APPROXIMATE ANALYTIC WAVE FUNCTION METHOD IN ELECTRON ATOM SCATTERING CALCULATIONS. Budi Santoso

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1 APPROXIMATE ANALYTIC WAVE FUNCTION METHOD IN ELECTRON ATOM SCATTERING CALCULATIONS Bud Satoso ABSTRACT APPROXIMATE ANALYTIC WAVE FUNCTION METHOD IN ELECTRON ATOM SCATTERING CALCULATIONS. Appoxmate aalytc wave fucto method the calculato of electo atom scatteg coss sectos have bee developed. Ths method assumes that the domat scatteg ampltude s cotbuted by the statc potetal of the scatteg pocess. The statc potetal s modeled such that the secod Bo off shell aalytc fom s avalable,.e. the fom of sceeed Coulomb potetals. The potetal stegth paametes ae detemed fom the vaatoal Hatee-Fock method (Salvatz et al 987). Appoxmate aalytc wave fucto descbg the scatteg pocess the statc model ca be obtaed employg the Fedholm tegal equato method developed by Holt ad Satoso (97). Usg ths wave fucto, ay scatteg pocess that s cosdeed to be petubg tems, ca be calculated staght fowad. Ths appoxmate wave fucto mpoves the Bo appoxmato that moe wave umbes as a esult of wave fucto speadg have bee take to accout. INTRODUCTION Scatteg pocesses have bee a subject of teest eve befoe the bth of quatum theoy, sce much fomato about the stuctue of matte ca be deved fom ths study. Wth the developmet of quatum theoy, moe poweful tools fo pedctg teacto model subatomc level ca be exploed. Wth the famewok of oelatvstc quatum mechacs, a umbe of elatvely smple potetal models have bee developed,.e. squae well, gaussa, sceeed coulomb ad expoetal potetals. Eve wth these smple models, exact calculatos caot be made eadly avalable. Futhe appoach eed to be developed. Elaboato usg dect umecal tegato the patal wave aalyss appoach has bee used as a efeece to othe appoaches. The patal wave appoach has the advatage of moe elable accuacy but wth the dawback of acqug umeous patal wave cotbutos have to be cluded the calculatos. A wdely smple appoach s the use of the Bo appoxmato, wth the ma dawback that the esults have uceta accuacy. Impovemet of the bo appoxmato has bee developed by Holt ad Satoso (974) by substtutg the umecal values of the few lowest patal wave cotbutos. The covegece s fast, demostatg that the accuacy obtaed by patal wave method ca be acheved by calculatg oly few lowest ode oly. Aothe appoach that s complemetay to the dffeetal method s the tegal method. I ths appoach the wave fucto s expaded the fom of foue tegal Nuclea Techology Assessmet Cete, Natoal Atomc Eegy Agecy (Bata)

2 temed as the Fedholm tegal method, developed by Holt ad Satoso (973). Ths method has the advatage of pesetg the wave fucto soluto a appoxmate aalytc fom, but wth the dawback of dealg wth elatvely lage dmeso of smultaeous equato. Besdes ths, the off shell Bo matx elemets eed to be gve aalytcally, othewse the computato becomes too tedous. Moeove the methods gve by Holt ad Satoso above wee lmted oly to cetal models that do ot epeset eal system. I the actual poblem, the statc potetal appoach s suffcet, because vaous atomc pocesses such as exctatos, eaagemets, ozatos, stppg ad othes caot be descbed usg the smple models. Oe has to stat by wtg hamltoa deved fom the ketc ad coulomb potetal eeges. Futhe appoxmato s possble whe oe deals wth smple system such as hydoge o helum taget atoms. Hghe atomc umbes wth moe electos volved ca be dealt wth statstcal model. Recet teest electo scatteg poblem have bee gve by Sekewcz et al (989), Macek et al (994) ad Faste (996). I the case of elastc scatteg whee a statc cetal potetal model ca be employed, a systematc model deved fom Hatee-Fock vaatoal techque has bee gve by Salvat et al (987). Combato of sceeed coulomb types wth paametes have bee gve fo all taget atoms wth atomc umbe (hydoge) to 9(uaum). Ths type of potetal ca be adopted ths method. THEORETICAL FORMULATION Let a beam of electos scatteed by a eutal taget atom wth the atomc umbe Z. The hamltoa ca be wtte as h { + Ha + Vt} Ψ(, R) = EΨ (, R) () m whee s the posto of the electo pojectle fom the taget ucleus (cosdeed as the cete of mass of the system), R =,,..., Z descbg the total coodates of the Z electo tagets, H a s the hamltoa of the taget atom gve by Z Z h Ze e Ha = {( ) + } () m = j= j satsfyg H ϕ ( R) = ε ϕ ( R) (3) a

3 ad V ( t R ) s the potetal teacto of electo pojectle ad the taget atom gve by Z Ze e Vt ( R) = + (4) = I pcple the taget wave fuctos ad the ege eeges ae kow. A well kow method of obtag these ae the Hatee-Fock vaatoal techques. It s theefoe coveet to expad Ψ tem of ϕ gve by Ψ( R) = F ( ) ϕ ( R) (5) so that afte some eaagemet, equato () ca be wtte as whee k { + k } F ( ) = U ( ) F ( ) m m m m = E h ( ε ) ad U m * m () = ϕ m ( RV ) t (, R) ϕ ( RdR ) h I statc appoxmato = ad U m = except fo U. Ths wll smplfy equato (6), { + k } ψ( ) = U( ) ψ( ) whee we have wtte Ψ=F ad U = U. I geeal, oe could ty to ( appoxmate Um () Fm () = U() F (), whee U ( s a complex m potetal. Soluto of equato (7) a tegal fom s gve by ψ( ) exp( = k. ) + G(, ') U( ') ψ( ') d' (8) Fo bevty t s wtte the Dac otato ψ = k + GUψ whee the gee fucto G (, ') s gve by dp exp( k ' G (, ') = ( ) = (9) 3 π p k + ε 4π ' a (6) (7) 3

4 Expadg ψ ( ) as ψ ( ) = c( p)exp( p. ) dp () ad substtute ths to equato (8), multplyg acoss wth oe obtas qu c ( p ) q U UGU p dp = q U k () Fo potetals of the fom (Salvat et al 987) 3 Z U ( ) = A exp( α ) = () whee A ad α ae paametes gve by Salvat et al (987), the elemets v qu p ad qugu p ca be evaluated aalytcally,.e. A qu p = Z 3 = p q + α (3) 3 3 AA j + C γ γ q UGU p = Z l = j= γ C γ γ C (4) whee γ = Λ{( p q) + α ( Λ + q + α j) + α j( Λ + p + α )} C = [( p q ) + ( α + α j) ].[ q + ( α + Λ) ].[ p + ( α + Λ) ] (5) Λ =k The oly ukow equato () s cp ( ), ad the tegato ca be appoxmated by pot quadatues otg that dp = p dp d cosϑ dϕ π pk. Sce the potetal teactos ae cetally symmetcal fucto, the scatteg ampltude s ot fucto of ϕ. Dect umecal tegato ca theefoe be doe ove ϕ, so that pot quadatues ae appled oly fo the othe two tegatos to gve a smultaeous lea equato 4

5 N = M j= A c = b klj j kl (6) whee N ad M ae the umbe of quadatues the p ad cos( ϑ pk ) tegatos espectvely ad π A = d ϕ p, x U UGU p, x klj k l j j k l b = p, x U k, (7) Soluto of c j gves the aalytc appoxmate soluto fo F ad hece Ψ whch s Ψ(, R) F ( ) ϕ ( R) (8) = Scatteg ampltude of ay tasto fom the goud state to a excted state ϕ ( R) ca be computed fom N M f( ϑ) = cj ks, xs U p, x j (9) 4π = j= the matx elemets ae udestood to be tegated ove dϕ pk. Ths fomulato mpoves the Bo appoxmato that the Bo takes oly oe c j =δ k δ j RESULTS AND DISCUSSIONS The method has bee appled to the calculatos of electo scatteg by oble gas atoms. The k -tegato quadatues have bee chose as k = k, k = k,, k 3 =.5k,, k 4 =.5k,, k 5 =.75k,, k 6 =3k,, k 7 = 4k ad so o, whle x -quadatues ae x =, x = -, x 3 =, x 4 =.75, x 5 = -.75, x 6 =.5, x 7 = -.5 ad so o. Table pesets the values of c j fo N = 3, M = 4, whle table pesets the values of total elastc scatteg coss sectos of electo scatteg at ev. As ca be see fom ths table, the values of the Bo appoxmatos ae fa fom the coveged values obtaed umecally tegatg the adal pats the phase shft scheme, ad hece they may ot be expected to gve good esults fo calculatg ay exctato pocess. The aalytc epesetato wave fucto 5

6 method developed these calculatos howeve stll gve a good ode of magtudes eve f oly few umbe of pot quadatues ae used. The daw back of the method developed ths calculato s that the dmeso of the lea equato to be solved escalates as N M. A othe poblem would be elated to the umecal stablty of the ll codtoed solvg the lea equato. REFERENCES. HOLT A R ad SANTOSO B, J.Phys B: Atom.Molec Phys., 4, (973) 9. HOLT A R ad SANTOSO B, J.Phys B: Atom.Molec.Phys., 9, (974) 3 3. SALVAT F, MARTINEZ J D, MAYOL R ad PARELLADA J,, J.Phys. B: Atom.Molec.Phys. 36, o (987) 4. SIENKIEWICZ J E ad BAYLIS W E, J.Phys. B: Atom.Molec.Phys., (989) MACEK J H ad BARRACHINA R O, Commets o Atom.Molec.Phys.,4, (994) FAINSTEIN P D, Gulyas L, MARTIN F ad SALIN A, Phys.Rev. A53, (996) 54 6

7 Table. The values c j (N=3, M=4) of ev e-he scatteg, gvg total elastc coss secto Q =.66 (a.u.) compaed to Q(Bo) =.46 ad Q(umec) =.74 j Table. Total elastc coss sectos ( a.u) of ev electo scatteg Methods Tagets He Ne A K Xe Numec Aal. (peset) (N=7, M=8) Bo Thd Ode