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THE SCATTERING LAW S (Q, ω) OF LIQUID
ALUMINUM
O. Eder, B. Kunsch, J. Suck, M. Suda
To cite this version:
JOURNAL DE PHYSIQUE CoZZoque C8, suppZSment
au
n08, Tome 41, aoCt1980,
pageC8-226
THE SCATTERING LAW s(Q,~) OF LIQUID ALUMINUM
*
O . J . Eder, B . Kunsch, J . B . Suck and M. Suda
Forschungszentrwn Seibersdorf, A-2444 Seibersdorf, Austria *~nstitut- Laue-Langevin, Grenob le, France
The dynamical s t r u c t u r e f a c t o r o f l i q u i d aluminum f o r c o n t a i n e r and background, they were normalized was a l r e a d y s u b j e c t t o both experimental and theo- t o Vanadium. These raw r e s u l t s were used a s i n p u t r e t i c a l t r e a t m e n t ( 1 , 2 ) . I n t h i s s h o r t c o n t r i b u - f o r m u l t i p l e s c a t t e r i n g (MS) c a l c u l a t i o n s using t h e t i o n we r e p o r t our f l r s t r e s u l t s of a neutron Monte Carlo program o f Copley ( 6 ) . However t h e r e i s s c a t t e r i n g experiment, which we have performed on a s y e t no p o s s i b i l i t y t o i n c l u d e t h e s i n g l e c r y s t a l t h e t i m e - o f - f l i g h t s p e c t r o m e t e r IN4 a t t h e I n s t i - c o n t a i n e r i n M S c o r r e c t i o n s . The f l n a l r e s u l t was t u t e Laue-Langevin, Grenoble. We b r i e f l y summarize o b t a i n e d a f t e r applying s m a l l c o r r e c t i o n s t o t h e t h e main parameters o f t h e experiment, a s e x p e r i - normalizing f a c t o r and MS and a f t e r i n f e r r i n g a mental d e t a i l s and a f u l l account on t h e d a t a ( Q , ) independent background c o r r e c t i o n i n o r d e r e v a l u a t i o n procedure w i l l be given elsewhere ( 3 ) . t o f i t t h e t h r e e lowest even moments o f S ( Q , t i a )
-
A n incoming neutron energy o f E = 5 5 . 5 m e V has been chosen and the d e t e c t o r s were p l a c e d a t a d i s t a n c e of 4m from t h e sample a t s c a t t e r i n g a n g l e s ranging from go t o € 3 8 . ~ ~ . T h i s corresponds t o
0 . 9
h
Q4
7.28-I and -904
nrc
6
45mV. The sample had a p u r i t y of 99.99%. I t was c o n t a i n e d i n an o r i e n t e d s i n g l e c r y s t a l s a p p h l r e tube of 14mm i n - n e r diameter which was s e a l e d a t b o t h ends by bo- ron n i t r i d e plugs. No e l a s t i c c o h e r e n t s c a t t e r i n g c o n t r i b u t i o n s o f t h e c o n t a i n e r were d e t e c t e d . The sample was heated i n an evacuated furnace by a t h i n walled vanadium tube ( 4 ) . Because o f t h e l a c k o f c o l l i m a t o r s i n f r o n t of t h e d e t e c t o r s some d l £ - f i c u l t i e s were encountered when t h e Debye-Scherrer l i n e s o f t h e furnace w a l l s had t o be s u b t r a c t e d .The usual s e t o f experimental r u n s was performed (sample, empty c o n t a i n e r , empty f u r n a c e , Vanadium and Cadmium). The sample was measured a t two tem- p e r a t u r e s 1018K and 1078K ( 5 ) . One sample run took a b o u t 4.5 days. A f t e r having c o r r e c t e d t h e d a t a
A very s a t i s f a c t o r y agreement could be achieved which i s shown i n f i g . 1.
4 Fig. la
-
lc: (do>,
(a2)
and (a2
f o r l i q u i d aluminum a t l018K ( c i r c l e s : exp. r e s u l t , f u l l l i n e : t h e o r y , dashed l i n e : VEL model, open s q u a r e s : molecular dynamics r e s u l t ( 2 ) ; f o r f u r t h e r d e t a i l s s e e t e x t ) .I n l a - c t h e f u l l l i n e s denote t h e experimental s t r u c t u r e f a c t o r a t lOlOK ( 7 1 , t h e second moment
2 2
ti
Q kT/m and t h e f o u r t h moment i n t h e approxima- t i o n o f Hubbard and Beeby (81, r e s p e c t i v e l y . Theparameters have been taken from Bansal ( 9 ) . The open c i r c l e s denote o u r experimental r e s u l t s . A s t h e y have n o t y e t been extended beyond t h e (Q,k*c) -
l i m i t s imposed by t h e experimental c o n d i t i o n s , we have included i n f i g . 1 , denoted by dashed l i n e s , t h e r e s u l t s t h a t a r e o b t a i n e d by i n t e g r a t i n g with- i n t h e same l i m i t s a model ? ( ~ , k w ) t h a t i s based on v i s c o e l a s t i c t h e o r y (VEL) and t h a t s a t i s f i e s t h e f i r s t t h r e e even moment r e l a t i o n s . The devia- t i o n i n t h e f o u r t h moment can be a t t r i b u t e d t o t h i s e f f e c t . The a n a l y t i c a l e x p r e s s i o n o f VEL i s given i n r e f e r e n c e 10, formulae 8.18 and 8.19.
F i g . 2b: Same a s f i g . 2 a a t Q = 2.67%'. P l e a s e note t h e change i n s c a l e by a f a c t o r o f l o .
Fig. 2c: Same a s f i g . 2 a a t Q = 4.02-I.
Fig. 2a: Symmetrical s c a t t e r i n g law S(q,t;o ) of l i q u i d aluminum a t T = 1018K and Q = 2.18". Open c i r c l e s : experimental p o i n t s with t h e i r s t a t i s - t i c a l e r r o r . F u l l l i n e : MD r e s u l t o f Ebbsjd f o r SGT-potential ( 2 a ) . Dashed l i n e : VEL model.
The parameters O and
d
have a g a i n been taken fromE
r e f . 9 and Z ( 0 ) h a s been c a l c u l a t e d from sound ve- l o c i t y (11,121 and v i s c o s i t y d a t a ( 5 ) . The o n l y f r e e parameter Ql was chosen t o be 28-l. r ( Q ) was modified a s d e s c r i b e d elsewhere ( 1 3 ) . T h i s model was used a l s o f o r e x t r a p o l a t i o n o f t h e experimen-
03-228
JOURNAL. DE PHYSIQUE d e s c r i b e d above. The h i g h e r temperature r e s u l t i sn o t shown s e p a r a t e l y t h e agreement being roughly t h e same.
In f i g . 2 a - d we compare o u r r e s u l t s with molecu- l a r dynamics (MD) c a l c u l a t i o n s which have been p u b l i s h e d very r e c e n t l y (2a) and o u r model VEL. EbbsjZ, e t a l . have performed MD-calculations around 975K u s i n g t h r e e p o t e n t i a l s which t h e y have denoted
AGV, SGT and TGT. They p r o v i d e t a b u l a t e d r e s u l t s a t t e n Q v a l u e s ( 2 b ) , e i g h t of them a r e covered i n o u r experiment. able 1 l i s t s t h e s t a n d a r d deviation o f t h e s e t h r e e t h e o r e t i c a l dynamical s t r u c t u r e f a c t o r s from o u r experimental r e s u l t s . 1018K 1078K AGV 1.94x10-~ 2.20x10-~ (meV
-
1 SGT 1 . 4 3 x 1 0 - ~ 1.63x10-~ TGT 1 . 4 5 ~ 1 0 - ~ 1 . 7 0 ~ 1 0 - ~ VEL ~ . ~ o x I o - ~ 2 . 6 8 ~ 1 0 - ~ -3 e r r o r 1 . 7 4 ~ 1 0 1 . 6 6 x 1 0 - ~ Table 1 We have a l s o i n c l u d e d i n t a b l e 1 t h e corresponding value f o r t h e v i s c o e l a s t i c (VEL) model d e s c r i b e dabove and f o r comparison a number which was ob- t a i n e d by r e p l a c i n g i n t h e c a l c u l a t i o n t h e devia- t i o n s by t h e s t a t i s t i c a l e r r o r s o f t h e experimen- t a l r e s u l t s . A s t h e agreement is b e s t f o r SGT we have i n c l u d e d t h e s e r e s u l t s i n o u r f i g u r e s t o g e t h e r w i t h VEL which i s s t i l l b e t t e r by almost one o r d e r of magnitude, c l o s e l y approaching t h e number c o r - responding t o t h e e r r o r . I n f i g . 2 a , 2b and 2d t h e
C
v a l u e s o f S(Q,O)SGT a r e i n s e r t e d numerically b e i n g o f f s c a l e . We conclude t h a t VEL i s an appro- p r i a t e model f o r MS-calculations and i t s a p p l i c a - b i l i t y f o r e x t e n d i n g the e x p e r i m e n t a l d a t a t o h i g h e r
60
v a l u e s w i l l be i n v e s t i g a t e d . The f i r s t t h r e e even moments o f S(Q,k*,)SCT a r e i n c l u d e d a s open s q u a r e s i n f i g . 1.Fig. 3a:
F ( Q , L ~
) o f l i q u i d aluminum a t Q = 2.1g-? Open and f i l l e d c i r c l e s denote r e s u l t s a t 1018K and 1078K r e s p e c t i v e l y .Fig. 3b: Same a s f i g . 3a a t Q = 2.672-l. P l e a s e n o t e t h e change i n s c a l e by a f a c t o r o f
lo.
F i g . 3d:
z ( ~ , k c r )
) o f l i q u i d aluminum a t 5.02-I. Open c i r c l e s : 1018K, f i l l e d c i r c l e s l078K. I n f i g . 3a-d w e show t h e v a r i a t i o n o f t h e e x p e r i - N m e n t a l S ( Q , % W ) w i t h t e m p e r a t u r e a f t e r h a v i n g t h e d a t a smoothed. As t h e q u e s t i o n o f t h e o c c u r r e n c e o f c o o p e r a t i v e phenomena i s o f g r e a t t h e o r e t i c a l i n t e r e s t , we i n c l u d e some c o n s i d e r a t i o n s i n o u r d i s c u s s i o n . F i g . 4 shows t h e low Q r e g i o n which was c o v e r e d by o u r e x p e r i m e n t . The f u l l l i n e c o r r e s p o n d s t o o u r go c o u n t e r , t h e d a s h e d l i n e s show t h e r e l a t i o nkc^
=%
cQ, c b e i n g t h e sound v e l o c i t y . The open s q u a r e s d e n o t e t h e p o s i t i o n s o f t h e s i d e p e a k s o f,..
S(Q,Gw )SGT. T h e r e f o r e a much h i g h e r incoming n e u t r o n e n e r g y t o g e t h e r w i t h the possibility t o measure a t l o w e r s c a t t e r i n g a n g l e s i s n e c e s s a r y t o answer t h i s i n t e r e s t i n g q u e s t i o n . Acknowledgements We a r e g r a t e f u l t o D r . E - S c h a c h i n g e r o f T e c h n i c a l U n i v e r s i t y Graz f o r w r i t i n g t h e i n t e r p o l a t i o n p r o - grams. Thanks a r e due t o M r . J.Rupp f o r c o n s t r u c - t i o n o f t h e f u r n a c e a n d c a r e f u l p r e p a r a t i o n o f t h e e x p e r i m e n t . M s . M . C . ~ a s t n e r h a s c o n t r i b u t e d to t h e f l n a l d a t a e v a l u a t i o n .R e f e r e n c e s
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