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POLYMORPHIC TRANSITIONS IN ALKALI
HALIDES, A MOLECULAR DYNAMICS STUDY
M. Parrinello, A. Rahman
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque C6, suppZ6ment au n o 12, Tome 42, ddcembre 1981 page c6-5 1 I
POLYMORPHIC TRANSITIONS I N A L K A L I HALIDES, A MOLECULAR DYNAMICS STUDY
M. P a r r i n e l l o and A. ~ahman*
U . o f P i e s t e , 2lrieste, I t a l y
*Argonne NationaZ Laboratory, Argonne, IL 60439, U . S. A.
i s found
.-
Using constant pressure molecular dynamics method, B1 + B2 t r a n s -K'c~' has been studied. The microscopic d e t a i l o f t h e t v a n s i t i o n t o be d i f f e r e n t from the one c o n j e c t u r e d about i n t h e l i t e r a t u r e .
1. I n t r o d u c t i o n . - I n Born and ~ u a n g l can be found a s h o r t sumnary on t h e s u b j e c t o f t h e r e l a t i v e s t a b i l i t y o f t h e zincblend, r o c k s a l t and cesium c h l o r i d e polynorphs o f a l k a l i ha1 ides. A more d e t a i l e d account was given b y Tosi and ~ r a i .2 The the- o r e t i c a l work has o n l y been concerned w i t h t h e e v a l u a t i o n o f t h e p o t e n t i a l energy o f t h e system using v a r i o u s s h o r t range i n t e r a c t i o n s between t h e ions.
I n t h e i r s t u d y Tosi and ~ u n i 3 used h i g h pressure data i n p a r a m e t r i z i n g t h e i r p o t e n t i a l and argued t h a t a t h i g h d e n s i t y t h e p o t e n t i a l s needed a r e d i f f e r e n t from t h e ones r e l e v a n t under more normal conditions; t h e y a l s o found i t necessary t o make t h e p a i r p o t e n t i a l s s t r u c t u r e dependent.
Cohen and ~ o r d o n 4 used t h e parameter f r e e p o t e n t i a l f u n c t i o n s developed b y Gordon and Kim5, t o s t u d y t h e r o c k s a l t t o cesium c h l o r i d e t r a n s i t i o n pressure and t h i s study was extended b y Boyer6 who i n c l u d e d t h e e f f e c t o f harmonic v i b r a t i o n s i n t h e c a l c u l a t i o n o f the f r e e energy; t h i s however d i d n o t m a t e r i a l l y a f f e c t t h e con- c l u s i o n s based on s t a t i c c a l c u l a t i o n s .
I n t h e work presented here we have used t h e
ord don-~im~
p o t e n t i a l s i n t h e para- m e t r i z e d form developed b y Boyer.6 The k e y element i n t h i s study i s the use o f new molecular dynamics methods b r i e f l y sumnarized i n t h e f o l l o w i n g s e c t i o n . These meth- ods make i t p o s s i b l e f o r a system o f c l a s s i c a l p a r t i c l e s t o rearrange i n t o new crys- t a l l i n e p a t t e r n s i f t h e ambient c o n d i t i o n s o f temperature and pressure are f a v o r - able f o r such rearrangement.2. P e r i o d i c Boundary Conditions Varying i n Time.- A system o f N p a r t i c l e s c o n f i n e d b y p e r i o d i c boundary c o n d i t i o n s i s i n c r y s t a l l o g r a p h i c language, e q u i v a l e n t t o a Bravais l a t t i c e d e f i n e d b y
a,b,c,
the u n i t c e l l having volunen
= a.(b A c ) and con- t a i n i n g N p a r t i c l e s w i t h f r a c t i o n a l coordinatessi
spreading t h e system i n t h e u n i t c e l l i n some prescribed fashion.P a r r i n e l l o and ~ahman7 introduced t h e p o s s i b i l i t y o f making t h e v e c t o r s
C6-5 12 JOURNAL DE PHYSIQUE
dynamical v a r i a b l e s i n a d d i t i o n t o t h e usual ones, namely a l l t h e
si.
L e t_h
=ta,b,c,l
be t h e tensor formed by t h e v e c t o r s a,b,c. Then a = deth
,
t h e p o s i t i o n vector i s1
=B
-
-
s and t h e d i s t a n c e square i s given by r2 =zOh*h
2
=2
5
where G i s t h e r e f o r e t h e m e t r i c tensor; t h e prime denotes a transpose i n t h e usual way.The equations o f motion generated b y t h e Lagrangian introduced7 are as f o l l o w s :
where t h e various symbols a r e d e f i n e d as:
2
={L%,
2 2 ,
&-bl
n; =
z
(mixi
l
i
+ fi -1 r . ) , (dyadic n o t a t i o n )(4)
,
( 6 )p, t h e e x t e r n a l 1 y a p p l i e d pressure, V, t h e p o t e n t i a l o f p a r t i c l e i n t e r a c t i o n s ,
W, has dimension o f mass; i t gives i n e r t i a t o t h e temporal change o f t h e hx,. Results based on these equations have been published7 a l r e a d y and a g e n e r a l i t a - t i o n from p t o a general a n i s o t r o p i c s t r e s s tensor has a l s o been made and t h e r e - s u l t s presented elsewhere.8 A d i s c u s s i o n o f t h e r o l e played by W i s a l s o given t h e r e
.*
Eq. ( 2 ) shows t h a t the v e c t o r s
a,k,c
i.e. the n i n e l e n g t h s hi, change i n t i m e because o f t h e imbalance between t h e e x t e r n a l pressure and t h e momentary, i n t e r n a l - l y generated, s t r e s s from p a r t i c l e moments and from i n t e r n a l f o r c e s .I n parenthesis i t i s worth mentioning t h a t the o l d molecular dynamics methods i m p l i c i t l y assumed t h e e x t e r n a l s t r e s s t o change w i t h t i m e i n such a way as t o bal- ance o u t
e
and g i v ek
= 0; t h i s coupled w i t h i n i t i a l c o n d i t i o nk
= 0 a u t o m a t i c a l l y gave a non-varying Bravais c e l la,b,c.
3. Molecular Dynamics Model f o r C r y s t a l l i n e KC1.- A n e u t r a l system o f 500 ions (250
K+
and 250 Cl-) was given a r o c k s a l t s t r u c t u r e a t genesis i n t h e f o l l o w i n g unusual manner (see s e c t i o n 5 below). I n a r e c t a n g u l a r p a r a l l e l o p i d e d formed b y v e c t o r sa,b,c
o f l e n g t h s 5S, 5S, 5 e S , a body-centered t e t r a g o n a l l a t t i c e w i t h 250 s i t e s (2x53) was constructed, t h e l a t t i c e constants being s,S,fiS.A
displacement o f (S/2,S/2,0) produced a s i m i l a r l a t t i c e o f 250 s i t e s f o r the o t h e r species i n a r o c k s a l t s t r u c t u r e .
t h e convenient parametrized form given b y ~ o ~ e r
.6
The l / r Coulomb p o t e n t i a l o f course makes it necessary t o use Ewald's sumnation; t h i s has t o t a k e proper account o f t h e ever changing v e c t o r s a,b,c which d e f i n e t h e p e r i o d i c a l l y r e p e a t i n g c e l l w i t h 500 p a r t i c l e s . The u n i t o f mass was taken t o be t h e reduced mass o f a K, C1 p a i r ; i n these u n i t s W was chosen t o be 20. The i n t e g r a t i o n was done i n steps o f 0.29 X 10-14s.4. Behavior i n Time o f a KC1 System under H i g h Pressure.- A r o c k s a l t KC1 system o f 500 i o n s was e q u i l i b r a t e d a t zero pressure and 300" K; t h e pressure was increased t o 44 kb w i t h i n one molecular dynamics i n t e g r a t i o n s t e p . The d e n s i t y and temper- a t u r e are shown i n Fig. 1 as f u n c t i o n s o f t i m e a f t e r t h e moment o f sudden pressure change.
-2
: Time h i s t o r y o f a compression and decompression r u n . The p o i n t s p l o t t e d are 5 ~t apart hence t h e non-smooth appearance. I n regions <I> t o< I V >
the p a i r c o r r e l a t i o n s were monitored. These have n o t been shown i n t h i s paper. S i g n i f i - cance o f t h e numbered v e r t i c a l arrows i s discussed i s sections 4 and 6.We note t h a t
i ) i n about 1.3 ps t h e system acquires a new s t a t e a t 925" K, 2.6 gcm-3
i i ) t h i s t u r n s o u t t o be a metastab'le s t a t e which l a s t s f o r 1.5 ps more, t h e r e - gion o f t i m e between arrows marked #l and #2 i n Fig. 1.
i i i ) t h e r e g i o n between arrows # 2 and # 3 i s c l e a r l y one o f r a p i d changes; t h e d e n s i t y r i s e s t o 2.85 g ~ m - ~ and t h e temperature t o 1250' K, w i t h i n about 0.3 PS.
C6-5 14 JOURNAL DE PHYSIQUE
The K-K, K-Cl, Cl-C1 p a i r c o r r e l a t i o n s a l l showed t h a t t h e s h e l l distances and c o o r d i n a t i o n numbers had changed from those o f t h e r o c k s a l t s t r u c t u r e t o those o f cesium c h l o r i d e : t h e l i k e ions changing from an f c c t o a simple cubic arrangement and t h e u n l i k e ones changing from an octahedral, 6 coordination, t o t h e cubic, 8 c o o r d i n a t i o n , o f u n l i k e nearest neighbors.
5 . Microscopic D e t a i l o f t h e Transformation.- A p a r t i c l e b y p a r t i c l e a n a l y s i s o f t h e t r a n s f o r m a t i o n j u s t described r e v e a l s ( F i g . 2) how t h e 81 + B2 t r a n s f o r m a t i o n
occurs.
Figure 2A shows a body-centered t e t r a g o n a l l a t t i c e , l a t t i c e v e c t o r s
a,b,c,
lengths a,a, fia r e s p e c t i v e l y . The atoms are i n d i c a t e d b y m. This i s an f c c l a t t i c e o f ions. The o t h e r species, shown as 0 , completes the r o c k s a l t struc-t ure
.
Operation # l : Uniform d i l a t a t i o n o f amount
fi
i n t h e d i r e c t i o n o f2
as i n - d i c a t e d by t h e t h i c k a r r o w i n Fig. 2A. The r e s u l t i s shown i n Fig. 26:a,b,c
be- come a,'b,c.Operation #2: A move o f a1 t e r n a t e planes i n t h e
c
d i r e c t i o n as i n d i c a t e d b y t h e f i n e arrows i n Fig. 28. The r e s u l t i s shown i n Fig. 2C; t h e center o f t h e square face formed b y 2 : ~ i s now occupied by 0.
An atom o f t h e same t y p e occu-p i e s t h e o p p o s i t e square face; it i s t h e shadowy f i l l e d c i r c l e i n Fig. 2C. Fig. 2C shows a simple c u b i c l a t t i c e o f l i k e ions, t h e l i k e and u n l i k e ions together forming a cesium c h l o r i d e s t r u c t u r e .
ROCK SALT
A
CESIUM
CHLORIDE
C
I n t h e conventional coordinate system used f o r B1 s t r u c t u r e s the transforma- t i o n i s a d i l a t a t i o n i n t h e (1,1,0) d i r e c t i o n and a transverse, ( 0 0 l ) , zone bound- a r y phonon w i t h p o l a r i z a t i o n vector i n t h e (-1,1,0) d i r e c t i o n . The r o c k s a l t + ce-
sium c h l o r i d e transformation described i n the previous section occurs according t o t h e p a t t e r n j u s t described.
The speculation i n t h e l i t e r a t u r e 9 i s t h e r e f o r e n o t substantiated b y our c a l c u l a t i o n . I t was conjectured t h a t t h e change occurred by a t r i g o n a l l a t t i c e o f angle 60' (=fee) becoming a t r i g o n a l l a t t i c e o f angle 90" which i s a simple cu- b i c l a t t i c e .
6. E f f e c t o f Decompression.- The r e g i o n o f t i m e between arrows # 3 and 4 i n Fig. 1
shows t h e h i g h pressure, h i g h temperature cesium c h l o r i d e s t r u c t u r e i n
a
s t a t e o f e q u i l i b r i u m . At the t i m e i n d i c a t e d by arrow #4 t h e pressure was dropped from the ambient value, 44 kb t o zero i n one molecular dynamics time step. Analysis o f t h e s t r u c t u r e between arrows # 4 and 5 shows t h a t t h e system was i n t h e process o f chang- i n g from a B2 t o a B1 s t r u c t u r e . Beyond arrow #5 t h e system i s a hot, zero pres- sure, s t a b l e r o c k s a l t s t r u c t u r e .7. Conclusion.- The above example shows t h a t w i t h constant pressure molecular dy- namics c a l c u l a t i o n s i t i s p o s s i b l e t o s t u d y t h e k i n e t i c d e t a i l o f p l p o r p h i c t r a n - s i t i o n s . I n t h e case of KC1 several o t h e r s i m i l a r c a l c u l a t i o n s have shown t h a t l a r g e pressures and h i g h temperatures are needed t o t r i g g e r t h e t r a n s i t i o n . Thus i n s t u d y i n g polynorphic t r a n s i t i o n s i t i s necessary n o t o n l y t o i n q u i r e i n t o t h e thermodynamic t r a n s i t i o n parameters b u t also, u s i n g e.g. the method e x e m p l i f i e d above, i n t o t h e h e i g h t o f t h e various b a r r i e r s t h a t hinder t h e t r a n s i t i o n .
References
M. Born and K. Huang, "Dynamical Theory o f C r y s t a l L a t t i c e s , " (Oxford, 1954).
M. Tosi and T. Arai, i n "Advances i n High Pressure Research," Vol.
I ,
1966. Ed. R. S. Bradley (Academic Press).
M. Tosi and F. Fumi, J. Phys. Chem. S o l i d s ,
g,
359 (1962). A. J. Cohen and R. G. Gordon, Phys. Rev. B E , 3228 (1975). R. G. Gordon and Y . S. Kim, J. Chem. Phys.-
56, 3122 (1972). L. L. Boyer, Phys. Rev. B g , 3673 (1981).M. P a r r i n e l l o and A. Rahman, Phys. Rev. L e t t .
