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TRANSPORT PROPERTIES IN LIQUID AND AMORPHOUS METALS
L. Roth, Vijay, A. Singh
To cite this version:
L. Roth, Vijay, A. Singh. TRANSPORT PROPERTIES IN LIQUID AND AMORPHOUS METALS.
Journal de Physique Colloques, 1980, 41 (C8), pp.C8-459-C8-462. �10.1051/jphyscol:19808114�. �jpa-
00220210�
JOURNAL DE PHYSIQUE
L . M . Roth and Vijay A. S i n g h f
Department of P h y s i c s , S t a t e U n i v e r s i t y o f New York a t Albany, Albany New York 12222, U.S.A
We have o b t a i n e d a r e s u l t f o r t h e c o n d u c t i v i t y of a l i q u i d m e t a l i n t h e e f f e c t i v e medium a p p r o x i - mation' (EMA), which r e d u c e s t o V e l i c k y ' s c o h e r e n t p o t e n t i a l a p p r o x i m a t i o n (CPA) r e s u l t f o r t h e c r y s - t a l l i n e a l l o y We u s e a method s i m i l a r t o t h a t which P o p i e l a w s k i used t o o b t a i n t h e Gyorffy- 4 Korringa-Mills (GKM) r e s u l t 5
.
T h i s v a r i a t i o n a l d e r i v a t i v e t e c h n i q u e i s g e n e r a l enough t o a l l o w u s t o d e r i v e e x p r e s s i o n s f o r t h e c o n d u c t i v i t y f o r a l l t h e a p p r o x i m a t i o n s c h a n e s proposed t o d e a l w i t h t h e one e l e c t r o n p r o p e r t i e s of l i q u i d m e t a l s . How- e v e r , t h o s e schemes which d o n o t r e n o r m a l i z e t h e e l e c t r o n p r o p a g a t o r l e a d t o s i n g u l a r i t i e s , and we s h a l l s p e c i f i c a l l y c o n s i d e r o n l y t h e GKM and EMA t h e o r i e s h e r e .The t h e o r y i s based on t h e Kubo f o r m c l a 2 f o r t h e c o n d u c t i v i t y
where j i s a c u r r e n t o p e r a t o r , a n d 9 ( w ) i s t h e
X
o n e e l e c t r o n Greens f u n c t i o n b e f o r e c o n f i g u r a t i o n a v e r a g i n g
--
t h e l a t t e r i s d e n o t e d by < > c . The o t h e r t r a n s p o r t c o e f f i c i e n t s obey s i m i l a r equa- t i o n s 6.
W e need t o c a l c u l a t e t h e q u a n t i t y*
Work s u p p o r t e d i n p a r t by NSF Grant No. DMR 75-18104.P r e s e n t Address : S o l a r Energy Research I n s t i t u t e Golden, Colorado, 80401
The most i n t e r e s t i n g c a s e w i l l be7 z = Z'*= w
+ i b . The o b j e c t i s t o o b t a i n an a p p r o x i m a t i o n t o n which is c o n s i s t e n t w i t h a g i v e n l i q u i d - m e t a l
1-1
t h e o r y f o r t h e Greens f u n c t i o n . The v a r i a t i o n a l d e r i v a t i v e method i s based on t h e f a c t t h a t a i s
11 e q u a l t o a c e r t a i n o r d e r e d v a r i a t i o n of t h e G r e e n ' s f u n c t i o n G = < >:
9
n = 6G
11 (3)
T h i s o r d e r e d v a r i a t i o n i s o b t a i n e d by l e t t i n g
GPO +
6G0, where G i s t h e f r e e e l e c t r o n Greens f u n c t i o n and where 6Go = Go j u G o f , and f i n d i n g t h e change i n G t o f i r s t o r d e r , b u t p u t t i n g primes on a l l q u a n t i t i e s t o t h e r i g h t o f Go. Here we w r i t e G0(z1) a s G o ' , e t c . Eq. (3) f o l l o w s from t h e Greens f u n c t i o n e q u a t i o ng =
Go + G o V 9 = Go + 9 V Go ( 4 ) where V =Zvi
i s t h e p o t e n t i a l d u e t o t h e i o n s .i
I n t r o d u c i n g t h e s e l f - e n e r g y 1, which from i t s d e f i n i t i o n obeys.
we f i n d
a = 6 G = G j G 1 + G 6 Z G '
11 U ( 6 )
T h i s w i l l be o u r fundamental e q u a t i o n .
To e v a l u a t e 61 we u s e t h e s e v e r a l . l i q u i d m e t a l t h e o r i e s . I n what f o l l o w s we
ell
r e l y h e a v i l y on a r e c e n t a r t i c l e 8 on t h e problem of a n a l y t i c i t y f o r t h e m u l t i p l e s c a t t e r i n g t h e o r i e s , which g i v e s much of t h e needed formalism. We d e f i n e d i n t h a t a r t - i c l e a s e l f -energy p a t h o p e r a t o r a(:,:',$,$*),
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19808114
c8-460 JOURNAL DE PHYSIQUE
which h a s t h e p r o p e r t y t h a t t h e s e l f energy o p e r a - t o r i n E q . (5) is g i v e n by,
where A
%
i s t h e c o m p l e t e l y d i a g o n a l o p e r a t o rand Gk i s a c-number. For t h e EMA we c a n w r i t e 8 Here r and r ' a r e e l e c t r o n c o o r d i n a t e s and R and R'
a r e i o n c o o r d i n a t e s c o r r e s p o n d i n g t o t h e f i r s t and l a s t s c a t t e r i n g s i n e a c h sequence c o n t r i b u t i n g t o a. We a l s o make u s e of a r e l a t i v e c o o r d i n a t e d e s c r i p t i o n which i s most c o n v e n i e n t l y g i v e n i n k-space :
A h
The f i r s t form f o r K and GZk depends on t h e medium p a t h o p e r a t o r which o b e y s
Where 0 i s t h e volume of t h e system t h i s i s p o s s i - b l e b e c a u s e of t r a n s l a t f . o n a 1 symmetry. I f we set
where P = = k , we have s i m p l y t h e s e l f e n e r g y i n a
k r e p r e s e n t a t i o n
The medium p a t h o p e r a t o r and t h e s e l f - e n e r g y p a t h o p e r a t o r a r e r e l a t e d by
For s e v e r a l l i q u i d m e t a l t h e o r i e s , a h a s a n i n t e g r a l e q u a t i o n form
The second e q u a l i t i e s i n Eq. (15) i n v o l v e G r a t h e r
A
t h a n Go. I n t h e d e f i n i t i o n of KO t h e i n t e g r a n d s c a n b e w r i t t e n i n r e v e r s e o r d e r of f a c t o r s . W e c a n
L
d e f i n e Glk a s t h e i n t e g r a n d , b u t i t i s n o t u n i q u e f o r t h e EMA.
As a consequence of t h e i n t e g r a l e q u a t i o n (Eq. 10) f o r a , we c a n r e a d i l y p r o v e t h e i m p o r t a n t Here we have s u p p r e s s e d t h e upper i n d i c e s . The
k e r n a l Qp' i s d e f i n e d i n t h e same way a s
tkpp'
a s a r e a l l " h a t t e d " o p e r a t o r s . We f u r t h e r w r i t e
r e s u l t
The second form of E q . (10) depends on t h e c o h e r e n t T-matrix which obeys t h e i n t e g r a l e q u a t i o n
Using t h i s and Eqs. (9) and (6) we c a n e x p r e s s n
v
i n t h e form
F o r t h e GKM we have
F o r t h e GKM we have s i m p l y s p a c e , u s i n g t h e r e s u l t t h a t K (RR') = ~ ( R - R ' ) G ~ ,
h ' 3 1
6%" = 871 6(p-p')
n
S (k-p) 5 ~ P (21)where S(k-k') = 1
+
n h ( k - k t ) i s t h e x-ray i n t e r - f e r e n c e f u n c t i o n s i n c e 6G i s j u s t n t h i s r e -P PP'
s u l t s i n a n i n t e g r a l e q u a t i o n f o r n
v
~ ( k , k ' , z , z ' ) = V k k l V i l k S ( k - k ' ) / n (22) where f o r GKM
"
kk' E. h kk'Vkk' = a k = t ( l
+
akWk)tcl (23)T h i s e q u a t i o n h a s t h e B e t h e - S a l p e t e r form proposed by Rubio'. W i s t h e v e r t e x f u n c t i o n and V is a n e f f e c t i v e p o t e n t i a l . T h i s r e s u l t was o b t a i n e d by
~ o ~ i e l a w s k i 4 who o b s e r v e d t h a t i t i s t h e same a s t h a t used by A s h c r o f t and s c h a i c h l 0 . A c t u a l l y A s h c r o f t and S c h a i c h r e p l a c e G by Go i n Eq. ( 1 3 ) . P r o v i d e d t h e l e v e l w i d t h i s s m a l l compared t o t h e F e r m i e n e r g y , t h e e f f e c t i v e p o t e n t i a l V k k l r e p l a c e s
1 0 t h e o r d i n a r y p o t e n t i a l v k k l i n t h e Ziman f o r m u l a
.
O t h e r w i s e , t h e i n t e g r a l e q u a t i o n , Eq. (22) i s n o t d i f f i c u l t t o s o l v e i t e r a t i v e l y .
For t h e EMA, we h a v e o b t a i n e d t h e f o l l o w i n g
A A A
set of i n t e g r a l e q u a t i o n s f o r 5Kk = 6K
+
6Wkh A h . A ? h 4 d k '
-
+
1 ~ ~ ~+ qlQk1 . t ~6Wk11Qkl 6 ~ ~2,') - 7 "4'
8a
" A
*
d k '-
+
z k . Q k I 5 w k l )7
Sn
I
Here n Ok = G j vk Gk. The p r o o f o f t h i s r e s u l t w i l l b e p u b l i s h e d e l s e w h e r e 1 2 , b u t we n o t e t h a t t h e r e s u l t f o r KO was e a s i e r t o p e r f o r m i n r e a l
where G~ c o r r e s p o n d s t o removing a n i o n a t R. I n t h e second term of Eq. (24) we h a v e used t h e iden- t i t y
and t h e unsymmetrical form o c c u r s b e c a u s e we must a v o i d o v e r m u l t i p l y i n g c e r t a i n t e r m s by t h e p a i r d i s t r i b u t i o n f u n c t i o n g ( R - R f ) . We can e x p r e s s a l l q u a n t i t i e s i n t e r m s of W, a, and G by w r i t i n g
k
and t h e a n a l o g o u s r e s u l t i n r e v e r s e .
We have shown12 t h a t i n t h e c a s e of t h e r a n - dom a l l o y t h e EMA r e s u l t r e d u c e s t o t h a t of V e l i c k y f o r t h e a l l o y CPA. T h i s i s a s i t s h o u l d be b e c a u s e t h e EMA r e d u c e s t o t h e CPA f o r t h i s case9''.
The r e s u l t e x p r e s s e d i n Eqs. (24) and (25)
)r f i
i s somewhat c o m p l i c a t e d b e c a u s e K and W h a v e k upper i n d i c e s , and b e c a u s e t h e r e a r e two coupled e q u a t i o n s . A s i m p l i f i c a t i o n which may b e u s e f u l i s r e l a t e d t o a n a p p r o x i m a t i o n i n t r o d u c e d by Schwartz 11
.
I f we a l l o w g2 i n Eq. (24) and n e g l e c t t h e l a s t two t e r m s i n Eq. (26)
,
we f i n dA h dk'
-
+
~ k l ~ k , 6 % v ~ i l ~ i l ]2
I f i n a d d i t i o n we n e g l e c t t h e f i r s t term i n Eq.(27), o u r r e s u l t reduced t o t h e form u s e d by Rubio7, ~ q .
( 2 2 ) , and now t h e e f f e c t i v e p o t e n t i a l i s g i v e n by
Comparing t h i s w i t h t h e GKM r e s u l t , we s e e t h a t t h e l a t t e r o m i t s t h e l a s t f a c t o r and i s n o t s y m m e t r i c a l i n k and k t , a s it s h o u l d be. A c t u a l l y i f we o m i t b o t h f a c t o r s we would t a k e a s t h e e f f e c t i v e poten- t i a l t h e c o h e r e n t T-matrix t
.
c8-462
J O U R N A L DE PHYSIQUEWe a r e i n v e s t i g a t i n g t h e s e v a r i o u s approxima- t i o n s u s i n g a s i m p l e model w i t h a s e p a r a b l e poten- t a i l . W e hope t o u s e our EMA r e s u l t t o i n v e s t i g a t e v e r t e x c o r r e c t i o n s , s a t u r a t i o n e f f e c t s , and d e v i a - t i o n s f r o m t h e Ziman f o r m u l a f o r c o n d u c t i v f t y .
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614 (1969).K. Levin, B. V e l i c k y and H. E h r e n r e i c h , Phys.
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BL,
3290 (1970);F. J . K o r r i n g a and R . L. M i l l s , Phys. Rev. B z , 1654 (1972).
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200 (1977).J. Rubio, J . Phys.
g,
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