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SCREENING OF THE SHORT-RANGE POTENTIAL
BY THE LOCAL FIELD CORRECTION IN THE
LATTICE DYNAMICS OF VANADIUM
B. Onwuagba, A. Animalu
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque C6, supplgment au n o 12, Tome 42, de'cembre 1982 page C6-401
SCREENING OF THE SHORT-RANGE POTENTIAL BY THE LOCAL FIELD CORRECTION
I N
THE LATTICE DYNAMICS OF VANADIUM
B.N. Onwuagba and A.O.E. Animalu
Department of Physics, university of Nigeria, Nsukka, Nigeria.
A b s t r a c t . - The d i e l e c t r i c f o r m u l a t i o n o f t h e l o c a l f i e l d c o r r e c t - Ton t o t h e dynamical m a t r i x f o r phonon d i s p e r s i o n r e l a t i o n i n d- band m e t a l s i n t h e t r a n s i t i o n - m e t a l model p o t e n t i a l a p p r o x i m a t i o n i s re-examined, w i t h a view t o e x p l o r e i t s i n f l u e n c e on t h e s h o r t - r a n g e (Born-Mayer; i n t e r a t o m i c f o r c e c o n t r i b u t i o n t o t h e dynamical m a t r i x . 1-t i s observed t h a t t h e o f f - d i a g o n a l components o f t h e i n v e r s e d i e l e c t r i c m a t r i x , which d e t e r m i n e t h e l o c a l f i e l d c o r r e c - t i o n , make c o l ~ t r i b u t i o n s t h a t t e n d t o c a n c e l o r "screen1' t h e Born- Mayer c o n t r i b u t i o n , j u s t a s t h e d i a g o n a l components of t h e i n v e r s e d i e l e c t r i c m a t r i x t e n d t o p r o v i d e s c r e e n i n g o f t h e long r a n g e
(Coulombic) c o n t r i b u t i r m . Numerical c a l c u l a t i o n f o r vanadium shows t h a t t h e c a n c e l l a t i o n o f t h e Born-Mayer c o n t r i b u t i o n t o t h e dynam- i c a l m a t r i x by t h e l o c a l f i e l d c o r r e c t i o n i s p h y s i c a l l y s i g n i f i c - a n t , and l e a d s t o a b e t t e r u n d e r s t a n d i n g o f t h e s o f t modes i n t h e phonon s p e c t r a o f body-centered c u b i c t r a n s i t i o n m e t a l s . 1. I n t r o d u c t i o n .
-
S e v e r a l y e a r s a g o , O l i and ~ n i m a l u ' developed a t h e o r y o f t h e l o c a l f i e l d c o r r e c t i o n t o t h e dynamical m a t r i x f o r t h e phonon d i s p e r s i o n r e l a t i o n i n d-band m e t a l s i n t h e t r a n s i t i o n - m e t & model p o t e n t i a l a p p r o x i m a t i o n , w i t h a view t o e l u c i d a t e t h e o c c u r r - e n c e o f s o f t modes i n t h e body-centered c u b i c t r a n s i t i o n m e t a l s , vanadium and niobium. The r e s u l t s of t h i s e f f o r t was s u b s e q u e n t l y r e p o r t e d 2 i n t h e 1977 Conference on T r a n s i t i o n M e t a l s . The i m p o r t a n t s t e p s i n t h e t h e o r y may be summarised b r i e f l y , a s f o l l o w s . The dynam- i c a l m a t r i x f o r phonon d i s p e r s i o n r e l a t i o n i n a B r a v a i s l a t t i c e i s s p l i t , a s i n t h e c a s e o f s i m p l e m e t a l s , i n t o a sum o f t h r e e t e r m s , namely t h e Coulombic c o n t r i b u t i o n due t o t h e long-range Coulombi n t e r a c t i o n between t h e b a r e i o n s o f t h e me-t.aL, a s h o r t - r a n g e (Born- Mayer) c o n t r i b u t i o n due t o t h e o v e r l a p o f t h e c o r e w a v e f u n c t i o n s c e n t e r e d on n e i g h b o r i n g i o n s , and t h e e l e c t r o n i c c o n t r i b u t i o n due t o t h e i n d i r e c t i o n - i o n i n t e r a c t i o n v i a t h e p o l a r i s a t i o n f i e l d o f t h e c o n d u c t i o n e l e c t r o n s . The e l e c t r o n i c c o n t r i b u t i o n i n v o l v e s , t h e r e f m e t h ~ t s c r e e n i n g a c t i o n o f t h e g a s o f c o n d u c t i o n e l e c t r o n s t h r o u g h t h e e l e c t r o n - p h o n o n i n t e r a c t i o n m a t r i x e l e m e n t s
,
g q ,,
which a r e p r o p o r t - i o n a l t o t h ~ s c r e e n e d p s e u d o - p o t e n t i a l o r model p o t e n t i a l form f a c t o r , V ( q ) . Because o f t h e v i r t u a l bound c h a r a c t e r o f t h e transition-me-traL d - e l e c t r o n s , l o c a l f i e l d c o r r e c t i o n s a r i s e i n t h e c o n v e n t i o n a ls c r e e n i n g c a l c u l a t i o n , a s t h e o f f - d i a g o n a l components o f t h e i n v e r s e d i e l e c t r i c m a t r i x . The purpose o f t h i s p a p e r i s t o show t h a t t h e
C6-402 JOURNAL DE PHYSIQUE c o n t r i b u t i o n t o t h e dynamical m a t r i x from t h e l o c a l f i e l d c o r r e c t i o n t e n d s t o c a n c e l t h e Born-Mayer c o n t r i b u t i o n , w i t h t h e r e s u l t t h a t t h e e f f e c t i v e s h o r t - r a n g e i n t e r a t o m i c f o r c e t u r n s o u t t o be a t t r a c t i v e r a t h e r t h a n r e p u l s i v e . 2 . 0 0 -
-
The phonon f r e q u e n c i e s i n a B r a v a i s l a t t i c e a r e determined by t h e s e c u l a r e q u a t i o n C R i s t h e dynamical m a t r i x , c o n s i s t i n g of where D a B Da + D a B + Da t h e t h r e e (Coulombic, R e p u l s i v e , and E l e c t r o n i c ) c o n t r i b u t i o n s , a s i n d i c a t e d . The e l e c t r o n i c c o n t r i b u t i o n , i n p a r t i c u l a r , h a s t h e form: E ( q + g ) $ + g f ) a g ' D a B ( q ) = - M U = ( E @ G ( ~ + ~ , ~ + ~ ' ) - E'2Q
G ( g , g r ) ) ( 2 ) g , g l lq+g12 g , g f lg12 where i s t h e energy-wave-number c h a r a c t e r i s t i c m a t r i x , - ~ ( q f g , q + g ' ) = [v(q+g)-'
[ 6 g ? g t-
E (q+g ,q+g'
)]
( 4 ) b e i n g t h e g e n e r a l i z e d susceptibility m a t r i x , v ( q + g ) t h e F o u r i e r t r a n s - 2 2form o f t h e elec'cron-electron i n t e r a c t i o n , u = ( 4 n z e /Mao)$ t h e i o n P
plasma f r e q u e n c y , z t h e chemical v a l e n c e and
no
t h e a t o m i c volume. The e x p r e s s i o n f o r G ( q + g , q + g f ) c a n be s e p a r a t e d f u r t h e r i n t o a d i a g o r r a 1 p a r t G o ( 9 + g ~ q + g ) 8 g , g ~ which r e p r e s e n t s t h e s c r e e n f n g of t h e Coul- ombic c o n t r i b u t i o n , and an o f f - d i a g o n a l p a r t G l l c ( q + g , q + g f ) which r e p r e s e n t s t h e l o c a l - f i e l d c o r r e c t i o n and t e n d s t o c a n c e l o r " s c r e e n " t h e r e p u l s i v e c o n t r i b u t i o n . E x p l i c i t l y ,i n which VH i s t h e t r a n s i t i o n - m e t a l model p o t e n t i a l , and aVN/aE c h a r a c t e r i z e s t h e d e p l e t i o n h o l e whose dependence on e n e r g y and on c o r e e l e c t r o n w a v e f u n c t i o n s i s t y p i c a l l y o f t h e H a r r i s o n form 3
Fig. 1: Phonon spectrum of vanadium in the principal directions.
3. Numerical Results.- The phonon frequency reduces to the form
v = (o /21r)(o:+w~-w~-o~~)< in which the Coulombic contribution P
(w:) is screened by the electronic contribution given by the diagonal part of G(q+g,q+gV) in the summation in Eq. (21, while the Born-Mayer contribution (w:) is screened by the local field correction which is well represented by w%c = Y.5w2exp(-aq/2r), for the longitudinal
r
branches and by
wit
= 0.020: for the transverse branches, the numerical factors being determined so as to fit the ab initio calculation of .t~ef. 1 and guarantee symmetry requirements at the points P and H of the Brillouin zone, as shown in Fig. 1. We see from Fig. 1 that the overall agreement between the present theory and experiment4 is quite good.References.
1. Oli, B.A. and A.O.E. Animalu. Phys. Rev. B
2,
2398 (1976) 2. Animalu, A.O.E., Transition Metals, EdM.J.G.
Lee. J.M. Perz andE.
Fawcett, (19771, p. 706.3. Harrison, W.A. Phys Rev.
