Agrawal, R M

Rathore, R P S

Saxena, S K

IACS Institutional Repository

Indian J. Phys. 70A (2), 259-264 (1996)I J P  A— an international journalT o ta l b in d in g  en erg y  via th e  b a n d  stru ctu re  en erg y  o iA d  g ro u p  tra n sit io n  m eta lsS  K  S a x e n a ,  R  M  A g r a w a l  a n d  R  P  S  R a t h o i e *Department of Physics and Computer Science, DayalMgh Educational Institute (Deemed University), Dayalbagh-282 005, Agra, India * Department of Physics. K B S College, Agra-282 002, IndiaReceived 2 May 1995. accepted 6 November 1995Abstract : The binding in metals provides a basis genesis to discuss the cohesive, elastic, lattice dynamical and other allied properties of the metals A thorough and comprehensive analysis with regard to (i) various energy terms contributing to total metallic bonding, (ii) forrgs of the model potentials incurring the band structure part of the binding, (ni) implication of s -d  hybridization and (iv) effect of electron screening, has prompted us to undertake the present study of binding m several complex metals which turn out to be superconducting at low temperatures and bear hep, bet and f a  configurations at room tempeiature / e. Yttrium (Y), Zirconium (Zr), Niobium (Nb). Molybdenum (Mo), Ruthenium (Ru), Rhodium (Rh) and Palladium (Pd)Keywords : Electron hopping, s- and c/-bands, pseudopotential PACS Nos. : 7 1 25 Pi, 71.45.NtW i l l s  a n d  H a r r i s o n  [ 1 ]  h a v e  d e a l t  w i t h  t h e  s-s  i n t e r a c t i o n s  w h i l e  H a r r i s o n  a n d  P r o y e n  [ 2 ]  h a v e  f o r m u l a t e d  t h e  i n t e r a c t i o n s  i m p l i c a t i n g  t h e  d - b a n d  e l e c t r o n s .  C h e l i k o w s k y  [ 3 )  h a s  c a l c u l a t e d  t h e  b i n d i n g  e n e r g y  o f  s i m p l e  m e t a l s  i n  T h o m  a s - F e r m i  a p p r o x i m a t i o n .  E m p t y  c o r e  p o t e n t i a l  d u e  t o  H e i n e  a n d  A b a r e n k o v  [ 4 ]  h a s  b e e n  d e p l o y e d  b y  S i n g h  e t a l  [ 5 ]  w h i l e  W i l l s  a n d  H a r r i s o n  [ 1 ]  h a v e  u s e d  a  s i m i l a r  p o t e n t i a l  t o  c a l c u l a t e  t h e  b i n d i n g  i n  m e t a l s .  T h e s e  p o t e n t i a l s  1 6 ]  b e s i d e s  b e i n g  d i s c o n t i n u o u s  i n  r e a l  s p a c e ,  d o  n o t  i m b i b e  t h e  s h o r t  r a n g e  r e p u l s i v e  c h a r a c t e r  e x p l i c i t l y  i n  t h e i r  f o r m u l a t i o n s .V a r i o u s  i n t e r a c t i o n  e n e r g i e s  a s s o c i a t e d  w i t h  t h e  e l e c t r o n s  a r e  i n f l u e n c e d  b y  s-d  h y b r i d i z a t i o n  w h i c h  s u b s t a n t i a l l y  a l t e r s  t h e  e f f e c t i v e  o c c u p a n c y  o f  m e t a l l i c  d -  a n d  e s t a t e s .  T h e  p h e n o m e n o n  o f  e l e c t r o n - s c r e e n i n g  a l s o  c o n t r i b u t e s  t o  t h e  b i n d i n g  o f  t h e  m e t a l s .T h e  s t u d i e s  u n d e r t a k e n  s o  f a r ,  i n v o l v i n g  p s e u d o p o t e n t i a l  t h e o r y  1 1 - 3 , 5 ] ,  h a v e  u s e d  t h e  f i x e d  v a l e n c e  f o r  t h e  t r a n s i t i o n  m e t a l s  u n d e r t a k e n  f o r  t h e  s t u d y ,  a n d  f i t  t h e  d i f f e r e n t©  1 9 9 6  I  A C S2 6 0 S  K  S a x en a , R  M  A g r a w a l  a n d  R  P  S  R a th o r em o d e l  r a d i u s  p a r a m e t e r  ( j )  t o  t h e i r  m o d e l  p o t e n t i a l  ( $ )  t o  a c h i e v e  t h e  r e q u i r e d  p u r p o s e .  T h e  a s s i g n m e n t  o f  a  c h o s e n  v a l e n c e  t o  p a r t i c u l a r  t r a n s i t i o n  m e t a l  m a y  b e  q u e s t i o n a b l e  w h i c h  l e d  u s  t o  i n v e s t i g a t e  t h e  p r o b l e m  a f r e s h  w i t h  a d d e d  t h o r o u g h n e s s .W e  h a v e  p u r p o s e l y  t r e a t e d  t h i s  p r o c e d u r e  f o r  t h e  t r a n s i t i o n  m e t a l s  j u s t  i n  a  r e v e r s e  m a n n e r  t o  t h e  c o n v e n t i o n a l  m e t h o d  i.e. l e a v i n g  t h e  v a l e n c e  Z  a s  a d j u s t a b l e  p a r a m e t e r  a n d  t h e  m o d e l  r a d i u s  (s) a s  f i x e d  ( d e t e r m i n e d  b y  f i t t i n g  t o  t h e  e q u i l i b r i u m  v o l u m e )  f o r  a l l  t h e  m e t a l s .P r e s e n t  s t u d y  h a v e  u t i l i z e d  t h i s  a p p r o a c h  f o r  c o m p u t i n g  t h e  b i n d i n g  e n e r g y  i n  b o t h  r e a l  a n d  F o u r i e r  s p a c e  b y  t a k i n g  t h e  f i x e d  m o d e l - r a d i u s  f o r  t h e  m e t a l .  T h e  a p p r o p r i a t e  v a l e n c e  ( Z )  c o r r e s p o n d i n g  t o  t h e  m e t a l ,  i s  t h e n  s o u g h t  t o  g e t  t h e  c o r r e s p o n d i n g  v a l u e  t o  t h e  m e t a l i c  e x p e r i m e n t a l  d a t a .A c c o r d i n g l y ,  t h e  i n t e r a c t i o n  s y s t e m  c o m p r i s i n g  o f  t e r m s  d u e  t o  f r e e  e l e c t r o n  g a s ,  c o r e ,  d-b a n d s  a n d  b a n d  s t r u c t u r e  c o m p o n e n t s ,  i s  c a r r i e d  o u t  f o r  t h e  c o m p l e x  m e t a l s  w h i c h  e x h i b i t  s u p e r c o n d u c t i n g  p r o p e r t i e s  a t  l o w  t e m p e r a t u r e .T h e  b i n d i n g  e n e r g y  c o m p r i s e s  o f  f o u r  c o n t r i b u t i o n s  [ 1 ]  i.e. f r e e  e l e c t r o n  e n e r g y  (Ek). d-b a n d  w i d t h  e n e r g y  ( £ / , ) ,  d-b a n d  c e n t e r  o f  g r a v i t y  s h i f t  e n e r g y  ( E( )  a n d  b a n d  s t r u c t u r e  e n e r g y  (Ehs).T h e  b o n d i n g  t e r m  d u e  t o  t h e  w i d t h  o f  d-b a n d s  g i v e s  r i s e  t o  a t t r a c t i v e  p a r t  o f  t h e  e n e r g y .  T h e  s h i f t  o f  c e n t e r  o f  g r a v i t y  o f  e f - b a n d s  l e a d s  t o  a  r e p u l s i v e  p a r t  o f  e n e r g y .F o l l o w i n g  W i l l s  a n d  H a r r i s o n  [ 1 ] ,  Zs f o r  t h e  m e t a l s  i s  e q u a t e d  t o  1 .5  a n d  t h e  v a l e n c e  d u e  t o  d - e l e c t r o n s  i.e. Zd (= Z -Z S) i s  d e t e r m i n e d  b y  t h e  k n o w l e d g e  o f  Z a n d  Z v.T h e  s u m  o f  a b o v e  t e r m s  m a y  n o w  b e  w r i t t e n  a sVd == Efe 1 )T h e  i n p u t  d a t a  n e e d e d  t o  c o m p u t e  Vd f o r  t h e  m e t a l s  u n d e r  s t u d y  a r e  g i v e n  i n  T a b l e  1 . T h e  c o m p u t e d  v a l u e s  o f  Vd f o r  t h e  m e t a l s  a r e  a l s o  s h o w n  i n  t h i s  t a b l e .Table 1. Input data and computed V(d) for the metals.Metal LatticeconfigurationLattice constant ( A )Coreradius (rc)( A )Estate radius (rd)( A )V(d) =Efe +  E-fr +  (ev / atom)a cY hep 3.65 3.75 1.09 1.58 ”9.21331Zr hep 3.23 5.15 1.06 1.41 -8.1536Nb bee' 3.30 — 1.01 1.28 -8.2022Mo bee 3.15 — 0.95 1.20 -8.4820Ru hep 2.71 4.28 0.72 1.05 -11.5329Rh fee 3.80 — 0.62 0.99 -13.3734Pd fee 3.89 — 0.52 0.94 -15.4489T o  c o m p u t e  t h e  b a n d  s t r u c t u r e  e n e r g y  (Ehs) f o r  t h e  c o m p l e x  s u p e r c o n d u c t i n g  m e t a l s  o f  p r e s e n t  c o n c e r n ,  w e  h a v e  m a d e  t w o - f o l d  c o m p u t a t i o n s .  T o  r e v e a l  t h e  n a t u r eTotal binding energy via the band structure energy etc 261o f  i n t e r a c t i o n s  i n  r e a l  s p a c e  w e  h a v e  a t  f i r s t  c o m p u t e d  t h e  e n e r g y  a t  F e r m i  w a v e  v e c t o r  (kf).T h e  e x p r e s s i o n  f o r  t h e  b a n d  s t r u c t u r e  e n e r g y  Erhs i n  r e a l  s p a c e  m a y  b e  w r i t t e n  o n  t h e  l i n e s  o f  H a r r i s o n  1 7 ]  i.e.E rhs =  9 n Z } / E f  W%(2kf ) cos(2kf rs)/(2kf r y , ( 2 )w h e r e  E f , k f , r v a n d  Wh a r c  t h e  F e r m i  e n e r g y ,  F e r m i  r a d i u s ,  e l e c t r o n  s e p a r a t i o n  a n d  b a r e  i o n  f o r m  f a c t o r  r e s p e c t i v e l y  [ 8 - 1 1 ] .  T h e  m o d e l  p o t e n t i a l s  d u e  t o  A s h c r o f t  [ 8 ] ,  V r a t i  et al [ 9 ] ,  I d r c s s  et al [  1 0 ]  a n d  K u l s h r e s t h a  et al[  1 1 ] ,  a r c  c h o s e n  t o  o b t a i n  t h e  b a n d  s t r u c t u r e  e n e r g y .I n  o r d e r  t o  i n v e s t i g a t e  t h e  p h e n o m e n o n  o f  h y b r i d i z a t i o n ,  t h e  d e p e n d e n c e  o f  E [)s o n  Z r i s  e x p l o r e d .  T h e  v a r i a t i o n  o f  £ £ v w i t h  Z r  i s  d e p i c t e d  i n  F i g u r e  1 f o r  a  r e p r e s e n t a t i v eThousandVALENCE ( Zr) -----------Figure 1. The dependence of band structure energy £ on valence Zr for Pd, as a representative case, in real space. The symbols [+, *, • , □ ], represent the computed findings for the Model potentials due to Ashcroft [8], Vrati et a l [9], Idress et a l [10] and Kulshrestha et a l [II] respectively.m e t a l  P d .  R e l e v a n t  v a l e n c e  Z r  f o r  w h i c h  t h e  a s s u m e  t h e  a p p r o p r i a t e l y  a t t r a c t i v e  c h a r a c t e r  f o r  d i f f e r e n t  M o d e l  p o t e n t i a l s  a r e  e n l i s t e d  i n  T a b l e  2 .F o r  a n a l y z i n g  t h e  c h a r a c t e r i s t i c  f e a t u r e s  o f  i n t e r a c t i o n s  a s s o c ia t e d  w i t h  t h e  £ ^ s , w e  h a v e  f u r t h e r  c o m p u t e d  t h e  b a n d  s t r u c t u r e  e n e r g y  i n  F o u r i e r  s p a c e  w h i c h  m a y  b e  e x p r e s s e da sE l  = 0 / ( 4 Jte2) ^ q 2W2h(q) ( e (9 ) - l ) / ( l  + { (e (9 ) - l ) ( l - / ( ? » } ) ,  (3)w h e r e  e(q) i s  t h e  u s u a l  H a r t r e e - F o c k  d i e l e c t r i c  f u n c t i o n , / f a ) ,  t h e  c o r r e c t i o n  f a c t o r  d u e  t o  e x c h a n g e  a n d  c o r r e l a t i o n  e f f e c t  a s s o c ia t e d  w i t h  t h e  e l e c t r o n s  a n d  W ^ f a ) ,  t h e  b a r e  i o n  f o r m  f a c t o r s  [ 8 - 1 1 ] .  P r e s e n t  i n v e s t i g a t i o n  e m p l o y s  a n  e f f i c i e n t  a n d  m o d e r a t e  s c h e m e  d u e  t o  V a s h i s h t a  a n d  S i n g w i  [ 1 2 ]  f o r  c o m p u t i n g / f a ) .262 S K Saxena, R M Agrawal and R P S Rathore 1I n  o r d e r  t o  s t u d y  t h e  p r o c e s s  o f  s-d  h y b r i d i z a t i o n ,  t h e  £JJV a r e  c o m p u t e d  f o r  d i f f e r e n t  Zq. T h e  c o m p u t e d  p l o t s  E%s vs Zq w i t h i n  t h e  f r a m e  w o r k  o f  t h e  s a id  p o t e n t i a l s  a r e  s h o w n  i nTable 2. Total binding energy computed by using eq. (4).Metal ReferencesValence in r-space ZrValence in <y-spacc ZfBinding energyPresent expY Ashcroft [8] 8 559 0.354 -17.122 -17.222Vrati et al [9] 8.550 0.640 -16.438Idress et al [10J 8.537 0.122 -17.221Kulshrestha et al[ 11] — — —Zr Ashcroft [8] 8.570 0.410 -19.568 -20.183Vrati etal[9] 8.562 0 747 -19.558Idress et al [ 10] 8.542 0.141 -20.008Kulshrestha et al [ 11 ] 8.544 0 150 -20.321Nb Ashcroft [8] 8.561 0.415 -20.295 -21.337Vrati et d/[9] 8.557 0.747 -21.033Idress et al [101 8.533 0 140 -21 171Kulshrestha etal[ 11] 8.535 0.054 -20.566Mo Ashcroft [8] 8 530 0.411 -21.785 -21 607Vrati etal[9] 8.521 0.730 -21.163Idress et al [10] 8.496 0 138 -21.814Kulshrestha et al [ 11 ] 8.500 0.200 -21.718Ru Ashcroft [8] 8.545 0.355 -21.823 -21.97Vrati et al[9] 8.520 0.650 -21 668Idress et al [10] 8.519 0.120 -2lt943Kulshrestha etal [11] 8 523 0.230 -22.082Rh Ashcroft [8] 8.575 0.309 -21.549 -21.457Vrati et a ll9] 8.572 0.546 -21.321Idress etal [10] 8.558 0.103 -21.332Kulshrestha etal [II] 8.559 0.295 -21.508Pd Ashcroft [8] 8.529 0.270 -20.578 -21.240Vrati etal [9] 8.520 0.475 -21.038Idress et <i/[10] 8.516 0.090 -21.069Kulshrestha etal [11] 8.516 0.310 -21.508F i g u r e  2  f o r  t h e  r e p r e s e n t a t i v e  m e t a l  P d .  T a b l e  2  e x h i b i t s  s o m e  o f  t h e  a p p r o p r i a t e  a n d  a t t r a c t i v e  c o m p u t e d  v a lu e s  o f  w i t h  t h e  c o r r e s p o n d i n g  a p p r o p r i a t e  v a le n c e  Zq.T h e  t o t a l  b i n d i n g  e n e r g y  f o r  t h e  m e t a l l i c  i o n s  o f  g i v e n  v a le n c e  a n d  o c c u p y i n g  t h e  f i n i t e  v o l u m e  w i t h i n  t h e  q u o t e d  m o d e l  p o t e n t i a l s  t r e a t e d  i n  s e c o n d  o r d e r  m a y  b e  w r i t t e n  a s ,£ bind =  V(d) + [E'bs +  E l]/2 . (4)T o ta l b in d in g  e n e r g y  v ia  th e  b a n d  s tr u c tu re  e n e r g y  e tc 263F o r  c o m p u t i n g  t h e  b i n d i n g  e n e r g y  i n  r e a l  s p a c e ,  w e  h a v e  s u b s t i t u t e d  t h e  E[?K a t  Zr a s  p e r  c o n t e n t s  o f  t h e  T a b l e  2 .  T h e  b i n d i n g  e n e r g y  i n  F o u r i e r  s p a c e  i s  c o m p u t e d  b y  p u t t i n g  t h e  E%s a t  Zq o b t a i n e d  f r o m  t h e  d a t a  g i v e n  i n  T a b l e  2 .Figure 2. The dependence of Band structure energy on valence Zq, Pd as a representative case, in Fourier space The symbols are present same as in Figure IT h e  c o m p u t e d  b i n d i n g  e n e r g y  i n  r e a l  a n d  F o u r i e r  s p a c e s  a r c  e n l i s t e d  i n  T a b l e  2 .  T h e  l a s t  r o w  o f  t h e  t a b l e s  e x h i b i t s  t h e  e x p e r i m e n t a l  b i n d i n g  e n e r g y  b e i n g  c o m p u t e d  b y  u s i n g  t h e  f o l l o w i n g  r e l a t i o n ,^ b m d  ( E x p e r i m e n t a l )  =  -  E cohcsive -  Ex -  1 / 4  ( £ ,  +  £ „ ) .  ( 5 )w h e r e  £ COheSjv c  *s  t h e  c o h e s i v e  e n e r g y ,  E\% t h e  f i r s t  i o n i z a t i o n  e n e r g y  a n d  Eu, t h e  s e c o n d  i o n i z a t i o n  e n e r g y .  T h e  /^cohesive* E\ a n d  £11 d a t a  a r e  t a k e n  f r o m  K i t t c l  [ 1 3 ] .P r e s e n t  i n v e s t i g a t i o n  a c c o u n t s  c o m p r e h e n s i v e l y  f o r  t h e  c o n t r i b u t i o n s  o f  a l m o s t  a l l  i n t e r a c t i o n s  p r e s e n t  i n  t h e  m e t a l s .  T h e  t h e o r y  b e s i d e s  b e i n g  e x t e n s i v e  a s  c o m p a r e d  t o  o t h e r s  [ 1 , 2 ] ,  p r e d i c t s  g o o d  a g r e e m e n t  w i t h  t h e  e x p e r i m e n t a l  v a lu e s .T h e  c o n t e n t s  o f  T a b l e  2  r e v e a l  t h a t  t h e  a c c o r d e d  v a lu e s  o f  t h e  b i n d i n g  e n e r g y  a r e  o b t a i n e d  f o r  w h i c h  t h e  a v e r a g e  v a l u e s  o f  t h e  v a le n c e s ,  Z r a n d  7q a r c  8 . 3 4 5  a n d  0 . 2 5 5  r e s p e c t i v e l y .  A s  t h e  m a x i m u m  o c c u p a n c y  n u m b e r  o f  t h e  d  a n d  s s u b s h e l l s  a r c  1 0  a n d  2  r e s p e c t i v e l y ,  s o  i n  r e a l  s p a c e  t h e r e  m a y  b e  m u c h  p r o b a b i l i t y  o f  t r a n s f e r r i n g  t h e  e l e c t r o n s  f r o m  d  — » s s t a t e s  w h i l e  i t  i s  i n v e r s e ,  f r o m  s — > d, i n  F o u r i e r  s p a c e .T h e  s t u d y  e s t a b l i s h e s  t h a t  a l l  t h e  c h o s e n  m o d e l  p o t e n t i a l s  i r r e s p e c t i v e  o f  t h e i r  s h a p e s ,  f o r m s  a n d  t h e o r e t i c a l  d e t a i l s  l e a d  t o  a l m o s t  i d e n t i c a l  p r e d i c t i o n s  o f  v a l e n c e  c o r r e l a t e d  b a n d  s t r u c t u r e  b o n d i n g .T h e  v a l e n c e  d e p e n d e n c e  o f  ( F i g u r e  1 )  e x h i b i t s  o n e  t y p e  o f  h y b r i d i z a t i o n  w h i c h  s u g g e s t s  h o p p i n g  o f  e l e c t r o n s  f r o m  d- b a n d s  t o  5 - b a n d s .  T h e  E%s vs Zq ( F i g u r e  2 )  s t u d y  r e v e a l s  o t h e r  t y p e  o f  h y b r i d i z a t i o n  w h i c h  g i v e s  r i s e  t o  t h e  h o p p i n g  f r o m  5 - s t a t e s  t o  J - s t a t e s .70A (2)-l I264 S K S a x e n a , R M A g r a w a la n d R P S R a th o r eThe two types of hybridization associated with unscreened and screened electron-ion interactions points toward the inverse correlation between valence and shift o f interaction energy due to the pseudopotcntial. The correlation may be attributed to the degree o f oscillations o f electron wave function occurring inside the core region.References[ 11 J M Wills and W A Harrison Pliys Rev. B28 4383 (1983)(2 j W  A Harrison and S Froyen Phys. Rev B21 2714 (1980)[3] J R Chclikowsky Phys. Rev. Lett. 47 387 (1982)[41 V Heine and I Abarenkov Phil Mag 9 451 (1984)[5] N Singh, N S Banger and S P Singh Phys Rev. B28 415 (1988)[6J N Singh, N S Banger and S P Singh Phys. Stat. Sol. K19 154 (1989); Phys. Rev. B39 3097 (1989); P ins Lett 135A 368 (1989)[7] W A Harrison Pseudopotential in the Theory o f  M etals (New Y o rk ; Benjamin) (1966)[8] N W Ashcroft Phys. Lett. 23 48 (1966)[9] S C Vrati, N Rani. D K Gupta and H C Gupta Phys. Lett. 24A 139 (1979)110| M ldress, Farid, A Khwaja and M S K Razami Solid State Commun 41469 (1982)1111 O P  Kulshrestha, H C Gupta and J C Upadhyaya Pliysica B84 236 (1976)112| P Vashistha and K S Smgwi Phys. Rev. B6 875 (1972)113) C Kittel Introduction to Solid Stale Physics 5th ed (New York : Wiley) (1976)

Total Binding Energy Via the Band Structure Energy of 4d Group Transition Metals

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/4883/1/Total%20Binding%20Energy%20Via_By%20S%20K%20Saxena.pdf

Total Binding Energy Via the Band Structure Energy of 4d Group Transition Metals

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