Haldar, S

Sarkar, A

Sen, D

IACS Institutional Repository

Indian J. Phys. 75A (1), 21-23 (2001)U P  Aan inicrnational journalOn the extraction of effective pair interaction of simple metals from third order pseudopotential energyD  S e n * , A  S a rk a r* *  an d  S H aid ar^Condensed Matter Physics Kcsca|Ch Centic. IX'partment of Physics. Jadavpur University. Calcutta-700 032. IndiaAbstract : Complete expression of the additional pair interaction duC to the third-order energy term in the elcction-ion interaction is lacking in the literature We present here the formulation of the pair interaction from third-order energy term An eslimuic ol the additional terms with localised ah mitio pseudopotcntial is made and found to produce good agreement with improved recent calculationsKeywords . Pseudopotential theory of metals, effective interaction, perturbation theoryPACS Nos. . 34 20.Cf, 7M !i.Hx. 63 10+a1. Introductionrh e  p se u d o p o te n tia l a p p ro a c h  is  m o s t  p o w e rfu l in  its a p p lic a tio n  to the  th e o ry  o f  m e ta ls  fo r  its  a m e n a b il i ty  to  a  p e r tu rb a tiv e  c a lc u la tio n . T h e re  a re  s e v e ra l in d ic a t io n s  f 1-4] th a t th ird -  an d  lo u rth -o rd c r te rm s  p lay  an  im p o rta n t ro le  in the  la ttice  m ech a n ica l p ro p e r tie s  o f  m e ta ls . T h e  to ta l e n e rg y  o f  a  m e ta l, a c c o rd in g  to  p se u d o p o te n tia l th e o ry , m a y  b e  e x p re s s e d  in  te rm s  o f  d e n s ity  d e p e n d e n t in te r io n ic  in te r a c t io n s  in  a d d it io n  to  th e  v o lu m e  d e p e n d e n t e n e rg y . A  re a l s p a c e  d e s c r ip t io n  o f  s y s te m s  in te rm s  o f  e f fe c tiv e  in te ra c t io n s  is  v e ry  u se fu l in  m a te r ia l  p h y s ic s  fo r severa l re a so n s . B e s id e s  p ro v id in g  im p ro v e d  in te rp rc ta b ility  an d  d ir e c t u n d e r s ta n d in g  o f  s t r u c tu r a l  s t a b i l i ty  a n d  r e la x a t io n  p h e n o m e n a  in  s im p le r  te rm s , it o f f e r s  v ia b le  c o m p u ta tio n a l sch e m e s  fo r  s im u la tio n  o f  c o m p le x  m a te r ia l sc ie n c e  p ro b le m s.In te rio n ic  in te ra c t io n  o f  s im p le  m e ta ls , w ith in  th e  fra m e w o rk  o f  l in e a r  s c re e n in g  th e o ry  d u e  to  th e  th i rd -o r d e r  p e r tu rb a tio n  te rm  in th e  e le c tro n - io n  p s e u d o p o te n tia l ,  w a s  d e v e lo p e d  e a r l ie r  by H a se g a w a  [5]. T h e  im p o r ta n c e  o f  th e  a d d itio n a l p a ir  p o te n tia l term  in d e te rm in in g  th e  s tru c tu ra l a n d  th e rm o d y n a m ic  p ro p e rtie s  o f  liq u id  m e ta ls  h a s  a ls o  b e e n  d is c u s s e d . H o w e v e r , H a se g a w a  h as  c o n s id e re d  th e  tw o  s y m m e tr ic  te rm s  o n ly  a n d  o m itte d  a’ Corresponding Author^^Barasat Govt. College, Barasat-743 201, West Bengal, India.B. K. Girls' College, Howrah-711 101, West Bengal, IndiaAsansol Engineering College, Asansol-4, West Bengal, India.E-inail : sandeep@juphys.emet.inth ird  te rm  o f  c o m p a r a b le  m a g n i tu d e . T h e  d e r iv a t io n  o f  th e  e x p re s s io n  Ib r th is  te rm  is so m e  w h a t c o m p lic a te d  lik e  th a t o f  th e  th re e  -  ion  in te ra c t io n , b u t a f te r  s im p lif ic a tio n  it tu rn s  o u t to  be  e x a c tly  e q u a l to  th e  .sy m m etric  te rm s . T h e  purpo.se o f  th e  p re s e n t  p a p e r  is to  d is c u s s  th e  fo r m u la t io n  [6) o f  th e  fu ll e x p re s s io n  o f  th e  a d d it io n a l p a ir  in te ra c t io n  V^{r) d u e  to  th e  th ird  o rd e r  te rm  in  e le c tro n - io n  p se u d o p o tc n tia l .2. FormulationThird-order enerf*y and interionic potentials within local approximation :T h e  th ird -o rd e r  b a n d  s tru c tu re  e n e rg y  is g iv e n  by  [7]Vj( 1)w h e re  = | ^ i | >  Ihe p r im e d  su m s  e x c lu d e  n u ll a n d  ^  is th e  io n ic  v o lu m e . T h e  s tru c tu re  fa c to r  5^ is d e f in e d  as1N  b e in g  th e  to ta l n u m b e r  o f  io n s . IT ie  s c re e n e d  p se u d o p o tc n tia l  fo rm  fa c to r  (O^  a rc  p la n e  w a v e  m a tr ix  e le m e n ts  o f  b a re  io n - e le c tro n  p se u d o p o tc n tia l  d iv id e d  b y  th e  d ie le c tr ic  fu n c tio n , g is©2001 lACS22 D Sen, A Sarkar and S Haidarth e  sy m m e tr ic  fu n c tio n  o f  th e  th re e  m o d u lii <72 ^ 3W ritin g  fo rm a lly  ^^ 3 ( ^ 1 ’ m ay  be w ritte n  asE, =  1 2 ' '  N-^ ^  ' X  ^ £ 3q^  khn= i2  “ N ’ I ' s .^.+ I £3 rk~l - m k~m^ /+ XH ^/ -  m A - / / w+ 2 (2)T lic firs t te rm  re p re se n t o n e -b o d y  p o te n tia l o r  th e  v o lu m e  d e p e n d e n t  p a r t  o f  th e  e n e r g y  a n d  is  o u t  o f  o u r  p r e s e n t  c o n s id e ra tio n . F ro m  sy m m e try  th e  s e c o n d  an d  th ird  te rm s  g iv e  id en tica l c o n tr ib u tio n s  to  th e  tw o  b o d y  (c e n tra l)  in te ra c t io n  an d  to g e th e r  re p re se n t <l>l(r) o f  H a s e g a w a ’s eq . (3 ), T h e  4 th  te rm  a lso  rep resen ts  (c e n tra l) p a ir  in te rac tio n  o f  co m p a ra b le  m ag n itu d e  an d  w as  m isse d  o u t in  H a se g a w a 's  w o rk . T h e  5 th  te rm  g iv e s  th e  th re e -b o d y  in te ra c tio n  o f  Ey2 a. Simplification o f two body part from the 2nd and 3rd terms:T h e  tw o  id e n tic a l te rm s  m a y  be e x p re s se d  asIININ(NQY 7r l'9i 'i /k! (.NQYFirst w e consider r as the polar axis and 9  ^ =  . The aboveintegral m ay be written as8;r^jg ^  dq, jq ^  dq2 | d c o s 0 2W e in c lu d e  n u ll q in  th e  s u m m a tio n  a n d  w rite  v - \ r )  =  — i - 5- yw h e re  v \ ; )  is a  p a r t  o f  tw o -b o d y  in te ra c t io n  fro m  3 rd  o rd e rp e r tu rb a tio n . R e p la c in g  V  b y  -  f  w c  g e t« { I n y  •', ,  . 2  (N£2 f  t t  ,f - u r c o s B ^ - j  ^J e  * “ ' d c o s 0 „ ,, ^ , f  V ' J< 7i d q .  j d  COS0 24 r t ^ ^ •' <7|TT h e  $ 2  in te g ra t io n  m a y  b e  p e r f o rm e d  u s in g  q  ^ a s  th e  p o la r  a x is  i.e. p u tt in g   ^ s 0  a n d  e x p re s s in g  £ ,  fo rm a lly  asfin a lly  w riteV  ^( r) =  y ^  f  f J  g   ^  ^  ^^ "  q} dq  ^ q\ dq2d co^ 9.2b. Contribution to V^(r) from the 4th term :T h e  fo u r th  te rm  s im ila r ly  g iv e s  r is e  to  a  p a ir  in te ra c t io n  o f  th e  formJ J 92. 0, , ,  *"-''dq,dq2 . (3)F irs t w e  c o n s id e r  th e  q  ^ in te g ra t io n  b y  ta k in g  q^  as th e  p o la r  a x is . W c n o teQ2-r = q2 rco% 6 ^^ ^=  q2f  (co.s _ c o s  e,!^ , -t-sin  s in  0^^, c o s  ^ ) .  (4)w h e re  0  is th e  a n g le  b e tw e e n  ( ^ |  , ^ 2   ^ phu^es. U s in gcq . (4 )  th e  in te g ra t io n  o v e r  0  c a n  b e  e x p re s s e d  in  te n n s  o f  th e  z c ro lh -o rd e r  B e s se l fu n c t io n  o f  th e  f ir s t  k in d  ;rlnJ^ ff0 c x p ( ~ / Q c o s 0 ) =  2 ; r 2 o(C ?)'w h e re  Q =  q2 r s in  9^ ^^  s in  9 ^^ ^^ . W e th e n  h a v e=  c x p ( - ' 9 i '- )  J g j  dq, j £ j ( 9 | . 9 2 ’0 , , « , )^ ( Q ) e x p (~iq2 r c o s  c o s 0 ,^ , ) d c o s  .F o r ^ j  in te g ra tio n  w e  n o w  t a k e r  a s  th e  p o la r  a x is . P ro c e e d in g  in  a  s im ila r  m a n n e r  w e  g e l('■) =  J 9 ?dq  ^ J d  c o s  6 ^^ , e x p  (~iq^r c o s  0 ,^ , )I  ^ 2dc/2 j  e . i ( 9 i . 92* 0 , , , 3 ) do(Q)On the extraction o f  eff ective pa ir interaction o f  sim ple m etals etcChanging a variable v = co s0 ^ ,, we getv ' V) =  7 ^ -  J J j f  3 I ' ^ 2  - , , )d cose <h‘hj'^ dx J o ( 0  cos [(i/ir + (?2r  )a] .T h e  l a s t  i n t e g r a l  c a n  b e  e v a l u a t e d  a n a l y t i c a l l y .  P u t t i n g  Q  =  s i n 0 < ,^ , ,7 l - A - - ) ‘'^  =  a ( l - A ‘ ) ‘'^  a n d/i =  (^ 1  +  9 2  ® i / i ) i nt egr a l  is 181s i n ( a ‘,  s i n ( a ’ +  p-f  , / ,1 2\i/2\ o j si iu + p  jy o ( a ( l - j c  ) ] cos Pxdx = ------- 5— - t - wTJo ’ (a-+p- r -Finally ,,, I f f f  -> ■> i ta"  V ( '• )= t - 4 ll| f.i(9 i-92-^)9r r i - — i— 7  IP4n- ( a - + j 3 “ )dc{^d(]2 dcos02 2 k l+92 k) , ,------------------------ J  COS 0  clq^  dcf')|9i +92|''T h e  in te g ra l is  a c tu a l ly  id e n t ic a l to  th a t fo r  V^{r) a n d  th e  to ta l th i rd -o r d e r  c o n tr ib u t io n  V^(r) to  th e  p a ir  in te ra c t io n  is, ih c rc lo rc , g iv e n  b yV \ r ) - 2 v ^ { r ) ’^ v'^{r)3 f f f  > s in (^ / ir )  , ^ ,=  7 ~ T  ^ / i ------------- t / c o s 0  (Iq,47T q^r3. Discussionriie  th ird  o rd e r  c o n tr ib u t io n  d e p e n d s  c o n s id e ra b ly  o n  th e  fo rm  fac to rs  at la rg e  q. T h e  e s t im a te  o f  th e  a d d it io n a l te rm s  w ith  alo c a l o r  an  ”o n -F c rm i-S p h e re "  a p p ro x im a te d  p se u d o p o le n tia l  fo rm  fac to r, h a v in g  sp u r io u s  la rg e  q o s c i l la t io n s ,  is th e re fo re  a lw a y s  su sp e c t. C a lc u la tio n  w ith  ah initio p s e u d o p o te n tia l  fo n n  fa c to r  is e x p e c te d  to  p ro v id e  a re l ia b le  e s t im a te  o f  th e  a d d itio n a l te r m . H o w e v e r ,  th e  p e r tu r b a t io n  s e r ie s  w i th  th e  ah initio p se u d o p o le n tia l  fo rm  fa c to r  is an  o sc i l la tin g  o n e  a n d  th e  e n e rg y  u p to  th i rd -o rd e r  te rm  is n o t q u ite  re l ia b le  fo r th e  d e te rm in a tio n  o f  th e  e q u il ib r iu m  la ttic e  p a ra m e te r s  (s )  |3 ] .  A p p lic a t io n  o f  c o n v e rg e n c e  a c c e le ra t io n  m e th o d  to  th e  p e r tu rb a t io n  s c r ie s  s e q u e n c e  is fo u n d  to  p ro d u c e  e x c e l le n t s ta b le  re s u lt  fo r  th e  to ta l  en e rg y . A lso , th e  a v e ra g e  v a lu e s  o l p h o n o n  f re q u e n c ie s  u p to  th i rd -o rd e r  re p ro d u c e  |3 ]  th e  o b s e rv e d  p h o n o n  d is p e rs io n  c u rv e s  a lm o st exac tly . wSimilarly in th is  ca.se, the  re su ltin g  av e ra g e  Ipair m le ra d io n  u p to  th ird -o rd e r  c o m p a re s  w e ll | 9 |  w ith  re c e n tly  d e v e lo p e d  c a lc u la tio n  [101 b a se d  o n  la rg e  se t o f  e x p e r im e n ta l lan d  ah initio d a ta  tAcknowledgmentO n e  o f  the a u th o rs  (S H ) a c k n o w le d g e s  fin an c ia l su p p o r t o f  U G C  (U n iv e rs ity  G ra n ts  C o m m iss io n ) , In d ia .R efer ciu’cs(I i CM Rcrioni, () Bisi, C Calamlia and F Ni/zoli / F5 40)(I97S)(2) J Haminerbcrii and N W Ashcroll Pliys Rex 41)9 (1974)|31 1) Sen rh\s Rev B 42 1217 (1990)14] B DuUa, i) Sen and I’) Roy PhvsKa B210 7^ (199S)[S| M Hasegawa J Rim F6 649 (1976)|6| n Sen CMPRf^  Inicinal Research Report 1996 97 (unpublished)[71 P Lloyd and C Sholl ./ Rhvs ('1 1620 (1968)18] IS Gradslueyn and 1 M Ry/hik Table ot Intcji^ ntls, Senes and PwiludK (New York • Academic) p 727 (1965)[9] S Haidar, A Sarkai and I) .Sen CMDAYS^^i^ (CMPRC, Jadavpur University, Calculta) Indian J PIm 75A 25 (2001)[10] Y Mishin, \) Faikas, M J Mchl and D A Papaconslaniopoulos PIm Rev B59 3.29,2 (1999)

On the Extraction of Effective Pair Interaction of Simple Metals from Third Order Pseudopotential Energy

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/5682/1/On%20the%20Extraction%20of%20Effective%20Pair_By%20D%20Sen.pdf

On the Extraction of Effective Pair Interaction of Simple Metals from Third Order Pseudopotential Energy

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