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CALCULATIONS OF THE FCC-BCC 4He PHASE TRANSITION AT HIGH PRESSURE. AN EXAMPLE :
THE CORRELATED CELL MODEL
P. Loubeyre
To cite this version:
P. Loubeyre. CALCULATIONS OF THE FCC-BCC 4He PHASE TRANSITION AT HIGH PRES- SURE. AN EXAMPLE : THE CORRELATED CELL MODEL. Journal de Physique Colloques, 1984, 45 (C8), pp.C8-19-C8-22. �10.1051/jphyscol:1984804�. �jpa-00224302�
JOURNAL DE PHYSIQUE
Colloque C8, supplément au n ° l l , Tome *5, novembre 198* page C8-19
CALCULATIONS OF THE FCC-BCC ^He PHASE TRANSITION AT HIGH PRESSURE.
AN EXAMPLE : THE CORRELATED CELL MODEL
P. Loubeyre
Physique des Milieux Très Condensés, Université Pierre et Marie Curie, T13 E4, 4, Place Jussieu, 752S0 Paris Cedex OS, France
Résumé - Les mesures récentes des propriétés thermo- dynamiques de 1 ' He aux hautes pressions semblent indiquer la stabilité de la structure bec le long de la courbe de fusion avec un point triple proche de 300 K. Trois calculs seront présentés et en particulier un modèle de cellules corrélées sera développé. Celui- ci met en évidence simplement la précision nécessaire à la description des phases solides et le rôle des modes mous 1 ] 0 de la phase bec qu'ils stabilisent par entropie.
Abs tract - The recent measurements on high pressure thermodynamic properties of He indicate a bcc stable phase before melting with a triple point around 300 K.
Three methods of calculation are presented and more particularly a correlated cell model will be develop- ped. It simply shows the accuracy needed for the des- cription of the solid phase and how bcc is stabilized by the large entropy of her 1.10 m o d e s .
I - Introduction
Recent measurements of the thermodynamic properties of He near melting at high pressure point to the existence of a solid fec- bec phase transition with a triple point near 300 K /l/.
Calculations of the phase diagram of the solid require a des- cription of the solid with an accuracy better than the energy difference. Near melting it is a many-body problem for an an- harmonic crystal. Helium has a well known hamiltonian and three body forces, the smallest of all rare g a s e s , will be neglected.
It is thus a model system to test the different methods for calculating solid thermodynamic properties. Three methods were tried .
1] The constantpressure molecular dynamics simulation on a s y s - tem of 432 particles in which the system evolues to its stabler structure /2 /. Essentially the lagrangian of the system is cons- tructed so tliat the cell chanqesshape and size, preventing spu- rious metastable phases induced by periodic conditions. Tran- sition of the system from fee to bcc before melting was obser- ved at 16 0 P a and 32 7 K. Other isobaiic runs, their thermody- namic analysis, the phonon spectrum obtained from velocity autocorrelation function and the phonons dispersion curves from S(K,w) will be described in a more extensive publication.
2) A self consistent phonon calculation using various approxi- mations : the free energy was calculated by a high temperature expansion ; momentum (<,w> , < w > , < w > ) were calculated by sums on the crystal lattice. The frequencies were renormalized indif- firently on the optimum Einstein or Debye spectrum.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984804
C8-20 J O U R N A L DE PHYSIQUE
T h e f r e q u e n c y s p e c t r u m w a s a p p r o x i m a t e d w i t h a s i x t h o r d e r p o l y n o m i a l b y a l e a s t - s q u a r e m e t t l o d . F u r t h e r m o r e t h e f i r s t a n h a r m o n i c o d d t e r n F 3 3 w a s a d d e d . T h i s f r e e e n e r g y c a l c u l a t i o n g i v e s t h e q u a l i t a t i v e b e h a v i o r o f t h e p h a s e d i a g r a m b u t w i t h a t r i p l e p o i n t w h i c h i s t o o h i g h . I t i s w e l l k n o w n t h a t s i n g u l a r i - t i e s i n t h e s p e c t r u m a r e r e l a t e d t o e x t r e m a o r s a d d l e p o i n t s i n t h e d i s p e r s i o n c u r v e s a n d t h e 1 1 0 b r a n c h o f b c c s t r u c t u r e h a s s h o r t w a v e l e n g t h s o f t m o d e s . S o o u r p o l y n o m i a l e x p a n s i o n i s t o o r e g u l a r a n d i n d u c e s e r r o r s i n t h e c a l c u l a t i o n o f (Lnw), a n i m - p o r t a n t t e r m i n t t i e f r e e e n e r g y . A s a m p l i n g o v e r t h e B r i l l o w i n z o n e i s b e i n q m a d e a n d t c n d s t o c o m p a r e w e l l w i t h e x p e r i m e n t . / 3 /
F i n a l l y a c e l l m o d e l w i t h c o r r e l a t i o n s w i l l b e d e s c r i - b e d i n m o r e d e t a i l i n t h e n e x t s e c t i o n . I t h a s t h e a d v a n t a g e o f p o i n t i n g o u t s i m p l y t h e f a c t o r s t h a t d e t e r m i n e t h e t r a n s i t i o n a n d g i v e s t h e r e q u i r e d a c c u r a c y f o r t h e d e s c r i p t i o n o f t h e s o l i d p h a - s e .
1 1 - C o r r e l a t e d c e l l m o d e l
I n t h i s m o d e l , p r e v i o u s l y d e s c r i b e d f o r t h e c a l c u l a t i o n o f t h e p h a s e d i a g r a m o f c o l l o i d a l s u s p e n s i o n s / 4 / , e a c h a t o m i s c o n s - t r a i n e d i n a v o l u m e V / K a n d m o v e s i n t h e m e a n f i e l d p o t e n t i a l o f t h e o t h e r a t o m s . T h e a L o m s i n t e r a c t t h r o u g h t h e m o d i f i e d A z i z p a i r p o t e n t i a l / 1 / a n d t h u s t h e p a r t i t i o n f u n c t i o n f a c t o r i z e s i n t o i t s k i n e t i c a n d c o n f i g u r a t i o n a l p o r t s :
o h e r e ( v ( r j ) , t h e m e a n f i e l d p o t e n t i a l i s
R i n o t e s t h e l a t t i c e e q u i l i b r i u m p o s i t i o n s . A l t h o u g h i t i s i m - p o r t a n t t o k e e p t e r m s o f h i g h e r o r d e r t h a n t h e h a r m o n i c o n e i n t h e m e a n f i e l d p o t e n t i a l , t h e a s y m m e t r y o f t h e a v e r a g e m o t i o n o f t h e i n t e r a c t i n g a t o m s i s a h i g h e r - o r d e r e f f e c t a n d we w i l l d o a s e l f - c o n s i s t a n t q a u s s i a n a v e r a g e :
~ ( r ) = ( E * ) ' " 2 e x p ( - 3 r 2 )
3 < u ) 2<u')
2 .
<u > I S t h e m e a n s q u a r e d i s p l a c e m e n t . Me e x p a n d t h e p a i r p o t e n t i a l t o f o u r t h o r d e r , t h e n d o l a t t i c e s u m s t a k i n g i n t o a c c o u n t t t i e c u b i c s y m m e t r y o f t h e l a t t i c e a n d p e r f o r m t t ~ e g a u s s i a n a v e r a g e . T h e m e a n f i e l d p o t e n t i a l i s t h e n :
2 9
~ ( r ) = V O + V 2 r + VbOr- v b l ( r x 4 + r y 4 + r z 4 w h e r e t h e p o t e n t i a l c o e f f i c i e n t s a r e e x p l i c i t l y :
A p p r o x i m a t i n g t h e c x p o n e n t i a l o f t t i e a r i h a r m o n i c t e r m s b y i t s f i r s t o r d e r e x p a n s i o n i n t h e c o n f i g u r a t i o n i n t e g r a l a n d s u b s t r a c t i n g h a l f t h e a v e r a q e p o t e n t i a l e n e r g y s i n c e i t i s a m e a n f i e l d c a l c u l a t i o n , t h e f r e e e n e r q y i s g i v e n b y
2 .
T h e m e a n s q u a r e d e v i a t i o n u 1s d e t e r m i n e d b y t h e s e l f - c o n s i s - t e n t e q u a t i o n
u 2 = 3KT
2V2
I n t h e f o l l o w i n g we w i l l 3 g i v e t h e r e s u l t s o f o u r a p p r o x i m a t i o n s a t T = 4 0 0 K a n d ~ 1 4 . 5 1 5 cm / m o l e , w h e r e b c c s h o u l d b e s t a b l e r t h a n f c c . T h e n u m e r i c a l r e s u l t s o f t h e a b o v e c e l l m o d e l y i e l d v a l u e s f o r t e r m s w h i c h a r e 5:'; o f t h e h a r m o n i c o n e s ; t h i s j u s - t i f i e s t h e e x p o n e n t i a l e x p a r l s i o r i , a n d t h e f r e e e n e r g i e s o f f c c s t r u c t u r e , B F f c c 4 . 9 5 a r i d b c c s t u r c t u r e , B F b c c = 5 . 0 9 .
I\: N
T h e y p r e d i c t a s t a b l e f c c p h a s e . T h i s i s n o t s u r p r i s i n g s i n c e t h e c e l l m o d e l h a s a s e r i o u s f l a w f r a m a t h e o r e t i c a l p o i n t o f v i e w i n t h a t c e l l c o r r e l a t i o n s i n t h e m o t i o n s b e t w e e n n e i g h b o u - r i n g a t o m s a r e n e g l e c t e d .
I n a f i r s t s t e p we a d d p a i r e c o r r e l a t i o n s a n d we o b t a i n B F f c c = 4 . 6 7 a n d B F b c c 1 4 . 7 5 w i t t i - B F c o r r e q u a l 0 . 2 3 f o r f c c
N N N
a n d 0 . 2 7 f o r b c c . We n o t e t h a t t h i s c o r r e l a t i o n e n e r g y i s v e r y c l o s e t o t h e Domb S a l t e r a p p r o x i m a t i o n ( 0 . 2 3 ) / 5 / . B e c a u s e t h e Domb s a l t e r r e s u l t i s k n o w n t o b e a g o o d r e p r e s e n t a t i o n o f f c c r a r e g a z c r y s t a l s i L v a l i d a t e s o u r c o r r e l a t i o n p a i r a p p r o - x i m a t i o n f o r f c c . B u t f c c i s s t i l l s t a b l e r t h a n b c c a n d h e n c e i t b e c o m e s c l e a r L h a t a n t i a r m o n i c c o l l e c t i v e m o d e s a r e e s s e n t i a l t o t h e s t a b i l i t y o f t h e b c c p h a s e .
F o r h a r d s p h e r e s t t i e b c c s t r u c t u r e i s u n s t a b l e w i t h r e s p e c t t o 1 1 0 s l i d i n g m o d e a n d f o r o u r p o t e n t i a l i t i s c e r t a i n l y l o w f r e - q u e n c y a n d w o u l d s t a b i l i z e b c c b y i t s l a r g e e n t r o p y . We c a n i n c o r p o r a t e t h i s m o d e b y a d d i n g t o o u r b c c c e l l m o d e l s t r o n g c o r r e l a t i o n s i n t h e 1 1 0 p l a n e / 6 / . To a c h i e v e t h i s we l e t a n a t o m a n d i t s f o u r 1 1 0 p l a n e n e a r e s t n e i g h b o r s m o v e a s a r i g i d u n i t . I f we r l e g l e c t c o r r e l a t i o n e f f e c t s o n t h e a n 9 a r m o n i c t e r m s , t h e 1 1 0 c o r r e l a t i o n s c h a n g e t h e t i a r m o n i c p a r t V2r o f t h e m e a n f i e l d p o t e l i l i a 3 i r i t o a ? a r l i s o t r o p i c o n e :
v r 2 + v r y r w h e r e x a n d y d i r e c t i o n s a r e i n t h e 1 1 0 p f 8 n ~ . ~ e c a x s e o ; V ~ 6 e ' g a u s s i a n a v e r a g e L h e d i f f e r e n c e i n t h e f r e e e n e r g y c o m e s f r o m t h e e n t r o p i c t e r m :
B F c o r r 1 1 0 = 1 I n ( V Z x V 2 )
N - 2 +
I n Ltie z d i r e c t i o n we t a k e i n t o a c c o u n t t h e p r e v i o u s p a i r c o r - r e l a t i o n s . T h e 1 1 0 c o r r e l a t i o n s a r c l i m i t e d t o a g r o u p o f f i v e a t o m s : a n a t o m h a s 1 p r o b a b i l i t y t o b e a t t h e c e n t e r a n d f?
L; c
a t h e e d g e o f t h i s f i v e a t o m 1 1 0 p l a n e r e c t a n g l e . W i t h t h i s ' w e i g h e d s u m V Z x a n d V a r e g i v e n b y :
2 ~
JOURNAL DE PHYSIQUE
w h e r e 0 a n d 1 , 2 , 3 , 4 l a b e l t h e c e n t e r a n d e d g e a t o m s . W i t h t h i s c o r r e l a t i o n c o n e c t i o n , t h e b c c f r e e e n e r g y , ( 3 F b c c = 4 . 6 2 8 , b e c o m e s s m a l l e r t h a n t h e f c c o n e i n t h e p a i r - c o r r e l a t e d c e l l N m o d e l . I t i s c l e a r t h a t t h e 1 1 0 c o r r e l a t i o n s s t a b i l i z e b c c a n d t h a t c o n f i r m s t h e i m p o r t a n c e o f c o l l e c t i v e a n h a r m o n 3 c m o d e s . Q u a n t u m c o r r e c t i o n s a r e t a k e n i n t o a c c o u n t b y t h e h t e r m o f t h e W i g n e r - k i r k w o o d e x p a n s i o n / 7 / .
F r o m t h e t o t a l f r e e e n e r g y f u n c t i o n ( s u m o f t h e c e l l m o d e l , c o r r e l a t i o n s a n d q u a n t u m p a r t s ) we c o n s t r u c t t h e t r a n s i t i o n l i n e b y a M a x w e l l d o u b l e t a n q e n t c o n s t r u c t i o n . T h e r e s u l t s g i v e s a t r i p l e p o i n t a t 2 2 0 K a n d a b c c s t a b l e p h a s e a b o v e a - l o n g t h e m e l t i n g c u r v e . T h e y c a n b e c o n s i d e r e d t o b e ~ n g o o d a g r e e m e n t w i t h e x p e r i m e n t , t a k i n g i n t o a c c o u n t t h e s e n s i t i v i t y o f t h e r e s u l t s t o a l t e r a t i o n s o f t h e m o d e l .
C o n l u s i o n :
C a l c u l a t i o n s o f s u c h p h a s e t r a n s i t i o n r e q u i r e s a n a c c u r a c y b e t t e r t h a n 2 / 1 0 0 U . C o m p a r i s o n w i t h e x p e r i m e n t g r e a t l y h e l p s i n t e s t i n g t h e v a r i o u s s i m p l i f y i n g h y p o t h e s i s i n t h e m o d e l s w h i c h c o u l d a f t e r w a r d s b e a p p l y w i t h c o n f i d e n c e t o o t h e r s y s - t e m s . The c e l l m o d e l c a l c u l a t i o n i l l u s t r a t e d h o a a l t e r a t i o n s o f a m o d e l c a n i n d i c a t e w h i c h f a c t o r s w i l l c a u s e a s o l i d p h a s e c h a n g e .
R e f e r e n c e s :
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/ 2 / D . LEVESUUE, J . J . WCIS, M.L. K L E I N P h y s . R e v . L e t t - 5 1 ( 1 9 8 3 ) 6 7 0 / 3 / P. LOUBEYRE
To b e p u b l i s h e d .
/ 4 / K . HONE, S. ALEXANDER, P . C H A I K I N , P . PINCUS J Chem. P h y s . 2 ( 1 9 8 3 3 1 4 7 4
/ 5 / R a r e Gas S o l i d s , e d i t e d b y M . L . K L E I N a n d J.A. VENABLES 3 2 0
/ 6 / L3.L H O L I A Y , e t A 1 Chem. P h y s . 2 ( 1 9 7 3 ) 5 4 4 4 / 7 / P . LOUBEYRC, J . P . HANSEN,
P h y s . L e t t K A ( 1 9 8 0 ) 1 8 1
