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THERMAL DEPENDENCE OF THE STRUCTURE
FACTOR IN AMORPHOUS SOLIDS
C. Tête, D. Boumazouza, G. Marchal, Ph. Mangin, J. Bouillot, C. Zeyen
To cite this version:
JOURNAL
DE
PHYSIQUEColloque C8, suppl6rnent a u n 0 i 2 , Tome 46, dCcernbre 1985 page C8-281
T H E R M A L DEPENDENCE OF T H E STRUCTURE FACTOR
I N
AMORPHOUS S O L I D SC . T @ t e , D . ~ o u m a z o u z a * , G . ~ a r c h a l * , Ph. ~ a n ~ i n * * , J . ~ o u i l l o ? * and C.M.E. zeyen**
EcoZe d e s Mines, P a r e d e S a u r u p t , 54042 Nancy Cedex, F r a n c e * ~ n i v e r s i t e ' Nancy I, C.O. 240, 54037 Nancy Cedex, F r a n c e
* * ~ n s t i t u t Laue-Langevin, B.P. 156 X, 38042 GrenobZe Cedex, F r a n c e
RBsumQ - Le f a c t e u r d e s t r u c t u r e t o t a l ST(K) d ' u n m a t Q r i a u amorphe e s t l a somme d ' u n e c o n t r i b u t i o n Q l a s t i q u e e t d ' u n e c o n t r i b u t i o n i n d l a s t i q u e . L o r s q u e l a t e m p d r a t u r e augmente, l a c o n t r i b u t i o n Q l a s t i q u e d Q c r o i t a l o r s que l a composante i n s l a s t i q u e augmente. AprGs un r a p p e l d e l a forme d e c e s deux c o n t r i b u t i o n s d a n s l 1 e s p a c e r G c i p r o q u e , nous p r Q s e n t o n s l l Q v o l u t i o n t h e r m i q u e du f a c t e u r d e s t r u c t u r e t o t a l ST(K) e t d e l a p a r t i e 6 l a s t i q u e s e u l e , pour l ' a l l i a g e FeglPgC1l amorphe. Les r Q s u l t a t s s o n t c o m p a t i b l e s a v e c un modPle d e Meisel-Cote-Debye oii l e s p r o c e s s u s 2*1 phonon s o n t t r a i t Q s e x a c t e m e n t .
A b s t r a c t
-
The t o t a l s t r u c t u r e f a c t o r o f a n amorphous s o l i d ST(K) i s t h e sum of a n e l a s t i c and a n i n e l a s t i c component. When t h e t e m p e r a t u r e i n c r e a s e s , t h e e l a s t i c c o n t r i b u t i o n d e c r e a s e s , w h i l e t h e i n e l a s t i c p a r t i n c r e a s e s . A f t e r d i s c u s s i n g t h e f o r m s o f t h e s e two c o n t r i b u t i o n s i n t h e r e c i p r o c a l s p a c e , we p r e s e n t r e s u l t s showing t h e t h e r m a l dependence of ST(K) and o f t h e e l a s t i c c o n t r i b u t i o n a l o n e i n t h e FeglPgC1l amorphous a l l o y . The r e s u l t s a r e con- s i s t e n t w i t h a Meisel-Cote-Debye model, when t h e o n e phonon p r o c e s s e s a r e t r e a t e d e x a c t l y .I
-
INTRODUCTIONThe t o t a l s t r u c t u r e f a c t o r of amorphous m a t e r i a l s ST(K) i s t h e sum o f two compo- n e n t s : t h e e l a s t i c component d u e t o t h e d i f f r a c t i o n of p a r t i c l e s w i t h o u t exchange of e n e r g y between t h e p a r t i c l e s and t h e sample and t h e i n e l a s t i c component d u e t o t h e d i f f r a c t i o n o f t h e i n c i d e n t beam w i t h e x c i t a t i o n o r d e - e x c i t a t i o n of v i b r a t i o n a l modes i n t h e sample. The t h e r m a l v a r i a t i o n of ST(K) i s t h e r e s u l t o f t h e s i m u l t a n e o u s d e c r e a s e o f t h e e l a s t i c c o n t r i b u t i o n and i n c r e a s e of t h e i n e l a s t i c one a s t h e temper- a t u r e i n c r e a s e s . The e l a s t i c c o n t r i b u t i o n i s r e l a t e d t o t h e e q u i l i b r i u m p o s i t i o n of atoms, i . e . i n a n amorphous m a t e r i a l t o t h e t o p o l o g i c a l d i s o r d e r , and t h e i n e l a s t i c c o n t r i b u t i o n i s t h e consequence o f t h e d i s p l a c e m e n t of atoms d u e t o t h e t h e r m a l a g i t a t i o n i n t h e m a t e r i a l . Thus t h r e e i m p o r t a n t problems a r e c o n n e c t e d : i ) t h e mixing of t h e t o p o l o g i c a l and t h e r m a l d i s o r d e r s i i ) t h e e l a s t i c and i n e l a s t i c c o n t r i b u t i o n s t o t h e t o t a l s t r u c t u r e f a c t o r i i i ) t h e t h e r m a l dependence of t h e t o t a l s t r u c t u r e f a c t o r .
The e x a c t t r e a t m e n t of t h e s e problems i s d i f f i c u l t a n d r e q u i r e s a knowledge of t h e t o t a l d y n a m i c a l s t r u c t u r e f a c t o r S(K,w) whose d e t e r m i n a t i o n h a s b e e n c a r r i e d o u t by Suck e t a l . / 1 / f o r s e v e r a l m e t a l l i c g l a s s e s . A more s i m p l e a p p r o a c h i s t o u s e t h e Debye model of v i b r a t i o n o f atoms and t o r e d u c e t h e d e s c r i p t i o n of t h e phenomena t o o n l y two p a r a m e t e r s : t h e Debye t e m p e r a t u r e OD and t h e Debye wave v e c t o r q ~ . T h i s model i s c e r t a i n l y poor f o r q u a n t i t i e s v e r y s e n s i t i v e t o t h e d e t a i l s of S(K,w) b u t c a n b e a good a p p r o x i m a t i o n f o r t h e d e s c r i p t i o n of phenomena r e s u l t i n g from mean v a l u e s o r i n t e g r a t e d q u a n t i t i e s . The t h e r m a l v a r i a t i o n of t h e s t r u c t u r e f a c t o r of amorphous m a t e r i a l i s c e r t a i n l y one of s u c h q u a n t i t i e s . I n a p r e v i o u s p a p e r / 2 / i n t h e f r a m e o f t h e Debye model and by u s i n g e x t e n s i v e l y t h e n o t i o n of phonons, we in- t r o d u c e d t h e n-phonon s t r u c t u r e f a c t o r s An(K) and showed what k i n d of a p p r o x i m a t i o n s l e a d t o t h e u s u a l r e l a t i o n [ST(K)
-
11 = [a(K)-
1I
e -2WT(K) / 3 / and t o t h e l e s s u s u a lC8-282 J O U R N A L
DE
PHYSIQUE-
2WT (K) r e l a t i o n i n t r o d u c e d by M e i s e l and Cote / 4 / [S (K)-
A ( K ) ] = [a(K)-A(K)]eT
a(K) i s trae s t a t i c s t r u c t u r e f a c t o r , A(K) i s t h e one-phonon s t r u c t u r e f a c t o r and WT(K) t h e Debye Waller f a c t o r . A f t e r r e c a l l i n g t h e a n a l y s i s of t h e c o n t r i b u t t o n t o S (K) and t h e a p p r o x i m a t i o n l e a d i n g t o t h e above r e l a t i o n s , we p r e s e n t d a t a
T
showlng t h e v a l i d i t y of t h e a n a l y s i s t a k i n g i n t o account t h e one phonon c o n t r i - b u t i o n . A f t e r a summary of t h e c a l c u l a t i o n s and of t h e a p p r o x i m a t i o n s , we p r e s e n t d a t a showing t h e t h e r m a l e v o l u t i o n of t h e t o t a l s t r u c t u r e f a c t o r and of i t s e l a s t i c p a r t a l o n e a s a f u n c t i o n of t h e t e m p e r a t u r e and compare them t o r e s u l t * expected from t h e d i f f e r e n t a p p r o x i m a t i o n s .I1 - SUMMARY OF THE APPROXIMATIONS
I f t h e e n e r g y of t h e p a r t i c l e s of t h e i n c i d e n t beam i s l a r g e compared t o t h e energy of phonons i n t h e m a t e r i a l
(X
r a y a p p r o x i m a t i o n ) where S (K) i s t h e i n s t a n t a n e o u s c o r r e l a t i o n f u n c t i o n +T R . ( t ) i s t h e p o s i t i o n of atom.
j
a t time t 1 f (K) i s t h e atomic form f a c t o r The F o u r i e r t r a n s f o r m of ST(X) g i v e s t h e reduced p a i r c o r r e l a t i o n f u n c t i o n g T ( r ) where 2 4.rrp0r g ~ ( r ) i s t h e d e n s i t y of p r o b a b i l i t y of f i n d i n g a n atom a t d i s t a n c e r from a g i v e n atom a t t h e same t i m e . % ( r ) c o n t a i n s b o t h t h e p o s i t i o n a l and t h e r m a l d i s o r d e r .I n s e p a r a t i n g t h e t o p o l o g i c a l d i s o r d e r w i t h r e s p e c t t o t h e mean+atomic p o s i t i o n
+
x from t h e t h e r m a l d i s o r d e r which r e s u l t s i n t h e d i s p l a c e m e n t ug from t h e e q u i l i -
b:ium p o s i t i o n + -f
+
R g ( t ) = X Q + u R ( t )
and i n d e s c r i b i n g t h e movements of atoms by phonon modes i n t h e Debye model, we o b t a i n e d
S (K) = e-2W [a(K) + 2W Al(K) + T
The F o u r i e r t r a n s f o r m of a(K) g i v e s t h e reduced p a i r c o r r e l a t i o n f u n c t i o n g ( r )
4.rrporLg ( r ) i s t h e d e n s i t y of p r o b a b i l i t y of f i n d i n g t h e q u i l i b r i u m p o s i t i o n a t d i s t a n c g
r
from t h e e q u i l i b r i u m p o s i t i o n of a g i v e n atom.A,(K) i s t h e two phonon s t r u c t u r e f a c t o r e t c . A , (K) c a n be c a l c u l a t e d from a(K) w i t h
where n i s t h e number of phonons of wave v e c t o r q.
4
A2(K) c a n b e deduced from A (K) by u s i n g t h e same r e l a t i o n e t c . . 1
F i g . 1 shows t h a t A (K) f o l l o w s t h e o s c i l l a t i o n s of a(K) b u t w i t h a s m a l l e r ampli- 1 tude,A2(K) o s c i l l a t e s l i k e Al(K) b u t i s c l o s e r t o I e t c . . . A,(K) i s an a v e r a g e of An-l(K) o v e r a s p h e r e of c e n t e r K and r a d i u s qD. I n t h e Debye model qD i s a f u n c t i o n of t h e a t o m i c d e n s i t y n : qD = (6n2n)1/3
.
So a measurement of n g i v e s qD. q c a n however be determined empirically from t h e ~ o s i t i o n Kl of t h e f i r s t maximum09
a(K).
By u s i n g t h e E h r e n f e s t r e l a t i o n Kl.Rl = 7.8 where R1 i s t h e mean i n t e r a t o m i c d i s - t a n c e and by assuming t h e packing f r a c t i o n i s a b o u t 0 . 6 8 , we o b t a i n : qD c 0.54K1. F o r t h e FeglP8C1 amorphous a l l o y qDo=
1.65 1-1 ( i n o u r p r e v i o u s paper 1 2 1 , we u n d e r e s t i m a t e d qD i n t a k i n g qD = 1 . 2 A. A s a consequence, we a l s o u n d e r e s t i m a t e dOD 1 7 1 . ) AI(K) i s dependent on t h e t e m p e r a t u r e b u t r e a c h e s v e r y e a r l y i t s h i g h t e m p e r a t u r e l i m i t .
C8-284 JOURNAL
DE
PHYSIQUEThis N a g e l - E i n s t e i n ( ~ - E ) approximation i s e q u i v a l e n t t o assuming t h a t t h e i n e l a s t i c components of ST(K) a r e s t r u c t u r e l e s s i n K space.
The o t h e r extreme approximation i s t o c a l c u l a t e A l @ ) e x a c t l y and t o w r i t e A2 (K) = A3 (K) =
. . .
A1 (K) = A (K) and then-2WT (K) [ sT(K) - A(K)] = [ ST(K) - A(K)l e
This tleisel-Cote-Debye (?I-C-D) approximation i s c e r t a i n l y good when W1, t h e s t a t i s t i c a l weight of A1 (K)
,
i s l a r g e r t h a n a l l o t h e r Wn, i . e . f o r low temperatures and low K v a l u e s .An i n t e r m e d i a t e approximation ( I ) r e q u i r i n g only t h e d e t e r m i n a t i o n of A I (K) would be t o assume A1 (K) = A(K) and A2(K) = A3(K) =
...=
1 , t h e n :For t h e two extreme c a s e s t h e d i f f e r e n c e s AST T (K) = ST (K) - ST (K) can be
w r i t t e n : 1 2 1 2
t h e d i f f e r e n c e s expected from t h e N-E, M-C-D and I models a r e presented i n f i g . 3 f o r T I < T2
.
Five o b s e r v a t i o n s can be made :1) AS has t h e s i g n of (a(K)-1) i n t h e N-E approximation and t h a t of (a(K)-A(K)) i n t h e M-C-D approximation. The comparison of t h e occurrence of t h e z e r o v a l u e s of AS(K) i n t h e K range of t h e second peak of a(K) i s a p a r t i c u l a r l y i n t e r e s t i n g t e s t .
2) The p a r t l o c a t e d between t h e f i r s t and t h e second peak of AS(K) i s more f l a t i n t h e M-C-D model.
2
3 ) A t low K, AS evolves a s -K i n t h e N-E model and a s -A(K)K2 ( t e n d i n g more r a p i d l y towards z e r o ) i n t h e W-C-D model.
4 ) For t h e same parameters (T,BD), AS(K) i s s m a l l e r i n t h e M-C-D approximation t h a n i n t h e N-E approximation. That c a n be understood i f we r e c a l l t h a t t h e damping of ST(K) i s due t o t h e spread of t h e i n t e r a t o m i c d i s t a n c e s about a mean v a l u e . I n t h e N-E model, atoms move independently and t h e s p r e a d o f t h e int'eratomic d i s t a n c e s i n c r e a s e s with t h e temperature.In theM-C-D model t h e displacement of atoms a r e c o r r e l a t e d by t h e phonon modes. Then when an atom moves i n a d i r e c t i o n ,
t h e neighbours move r a t h e r i n t h e same d i r e c t i o n and t h e spread of t h e i n t e r - atomic d i s t a n c e s i n c r e a s e s much l e s s .
5) The d i f f e r e n c e s expected from t h e i n t e r m e d i a t e model have t h e f e a t u r e of t h o s e expected from t h e M-C-D a t low T and f o r small K v a l u e s and t h o s e of t h e N-E
model a t high temperature and f o r t h e l a r g e r K v a l u e s .
A s
I
(M-C-D)
1
F i g . 4 - D i f f e r e n c e between s t r u c t u r e f a c t o r s measured a t T1 = 4.2K and T2 = 300K from t h e Fe81P8C!1 amorphous a l l o y . T h i s e x p e r i m e n t a l difference i s t o be compared t o t h e t h e o r e t i c a l d i f f e r e n c e s shown i n f i g . 3. U.J
A s
0.2
A measured o calculated1
total-
(M-C-D) Measured (Fe81P8C11) - F i g . 3-
T h e o r e t i c a l d i f f e r e n c e s between F i g . 5-
Thermal e v o l u t i o n of S T , S A E t h e t o t a l s t r u c t u r e f a c t o r s o b t a i n e d a t a t K, = 3.08 1-1. The e v o l u t i o n s T T1=4.2K and a t T2=300K expected from N-E c a l c u l a t e d i n t h e M-C-D ( o r I ) model model,
M-C-D model,
I model w i t h q ~ = 1 . 6 5A-1
and QD = 280 K AST (K) = ST (K)-
S T ~ ( K ).
f i t v e r y w e l l t h e e x p e r i m e n t a l d a t a .1
0.1
-111 - EXPERIMENTAL PROCEDURE
C8-286 JOURNAL DE PHYSIQUE
S (K) o b t a i n e d on D2 f o r T = 4.2K is t h e s t r u c t u r e f a c t o r we took a s a (K) i n f i g . 1 ( z l t h o u g h t h e e q u i v a l e n c e i s n o t q u i t e exact).AS(K) shown i n f i g u r e 4 i s t h e
d i f f e r e n c e between s t r u c t u r e f a c t o r s measured a t TI = 4.2K and T2 = 300K. T h i s c u r v e i s q u i t e s i m i l a r t o t h a t of f i g u r e s 3b and 3 c . It d e m o n s t r a t e s the poor v a l u e o f t h e N-E model compared t o t h a t of t h e 14-C-D and I models.
The e v o l u t i o n of t h e i n t e n s i t y of t h e f i r s t peak of ST(K) a s a f u n c t i o n of t h e t e m p e r a t u r e , by t a k i n g ST(K) o b t a i n e d a t 4.2K a s r e f e r e n c e , i s shown i n f i g . 5. The e x p e r i m e n t a l d a t a a r e compared w i t h t h e c a l c u l a t e d d i f f e r e n c e s from t h e M-C-D model w i t h qp = 1.65 A-1 and OD = 280K. OD h a s been chosen t o o b t a i n t h e b e s t f i t
t o t h e e x p e r m e n t a l d a t a .
The sound v e l o c i t y c a l c u l a t e d from OD and i s 2220 m/s. It i s v e r y c l o s e t o t h e mean sound v e l o c i t y o b t a i n e d from m e c h a n i c 3 measurements
/
81.
As e x p e c t e d , t h e i n t e n s i t y of t h e f i r s t peak of t h e e l a s t i c p a r t of ST(K) measured on Dl0 d e c r e a s e s more q u i c k l y t h a n t h e i n t e n s i t y of t h e f i r s t peak of t h e t o t a l s t r u c t u r e f a c t o r ST(K) ( f i g . 5 ) . AE TO t a k e i n t o a c c o u n t t h e r e s o l u t i o n of t h e i n s t r u m e n t , we r e p l a c e AI(K) by Al (K) i n t h e M-C-D r e l a t i o n s where
J
q ( 2 n q + l ) d q qD 0 w i t h q - 1 im - a E ~ . The e v o l u t i o n of "A: c a l c u l a t e d w i t h ALE (K) f i t s v e r y w e l l e x p e r i m e n t a l d a t a o b t a i n e d o n D10. 1I V
-
X-RAYS AND NEUTRONSThe f o r m u l a .of s e c t i o n I1 a r e e x a c t o n l y i n t h e X-ray approximation. They a r e no l o n g e r v a l i d when t h e energy of t h e i n c i d e n t p a r t i c l e s i s n o t l a r g e com a r e d w i t h t h e Debye energy. The k i n e t i c e n e r g i e s of n e o t r o n s of wavelength 1.215
fl
and 0.93 a r e 53.10 mev and 90.64 mev r e s p e c t i v e l y and a r e l a r g e compared w i t h t h e Debye e n e r g y 24.15 mev. When t h e e n e r g y of i n c i d e n t n e u t r o n i s comparable t o ED o r when it i s l o w e r , two e f f e c t s c o m p l i c a t e t h e problem :i ) The e n e r g y of t h e i n c i d e n t n e u t r o n s c a n b e t o o s m a l l t o c r e a t e phonons of t h e h i g h e r e n e r g i e s
i i ) For g e o m e t r i c a l r e a s o n s due t o t h e c o n s e r v a t i o n of energy, t h e f i r s t peak of i n e l a s t i c component does n o t a p p e a r j u s t u n d e r t h e f i r s t peak of t h e e l a s t i c s t r u c t u r e f a c t o r . A s a consequence, a t K 1 , t h e i n c r e a s e of t h e i n e l a s t i c c o n t r i b u t i o n i s s m a l l e r t h a n i n t h e X-ray a p p r o x i m a t i o n and t h e d e c r e a s e of ST
(K1)
i s l a r g e r .F i r s t r e s u l t s o b t a i n e d w i t h A=2.5188 (E=12.3 mev) a r e i n agreement w i t h t h e s e remarks. REFERENCES
Suck, J . B . , Rudin, H . , G u n t h e r o d t , H . J . , and Beck, H . , J . Phys.C. 14 (1981) 2305. Mangin, Ph., TBte, C. and Marchal, G . , S o l i d S t a t e Comm.
50
( 1 9 8 4 ) 1 0 7 3 .C a r g i l l , G.S., J.A.P.
41
(1970) 12. Nagel, S.R., Phys. Rev. B16 - (1977) 1694.Cote, P.S., Meisel,L.V., Phys. Rev. L e t t .
