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Submitted on 1 Jan 1980
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EFFECTIVE PAIR POTENTIALS FROM
STRUCTURE FACTORS
R. Mountain
To cite this version:
JOURNAL DE PHYSIQUE CoZZoque: C8, supple'ment a u n08, Tome 41, aogt 1980, page
C8-297
EFFECTIVE P A I R POTEt%rnLA'lLS
FROMSTRUCTURE FACTORS
R.D. MountainNational Bureau o f Standards, Washington,;E.C.~ 20234 U.S.A.
I n t h i s paper we examine t h e problem of e x t r a c t i n g e f f e c t i v e p a i r p o t e n t i a l s f o r expanded f l u i d rubidium from s t r u c t u r e f a c t o r d a t a . Our approach t o t h i s t a s k i s based on t h e o b s e r v a t i o n , which i s d i s c u s s e d i n d e t a i l below, t h a t t h e 1 Percus-Yevick e q u a t i o n p r o v i d e s an a c c u r a t e connection between t h e p a i r c o r r e l a t i o n f u n c t i o n , g 2 ( r ) , and an e f f e c t i v e p a i r p o t e n t i a l , $ ( r ) , over an i n t e r e s t i n g range of d e n s i t i e s and t e m p e r a t u r e s f o r expanded f l u i d rubidium. T h i s i s q u i t e u n l i k e t h e c a s e f o r t h e r a r e g a s f l u i d s 2 where Percus- Yevick t h e o r y i s a c c u r a t e only f o r t h e low d e n s i t y , h i g h t e m p e r a t u r e g a s regime.
F i r s t we compare t h e r e s u l t s of Monte Carlo and Percus-Yevick c a l c u l a t i o n s of g ( r )
2
o b t a i n e d u s i n g t h e e f f e c t i v e p o t e n t i a l developed by P r i c e , e t a13 f o r d e n s i t i e s and t e m p e r a t u r e s o f expanded f l u i d rubidium where s t r u c t u r e f a c t o r
4
measurements have been made by Freyland.
From t h e s e c a l c u l a t i o n s we conclude t h a t t h e Percus-Yevick e q u a t i o n i s u n s a t i s f a c t o r y f o r t e m p e r a t u r e s below 1200 K and f o r d e n s i t i e s3
g r e a t e r t h a n 1.10 g/cm.
Then we u s e t h e Percus- Yevick e q u a t i o n t o examine t h e e f f e c t on t h e s t r u c t u r e f a c t o r , s ( Q ) , of modifying t h e shape and range of t h e e f f e c t i v e p a i r p o t e n t i a l o f P r i c e , e t a l . We f i n d t h a t t h e r e p u l s i v e p a r t must b e- -
$ ( r ) i s b r i e f l y d i s c u s s e d and p o s s i b l e compli- c a t i o n s a r i s i n g from c r i t i c a l p o i n t e f f e c t s a r e mentioned.The Monte Carlo method of M e t r o p o l i s ,
a15
i s a w e l l known method f o r e s t i m a t i n g t h e e q u i l i b r i u m p r o p e r t i e s of models of dense f l u i d s w i t h an accuracy of about 1%. During t h e c o u r s e o f an e a r l i e r Monte Carlo s t u d y of t h e s t r u c t u r e6
of expanded f l u i d rubidium
,
it was noted ( b u t not r e p o r t e d ) t h a t t h e Percus-Yevick e q u a t i o n y i e l d e d r e s u l t s which a r e i n c l o s e agreement w i t h Monte Carlo e s t i m a t e s f o r g 2 ( r ) . F i g . l a d i s p l a y s t h e Monte Carlo r e s u l t s u s i n g t h e P r i c e e f f e c t i v e p o t e n t i a l f o r rubidium a t 1200 K and p = 1 . 1 03
g/cm
.
On t h i s s c a l e , t h e Monte Carlo and Percus- Yevick r e s u l t s a r e i n d i s t i n g u i s h a b l e . The devia-PY MC t i o n p l o t found i n F i g . l b shows how g2
-g2 v a r i e s a s a f u n c t i o n of r. The u n c e r t a i n t i e s i n t h e Monte Carlo r e s u l t s a r e oS t h e same magnitude a s t h e d i f f e r e n c e s shown h e r e . The r e l a t i v e l y l a r g e d i f f e r e n c e i n t h e v i c i n i t y of t h e f i r s t maximum i s an a r t i f a c t of t h e 0.050 r e s o l u t i o n used i n t h e Monte Carlo program. The Monte Carlo
5
r e s u l t s a r e f o r a sample based on 5x10 t r i a l s a s d e s c r i b e d elsewhere6 and t h e Percus-Yevick r e s u l t s were o b t a i n e d u s i n g an i t e r a t i v e procedure devel-
7
oped by Broyles
.
At lower t e m p e r a t u r e s and s o f t e n e d and t h e range o f t h e a t t r a c t i v e p a r t must h i g h e r d e n s i t i e s we f i n d s y s t e m a t i c d i f f e r e n c e s b e extended i f agreement i s t o be o b t a i n e d w i t h between t h e Monte Carlo and Percus-Yevick r e s u l t s t h e measurements of F r e y l a n d , e t a l . The e x t r a c t i o n which a r e g r e a t e r t h a n t h e u n c e r t a i n t y i n g MC2 . o f t h e energy-wave nwnber c h a r a c t e r i s t i c from This i s i l l u s t r a t e d i n Fig. 2 f o r T=900 K and
21
C8-298
JOURNAL DE PHYSIQUE1.0
r
/CT2.0
F i g . 1. ( a ) Monte Carlo p a i r f u n c t i o n f o r t h e P r i c e p o t e n t i a l f o r rubidium a t T=1200K and
3
p=l.lOg/cm
.
( b ) Deviation of t h e Percus-Yevick s o l u t i o n from t h e Monte Carlo r e s u l t s .3
p = 1.22 g/cm
.
From t h i s we conclude t h a t t h e Percus-Yevick equation can be used t o e s t i m a t e g 2 ( r ) t o w i t h i n ?2% f o r rubidium f o r T>_
1200 K3
and p- 1.10 g/cm.
I n p r i n c i p l e , it should b e p o s s i b l e t o e x t r a c t e f f e c t i v e p a i r p o t e n t i a l s from s t r u c t u r e f a c t o r d a t a u s i n g t h e Percus-Yevick e q u a t i o n where 1/@ = k T, Boltzmann's c o n s t a n t t i m e s t h eB
a b s o l u t e temperature and ~ ( r ) i s t h e d i r e c t c o r r e - l a t i o n f u n c t ionL. However, i f s e n s i b l e r e s u l t s a r e t o be o b t a i n e d , more e x t e n s i v e s t r u c t u r e f a c t o r d a t a a r e r e q u i r e d t h a n a r e c u r r e n t l y a v a i l - a b l e . We have i n s t e a d chosen t o i l l u s t r a t e t h e method of e x t r a c t i n g e f f e c t i v e p a i r p o t e n t i a l s by a l e s s d i r e c t procedure d e s c r i b e d below.3
The e f f e c t i v e p a i r p o t e n t i a l of P r i c e ,&t
f o r rubidium a t T = 1400 K and p = 0.96 g/cm3 i s shown i n F i g . 3 a s t h e dashed curve. The s o l i d curve r e p r e s e n t s a modified p o t e n t i a l which l e a d s t o a s t r u c t u r e f a c t o r i n agreement w i t h t h e4
measurements o f Freyland
& &
.
The s o l i d curve was o b t a i n e d by s o f t e n i n g t h e r e p u l s i v e p o r t i o n and by s t r e t c h i n g t h e a t t r a c t i v e p o r t i o n of t h e P r i c e p o t e n t i a l . The amount of s o f t e n i n g and s t r e t c h i n g needed was determined by s o l v i n g t h e Percus-Yevick equation f o r v a r i o u s t e s t p o t e n t i a l s and t h e n r e f i n i n g t h e m o d i f i c a t i o n s u n t i l t h e p r e d i c t i o n s f o r S ( O ) and t h e v a l u e o f S ( Q ) a t t h eI
,
,
,
p ~ i n c i p a l maximum were w i t h i n 10% of t h e e x p e r i -1.0
2.0
mental v a l u e s . The c a l c u l a t e d s t r u c t u r e f a c t o r i sr
/(TF i g . 2. D e v i a t i o n of t h e Percus-Yevick s o l u t i o n shown i n F i g . 4 along w i t h two experimental from t h e Monte Carlo r e s u l t s f o r t h e P r i c e p o i n t s i n d i c a t e d by X ' s .
3
maximum and t h e s t r e t c h i n g of t h e a t t r a c t i v e r e g i o n was needed t o o b t a i n S ( 0 ) . The p r e d i c t e d v a l u e of S ( O ) t u r n s o u t t o b e r a t h e r s e n s i t i v e t o t h e d e t a i l s of t h e a t t r a c t i v e p a r t o f t h e p o t e n t i a l once t h e r e p u l s i v e p a r t i s f i x e d . No c l a i m f o r t h i s p o t e n t i a l a s b e i n g t h e c o r r e c t one can be made s i n c e o n l y t h e g r o s s f e a t u r e s o f S(Q) were c o n s i d e r e d and only simple v a r i a t i o n s i n t h e shape o f t h e p o t e n t i a were examined. However, t h i s
I
example does s h o w t h a t g e n e r a l t r e n d s i n t h e v a r i a - t i o n of t h e p o t e n t i a l w i t h d e n s i t y can b e e x t r a c t e d from S ( Q ) d a t a i n t h e r e g i o n of d e n s i t i e s l e s s 3 t h a n p = 1 . 1 0 g/cm f o r rubidium. F u r t h e r e f f o r t s t o e x t r a c t p o t e n t i a l s should b e g i n1.0
2.0
by employing p h y s i c a l arguments t o develop systema-r /n- ' I V
t i c procedures f o r modifying e x i s t i n g p o t e n t i a l s . F i g . 3. P a i r p o t e n t i a l s f o r rubidium a t
3 It i s i n p r i n c i p l e p o s s i b l e t o i n v e r t a p=0.96g/cm
.
Dashed curve: P r i c e . S o l i d curve:p o t e n t i a l f u n c t i o n @ ( r ) and o b t a i n t h e energy-wave modif'ied p o t e n t i a l d e s c r i b e d i n t e x t . number c h a r a c t e r i s t i c F ( Q ) u s i n g t h e r e l a t i o n
8
F i g . 4. S t r u c t u r e f a c t o r c a l c u l a t e d using t h e modified p o t e n t i a l o f F i g . 3 . The X's a r e experimental v a l u e s chosen t o i n d i c a t e t h a t t h e degree of agreement achieved i s on t h e o r d e r of2
l ow.
I n p r a c t i c e t h i s r e q u i r e s $ ( r ) t o b e known a c c u r a t e l y over a l a r g e range of r v a l u e s , i n c l u - ding t h e p h y s i c a l l y i n a c c e s s i b l e r e g i o n of s m a l l r . The procedure d i s c u s s e d h e r e f o r c o n s t r u c t i n g $ ( r ) i s u n r e l i a b l e i n t h i s r e g i o n s o a t t e m p t s t o c o n s t r u c t F ( Q ) v i a Eq. ( 2 ) a r e u n l i k e l y t o l e a d t o p h y s i c a l l y s i g n i f i c a n t r e s u l t s . For example, t h e simple m o d i f i c a t i o n s employe&' i n determining t h e e f f e c t i v e p o t e n t i a l of F i g . 3 a p p a r e n t l y over- emphasized t h e magnitude o f t h e s m a l l r p o r t i o n a s t h e r e s u l t i n g F(Q) was n e g a t i v e .JOURNAL DE PHYSIQUE t o t h e liquid-vapor c r i t i c a l p o i n t , t h e n t h e long- r a n g e s p a t i a l c o r r e l a t i o n s a s s o c i a t e d w i t h t h e c r i t i c a l p o i n t would confuse t h e a n a l y s i s of t h e 9 d a t a . For rubidium, t h e c r i t i c a l d e n s i t y i s 3 p = 0.347 f .002 gfcm s o c r i t i c a l p o i n t e f f e c t s should not b e
a
problem f o r d e n s i t i e s g r e a t e r t h a n3 0.4 g/cm
.
References
P.A. E g e l s t a f f , An I n t r o d u c t i o n t o t h e Liquid S t a t e , (Academic P r e s s , New York, 1 9 6 7 ) , p 57. P.G. Mikolaj and C . J . P i n g s , J.Chem.Phys.
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Ber.Eunsenges
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