Comment on “lattice energy in alkali halide crystals”

Solid State Communications, Vol.53,No.8 pp.671-672, 1985. Printed in Great Britain.

COMMENT

ON " L A T T I C E

ENERGY M.M.

Instituto UNICAMP,

IN A L K A L I

0038-]098/85 $3.00 + .00 Pergamon Press Ltd.

HALIDE

CRYSTALS"

Shukla

de F { s i c a " G l e b W a t a g h i n " 1 3 1 0 0 C a m p i n a s SP B r a s i l

(Accepted for publication 7 December 1984 by S. Amelinckx) R e c e n t l y Y a d a v has p r o p o s e d a n e w r e p u l s i v e t e r m in the i o n i c i n t e r a c t i o n p o t e n t i a l w h i c h has t h r e e u n k n o w n p a r a m e t e r s . In o r d e r to d e t e r m i n e u n i q u e l y the m o d e l p a r a m e t e r s he p r o p o s e d a c o n d i t i o n w h i c h is f o u n d to be physically unconsistent.

In a r e c e n t c o m m u n i c a t i o n Y a d a v [ i] has p r o p o s e d a n e w r e p u l s i v e t e r m in the i o n i c i n t e r a c t i o n p o t e n t i a l , i.e. ~ = A r -n e -r/h , and has u t i l i z e d this to d e t e r m i n e the c o h e s i v e e n e r g y of s e v e r a l a l k a l i h a l i d e c r y s t a l s . In g o i n g t h r o u g h this w o r k some d i s c r e p a n c i e s are f o u n d in the e v a l u a t i o n of the p o t e n t i a l p a r a m e t e r s w h i c h m a y lead the t h e o r e t i c a l r e s u l t s meaningless. We s t a r t w i t h the f o r m of repulsive p o t e n t i a l g i v e n by

T a k i n g the f i r s t and s e c o n d d e r i v a t i v e s of ~(r), e q u a t i o n s (3) (4) w o u l d give

~(r)

= A r

e

-r/l

r

2 e ZlZ2C~ 2

r + ~)e

A(n

o ---n~ 1 l r

=

0

ro

(5)

2e2azlz2 3

-n

and

+ ~

ro

r2 2nr o + __~ %2

ro + n(n+l)]e I

-n-2 r°

(I) 9Kr

and the total alkali halide

2 e ZlZ2 ~ -n -r/l + A r e r

~(r)

O

p o t e n t i a l e n e r g y of the c r y s t a l s is g i v e n by

T h e r e are three u n k n o w n p a r a m e t e r s , n a m e l y , A , n and I w h i c h can not be o b t a i n e d by s o l v i n g the a b o v e two e q u a t i o n s . If one is, h o w e v e r , i n t e r e s t e d in d e t e r m i n i n g u n i q u e l y all the t h r e e p a r a m e t e r s , one m o r e i n d e p e n d e n t e q u a t i o n s h o u l d be u t i l i z e d . V a r s h n i [ 2] and V a r s h n i and S h u k l a [ 3] h a v e used experimental vibrational energy of c r y s t a l in c o m p u t i n g the u n k n o w n p a r a m e t e r s . Y a d a v [ i] , on the o t h e r h a n d , has i n t r o d u c e d an a r b i t r a r y c o n d i t i o n

(2)

The u s u a l p r o c e d u r e a d a p t e d by v a r i o u s w o r k e r s in this f i e l d in the e v a l u a t i o n of m o d e l p a r a m e t e r s is to m a k e use of the c o n d i t i o n s :

d~,r,)~( ~ dr

= 0

(6)

B

(3)

r=r o

d~(r) dv = 9Kr o

d2~(r)) dr

2

(4)

But

d~(r) dv

= A r -3

r d~(r) 3V dr

(7)

(8)

r=r

0

and the e x p l i c i t f o r m of a b o v e e q u a t i o n (7), w i t h the h e l p of (8) and (2),

In the a b o v e e q u a t i o n s 8 is the c o m p r e s s i b i l i t y of the c r y s t a l and K is the s t r u c t u r a l c o n s t a n t r e l a t i n g the e q u i l i b r i u m d i s t a n c e (ro) b e t w e e n n e a r e s t n e i g h b o u r s to v o l u m e of the p r i m i t i v e cell in the f o l l o w i n g m a n n e r , i.e. V = Kr 3 . O

2 d~(r) dv

r = -3V [-

e z iz2(~ A -n-i 2 - y r r

r e-~(n+

(9)

o

671

)1

672

Equation (7) a n d (9) a r e i d e n t i c a l and a comparison of t h e s e e q u a t i o n s ~w~ t h a t w h i l e the e x p l i c i t f o r m of dV contains all the t h r e e p a r a m e t e r s , A , n a n d %, Y a d a v [ i] s u p p r e s s e s % a n d n. He c o u l d b a d c h o s e n dq~(r)dv = A r -m, with m = 1,2,3,4 or any i n t e g e r . The equation (7) t h u s c o n s t i t u t e s a serious restriction on the f o r m of potential function and is n o t w a r r a n t e d by a n y s t a b i l i t y condition a b o u t the lattice. Moreover s i n c e the e q u a t i o n (7) is n o t w r i t t e n at the e q u i l i b r i u m it s h o u l d be t r u e for a n y v a l u e of r. On t h e o t h e r h a n d if the e q u a t i o n (7) were written f o r the e q u i l i b r i u m value, the r i g h t h a n d s i d e of this e q u a t i o n s h o u l d be zero. T h i s p o i n t has b e e n m i s s e d by Y a d a v [ i] and in o r d e r to evaluate the model parameter he h a s d~(r)~ =A r -3 and not zero. To prove dV " r=r o t h i s is t r i v i a ~ b u t g i v e n h e r e in o r d e r to c l a r i f y this point. us

write

d~(r)) dV r=r

= A

d~(r)) dr

or

thus,

r=r

and

3 AK r o

o

d2~(r)) dr 2

now

-3 r=r

r 2

o

equating

KA o

=

~"(r)r=r

9Kr ~

____£o

from

o

equation (6) (2) of Y a d e v of

Yadev

[ I]

of t h i s p a p e r or [ i] , we h a v e the 3 r3 -A = o.

equation result

B

The present study thus shows Y a d a v [ i] has u s e d ~' ( r ) r = r ~ O a n d -3

used

Let

Vol. 53, No. 8

COMMENT ON "LATTICE ENERGY IN ALKALI HALIDE CRYSTALS"

AK r

at

the

same

time

to

that

o determine

o model parameters which makes theoretical results unavalid.

the

his

r -3 o ' thus

o d~(r)) dr

r=r

_d~(r)) dV o

r=r

mdV = A r-3.3Kr 2 "dr o o o

The authors Professor R a m S. comments and the the m a n u s c r i p t .

wishes to t h a n k Katiyar for h i s critical reading

of

REFERENCES I. 2.

R.B. Y a d a v , S o l i d St. Comm. 46, 341 (1983). Y.P. V a r s h n i , Rev. Mod. P h y s . 29,

3.

664 (1957). Y.P. V a r s h n i and R.C. S h u k l a , Mod. P h y s . 35, 130 ( 1 9 6 3 ) .

Rev.