CALCULATION OF DEFECT ENTROPIES IN IONIC CRYSTALS IN THE QUASIHARMONIC APPROXIMATION : APPLICATION TO FLUORITES

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CALCULATION OF DEFECT ENTROPIES IN IONIC

CRYSTALS IN THE QUASIHARMONIC

APPROXIMATION : APPLICATION TO FLUORITES

P. Jacobs, M. Nerenberg, J. Govindarajan, T. Haridasan

To cite this version:

JOURNAL DE PHYSIQUE

CoZZoque C6, supple'ment au n o 12, Tome 42, de'cembre 1981 page C6-914

CALCULATION OF DEFECT ENTROPIES IN IONIC CRYSTALS IN THE QUASIHARMONIC

APPROXIMATION

:

APPLICATION TO FLUORITES

P.W.M. Jacobs, M.A.H. Nerenberg,

J.

Govindarajan and T.M. Haridasan

University of Western Ontario, London, Ontario, Canada N6A 5B7

1. Outline of Theory The dynamical matrix for a crystal containing an inter- stitial can be written in the form

where L is the dynamical matrix of the perfect crystal, h is that part of L that refers exclusively to the interstitial and

A is the matrix that connects the inter-

stitial coordinates with those of the rest of the lattice. A~ is the transpose of A.

6L

is the matrix that describes the force constant changes which occur because of the relaxations of the lattice ions due to the presence of the interstitial. The Green matrix

G

= i-'may also be written in partitioned form like that in (I), whence it may be shown that the determinant of is

1

= I I - G ~ L ' I I G l l A ) - ' (2)

T

where 6L1 =

6L

+

A A-I A and I is the 3N x 3N unit matrix,

N being the number of

atoms in the crystal. Since static lattice calculations show that relaxations of the lattice ions are small except for the first and second neighbours, the non-zero elements of 61, will be restricted to a small sub-space comprising these n ions that are perturbed appreciably by the interstitial. It then follows that

I

I

-

G6L'

I

=

11

-

g8~'I (3)

where I on the RS of (3) is now the 3n

x

3n unit matrix. 8R1 = 611

+

A-'

aT

where

and the dimensions of the matrix a are 3n x 3.

The entropy change associated with the introduction of an anion from into an interstitial position in the high-temperature approximation may then be written in the form

s = -$k Rn Rim

I

I

-

g8R1

[

+

3k (1

-

Rn (hwI/kT)

i ( 6 )

W O

where

wI

is the hypothetical frequency of vibration of an interstitial in a lattice in which all the host-lattice ions are frozen at their (perfect lattice) equilibrium positions. In similar fashion the dynamical matrix of a crystal with a vacancy is

-

Lv = Lt

-

.6R (7)

where Lt i s t h e dynamical m a t r i x L o f t h e c r y s t a l w i t h o u t a vacancy, t r u n c a t e d by

removing t h e t h r e e rows and columns b e l o n g i n g t o t h e i o n which is n o l o n g e r p r e s e n t .

6R i s t h e m a t r i x of f o r c e c o n s t a n t changed due t o t h e r e l a x a t i o n s of t h e i o n s c a u s e d

by t h e vacancy. I t i n c l u d e s a s w e l l t h e changes i n t h e " s e l f - t e r m s " of t h e i o n s

b e c a u s e o f t h e removal from t h e s e sums t h e c o n t r i b u t i o n s due t o t h e i o n which h a s

now been removed. By p a r t i t i o n i n g

iv,

and a l s o t h e p e r f e c t c r y s t a l dynamical m a t r i x

L, i n t o b l o c k s s u c h t h a t one o f t h e d i a g o n a l b l o c k s o f

i

r e p r e s e n t s t h e ' d e f e c t

v

s p a c e ' ( t h a t i s t h o s e i o n s t h a t a r e p e r t u r b e d by t h e p r e s e n c e o f t h e vacancy

-

i n

p r a c t i c e t h i s means i t s f i r s t and second n e i g h b o u r s ) a n d t h e c o r r e s p o n d i n g b l o c k

o f L r e p r e s e n t s t h e d e f e c t space-to-be ( i n c l u d i n g t h e i o n t h a t w i l l become a

vacancy) i t may b e shown t h a t t h e e n t r o p y change a s s o c i a t e d w i t h t h e removal o f a

l a t t i c e a n i o n t o

-

i s s v = -&k Rn Ilim {m3

I

gv

]

I

( g i l l t

-

651

1

( k ~ / h ) ~ }

-

3k h P 0

-

(8) The e n t r o p y of f o r m a t i o n of a F r e n k e l d e f e c t i s t h u s g i v e n by = -+k Ln R i m { I g v [

1

(g;')t

-

6RI

11

-

g 6R9Im3

1

w o (9) I n t h e s e e x p r e s s i o n s m- i s t h e mass o f a n a n i o n i s t h e p e r f e c t - l a t t i c e Green gv m a t r i x f o r t h e d e f e c t space-to-be and ( g -') i s t h e t r u n c a t e d v e r s i o n o f t h e v t

r e c i p r o c a l o f g ( t h a t is, w i t h t h e 3 rows and columns d e l e t e d t h a t c o r r e s p o n d t o

t h e i o n which i s t o b e removed when t h e vacancy i s f o u n d ) . We may n o t m u l t i p l y o u t

t h e f i r s t p r o d u c t of d e t e r m i n a n t s i n (9) s i n c e t h e o p e r a t i o n s o f t r u n c a t i o n and i n v e r s i o n do n o t , i n g e n e r a l , commute.

2. C a l c u l a t i o n The n e c e s s a r y Green f u n c t i o n s were e v a l u a t e d u s i n g s h e l l model

p a r a m e t e r s f i t t e d t o t h e e x p e r i m e n t a l phonon d i s p e r s i o n . These p a r a m e t e r s were

t a k e n from t h e work of Elcombe and p r y o r 1 and I3lcombe2. F o r t h e c a l c u l a t i o n o f 611

we r e q u i r e t h e c o o r d i . n a t e s o f t h e i o n s i n t h e i m p e r f e c t c r y s t a l which were o b t a i n e d

from t h e HADES program.3 The changes i n t h e s h o r t - r a n g e i n t e r a c t i o n w e r e r e s t r i c t e d

t o second n e i g h b o u r s , b u t i n t h e c a l c u l a t i o n o f t h e s e l f - t e r m s i n t h e 6R m a t r i x , i t was n e c e s s a r y t o e n s u r e t h a t e a c h i o n had i t s f u l l complement of f i r s t and second

n e i g h b o u r s , e v e n though t h e s e n e i g h b o u r s might l i e o u t s i d e t h e d e f e c t s p a c e . The

s h o r t - r a n g e f o r c e c o n s t a n t s were c a l c u l a t e d u s i n g t h e two-body c e n t r a l p o t e n t i a l of

Catlow, N o r g e t t and ~ o s s ~ . The change i n t h e coulomb i n t e r a c t i o n was c a l c u l a t e d

o n l y between t h e i o n a t t h e o r i g i n (which l a t e r became t h e s i t e o f t h e vacancy) and

i t s f i r s t and second n e i g h b o u r s ( ~ e n e d e k ' ) . The d e f e c t s p a c e c o m p r i s i n g t h e i n t e r -

s t i t i a l and i t s f i r s t and second n e i g h b o u r s c o n s i s t e d o f 1 5 i o n s , b u t 95 i o n s had

t o b e t a k e n i n t o a c c o u n t w h i l e c a l c u l a t i n g t h e d i a g c n a l t e r m s o f 6L. The change

i n t h e coulomb f o r c e between t h e i n t e r s t i t i a l and o n l y i t s f i r s t and second n e i g h -

b o u r s was i n c l u d e d . A d d i t i o n a l l y , t h e p r e s e n c e o f t h e i n t e r s t i t i a l is f e l t t h r o u g h

t h e t e r m s a aT, where I\-' was c a l c u l a t e d from t h e s e l f term o f t h e i n t e r s t i t i a l .

The c a l c u l a t i o n i s a q u a s i h a r m o n i c one i n t h a t t h e a c t u a l l a t t i c e p a r a m e t e r a t

e a c h T was u s e d i n d e t e r m i n i n g t h e r e l a x a t i o n s o f t h e i o n s . The Green m a t r i x a t ,

C6-9 16 JOURNAL DE PHYSIQUE

v

F r e n k e l d e f e c t f o r m a t i o n a t c o n s t a n t volume, sF. For comparison w i t h e x p e r i m e n t we

need t h e c o r r e s p o n d i n g e n t r o p y a t c o n s t a n t p r e s s u r e sP. S i n c e t o f i r s t o r d e r i n

2 ,

t h e volume change on forming a F r e n k e l d e f e c t a t c o n s t a n t p r e s s u r e , t h e Gibbs e n e r g y v

change gP i s e q u a l t o t h e Helmholtz e n e r g y change f V

',

sP i s _{r e l a t e d t o by s by}

sp

-

sv =

-

(af")

"v (10)

where v i s t h e volume p e r molecule (=2r:, r, b e i n g t h e

F-

F- n n d i s t a n c e ) and

BP

t h e

e x p a n s i v i t y . S i n c e b o t h t h e e n e r y and e n t r o p y of d e f e c t f o r m a t i o n a t c o n s t a n t

volume uV and sV a r e d e t e r m i n e d by o u r c a l c u l a t i o n s , f V and sP may b e found and

compared w i t h e x p e r i m e n t a l v a l u e s . 3. R e s u l t s CaF2

T/K

300 1000 1500 s V / k 0.73 2.88 5.30 s P / k 8.83 13.30 18.42 s P / k 5.74 5.68 a _{Based on e x t r a p o l a t e d v a l u e s .}

F o r caFz8 t h e mean c a l c u l a t e d v a l u e o f sP i n t h e r a n g e 550

-

1000 K i s -/Ok,

and t h e e x p e r i m e n t a l v a l u e o f sP i s a b o u t 5 k . ' ~ o r SrF2 sP ( e x p t ) = 4 . 1 k i n t h e

r a n g e -570

-

1100 K , and t h e c a l c u l a t e d v a l u e o f sP i s a b o u t 5 k. The quasiharmonic

c a l c u l a t i o n depends c r i t i c a l l y on a knowledge of t h e t e m p e r a t u r e dependence o f t h e

l a t t i c e c o n s t a n t .

Acknowledgement: We acknowledge s u p p o r t o f t h i s r e s e a r c h by t h e N a t u r a l S c i e n c e s

and E n g i n e e r i n g Research C o u n c i l o f Canada. T.M.H. i s g r a t e f u l t o t h e C e n t r e f o r

I n t e r - d i s c i p l i n a r y s t u d i e s i n Chemical P h y s i c s f o r t h e award o f a F e l l o w s h i p and f o r f u r t h e r f i n a n c i a l s u p p o r t , and t o Madurai Kamaraj U n i v e r s i t y f o r l e a v e o f

absence. We a r e most g r a t e f u l t o D r . A.B. L i d i a r d and D r . M. Hutchings f o r d a t a

on t h e l a t t i c e c o n s t a n t of CaF2. R e f e r e n c e s

1. Elcombe M.M. and P r y o r A.W., J - P h y s . C ( S o l i d S t a t e )

3,

492 (1970).

2. Elcombe M.M., J.Phys. C ( S o l i d S t a t e )

2,

2702 (1972).

3. N o r g e t t M . J . , AERE Harwell R e p o r t R 7650 (1974).

4. Catlow C.R.A., N o r g e t t M . J . and Ross T.A., J.Phys. C 1 0 , 1627 (1977).

5. Benedek G., P h y s i c s o f I m p u r i t y C e n t r e s i n C r y s t a l s ( T a l l i n n ) , P.182 (1972).

6. H a r i d a s a n T.M., G o v i n d a r a j a n J., Nerenberg M.A.H. and J a c o b s P.W.M., J.Phys.

C l2, 5371 (1979).

7. G i l l a n M . J . , P h i l . M a g s , 301 (1981).

8. Ong S.H. a n d J a c o b s P.W.M., J.Phys. ( P a r i s ) 37, C7-331 (1976).