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PAIR INTERCHANGE ENERGIES AND THEIR STRUCTURE DEPENDENCE IN Cu-Zn
M. Ahlers
To cite this version:
M. Ahlers. PAIR INTERCHANGE ENERGIES AND THEIR STRUCTURE DEPENDENCE IN Cu-Zn. Journal de Physique Colloques, 1982, 43 (C4), pp.C4-115-C4-120. �10.1051/jphyscol:1982410�.
�jpa-00221954�
JOURNAL DE PHYSIQUE
CoZZoque C4, suppldment au n o 12, Tome 43, de'cembre 1982 page C4-115
P A I R INTERCHANGE E N E R G I E S AND T H E I R STRUCTURE DEPENDENCE I N Cu-Zn
M. A h l e r s
Centro ~ t h i c o RariZoche, Comisi6n NacionaZ de ~ n e r ~ & z Athnica, 8 4 0 0 - S . C. de Bariloche, Argentina
( R e v i s e d t e x t a c c e p t e d 2 3 S e p t e m b e r 1982)
A b s t r a c t . - The e n t h a l p y of t h e o r d e r d i s o r d e r t r a n s i t i o n f o r f a c e c e n t e r e d Cu-Zn h a s been d e d u c e d . A l t h o u g h t h e l o n g r a n g e o r d e r e d a p h a s e i s n o t s t a b l e , i t c a n be o b t a i n e d by a s e q u e n c e o f s t r e s s i n d u c e d m a r t e n s i t i c t r a n s f o r m a - t i o n s , s t a r t i n g from t h e o r d e r e d 0' p h a s e . I t i s shown t h a t t h e r e s u l t s a r e c o m p a t i b l e w i t h t h e h y p o t h e s i s t h a t t h e h e a t of f o r m a t i o n of t h e d i s o r d e r e d and o r d e r e d f c c and bcc p h a s e s w i t h r e s p e c t t o t h e p u r e m e t a l s of t h e same s t r u c t u r e c o n s i s t o n l y of a c o n f i g u r a t i o n a l c o n t r i b u t i o n and t h a t t h e f i r s t n e i g h b o r i n t e r c h a n g e e n e r g y is i n d e p e n d e n t of s t r u c t u r e f o r f c c and b c c .
I . I n t r o d u c t i o n . - The s t a b i l i t y of t h e e q u i l i b r i u m p h a s e s i n n o b l e m e t a l a l l o y s d e p e n d s p r i n c i p a l l y on t h e e l e c t r o n c o n c e n t r a t i o n and on t h e d i s t r i b u t i o n of t h e d i f f e r e n t a t o m s on t h e l a t t i c e s i t e s , i - e . on t h e c o n f i g u r a t i o n . T h e r e i s s t i l l no a g r e e m e n t on t h e i m p o r t a n c e of e a c h of t h e two f a c t o r s i n t h e r e l a t i v e s t a b i l i t y of d i f f e r e n t p h a s e s . I n t h e v i r t u a l c r y s t a l a p p r o x i m a t i o n ( 1 ) i t had been assummed t h a t t h e c o n f i g u r a t i o n a l e n t h a l p y c o n t r i b u t i o n i s t h e same f o r d i f f e r e n t p h a s e s of t h e same c o m p o s i t i o n and t h a t t h e r e f o r e t h e i n t e r a c t i o n of t h e Fermi s u r f a c e w i t h t h e B r i l l o u i n zone p r e s e n t s t h e most i m p o r t a n t c o n t r i b u t i o n ( f o r a r e v i e w s e e ( 2 ) ) . D o u b t s have been e x p r e s s e d however, w h e t h e r t h e e l e c t r o n e n e r g y and i t s v a r i a t i o n w i t h c o m p o s i t i o n a l o n e s u f f i c e s t o a c c o u n t f o r t h e o b s e r v e d p h a s e d i a g r a m s of t h e n o b l e m e t a l a l l o y s ( s e e 3 ) . R e c e n t l y i t h a s been shown ( 4 ) , t h a t t h e a / B p h a s e b o u n d a r y of s e v e r a l Cu b a s e d b i n a r y a l l o y s c a n be w e l l a c c o u n t e d f o r by a c o n f i g u r a t i o n a l c o n t r i b u t i o n which h a s been d e r i v e d u n d e r t h e a s s u m p t i o n t h a t t h e p a i r i n t e r c h a n g e e n e r g i e s depend on t h e p a i r d i s t a n c e , b u t n o t on t h e l a t t i c e s t r u c t u r e , which i m p l i e s t h a t t h e n e a r e s t n e i g h b o r i n t e r c h a n g e is t h e same i n both p h a s e s , t h a t t h e s e c o n d n e i g h b o r v a l u e f o r f c c is z e r o and t h a t t h e t h i r d n e i g h b o r e n e r g y f o r f c c i s n o t z e r o . The same a p p r o a c h had a l s o been used t o e v a l u a t e t h e e n e r g y d i f f e r e n c e s between t h e o r d e r e d 0' p h a s e , t h e o r t h o r h o m b i c 9R m a r t e n s i t e w i t h t h e s t a c k i n g s e q u e n c e ABCBCACAB and t h e f a c e c e n t e r e d s t r e s s i n d u c e d p h a s e ( 4 , 5 ) .
A l t h o u g h t h e p s e u d o p o t e n t i a l t h e o r y f o r s i m p l e m e t a l s p r e d i c t s p a i r i n t e r c h a n g c e n e r g i e s t h a t t h e n a r e s t r u c t u r e i n s e n s i t i v e , i t would be d e s i r a b l e t o h a v e a n e x p e r i m e n t a l proof f o r t h i s s u g g e s t i o n f o r t h e n o b l e m e t a l s a l l o y s i n o r d e r t o j u s t i f y b e t t e r t h e a p p r o a c h which we have u s e d . T h i s w i l l be a t t e m p t e d i n t h i s p a p e r .
P a i r i n t e r c h a n g e e n e r g i e s c a n i n g e n e r a l be d e t e r m i n e d from o r d e r - d i s o r d e r t r a n s i t i o n s . For an e v a l u a t i o n of t h e i r s t r u c t u r e s e n s i t i v i t y , s a y between t h e f c c and t h e bcc s t r u c t u r e s , o r d e r t r a n s f o r m a t i o n s s h o u l d be a v a i l a b l e f o r b o t h s t r u c t u r e s . The 0 p h a s e i n Cu-Zn o r d e r s a t low t e m p e r a t u r e s t o B ' , and p a i r i n t e r c h a n g e e n e r g i e s have been d e r i v e d ( 7 ) . The e q u i l i b r i u m a p h a s e d o e s n o t o r d e r . However, a n o r d e r e d f a c e c e n t e r e d s t r u c t u r e c a n be o b t a i n e d , s t a r t i n g from B ' , t r a n s f o r m i n g i t f i r s t t o t h e m a r t e n s i t i c 9 R and t h e n s t r e s s i n d u c i n g t h e f c c c e n t e r e d p h a s e . The e n t h a l p y d i f f e r e n c e s f o r t h e s e t r a n s i t i o n s have been d e t e r m i n e d (6,8). T h e r e f o r e i n p r i n c i p l e i t i s p o s s i b l e t o d e r i v e t h e i n t e r c h a n g e e n e r g y f o r t h e f c c l a t t i c e , o n c e t h e e n t h a l p y d i f f e r e n c e between d i s o r d e r e d f c c and o r d e r e d B '
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982410
C4-116
JOUIWAI. DL PHYSIQUE
i s known. I n t h e f o l l o w i n g , p a i r i n t e r c h a n g e e n e r g i e s f o r t h e bcc l a t t i c e o f CuZn w i l l be d e r i v e d and compared w i t h t h o s e f o r b c c .
Once t h e c o n f i g u r a t i o n a l e n t h a l p y c o n t r i b u t i o n i s known w i t h s u f f i c i e n t p r e c i s i o n t h e e l e c t r o n i c p a r t c a n be d e t e r m i n e d a n d t h e q u e s t i o n c a n be a n s w e r e d t o w h a t e x t e n t t h e F e r m i s u r f a c e i n t e r a c t i o n w i t h t h e B r i l l o u i n z o n e a f f e c t s t h e r e l a t i v e s t a b i l i t y of d i f f e r e n t p h a s e s . T h i s p o i n t w i l l be c o n s i d e r e d i n t h e l a s t p a r t o f t h e p a p e r .
2 . The f o r m a t i o n e n t h a l p y o f t h e o r d e r e d f a c e c e n t e r e d c u b i c CuZn.- I t s d e t e r m i n a t i o n i s decomposed i n t o f o u r s t e p s : a ) S t a r t from t h e e n t h a l p y o f f o r m a t i o n f o r t h e o r d e r e d B'bcc. b ) Add t h e e n t h a l p y d i f f e r e n c e b e t w e e n t h e m a r t e n s i t i c 9R and t h e o r d e r e d D', w h i c h h a s been m e a s u r e d (8). c ) Add t h e e n t h a l p y d i f f e r e n c e f o r t h e t r a n s f o r m a t i o n f r o m 9R t o t h e f c t 3R l a t t i c e . I t h a s b e e n d e d u c e d from t h e c r i t i c a l s t r e s s n e c e s s a r y t o i n d u c e 3R ( 6 ) . d ) The r e s u l t i n g 3R p h a s e h a s a t e t r a g o n a l o r d e r i n d u c e d d i s t o r t i o n . An a d d i t i o n a l e n e r g y i s r e q u i r e d t o d i s t o r t i t t o t h e c u b i c f c c o r d e r e d s t r u c t u r e , whose f o r m a t i o n e n e r g y c a n t h e n be c o m p a r e d w i t h t h a t o f t h e d i s o r d e r e d a f c c .
F i g u r e 1: E n t h a l p i e s of f o r m a t i o n ( i n d e g r e e s K e l v i n ) v e r s u s Zn c o n c e n t r a t i o n i n Cu-Zn: X m e a s u r e m e n t s a t 573 K ,
8 e q u i l i b r i u m v a l u e d e r i v e d f o r o r d e r e d 0 ' a t OK, - t l E r d ( O ~ ) , v a l u e f o r o r d e r e d 9R m a r t e n s i t e ( e ) and t h a t f o r t h e f c t s t r u c t u r e (a ),
-
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e x t r a p o l a t i o n f o r d i s o r d e r e d a p h a s e , Q v a l u e f o r d i s o r d e r e d h y p o t h e t i c a l 9R m a r t e n s i t e .The e n t h a l p y of f o r m a t i o n of t h e o r d e r e d B ' b c c i s now r e d e r i v e d f o r a 4 7 . 8 a t % Zn a l l o y a n d s u b s e q u e n t l y is e x t r a p o l a t e d t o o t h e r c o m p o s i t i o n s ( s e e a l s o ( 7 ) ) . The h e a t s of f o r m a t i o n t 1 ° ' ( 5 7 3 ~ ) o f o r d e r e d B'CuZn h a v e been m e a s u r e d by B l a i r a n d Downie ( 9 ) , and a r e shown i n t h e f i g u r e , i n w h i c h v a r i o u s e n t h a l p i e s of f o r m a t i o n a r e p l o t t e d v e r s u s Zn c o m p o s i t i o n .
A t 573K, some d i s o r d e r is p r e s e n t . I n o r d e r t o g e t t h e v a l u e f o r t h e o r d e r e d g r o u n d s t a t e , i t i s n e c e s s a r y t o add t h e c o n t r i b u t i o n r e s u l t i n g from t h e d i f f e r e n c e i n o r d e r a t 573K and i n e q u i l i b r i u m a t OK, which w i l l be c a l c u l a t e d f i r s t . From n e u t r o n d i f f r a c t i o n e x p e r i m e n t s ( 1 0 ) t h e l o n g r a n g e o r d e r p a r a m e t e r had been d e t e r m i n e d a s a f u n c t i o n o f
,
t e m p e r a t u r e f o r a 47.8 a t % Zn a l l o y . The c o r r e s p o n d i n g e n t h a l p y d i f f e r e n c eAH^,^
( T ) b e t w e e n t h e d i s o r d e r e d s t a t e a t t h e c r i t i c a l t e m p e r a t u r e Tc and a t e m p e r a t u r e T i T c had a l s o b e e n m e a s u r e d f o r p r a c t i c a l l y t h e same c o m p o s i t i o n ( 4 7 , 5 a t % Zn) ( 1 ) . D e n o t e by 2XcU t h c d i f f e r e n c e of t h e e x p e r i m e n t a l l y d e t e r m i n e d o c c u p a t i o n p r o b a b i l i t y qf Cu a t o m s b e t w e e n t h e two s u b l a t t i c e s o f t h e B2 s t r u c t u r e . Then a p l o t of A H ~ , ~ ( T ) v e r s u sxEU
i s e x p e c t e d t o g i v e a s t r a i g h t l i n e , a c c o r d i n g t o t h e B r a g g - W i l l i a m s t h e o r y , and i n d e e d d o e s s o e x c e p t a t t h e h i g h e s tXcU
( i . e . l o w e s t t e m p e r a t u r e s ) , a t which t h e o r d e r i n gbecomes s l u g g i s h . Due t o t h e s h o r t r a n g e o r d e r c o n t r i b u t i o n , w h i c h i s n o t c o n s i d e r e d i n t h e B r a g g - W i l l i a m s t h e o r y , t h e d a t a h a v e t o be c o r r e c t e d . T h i s is d o n e by a s c a l i n g f a c t o r of 1 / 0 . 6 7 ( 7 ) . The r e s u l t i s :
AH^:^
( T ) = - 2 2 1 5 ~ : , [ ~ ] ( e q . 1 ) (The B o l t z m a n n c o n s t a n t h a s been c h o s e n a s t h e a d i m e n s i o n a l u n i t k g = l , i m p l y i n g t h a t t h e e n e r g i e s a r e e x p r e s s e d i n K. I n o r d e r t o t r a n s f o r m t h e e n e r g i e s t o t h e u s u a l J / m o l y u n i t s , t h e y h a v e t o be m u l t i p l i e d by t h e m o l a r g a s c o n s t a n t R O = 8 . 3 1 J K-' m o l e - ' ) . E x t r a p o l a t i n g t o T=OK, i . e . XCu=CZn, t h e o r d e r - d i s o r d e r e n t h a l p y i s o b t a i n e q i n a g r e e m e n t w i t h t h a t d e r i v e d from t h e c r i t i c a l t e m p e r a t u r e ( 7 ) . By a d d i n g h H t r d ( 0 ) - ~ ~ E r d ( 5 7 3 ) t o t h e h e a t of f o r m a t i o n m e a s u r e d by B l a i r and Downie, t h e e n t h a l p y o f f o r m a t i o n o f t h e o r d e r e d g r o u n d s t a t e l a t 47.5 a t % Zn is o b t a i n e d , w h i c h i s shown i n t h e f i g u r e ( @ ) a n d is d e n o t e d by ~ t ~ ~ ( 0 . 4 7 5 , T=OK).I t i s now n e c e s s a r y t o e x t r a p o l a t e
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t o o t h e r c o m p o s i t i o n s . T h i s w i l l be d o n e i n t h e f o l l o w i n g way: a ) The e q u a t i o n 1 i s assumed t o be v a l i d i n t h e w h o l e c o m p o s i t i o n r a n g e ; b ) The e n t h a l p y of f o r m a t i o n o f t h e d i s o r d e r e d 6 w i t h r e s p e c t t o t h e hcc s t a t e s of t h e p u r e m e t a l s i s a p p r o x i m a t e d by a r e g u l a r s o l u t i o n e q u a t i o n , A'CCuCZn, w i t h A i n d e p e n d e n t o f c o m p o s i t i o n i n t h e same c o m p o s i t i o n r a n g e . B o t h h y p o t h e s e s a r e w e l l j u s t i f i e d f o r many s o l i d s o l u t i o n s . The e n t h a l p y of f o r m a t i o n o f t h e b c c p u r e Cu and Zp had b e e n d e r i v e d by Kaufman ( 1 2 ) . The r e s u l t i n g c o m p o s i t i o n d e p e n d e n c e of H : ~ ~ h a s been p l o t t e d i n t h e f i g u r e . I n t h e f o l l o w i n g t h e v a r i o u s e n t h a l p i e s w i l l be compared a t 40 a t % Zn, b e c a u s e t h e m a r t e n s i t i c t r a n s f o r m a t i o n h a s been o b s e r v e d i n t h i s r a n g e .The e n t h a l p y d i f f e r e n c e b e t w e e n t h e o r d e r e d D' a n d 9R m a r t e n s i t e had ben d e d u c e d e l s e w h e r e ( 8 ) , t h a t between 3R and 9R was d e r i v e d from t h e c r i t i c a l s t r e s s t o i n d u c e 3R. Due t o a h y s t e r e s i s i n s t r e s s on l o a d i n g and u n l o a d i n g ( 6 ) , a s m a l l u n c e r t a i n t y o f 12K r e s u l t s f o r t h e e n t h a l p y d i f f e r e n c e . I f t h e u p p e r s t r e s s ( o n l o a d i n g ) is t a k e n , t h e e n t h a l p y of f o r m a t i o n of t h e f c t p h a s e is e q u a l to t h a t f o r t h e D' p h a s e a t 40 a t % Zn.
The t e t r a g o n a l o r d e r i n d u c e d d i s t o r t i o n of t h e f a c e c e n t e r e d s t r u c t u r e d e p e n d s o n c o m p o s i t i o n ( 1 4 ) . The c o r r e s p o n d i n g d i s t o r t i o n e n e r g y is d i f f i c u l t t o e v a l u a t e . I n p r i n c i p l e i t c a n be c a l c u l a t e d by f i r s t d i s t o r t i n g t h e d i s o r d e r e d f c c t o f c t , and t h e n by a d d i n g t h e c h a n g e i n c o n f i g u r a t i o n a l o r d e r c o n t r i b u t i o n . I n p r a c t i c e , h o w e v e r , t h e s e c o n t r i b u t i o n s c a n n o t be c a l c u l a t e d w i t h s u f f i c i e n t a c c u r a c y , b u t a r o u g h u p p e r l i m i t e s t i m a t e c a n be made: I f t h e c o n f i g u r a t i o n a l c o n t r i b u t i o n v a r i e s l i n e a r l y w i t h t h e d i s t o r t i o n ( 1 - c / a ) , t h e n i t d o e s n o t a f f e c t t h e e l a s t i c c o n s t a n t s , s i n c e t h e l a t t e r d e p e n d o n t h e s q u a r e of t h e s t r a i n
(2
t h e a x i a l r a t i o o f t h e f c t l a t t i c e ) . A l t h o u g h no m e a s u r e m e n t s f o r o r d e r e d f c t e x i s t , s o u n d v e l o c i t i e s h a v e b e e n o b t a i n e d f o r 9R m a r t e n s i t e s from n e u t r o n d i f f r a c t i o n a n d u l t r a s o n i c m e a s u r e m e n t s (15), They i n d i c a t e t h a t t h e s m a l l e s t e l a s t i c c o n s t a n t s a r e i n d e e d c l o s e t o t h o s e f o r (C,l-C,2)/2 of t h e d i s o r d e r e d a p h a s e . U s i n g t h e e x t r a p o l a t e d Cll-C12=3.1010 ~ / f o r a CuZn a t 40 a t m ~ % Zn, and a d i s t o r t i o n ( 1 - c / a ) = 8 . 7 % ( 1 4 ) , t h e e l a s t i c d i s t o r t i o n e n e r g y Edist a s a n u p p e r l i m i t a m o u n t s t o 70K.3. The f o r m a t i o n e n t h a l o f d i s o r d e r e d f c c a h a s e Cu-Zn.- The h e a t of f o r m a t i o n H~ h a s b e e n d e t e r m i n e d P Y b y v a r i o u s a u t h o r s i A h e r a n g e b e t w e e n 293 a n d 5 7 3 K(16).
Below 30 a t % Zn t e m p e r a t u r e h a s l i t t l e e f f e c t , i n d i c a t i n g t h a t s h o r t r a n g e c o n t r i b u t i o n s a r e n o t i m p o r t a n t . Above 31 a t % t h e r e a r c s t r o n g d e v i a t i o n s from a p a r a b o l a which have b e e n a s s o c i a t e d w i t h s h o r t r a n g e o r d e r . W i t h i n 20 t o 30 a t % Zn t h e c o m p o s i t i o n d e p e n d e n c e of Ha c a n w e l l be a d j u s t e d by a p a r a b o l a a n d a n e x t r a p o l a t i o n t o 40 a t % Zn s e e m s w e l l j u s t i f i e d . T h i s e x t r a p o l a t e d c u r v e is shown i n t h e f i g u r e . I n t h e f i g u r e i s a l s o d r a w n t h e e n t h a l p y of f o r m a t i o n of a h y p o t h e t i c a l c l o s e p a c k e d d i s o r d e r e d 9R m a r t e n s i t e by a d d i n g s t a c k i n g f a u l t s of d e n s i t y 1 1 3 . The s t a c k i n g f a u l t e n e r g i e s a r e t h o s e d e r i v e d from node m e a s u r e m e n t s i n t h e a p h a s e ( 1 7 ) .
CL-118 JOURNAL DE P ~ I Y S I Q U E
4. D i s c u s s i o n . - The e n t h a l p y d i f f e r e n c e between e o r d e r e d and d i s o r d e r e d f c c p h a s e s i s r e l a t e d t o t h e p a i r i n t e r c h a n g e e n e r g i e s
dh
b e t w e e n i - t h n e i g h b o r s byHfcc H f c c =
-
[ 2 m i 1 ) - 3 m L 2 ) + 4 m i 3 ) ] ~ 2 C uo r d
-
d i s ( e q . 2 )~ e r e , r n $ ~ ) i s r e l a t e d t o t h e i n t e r a c t i o n e n e r g i e s V ~ ) , V $ ; ) and ~ 2 betwee ) AA B a n d & atom p a i r s i n i - t h n e i g h b o r p o s i t i o n i n t h e u s u a l way by tn(i)= -2V?i)+VtiB+
+
V B B.
T h e i n t e r c h a n g e e n e r g i e s f o r f o u r t h and h i g h e r o r d e nePgh&:s g v e e e n n e g l e c t e d . F o r 40 a t % I n , a c c o r d i n g t o t h e p r e s e n t r e s u l t s , iz:;-
Hdis i s-250K f o r Edist = OK -180K f o r Edist = 70K Thus
780'120K f o r Edist = OK ( 1 ) - 1 . 5 ,:2)+ ~ ~ $ 3 ) =
ma 560'120K f o r Edist = 70K
The e s t i m a t i o n of t h e e r r o r s was made by assumming t h a t t h e y a r e due s o l e l y t o t h e s c a t t e r of t h e measured h e a t s of f o r m a t i o n .
We w i l l now show t h a t t h e r e s u l t s a r e c o n s i s t e n t w i t h t h e f o l l o w i n g s u p p o s i t i o n s :
A . The h e a t of f o r m a t i o n of t h e d i s o r d e r e d and o r d e r e d f c c and bcc p h a s e s w i t h r e s p e c t t o t h e p u r e m e t a l s of t h e same s t r u c t u r e s c o n s i s t o n l y of a c o n f i g u r a t i o n a l c o n t r i b u t i o n .
B. The f i r s t n e i g h b o r i n t e r c h a n g e e n e r g y is i n d e p e n d e n t of s t r u c t u r e f o r f c c and bcc.
The c o n f i g u r a t i o n a l c o n t r i b u t i o n i n t h e d i s o r d e r e d a p h a s e w i t h r e s p e c t t o t h e p u r e f c c m e t a l s y i e l d s ( s e e Ref . 4 ) :
6mi1)
+
3mi2)+
12mL3) = 4410K ( e q - 4 )The c o r r e s p o n d i n g c o n t r i b u t i o n i n t h e d i s o r d e r e d b c c is ( s e e R e f . 4 ) :
4wh1)
+
3wh2)+
6wh3) = 5235K ( e q . 5 )The D p h a s e o r d e r c o n t r i b u t i o n g i v e s
Here t h e w6') a r e t h e i n t e r c h a n g e e n e r g i e s f o r i - t h n e i g h b o r s i n t h e B p h a s e . The n u m e r i c a l v a l u e s a r e t h o s e d e r i v e d i n t h i s p a p e r , a f t e r s u b t r a c t i n g t h e c o n t r i b u t i o n of t h e p u r e e l e m e n t s .
From e q u a t i o n s 3 t o 6 f o l l o w
mh2)=
-
22K, 4 1 )+
2mL3) = 7 4 7 ~ f o r E~~~~ =o
m i 2 ) =
+
87K, mL1)+
2mL3) = 690K f o r Edist = 70K - 9 3 K , 4 2 )+
2w63) = 5 0 3 ~h e w a r e s i m i l a r t o I n d e n ' s v a l u e s ( 7 w h i c h w e r e d e r i v e d w i t h t h e s a m e a s s u m p t i o n A , i . e . w61) = 955K and w62) = 535K).
From s u p p o s i t i o n B f o l l o w s
mA3) = -92K f o r Edist = OK mi3) = -120K f o r Edist = 70K
I n s p i t e o f t h f l f a r g e u r v r t a i n t y i n Edi t , t h e r e s u l t shows t h a t mi2) is s m a l l , c o m p a r e d t o na ())The ma i s a l s o s m a l l , g u t d e f i n i t e l y n o t z e r o . I n s p i t e o f t h e s m a l l n e s s of ma
,
r d n e i g h b o r c o n t r i b u t i o n t o t h e c o n f i g u r a t i o n e n t h a l p y i s c o n s i d e r a b l e ,si',:
e n t e r s w i t h a h i g h m u l t i p l i c i t y i n e q u a t i o n 4 , d u e t o t h e f a c t t h a t e a c h atom h a s 24 t h i r d n e i g h b o r s of t y p e < l 1 2 > ( c o m p a r e d t o 6 f o r s e c o n d n e i g h b o r s ) .T h e mLi) w h i c h h a v e b e e n d e r i v e d h e r e w i t h t h e s u p p o s i t i o n s A a n d B a r e s u p p o r t e d a l s o by t h e e v a l u a t i o n of t h e e n t h a l p y d i f f e r e n c e between 6 and 9R m a r t e n s i t e : u s i n g s u p p o s i t i B , i t was f o u n d t h a t t h e s t r o n g t e m p e r a t u r e d e p e n d e n c e o f M s i s d u e t o = - 2 5 ( 5 . I n a d d i t i o n , f r o m t h e c o m p o s i t i o n d e p e n d e n c e of e s t r e s s , which i s n e c e s s a r y t o i n c e t h e o r d e r e d a p h a s e from 9R i n C u i n , a
mi'y=
- 1 l O X was deduced ( 6 , 8 ) . Both a r e c l o s e t o t h a t one d e r i v e d h e r e . A t h i r d r e s u l t which a l s o s e r v e s a s s u p p o r t , i s t h a t t h e p h a s e s a i l i t y b e t w e e n t h e e q u i l i b r i u m a a n d D c a n be w e l l a c c o u n t e d f o r by u s i n g t h e mLiy, n o t o n l y f o r Cu-Zn b u t a l s o f o r o t h e r Cu b a s e a l l o y s ( 4 ) .Two r e m a r k s r e l a t e d t o o r d e r i n g i n f c c a r e w o r t h w h i l e making: The o r d e r - d i s o r d e r e a l p y is one t h i r d of t h e c o n f i g u r a t i o n a l c o n t r i b u t i o n f o r d i s o r d e r e d f c c , i f m:jP c a n be n e g l e c t e d . For Cu-ln t h i s r e l a t i o n h o l d s q u i t e w e l l , but f o r many o t h e r s y s t e m s t h i s is not t r u e ( 1 8 ) , a n example b e i n g AuCu, which i n d e e d h a s a l a r g e d 2 ) ( 1 9 ) of a b o u t 0.2m61).
From t h e o r d e r - d i s o r d e r e n t h a l p y i s s h o u l d be p o s s i b l e t o o b t a i n t h e o r d e r i n g t e m p e r a t u r e , i f t h e c o r r e c t i o n f a c t o r K due t o s h o r t r a n g e o r d e r is known. I f o n l y f i r s t n e i g h b o r i n t e r c h a n g e e n e r g i e s a r e c o n s i d e r e d , a c o m p a r i s o n between t h e c l u s t e r v a r i a t i o n and t h e Bragg W i l l i a m s method y i e l d s K = 0.5 and t h e o r d e r i n g t e m p e r a t u r e from A 1 t o t y p e L12 s t r u c t u r e is ( 1 8 ) .
( t h e t h i r d n e i g h b o r term added t o I n d e n s r e s u l t ( 1 8 ) ) . F o r Czn = 0.40 t h i s i m p l i e s , u s i n g a l s o h e r e K = 0 . 5 , T ; ~ z - ~ ' = 374K f o r Edist= 0 and 270 K f o r Edist= 70K ( e q . 3 ) . The o r d e r i n g t e m p e r a t u r e f o r t h e a p h a s e t h e r e f o r e is e x p e c t e d t o be low. I t i s u n l i k e l y t h a t a t t h e t e m p e r a t u r e s f o r which Orr and A r g e n t measured t h e h e a t s of f o r m a t i o n of t h e a p h a s e a l l o y s any d e g r e e o f l o n g r a n g e o r d e r e x i s t e d .
The o r d e r - d i s o r d e r t e m p e r a t u r e h a s a l s o been e s t i m a t e d by C h a n d r a s e k a r a n and Miodownik ( 2 0 , 2 1 ) f o r t h e f c c s t r u c t u r e a t a c o m p o s i t i o n of 3 0 a t % Zn. I n t h e a b s e n c e of a d e f i n i t e e x p e r i m e n t a l v a l u e , t h e y u s e ~ : ' 2 - ~ ' = 2 7 5 K ( 2 1 ) , which c o r r e s p o n d s t o 315 K f o r t h e 4 0 a t % Zn ~ f l( e q . 7 ) , and l i e s i n t h e r a n g e e x p e c t e d ~ ~ l from o u r r e s u l t s . From t h e e s t i m a t e d Tc 2
,
t h e measured h e a t s of f o r m a t i o n and a p o s t u l a t e d s m a l l e l e c t r o n c o n t r i b u t i o n , t h e y a l s o o b t a i n e d p a i r i n t e r c h a n g e e n e r g i e s . Comparing t h e i r r e s u l t s w i t h t h e p r e s e n t o n e s i t is t o be n o t e d t h a t t h e i r m i 2 ) = 1 5 0 K i s h i g h e r t h a n o u r s , b u t t h i s is due i n l a r g e p a r t t o t h e h i g hK = 0.82 ( s e e e q . 7 ) which t h e y have u s e d . I f a r = 0.5 had been a p p l i e d , a c c o r d i n g t o R e f . 1 8 , m, w e r e o n l y 20 K, i n b e t t e r a g r e e m e n t w i t h t h e p r e s e n t r e s u l t s .
C4-120 JOUKXAL DE PHYSIQUE
A c c o r d i n g t o e q s . 3 a n d 4 t h e c o m b i n a t i o n mA1)
+
h i 3 ) i s d e d u c e d , f o r which t h e y g e t 625 K ( w i t h K = 0 . 8 2 ) , which is c h a n g e t o 690 K f o r K = 0.5. C o n s i d e r i n g t h e u n c e r t a i n t y i n t h e i r e s t i m a t i o n of TklI-A' and K , t h e i r $ I ) + 2mi3) is q u i t e c l o s e t o o u r s . The main d i f f e r e n c e t o o u r r e s u l t s s t e m s from r e p l a c i n g s u p p o s i t i o n B, i . e . t h a t t h e n e a r e s t e i g h b o r p a i r i n t e r c h a n g e e n e r g i e s a r e s t r u r e s e n s i t i v e , b y t h e a s s u m p t i o n m;$=O. However, on t h e o r e t i c a l g r n d s a n mlt5= and from t h e a n a 1 y s i s of t h e 1811 t o f c c t r a n s f o r m a t i o n a n m!')* 0 would be e x p e c t e d , a s d i s c u s s e d a b o v e .T h u s , a l t h o u g h i t must be r e a l i z e d t h a t t h e p r e s e n t r e s u l t s do n o t p r e s e n t a p r o o f f o r s u p p o s i t i o n s A and B, t h e r e is good e v i d e n c e t h a t , t a k i n g a l l r e s u l t s t o g e t h e r , t h e two s u p p o s i t i o n s a r e l i k e l y t o be c o r r e c t . T h i s would mean t h a t t h e r e l a t i v e s t a b i l i t i e s between d i f f e r e n t a and f3 p h a s e s c a n be a c c o u n t e d f o r by c o n f i g u r a t i o n a l i n t e r c h a n g e e n e r g i e s , w i t h o u t n e e d i n g t o u s e c o n c e p t s r e l a t e d t o t h e i n t e r a c t i o n of t h e Fermi s u r f a c e w i t h t h e B r i l l o u i n zone (when r e f e r r e d t o t h e p u r e e l e m e n t s of t h e same s t r u c t u r e s ) . The l a t t e r c o n c l u s i o n r e m a i n s v a l i d , even i f s u p p o s i t i o n B would n o t h o l d .
D i s c u s s i o n s and a c r i t i c a l r e a d i n g of a n e a r l i e r v e r s i o n of t h i s p a p e r by A.P.
Miodownik a r e g r a t e f u l l y a c k n o w l e d g e d . T h i s work was p a r t i a l l y s u p p o r t e d by t h e A r g e n t i n i a n C o n s e j o N a c i o n a l d e I n v e s t i g a c i o n e s C i e n t i f i c a s y T P c n i c a s .
R e f e r e n c e s
JONES, H. P r o c . Phys.Soc.
A49
( 1 9 3 7 ) 250.MASSALSKI, T.B., MIZUTAMI, V., P r o g r . M a t . S c i .
2
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4
(1970) 250.AHLERS, M., J. Phys.F
2
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BARCELO, G., AHLERS, M.,
GACIOLI,
R., Z - M e t a l l k d e .70
(1979) 732.INDEN, G . , 2. M e t a l l k d e .
66
( 1 9 7 5 ) 648.ARNEODO, W . , AHLERS, M., Acta Met.
22
(1974) 1475.BLAIR, G.R., DOWNIE, D.B., M e t a l S c i . J . 4 ( 1 9 7 0 ) 1.
CROMBIE, R., DOWNIE, D.B., A c t a Met.
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(1971) 1227.HULTGREN, R . , ORR, R.L., ANDERSON, P.D. and KELLEY, K . K . ( e d ) . S e l e c t e d V a l u e s o f Thermodynamic P r o p e r t i e s of M e t a l s and A l l o y s , New York Wiley 1963.
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