim
- - ~oting that (ii.5.i) contains an i n f i n i t e geometric series,
we sum it and (11.5.7) simplifies to
2
k; wl + I 0 (1) dk
g 1- - 7
I' ZkF ki W1
(11.5.9)
It can be shown that this divergent term cancels the
++ divergence from IK d3k-. Bringer and Wagner (1973) have
derived the identity
+ -+. 1 K++ -+ $ 0 VkG ( k , E ) = (k,E) (11.5.10)
++ -+ Hence, j k 2 ~ e K (k,EF)dk=
+ G being given by (II.5.2a), we have, on differentiating with
respect to k,
ar aA aG+ 2 k - i - f - - - - ak ak ak (E*-~*-A) 2-I'2+2i~ ( E ~ - ~ ~ - A )
,
Taking the real part of (11.5.12) and taking the limit
as r -t 0, we have
a G+ -k2-~) 2-r2-2i~ (E~-~~-A)] ak
- Lim Re Re - - Lim r+0 r+o
[(EF-k2-A) 2-T'2] 2+41'2 (EFk2-A)
- - -
+- 1 Now, K falls off as r~ for large values of k as seen
from (11.5.2~) and (11.5.6) and so does Re K++, as seen from
1 1 . 5 1 3 ) Hence, on combining (11.5.9) and (11.5.11), we
obtain
The two divergences cancel, leaving behind only the first
term in (11.5.14). Hence, (11.5.1) reduces to
Generalizing the definition (11.4.7) of W1, we define 71
where 0 is the angle between and 2' and Pe (Cos0) are
Legendre polynomials.
Next, we z e d to obtain a relation between r and W,
Starting from the fundamental definitions for G ( z ) given by
(II.2.1), we have
Taking the ensemble average, we obtain
and so
(11.5.18) reduces to
3 + Defining (11.4.1) as ~ ( k , k',zlrz2) which has a diagrammatic
expansion and satisfies an integral equation like (11.4.39,
we obtain
Recalling (11.5.171, we have
Hence, we obtain
From (115.21) taking zl=E + io, z2 = E - io, we obtain
+ +,, T ( k , E ) = L (k,k ) r (k" ,E) (11.5.22) gs I (E-k' ' 2 - ~ ) 2+r2 (kt ' ,E)
When the r on the right-hand side of (11.5.22) is small com-
pared to EF, we have approximately
The last line is obtained by making use of the definition
W, given by (11.5.16) .
L e t us define g r1
= n-- k 2 F W1•‹ S i n c e I? i s given by
(11.5.23), we have
ne2T Now, comparing (11.5.15) with o = - m given by the nearly-
free-electron model, it becomes clear that 7 is related to
the relaxation time T.
The o n l y a s s u n ~ y t i . o n s underlying the derivation of the
Ziman formula are that I? is small in the self-energy term,
which implies we are assuming a nearly-free-electron model,
and the use of Born approximation. In other words, in the
expansion for the pseudopotential, W, we only retain the
lowest second order term. Diagrammatically, this may be
expressed as
W = =?= 0 *
Hence, substituting (11.5.25) in (11.5.24) gives us
Introducing (11.5.26) in (115.15) and a simple change of
variable in (II.5.26), as in Chapter I, leads to the Ziman
formula (1.1.1).
We have thus derived the Ziman formula by a general method
and this exemplifies how we may extend this to higher orders
in the Kubo-Greenwood formalism. In order to make the theory
applicable to higher orders, all that has to be done is to
replace W by as many terms as we want to include from (II.4.2),
instead of just (11.5.25) .
CHAPTER 111: CORRELATION FUNCTIONS
"A good notation has a ~ u b t Z e t y and suggestiveness which a t times
maka it seem almost l i k e a Zive teacher."
--Bertrand Russell
The equilibrium structure of a liquid may be described
in terms of acomplete set of n-body atomic distribution
functions. In this chapter, we introduce the notation
and describe systematically the general properties of
the correlation functions. Here, in some parts of the
chapter, we follow closely the treatment in Ballentine (1974).
The s-particle distribution function is defined such -+
that ns (R1, . ,,, 8 ) d3 R1 . . . d 3 ~ s is the probability of finding S
a particle in the volume element d\l centred on another
particle in d31Z2 centred on R2, etc. It is normalized so
that f...f~~~(%~, ...% s )d3~R1..d3~s = N(N - 1) ... (N - s + 1)
(111.1.1)
where N is the total number of atoms in the large but finite
volume of the system.
It is convenient to introduce dimehsionless distribution
functions,
which are normalized such that gs = 1 for a perfect gas.
N Here, n is the number density given by n = . We next introduce the set of dimensionless cluster
functions, which are defined so that
when t h e set of eo-ord ina tes { B ~ . . .st) a r e f a r removed
from t h e set . . . ss ) . The r e l a t i o n between t h e
d i s t r i b u t i o n f u n c t i o n s and t h e c l u s t e r f u n c t i o n s i s :
A s s t a t e d e a r l i e r i n Chapter I , it would be i n t e r e s t i n g
t o examine t h e n a t u r e of t h e s e f u n c t i o n s i n momentum
r e p r e s e n t a t i o n because of t h e t r a n s l a t i o n a l i n v a r i a n c e of
ensemble averages . Reca l l i ng t h e exp res s ion (1 .1 .3) f o r t h e
t h e s t r u c t u r e f a c t o r ,
w e may d e f i n e b2 ( q ) such t h a t
S2 ( 4 ) = 1 + b2 ( 4 ) (111.1.6)
Genera l iz ing t h e above, we d e f i n e t h e b - func t ions , t h e
F o u r i e r t rans forms of t h e c l u s t e r f u n c t i o n s
+- -+ The Kronecker d e l t a occu r s because hs(R1, ... Rs) i s
unchanged by a s imul taneous d i sp lacement of a l l of i t s -+ -+
arguments and s o t h e f u n c t i o n b s ( q l l . . .q 2 ) i s de f ined -f -+ -+
on ly when ql + q2 + ... + qs = 0 .
c o r r e l a t i o n s ~ f t h e c o l l e c t i v e v a r i a b l e s p+, de f ined g
i n Chapter I . The ensemble average of a produc t of
t h e s e c o l l e c t i v e v a r i a b l e s
< p-+, p-t ... p+ > i s t h e i n t e r e s t i n g q u a n t i t y , a s it 9 1 92 9s
c o n t a i n s t h e necessary in format ion about s - p a r t i c l e
c o r r e l a t i o n f u n c t i o n s .
Reca l l i ng t h e exp res s ion (1.1.9) f o r t h e t r i p l e t 3 3 +
s t r u c t u r e f u n c t i o n S (ql , q 2 , q3) , w e can break it up 3
a s fo l lows :
where t h e b ' s a r e de f ined by (111 .1 .7) . The above
terms are ob ta ined a s fo l lows :
The t e r m s w i t h a = ' B = y y i e l d
3 3 -+ because ql + q2 + q3 = 0 .
N 3 3
1 Those w i t h a = B y i e l d e i (ql+q2 (ga-Sy) and on LA a ,
t a k i n g t h e ensemble average , w e o b t a i n b 2 ( q l ) , and 1
s i m i l a r l y f o r b2 (q2 ) and b 2 ( q 3 ) . hose w i t h a # B # y
A s s t a t e d i n Chapter I , (I. 1 . 8 ) , t h e t h i r d o r d e r t e r m
i n t h e p e r t u r b a t i o n expansion invo lves t h e t r i p l e t s t r u c t u r e
- + 3 +
function S3(ql, q2, q 3 j . Enough experimental and
theoretical facts are known about the structure factor and
b2(q). So it is necessary only to examine the properties -+ -+
of b3(bl, q2, q3) in our study of the triplet correlation
function.
We first study the behaviour of the triplet distribution -+ +
functi~ng~(;~~! r23, rjl) under certain limiting conditions.
We then examine what constraints are imposed upon -+ -+
b3 (Gl r q2 t q3) under these conditions.
It is known that two.particles can never overlap.
Hence,
Writing the above in terms of cluster functions as in
(111.1.4) , we have
But, g2(r) = 0
lim r+O
Hence, h2(0) + 1 = 0
(111.1.10)~ thus, reduces to
The equivalent of the above equation in momentum space
is given by
J
Furthermore, we know that when particle 3 is greatly
separated from particles 2 and 1, we obtain I
h 3 q 2 ' "; ) = 0
and h2 (a) = 0 (111.1.13) 3 3 3
So, g3 (r12, = h3 (r12 , -, a) + 2h2 (a) + h2 (r12) + 1
= g (g 1 2 12 ( 1 1 1 . 1 . 1 4 )
3 3 3 3
If b3 (ql t q2 , -ql, -q2) i s a smooth f u n c t i o n , t h e
Reimann-Lesbegue Theorem (see Hartmann 1962) , ensu res t h a t
c o n d i t i o n (111.1.13) i s s a t i f i e d .
g3(123) can never be n e g a t i v e , as it i s a p r o b a b i l i t y
d i s t r i b u t i o n f u n c t i o n . However, it i s d i f f i c u l t t o
v i s u a l i z e i t s equiva lence i n q-space. Hence, it i s
d i f f i c u l t t o s a t i s f y t h i s c o n d i t i o n .
W e have o u t l i n e d a few p r o p e r t i e s of t r i p l e t c o r r e l a t i o n
f u n c t i o n s i n bo th r and q space. I n Chapter V , w e s h a l l 3 -+ 3
i n v e n t an approximation f o r b3(q l , q 2 , q3 ) g i v i n g t h e
c o r r e c t long-wavelength l i m i t (see Chapter I V ) and
s a t i s f y i n g t h e p r o p e r t i e s (111.1.12) o r (111.1.11) and
(111.1.13). We need a t r a c t a b l e and a c c u r a t e approximation
f o r t h e t r i p l e t c o r r e l a t i o n f u n c t i o n , i n t h e c a l c u l a t i o n o f
t h e t h i r d o r d e r t e r m i n t h e Kubo-Greenwood formalism.
CHAPTER IV
TRIPLET STRUCTU3XE FUNCTION IN THE LONG WAVELENGTH LIMIT
In this chapter, we make use of the thermodynamic fluctu-
ation theory to derive an exact expression for triplet and
higher order structure functions in the long-wavelength limit.
Sec. (4.1) : Fluctuation Theory
Chapter I,
(IV. 1. la)
+ +; +< = 0 but no smaller subset of the q vectors sums if 2 3
to zero, it is apparent that structure functions represent
c~rrelations cf density fluctuations. A long wavelength
fluctnation may be treated thermodynamically. Callen (1960,
Chapter 15) has set up a general thermodynamic fluctuation
theory, which is applicable to the problem of a liquid metal
subject to density fluctuations.
As is commonly known, a macroscopic system such as a
liquid metal undergoes rapid and incessant transitions
amongst its microstates and as a consequence, the extensive
parameters of systems in contact with reservoirs undergo
macroscopic fluctuations.
Parameters that have values in a composite system equal
to the sum of the values in each of the subsystems are called
extensive parameters. These play a key role in thermodynamic
fluctuation theory. Examples of extensive parameters, denoted
by Xk, are the internal energy U, the macroscopic volume Q ,
and the number of particles, N. Entropy, in general, is a
function of the extensive parameters, X,_. Mathematically,
this may be expressed as
s = s (X,)
The derivative of the entropy with respect
sive parameter, holding all the other extensive
constant, is
Temperature,
of intensive
Suppose
with entropy
defined as an intensive parameter,
to an exten-
parameters
denoted by Fk.
pressure and the chemical potentla1 are examples
parameters.
a system with entropy Sl connected to a reservoir
S2, by a diathermal, movable, permeable piston,
through which the extensive parameters Xk can flow.
Then, the entropy of the composite system is given by
S (Xk) = S1+S2 (IV. 1. lb)
Macroscopically, entropy is defined as
S = kB log W (IV. 1.2)
where k is Boltzmann's constant and W here is the proba- B
bility distribution. In other words, we have
W = Ce S/kg
where C is a constant, in the sense that it is independent of
the fluctuating variables, namely, the extensive parameters,
but it can depend on the intensive parameters.
(2) in the two and Xk An extensive parameter has values Xk
subsystems composing the composite system. The instantaneous
value of entropy is a function of the extensive parameters.
(1) ) S1 = S(Xk
(2 ) ) S2 = S(Xk (IV. 1.4)
rough this diathermal,
movable, permeable piston.
Although the extensive parameters in each subsystem can
fluctuate, those of the composite system remain constant at
every instant (provided the composite system is isolated).
' * ) = F~ = constant Xk + Xk (IV.l.5)
(IV. 1.6)
where the bar in (fV.1.5) indicates an equilibrium value.
Since entropy is maximum at equilibrium,the equilibrium
and xi2), if unconstrained, are determined values of Xk
by the vanishing of the quantity as as or- where the ax:*) '
derivative is to be evaluated at equilibrium. Our condition
may be expressed as
(1) From (IV.1.6), we have dXk (2) = -dXk . reduces to
which means
(IV. 1.7)
Hence (IV.1.7)
The bar in (IV.1.9) indicates an equilibrium value, as it will
in all of the following discussion,
In the limit of an infinite system, when thermodynamics
applies, the distribution function is sharply peaked about the
equilibrium value. Hence only those values of the extensive
parameters near the equilibrium value are important. Assuming
now that the reservoir is very much larger than the system,
1 << lzi2) /and hence << IEkl. we have Ixk
Expanding in a Taylor's series about equilibrium, we obtain
(IV. 1.11)
(IV. 1.12)
Substituting (IV. 1.11) and (IV. 1.12) in (IV. 1.10) , we have
(IV. 1.13)
Higher order terms in the Taylor's series contribute only
negligibly, as only the values near equilibrium are expected t l \
to be important and I xi'' I << 1 Xk 1 . Taking the sum of S1 and S2, we obtain
Substituting (IV. 1.14) in (IV. 1.3) , we obtain
(IV. 1.15)
We, now, apply the above result to derive expressions
for the moments of fluctuating variables. The deviation from
equilibrium of a fluctuating variable is given by - sx = x - x .
j j j
The second order moment is, by definition,
(IV. 1.16)
On differentiating (IV.1.15 with respect to Fk, we obtain
(IV. 1-17)
Substituting (IV. 1.17) in (IV. 1.16) , we have
a - But 2- ( 6 X j ) = - aFk a% ( X j - X j )
In obtaining (IV.1,19), we have made use of the fact that X j '
being an extensive parameter, is independent of Fk. However, -
th.e equilibrium value of the extensive parameter, X . is I
determined by the vanishing of (IV.P.7) and hence upon the
existence of Fkk'
We note that I GX.WdXo....dXs = <6X.> is zero. This is 3 3 -
because X j
= X in the limit of an infinite system, when j '
the distribution function is sharply peaked about the equi-
librium value. Hence, on substituting (IV. 1.19) in (IV. 1.18) ,
(IV.1.18) reduces to
(IV. 1.20)
where the partial derivative is to be evaluated holding all
the other intensive parameters, Fo, F1# .... Fk+lt...m
Fs constant.
Extending the same argument to more variables, w e can
obtain expressions for third and higher order moments. In
the third order case, have
Treating the quantity in parenthesis as a unit and proceeding
as in (IV.1.20), (IV.1.21) may be expanded to
a a I6x.GX.WdXO....dXs+kB .f W - -k - ( 6 ~ ~ 6 ~ . ) d ~ . . . .dxs " aFk 1 3 aFk 3 0
(IV. 1.22)
(IV. 1.23)
as <6xi> and <6X :, vanish. j
We have thus been able to express the third order moment
in terms of the second order moment.
Xn a ai~kti lar manner, we can extend the above discussion
to the case of the nth order moment.
<6x16x26x3.. . 6x,> = a j ( 6 ~ ~ 6 ~ 2... 6xn I)~d~o.. .dxS -kB aFn
-
a +kg I w - (6x16x2.. . 6 x,- ) dxo . . . dxs a i? n
axn- =? - k -- a ~ 6 x 1. a * 6Xn-1 > - kg <6x1.. . &Xn-2> ...
B ag n
a% a",-2 - k <SP16X2.. . ..... (IV.1.24) B ai? n
(IV. 1.24) contains one (n-1) th order moment and (n-1)
terms of (n-2) '' order moments. The (n-2) th order moments
may be simplified and written in terms of (n-3Ith order
moments, and so on.
In other words, we may express higher moments in terms
of lower order moments.
Sec. (4.2): Application of Fluctuation Theory to the
liquid structure functions:
The problem of liquids is characterized by density -t 3
ik'% fluctuations. The quantity p, = C e , introduced in
k N
Chapter I, is the Fourier transform of the particle number
distribution, and is subject to fluctuations. Furthermore,
it is an extensive parameter. The intensive parameter corre-
sponding to p+ is also sinusoidally varying and is denoted by .I k i
(- =) V,- In the long-wavelength limit, as k -+ 0, u, T k k
approaches the ordinary chemical potential, p.
This may be derived as follows:
From thermodynamics of a single-component, homogeneous system,
we have
from which we obtain
From (PV.2.P), it fviiows t h a t the intensive parameters
corresponding to the extensive parameters U, Q, N are
1 P l-' ?;' T and - .
Generalizing (IV.2.1) for an inhomogeneous system, we
have .,
(IV. 2.3)
We, hence, obtain 1-I-b
Fourier components of density fluctuations are defined
Considering the second order moment of density fluctuations,
we have, from (IV.1.20) I
ab k,
where the derivative is to be evaluated holding the other in-
tensive parameters constant, in this case T, and also the
extensive parameter R .
We may, hence, write
where, in the first line, we have introduced
lim k + O
For a homogeneous, single-component system, we have the
thermodynamic relation
pdS2 + S2dp = SdT + Ndv: Hence, we obtain N = S2 (%) Tf S2
from which, we have
(IV. 2.5)
The above is familiarly known as the Gibbs-Duhem relation.
Substituting (XV, 2.6) in (IV. 2.4) , we obtain
(IV. 2.7)
where kT is the isothermal compressibility, and T, is the
absolute temperature. Hence, from (IV.2.7), we verify that
the long wavelength limit structure factor S2(0) is given by
n k TkT, as stated earlier in Chapter I. no is the number density. 0 B
In a similar manner, we proceed to derive exact expres-
sions for higher order structure functions in the long-
wavelength limit.
Considering the triplet case, where we have kl+k2+k3 = 0,
we may write, using (IV.1.23),
a <P+ P-+ > 'P, P, P, > = -ks kl k2 k3 k3 kl k2
(IV. 2.8)
This is a result we have derived originally, by principles
that are fundamentally simple. Hence, we arrive at the
important result that
Extending the same argument to the n-particle case,
using (IV. 1.24) , we obtain
( I V . 2.10)
-%
In t h e above, it is assumed t h a t Sl+ik2+. . . .+kn = 0 b u t no 3 3
s u b s e t o f t h e remaining kl, .... kn-l sums t o zero. Hence
, , t h the { n - ~ , order aoxients in (IV. 2.10) vanish, due to the
e x i s t e n c e of t h e d e l t a - f u n c t i o n (see B a l l e n t i n e 1974) . So,
on ly t h e f i r s t term in (IV.2.10) remains. On t a k i n g t h e
long wavelength l i m i t i n t h e above, it s i m p l i f i e s t o
( I V . 2. l l a )
W e , hence, o b t a i n t h e impor tan t r e s u l t
( I V . 2.11b)
In the above discussion, it is assumed that we take the Limit
as kn+O. but not kn= 0. in <p+ p+ ....p+ >, which is a
kl k2 k n -t +
discontinuous function. sn (gl. g2 , . . . . . kn-l ,k ) is the con- n
tinuous part of the above discontinuous function, recalling
Chapter 111.
We have, thus, obtained a generalization of(IV.2.7).
that is of fundamental importance to the study of higher order
terms in the liquid metal problem. This result shows that if
the liquid structure is sufficiently resistant to compression,
any partial long wavelength limit of any order structure
function will be small, and so will cancel the large value of
the screened potential in that limit.
Equatinn !LV.2,9) is of significant importance to svbse-
quent portions of this thesis.
Sec. ( 4 . 3 ) : -- RePatidan to other Work
Egelstaff et a1.,(1971) have derived a relation con-
necting the triplet distribution function in real space with
the pair distribution function, using the theory of Schofield
(1956).
We can Fourier transform Egelstaff's equation and obtain
the same result (IV.2.9). This also serves as a means of
checking our result.
Egelstaff's paper has the following equation:
Writing the g3 in terms of h and h2's as ir. Chapter 111, 3
and Fourier transforming the right-hand side of (IV.3.1)
yields
(IV. 3.2)
where we have used the Dirac delta function. The notation
here is the same as we previously introduced. (IV.3.2)
simplifies to
The left-hand side of (IV.3.1) may be simplified to
d n k T -[n2h2 (r)+n2], making use of the relation between o B ap
the distribution functions and cluster functions. On per-
forming the differentiation in the above, we have
We Fourier transform the above to obta in
which simplifies to
In the
made use of
(IV. 3.5)
above discussion of Fourier transforms, we have
the fact that the b-functions are the Fourier
transforms of the h-functions as defined in Chapter 111.
Now, equating ( Iv .3 .5) and (IV.3.3), we get
NOW, recalling that
we may obtain (IV. 2.9) from (IV. 3.6).
3 + In (IV.3.7j, when q3+0, and ql= -q2, all the terms are
of the order of unity and it is not at all obvious that the
various terms cancel each other, leaving a very small number.
But our method of deriving (IV.2.9), using fluctuation theory,
makes the answer immediately transparent. It is at once clear
that higher order moments in the long-wavelength limit are
apt to be smaller than lower order moments, involving
successive order pressure derivatives, as they do. The magni-
tudes of the various order terms are not apparent in the
second approach. Further, it is not easy to extend the
second approach to higher orders.
Yvon (1969) has calculated higher order moments in the
framework of the canonical distribution. He has, however,
only considered the case when all the k's approach zero.
1 He defines B g = -- ?J p+ and a = - . His results are r k
(IV. 3.8a)
(IV. 3.8~)
Equivalent results are easily obtained by our theory as
follows :
Recalling (IV.2.11), we note that <P+ P+ .. . .P, >
kl k2 k n
lim kn+O
may be expressed in terms of the derivative with respect to
the chemical potential of the (n-l)th order moment. NOW,
letting kn - 1+0, the (n-llth order long-wavelength limit moment may be expressed in terms of the (r1-2)~~ order moment
and so on. We obtain
lim kl,k2,. . .k lim kl, k2,. . . knml +O n+O
lim kl, k2,. . . kn-2+0
lim k+O
Yvon's results follow immediately by identifying his
variables in terms of ours.
CHAPTER V: LONG WAVELENGTH STRUCTURE FUNCTIONS:
HARD SPHERE MODEL AND COMPARISON WITH
EXPERIMENT
In this chapter, we use the hard sphere model and
experimental data to illustrate the theoretical results we
obtained in Chapter IV.
Section (5.1) : Hard Sphere Model
Our next task is to compare the orders of magnitude of
the various order structure functions in the long wavelength
limit. From (IV.2.10), it is apparent that we have to eva-
luate higher order pressure derivatives, when we are to
evaluate higher order structure functions in the long wave-
length limit. The hard sphere model readily allows us to
carry out higher order differentiations, and hence permits
us to compare S2, S3, S4, etc. We may also use it as a theo-
retical model for comparison with experiment.
Ashcroft and Lekner (1966) cite an expression for the
structure factor, using the hard sphere model. We quote it
here.
where we have the structure factor as a function of a dimen-
sionless variable ka. 0 is the hard sphere 'iameter and n
is the packing fraction, defined as ?no3, where n is the
number density. The terms C1, C 2 and C3 are given by
Here, a = (1+2rl) 1
(1-11)
The Percus-Yevick approximation gives an expression for
the structure factor in the long wavelength limit.
Consfdering the triplet case, we note from (IV.2.9)
that we need to obtain an expression for the pressure deriva-
tive of the structure factor in terms of the hard sphere
parameters. We qeglect any variations of o with respect to
pressure and treat it to be rigid. Now, nkgT a s 2 ( k ) may be
ap
written as
as2 (k) = TI S2(0) (V. 1.3)
ar l
-> -3. as2 (k) Hence, S3 (kt -ky 0) = cS2 ( 0 ) + S 2 (0 ) S2 (k) (V.1.4)
aTI
We may, thence, use (V.1.4) to plot the triplet structure
function in the long wavelength limit, as a function of the
dimensionless variable ko, for comparison with S g ( k ) .
Similarly, it is possible to go to higher orders. For
the quadruplet case, we have, from (IV.2.10) ,
Since S3 (Sl I%3) is known only when one of its arguments -+
vanishes, we can evaluate (V.1.5) only by letting k j also -+
approach zero. Hence, letting k3 also approach zero, we
have
In the above, as in earlier cases, the momentum vectors form
the sides of a closed polygon and no smaller subset of the
% vectors sums to zero.
We, then, introduce the hard sphere model in (V.1.6)
to obtain
A s h c m f t and Lekner found that n = .45 was a good value
for most liquid metals, at the melting point. We quote a
few numbers to present a comparison. For n = .45,
Here, w e note that S3. S 4 are successively smaller than
s2 (0 )
~,(~,-~,0,0), S3($,-%,0) and S 2 ( k ) are all plotted as
functions of ka and the results are presented in the adjoin-
ing graph. S3 (%,-2.0) is found to be much smaller, almost
F i g u r e I: Hard Sphere Model: SZ (q) ,S3 (q,-q. 0)
-+ ten times smaller, than S (2) and S (~,-$,o,o) is found to 2 4
be even smaller than s 3 (c ,-g, 0) .
Section (5.2) ; Experimental Data
Tsuji, Endo, Minomura and Asamui (1972) have measured as (q)
2 S2h) and ( ap )T from X-ray scattering experiments, for
sodium and potassium at 140•‹C and 120•‹C respectively. Egelstaff
et ai.,(1971) have studied rubidium, using neutron diffraction as
2 techniques, and report experimental measurements of nok,T(ap)T.
3 3 Using these experimental data, we calculate Sj(q,-q,O),
and compare with our hard sphere results forn = 0.45, reported
to be a good value for most liquid metals at the melting point, Ill
(see Ashcraft and Lekner, 1966) and witha = 6.2 Bohr radii for ( I Ill
Ill Na, 7.68 for K, and 8.14 for Rb. The hard sphere diameter Il,
I Ill
values are also from Ashcroft and Lekner (1966). We present I I Ill
I I
our results graphically in figs. 11, I11 and IV.
For Rb, we note the fairly close agreement, although the
experimental values of the pressure derivative approach zero
for q+m and approach the q+O limit, far more rapidly than do
the corresponding hard sphere values. - .
The experimental data for Na disagrees greatly with the
hard-sphere data. Although the general behaviour is similar,
the peak value in the experimental case is almost thrice as
large as in the hard sphere model. There is a great difference
-t -t Rb: Hard Sphere and ~xperimental Sj(q,-q.0) vs q
Figu r Na: Hard Sphere and Experimental Sg (;,-;, 0)
-+ 3 Figure 1V:~:Hard Sphere and Experimental Sj(qt-q.0) vs q
- Ex per imenta l
- - - - - - Hard Sphere
between experimental and hard sphere data for K also. Tsuji
et al., report that a fixed packing fraction model describes
adequately the height of the first peak in the structure factor
of liquid Na under pressure while a fixed diameter model is more
appropriate to account for the behaviour of liquid potassium
around the first peak in the structure factor. However, from
our graphs,it is apparent that the fixed diameter model does
not represent the behaviour of liquid K. The experimental 3 -t
S3(q,-q,0) is as much as eight times as large as the fixed
diameter model value. The first peak in the structure factor, 0 - i
as measured by Tsuji et al., occurred at q = l.61A and from
our graphs, we notice that the experimental data does indeed
coincide with the hard sphere T C I O ~ ~ , rather f~rtci tsasly.
Hence, their conclusion is not valid. 3 3
For Rb. the experimental Sj(q,-q.0) is small compared to
unity, whereas for Na and K, it is larger than, almost twice
as large as unityewhile the hard-sphere model adequately fits
the experimental structure factor curve for all the alkali
metals (see Ashcroft and Lekner, 1966), the fact that it does 3 3
not, for Sj(q,-q,O), for Na and Kt make this experimental data
appear somewhat dubious. It is surprising that Na and K
behave dissimilarly from Rb. Hence, we shall use only the
experimental data for Rb, in our calculations in Chapters VI
and VII.
Section 5.3) : Structure unctions when aPP q's approach zero
We, next, specialize to the case when all the q's are
zero. S3 (0,0,0) and S4 (0,O. 0.0) are evaluated for mercury, as
experimental data exists o n l y for Hg- The measurements of
pressure versus volume by ~ridgmann(l911) and by Gordon and
Davis (1967) are used in this calculation.
The is~thermzl compressibility is
The structure functions when all the Cl' s approach zero are
given by
i the general case Sn (0,O , . . . . , 0) being given by (IV. 3.9) .
carry this out by fitting polynomials of R as a function of p
1 and as a check, as a function of P . We start with lower order
polynomials, calculate coefficients and proceed to successively
higher order polynomials, When the coefficients of two con-
secutive polynomials begin to disagree, we stop and choose the
order of the polynomial that is appropriate. A third order
polynomial is suitable in both cases. We, then, calculate
S3(0,0,0) and S4 (0,0,0,0) using 07.3.2) and present the results
in tabular form. The long wavelength limit structure factors
agree very closely in the two cases, that is, using ~ridgmann's
data and Gordon and Davis' data. In the case of S3(0,0,0,),
there is about 40% disagreement, whereas for S4(0,0,0,0), there
is disagreement even regarding the sign. In (V. 3.2) , it is the
highest order pressure derivatives of the compressibility that
make the greatest contributions and the sign that they have
affects the results vastly. Hence the disagreement in sign in
Sq(O,OIOIO) using the two sets of data. [For a presentation
and comparison of the data, see Ross and Greenwood 1969).
r. .-, * I - ' 0 I 4 0 x rl !n X Ul I-
" cn co I P.7
N N N d d 3 - I
I 1 1 0 0 0 r l r l r l X X X
I I I 0 0 0 r i r l r l X X X
rl rl r-l X X X !-A m r! co cn m
( U C V C \ I I l l
X X X cc! '2.2 L? m m cn
B
C V ( U w I I I
['- a r g w 0 0 0 0
0 0
i Section (5.4)
1 When the
A thermodynamic method, using the pressure derivative of sound velocity
equation of state is not known, a thermodynamic
calculation for S3 (0,0,0) is possible using the pressure
derivative of sound velocity.
S3(0,0,0) is given by
The velocity of sound vs is related to the adiabatic
compressibility by
where y is the ratio of specific heats 3 , c,
and P is the density of the material.
Differentiating (V. 4.2) with
from which we obtain
(V. 4.2)
respect to pressure, we have
Now, we may express Y in terms of the expansivity a , the
absolute temperature T and the specific heat at constant
pressure, C P
Hence, we obtain
The expansivity a is, by definition,
aa a 1 a n Hence, - 1 = --I- (-1 1 a? - ap n aT p T
'I'
Differentiating (V.4.2) with respect to T, we obtain
(V. 4.5)
(V. 4.6)
Using (V.4.7), (V.4.6) and (V.4.5) and (V.4.3). we obtain
(V. 4.8)
In (V.4.89, the last two terms contribute only of the order of
about 5x10- 3 , the second term makes the major contribution,
of the order of about 7.
1 ac
In (V. 4.8) , - KK, $ may be written as P *
The above has been obtained by using ordinary thermodynamic
M relations. S is the entropy and V is the molar volume, - P
where M is the atomic weight and P is the density of the substance.
W e use t h e vs(p) and vs(T) experimental d a t a of Coppens
e t a l . ! (1967) f o r Hg a t a temperature T = 303OK. i n our calcu-
l a t i o n of S3 (0.0,O) . I
I n t h e evalua t ion of (V.4.9), t h e expans iv i ty a may be
c a l c u l a t e d from t h e dens i ty d a t a from t h e Handbook of Chemistry
D ~ ~ ~ e i ~ c (1966). at 30•‹C is l.88xl0-' (deg)- ' and a t 40•‹C u r l u A a.1 -A'-
i s 1 .81~10- ' (deg)- ' , hence g iv ing aT = 0 . 0 7 ~ 1 0 - ~ (deg)-2.
The sound v e l o c i t y vs a t T = 303OK is 144600 cms (set)-'
(Coppens e t a l . . l 967) . C (T ) d a t a from Hultgren (1963) is used ac P
i n eva lua t ing & = 1.73~10- 3x4. 186~10 'e rgs (deg)-2 (gm)- '
= 6 . 6 9 ~ 4 . 1 8 6 ~ 1 0 ~ e r g s (g-rn)-' (deg)- '
From ( V . 4 . 4 ) , y = 1.0008
Hence, using (V.4.8)1 we o b t a i n
These values are in agreement with those calculated in Sec.
( 5.3) , using Gordon and Davis ' data.
Section 5.5: Comparison with other theories
Egelstaff et a1.,(1971) and Toombs (1970) report that
the Kirkwood superposition approximation yields about +4.5 for
asz (9)
nok~T ap and + 4 . 6 for S3(0,0,0) for Rb.
Egelstaff et aP.,(1971) have roughly a s2 (9)
magnitude for nokgT 8~
near the triple
estimated an order of
point to be
approximately -4x10-', in agreement with our evaluation of the
same, using the hard sphere model. From the hard sphere model.
we obtain S3 (0.0,O) to be of the order of -2xl0-~ . Our
calculation of the same for Hg using experimental data has
yielded a value of - 2x 10- * . The superposition approximation
is neither in agreement with the sign nor the magnitude, and
hepce is a poor approximation for our purposes.
Toombs ( 1970 ) , has used a co l l ec t ive movement theory t o
est imate S3(0,0,0) for Rb t o be approximately - 3 x 1 0 - ~ .
According t o our estimate, h i s value i s much too small.
BISDEZS FOX TEE TRIPLET STRUCTUIiE FUNCTION
I n t h i s chapter , we l i s t some of t h e phys ica l cond i t ions
t h a t any model t r i p l e t s t r u c t u r e func t ion should s a t i s f y ,
bes ides g iv ing t h e c o r r e c t long-wavelength l i m i t , (IV.2.9).
We, then , s e t f o r t h some models f o r t h e t r i p l e t b-function, + + 3
b3 (41 , •÷2~•÷3) which forms p a r t of t h e t r i p l e t s t r u c t u r e funct ion .
A s s t a t e d i n Chapter 111, it i s s u f f i c i e n t t o examine t h e 3 3 -F -+ 3 -+
p r o p e r t i e s of b3 (ql,q2,q3) , i n our s tudy of S3 (ql,q2.q3) I a s
enough t h e o r e t i c a l and experimental f a c t s a r e known about t h e
structure factor and hence of b 2 ( q ) .
Sec t ion ( 6 . 1 ) : Conditions t o be s a t i s f i e d by any model
and d a t a used i n t h e c a l c u l a t i o n
The va r ious condi t ions t o be s a t i s f i e d i n momentum space
a r e a s follows:
(i)
(ii)
(iii)
i
I n o t h e r words, ql,q2,q3 should form t h e s i d e s of a
t r i a n g l e . 3 3 3
The model funct ion f o r b3(ql,q2,q3) should be symmetric
+ + as, (q) b3(q,-q,0) = nokgT a p + [S2 ( 0 ) -21 is2 (q) - 1 1
( i v ) b3 (m,m,q) = 0.
This cond i t ion is necessary s o t h a t i ts Four ier t r ans -
form w i l l be convergent.
(v) W e r e c a l l equat ion (111.1.12) from Chapter 111.
( V I . 1.2)
3 3 3
( v i A s s t a t e d i n Chapter 111, b3(ql,q2,q3) should be a
smooth funct ion .
We, next , l i s t below t h e va r ious condi t ions t o be s a t i s f i e d
i n r e a l space. We merely r e c a l l Chapter I11 here.
( v i i ) From t h e f a c t t h a t no two p a r t i c l e s may over lap , w e
obta ined i n Chapter I11 t h e cond i t ion
( V I . 1.3)
A s noted i n Chapter 111, ( V I . 1 . 2 ) is merely t h e
equiva lent of ( V I -1 .3 ) i n momentum space.
A s d iscussed i n Chapter 111, on looking a t t h e
equiva lent of t h e above i n momentum space, w e o b t a i n t h e con-
d i t i o n t h a t b3 should be a smooth func t ion , which we have a l ready
s e t f o r t h i n t h e l i s t of condi t ions t o be s a t i s f i - e d i n q-space.
I n any c a l c u l a t i o n f o r t e s t i n g t h e s u i t a b i l i t y of a
model, we need information on t h e p ressu re d e r i v a t i v e of t h e -t 3 '
s t r u c t u r e f a c t o r , ( s e e V I . 1.1) . I n o rde r t o form b3 (q,-q, 0) ,
we use the experimental data of Egelstaff et a1.,(1971) for as, (s?
L the pressure derivative of the structure factor, nokBTap I T at 60•‹C and the structure factor SZ(q) for Rb. We use a grid
of 150 points, for q ranging between O and 7.5 (A0)-', in
I steps of 0.05 (A") -' . When working in r-space, we evaluate
Fourier transforms at 150 corresponding points in r space,
'IT such that Ar = EP-== where Aq is the spacing in q-space. The experimental data ranges only from 1 (A0)-I to 2.5 (A0) -1.
As experimental data does not exist for the q+O limit, we use
the hard sphere value for this point. Intermediate values,
from q=O to q=1 (Ao)-', are obtained by interpolation. From as2 (4)
the experimental data it is apparent that nokgTap is
very small even at 4=2. 5 (A0)-'. Hence, we extrapolate to
zero, beyond c~=2.5 (A')-~. h 2 (r) is obtained by Fourier
transforming b2(q) For values of r<o, the hard-sphere
diameter, h2(r), instead of being equal to -1, as it should,
behaved in a random, oscillatory fashion. This is due to
accumulation of experimental error, as is common in all
experimental data. In order to make h2(r) behave smoothly
within rca, Ballentine adopted the following approach.
In the small r region (r<o), the value of h2(r) is
truncated to -1. h2(r) is then Fourier transformed to
o b t a i n b 2 ( q ) . This b2(q) i s compared wi th t h e experimental
curve and wherever it departed from t h e experimental curve
beyond t h e l i m i t s of experimental e r r o r , b 2 (q) i s s e t equal
t o t h e experimental va lue p l u s t h e maximum e r r o r . Now
b (q) i s Four ier transformed once again t o o b t a i n h 2 ( r ) . 2
This i t e r a t i v e procedure i s t o be continued u n t i l convergence
i s obtained. But it was not p o s s i b l e t o o b t a i n abso lu te
convergence. Using t h e o r i g i n a l d a t a , t h e maximum dev ia t ion
of h2 ( r ) from -1 i n t h e region r < o , was 1.07. Af te r t h e
i t e r a i i v a procedur~, t h e dev ia t ion is only 0.22, an improve-
ment of 80%.
The same corrective procedure i s adopted f o r t h e
pressure d e r i v a t i v e of t h e s t r u c t u r e f a c t o r . Using (111.1.51, as, (s>
4 - t h e Four ier t ransform of n k T i s given by 0 13 ap
For r < a I t h e above reduces t o -S2(0) . The i t e r a t i v e procedure
is performed. The maximum dev ia t ion of t h e Four ier t ransform ah2 (4)
of nokgT ap
from - S 2 ( 0 ) , using t h e o r i g i n a l d a t a , w a s 0.182.
A f t e r +he i t e r a t i v e procedure has been c a r r i e d o u t , t h e devi-
a t i o n i s 0.062. I n t h i s case, t h e inprove6 d a t a is no t q u i t e
a s good a s i n t h e previous case.
Sec t ion (6.2) : Models f o r b3 (qlrq2,q3) :-
We s h a l l f i r s t work i n q-space, t o inven t a model and t o
t e s t whether it s a t i s f i e s t h e va r ious phys ica l condi t ions set
f o r t h e a r l i e r .
Let us d e f i n e f (q) = b 3 (q,-q,O) given by P1.1.1). - The following approximation f o r b3 (qlt q2 q3) -
+ w ~ q , ) f ( q , ) l ".- -.--- ..-" - - -.- .-.em-
w (q I ) +W Jq2) +W (q3)
g ives t h e corcect long wavelength l i m i t . Here, W(q) is a
n func t ion slxch t h a t W ( O ) = 0 and W ( m ) = We choose W (q)=q
where n>3. I n t h e case when n<3, t h e i n t e g r a t i o n (VI. 1 . 2 )
w0111.d diverge.
The
b3 '
approximation
j 3 - f
'(ql I q2 1 q3) = (VI. 2 . 1 )
where n 2 4, satisfies conditions (i), (ii), (iii), (ivj and
(vi). But it fails to satisfy condition (v). On integrating 3 3 3 - f 3
b3(qltq2, -q1,q2) with respect to ql, we obtain a function of + q2, oscillatory in its behaviour, the amplitude of oscillation
3 increasing with q2, for all values of n > 3. The difficulty
comes from the second term, which yields
91 where x = - . C42
3 For any value of n>3, the above reduces to 2n q 2 f(q2) C4,
where C4 is a constant, independent of q2. In the hard-sphere Cos (q2)
model, f (q2) behaves like .- Hence, the integral behaves 4-1
L
like a cosine function, with the amplitude of oscillation in-
creasing with q2. We tested this for Na, for n = 4, using the
hard-sphere model. This shows that the integral does not even
remotely resemble -2b2(q2). i
Condition (v) not being satisfied by (VI. 2.1) , this
approximation is not to be relied upon. Although it incorporates
the feature of the correct long-wavelength limit, it is really
unphysical.
The first model having failed, we proceed ts examine
other possible approximations, in our choice of a suitable 3 -, s
b3 (q11q2!q3) that is neither tedious nor cumbersome to work
with and hence practical for computational purposes.
The approximation t (ql) +t (q2) +t (q3) for b3 where t is
an unknown function of q, may be made to satisfy the long-
wavelength limit (iii), but it will not satisfy condition (iv),
because t(q) is finite, when we take b3(mIm,q) and will not let
b3 (a, a, q) be zero.
The approximation t (ql) t (q2) t (q3) will satisfy (i) , (ii) 3 -%
and (iv) , hut it cannot be made to satisfy (iii) as b3 (4,-q, 0)
is sometimes negative and hence solving t(q) would involve
taking the square root of a negative quantity.
The approximation
is n o t s u i t a b l e , a s it cannot be made t o s a t i s f y (iii). I t
involves tak ing t h e square r o o t of a nega t ive q u a n t i t y , when
so lv ing f o r t (q) i n terms of b j (q,-q, 0 ) . Let us choose
where both t and u a r e unknown. The only proper ty known about
them IS t h a t
t (q) -+O a s q+a
and u ( q ) + O as q+M, SO as to satisfy ( i v ) . This model
s a t i s f i e s cond i t ions (i) , (ii) , ( i v ) , ( v i ) . W e s h a l l determine
t (q) and u ( q ) by requ i r ing (VI.2.2) t o s a t i s f y cond i t ions (iii)
and ( v i i ) .
Sect ion ( 6 . 3 ) : --.- Solving for unknown func t ions t (q) and u (q)
W e wish t o s o l v e f o r t (q) and u (q ) by making ( V I . 2.2)
s a t i s f y c o n d i t i o i ~ s (xii) and ( v i i ) . However, t h i s poses a
problem, a s condi t ion (iii) is i n q-space, whi le ( v i i ) is
i n r-space,
We begin by r e c a l l i n g some of t h e d e f i n i t i o n s i n Chapter
111.' The r e l a t i o n s h i p between t h e b and h-funct ions is a s
follows :
where T (r13) and T (r23) are defined by (VI .3.5) . Hence, we obtain
We have thus established the relation between
T (r) and t (q) n
and a similar relationship holding for v(r) and u(q).
We shall, now, make (VI. 2.2) satisfy condition (iii) . In
(VI.2.2), letting one of the q's approach zero, we obtain
-+ + b3(q,-q.0) = 2t(0) t ( q ) + t (q) '-u(q) '-2u(O)u(q)
(VI.3.10)
-+ -+ b (q,-q,O) is a known quantity, given by (VI.1.1). We may, 3
hence, solve for t (q) in terms of b3 (GI -G, 0) and u (q) , choosing an approximation for u(q) .
where t ( 0 ) = ( V I . 3. l i b )
Here t h e negat ive s i g n is inadmiss ib le , f o r t h e fol lowing
mathematicai reason. S u b s t i t u t i n g f o r t ( O j from jVI.3.llb) i n
( V I . 3. lla) , and t ak ing t h e va lue of t (q) when q-4, w e o b t a i n
t ( 0 ) aga in , from which we may check i f w e o b t a i n (VI.3. l lb)
again. While doing so , it i s found t h a t only t h e p o s i t i v e s i g n
i s admissible .
' 1 I Ill Turning our a t t e n t i o n now t o r e a l space, from Chapter 111,
III I
we have
h3 (123) = g 3 (123)-h2 (12)-h2 (23)-h2 (31)-1
From (VI.3.12), we may s o l v e f o r ~ ( r ) i n terms of r ( r ) and
hZ (r) t o o b t a i n
where ~ ( 0 ) = i
H e r e , aga in t h e negat ive s i g n is inadmiss ib le f o r reasons
s i m i l a r t o t h e one s t a t e d above f o r (VI.3.11).
3 3 f o r t (q) i n terms of u (q) and b j (q , -q, 0) . W e , then , Four ier
t ransform t ( q ) on t h e computer t o o b t a i n -r ( r) . W e s o l v e eqn.
( V I . 3.13) f o r ii ( r ) i n t e r m s of 7 (r) and h2 ( r ) . W e once again
Four ier t ransform P ( r ) which should g i v e u s u ( q ) . W e check
whether t h e u ( q ) obtained a s ou tpu t i s c l o s e t o t h e u ( q ) we
fed i n a t t h e s t a r t . I f they do no t agree , w e use t h e u ( q )
obta ined as output a s t h e new i n p u t and cont inue t h e i t e r a t i v e
procedure.
However, t h i s i t e r a t i v e procedure f a i l e d t o converge and,
i n f a c t , it diverged. From t h i s method, w e do n o t even know
how bad t h e approximation is, i n terms of o rde r of magnitude.
W e , hence, c o n t r i v e another approach.
Sec t ion ( 6 . 4 ) : Least Squares Minimisation Method
We choose an approximation for u(q) wi th some a d j u s t a b l e
parameters i n it, t h a t would al low us t o a l t e r t h e shape and
n a t u r e of t h e funct ion . W e s o l v e f o r t ( q ) a s previously.
u ( q ) and t ( q ) a r e Four ier transformed s e p a r a t e l y t o o b t a i n
ii(r) and r ( r ) . We then form h 3 ( r f r f 0 ) us ing (VI.3.7):
According to (VI. 3.13) , this should equal -2h2 (r) . When the h (r,r,O) we have formed, as in (VI.4.1), does 3
not agree with the known -2h,(r), a least squares minimisation L.
technique is adopted to minimise the error
It is the weighted sum of the squares
between h3(r.r.0) and -2h2(r).
E, defined as
of the difference
- iu where W ( 1 ) = --- 1. 2 is the weight function and rI is the
1+ im)
Ith point in the grid of 150 points in r-space, introduced in
section (6.1). As there is some uncertainty about the experi-
mental data in the small r region, the weight function we have
chosen does not emphasize this region. Maximum emphasis is
given to the region around the first peak and around the minima
of -2h2 (r) . In the region of large r, -2h2 (r) is very small
and hence this region is given least weighting. Further,
there is close agreement between hg (r, r, 0) and -2h2 (r) in this
region. It is not clear whether this is a weakness or strength
of the model.
The least squares minimisation technique was performed
using a subroutine called FMCG, from the Scientific Subroutine
Package a t t h e Computing Cente r . The method i s a s
fo l lows: F u r one set of va lues of t h e a d j u s t a b l e
parameters i n u ( q ) , t h e e r r o r E i s c a l c u l a t e d . Next, t h e
a d j u s t a b l e parameters a r e a u t o m a t i c a l l y changed by a
smal l amount A , such t h a t t h e new p o i n t would be a long
t h e l i n e of s t e e p e s t d e s c e n t o r down t h e nega t ive of
t h e g r a d i e n t c a l c u l a t e d a t t h e p rev ious p o i n t . The e r r o r
c a l c u l a t e d i n t h i s c a s e i s hence smaller than t h e prev ious
e r r o r . The i t e r a t i v e p roces s i s cont inued u n t i l two
consecu t ive v a l u e s of t h e e r r o r d i f f e r by an amount w i t h i n
t h e accuracy s p e c i f i e d .
Scvc.rill a ~ p r o x i m a t i o n s f o r u ( q ) were chosen and the
same procedure was performed. The r e s u l t s a r e t a b u l a t e d
i n Table 11.
By t h i s method, we have been a b l e t o s a t i s f y c o n d i t i o n s
(iii) and ( v i i ) l i s t e d i n s e c . ( 6 . 1 ) t o t h e e x t e n t t o
which w e have been a b l e t o minimise t h e e r r o r .
We l i s t below a few of t h e approximat ions t r i e d .
The f i r s t approximation f o r u ( y ) i n Table 11, approaches
- 4 ze ro a s q , f o r l a r g e va lues of q. Powers of q l e s s
t han 4 were n o t t r i e d a t a l l , because they would cause
t h e i n t e g r a l ( V 1 . 1 . 2 ) t o d ive rge . L e t t i n g u (q ) approach
-5 ze ro a s q a s i n ( 2 ) , makes t h e minimum e r r o r l a r g e r
t han i n t h e p rev ious c a s e . Hence, it i s r easonab le t o
l e t u ( q ) approach ze ro a s q-4, f o r l a r g e v a l u e s of q.
M 8 8 r-i (U w w 0 u'
R k w w rd o N
d & w 0 w m c w m rl m 0
2 . 6
rl N N 8
m I. m a3 a3 -
w 03 r-i u'
I
i 0 m I. m rl U) cn €n €n N a3 N rl r-i cn Ln N en 0 N Ln Ln N UJ
rd 0 0 0 rl w N N rl rl rl N
0 rl Ln I. m rt w i. I. cn 0 cn i. I. rl rl m u' N N 0 CV I. W
en en w w rl TP N u' rl rl r-i I
u' a3 u' r- 0 N \S) M rl OI cn cU r-i In M W
m v In en rd rl tU rl
In (3) , u (q) falls off as q-4 and for small values of q,
linear and quadratic terms are introduced in the numerator,
However, the error in this case is poorer. We, next,
introduce linear and quadratic terms in the denominator,
as in (4). The error is vastly reduced now. However, + + +
on plotting S3(q,q,q), the curve behaves linearly in the -+
small q region. From our knowledge of S2(q) and the j. -+
hard sphere S3(q,-q,O), they remain fairly flat in the
small q-region before they begin to rise. Hence, it is + + +
reasonable to believe that S3(q,q,q) being a correlation
of density fluctuations should not abruptly rise in the -+
small q region. This is because the mean fluctutation
over q is bound to be the same in S2(q), s~(~,-Q,o) and
+ + -+ Sj (q,q,q) . In this sense, (4) is undesirable. In (5 ) ,
+ -f + (6) and (7) , we allow S3 (q,q,q) to behave cubically and
as the fifth and seventh powers of q respectively in 3
the smaller q region. Although the error is not as small -4 -+ -f.
as in (4) , S 3 (q ,q,q) behaves more as we believe it ought to
Approximation (8) is, by far, the best chosen, in the
sense that it gives the smallest error, of all the approximations
-+ -+ -+ j.
tried. S j ( q , q , q ) behaves as q6 in the small q region.
In each of the above approximations, we used different
starting values for the adjustable parameters in the
iterative procedure. In the approximations (1) to (7) ,
no matter where we started, the final solution was the
same. For instance, in approximation (3), the two
starting values given by a = 2.79024, f3 = 18.41547,
rl = 25.02814, 8 = 18.50803 and a = -5.28582,
f3 = 1.90788, q = 12.53621 and 8 = 15.72656 led to the
same final solution as given in Table 11. Thus, the
solution appears to be unique for the approxiamtions (1)
to (7). But in approximation (8), the solution is
distinctly different from the previous solutions, suggesting
non-uniqueness. Other local minima, with somewhat larger
errors, have been found for approximation (8).
We have not used all of the approximations listed in
Table I1 for our calculation of resistivity in Chapter VII
because it would use too much computing time. We choose
those that are distinctly representative in their behaviour.
Approximation (5) , giving an error smaller than (1) ,
(2), (6) or ( 7 ) , represents the general behaviour of the
functions (1) to (7). Approximation (8) is distinctly
different in that it allows both u and t to assume negative
shown in figure VI and V respectively, for these two
approximations.
While the overall behaviour of the two u-functions
is similar, they are very different in the region between
O -1 q L = 0 and q = 2(A) . The t-function is very close to
the u-function. Hence there is potential cause for
-+ -+ -t concern due to round-off error. The S3(q,q,q) is quite
Figure V: Graphs of the u- and t- functions vS q,
for 2 approximations, the 4 and 6 parameter u(q)
given by (5) and (8) in Table 11.
+ + + Figu re VI: S g (q,q,q) for approximations (5) and (8) in
Table I1 and for the Greenwood approximation.
Figure VI1 : g (r,r,r) for approximation (8) in Table I1 3
and comparison with gg (r, r, r) from the Super-
position approximation.
O -1 different in the two cases, for q = 0 to q = 1.2(A) . Beyond that, there is close agreement around the major
peak of S3(qlq,q), no matter what approximation is used for
I u(q), as b3 is small in that region.
In Fig. VII, we show a graph of g3(r,r,r) for the
u-function numbered (8) in Table 11. We compare it with
that obtained using the superposition approximation.
In the region of large r the two agree. In the region of
small r, our mode1 is not very accurate because the data
is not accurate there. In the intermediate region, the
superposition approximation gives large values for 93(r,r,r). -f
Hence S j ( O , O , V ) , belng an integral over r, is iarge in
the superposition approximation, making it unsuitable for
our purposes. Our model qives a small value for S,(O,O,O). 3
However, our model is not entirely flawless, as it is not
positive definite in the region of the first minimum of g (r), 3
0
r 6.65 (A) . However, for our purposes, the conditions in
q-space are more important and hence we have disregarded
this fact.
CHAPTER VIIt CALCULATION OF RESISTIVTTY
I n t h i s chap te r , w e d e r i v e expres s ions f o r W ( 3 ) ( 2 ) andW ,
def ined i n (II.4.2), and use them i n ou r c a l c u l a t i o n of P (2)
(3) and P , t h e second and t h i r d o r d e r terms i n t h e r e s i s t i v i t y
formula. W e compare our t h e o r e t i c a l r e s u l t s w i th exper imenta l
r e s u l t s f o r Rb.
Sec t ion ( 7 . 1 ) : Express ions f o r W (3) (2) and W .
Recal l ing equatioff (11.4.2) , w e w r i t e -4 -4 w k** I
which we may expand t o
The atomic volume n o is def ined such t h a t fi= NQo.
(VII.1.3)
Considering each term i n (VII.1.2) i n d i v i d u a l l y , and
using (VII .1 .3) , w e ob ta in
u3 -+t
where k t k i s defined t o be equal t o <$ lv / 2' > as i n Chapter I. Ti-
On using (VIP. l .3) , t h e second term i n ( V I I . 1 . 2 ) becomes
3 + -+ -+ -+ -4-1 -p. -+ -t where q = k-kl, q2 1
= kl-k , q =k'-k. W e have used t h e 3
approximation E-k; f o r t h e r e a l p a r t of ~ ( k ~ ) . r is
2 2 (E-kl) + r 2
t h e imaginary p a r t of t h e self-energy term. We inc lude it i n
o rde r t o avoid any s i n g u l a r i t i e s . For t h e r e a l g a r t of t h e h k?
( ~ u t E2=2m=3 s e l f energy, A, we have used t h e f r e e e l e c t r o n 2m
for t h e sake of convenience.
3 -f
L e t u s assume k and k' t o be f i x e d i n t h e y-z p lane , each
making an ang le of el w i th t h e z-axis .
+?
The magnitudes of k and k' a r e equa l t o kF, t h e Fermi momentum
-f + I k I = / k' 1 = kF. This i s s o because of t h e presence of t h e d e l t a
" '4 function T - F(k-kF) i n t h e s p e c t r a l f u n c t i o n t h a t e n t e r s 2 k ~
the r e s i s t i v i t y formula.
3
k 1s allowed t o vary , a s shown i n t h e diagram. 1 3
0 and @ a r e t h e p o l a r ang les of kl. From t h e geometry
of t h e diagram, w e have
& A h
where i , j , k a r e t h e u n i t vec to r s .
W e hence o b t a i n
(VII. 1.6)
Expanding t h e three-dimensional i n t e g r a l , (VII.1.5) reduces t o
2 n/ 2
where we t a k e / d$= 4 d$, a s i s c l e a r from ./ 0 t h e diagram.
Sect ion (7 .2 ) : The pseudopotent ia l and screening:
t h e symmetry of
A pseudopotent ia l is a way of represent ing t h e i n t e r a c t i o n
of an o u t e r e l e c t r o n wi th t h e ion and i t s surrounding cloud of
e l e c t r o n s . This concept of a va lence e l c t r o n moving i n t h e
s e l f - c o n s i s t e n t p o t e n t i a l due t o t h e i o n c o r e s and o t h e r
valence e l e c t r o n s has been introduced i n Chapter I. This t o t a l
p o t e n t i a l i s w r i t t e n a s
The p o t e n t i a l due t o t h e n u c l e i
i n t h e form ( V 1 1 . 2 . 1 ) s i n c e t h e co re
c e r t a i n l y can be w r i t t e n
e l e c t r o n s are s o t i g h t l y
bound t h a t it i s an e x c e l l e n t approximation t o w r i t e t h e
p o t e n t i a l of t h e bare ion cores a s
so it is not obvious that the screening potential which they
produce can be written in this form. However, if the screening
charge distribution is treated as a linear response to the self-
consistent potential, then it follows that the sum of the bare
ion potential and the screening potential does indeed have the
form (VII. 2.1) . If vb (;) is a local potential, then 3 -+ 3 -+ <klvb1k+q> - V b ( G ) is independent of k and is just the Fourier
-f transform of Vb(r) The Fourier transform of the self-consistent
potential is given by
where B (q) is a dielectric screening function. (VII. 2.3) is
the Fourier transform of (VII.2.1), if we identify the ef- -t -+
fective potential centered on atom j,u(r-ri), as the inverse J
Fourier transform of e iG.6 j ub (9) / a (q) . We now give a brief account of the pseudo-potential, fol-
lowing the treatment of Ballentine and Gupta(l971).
The self-consistenuy screened model potential in a metal,
W = v +v +v +v M SC XC d
is the sum of the bare model potential VM. the screening
potential VSC due to the redistribution of the conduction
electron charge and Vd due to the depletion charge and the
exchange and c o r r e l a t i o n e f f e c t i v e p o t e n t i a l VXC of t h e con-
duct ion e lec t rons . I )
V i s a sum of s i m i l a r model p o t e n t i a l s vM centered M
around each ion, of t h e form
.. Gi
where pR is t h e p r o j e c t i o n opera to r f o r angular momentum
quantum number R . The above form i s o r i g i n a l l y due t o Heine
and ASarenkov ( 1 9 6 4 ) and A ~ i m a l u and ~ e i n e ( 1 9 6 5 ) . The model
RM is chosen r a t h e r a r b i t r a r i l y wi th in t h e range between t h e
i o n i c co re r a d i u s Rr and Wigner S e i t z r a d i u s of t h e atom i n - a s o l i d . For Rc2 t h e depth parameter AQ(E) i s ad jus ted s o t h a t
f o r energy E t h e logar i thmic d e r i v a t i v e of t h e pseudo-wave-
funct ion is t h e same a s t h a t f o r t h e t r u e wave funct ion a t
r = . This is done by f i t t i n g t h e spec t roscopic term values
of t h e s i n g l e ion and l i n e a r l y e x t r a p o l a t i n g t o o t h e r energies .
For 2 > 2 , AQ i s s e t equal t o A2. VSC i s r e l a t e d t o t h e screening
charge d e n s i t y p by Poisson ' s equat ion SC
-f 4 n e 2 + and Vd(q) = --
- 2 Pd (9) -
First order perturbation gives
where M and ME are components of the usual effective mass m* I(
(Weaire 1967) . Ballentine and Gupta (referred to as BG) have obtained
VXC, the exchange and correlation effective potential of the
valence electrons as follows: In analogy to VSC(q), VXC(q)
may be written as Co(q) pSC (q), where Co(q) is evaluated by
applying the formalism of Hokenberg and ~ohn(1964) and m h n
and Sham(1965). Both VSC and VXC are calculated self-
consistently. The final result for the momentum space matrix
element of the screened model potential for a single ion is
normalized over an atomic volume no. It is
where ~ ( q ) is the dielectric screening function and is given
4 where )( (q) = ------- 2md k 3 -9
(2T) ' k2- (k+q)
M~ In (VII.2.7), it is assumed that is independent of k.
The inhomogeniety correction potential vic(q) arises from
the fact that the potentiai corresponding io an exhaage azd
correlation charge density C (q) pSC(q) is an overestimate of 0
vxc inside the ion core.
The depletion potential vd(q) arises due to a depletion
charge. This depletion charge is due to the fact that the
pseudo-wave-functions are normalized to be equal to the true
wave functions outside the spheres of radius %, but are quite different within the spheres. The pseudo-wavefunctions have no
nodes within the cores, whereas the true wavefunction must have
several nodes in order to be orthogonal to the core wave-
functions. This causes a depletion charge. Formulas for
evaluating co(q), vd (q) and vic(q) can be found in section 3 of
Be.
To simplify the screening calculation BG have neglected the -+ k dependence of the model potential (but not its E dependence)
in evaluating the screening integral in (VI.2.6). They have 3 + 3
chosen k and k+q on the Fermi sphere for q<2kF, and antiparallel
for q>2kF Equation (VII. 2.6) , becomes, a•’ ter some manipulation
Here v and vnl represent the local and nonlocal parts of vM loc
respectively.
Values of the model potential parameters for Rb are
listed in BG.
The Ashcroft potential, (Ashcroft 1968) also known as the
empty core approximation, may be derived from (VII.2.6), by
letting
The Ashc ro f t poten2i.dl is a local potential and is given by
in our numerical calculations, we have used "semi-atomic"
units so that energies are in rydbergs and momenta in atomic - 0
R 2 is the Bohr radius. We choose 2m=1, units, a 'where a. = - me
e2= 2, and ?i = 1.
Section (7.3) : Calculation of W (2) and W (3) and resistivity
Using (1.1.1) and (1.1.15), the second and third order con-
tributions to resistivity are given by
9 1 ~ ~ ~ r n ~ *(3) and - R 9 3d9 ~;4e* h3kg o
The appearance of M; and $ in the above is due to a correct normalization of plane waves. This leads to about 20% higher
calc according to Greenfield and Wiser (1973). The norma-
lisation factor may be obtained as follows:
The true wave-function gives
<$ 1 = 1
The pseudo-wavefunction is related to the true wave-function
such that
< @ I $ > =, 1
However < @ I @ > + 1, in fact < @ I @ > = - > 1 (see Shaw and Y E
Harrison, 1967). In the nearly-free-electron model,
19>=c5/k>, where C5 is the normalization constant. Hence
<@/g>=c;<klk>= C; = - . Hence, the normalization M~
I constant = - . %
In the second order term, p ( 2 ) , the wave-f unction occurs 4 1 times (see VII.1.2), hence we have - . In p ( ) , the wave-
5 % I function occurs 6 times, hence - . M:
(3) In the case when q3+0, '- from (VII. 1.71, simply
Ro becomes
where 4 (k2+k;-2kkl~os%) and may be easily evaluated
as S3(q1-ql,O) is known, and does not involve any approximation + + +
for 3 (ql,q2,q3) It is found to be 2.66x10-', much less than - -
w (21 the value of .K--- in the same limit (1.28~10-~ 1 .
b L 0
For q3f 0, the calculation of w ( ~ ) is more complicated.
We need to evaluate the 3-dimensional integral (VII.1.7).
The multiple integration was done by a Gauss-Legendre numerical
quadrature method. It was done by means of a multiple inte-
gration subroutine GLINT (by C. Moore, from the university of
Michigan Computing Centre). Each of the 4 . 0 and kl integrations
were performed over a number of regions into which the entire
was subdivided i n t o 2N s u b i n t e r v a l s and an M~~ o rde r Gauss-
Legendre i n t e g r a t i o n was performed over each sub in te rva l .
N I Y f o r t h e t h r e e i n t e g r a t i o n s were optimized t o s u i t a b l e
va lues , such t h a t when N o r M was increased , t h e answer ob-
t a ined d id no t d i f f e r by more than t h e requi red accuracy from
t h e answer obtained by using a smal ler va lue f o r N o r M. For
t h e @- in tegra t ion , N = l , M=4, f o r t h e 8 - in teg ra t ion N=l, M=5 and
f o r t h e k l - in tegra t ion N = l , M=8 were found t o be appropr ia te .
The k l - in tegra t ion was performed from 0 t o 10 kF because t h e
pseudopotent ial i s small beyond t h a t . W e s p l i t t h e kl-inte-
g r a t i o n i n t o s e v e r a l reg ions , from 0 t o 0.85 kF, and a region
e q u i d i s t a n t about kF t o eva lua te a p r i n c i p a l va lue i n t e g r a l
i n t h a t region 0.85 kF t o 1.15 kF, a ~ d t h e region beyond t h a t
being subdivided i n t o t h r e e regions 1.15 t o 2.15 kF, 2.15 t o
3.15 and 3.15 t o 10 kF.
A s a check on t h e c a l c u l a t i o n , w e may perform t h e e n t i r e IT
i n t e g r a t i o n i n another coordina te system when €I1= 2 . The
@- in tegra t ion t r i v i a l l y y i e l d s IT.
lal 1 w i l l now be given by (kZ+k;-2kkl~osf3Z) % , ]Q2 J by
I l l
I ' I !
and kl- i n t e g r a t i o n s need t o be performed. When t h e calcu-
l a t i o n s w e r e performed i n t h e two co-ordinate systems, t h e
r e s u l t s agreed.
Using t h e prel iminary model V 1 . 2 . 1 involving hard T.7 ( 3 1 YY
spheres and t h e Ashcroft p o t e n t i a i , w e c a l c u l a t e d and 0
- R f o r Na. (VII.1.7) was evaluated f o r va r ious o r i e n t a t i o n s " V
0
8 and W '3 ' and W ( 2 ) 1 - '2) c a l c u l a t e d f o r each ang le and p -
Ro % and p ' 3 ' were evaluated using (VII.3.1).
be 1 0 t imes smal ler than t h e second order con t r ibu t ion . However,
t h i s r e s u l t i s not t o be r e l i e d upon, because t h e approximation 3 -+ -f
chosen f o r b3 (qL,q2,qj) does n o t s a t i s f y cond i t ion (v) i n
Chapter V I . -+ -+ -+
The second model cons t ruc ted i n Sec. ( 6 . 3 ) f o r b3 (ql,Cj2,q3)
given by 071.3.4) was next used i n t h e r e s i s t i v i t y c a l c u l a t i o n
f o r Kb. The pseudo-potential d iscussed i n t h e e a r l i e r p a r t
of Sec.(7.2) was used, a s t h e empty c o r e approximation i s n o t
s u i t a b l e f o r Rb. This is because Rb i s a t r a n s i t i o n - l i k e metal
and i ts d - s t a t e s a r e important. (See Cohen and Heine, 1970).
( 3 ) (2) and w ( ~ ) and W , p ( 3 ) were c a l c u l a t e d , a s descr ibed
e a r l i e r . using t h e two approximations numbered (5 ) and (8) i n
Table 11.
In order to examine which regions of q are important in
the resistivity calculation, we calculate P '3) using several + + - +
approximations for S3 (ql ,q2, q3)
We calculated P (3) for a random system (Sj=l). + + +
An approximation for s3 (qlr q2, 63) suggested by Greenwood
was next used. This approximation violates the long-wavelength
limit. It gives S3(0,0,0) = S2(0)' while it should be as, (01
~ ~ ( 0 ) ~ + n k T 3- In the limit when one of the q's o B F
1 approaches zero, it gives -[S (q) 2+ 2S2 (0) S2 (q) 1 . In this I 3 2 I
I sense, it is a poor approximation.
Bringer and Wagner (1971) have calculated. p(3) by re-
taining only those terms in the third order which involve just I ,
I
two scattering centers. Their approximation is equivalent to 1 I
- + - + - + Their S3(y,q,q) - l+3b 2 (q) has a large negative part, all the way to q = O . It badly violates the long-wavelength limit, and
(3) it is interesting to see how much this contributes to p . Another approximation for S3 is to truncate the peak of
b2(q) to zero beyond the very first value of q for which it is -+ -b -+
zero. S3(q ,q,q) has a large positive peak, caused by the peak
---- Greenwood S3
- - - 4 parameter S3
- - - - - - - 6 parameter S3
- Bringer & Wagner S3
---- truncated peak, in b2(q), using 6 parameter S3 - S3 = 1
TABLE I11
R e s i s t i v i t y , P in i l R c r n
Including -I-- Normaliza- - 8. 71 I Constant I Without normaliza- I tion I Constant
0 (2) , p ' 3 ) are our theoretical values at 60•‹C using approximation (8) of Table II in S 3 .
Pexp is the experimentla value at 40•‹C, quoted by Ashcroft and
Lekner . psf P~~ are Sundstrom's (1965) and Ashcroft and Leknerfs(1966)
theoretical values.
Ratio of D ( 3 ) to D (2)
No. Case
4 parameter S 3
6 parameter S 3
Greenwood S 3
Truncated approxi- mation
Including without ME
in b2(q). Hence, by truncating the peak, we can examine the 3 3 3 (3)
contribution of the large positive peak in S,(q,q,q) to p . The graphs of W (3)
7 q3 and
fig VII. The numerical results
In all of the cases tried,
(2 ) than p .
w(~) q3 versus
are presented
we found that
-f 3 3
We examine the behaviour of S j (q,q,q) to
of 6 contribute dominantly. We distinguish 3
model, the long-wavelength region from q=O to
q are shown in
in Table 111.
P '3 ' is larger
see which regions
regions for our
O- 1 q=0.4 A I
O- 1 O - 1 the intermediate wavelength region from 0.4 A to 1.3 A
(2kF for R b ) , and the large q region beyond that. In the
small q region, we compare our approximation (8) with the
approximation due to Bringer and Wagner. The latter gives a 1
large negative value for S3 in this region and violates the long- I
(3 wavelength limit badly. It yields a larger value for p I
I 'I
I I
than does our model. The long-wavelength region is important
and it contributes 1.2% to p ' 3 ) . In the large q region, the + + +
S3(qfq,q) is almost the same, no matter what approximation is
used, as b3 is small in this region. When we compare the con-
tributions to p (3) due to (8) and the truncated approximation
we find that the large q region contributes about 30%. To
examine the intermediate region, we compare our apprnxi-nation
(8) with (5) which are very different in this region. (See
+ + 3 Fig. VI for S3(q,q,q) curves for both). This difference
causes a difference of 50% in p(3). Greenwood's S3 is small
in that region. Hence Greenwood's P ( 3 ) is smaller than the
P ( 3 ) from approximation (8) or (5). Thus the intermediate
wavelength region contributes the dominant part.
In order to exqmine whether making S3 large and positive
in the long-wavelength region would change the order of magni-
tude of p ( 3 ) , we calculated P ( 3 ) for a random system (S3=1).
The large negative contribution tends to cancel out the
positive part, heaving P (3) to be just as large as P(2). The
superposition approximation gives a large value for S3 in this
region and is known to be a poor approximation. ' 1
In the preliminary calculation, using (VI .2.1) for 3 3 -+
,q2 ,q3) , Sj (q,q!q) gave the correct long-wavelength limit b3(G1 + + 1
and then remained small, as q got larger up to 2kF his I
, 1 1
explains the small value of P ( ~ ) obtained in that case.
However this model was of an ad hoe nature, and cannot be
relied upon.
From Table 111, we note that Sundstrom's (9965) P ( 2 ) is
more than twice the size of our P(2). She used experimental
structure factors and the Heine-Abarenkov potential which is only
slightly different from the pseudopotential we have used.
The resistivity is very sensitive to the form of the pseudo-
potential used. Ashcroft and Lekner(l966) used a hard-sphere
model f o r S (q) and t h e Ashcrof t p o t e n t i a l i n t h e i r c a l c u l a t i o n 2
of P ( 2 ) . The r e s u l t s a r e a l l shown i n Table 111. Although
Sundstrom and Ashcrof t and Lekner d i d n o t c a l c u l a t e P (2 ) us ing
t h e c o r r e c t no rma l i za t ion c o n s t a n t , w e have done s o w i t h t h e i r
P ( 2 ) s o w e may compare o u r r e s u l t s w i t h t h e i r s .
Our c a l c u l a t i o n i s t h e on ly one t h a t has ever been done
t h a t i n c l u d e s t h e f e a t u r e of t h e c o r r e c t long-wavelength
l i m i t . So any f u t u r e work t h a t might be done would have t o
i n c o r p o r a t e ou r r e s u l t s i n t o it.
- 129 - CHAPTER VIII
CONCLUSIONS
A major result of this thesis is (IV.2.11), the
exact expression for the nth order correlation function in
the long-wavelength limit. We have derived this result ri-
gorously from fluctuation theory. From the nature of the
expression, it is evident that if the liquid structure is
sufficiently resistant to compression, any partial long-
wavelength limit of any order structure function is small.
We have verified this for the hard sphere model and for
R b , using existing experimental data. We have also
verified this For Bg using experimental data, in the case
when all q's approach zero. We have thus proved true
Ballentine's (1966) conjecture that the small value of
the long-wavelength limit structure functions cancels
the large value of the screened potential in that limit.
, q ) which We have devised an approximation for b3(q1,q2
gives the correct long-wavelength limit and satisfied the
condition that two particles may not coincide.
We distinguish three regions of q-space for our model;
O-1 the region of small q extends from q = 0 to q = 0.4A . The
"-1 intermediate q region extends from q = 0.4;-I to 1.3A ,
(about 2kF for Rb). The large q region is the region beyond
1.3;-I.
The superposition approximation violates the long-
wavelength limit and gives a l a r y e value for S (0,6, O ) , and 3
is hence unsuitable for our purposes.
We calculated p (3) , the third order contribution and
found it to be surprisingly large. We used several different
approximations for S3 in order to examine which regions of
q contribute dominantly. We examined the long-wavelength
region by comparing p ( 3 ) obtained from our approximation (8)
and from the approximation due to Bringer and Wagner. The
latter violates the long-wavelength limit very badly and
makes S3 large and negative in that region thus making P (3)
larger. The long-wavelength region is important and contri-
butes about 128 to the resistivity. We examined the large
q region by comparing (8) with the truncated peak approxi-
mation. We find that the large q region contributes about
30%. Our model is unambiguclls in these two regions. We
examined the intermediate wavelength region by comparing with
the 4-parameter approximation (5) of Table 11, as these are
very different in that region. We find that this difference
causes a difference of 50% in P (3) . Thus this region
contributes the dominant part, but unfortunately our
model is uncertain in this region.
In order to obtain more information about this region, -+ -+ -+
we attempted a Monte Carlo calculation of S3 (ql ,q2 ,q3)
using the fundamental definition of S3 that it is an ensemble
average over configurations. (For details of the methbd,
considering 256 particles in i00,000 configurations in a
hard sphere model. We used the configuration data generated
by D. Card and J. Walkley of the ~hem~stry Department at SFU.
The S2(q) we calculated however, depended upon the orientation
of q.This is undesirable, necessitating averaging over
orientations and in the triplet case, makes it rather cumber-
some. This is because 256 particles and 100,000 configura-
tions are not enough to simulate an infinite system. We quote
a few values of 8 ( q j . For q = 6.5, when q 2 X = 5 f q~ = 2 1
qz = 3.64, S2(q) = 1.56; when q X
- 1.5, = 6, qy = 2, q3 -
- S2(q) = 0.798. For qx = 4, qy - 3, qZ = 4.15, S 2 (q) = 1.35,
all for the same values of y. Card and Walkley suspect
that their system has not yet settled down to equilibrium.
Hence, the S3 we calculated using these configurations
is only of dubious significance. But there is hope for the
future in this direction.
Another fact worth mentioning is that the resistivity
calc~lation is very sensitive to the nature of the pseudo-
potential used. Sundstrom (1965) used the Heine-Abarenkov
potential which is only slightly different from the form
( * ) for ~b is almost twice as we have used, and yet her p ,
large as ours.
It would be interesting to calculate P ( 3 ) for a weaker
scatterer like Na. But experimental data, which we consider
useful for our purposes, exists only for Rb.
A calculation of p ( 3 ) for a polyvalent metal would be
interesting because kF is larger and hence the region of
large q, which is not uncertain in our model, would be
more important than for R h .
In spite of inadequacies of our model in the
intermediate wavelength region, our S 3 is the best approxi-
mation and is far superior to any of the approximations
used in the past, as it gives the correct long-wavelength
limit. Ours is the only calculation of p ( 3 ) that has
examined the long-wavelength region and we hope we have
shed light in this direction. Any future work would have
to incorporate our results into it.
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