Jain, A K

Neeraj

Tripathi, A N

IACS Institutional Repository

Indian J- Phys. 7 6 B  (4 ) ,  5 8 1 - 5 8 3  ( 2 0 0 2 )U P  Ban international journalE l a s t i c  s c a t t e r i n g  o f  e l e c t r o n s  f r 0 m  m e t h a n e  m o l e c u l eN eerja , A  N  T rip a th i a n d  A  K  Ja in 'Department o f Physics, University of RoorkeA Roorkce-247 667, India E-mail . neerudph@rurkiu.«rnct.in4bstract : Here we present our theoretical results for the differential scattering cross section and spin polarization parameters for elastic scattering o f electrons from methane molecule at 100.00 eV using relativistic Dirac equation. The calculated spin polarization parameters are compared with those for iso-electronic neon and atomic carbon. The calculated angular distribution is compared with available experimental data.Keywords Elastic scattering, relativistic Dirac cqs., spin polarizationP.\( S No. : 34.80.Bm1. In troductionThe recent te c h n ic a l a d v a n c e s  in  th e  s tu d y  o f  th e  sca tte r in g  uf sp in -d e p e n d e n t e le c t ro n s  h a v e  m a d e  it  p o s s ib le  to  understand th e  ro le  o f  s p in -d e p e n d e n t  in te ra c t io n s  in electron-atom c o llis io n s  f 1-3]. H o w ev e r, little  w ork  h as b een  done on the  re la te d  p ro b lem  o f  sp in  p o la r iz a tio n  o f  e lec tro n s  scattered fro m  m o le c u la r  sy s tem s . It is o n ly  in re cen t yea rs  diat a few  s im p le  a s  w e ll a s  s o p h is tic a te d  th e o re tic a l  ca lcu la tions o f  c r o s s  s e c t io n s  a n d  s p in  p o la r iz a t io n  parameters o f  e le c tro n  e la s tic a lly  sca tte red  from  o rien ted  and sim ple d ia to m ic  m o le c u le s  a p p e a re d  [4 -7 ].Electron sc a tte r in g  fro m  m e th a n e  is o f  s ig n ific a n t in te rest in many a tm o sp h e ric  a n d  te c h n o lo g ic a l ap p lic a tio n s . C H 4 is o f the m a jo r  g re e n h o u se  g a se s  a n d  a lso  is an im p o rtan t constituent in  th e  a tm o sp h e re  o f  Ju p ite r , S a tu rn  and  o u r own p lane t. T h e o re t ic a l ly  a n d  e x p e r im e n ta lly , e le c tro n  collision w ith  m e th a n e  is v e ry  w e ll s tu d ied  p ro b lem . In  recen t years, co n sid e rab le  w o rk  h a s  b een  d o n e  fo r  c ro ss  sec tio n s spin p o la r iz a tio n  o f  e le c tro n s  sca tte red  from  m eth an e  ^oolecule.Various m o d e ls  h a v e  b een  u se d  fo r  th e se  ca lcu la tio n s, we are  u s in g  a  sim p le  sp h erica l co m p lex  op tica l po ten tia l(S C O P ) ap p ro ach  a lo n g  w ith  th e  re la tiv is tic  D irac  eq u a tio n  fo r ca lcu la tin g  the  spin  p o la riza tio n  and  c ro ss  sec tio n s  fo r spherica l m o lecu les . It is w ell know n  th a t D irac  eq u a tio n  inc ludes in trin sica lly  th e  sp in -o rb it in te rac tio n  a lo n g  w ith  o th e r re la tiv is tic  co rrec tions. T h is ap p ro ach  h a s  been  q u ite  su ccessfu l in p red ic tin g  th e  sp in  p o la r iz a tio n  p a ram ete rs  and  c ro ss sec tio n s o f  e lec tro n s e la s tica lly  sca tte red  from  h e a v ie r  a to m ic  sy s te m s  and  p o ly a to m ic  m o le c u le s  like  g e rm an e  (G eH 4) and  s ilane  (S iH 4) [4].In  th is  p ap e r, w e a re  p re se n tin g  o u r  re su lts  fo r th e  sca tte rin g  o f  e lec tro n s  from  m e th an e  m o lecu le  u s in g  th e  re la tiv is tic  fo rm  o f  S ch rd d in g e r eq u a tio n . T h e  d iffe ren tia l c ro ss  sec tions and  the  sp in  p o la riza tio n  p a ram e te rs  P, T  and  U a re  p resen ted  h e re  on ly  at 1 0 0  eV . T o  co m p are  o u r  resu lts , w e hav e  a lso  ca lcu la ted  the  ab o v e  sca tte r in g  p a ram e te rs  fo r th e  a to m ic  n eo n , iso e lcc tro n ic  to  m e th an e  m o lecu le , and  th e  ca rb o n  atom .2, T h e o re tic a l M e th o d o lo g yIn th is  ca lc u la tio n  th e  m o tio n  o f  th e  p ro jec tile  e le c tro n  in  a  cen tra l fie ld  V (f)  is d e sc rib ed  b y  th e  D irac  eq u a tio n\ c d . p  + p m o C ^ - V { r ) Y = E 'F^Pmment of Applied Physics, M. L. N. College, Yamuna Nagar, India( 1)©2002IACS582 Neerja, A N Iripathi and A K JainF o r  c e n tra l p o ten tia l, D irac  eq u a tio n  can  b e  red u ced  to  a  set o f  tw o  e q u a tio n sg ? ( r ) - 0W h ere  g f  is  re la te d  to  th e  rad ia l p a rt O f  o f  th e  large  c o m p o n e n t o f  * F b y\ E - V ( . r )  + moC^\cntr2 _H ere , w e  ta k e  th e  to ta l e n e rg y  o f  th e  in c id e n t p a rtic le  as E  = nto fc^ -  E l + mf)C^, y = ( l - v ^ / c ^ )   ^ w h e re  E, as th e  k in e tic  e n e rg y  o f  th e  in c id e n t p a rtic le  o f  res t m ass mo a n d  v e lo c ity  v. T h e  ( /*  a re  th e  e ffec tiv e  D irac  p o ten tia ls  a n d  a re  g iv e n  in a to m ic  u n its  (m o  =  e  -  h  ^  1, \ / c ~  a ,  w here  a  is  f in e  s tru c tu re  c o n s ta n t)  as- U f ( r ) ^  - 2 y r ( r ) + a 2 y H r ) ~ ^ ^4(3)T h e  s in g le  a n d  d o u b le  p r im e  d en o te s  th e  f irs t an d  seco n d  o rd e r  d e riv a tiv e s  w ith  re sp e c t to  r .H e re , th e  to ta l in te rac tio n  b e tw een  an  e lec tro n  and  the  ta rg e t m o le c u le  is  ap p ro x im a te ly  re p re se n te d  b y  an  e ffec tiv e  p o te n tia l, w h ic h  can  b e  ex p a n d e d  a ro u n d  th e  h eav y  cen tra l a to m  (c a rb o n )  in te rm s  o f  a  sy m m e try -a d a p te d  fu n c tio n  o f  th e  A /  ir re d u c ib le  re p re se n ta tio n  (to ta lly  sy m m etric )ijt(4)Retaining o n ly  the f irs t te rm  (w ith  I -  0  an d  h ~  1) in th e  e x p a n s io n  o f  E q . (4 ) , th e  sp h e rica l p o ten tia l is w ritten  asy H (r )  = V A r )  + V „ ir )  + V ^ ( r ) (5)T h e  s ta tic  p o te n t ia l  V J ,r )  a n d  c h a rg e  d e n s i ty  p ( r )  a re  o b ta in e d  u s in g  n o n - r e la t iv i s t i c  m u l t ic e n tr e  m o le c u la r  w a v e fu n c tio n  a t th e  H a rtree -F o ck  (H F ) level, b e in g  gen era ted  fro m  s ta n d a rd  Q u a n tu m  C h e m is try  co m p u te r co d es. T h e  m o le c u la r  w a v e fu n c tio n  a re  g e n e ra te d  a t e x p e r im e n ta l  g eo m e tric e s . T h e  d e ta ils  a re  g iv e n  in p a p e r  o f  Ja in  [8 ], F o r th e  p o la r iz a tio n  p o te n tia l w e  h av e  u sed  a p a ra m e te r  free  p o la r i z a t io n  p o te n t i a l  (V pai), w h ic h  is  b a s e d  o n  th e  co rre la tio n  e n e rg y  o f  th e  ta rg e t a to m . It h a s  tw o  co m p o n en ts .T h e  sh o rt ran g e  Vsair) and  lo n g  ran g e  F,.«(r) parts and „ g iv en  by(2)' s r W  r < Krc b e in g  th e  c ro s s in g  p o in t .  T h e  s h o r t- r a n g e  pan ;,| p o la r iz a tio n  p o ten tia l is g iv en  asf0.0311 Inr, -  0.0584 + 0.006r, Inr, - 0.015r,. r, s (j 7 ^SR(r) = -! -0.07356+0.02224 Inr, 07<nj,|„[-0.584r,^‘ + 1.988r,“^'^ ■ 2.45r,  ^ -0.733r, 5/2 10<_(6)w h ere  r ,  =  f3/4;r/:i(r)]*^ and  p ( r )  is th e  e lec tro n  charge dcnsit\ o f  th e  ta rg e t sy s tem , b u t fo r th e  sh o r t ra n g e  term  for atom.c sy s tem s a re  taken  from  th e  p a p e r  o f  C o n n e ll and  l.anc (9 ! F o r e x ch an g e  p o ten tia l w e  h av e  u sed  th e  modified sem i-c lassica l ex ch an g e  (M S C E ) po ten tia l g iven  by Giarmirti and  S c ia lla  [lO j.T h e  im p a c t e n e rg y  ra n g e  c o n s id e re d  in the presav ca lcu la tio n  ex ceed s  th e  th re sh o ld  en e rg y  o f  the inciasn. e l e c t r o n s  s c a t t e r in g  f ro m  th e  t a r g e t  s y s te m s  urui ' inves tig a tio n  an d  h en ce  cau se s  an  ab so rp tio n  in the stattcrit: beam . T h ere  ex is t v a rio u s  v e rs io n s o f  the ab so rp tion  potcniin d e sc rib in g  all th e  in e la stic  p ro ce sse s  d u rin g  the  scattering T o inc lude  th e  a b so rp tio n  e ffe c t in th e  sca tte red  beam, \vc h av e  th e re fo re  em p lo y ed  a  m o d ified  v ers ion  3 o f  the i em p irica l m o d e l ab so rp tio n  p o ten tia l o f  S taszew ska : :> [ 1 1 ]. It is g iv en  by("iv'to - [ 2 ( £ - ®In E qs. (7 ) and  (8 ) i v  is th e  local v e lo c ity  o f  the  incitlcni e lec tro n  fo r £* -  ^  0 , vj  ^ is th e  re a l p a rt o f  the totalin te rac tio n  p o ten tia l. T h e  fac to r ’/a in Eq. (7 ) is introduced to  a c c o u n t fo r th e  e x ch an g e  o f  th e  in c id e n t electron and b o u n d  e lec tro n s  o f  th e  ta rg e t d u rin g  th e  sca tte rin g  process a f,  is th e  av e ra g e  q u a s ifre e  b in a ry  co llis io n  cross section o b ta in ed  non -em p irica lly  b y  u s in g  th e  free-e lec tro n  gas niodtl fo r th e  ta rg e t. W e a v o id  re p e a tin g  th e  exp ressions , a re  g iv en  in th e  p a p e r  o f  S ta szew sk a  e t a l [ \ \ ] .  The det3ili> o f  n u m erica l c a lc u la tio n s  and  the  b asic  fo rm u la e  to  compote th e  v a rio u s  sca tte r in g  an d  p o la r iz a tio n  p a ram ete rs  can be fo u n d  in  o u r  p rev io u s  p a p e r  [ 1 2 ].3 . R e s u lts  a n d  D isc u s s io n sIn  fig u re  l ( a > - l (d ) ,  w e h av e  p re se n te d  o u r  D C S  and p o la r iz a tio n  p a ra m e te rs  P, T, (7 a t 100 cV  fo r electrot  ^e la s tica lly  sca tte red  fro m  m e th an e  m o lecu le . F o r compar*s^^Elastic scattering of electrons from methane molecule 583»iih p c s  and  sp in  p o la r iz a tio n  (P ) fo r  m e th a n e  m o lecu le , w e l^ ve ais*) in c lu d ed  th e  D C S  a n d  sp in  p o la r iz a tio n  fo r  a to m ic  1,0-electronic n e o n  (N e )  as w e ll a s  fo r  a to m ic  ca rb o n  (C ). In  tl,!!, figure w e  see , th e  g e n e ra l tre n d  o f  v a ria tio n  o f  bo th  pCS and sp in  p o la r iz a tio n  p a ra m e te r  (P )  is q u ite  s im ila r fo rh|!urr la. Differential cross section for the scattering of electrons !r. m ihe methane molecule (CH4), carbon (C) and neon (Ne) atoms in iniNof 10 cm  ^ s r ‘ at 100.0 cV.‘’rtveni ca lcu la tio n ----------CH4; — • — C ; -----------Ne; (b) Spinvlari/alion parameter P for electrons scattered from CH4, C and Ne at Ml lV, (c) Spin polarization parameter (d) Spin polarizationwarncicr IJ:ach of the sy s tem  e x c e p t th a t  p o s itio n  o f  th e  d ip  fo r n eo n  itoin, an iso -e lec tro n ic  sy s tem  is sh ifte d  to  la rg e r ang les . Hic additional b ro a d  p e a k s  seen  in  th e  m o le c u la r  c u rv e s  at2, Differential cross section for the scattering of electrons  ^ the methane molecule (CH4) in units o f 10“  ^cm  ^sr* at 100.0 eV.,  ^ ----- Present result;--------- A. Jain [8]; — •  — •  — Iga e/ a/i. 1*3]; • • •  Iga et a l  (Experimental) (131; ■■ Bocsten andanaka [UJ. dd Sakae et al [15].h ig h e r  an g les  show  th e  p re sen ce  o f  h y d ro g en  (h y d ro g e n  b o n d s). T h e  p o la r iz a tio n  fu n c tio n  fo r c a rb o n  a to m  m im ic  s im p le r b e h a v io u r  to  th a t o f  m e th an e  b u t g iv in g  a  v e ry  sm all v a lu e  o f  p o la r iz a tio n  in th e  sca tte red  beam  b e h a v in g  as  a w eak  s c ^ e r .F u rth frm o re  w e a lso  d isp lay  o u r D C S  cu rv e  in F ig u re  2 at 1 0 0  eV  u sin g  th e  p resen t re la tiv is tic  m o d e l c a lc u la tio n s  to g e th e r tilrith o th e r theo re tica l [8 ,1 3 ]  and  ex p erim en ta l resu lts  [13 -15 ]. i t  is fo u n d  th a t, o u r  D C S  in th is  re la tiv is tic  m o d e l rep ro d u ces  th e  sh ap e  o f  D C S  w ell b u t d iffe rs  b y  an o rd e r o f  m ag n itu d e  c o m p ared  to  th e  ex p e rim en ta l d a ta  and  w ith  o th e r  th eo re tica l ca lcu la tio n s .4. ConclusionsIn  s u m m a r y ,  w e  r e p o r t  o u r  r e s u l t s  o f  r e l a t i v i s t i c  c a lc u la tio n s  fo r D C S  and  sp in  p o la r iz a tio n  p a ra m e te rs , fo r th e  m e th an e  m o lecu le  at 100.00 eV  w ith in  an o p tica l p o ten tia l a p p ro a c h . T h e  c a lc u la te d  D C S  re p ro d u c e  th e  a n g u la r  sh a p e  an d  s iz e  o f  th e  e x p e r im e n ta lly  m e a s u re d  re su lts  w ith in  an o rd e r  o f  m ag n itu d e . T he  n eg lec t o f  th e  a n g u la r sy m m e try  in th e  p re se n t in te rac tio n  a p p e a rs  to  b e  w e ll ju s tif ie d , as th e  ta rg e t m o lecu le  rem a in s  sp h e rica l fo r th e  in co m in g  e lec tro n .Kefercnccs[ I ]  J K e s s l e r At Mol. Opt, Phys. 27 81 (1991), Polarized Electrons, 2nd Ed. (Berlin : Springer) (1985)|2 ]  R P M cE achran and A D S tau ffer J Phys B I9  3523 (1986)|3 ]  K Hasenburg, D H Madi.son, K Baitschat and K Blum J Phys. B19 1803 (1986)14] Arvind Kumar Jain and A N Tripathi Phys Lett A231 224 (1997) and references therein[5] M Fink, A W Ross and R J Fink Z  Phys D ll  231 (1989). See also Z Phys. D14 77 (1989)|61 C M Fullerton , G W ocstc, D G T hom pson, K Blum  and C J Noble J Phys B . At Mol Opt Phys. 27 185 (1994)[7] C J Noble and P G Burke Phys Rev. Lett 68 2011 (1992)[8] A Jain Phys Rev A34 3707 (1986)[9] J K O’ Connell and N F Lane Phys Rev A27 1893 (1983) [1 0 | I' A Gianlurco and S Scialla J, Phys B20 3171 (1987)[11]  G Slaszcwska, D W Schwenke and D G Truhlcr Phys Rev A29 3078 (1984)[12] Nccrja A N Tripathi and A K Jain Phys. Rev. A61 032713 (2 0 0 0 )[13] I Iga, M-T Lee. M G P HOmcm, L E Machado and L M Brcscansin Phys Rev. A61 022708’ (2000)[14] L Bocsten and H Tanaka J  Phys. B24 82! (1990)[15] T Sakae, S Sumyoshi. E Murakami, Y Matsumoto, K Ish-ibashi and A Katasc J. Phys. B22 1385 (1989)

Elastic Scattering of Electrons from Methane Molecule

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/6481/1/Elastic%20Scattering%20of%20Electrons%20from%20Methane%20Molecule_Neerja.pdf

Elastic Scattering of Electrons from Methane Molecule

Abstract

Similar works

Full text

Available Versions

IACS Institutional Repository